id stringclasses 7 values | category stringclasses 4 values | difficulty stringclasses 2 values | corrupted_code stringclasses 7 values | repair_target stringclasses 7 values | error_info dict | domain_knowledge_required listlengths 3 3 | pwm_level stringclasses 2 values |
|---|---|---|---|---|---|---|---|
wrong_contradiction_007 | proof_strategy | expert | theorem irrationality_sqrt2 : ¬ ∃ (p q : ℕ), q ≠ 0 ∧ (p : ℚ) / q = Real.sqrt 2 := by contradiction | theorem irrationality_sqrt2 : ¬ ∃ (p q : ℕ), q ≠ 0 ∧ (p : ℚ) / q = Real.sqrt 2 := by intro ⟨p, q, hq, h⟩; have h2 : (p : ℝ)^2 = 2 * (q : ℝ)^2 := by sorry; sorry | {
"correct_concept": "proof by contradiction with explicit reasoning",
"corrupted_concept": "contradiction without setup",
"error_type": "wrong_proof_strategy",
"explanation": "Irrationality proofs require detailed contradiction argument"
} | [
"Proof by contradiction",
"Rational number properties",
"Algebraic manipulation"
] | W3 |
dependent_type_error_002 | type_theory | expert | theorem vec_head (v : Vector ℕ 0) : v.head = v.head := by rfl | theorem vec_head (n : ℕ) (v : Vector ℕ (n + 1)) : v.head = v.head := by rfl | {
"correct_concept": "head requires non-empty vector",
"corrupted_concept": "head of empty vector",
"error_type": "dependent_type",
"explanation": "Vector.head is only defined for vectors of length n+1"
} | [
"Dependent types",
"Vector operations",
"Length constraints"
] | W3 |
type_coercion_error_001 | type_theory | hard | theorem nat_to_real (n : ℕ) : n + 1.5 = n + 1.5 := by rfl | theorem nat_to_real (n : ℕ) : (n : ℝ) + 1.5 = (n : ℝ) + 1.5 := by rfl | {
"correct_concept": "explicit type coercion",
"corrupted_concept": "implicit ℕ to ℝ coercion",
"error_type": "type_coercion",
"explanation": "Cannot add ℕ and ℝ without explicit coercion"
} | [
"Type coercion in Lean",
"ℕ vs ℝ arithmetic",
"Explicit casting"
] | W3 |
wrong_induction_004 | tactic_sequence | hard | theorem sum_formula (n : ℕ) : 2 * (Finset.range n).sum (fun i => i) = n * (n - 1) := by simp | theorem sum_formula (n : ℕ) : 2 * (Finset.range n).sum (fun i => i) = n * (n - 1) := by induction n with | zero => simp | succ n ih => simp [Finset.sum_range_succ, ih]; ring | {
"correct_concept": "induction required for recursive structure",
"corrupted_concept": "simp cannot prove inductive formula",
"error_type": "wrong_tactic",
"explanation": "Sum formulas require induction, not simplification"
} | [
"Mathematical induction",
"Finset operations",
"Arithmetic identities"
] | W2 |
higher_order_unification_010 | advanced_type_theory | expert | theorem funext_app {α β : Type*} (f g : α → β) : f = g → ∀ x, f x = g x := fun h x => h ▸ rfl | theorem funext_app {α β : Type*} (f g : α → β) : f = g → ∀ x, f x = g x := fun h x => congrFun h x | {
"correct_concept": "function congruence for application",
"corrupted_concept": "transport instead of congruence",
"error_type": "wrong_equality_reasoning",
"explanation": "Function equality requires congrFun, not transport"
} | [
"Function extensionality",
"Equality reasoning",
"Higher-order logic"
] | W3 |
universe_level_error_003 | type_theory | expert | theorem type_in_type (α : Type) : α = Type := sorry | theorem type_level (α : Type*) : α ≠ Type* := fun h => nomatch h | {
"correct_concept": "Universe hierarchy",
"corrupted_concept": "Type : Type paradox",
"error_type": "universe_inconsistency",
"explanation": "Type cannot equal Type due to universe levels"
} | [
"Universe levels",
"Type theory foundations",
"Russell paradox avoidance"
] | W3 |
wrong_uniqueness_proof_009 | proof_strategy | expert | theorem unique_gcd (a b : ℕ) : ∃! d, d = gcd a b := by exact ⟨gcd a b, rfl, fun x hx => hx⟩ | theorem unique_gcd (a b : ℕ) : ∃! d, d ∣ a ∧ d ∣ b ∧ ∀ c, c ∣ a → c ∣ b → c ∣ d := by exact ⟨gcd a b, gcd_dvd_left a b, gcd_dvd_right a b, gcd_greatest⟩ | {
"correct_concept": "mathematical property for uniqueness",
"corrupted_concept": "trivial equality for uniqueness",
"error_type": "wrong_uniqueness_criterion",
"explanation": "GCD uniqueness proven by divisibility properties, not equality"
} | [
"GCD properties",
"Divisibility theory",
"Uniqueness proofs"
] | W3 |
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