benchmark.csv

function_signature,docstring,test_cases,problem_spec_nl,problem_spec_formal_ground_truth,problem_spec_formal_generated,isomorphism_theorem,isomorphism_proof,implementation_signature,implementation,test_cases_lean,correctness_theorem,correctness_proof,helper_definitions,isomorphism_helper_lemmas,correctness_helper_lemmas
"def has_close_elements(numbers: List[float], threshold: float) -> bool","Check if in given list of numbers, are any two numbers closer to each other than given threshold.","[{""input"": [[1.0, 2.0, 3.0], 0.5], ""expected_output"": false}, {""input"": [[1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3], ""expected_output"": true}]","def has_close_elements(numbers: List[float], threshold: float) -> bool
""""""Check if in given list of numbers, are any two numbers closer to each other than given threshold.""""""","def problem_spec
-- function signature
(impl: List Rat → Rat → Bool)
-- inputs
(numbers: List Rat)
(threshold: Rat) :=
-- spec
let numbers_within_threshold :=
(∃ i j, i < numbers.length ∧ j < numbers.length ∧
i ≠ j ∧ |numbers.get! i - numbers.get! j| < threshold);
let spec (res: Bool) :=
numbers.length > 1 →
if res then numbers_within_threshold else ¬numbers_within_threshold;
-- program terminates
∃ result, impl numbers threshold = result ∧
-- return value satisfies spec
spec result
-- if result then spec else ¬spec","def generated_spec
-- function signature
(impl: List Rat → Rat → Bool)
-- inputs
(numbers: List Rat)
(threshold: Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ numbers threshold, problem_spec impl numbers threshold) ↔
(∀ numbers threshold, generated_spec impl numbers threshold) :=",sorry,def implementation (numbers: List Rat) (threshold: Rat) : Bool :=,"match numbers with
| []       => false
| (x::xs)  => (xs.any (fun y => (|x - y| < threshold))) || implementation xs threshold","#test implementation ([1, 2, 3]: List Rat) 0.5 = false
#test implementation ([1, 2.8, 3, 4, 5, 2]: List Rat) 0.3 = true","theorem correctness
(numbers: List Rat)
(threshold: Rat)
: problem_spec implementation numbers threshold  :=","by
unfold problem_spec
let result := implementation numbers threshold
use result
simp [result]
induction numbers
simp [implementation]
rename_i head tail ih
simp at *
by_cases result = false
rename_i h_result_false
simp [result] at h_result_false
simp [h_result_false]
simp [implementation] at h_result_false
have h_tail_impl_false := And.right h_result_false
have h_head_in_threshold := And.left h_result_false
simp [h_tail_impl_false] at ih
by_cases h_1_lt_tail: 1 < tail.length
have h_0_lt_tail_len : 0 < tail.length := by linarith
simp [h_0_lt_tail_len, h_1_lt_tail]
simp [h_1_lt_tail] at ih
intros i h_i j h_j h_i_neq_j
by_cases i = 0
rename_i h_i_eq_0
simp [h_i_eq_0]
simp [h_i_eq_0] at h_i_neq_j
have h_0_lt_j : 0 < j := by
  contrapose! h_i_neq_j
  linarith
let j' := j - 1
have h_j'_tail : (head::tail)[j' + 1]! = tail[j']! := by
  rw [List.getElem!_cons_succ]
have h_j'_j : j' + 1 = j := by
  simp [j']
  rw [Nat.sub_add_cancel h_0_lt_j]
rw [←h_j'_j]
simp [h_j'_tail]
have h_j'_lt_tail_len : j' < tail.length := by
  linarith
simp [h_j'_lt_tail_len]
apply h_head_in_threshold
simp
rename_i h_i_neq_0
have h_0_lt_i : 0 < i := by
  contrapose! h_i_neq_0
  linarith
let i' := i - 1
have h_i'_tail : (head::tail)[i' + 1]! = tail[i']! := by
  rw [List.getElem!_cons_succ]
have h_i'_i : i' + 1 = i := by
  simp [i']
  rw [Nat.sub_add_cancel h_0_lt_i]
rw [←h_i'_i]
simp [h_i'_tail]
have h_i'_lt_tail_len : i' < tail.length := by
  linarith
simp [h_i'_lt_tail_len]
by_cases 0 < j
rename_i h_0_lt_j
let j' := j - 1
have h_j'_tail : (head::tail)[j' + 1]! = tail[j']! := by
  rw [List.getElem!_cons_succ]
have h_j'_j : j' + 1 = j := by
  simp [j']
  rw [Nat.sub_add_cancel h_0_lt_j]
rw [←h_j'_j]
simp [h_j'_tail]
have h_j'_lt_tail_len : j' < tail.length := by
  linarith
simp [h_j'_lt_tail_len]
have h_i'_neq_j' : i' ≠ j' := by
  contrapose! h_i_neq_j
  linarith
have h_final := ih i'
simp [h_i'_lt_tail_len] at h_final
have h_final' := h_final j' h_j'_lt_tail_len
simp [h_i'_neq_j'] at h_final'
simp [h_j'_lt_tail_len] at h_final'
assumption
rename_i h_j_le_0
have h_j_eq_0 : j = 0 := by
  linarith
simp [h_j_eq_0]
have h_head_minus_tail : tail[i'] - head = -(head - tail[i']) := by
  simp
rw [h_head_minus_tail]
have h_head_minus_tail_lt_threshold : |-(head - tail[i'])| = |head - tail[i']| := by
  rw [abs_neg]
rw [h_head_minus_tail_lt_threshold]
apply h_head_in_threshold
simp
have h_tl_eq_0 : tail.length ≤ 1 := by
  linarith
by_cases tail.length = 1
rename_i h_tl_eq_1
simp [h_tl_eq_1]
intros i h_i j h_j h_i_neq_j
by_cases i = 0
rename_i h_i_eq_0
simp [h_i_eq_0]
by_cases j = 0
rename_i h_j_eq_0
rw [h_j_eq_0] at h_i_neq_j
contradiction
by_cases j = 1
rename_i h_j_eq_1
simp [h_j_eq_1]
simp [h_tl_eq_1]
apply h_head_in_threshold
simp
rename_i h_j_neq_1 h_j_neq_0
have h_0_lt_j : 0 < j := by
  contrapose! h_j_neq_0
  simp at h_j_neq_0
  contradiction
have h_1_lt_j : 1 < j := by
  contrapose! h_j_neq_1
  have h_j_eq_1' : j = 1 := by
    linarith
  contradiction
linarith
rename_i h_i_neq_0
by_cases i = 1
rename_i h_i_eq_1
simp [h_i_eq_1]
simp [h_tl_eq_1]
by_cases j = 0
rename_i h_j_eq_0
simp [h_j_eq_0]
have h_head_minus_tail : tail[0] - head = -(head - tail[0]) := by
  simp
rw [h_head_minus_tail]
rw [abs_neg]
apply h_head_in_threshold
simp
by_cases j = 1
rename_i h_j_neq_0 h_j_eq_1
rw [h_j_eq_1] at h_i_neq_j
contradiction
rename_i h_j_neq_1 h_j_neq_0
have h_0_lt_j : 0 < j := by
  contrapose! h_j_neq_0
  simp at h_j_neq_0
  contradiction
have h_1_lt_j : 1 < j := by
  contrapose! h_j_neq_1
  have h_j_eq_1' : j = 1 := by
    linarith
  contradiction
linarith
rename_i h_i_neq_1
have h_0_lt_i : 0 < i := by
  contrapose! h_i_neq_1
  simp at h_i_neq_1
  contradiction
have h_1_lt_i : 1 < i := by
  contrapose! h_i_neq_0
  have h_i_eq_1' : i = 1 := by
    linarith
  contradiction
linarith
rename_i h_tl_neq_1
rw [Nat.le_one_iff_eq_zero_or_eq_one] at h_tl_eq_0
by_cases tail.length = 0
rename_i h_tl_eq_0
simp [h_tl_eq_0]
rename_i h_tl_neq_0
cases h_tl_eq_0
contradiction
contradiction
-- case when tail has two elements that are close to each other
rename_i h_result_true
simp at h_result_true
simp [result] at h_result_true
simp [h_result_true]
simp [implementation] at h_result_true
by_cases h_0_lt_tail_len : 0 < tail.length
rw [List.exists_mem_iff_getElem] at h_result_true
simp [h_0_lt_tail_len]
cases h_result_true
use 0
simp
rename_i h_head_in_threshold
obtain ⟨ j, h_j_lt_len, h_j_in_threshold ⟩ := h_head_in_threshold
use j + 1
simp [h_j_lt_len]
assumption
rename_i h_tail_has_close_elements
simp [h_tail_has_close_elements] at ih
by_cases h_1_lt_tail : 1 < tail.length
simp [h_1_lt_tail] at ih
obtain ⟨ i, h_i_lt_tail_len, h_i_in_threshold ⟩ := ih
use i + 1
have h_i_lt_tail_len' : i + 1 < tail.length + 1 := by linarith
simp [h_i_lt_tail_len']
obtain ⟨ j, h_j_lt_tail_len, h_i_neq_j, h_j_in_threshold ⟩ := h_i_in_threshold
use j + 1
simp [h_j_lt_tail_len]
simp [h_i_neq_j]
simp [h_i_lt_tail_len, h_j_lt_tail_len] at h_j_in_threshold
assumption
use 0
simp
use 1
simp
unfold implementation at h_tail_has_close_elements
by_cases h_1_tail: tail.length = 0
have h_tail_empty := List.eq_nil_of_length_eq_zero h_1_tail
simp [h_tail_empty] at h_tail_has_close_elements
simp at h_1_lt_tail
have h_0_lt_tail_len : tail.length = 1 := by linarith
have h_tail_singleton: ∃ x, tail = [x] := by
  rw [List.length_eq_one] at h_0_lt_tail_len
  assumption
obtain ⟨ x, h_tail_singleton ⟩ := h_tail_singleton
simp [h_0_lt_tail_len]
simp [h_tail_singleton]
try (simp [h_tail_singleton] at h_tail_has_close_elements)
unfold implementation at h_tail_has_close_elements
simp at h_tail_has_close_elements
simp at h_0_lt_tail_len
have h_tail_ln_eq_0 : tail.length = 0 := by
  rw [h_0_lt_tail_len]
  simp
simp [h_tail_ln_eq_0]",,,
def separate_paren_groups(paren_string: str) -> List[str],"Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
separate those group into separate strings and return the list of those.
Separate groups are balanced (each open brace is properly closed) and not nested within each other
Ignore any spaces in the input string.
","[{""input"": ""( ) (( )) (( )( ))"", ""expected_output"": [""()"", ""(())"", ""(()())""]}]","def separate_paren_groups(paren_string: str) -> List[str]
""""""Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
separate those group into separate strings and return the list of those.
Separate groups are balanced (each open brace is properly closed) and not nested within each other
Ignore any spaces in the input string.
""""""","def problem_spec
-- function signature
(impl: String → List String)
-- inputs
(paren_string: String) :=
-- spec
let paren_string_filtered := (paren_string.toList.filter (fun c => c == '(' ∨  c == ')')).asString;
let forward_spec (result_list: List String) :=
-- every substring of the input string
-- that is a balanced paren group is in
-- the result list
∀ i j, j < paren_string_filtered.length → i ≤ j →
let substr_ij := (paren_string_filtered.take (j + 1)).drop i;
balanced_paren_non_computable substr_ij '(' ')' →
count_paren_groups substr_ij = 1 →
substr_ij ∈ result_list;
let backward_spec (result_list: List String) :=
-- every string in the result list is a
-- balanced paren group and is a substring
-- of the input string i.e. there is no
-- extra substring in the result list
-- that is not from the input string
∀ str, str ∈ result_list →
paren_string_filtered.containsSubstr str = true ∧
balanced_paren_non_computable str '(' ')' ∧
count_paren_groups str = 1;
let spec (result_list: List String) :=
forward_spec result_list ∧ backward_spec result_list;
-- program terminates
∃ result, impl paren_string = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → List String)
-- inputs
(paren_string: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ paren_string, problem_spec impl paren_string) ↔
(∀ paren_string, generated_spec impl paren_string) :=",sorry,def implementation (paren_string: String) : List String :=,"-- remove space or any other characters that are not '(' or ')'
let filtered := paren_string.toList.filter (fun c => c == '(' ∨  c == ')');
-- auxiliary recursive function:
let rec aux (cs : List Char) (cur : List Char) (balance : Int) (acc : List String) : List String :=
  match cs with
  | []      => acc.reverse  -- when finished, return accumulated groups in original order
  | c::rest =>
    let new_cur    := cur ++ [c]
    let new_balance:= if c = '(' then
                      balance + 1
                      else
                      balance - 1
    if new_balance = 0 then
      let group := new_cur.asString
      aux rest [] 0 (group :: acc)
    else
      aux rest new_cur new_balance acc;
aux filtered [] 0 []","#test implementation ""( ) (( )) (( )( ))"" = [""()"", ""(())"", ""(()())""]","theorem correctness
(paren_string: String)
: problem_spec implementation paren_string :=","by
unfold problem_spec
let result := implementation paren_string
use result
simp [result]
sorry","/--
name: string_is_paren_balanced_helper
use: |
  Helper function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced_helper
(paren_string: String) (num_open: Int): Bool
:=
-- Recursively check if the string is balanced
if paren_string.isEmpty then
  num_open = 0
else
  let c := paren_string.get! 0
  if c == '(' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open + 1)
  else if c == ')' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open - 1)
  else
    string_is_paren_balanced_helper (paren_string.drop 1) num_open
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: string_is_paren_balanced
use: |
  Function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced
(paren_string: String): Bool
:=
string_is_paren_balanced_helper paren_string 0

/--
name: count_paren_groups_helper
use: |
  Helper to count the number of groups of parentheses in a string.
problems:
  - 1
-/
def count_paren_groups_helper
(paren_string: String) (num_open: Int) (num_groups: Nat): Nat :=
-- Recursively count the number of paren groups
if paren_string.isEmpty then
  num_groups
else
  let c := paren_string.get! 0
  if c == '(' then
    count_paren_groups_helper (paren_string.drop 1) (num_open + 1) num_groups
  else if c == ')' then
    let new_num_groups :=
    if num_open == 1 then num_groups + 1 else num_groups
    count_paren_groups_helper (paren_string.drop 1) (num_open - 1) new_num_groups
  else
    count_paren_groups_helper (paren_string.drop 1) num_open num_groups
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: count_paren_groups
use: |
  Function to count the number of groups of parentheses in a string.
problems:
  - 1
-/
def count_paren_groups
(paren_string: String): Nat :=
count_paren_groups_helper paren_string 0 0",,
def truncate_number(number: float) -> float,"Given a positive floating point number, it can be decomposed into
and integer part (largest integer smaller than given number) and decimals
(leftover part always smaller than 1).
","[{""input"": 3.5, ""expected_output"": 0.5}]","def truncate_number(number: float) -> float
""""""Given a positive floating point number, it can be decomposed into
and integer part (largest integer smaller than given number) and decimals
(leftover part always smaller than 1).
""""""","def problem_spec
-- function signature
(impl: Rat → Rat)
-- inputs
(number: Rat) :=
-- spec
let spec (res) :=
0 ≤ res ∧
res < 1 ∧
number.floor + res = number;
number > 0 →
-- program terminates
(∃ result, impl number = result ∧
-- return value satisfies spec
spec result)","def generated_spec
-- function signature
(impl: Rat → Rat)
-- inputs
(number: Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ number, problem_spec impl number) ↔
(∀ number, generated_spec impl number) :=",sorry,def implementation (number: Rat) : Rat :=,number - number.floor,#test implementation 3.5 = 0.5,"theorem correctness
(number: Rat)
: problem_spec implementation number :=","by
unfold problem_spec
let result := implementation number
simp [result]
simp [implementation]
have h1: ∀ x: Rat, x.floor ≤ x := by
  intro x
  rw [←Rat.le_floor]
intro npos
apply And.intro
apply h1
have h2 := h1 number
sorry",,,
def below_zero(operations: List[int]) -> bool,"You're given a list of deposit and withdrawal operations on a bank account that starts with
zero balance. Your task is to detect if at any point the balance of account fallls below zero, and
at that point function should return True. Otherwise it should return False.
","[{""input"": [1, 2, 3], ""expected_output"": false}, {""input"": [1, 2, -4, 5], ""expected_output"": true}]","def below_zero(operations: List[int]) -> bool
""""""You're given a list of deposit and withdrawal operations on a bank account that starts with
zero balance. Your task is to detect if at any point the balance of account fallls below zero, and
at that point function should return True. Otherwise it should return False.
""""""","def problem_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(operations: List Int) :=
-- spec
let below_zero_condition := ∃ i, i ≤ operations.length ∧
(operations.take i).sum < 0;
let spec (result: Bool) :=
if result then below_zero_condition else ¬below_zero_condition;
-- program terminates
∃ result, impl operations = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(operations: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ operations, problem_spec impl operations) ↔
(∀ operations, generated_spec impl operations) :=",sorry,def implementation (operations: List Int) : Bool :=,"let rec check (ops : List Int) (acc : Int) : Bool :=
  match ops with
  | []        => false
  | op :: ops' =>
    let new_acc := acc + op
    if new_acc < 0 then true else check ops' new_acc
check operations 0","#test implementation [1, 2, 3] = false
#test implementation [1, 2, -4, 5] = true","theorem correctness
(operations: List Int)
: problem_spec implementation operations
:=","by
-- sometimes we have to create a temporary variable to use in the proof
unfold problem_spec
let result := implementation operations
use result
simp [result]
simp [implementation]
-- induction at this point will not work
-- because the induction hypothesis is not strong enough
-- we need to prove a stronger inductive hypothesis
have h_harder_inductive_hypothesis_true:
∀ h, ∀ ops,
implementation.check ops h = true →
∃ i, i ≤ ops.length ∧ h + (ops.take i).sum < 0 := by
  intros h ops h_check
  induction' ops generalizing h
  simp [implementation.check] at h_check
  rename_i head tail ih
  simp [implementation.check] at h_check
  by_cases h_head_lt_0 : h + head < 0
  use 1
  simp
  assumption
  simp at h_head_lt_0
  cases h_check
  linarith
  rename_i h_tail_sum_gt_0
  specialize ih (h + head)
  simp [h_tail_sum_gt_0] at ih
  obtain ⟨i, h_i_lt_tail_len, h_tail_sum_lt_0⟩ := ih
  use i + 1
  simp [h_i_lt_tail_len]
  linarith
by_cases h_if_pos: (implementation.check operations 0 = true)
simp [h_if_pos]
obtain ⟨i, h_i_lt_tail_len, h_tail_sum_lt_0⟩ := h_harder_inductive_hypothesis_true 0 operations h_if_pos
simp at h_tail_sum_lt_0
use i
have h_harder_inductive_hypothesis_false:
∀ h, ∀ ops, h ≥ 0 →
implementation.check ops h = false →
∀ i, i ≤ ops.length → h + (ops.take i).sum ≥ 0 := by
  intros h ops h_ge_0 h_check
  induction' ops generalizing h
  intros i h_i_lt_tail_len
  simp [implementation.check] at h_check
  simp
  linarith
  rename_i head tail ih
  simp [implementation.check] at h_check
  by_cases h_head_lt_0 : h + head < 0
  simp at h_head_lt_0
  cases h_check
  linarith
  simp at h_head_lt_0
  apply And.right at h_check
  specialize ih (h + head)
  simp at ih
  simp [h_head_lt_0, h_check] at ih
  intros i h_i_lt_tail_len
  simp at h_i_lt_tail_len
  simp
  specialize ih (i - 1)
  by_cases h_i_1_lt_i: 1 ≤ i
  let i' := i - 1
  have h_i'_lt_tail_len: i = i' + 1 := by
    simp [i']
    rw [Nat.sub_add_cancel]
    linarith
  simp [h_i'_lt_tail_len]
  rw [←Int.add_assoc]
  have h_i_minus_1_lt_tail_len: i - 1 ≤ tail.length := by
    linarith
  simp [h_i_minus_1_lt_tail_len] at ih
  simp [i']
  assumption
  simp at h_i_1_lt_i
  simp [h_i_1_lt_i] at ih
  simp [h_i_1_lt_i]
  linarith
simp at h_if_pos
simp [h_if_pos]
have h_harder' := h_harder_inductive_hypothesis_false 0 operations
simp at h_harder'
simp [h_if_pos] at h_harder'
intros i h_i_lt_tail_len
apply h_harder'
assumption",,,
def mean_absolute_deviation(numbers: List[float]) -> float,"For a given list of input numbers, calculate Mean Absolute Deviation
around the mean of this dataset.
Mean Absolute Deviation is the average absolute difference between each
element and a centerpoint (mean in this case):
MAD = average | x - x_mean |
","[{""input"": [1.0, 2.0, 3.0, 4.0], ""expected_output"": 1.0}]","def mean_absolute_deviation(numbers: List[float]) -> float
""""""For a given list of input numbers, calculate Mean Absolute Deviation
around the mean of this dataset.
Mean Absolute Deviation is the average absolute difference between each
element and a centerpoint (mean in this case):
MAD = average | x - x_mean |
""""""","def problem_spec
-- function signature
(implementation: List Rat → Rat)
-- inputs
(numbers: List Rat) :=
-- spec
let spec (result: Rat):=
0 < numbers.length →
0 ≤ result ∧
result * numbers.length * numbers.length =
(numbers.map (fun x => |x * numbers.length - numbers.sum|)).sum;
-- program terminates
∃ result, implementation numbers = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(implementation: List Rat → Rat)
-- inputs
(numbers: List Rat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Rat) : Rat :=,"if numbers.length = 0 then 1.0
else
  let mean := numbers.sum / numbers.length;
  let abs_diffs := numbers.map (fun x => |x - mean|);
  abs_diffs.sum / abs_diffs.length","#test implementation [1.0, 2.0, 3.0, 4.0] = 1.0","theorem correctness
(numbers: List Rat)
: problem_spec implementation numbers
:=","by
-- sometimes we have to create a temporary variable to use in the proof
unfold problem_spec
let result := implementation numbers
use result
simp [result]
simp [implementation]
by_cases (numbers.length = 0)
rename_i h
simp [h]
rename_i h
simp at h
have h1: 0 < numbers.length := by
  by_contra
  rename_i h2
  simp at h2
  contradiction
simp [h1]
simp [h]
sorry",,,
"def intersperse(numbers: List[int], delimeter: int) -> List[int]","Insert a number 'delimeter' between every two consecutive elements of input list `numbers'
","[{""input"": [[], 4], ""expected_output"": []}, {""input"": [[1, 2, 3], 4], ""expected_output"": [1, 4, 2, 4, 3]}]","def intersperse(numbers: List[int], delimeter: int) -> List[int]
""""""Insert a number 'delimeter' between every two consecutive elements of input list `numbers'
""""""","def problem_spec
-- function signature
(implementation: List Int → Int → List Int)
-- inputs
(numbers: List Int)
(delimeter: Int) :=
-- spec
let spec (result: List Int) :=
(result.length = 0 ∧ result = numbers) ∨
(result.length = 2 ∧ numbers.length = 1 ∧
result[0]! = numbers[0]! ∧ result[1]! = delimeter) ∨
(result.length = 2 * numbers.length - 1 ∧
∀ i, i < numbers.length →
result[2 * i]! = numbers[i]! ∧
(0 < 2*i - 1 → result[2 * i - 1]! = delimeter));
-- program termination
∃ result, implementation numbers delimeter = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → Int → List Int)
-- inputs
(numbers: List Int)
(delimeter: Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers delimeter, problem_spec implementation numbers delimeter) ↔
(∀ numbers delimeter, generated_spec implementation numbers delimeter) :=",sorry,def implementation (numbers: List Int) (delimeter: Int) : List Int :=,numbers.intersperse delimeter,"#test implementation [1, 2, 3] 4 = [1, 4, 2, 4, 3]
#test implementation [] 4 = []
#test implementation [1] 4 = [1]","theorem correctness
(numbers: List Int)
(delimeter: Int)
: problem_spec implementation numbers delimeter
:=","by
unfold problem_spec
let result := implementation numbers delimeter
use result
simp [result]
sorry",,,
def parse_nested_parens(paren_string: str) -> List[int],"Input to this function is a string represented multiple groups for nested parentheses separated by spaces.
For each of the group, output the deepest level of nesting of parentheses.
E.g. (()()) has maximum two levels of nesting while ((())) has three.
","[{""input"": ""(()()) ((())) () ((())()())"", ""expected_output"": [2, 3, 1, 3]}]","def parse_nested_parens(paren_string: str) -> List[int]
""""""Input to this function is a string represented multiple groups for nested parentheses separated by spaces.
For each of the group, output the deepest level of nesting of parentheses.
E.g. (()()) has maximum two levels of nesting while ((())) has three.
""""""","def problem_spec
-- function signature
(implementation: String → List Int)
-- inputs
(paren_string: String)
:=
-- spec
let spec (result: List Int) :=
let paren_space_split := paren_string.split (fun x => x = ' ');
result.length = paren_space_split.length ∧
∀ i, i < result.length →
let group := paren_space_split[i]!;
balanced_paren_non_computable group '(' ')' →
0 < result[i]! ∧ count_max_paren_depth group = result[i]!.toNat;
-- program termination
∃ result, implementation paren_string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → List Int)
-- inputs
(paren_string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ paren_string, problem_spec implementation paren_string) ↔
(∀ paren_string, generated_spec implementation paren_string) :=",sorry,def implementation (paren_string: String) : List Int :=,(paren_string.split (fun x => x = ' ')).map (fun x => count_max_paren_depth x),"#test implementation ""(()()) ((())) () ((())()())"" = [2, 3, 1, 3]","theorem correctness
(paren_string: String)
: problem_spec implementation paren_string
:=","by
unfold problem_spec
let result := implementation paren_string
use result
simp [result]
sorry","/--
name: string_is_paren_balanced_helper
use: |
  Helper function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced_helper
(paren_string: String) (num_open: Int): Bool
:=
-- Recursively check if the string is balanced
if paren_string.isEmpty then
  num_open = 0
else
  let c := paren_string.get! 0
  if c == '(' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open + 1)
  else if c == ')' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open - 1)
  else
    string_is_paren_balanced_helper (paren_string.drop 1) num_open
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: string_is_paren_balanced
use: |
  Function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced
(paren_string: String): Bool
:=
string_is_paren_balanced_helper paren_string 0

/--
name: count_max_paren_depth_helper
use: |
  Helper to count the maximum depth of parentheses in a string.
problems:
  - 6
  - 132
-/
def count_max_paren_depth_helper
(paren_string: String) (num_open: Int) (max_depth: Nat): Nat :=
-- Recursively count the maximum depth of parentheses
if paren_string.isEmpty then
  max_depth
else
  let c := paren_string.get! 0
  if c == '(' then
    let new_num_open := num_open + 1
    count_max_paren_depth_helper (paren_string.drop 1) (new_num_open) (max_depth.max new_num_open.toNat)
  else if c == ')' then
    count_max_paren_depth_helper (paren_string.drop 1) (num_open - 1) max_depth
  else
    count_max_paren_depth_helper (paren_string.drop 1) num_open max_depth
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: count_max_paren_depth
use: |
  Function to count the maximum depth of parentheses in a string.
problems:
  - 6
  - 132
-/
def count_max_paren_depth
(paren_string: String): Nat :=
count_max_paren_depth_helper paren_string 0 0",,
"def filter_by_substring(strings: List[str], substring: str) -> List[str]","Filter an input list of strings only for ones that contain given substring
","[{""input"": [[], ""a""], ""expected_output"": []}, {""input"": [[""abc"", ""bacd"", ""cde"", ""array""], ""a""], ""expected_output"": [""abc"", ""bacd"", ""array""]}]","def filter_by_substring(strings: List[str], substring: str) -> List[str]
""""""Filter an input list of strings only for ones that contain given substring
""""""","def problem_spec
-- function signature
(implementation: List String → String → List String)
-- inputs
(strings: List String)
(substring: String)
:=
-- spec
let spec (result: List String) :=
(∀ i, i < result.length → result[i]!.containsSubstr substring →
∀ j, j < strings.length ∧ strings[j]!.containsSubstr substring → strings[j]! ∈ result →
∀ j, j < result.length → result.count result[j]! = strings.count result[j]!);
-- program termination
∃ result, implementation strings substring = result ∧
spec result","def generated_spec
-- function signature
(implementation: List String → String → List String)
-- inputs
(strings: List String)
(substring: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ strings substring, problem_spec implementation strings substring) ↔
(∀ strings substring, generated_spec implementation strings substring) :=",sorry,def implementation (strings: List String) (substring: String): List String :=,strings.filter (fun x => x.containsSubstr substring),"#test implementation [] ""a"" = []
#test implementation [""abc"", ""bacd"", ""cde"", ""array""] ""a"" = [""abc"", ""bacd"", ""array""]","theorem correctness
(strings: List String)
(substring: String)
: problem_spec implementation strings substring
:=","by
unfold problem_spec
let result := implementation strings substring
use result
simp [result]
sorry",,,
"def sum_product(numbers: List[int]) -> Tuple[int, int]","For a given list of integers, return a tuple consisting of a sum and a product of all the integers in a list.
Empty sum should be equal to 0 and empty product should be equal to 1.
","[{""input"": [], ""expected_output"": ""(0, 1)""}, {""input"": [1, 2, 3, 4], ""expected_output"": ""(10, 24)""}]","def sum_product(numbers: List[int]) -> Tuple[int, int]
""""""For a given list of integers, return a tuple consisting of a sum and a product of all the integers in a list.
Empty sum should be equal to 0 and empty product should be equal to 1.
""""""","def problem_spec
-- function signature
(implementation: List Int → (Int × Int))
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: (Int × Int)) :=
let (sum, prod) := result;
(numbers = [] → sum = 0 ∧ prod = 1) ∧
(numbers ≠ [] →
(let (sum_tail, prod_tail) := implementation numbers.tail;
sum - sum_tail = numbers[0]! ∧
sum_tail * prod_tail + prod = sum * prod_tail));
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → (Int × Int))
-- inputs
(numbers: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Int) : (Int × Int) :=,"(numbers.sum, numbers.prod)","#test implementation [] = (0, 1)
#test implementation [1, 2, 3, 4] = (10, 24)","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
simp [implementation]
sorry",,,
"def sum_product(numbers: List[int]) -> Tuple[int, int]","From a given list of integers, generate a list of rolling maximum element found until given moment
in the sequence.
","[{""input"": [1, 2, 3, 2, 3, 4, 2], ""expected_output"": [1, 2, 3, 3, 3, 4, 4]}]","def sum_product(numbers: List[int]) -> Tuple[int, int]
""""""From a given list of integers, generate a list of rolling maximum element found until given moment
in the sequence.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: List Int) :=
result.length = numbers.length ∧
∀ i, i < numbers.length →
(result[i]! ∈ numbers.take (i + 1) ∧
∀ j, j ≤ i → numbers[j]! ≤ result[i]!);
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Int) : List Int :=,"let rec rolling_max (numbers: List Int) (results: List Int) (acc: Int) : List Int :=
  match numbers with
  | [] => results
  | n :: ns =>
    let new_acc := max acc n
    let new_results := results ++ [new_acc]
    rolling_max ns new_results new_acc
rolling_max numbers [] 0","#test implementation [1, 2, 3, 2, 3, 4, 2] = [1, 2, 3, 3, 3, 4, 4]","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
simp [implementation]
apply And.intro
induction numbers
simp
simp [implementation.rolling_max]
rename_i head tail ih
unfold implementation.rolling_max
simp
-- At this point we will have to
-- strengthen the induction hypothesis
-- to prove the correctness of the implementation
sorry
sorry",,,
def make_palindrome(string: str) -> str,"Find the shortest palindrome that begins with a supplied string.
Algorithm idea is simple:
- Find the longest postfix of supplied string that is a palindrome.
- Append to the end of the string reverse of a string prefix that comes before the palindromic suffix.
","[{""input"": """", ""expected_output"": """"}, {""input"": ""cat"", ""expected_output"": ""catac""}, {""input"": ""cata"", ""expected_output"": ""catac""}]","def make_palindrome(string: str) -> str
""""""Find the shortest palindrome that begins with a supplied string.
Algorithm idea is simple:
- Find the longest postfix of supplied string that is a palindrome.
- Append to the end of the string reverse of a string prefix that comes before the palindromic suffix.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(string: String) :=
-- spec
let spec (result: String) :=
is_palindrome result ∧
result.length ≥ string.length ∧
string.isPrefixOf result ∧
-- comprehensive check that the result is the shortest palindrome
(∀ (possible_palindrome: String),
string.isPrefixOf possible_palindrome →
is_palindrome possible_palindrome →
result.length ≤ possible_palindrome.length);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → String)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String) : String :=,sorry,"-- #test implementation """" = """"
-- #test implementation ""cat"" = ""catac""
-- #test implementation ""cata"" = ""catac""","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry","/--
name: is_palindrome
use: |
  Helper to check if a string is a palindrome.
problems:
  - 10
  - 48
-/
def is_palindrome
(s: String): Bool :=
s = s.toList.reverse.asString",,
"def string_xor(a: str, b: str) -> str","Input are two strings a and b consisting only of 1s and 0s.
Perform binary XOR on these inputs and return result also as a string.
","[{""input"": [""010"", ""110""], ""expected_output"": ""100""}]","def string_xor(a: str, b: str) -> str
""""""Input are two strings a and b consisting only of 1s and 0s.
Perform binary XOR on these inputs and return result also as a string.
""""""","def problem_spec
-- function signature
(implementation: String → String → String)
-- inputs
(a b: String) :=
-- spec
let spec (result: String) :=
  a.all (fun c => c = '0' ∨ c = '1') →
  b.all (fun c => c = '0' ∨ c = '1') →
  a.length = b.length →
  result.length = a.length ∧
  result.all (fun c => c = '0' ∨ c = '1') ∧
  (∀ i, i < a.length →
  let i_pos := String.Pos.mk i;
  (a.get i_pos = b.get i_pos → result.get i_pos = '0') ∧
  (a.get i_pos ≠ b.get i_pos → result.get i_pos = '1'));
-- program termination
∃ result, implementation a b = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → String → String)
-- inputs
(a b: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ a b, problem_spec implementation a b) ↔
(∀ a b, generated_spec implementation a b) :=",sorry,def implementation (a b: String) : String :=,sorry,"-- #test implementation ""010"" ""110"" = ""100""","theorem correctness
(a b: String)
: problem_spec implementation a b
:=","by
unfold problem_spec
let result := implementation a b
use result
simp [result]
sorry",,,
def longest(strings: List[str]) -> Optional[str],"Out of list of strings, return the longest one. Return the first one in case of multiple
strings of the same length. Return None in case the input list is empty.
","[{""input"": [], ""expected_output"": ""None""}, {""input"": [""a"", ""b"", ""c""], ""expected_output"": ""a""}, {""input"": [""a"", ""bb"", ""ccc""], ""expected_output"": ""ccc""}]","def longest(strings: List[str]) -> Optional[str]
""""""Out of list of strings, return the longest one. Return the first one in case of multiple
strings of the same length. Return None in case the input list is empty.
""""""","def problem_spec
-- function signature
(implementation: List String → Option String)
-- inputs
(strings: List String) :=
-- spec
let spec (result: Option String) :=
  (result = none ↔ strings.length = 0) ∨
  (∃ longest, result = some longest ∧
  longest ∈ strings ∧
  ∀ s, s ∈ strings → s.length ≤ longest.length →
  (∃ i, i < strings.length ∧
  strings[i]! = longest ∧ ∀ j < i, strings[j]!.length < longest.length));
-- program termination
∃ result, implementation strings = result ∧
spec result","def generated_spec
-- function signature
(implementation: List String → Option String)
-- inputs
(strings: List String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ strings, problem_spec implementation strings) ↔
(∀ strings, generated_spec implementation strings) :=",sorry,def implementation (strings: List String) : Option String :=,"if strings = [] then none
else
  let longest := (strings.map (λ s => s.length)).maximum.get!
  (strings.find? (λ s => s.length = longest)).get!","#test implementation [""a"", ""b"", ""c""] = some ""a""
#test implementation [""a"", ""bb"", ""ccc""] = some ""ccc""
#test implementation [] = none","theorem correctness
(strings: List String)
: problem_spec implementation strings
:=","by
unfold problem_spec
let result := implementation strings
use result
simp [result]
sorry",,,
"def greatest_common_divisor(a: int, b: int) -> int","Return a greatest common divisor of two integers a and b
","[{""input"": [25, 15], ""expected_output"": 5}, {""input"": [3, 5], ""expected_output"": 1}]","def greatest_common_divisor(a: int, b: int) -> int
""""""Return a greatest common divisor of two integers a and b
""""""","def problem_spec
-- function signature
(implementation: Int → Int → Int)
-- inputs
(a b: Int) :=
-- spec
let spec (result: Int) :=
(result ∣ a) ∧
(result ∣ b) ∧
(∀ (d': Int),
(d' > 0) → (d' ∣ a) → (d' ∣ b) →
d' ≤ result);
-- program termination
∃ result, implementation a b = result ∧
spec result","def generated_spec
-- function signature
(implementation: Int → Int → Int)
-- inputs
(a b: Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ a b, problem_spec implementation a b) ↔
(∀ a b, generated_spec implementation a b) :=",sorry,def implementation (a b: Int) : Int :=,(Int.gcd a b),"#test implementation 25 15 = 5
#test implementation 3 5 = 1","theorem correctness
(a b: Int)
: problem_spec implementation a b
:=","by
unfold problem_spec
let result := implementation a b
use result
simp [result]
sorry",,,
def all_prefixes(string: str) -> List[str],"Return list of all prefixes from shortest to longest of the input string
","[{""input"": ""abc"", ""expected_output"": [""a"", ""ab"", ""abc""]}]","def all_prefixes(string: str) -> List[str]
""""""Return list of all prefixes from shortest to longest of the input string
""""""","def problem_spec
-- function signature
(implementation: String → List String)
-- inputs
(string: String) :=
-- spec
let spec (result: List String) :=
result.length = string.length ∧
∀ i, i < result.length →
result[i]! = string.take (i + 1);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → List String)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String) : List String :=,"let rec aux (chars: List Char) (result: List String) (i: Nat) :=
if i < chars.length then
  aux chars (result ++ [(chars.take (i + 1)).asString]) (i + 1)
else
  result;
aux string.toList [] 0","#test implementation ""abc"" = [""a"", ""ab"", ""abc""]","theorem correctness
(string: String)
: problem_spec implementation string
:=","by
unfold problem_spec
let result := implementation string
use result
simp [result]
sorry",,,
def string_sequence(n: int) -> str,"Return a string containing space-delimited numbers starting from 0 upto n inclusive.
","[{""input"": 0, ""expected_output"": ""0""}, {""input"": 5, ""expected_output"": ""0 1 2 3 4 5""}]","def string_sequence(n: int) -> str
""""""Return a string containing space-delimited numbers starting from 0 upto n inclusive.
""""""","def problem_spec
-- function signature
(implementation: Nat → String)
-- inputs
(n: Nat) :=
-- spec
let spec (result: String) :=
let result_nums := result.splitOn "" "";
result_nums.length = n + 1 ∧
∀ i, i < n + 1 → result_nums[i]! = i.repr;
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(implementation: Nat → String)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat) : String :=,"(String.join (List.map (fun i => String.append i.repr "" "") (List.range n))).append (n.repr)","#test implementation 0 = ""0""
#test implementation 5 = ""0 1 2 3 4 5""","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
sorry",,,
def count_distinct_characters(string: str) -> int,"Given a string, find out how many distinct characters (regardless of case) does it consist of
","[{""input"": ""xyzXYZ"", ""expected_output"": 3}, {""input"": ""Jerry"", ""expected_output"": 4}]","def count_distinct_characters(string: str) -> int
""""""Given a string, find out how many distinct characters (regardless of case) does it consist of
""""""","def problem_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) :=
-- spec
let spec (result: Nat) :=
let string_idx := {i: Nat | i < string.length}.toFinset
let characters := string_idx.image (fun i => string.toList.get! i)
let lowercase_characters := characters.image (fun c => c.toLower)
result = lowercase_characters.card;
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String) : Nat :=,"let char_set := string.toLower.toList.toFinset
char_set.card","#test implementation ""xyzXYZ"" = 3
#test implementation ""Jerry"" = 4","theorem correctness
(string: String)
: problem_spec implementation string
:=","by
unfold problem_spec
let result := implementation string
use result
simp [result]
simp [implementation]
rcases string with ⟨data⟩
induction data
simp
have h1: """".toLower = """" := by
  unfold String.toLower
  unfold Char.toLower
  unfold String.map
  unfold String.mapAux
  simp
  intros
  contradiction
simp [h1]
rename_i head tail ih
unfold String.toLower
rw [String.map_eq]
simp
sorry",,,
def parse_music(music_string: str) -> List[int],"Input to this function is a string representing musical notes in a special ASCII format.
Your task is to parse this string and return list of integers corresponding to how many beats does each
not last.

Here is a legend:
'o' - whole note, lasts four beats
'o|' - half note, lasts two beats
'.|' - quater note, lasts one beat
","[{""input"": ""o o| .| o| o| .| .| .| .| o o"", ""expected_output"": [4, 2, 1, 2, 2, 1, 1, 1, 1, 4, 4]}]","def parse_music(music_string: str) -> List[int]
""""""Input to this function is a string representing musical notes in a special ASCII format.
Your task is to parse this string and return list of integers corresponding to how many beats does each
not last.

Here is a legend:
'o' - whole note, lasts four beats
'o|' - half note, lasts two beats
'.|' - quater note, lasts one beat
""""""","def problem_spec
-- function signature
(implementation: String → List Nat)
-- inputs
(string: String) :=
-- spec
let not_map := fun
  | ""o"" => 4
  | ""o|"" => 2
  | "".|"" => 1
  | _ => 0;
let spec (result: List Nat) :=
let space_split := string.splitOn "" "";
space_split.length = result.length ∧
∀ i < result.length, not_map (space_split.get! i) = result.get! i;
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → List Nat)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String) : List Nat :=,"(string.splitOn "" "").map (fun
  | ""o"" => 4
  | ""o|"" => 2
  | "".|"" => 1
  | _ => 0)","#test implementation ""o o| .| o| o| .| .| .| .| o o"" = [4, 2, 1, 2, 2, 1, 1, 1, 1, 4, 4]","theorem correctness
(string: String)
: problem_spec implementation string
:=","by
unfold problem_spec
let result := implementation string
use result
simp [result]
simp [implementation]
intro i
by_cases h_i_lt_len: (i < (string.splitOn "" "").length)
simp [h_i_lt_len]
simp [h_i_lt_len]",,,
"def how_many_times(string: str, substring: str) -> int","Find how many times a given substring can be found in the original string. Count overlaping cases.
","[{""input"": ["""", ""a""], ""expected_output"": 0}, {""input"": [""aaa"", ""a""], ""expected_output"": 3}, {""input"": [""aaaa"", ""aa""], ""expected_output"": 3}]","def how_many_times(string: str, substring: str) -> int
""""""Find how many times a given substring can be found in the original string. Count overlaping cases.
""""""","def problem_spec
-- function signature
(implementation: String → String → Nat)
-- inputs
(string substring: String) :=
-- spec
let spec (result: Nat) :=
(string.length < substring.length → result = 0)
∧
(string.length = substring.length →
((string = substring ↔ result = 1) ∧
(substring ≠ string ↔ result = 0)))
∧
(substring.length < string.length  →
let subtring_start_idx := {i: Nat | i ≤ string.length - substring.length};
let substring_occurrences := {i ∈ subtring_start_idx | (string.take (i + substring.length)).drop i = substring };
result = substring_occurrences.toFinset.card);
-- program termination
∃ result, implementation string substring = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → String → Nat)
-- inputs
(string substring: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string substring, problem_spec implementation string substring) ↔
(∀ string substring, generated_spec implementation string substring) :=",sorry,def implementation (string: String) (substring: String) : Nat :=,(string.findAllSubstr substring).size,"#test implementation ""aaa"" ""a"" = 3
#test implementation ""aaaa"" ""aa"" = 3
#test implementation """" ""a"" = 0
#test implementation ""a"" """" = 1
#test implementation ""a"" ""a"" = 1","theorem correctness
(string: String)
(substring: String)
: problem_spec implementation string substring
:=","by
unfold problem_spec
let result := implementation string substring
use result
simp [result]
sorry",,,
def sort_numbers(numbers: str) -> str,"Input is a space-delimited string of numberals from 'zero' to 'nine'.
Valid choices are 'zero', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight' and 'nine'.
Return the string with numbers sorted from smallest to largest
","[{""input"": ""three one five"", ""expected_output"": ""one three five""}]","def sort_numbers(numbers: str) -> str
""""""Input is a space-delimited string of numberals from 'zero' to 'nine'.
Valid choices are 'zero', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight' and 'nine'.
Return the string with numbers sorted from smallest to largest
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(numbers: String) :=
-- spec
let word_to_number_map : String → Int := fun word =>
  match word with
  | ""zero"" => 0
  | ""one"" => 1
  | ""two"" => 2
  | ""three"" => 3
  | ""four"" => 4
  | ""five"" => 5
  | ""six"" => 6
  | ""seven"" => 7
  | ""eight"" => 8
  | ""nine"" => 9
  | _ => -1;
let is_sorted_asc : List Int → Bool := fun numbers =>
let rec is_sorted_asc_helper : List Int → Bool → Bool := fun numbers is_sorted =>
  match numbers with
  | [] => is_sorted
  | [x] => is_sorted
  | x::y::rest => if x <= y then is_sorted_asc_helper (y::rest) true else false;
is_sorted_asc_helper numbers false;
let spec (result: String) :=
let result_split := result.splitOn "" "";
let numbers_split := numbers.splitOn "" "";
let result_mapped_to_numbers := result_split.map word_to_number_map;
let numbers_as_str_mapped_to_numbers := numbers_split.map word_to_number_map;
¬ -1 ∈ numbers_as_str_mapped_to_numbers →
result_split.length = numbers_split.length ∧
(∀ n, n ∈ numbers_as_str_mapped_to_numbers →
∃ m, m ∈ result_mapped_to_numbers) ∧
(∀ n, n ∈ result_mapped_to_numbers →
∃ m, m ∈ numbers_as_str_mapped_to_numbers) ∧
(∀ n, numbers_as_str_mapped_to_numbers.count n = result_mapped_to_numbers.count n) ∧
is_sorted_asc result_mapped_to_numbers = true;
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → String)
-- inputs
(numbers: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: String) : String :=,"let word_to_number_map : String → Int := fun word =>
  match word with
  | ""zero"" => 0
  | ""one"" => 1
  | ""two"" => 2
  | ""three"" => 3
  | ""four"" => 4
  | ""five"" => 5
  | ""six"" => 6
  | ""seven"" => 7
  | ""eight"" => 8
  | ""nine"" => 9
  | _ => -1;
let number_to_word_map : Int → String := fun number =>
  match number with
  | 0 => ""zero""
  | 1 => ""one""
  | 2 => ""two""
  | 3 => ""three""
  | 4 => ""four""
  | 5 => ""five""
  | 6 => ""six""
  | 7 => ""seven""
  | 8 => ""eight""
  | 9 => ""nine""
  | _ => ""invalid"";
let numbers_split := numbers.splitOn "" "";
let numbers_mapped_to_numbers := numbers_split.map word_to_number_map;
let sorted_numbers := numbers_mapped_to_numbers.mergeSort;
let sorted_numbers_mapped_to_words := sorted_numbers.map number_to_word_map;
let join: List String → String := fun words =>
  let head := words.get! 0;
  let tail := words.drop 1;
  tail.foldl (fun acc word => acc ++ "" "" ++ word) head;
join sorted_numbers_mapped_to_words","#test implementation ""three one five"" = ""one three five""","theorem correctness
(numbers: String)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
"def find_closest_elements(numbers: List[float]) -> Tuple[float, float]","From a supplied list of numbers (of length at least two) select and return two that are the closest to each
other and return them in order (smaller number, larger number).
","[{""input"": [1.0, 2.0, 3.0, 4.0, 5.0, 2.2], ""expected_output"": ""(2.0, 2.2)""}, {""input"": [1.0, 2.0, 3.0, 4.0, 5.0, 2.0], ""expected_output"": ""(2.0, 2.0)""}]","def find_closest_elements(numbers: List[float]) -> Tuple[float, float]
""""""From a supplied list of numbers (of length at least two) select and return two that are the closest to each
other and return them in order (smaller number, larger number).
""""""","def problem_spec
-- function signature
(implementation: List Rat → (Rat × Rat))
-- inputs
(numbers: List Rat) :=
-- spec
let spec (result: (Rat × Rat)) :=
2 ≤ numbers.length →
(let (smaller, larger) := result;
let abs_diff := |larger - smaller|;
smaller ≤ larger ∧
smaller ∈ numbers ∧
larger ∈ numbers ∧
(∀ x y, x ∈ numbers → y ∈ numbers →  abs_diff ≤ |x - y|) ∧
(smaller = larger → 1 ≤ (numbers.filter (fun z => z = smaller)).length));
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Rat → (Rat × Rat))
-- inputs
(numbers: List Rat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Rat): (Rat × Rat) :=,"let n := numbers.length;
let sorted_numbers := numbers.mergeSort;
let min_diff := sorted_numbers.get! 1 - sorted_numbers.get! 0;
let min_pair := (sorted_numbers.get! 0, sorted_numbers.get! 1);
let rec loop (i: Nat) (min_diff: Rat) (min_pair: (Rat × Rat)): (Rat × Rat) :=
  if i < n - 1 then
    let diff := sorted_numbers.get! (i + 1) - sorted_numbers.get! i;
    if diff < min_diff then
      loop (i + 1) diff (sorted_numbers.get! i, sorted_numbers.get! (i + 1))
    else
      loop (i + 1) min_diff min_pair
  else
    min_pair;
loop 1 min_diff min_pair","#test implementation [1.0, 2.0, 3.0, 4.0, 5.0, 2.2] = (2.0, 2.2)
#test implementation [1.0, 2.0, 3.0, 4.0, 5.0, 2.0] = (2.0, 2.0)","theorem correctness
(numbers: List Rat)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def rescale_to_unit(numbers: List[float]) -> List[float],"Given list of numbers (of at least two elements), apply a linear transform to that list,
such that the smallest number will become 0 and the largest will become 1
","[{""input"": [1.0, 2.0, 3.0, 4.0, 5.0], ""expected_output"": [0.0, 0.25, 0.5, 0.75, 1.0]}]","def rescale_to_unit(numbers: List[float]) -> List[float]
""""""Given list of numbers (of at least two elements), apply a linear transform to that list,
such that the smallest number will become 0 and the largest will become 1
""""""","def problem_spec
-- function signature
(implementation: List Rat → List Rat)
-- inputs
(numbers: List Rat) :=
-- spec
let spec (result: List Rat) :=
2 ≤ numbers.length →
let min_elem := numbers.min?.get!;
let max_elem := numbers.max?.get!;
let range := max_elem - min_elem;
result.length = numbers.length ∧
∀ i, i < numbers.length →
(min_elem ≠ max_elem →
result[i]! = (numbers[i]! - min_elem) / range) ∧
(min_elem = max_elem →
result[i]! = 0);
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Rat → List Rat)
-- inputs
(numbers: List Rat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Rat): List Rat :=,sorry,"-- #test implementation [1.0, 2.0, 3.0, 4.0, 5.0] = [0.0, 0.25, 0.5, 0.75, 1.0]","theorem correctness
(numbers: List Rat)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def strlen(string: str) -> int,"Return length of given string
","[{""input"": """", ""expected_output"": 0}, {""input"": ""abc"", ""expected_output"": 3}]","def strlen(string: str) -> int
""""""Return length of given string
""""""","def problem_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) :=
-- spec
let spec (result: Nat) :=
-- every character in the string is counted once
result = 0 ↔ string.isEmpty ∧
(0 < result → result - 1 = implementation (string.drop 1))
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String): Nat :=,string.length,"#test implementation """" = 0
#test implementation ""abc"" = 3","theorem correctness
(string: String)
: problem_spec implementation string
:=","by
unfold problem_spec
let result := implementation string
use result
simp [result]
sorry",,,
def largest_divisor(n: int) -> int,"For a given number n, find the largest number that divides n evenly, smaller than n
","[{""input"": 15, ""expected_output"": 5}]","def largest_divisor(n: int) -> int
""""""For a given number n, find the largest number that divides n evenly, smaller than n
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
0 < n → 0 < result → result ∣ n → ∀ x, x ∣ n → x ≠ n → x ≤ result;
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat) : Nat :=,"let possible_divisors := (List.range (n / 2 + 1)).drop 1
let reversed_possible_divisors := List.reverse possible_divisors;
Id.run do
  for i in reversed_possible_divisors do
    if n % i = 0 then
      return i
  return 1",#test implementation 15 = 5,"theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
sorry",,,
def factorize(n: int) -> List[int],"Return list of prime factors of given integer in the order from smallest to largest.
Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
Input number should be equal to the product of all factors
","[{""input"": 8, ""expected_output"": [2, 2, 2]}, {""input"": 25, ""expected_output"": [5, 5]}, {""input"": 70, ""expected_output"": [2, 5, 7]}]","def factorize(n: int) -> List[int]
""""""Return list of prime factors of given integer in the order from smallest to largest.
Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
Input number should be equal to the product of all factors
""""""","def problem_spec
-- function signature
(implementation: Nat → List Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: List Nat) :=
2 ≤ n →
(result.prod = n ∧
List.Sorted Nat.le result ∧
result.all (λ i => n % i = 0 ∧ Nat.Prime i));
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(implementation: Nat → List Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat) : List Nat :=,"Id.run do
let mut result := []
for num in List.range' 2 n.sqrt do
  let mut n' := n
  while n' % num = 0 do
    n' := n' / num
    result := result ++ [num]
result","#test implementation 8 = [2, 2, 2]
#test implementation 25 = [5, 5]
#test implementation 70 = [2, 5, 7]","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
simp [implementation, Id.run]
unfold List.foldl
intro h_2_lt_n
sorry",,,
def remove_duplicates(numbers: List[int]) -> List[int],"From a list of integers, remove all elements that occur more than once.
Keep order of elements left the same as in the input.
","[{""input"": [1, 2, 3, 2, 4], ""expected_output"": [1, 3, 4]}]","def remove_duplicates(numbers: List[int]) -> List[int]
""""""From a list of integers, remove all elements that occur more than once.
Keep order of elements left the same as in the input.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: List Int) :=
(∀ i: Int, i ∈ result → numbers.count i = 1) ∧
(∀ i: Int, i ∈ numbers → numbers.count i = 1 → i ∈ result) ∧
(∀ i j : Nat, i < result.length → j < result.length → i < j →
∃ ip jp : Nat, ip < jp ∧ result[i]! = numbers[ip]! ∧ result[j]! = numbers[jp]!)
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Int) : List Int :=,"Id.run do
  let mut result := []
  for number in numbers do
    if numbers.count number = 1 then
      result := result ++ [number]
  return result","#test implementation [1, 2, 3, 2, 4] = [1, 3, 4]","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def flip_case(string: str) -> str,"For a given string, flip lowercase characters to uppercase and uppercase to lowercase.
","[{""input"": ""Hello"", ""expected_output"": ""hELLO""}]","def flip_case(string: str) -> str
""""""For a given string, flip lowercase characters to uppercase and uppercase to lowercase.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(string: String) :=
-- spec
let spec (result: String) :=
let chars_in_result := result.toList;
let chars_in_string := string.toList;
chars_in_result.length = string.length ∧
(∀ i, i < chars_in_result.length →
  let c := chars_in_result.get! i;
  let c' := chars_in_string.get! i;
  (c.isUpper → c'.isLower) ∧
  (c.isLower → c'.isUpper) ∧
  ((¬ c.isUpper ∧ ¬ c.isLower) → c = c')
);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(implementation: String → String)
-- inputs
(string: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ string, problem_spec implementation string) ↔
(∀ string, generated_spec implementation string) :=",sorry,def implementation (string: String) : String :=,string.map (λ c => if c.isUpper then c.toLower else c.toUpper),"#test implementation ""Hello"" = ""hELLO""","theorem correctness
(string: String)
: problem_spec implementation string
:=","by
unfold problem_spec
let result := implementation string
use result
simp [result]
sorry",,,
def concatenate(strings: List[str]) -> str,"Concatenate list of strings into a single string
","[{""input"": [], ""expected_output"": """"}, {""input"": [""a"", ""b"", ""c""], ""expected_output"": ""abc""}]","def concatenate(strings: List[str]) -> str
""""""Concatenate list of strings into a single string
""""""","def problem_spec
-- function signature
(implementation: List String → String)
-- inputs
(strings: List String) :=
-- spec
let spec (result: String) :=
let result_chars := result.toList;
result_chars.length = (strings.map (λ s => s.length)).sum ∧
∀ i, i < strings.length →
(let string_in_result := strings.get! i;
let end_idx := ((strings.take (i + 1)).map (λ s => s.length)).sum;
let start_idx := end_idx - string_in_result.length;
let corresponding_string_in_result := ((result_chars.take end_idx).drop start_idx).asString;
corresponding_string_in_result = string_in_result);
-- program termination
∃ result, implementation strings = result ∧
spec result","def generated_spec
-- function signature
(implementation: List String → String)
-- inputs
(strings: List String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ strings, problem_spec implementation strings) ↔
(∀ strings, generated_spec implementation strings) :=",sorry,def implementation (strings: List String) : String :=,String.join strings,"#test implementation [] = """"
#test implementation [""a"", ""b"", ""c""] = ""abc""","theorem correctness
(strings: List String)
: problem_spec implementation strings
:=","by
unfold problem_spec
let result := implementation strings
use result
simp [result]
apply And.intro
simp [implementation]
repeat sorry",,,
"def filter_by_prefix(strings: List[str], prefix: str) -> List[str]","Filter an input list of strings only for ones that start with a given prefix.
","[{""input"": [[], ""a""], ""expected_output"": []}, {""input"": [[""abc"", ""bcd"", ""cde"", ""array""], ""a""], ""expected_output"": [""abc"", ""array""]}]","def filter_by_prefix(strings: List[str], prefix: str) -> List[str]
""""""Filter an input list of strings only for ones that start with a given prefix.
""""""","def problem_spec
-- function signature
(implementation: List String → String → List String)
-- inputs
(strings: List String)
(pref: String) :=
-- spec
let spec (result: List String) :=
result.all (λ s => s.startsWith pref) ∧
result.all (λ s => s ∈ strings) ∧
strings.all (λ s => s.startsWith pref → s ∈ result) ∧
∀ s : String, s ∈ result → result.count s = strings.count s;
-- program termination
∃ result, implementation strings pref = result ∧
spec result","def generated_spec
-- function signature
(implementation: List String → String → List String)
-- inputs
(strings: List String)
(pref: String) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ strings pref, problem_spec implementation strings pref) ↔
(∀ strings pref, generated_spec implementation strings pref) :=",sorry,def implementation (strings: List String) (pref: String) : List String :=,strings.filter (λ s => s.startsWith pref),"#test implementation [] ""a"" = []
#test implementation [""abc"", ""bcd"", ""cde"", ""array""] ""a"" = [""abc"", ""array""]","theorem correctness
(strings: List String)
(pref: String)
: problem_spec implementation strings pref
:=","by
unfold problem_spec
let result := implementation strings pref
use result
simp [result]
repeat sorry",,,
def get_positive(l: list),"Return only positive numbers in the list.
","[{""input"": [-1, 2, -4, 5, 6], ""expected_output"": [2, 5, 6]}, {""input"": [5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10], ""expected_output"": [5, 3, 2, 3, 9, 123, 1]}]","def get_positive(l: list)
""""""Return only positive numbers in the list.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: List Int) :=
  result.all (λ x => x > 0 ∧ x ∈ numbers) ∧
  numbers.all (λ x => x > 0 → x ∈ result) ∧
  result.all (λ x => result.count x = numbers.count x);
-- program termination
∃ result,
  implementation numbers = result ∧
  spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ numbers, problem_spec implementation numbers) ↔
(∀ numbers, generated_spec implementation numbers) :=",sorry,def implementation (numbers: List Int): List Int :=,numbers.filter (λ x => x > 0),"#test implementation [(-1), 2, (-4), 5, 6] = [2, 5, 6]
#test implementation [5, 3, (-5), 2, (-3), 3, 9, 0, 123, 1, (-10)] = [5, 3, 2, 3, 9, 123, 1]","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
repeat sorry",,,
def is_prime(n),"Return true if a given number is prime, and false otherwise.
","[{""input"": 6, ""output"": false}, {""input"": 101, ""output"": true}, {""input"": 11, ""output"": true}, {""input"": 13441, ""output"": true}, {""input"": 61, ""output"": true}, {""input"": 4, ""output"": false}, {""input"": 1, ""output"": false}]","def is_prime(n)
""""""Return true if a given number is prime, and false otherwise.
""""""","def problem_spec
-- function signature
(implementation: Nat → Bool)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Bool) :=
  result ↔ ¬ (∃ k, 2 ≤ k ∧ k < n ∧ n % k = 0);
-- program termination
∃ result,
  implementation n = result ∧
  spec result","def generated_spec
-- function signature
(implementation: Nat → Bool)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat): Bool :=,Nat.Prime n,"#test implementation 6 = false
#test implementation 101 = true
#test implementation 11 = true
#test implementation 13441 = true
#test implementation 61 = true
#test implementation 4 = false
#test implementation 1 = false","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
simp [implementation]
sorry",,,
def find_zero(xs: list),"xs are coefficients of a polynomial.
find_zero find x such that poly(x) = 0.
find_zero returns only only zero point, even if there are many.
Moreover, find_zero only takes list xs having even number of coefficients
and largest non zero coefficient as it guarantees a solution.
Note(George): This problem has been modified from the original HumanEval spec because of Real is not a computable type, but a zero does not necessarily exist over the rationals.
","[{""input"": [1, 2], ""output"": -0.5}, {""input"": [-6, 11, -6, 1], ""output"": 1.0}]","def find_zero(xs: list)
""""""xs are coefficients of a polynomial.
find_zero find x such that poly(x) = 0.
find_zero returns only only zero point, even if there are many.
Moreover, find_zero only takes list xs having even number of coefficients
and largest non zero coefficient as it guarantees a solution.
Note(George): This problem has been modified from the original HumanEval spec because of Real is not a computable type, but a zero does not necessarily exist over the rationals.
""""""","def problem_spec
-- function signature
(implementation: List Rat → Rat)
-- inputs
(xs: List Rat) :=
-- spec
let spec (result: Rat) :=
  let eps := (1: Rat) / 1000000;
  xs.length ≥ 1 → xs.length % 2 = 0 →
  ∀ poly : Polynomial Rat,
    poly.degree = some (xs.length - 1) →
    (∀ i, i ≤ xs.length - 1 → poly.coeff i = xs.get! i) →
    |poly.eval result| ≤ eps;
-- program termination
∃ result,
  implementation xs = result ∧
  spec result","def generated_spec
-- function signature
(implementation: List Rat → Rat)
-- inputs
(xs: List Rat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ xs, problem_spec implementation xs) ↔
(∀ xs, generated_spec implementation xs) :=",sorry,def implementation (xs: List Rat) : Rat :=,"let rec poly (xs: List Rat) (x: Rat) := xs.reverse.foldl (λ acc a => acc * x + a) 0;
let rec poly' (xs: List Rat) (x: Rat) := (xs.drop 1).reverse.foldl (λ acc a => acc * x + a) 0;
let rec eps := (1: Rat) / 1000000;
let rec find_zero (xs: List Rat) (guess: Rat) (fuel: Nat) :=
let eval := poly xs guess;
let eval' := poly' xs guess;
if eval ≤ eps ∨ fuel = 0 then (guess, fuel)
else
let guess' := (eval' * guess - eval) / eval';
find_zero xs guess' (fuel - 1);
(find_zero xs 1.0 1000000).1
-- Note: The above implementation can fail to converge in some cases. For example,
-- on the test case [-6, 11, -6, 1] as the derivative of the polynomial
-- can be zero at if the guess is 0. In such cases, the implementation will not
-- converge. The implementation can be improved by using a better guess.","#test implementation [1, 2] = -0.5
#test implementation [-6, 11, -6, 1] = 1.0","theorem correctness
(xs: List Rat)
: problem_spec implementation xs
:=","by
unfold problem_spec
let result := implementation xs
use result
simp [result]
intros h1 h2 poly h3 h4
repeat sorry",,,
def sort_third(l: list),"This function takes a list l and returns a list l' such that
l' is identical to l in the indicies that are not divisible by three, while its values at the indicies that are divisible by three are equal
to the values of the corresponding indicies of l, but sorted.
","[{""input"": [1, 2, 3], ""output"": [1, 2, 3]}, {""input"": [5, 6, 3, 4, 8, 9, 2], ""output"": [2, 6, 3, 4, 8, 9, 5]}]","def sort_third(l: list)
""""""This function takes a list l and returns a list l' such that
l' is identical to l in the indicies that are not divisible by three, while its values at the indicies that are divisible by three are equal
to the values of the corresponding indicies of l, but sorted.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) :=
-- spec
let spec (result: List Int) :=
  l.length = result.length ∧
  let every_third_idx := (List.range l.length).filter (λ i => i % 3 = 0);
  let every_third_val_in_result := every_third_idx.map (λ i => result.get! i);
  let every_third_val := every_third_idx.map (λ i => l.get! i);
  (∀ i, i < l.length → (i % 3 ≠ 0 → l.get! i = result.get! i)) ∧
  List.Sorted Int.le every_third_val_in_result ∧
  every_third_val.all (λ x => every_third_val_in_result.count x = every_third_val.count x);
-- program termination
∃ result, implementation l = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ l, problem_spec implementation l) ↔
(∀ l, generated_spec implementation l) :=",sorry,def implementation (l: List Int) : List Int :=,"let every_third_idx := (List.range l.length).filter (λ i => i % 3 = 0);
let every_third_val := every_third_idx.map (λ i => l.get! i);
let every_third_val_sorted := List.mergeSort every_third_val;
let result := l.mapIdx (λ i v =>
  if i % 3 = 0 then every_third_val_sorted.get! (i / 3)
  else v);
result","#test implementation [1, 2, 3] = [1, 2, 3]
#test implementation [5, 6, 3, 4, 8, 9, 2] = [2, 6, 3, 4, 8, 9, 5]","theorem correctness
(l: List Int)
: problem_spec implementation l
:=","by
unfold problem_spec
let result := implementation l
use result
simp [result]
repeat sorry",,,
def unique(l: list),"Return sorted unique elements in a list.
","[{""input"": [5, 3, 5, 2, 3, 3, 9, 0, 123], ""output"": [0, 2, 3, 5, 9, 123]}]","def unique(l: list)
""""""Return sorted unique elements in a list.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) :=
-- spec
let spec (result: List Int) :=
  (∀ x, x ∈ result ↔ x ∈ l ∧ result.count x = 1) ∧
  List.Sorted Int.le result
-- program termination
∃ result,
  implementation l = result ∧
  spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ l, problem_spec implementation l) ↔
(∀ l, generated_spec implementation l) :=",sorry,def implementation (l: List Int) : List Int :=,l.eraseDup.mergeSort,"#test implementation [5, 3, 5, 2, 3, 3, 9, 0, 123] = [0, 2, 3, 5, 9, 123]","theorem correctness
(l: List Int)
: problem_spec implementation l
:=","by
unfold problem_spec
let result := implementation l
use result
simp [result]
simp [implementation]
apply And.intro
intro x
apply Iff.intro
intro h
apply And.intro
repeat sorry",,,
def max_element(l: list),"Return maximum element in the list.
","[{""input"": [1, 2, 3], ""output"": 3}, {""input"": [5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10], ""output"": 123}]","def max_element(l: list)
""""""Return maximum element in the list.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(l: List Int) :=
-- spec
let spec (result: Int) :=
  l.length > 0 →
  ((∀ i, i < l.length → l.get! i ≤ result) ∧
  (∃ i, i < l.length ∧ l.get! i = result));
-- program termination
∃ result, implementation l = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(l: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ l, problem_spec implementation l) ↔
(∀ l, generated_spec implementation l) :=",sorry,def implementation (l: List Int) : Int :=,"match List.maximum l with
| none => panic! ""empty list""
| some x => x","#test implementation [1, 2, 3] = 3
#test implementation [5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10] = 123","theorem correctness
(l: List Int)
: problem_spec implementation l
:=","by
unfold problem_spec
let result := implementation l
use result
simp [result]
simp [implementation]
by_cases h_l_not_nil: l ≠ []
have h_max_not_none := List.maximum_ne_bot_of_ne_nil h_l_not_nil
intro h_len_gt_zero
apply And.intro
have h_l_max : ∃ r, l.maximum = some r := by
  have h_l_max' : l.maximum ≠ none := h_max_not_none
  rw [Option.ne_none_iff_exists] at h_l_max'
  obtain ⟨r, h_l_max⟩ := h_l_max'
  use r
  simp [h_l_max]
obtain ⟨r, h_l_max ⟩ := h_l_max
simp [h_l_max]
induction l
intro i h_i_lt_len
simp
simp at h_len_gt_zero
rename_i head tail ih
rw [List.maximum_cons] at h_l_max
by_cases h_tail_nil: tail = []
simp [h_tail_nil] at *
have h_head_eq_r : head = r := by
  simp [WithBot.some] at h_l_max
  rw [Option.some_inj] at h_l_max
  exact h_l_max
rw [h_head_eq_r]
simp [h_tail_nil] at ih
have h_0_lt_len: 0 < tail.length :=
   List.length_pos_of_ne_nil h_tail_nil
simp [h_0_lt_len] at ih
repeat sorry",,,
def fizz_buzz(n: int),"Return the number of times the digit 7 appears in integers less than n which are divisible by 11 or 13.
","[{""input"": 50, ""output"": 0}, {""input"": 78, ""output"": 2}, {""input"": 79, ""output"": 3}]","def fizz_buzz(n: int)
""""""Return the number of times the digit 7 appears in integers less than n which are divisible by 11 or 13.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
  (n = 0 → result = 0) ∧
  (0 < n → result = implementation (n - 1) →
    (n % 11 ≠  0 ∧  n % 13 ≠  0) ∨ n.repr.count '7' = 0) ∧
  (0 < n → result ≠ implementation (n - 1) →
    (n % 11 = 0 ∨  n % 13 = 0) ∧
    result - implementation (n - 1) = n.repr.count '7')
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat) : Nat :=,"let all_numbers_lt_n := {x | x < n};
let multiples_of_11_or_13 := all_numbers_lt_n ∩ {x | x % 11 = 0 ∨ x % 13 = 0};
let possible_multiple_with_7 := multiples_of_11_or_13 ∩ {x | x.repr.contains '7'};
possible_multiple_with_7.toFinset.sum (λ x => x.repr.count '7')","#test implementation 50 = 0
#test implementation 78 = 2
#test implementation 79 = 3","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
simp [implementation]
sorry",,,
def sort_even(l: list),"This function takes a list l and returns a list l' such that
l' is identical to l in the odd indicies, while its values at the even indicies are equal
to the values of the even indicies of l, but sorted.
","[{""input"": [1, 2, 3], ""output"": [1, 2, 3]}, {""input"": [5, 6, 3, 4], ""output"": [3, 6, 5, 4]}]","def sort_even(l: list)
""""""This function takes a list l and returns a list l' such that
l' is identical to l in the odd indicies, while its values at the even indicies are equal
to the values of the even indicies of l, but sorted.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) :=
-- spec
let spec (result: List Int) :=
  l.length = result.length ∧
  let even_idx := (List.range l.length).filter (λ i => i % 2 = 0);
  let even_val_in_result := even_idx.map (λ i => result.get! i);
  let even_val := even_idx.map (λ i => l.get! i);
  (∀ i, i < l.length → (i % 2 ≠ 0 → l.get! i = result.get! i)) ∧
  List.Sorted Int.le even_val_in_result ∧
  even_val.all (λ x => even_val_in_result.count x = even_val.count x);
-- program termination
∃ result, implementation l = result ∧
spec result","def generated_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(l: List Int) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ l, problem_spec implementation l) ↔
(∀ l, generated_spec implementation l) :=",sorry,def implementation (l: List Int) : List Int :=,"let even_idx := (List.range l.length).filter (λ i => i % 2 = 0);
let even_val := even_idx.map (λ i => l.get! i);
let even_val_sorted := List.mergeSort even_val;
let result := l.mapIdx (λ i v =>
  if i % 2 = 0 then even_val_sorted.get! (i / 2)
  else v);
result","#test implementation [1, 2, 3] = [1, 2, 3]
#test implementation [5, 6, 3, 4] = [3, 6, 5, 4]","theorem correctness
(l: List Int)
: problem_spec implementation l
:=","by
unfold problem_spec
let result := implementation l
use result
simp [result]
sorry",,,
def encode_cyclic(s: str) -> str,"Returns an encoded string by cycling each group of three consecutive characters.
Specifically, each group of exactly three characters 'abc' is transformed to 'bca'.
Groups of fewer than three characters at the end of the string remain unchanged.
","[{""input"": ""abcdef"", ""expected_output"": ""bcaefd""}, {""input"": ""abcde"", ""expected_output"": ""bcade""}, {""input"": ""ab"", ""expected_output"": ""ab""}]","def encode_cyclic(s: str) -> str
""""""Returns an encoded string by cycling each group of three consecutive characters.
Specifically, each group of exactly three characters 'abc' is transformed to 'bca'.
Groups of fewer than three characters at the end of the string remain unchanged.
""""""","def problem_spec
(impl: String → String)
(s: String) :=
let n := s.length;
let extract (chars: List Char) (start_index: ℕ) (end_index: ℕ) :=
  (chars.drop start_index).take (end_index - start_index + 1);
let spec (result: String) :=
  let encoded_chars := result.toList;
  let original_chars := s.toList;
  encoded_chars.length = n ∧
  (∀ i : ℕ, i * 3 + 3 ≤ n →
    extract encoded_chars (i * 3) (i * 3 + 2) =
      [original_chars.get! (i * 3 + 1), original_chars.get! (i * 3 + 2), original_chars.get! (i * 3)]) ∧
  (n % 3 ≠ 0 → extract encoded_chars (n - n % 3) (n - 1) =
    extract original_chars (n - n % 3) (n - 1));
-- program termination
∃ result,
  impl s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""abcdef"" = ""bcaefd""
-- #test implementation ""abcde"" = ""bcade""
-- #test implementation ""ab"" = ""ab""","theorem correctness
(s: String)
: problem_spec implementation s :=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry",,,
def prime_fib(n: int),"prime_fib returns n-th prime Fibonacci number.
Note(George): A proof of this problem requires the resolution of the open conjecture: there are infinitely many prime Fibonacci numbers.
","[{""input"": 1, ""output"": 2}, {""input"": 2, ""output"": 3}, {""input"": 3, ""output"": 5}, {""input"": 4, ""output"": 13}, {""input"": 5, ""output"": 89}]","def prime_fib(n: int)
""""""prime_fib returns n-th prime Fibonacci number.
Note(George): A proof of this problem requires the resolution of the open conjecture: there are infinitely many prime Fibonacci numbers.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
  n > 0 →
    (∃ i, Nat.fib i = result ∧ Nat.Prime result ∧
      (∃! S : Finset Nat, S.card = n - 1 ∧
      (∀ y ∈ S, (∃ k, y = Nat.fib k) ∧ y < result ∧ Nat.Prime y))
    )
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ implementation,
(∀ n, problem_spec implementation n) ↔
(∀ n, generated_spec implementation n) :=",sorry,def implementation (n: Nat) : Nat :=,"let rec fib_prime (n: Nat) (i: Nat) : Nat :=
  if Nat.Prime (Nat.fib i) then
    if n = 1 ∨ n = 0
    then Nat.fib i
    else fib_prime (n - 1) (i + 1)
  else fib_prime n (i + 1)
termination_by n
decreasing_by
  -- Really hard to prove termination here
  sorry
  sorry
fib_prime n 0","-- NOTE we changed to #eval! instead of #test
-- because we can't use #test for implementation
-- without a proof of termination
#eval! implementation 1 = 2
#eval! implementation 2 = 3
#eval! implementation 3 = 5
#eval! implementation 4 = 13
#eval! implementation 5 = 89","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
repeat sorry",,,
def triples_sum_to_zero(numbers: List[int]) -> Bool,"triples_sum_to_zero takes a list of integers as an input.
it returns True if there are three distinct elements in the list that
sum to zero, and False otherwise.
","[{""input"": [1, 3, 5, 0], ""expected_output"": false}, {""input"": [1, 3, -2, 1], ""expected_output"": true}]","def triples_sum_to_zero(numbers: List[int]) -> Bool
""""""triples_sum_to_zero takes a list of integers as an input.
it returns True if there are three distinct elements in the list that
sum to zero, and False otherwise.
""""""","def problem_spec
-- function signature
(implementation: List Int → Bool)
-- inputs
(numbers: List Int) :=
let sum_i_j_k (i j k: Nat) : Bool :=
  numbers[i]! + numbers[j]! + numbers[k]! = 0;
let exists_zero := 3 ≤ numbers.length ∧ (∃ i j k, i ≠ j ∧ i ≠ k ∧ j ≠ k ∧
 i < numbers.length ∧ j < numbers.length ∧ k < numbers.length ∧
 sum_i_j_k i j k)
-- spec
let spec (result: Bool) :=
result ↔ exists_zero
-- -- program termination
∃ result, implementation numbers = result ∧
spec result
--","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(x: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Int) : Bool :=,sorry,"-- #test implementation [1, 3, 5, 0] = false
-- #test implementation [1, 3, -2, 1] = true
-- #test implementation [1, 2, 3, 7] = false
-- #test implementation [2, 4, -5, 3, 9, 7] = true
-- #test implementation [1] = false","theorem correctness
(numbers : List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def car_race_collision(x: Nat) -> Nat,"Imagine a road that's a perfectly straight infinitely long line.
n cars are driving left to right;  simultaneously, a different set of n cars
are driving right to left.   The two sets of cars start out being very far from
each other.  All cars move in the same speed.  Two cars are said to collide
when a car that's moving left to right hits a car that's moving right to left.
However, the cars are infinitely sturdy and strong; as a result, they continue moving
in their trajectory as if they did not collide.
This function outputs the number of such collisions.
","[{""input"": 0, ""expected_output"": 0}, {""input"": 5, ""expected_output"": 25}]","def car_race_collision(x: Nat) -> Nat
""""""Imagine a road that's a perfectly straight infinitely long line.
n cars are driving left to right;  simultaneously, a different set of n cars
are driving right to left.   The two sets of cars start out being very far from
each other.  All cars move in the same speed.  Two cars are said to collide
when a car that's moving left to right hits a car that's moving right to left.
However, the cars are infinitely sturdy and strong; as a result, they continue moving
in their trajectory as if they did not collide.
This function outputs the number of such collisions.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(x : Nat) :=
-- spec
let spec (result: Nat) :=
  ∃ x_list : List Nat, x_list.length = x ∧ x_list.all (fun i => i = x)
  ∧ x_list.sum = result
-- -- program termination
∃ result, implementation x = result ∧
spec result
--","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(x: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (x : Nat) : Nat :=,sorry,"-- #test implementation 0 = 0
-- #test implementation 5 = 25","theorem correctness
(x : Nat)
: problem_spec implementation x
:=","by
unfold problem_spec
let result := implementation x
use result
simp [result]
sorry",,,
def incr_list(numbers: List[Int]) -> List[Int],"incr_list takes a list of integers as input and returns a new list
where each element is incremented by 1.
","[{""input"": [], ""expected_output"": []}, {""input"": [1, 3, -2, 1], ""expected_output"": [2, 4, -1, 2]}]","def incr_list(numbers: List[Int]) -> List[Int]
""""""incr_list takes a list of integers as input and returns a new list
where each element is incremented by 1.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: List Int) :=
  (result.length = numbers.length) ∧
  ∀ i, i < numbers.length →
  result[i]! = numbers[i]! + 1
-- -- program termination
∃ result, implementation numbers = result ∧
spec result
--","def generated_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(x: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Int) : List Int :=,sorry,"-- #test implementation [1, 2, 3] = [2, 3, 4]
-- #test implementation [5, 3, 5, 2, 3, 3, 9, 0, 123] = [6, 4, 6, 3, 4, 4, 10, 1, 124]","theorem correctness
(numbers : List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def pairs_sum_to_zero(numbers: List[int]) -> Bool,"pairs_sum_to_zero takes a list of integers as an input.
it returns True if there are two distinct elements in the list that
sum to zero, and False otherwise.
","[{""input"": [1, 3, 5, 0], ""expected_output"": false}, {""input"": [1, 3, -2, 1], ""expected_output"": false}, {""input"": [1], ""expected_output"": false}, {""input"": [2, 4, -5, 3, 5, 7], ""expected_output"": true}]","def pairs_sum_to_zero(numbers: List[int]) -> Bool
""""""pairs_sum_to_zero takes a list of integers as an input.
it returns True if there are two distinct elements in the list that
sum to zero, and False otherwise.
""""""","def problem_spec
-- function signature
(implementation: List Int → Bool)
-- inputs
(numbers: List Int) :=
let sum_i_j (i j: Nat) : Bool :=
  numbers[i]! + numbers[j]! = 0;
let exists_zero := 2 ≤ numbers.length ∧ (∃ i j, i ≠ j ∧
 i < numbers.length ∧ j < numbers.length ∧
 sum_i_j i j)
-- spec
let spec (result: Bool) :=
result ↔ exists_zero
-- -- program termination
∃ result, implementation numbers = result ∧
spec result
--","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(x: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Int) : Bool :=,sorry,"-- #test implementation [1, 3, 5, 0] = false
-- #test implementation [1, 3, -2, 1] = false
-- #test implementation [1, 2, 3, 7] = false
-- #test implementation [2, 4, -5, 3, 5, 7] = true
-- #test implementation [1] = false","theorem correctness
(numbers : List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
"def change_base(x: Nat, base: Nat) -> String","Change numerical base of input number x to base.
return string representation after the conversion.
base numbers are less than 10.
","[{""input"": ""(8, 3)"", ""expected_output"": ""22""}, {""input"": ""(8, 2)"", ""expected_output"": ""1000""}, {""input"": ""(7, 2)"", ""expected_output"": ""111""}]","def change_base(x: Nat, base: Nat) -> String
""""""Change numerical base of input number x to base.
return string representation after the conversion.
base numbers are less than 10.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat -> String)
-- inputs
(x base: Nat) :=
-- spec
let spec (result: String) :=
let result_array := result.toList.map (fun c => c.toNat - '0'.toNat);
let pow_array := (List.range result_array.length).map (fun i => base^(result_array.length - i - 1) * result_array[i]!);
let pow_sum := pow_array.sum;
(0 < base ∧ base ≤ 10) ∧
(∀ i, i < result_array.length →
result_array[i]! < base ∧ 0 ≤ result_array[i]! →
pow_sum = x);
-- program termination
∃ result, implementation x base = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → String)
-- inputs
(x base: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (x base: Nat) : String :=,sorry,"-- #test implementation 8 3 = '22'
-- #test implementation 8 2 = '1000'
-- #test implementation 7 2 = '111'","theorem correctness
(x base : Nat)
: problem_spec implementation x base
:=","by
unfold problem_spec
let result := implementation x base
use result
simp [result]
sorry",,,
"def triangle_area(a: float, h: float) -> float","Given length of a side and high return area for a triangle.
","[{""input"": ""(5, 3)"", ""expected_output"": 7.5}, {""input"": ""(8, 2)"", ""expected_output"": 8.0}]","def triangle_area(a: float, h: float) -> float
""""""Given length of a side and high return area for a triangle.
""""""","def problem_spec
-- function signature
(implementation: Rat → Rat -> Rat)
-- inputs
(a h: Rat) :=
-- spec
let spec (result: Rat) :=
  a = 0 → result = 0 ∧
  (a ≠ 0 → (2 * result) / a = h);
-- -- program termination
∃ result, implementation a h = result ∧
spec result
--","def generated_spec
-- function signature
(impl: Rat → Rat → Rat)
-- inputs
(a h: Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (a h: Rat) : Rat :=,0.5 * a * h,"-- #test implementation 5 3 = 7.5
-- #test implementation 8 2 = 8.0","theorem correctness
(a h : Rat)
: problem_spec implementation a h
:=","by
unfold problem_spec
let result := implementation a h
use result
simp [result]
sorry",,,
def fib4(n: int),"The Fib4 number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
fib4(0) -> 0
fib4(1) -> 0
fib4(2) -> 2
fib4(3) -> 0
fib4(n) -> fib4(n-1) + fib4(n-2) + fib4(n-3) + fib4(n-4).
Please write a function to efficiently compute the n-th element of the fib4 number sequence.  Do not use recursion.
","[{""input"": 5, ""output"": 4}, {""input"": 6, ""output"": 8}, {""input"": 7, ""output"": 14}]","def fib4(n: int)
""""""The Fib4 number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
fib4(0) -> 0
fib4(1) -> 0
fib4(2) -> 2
fib4(3) -> 0
fib4(n) -> fib4(n-1) + fib4(n-2) + fib4(n-3) + fib4(n-4).
Please write a function to efficiently compute the n-th element of the fib4 number sequence.  Do not use recursion.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
fibonacci_non_computable_4 n result
-- program terminates
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(x: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 5 = 4
-- #test implementation 6 = 8
-- #test implementation 7 = 14","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
repeat sorry","/--
name: fibonacci_non_computable_4
use: |
  Non-computable definition to check if a number is a Fibonacci number such that
  fib(n) = fib(n - 1) + fib(n - 2) + fib(n - 3) + fib(n - 4).
problems:
  - 46
-/
inductive fibonacci_non_computable_4 : ℕ → ℕ → Prop
| base0 : fibonacci_non_computable_4 0 0
| base1 : fibonacci_non_computable_4 1 0
| base2 : fibonacci_non_computable_4 2 2
| base3 : fibonacci_non_computable_4 3 0
| step : ∀ n f₁ f₂ f₃ f₄, fibonacci_non_computable_4 n f₁ →
fibonacci_non_computable_4 (n + 1) f₂ →
fibonacci_non_computable_4 (n + 2) f₃ →
fibonacci_non_computable_4 (n + 3) f₄ →
fibonacci_non_computable_4 (n + 4) (f₁ + f₂ + f₃ + f₄)",,
def median(numbers: List[float]) -> float,"Return median of elements in the list l
","[{""input"": [3, 1, 2, 4, 5], ""output"": 3}, {""input"": [-10, 4, 6, 1000, 10, 20], ""output"": 15.0}]","def median(numbers: List[float]) -> float
""""""Return median of elements in the list l
""""""","def problem_spec
-- function signature
(implementation: List Rat → Rat)
-- inputs
(numbers: List Rat) :=
-- spec
let spec (result: Rat) :=
  0 < numbers.length →
  let less_eq := (numbers.filter (fun x => x ≤ result));
  let more_eq := (numbers.filter (fun x => result ≤ x));
  let max_more_eq := more_eq.max?;
  let min_less_eq := less_eq.min?;
  let less_eq_count := less_eq.length;
  let more_eq_count := more_eq.length;
  let eq_count := (numbers.filter (fun x => x = result)).length;
  (less_eq_count + more_eq_count - eq_count = numbers.length →
  numbers.length ≤ 2 * less_eq_count →
  numbers.length ≤ 2 * more_eq_count) ∧
  ((numbers.length % 2 = 1 →
    result ∈ numbers) ∧
    (numbers.length % 2 = 0 → max_more_eq.isSome ∧
    min_less_eq.isSome ∧
    2 * result = max_more_eq.get! + min_less_eq.get!));
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(impl: List Rat → Rat)
-- inputs
(x: List Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Rat) : Rat :=,sorry,"-- #test implementation [3, 1, 2, 4, 5] = 3
-- #test implementation [-10, 4, 6, 1000, 10, 20] = 15.0","theorem correctness
(numbers: List Rat)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
repeat sorry",,,
def is_palindrome(string: str) -> Bool,"Checks if given string is a palindrome
","[{""input"": """", ""expected_output"": true}, {""input"": ""aba"", ""expected_output"": true}, {""input"": ""aaaaa"", ""expected_output"": ""True""}, {""input"": ""zbcd"", ""expected_output"": ""False""}]","def is_palindrome(string: str) -> Bool
""""""Checks if given string is a palindrome
""""""","def problem_spec
-- function signature
(implementation: String → Bool)
-- inputs
(string: String) :=
-- spec
let spec (result: Bool) :=
result ↔ is_palindrome string
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(x: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (string: String) : Bool :=,sorry,"-- #test implementation """" = true
-- #test implementation ""aba"" = true
-- #test implementation ""aaaaa"" = true
-- #test implementation ""zbcd"" = false","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry","/--
name: is_palindrome
use: |
  Helper to check if a string is a palindrome.
problems:
  - 10
  - 48
-/
def is_palindrome
(s: String): Bool :=
s = s.toList.reverse.asString",,
"def modp(n: Nat, p: Nat) -> Nat","Return 2^n modulo p (be aware of numerics).
","[{""input"": [3, 5], ""expected_output"": 3}, {""input"": [1101, 101], ""expected_output"": 2}, {""input"": [0, 101], ""expected_output"": 0}, {""input"": [100, 101], ""expected_output"": 1}]","def modp(n: Nat, p: Nat) -> Nat
""""""Return 2^n modulo p (be aware of numerics).
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat → Nat)
-- inputs
(n p: Nat) :=
-- spec
let spec (result: Nat) :=
0 < p ∧
result < p ∧
(∃ k : Nat, p * k + result = Nat.pow 2 n)
-- program termination
∃ result, implementation n p = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → Nat)
-- inputs
(n p: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (n p: Nat) : Nat :=,sorry,"-- #test implementation 3 5 = 3
-- #test implementation 1101 101 = 2
-- #test implementation 0 101 = 1
-- #test implementation 3 11 = 8
-- #test implementation 100 101 = 1","theorem correctness
(n p: Nat)
: problem_spec implementation n p
:=","by
unfold problem_spec
let result := implementation n p
use result
simp [result]
sorry",,,
def encode_shift(s: String) -> String,"returns encoded string by shifting every character by 5 in the alphabet.
","[{""input"": ""abc"", ""expected_output"": ""fgh""}, {""input"": ""xyz"", ""expected_output"": ""cde""}, {""input"": ""aaa"", ""expected_output"": ""fff""}]","def encode_shift(s: String) -> String
""""""returns encoded string by shifting every character by 5 in the alphabet.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(s : String) :=
let isAlphabetic (string: String) : Bool :=
∀ i, i < string.length →
let c := string.get! ⟨i⟩;
('a'.toNat ≤ c.toNat ∧ c.toNat ≤ 'z'.toNat) ∨
('A'.toNat ≤ c.toNat ∧ c.toNat ≤ 'Z'.toNat)
-- spec
let spec (result: String) :=
isAlphabetic result ∧ isAlphabetic s ∧
result.length = s.length ∧
∃ k : Nat, k < 26 ∧
∀ i : Nat, i < s.length →
((s.get! ⟨i⟩).toNat + k) % 26 = (result.get! ⟨i⟩).toNat
→ k = 5
-- program termination
∃ result, implementation s = result ∧
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""abc"" = ""fgh""
-- #test implementation ""xyz"" = ""cde""
-- #test implementation ""aaa"" = ""fff""","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry",,,
def remove_vowels(string: str) -> string,"remove_vowels is a function that takes string and returns string without vowels.
","[{""input"": """", ""expected_output"": """"}, {""input"": ""abcdef\nghijklm"", ""expected_output"": ""bcdf\nghjklm""}, {""input"": ""abcdef"", ""expected_output"": ""bcdf""}, {""input"": ""aaaaa"", ""expected_output"": """"}, {""input"": ""aaBAA"", ""expected_output"": ""B""}]","def remove_vowels(string: str) -> string
""""""remove_vowels is a function that takes string and returns string without vowels.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(string: String) :=
let is_consonant (c: Char): Bool :=
  let vowels := ""aeiouAEIOU""
  not (vowels.contains c);
-- spec
let spec (result: String) :=
result.all (λ c => is_consonant c) ∧ result.length ≤ string.length ∧
∀ c, result.contains c → string.contains c ∧
∀ c , string.contains c ∧ is_consonant c → (result.contains c);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(string : String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ string, problem_spec impl string) ↔
(∀ string, generated_spec impl string) :=",sorry,def implementation (string: String) : String :=,sorry,"-- #test implementation """" = """"
-- #test implementation ""cat"" = ""catac""
-- #test implementation ""cata"" = ""catac""","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry",,,
"def below_threshold(numbers: List[Int], threshold: Int) -> bool","Return True if all numbers in the list l are below threshold t, and False otherwise.","[{""input"": [[1, 2, 4, 10], 100], ""expected_output"": true}, {""input"": [[1, 20, 4, 10], 5], ""expected_output"": false}]","def below_threshold(numbers: List[Int], threshold: Int) -> bool
""""""Return True if all numbers in the list l are below threshold t, and False otherwise.""""""","def problem_spec
-- function signature
(impl: List Int → Int → Bool)
-- inputs
(numbers: List Int)
(threshold: Int) :=
-- spec
let numbers_below_threshold :=
  ∀ i, i < numbers.length → numbers[i]! < threshold;
let spec (res: Bool) :=
(numbers.length = 0 → res) ∧
(res ↔ numbers_below_threshold)
-- program terminates
∃ result, impl numbers threshold = result ∧
-- return value satisfies spec
spec result
-- if result then spec else ¬spec","def generated_spec
-- function signature
(impl: List Int → Int → Bool)
-- inputs
(numbers : List Int)
(threshold : Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ numbers threshold, problem_spec impl numbers threshold) ↔
(∀ numbers threshold, generated_spec impl numbers threshold) :=",sorry,def implementation (numbers: List Int) (threshold: Int) : Bool :=,sorry,"-- #test implementation ([1, 2, 4, 10]: List Int) 100 = true
-- #test implementation ([1, 20, 4, 10]: List Int) 5 = false","theorem correctness
(numbers: List Int)
(threshold: Int)
: problem_spec implementation numbers threshold  :=","by
unfold problem_spec
let result := implementation numbers threshold
use result
simp [result]
sorry",,,
"def add(x: Int, y: Int) -> Int",Add two numbers x and y,"[{""input"": [2, 3], ""expected_output"": 5}, {""input"": [5, 7], ""expected_output"": 12}]","def add(x: Int, y: Int) -> Int
""""""Add two numbers x and y""""""","def problem_spec
-- function signature
(impl: Int → Int → Int)
-- inputs
(x y: Int) :=
-- spec
let spec (res: Int) :=
  res - x - y = 0
-- program terminates
∃ result, impl x y = result ∧
-- return value satisfies spec
spec result
-- if result then spec else ¬spec","def generated_spec
-- function signature
(impl: Int → Int → Int)
-- inputs
(x y: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (x y: Int) : Int :=,sorry,"-- #test implementation 2 3 = 5
-- #test implementation 5 7 = 12","theorem correctness
(x y: Int)
: problem_spec implementation x y  :=","by
unfold problem_spec
let result := implementation x y
use result
simp [result]
sorry",,,
"def same_chars(s0: string, s1: string) -> Bool",Check if two words have the same characters.,"[{""input"": [""eabcdzzzz"", ""dddzzzzzzzddeddabc""], ""expected_output"": true}, {""input"": [""eabcd"", ""dddddddabc""], ""expected_output"": false}]","def same_chars(s0: string, s1: string) -> Bool
""""""Check if two words have the same characters.""""""","def problem_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(s0 s1: String) :=
-- spec
let spec (res: Bool) :=
  res ↔ (∀ c : Char, c ∈ s0.toList ↔ c ∈ s1.toList)
-- program terminates
∃ result, impl s0 s1 = result ∧
-- return value satisfies spec
spec result
-- if result then spec else ¬spec","def generated_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(s0 s1: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s0 s1, problem_spec impl s0 s1) ↔
(∀ s0 s1, generated_spec impl s0 s1) :=",sorry,def implementation (s0 s1: String) : Bool :=,sorry,"-- #test implementation 'eabcdzzzz' 'dddzzzzzzzddeddabc' = true
-- #test implementation 'abcd' 'dddddddabc' = true
-- #test implementation 'dddddddabc' 'abcd' = true
-- #test implementation 'eabcd' 'dddddddabc' = false
-- #test implementation 'abcd' 'dddddddabce' = false
-- #test implementation 'eabcdzzzz' 'dddzzzzzzzddddabc' = false","theorem correctness
(s0 s1: String)
: problem_spec implementation s0 s1  :=","by
unfold problem_spec
let result := implementation s0 s1
use result
simp [result]
sorry",,,
def fib(n: int) -> int,"Return n-th Fibonacci number.
","[{""input"": 10, ""expected_output"": 55}, {""input"": 1, ""expected_output"": 1}, {""input"": 8, ""expected_output"": 21}]","def fib(n: int) -> int
""""""Return n-th Fibonacci number.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
fibonacci_non_computable n result
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(x : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 10 = 55
-- #test implementation 1 = 1
-- #test implementation 8 = 21","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
repeat sorry","/--
name: fibonacci_non_computable
use: |
  Non-computable definition to check if a number is a Fibonacci number.
problems:
  - 55
sample_problems:
  - 3
-/
inductive fibonacci_non_computable : ℕ → ℕ → Prop
| base0 : fibonacci_non_computable 0 0
| base1 : fibonacci_non_computable 1 1
| step  : ∀ n f₁ f₂, fibonacci_non_computable n f₁ →
fibonacci_non_computable (n + 1) f₂ →
fibonacci_non_computable (n + 2) (f₁ + f₂)",,
def correct_bracketing(brackets: str) -> Bool,"brackets is a string of ""<"" and "">"".
return True if every opening bracket has a corresponding closing bracket, i.e., (each open bracket is properly closed)
","[{""input"": ""<"", ""expected_output"": false}, {""input"": ""<>"", ""expected_output"": true}, {""input"": ""<<><>>"", ""expected_output"": true}, {""input"": ""><<>"", ""expected_output"": false}]","def correct_bracketing(brackets: str) -> Bool
""""""brackets is a string of ""<"" and "">"".
return True if every opening bracket has a corresponding closing bracket, i.e., (each open bracket is properly closed)
""""""","def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(brackets: String) :=
-- spec
let spec (result: Bool) :=
  brackets.data.all (fun c => c = '<' ∨ c = '>') →
  (result ↔ balanced_paren_non_computable brackets '<' '>')
-- program terminates
∃ result, impl brackets = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(brackets : String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ brackets, problem_spec impl brackets) ↔
(∀ brackets, generated_spec impl brackets) :=",sorry,def implementation (brackets: String) : Bool :=,sorry,"-- #test implementation ""<"" = false
-- #test implementation ""<>"" = true
-- #test implementation ""<<><>>"" = true
-- #test implementation ""><<>"" = false","theorem correctness
(brackets: String)
: problem_spec implementation brackets :=","by
unfold problem_spec
let result := implementation brackets
use result
simp [result]
sorry",,,
def monotonic(numbers: List[int]) -> Bool,"Return True if list elements are monotonically increasing or decreasing.
","[{""input"": [1, 2, 4, 20], ""expected_output"": true}, {""input"": [1, 20, 4, 10], ""expected_output"": false}, {""input"": [4, 1, 0, -10], ""expected_output"": true}]","def monotonic(numbers: List[int]) -> Bool
""""""Return True if list elements are monotonically increasing or decreasing.
""""""","def problem_spec
-- function signature
(implementation: List Int → Bool)
-- inputs
(numbers: List Int) :=
let non_ordered := ∃ i j,
i < numbers.length - 1 ∧
j < numbers.length - 1 ∧
(numbers[i]! < numbers[i+1]!) ∧
(numbers[j+1]! < numbers[j]!);
-- spec
let spec (result: Bool) :=
  1 < numbers.length →
  result ↔ ¬non_ordered;
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(numbers : List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ numbers, problem_spec impl numbers) ↔
(∀ numbers, generated_spec impl numbers) :=",sorry,def implementation (numbers: List Int) : Bool :=,sorry,"-- #test implementation [1, 2, 4, 20] = true
-- #test implementation [1, 20, 4, 10] = false
-- #test implementation [4, 1, 0, -10] = true","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
"def common(l1: List[Int], l2: List[Int]) -> List[Int]","Return sorted unique common elements for two lists.
","[{""input"": [[1, 4, 3, 34, 653, 2, 5], [5, 7, 1, 5, 9, 653, 121]], ""expected_output"": [1, 5, 653]}, {""input"": [[5, 3, 2, 8], [3, 2]], ""expected_output"": [2, 3]}]","def common(l1: List[Int], l2: List[Int]) -> List[Int]
""""""Return sorted unique common elements for two lists.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int → List Int)
-- inputs
(l1 l2: List Int) :=
let is_unique (result: List Int) :=
  ∀ i j, i < result.length → j < result.length →
  i ≠ j → result[i]! ≠ result[j]!;
let is_sorted (result: List Int) :=
  ∀ i, i < result.length - 1 →
  result[i]! ≤ result[i + 1]!;
-- spec
let spec (result: List Int) :=
  is_unique result ∧
  is_sorted result ∧
  (∀ i : Int, i ∈ result ↔ i ∈ l1 ∧ i ∈ l2)
-- program termination
∃ result, implementation l1 l2 = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → List Int → List Int)
-- inputs
(x y: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (l1 l2: List Int) : List Int :=,sorry,"-- #test implementation [1, 4, 3, 34, 653, 2, 5] [5, 7, 1, 5, 9, 653, 121] = [1, 5, 653]
-- #test implementation [5, 3, 2, 8] [3, 2] = [2, 3]","theorem correctness
(l1 l2: List Int)
: problem_spec implementation l1 l2
:=","by
unfold problem_spec
let result := implementation l1 l2
use result
simp [result]
sorry",,,
def largest_prime_factor(n: Nat) -> Nat,"Return the largest prime factor of n. Assume n > 1 and is not a prime.
","[{""input"": 13195, ""expected_output"": 29}, {""input"": 2048, ""expected_output"": 2}]","def largest_prime_factor(n: Nat) -> Nat
""""""Return the largest prime factor of n. Assume n > 1 and is not a prime.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
  1 < n ∧ ¬ Nat.Prime n →
  (Nat.Prime result ∧ result ∣ n ∧
  ∀ i, i < n ∧ i ∣ n ∧ Nat.Prime i → i ≤ result);
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(x : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 13195 = 29
-- #test implementation 2048 = 2","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
sorry",,,
def sum_to_n(n: Nat) -> Nat,"sum_to_n is a function that sums numbers from 1 to n.
","[{""input"": 30, ""expected_output"": 465}, {""input"": 100, ""expected_output"": 4950}]","def sum_to_n(n: Nat) -> Nat
""""""sum_to_n is a function that sums numbers from 1 to n.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n : Nat) :=
-- spec
let spec (result: Nat) :=
  0 < n →
  (result = 1 ↔ n = 1) ∧
  (∀ i, implementation (i + 1) - (implementation i) = i + 1)
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 30 = 465
-- #test implementation 100 = 5050
-- #test implementation 5 = 15
-- #test implementation 10 = 55
-- #test implementation 1 = 1","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
sorry",,,
def correct_bracketing(brackets: str) -> Bool,"brackets is a string of ""("" and "")"".
return True if every opening bracket has a corresponding closing bracket.
","[{""input"": ""("", ""expected_output"": false}, {""input"": ""()"", ""expected_output"": true}, {""input"": ""(()())"", ""expected_output"": true}, {""input"": "")(()"", ""expected_output"": false}]","def correct_bracketing(brackets: str) -> Bool
""""""brackets is a string of ""("" and "")"".
return True if every opening bracket has a corresponding closing bracket.
""""""","def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(brackets: String) :=
-- spec
let spec (result: Bool) :=
  brackets.data.all (fun c => c = '(' ∨ c = ')') →
  (result ↔ balanced_paren_non_computable brackets '(' ')')
-- program terminates
∃ result, impl brackets = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(x : String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (paren_string: String) : Bool :=,sorry,"-- #test implementation ""("" = false
-- #test implementation ""()"" = true
-- #test implementation ""(()())"" = true
-- #test implementation "")(()"" = false","theorem correctness
(brackets: String)
: problem_spec implementation brackets :=","by
unfold problem_spec
let result := implementation brackets
use result
simp [result]
sorry",,,
def derivative(xs: List Int) -> List Int,"xs represent coefficients of a polynomial.
xs[0] + xs[1] * x + xs[2] * x^2 + ....
 Return derivative of this polynomial in the same form.
","[{""input"": [3, 1, 2, 4, 5], ""expected_output"": [1, 4, 12, 20]}, {""input"": [1, 2, 3], ""expected_output"": [2, 6]}]","def derivative(xs: List Int) -> List Int
""""""xs represent coefficients of a polynomial.
xs[0] + xs[1] * x + xs[2] * x^2 + ....
 Return derivative of this polynomial in the same form.
""""""","def problem_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(xs: List Int) :=
-- spec
let spec (result: List Int) :=
  result.length = xs.length - 1 ∧
  result = (check_derivative xs.reverse).reverse
-- program terminates
∃ result, impl xs = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(xs : List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ xs, problem_spec impl xs) ↔
(∀ xs, generated_spec impl xs) :=",sorry,def implementation (xs: List Int) : List Int :=,sorry,"-- #test implementation [3, 1, 2, 4, 5] : List Int = [1, 4, 12, 20]
-- #test implementation [1, 2, 3] : List Int = [2, 6]","theorem correctness
(xs: List Int)
: problem_spec implementation xs :=","by
unfold problem_spec
let result := implementation xs
use result
simp [result]
sorry",,,
def fibfib(n: int),"The FibFib number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
fibfib(0) == 0
fibfib(1) == 0
fibfib(2) == 1
fibfib(n) == fibfib(n-1) + fibfib(n-2) + fibfib(n-3).
Please write a function to efficiently compute the n-th element of the fibfib number sequence.
","[{""input"": 1, ""output"": 0}, {""input"": 5, ""output"": 4}, {""input"": 8, ""output"": 24}]","def fibfib(n: int)
""""""The FibFib number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
fibfib(0) == 0
fibfib(1) == 0
fibfib(2) == 1
fibfib(n) == fibfib(n-1) + fibfib(n-2) + fibfib(n-3).
Please write a function to efficiently compute the n-th element of the fibfib number sequence.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
fibonacci_non_computable_3 n result
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(x : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 1 = 0
-- #test implementation 5 = 4
-- #test implementation 8 = 24","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
unfold problem_spec
let result := implementation n
use result
simp [result]
repeat sorry","/--
name: fibonacci_non_computable_3
use: |
  Non-computable definition to check if a number is a Fibonacci number such that
  fib(n) = fib(n - 1) + fib(n - 2) + fib(n - 3).
problems:
  - 63
-/
inductive fibonacci_non_computable_3 : ℕ → ℕ → Prop
| base0 : fibonacci_non_computable_3 0 0
| base1 : fibonacci_non_computable_3 1 0
| base2 : fibonacci_non_computable_3 2 1
| step : ∀ n f₁ f₂ f₃, fibonacci_non_computable_3 n f₁ →
fibonacci_non_computable_3 (n + 1) f₂ →
fibonacci_non_computable_3 (n + 2) f₃ →
fibonacci_non_computable_3 (n + 3) (f₁ + f₂ + f₃)",,
def remove_vowels(string: str) -> Nat,"Write a function vowels_count which takes a string representing
a word as input and returns the number of vowels in the string.
Vowels in this case are 'a', 'e', 'i', 'o', 'u'. Here, 'y' is also a
vowel, but only when it is at the end of the given word.
","[{""input"": ""abcde"", ""expected_output"": 2}, {""input"": ""ACEDY"", ""expected_output"": 3}]","def remove_vowels(string: str) -> Nat
""""""Write a function vowels_count which takes a string representing
a word as input and returns the number of vowels in the string.
Vowels in this case are 'a', 'e', 'i', 'o', 'u'. Here, 'y' is also a
vowel, but only when it is at the end of the given word.
""""""","def problem_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) :=
let isVowel (c : Char) :=
  let vowels := ""aeiouAEIOU""
  vowels.contains c
let isY (c : Char) := c = 'y' ∨ c = 'Y'
-- spec
let spec (result: Nat) :=
string.data.all (fun c => c.isAlpha) →
if string.length = 1 then
  result = if isVowel string.data[0]! ∨ isY string.data[0]! then 1 else 0
else
  result = (if isVowel string.data[0]! then 1 else 0) + implementation (string.drop 1);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(impl: String → Nat)
-- inputs
(x : String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (string: String) : Nat :=,sorry,"-- #test implementation ""abcde"" = 2
-- #test implementation ""ACEDY"" = 3","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry",,,
"def circular_shift(x: Int, shift: Int) -> String","Circular shift the digits of the integer x, shift the digits right by shift
and return the result as a string.
If shift > number of digits, return digits reversed.
","[{""input"": [12, 1], ""expected_output"": 21}, {""input"": [12, 2], ""expected_output"": 12}]","def circular_shift(x: Int, shift: Int) -> String
""""""Circular shift the digits of the integer x, shift the digits right by shift
and return the result as a string.
If shift > number of digits, return digits reversed.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat → String)
-- inputs
(x shift: Nat) :=
let isReverse (s: String) : Bool :=
  let n := s.length;
  ∀ i, i < n / 2 → s.get! ⟨i⟩ = s.get! ⟨n - 1 - i⟩;
-- spec
let spec (result: String) :=
let x_str := Nat.repr x;
result.length = x_str.length ∧
(x_str.length < shift → isReverse x_str) ∧
(shift ≤ x_str.length →
  x_str.take shift = result.drop (x_str.length - shift) ∧
  x_str.drop shift = result.take (x_str.length - shift));
-- program termination
∃ result, implementation x shift = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → String)
-- inputs
(x y: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (x shift: Nat) : String :=,sorry,"-- #test implementation (12 : Int) (1 : Int) = ""21""
-- #test implementation (12 : Int) (2 : Int) = ""12""","theorem correctness
(x shift: Nat)
: problem_spec implementation x shift
:=","by
unfold problem_spec
let result := implementation x shift
use result
simp [result]
sorry",,,
def digitSum(string: str) -> Nat,"Write a function that takes a string as input and returns the sum of the upper characters only'
ASCII codes.
","[{""input"": """", ""expected_output"": 0}, {""input"": ""abAB"", ""expected_output"": 131}, {""input"": ""helloE"", ""expected_output"": 69}]","def digitSum(string: str) -> Nat
""""""Write a function that takes a string as input and returns the sum of the upper characters only'
ASCII codes.
""""""","def problem_spec
-- function signature
(implementation: String → Nat)
-- inputs
(string: String) :=
let isUpper (c : Char) :=
  65 ≤ c.toNat ∧ c.toNat ≤ 90
-- spec
let spec (result: Nat) :=
if string.length = 1 then
  result = if isUpper string.data[0]! then string.data[0]!.toNat else 0
else
  result = (if isUpper string.data[0]! then string.data[0]!.toNat else 0) + implementation (string.drop 1);
-- program termination
∃ result, implementation string = result ∧
spec result","def generated_spec
-- function signature
(impl: String → Nat)
-- inputs
(x: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (string: String) : Nat :=,sorry,"-- #test implementation """" = 0
-- #test implementation ""abAB"" = 131
-- #test implementation ""abcCd"" = 67
-- #test implementation ""helloE"" = 69
-- #test implementation ""woArBld"" = 131
-- #test implementation ""aAaaaXa"" = 153","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
sorry",,,
"def fruit_distribution(string: str, Nat n) -> Nat","In this task, you will be given a string that represents a number of apples and oranges
that are distributed in a basket of fruit this basket contains
apples, oranges, and mango fruits. Given the string that represents the total number of
the oranges and apples and an integer that represent the total number of the fruits
in the basket return the number of the mango fruits in the basket.
","[{""input"": [""5 apples and 6 oranges"", 19], ""expected_output"": 8}, {""input"": [""0 apples and 1 oranges"", 3], ""expected_output"": 2}, {""input"": [""2 apples and 3 oranges"", 100], ""expected_output"": 95}, {""input"": [""100 apples and 1 oranges"", 120], ""expected_output"": 19}]","def fruit_distribution(string: str, Nat n) -> Nat
""""""In this task, you will be given a string that represents a number of apples and oranges
that are distributed in a basket of fruit this basket contains
apples, oranges, and mango fruits. Given the string that represents the total number of
the oranges and apples and an integer that represent the total number of the fruits
in the basket return the number of the mango fruits in the basket.
""""""","def problem_spec
-- function signature
(implementation: String → Nat → Nat)
-- inputs
(string: String)
(n : Nat) :=
-- spec
let spec (result: Nat) :=
∃ x y : Nat, x + y = n - result
∧ (String.join [x.repr, "" apples and "", y.repr, "" oranges""] = string)
-- program termination
∃ result, implementation string n = result ∧
spec result","def generated_spec
-- function signature
(impl: String → Nat → Nat)
-- inputs
(x: String) (y: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (string: String) (n: Nat) : Nat :=,sorry,"-- #test implementation ""5 apples and 6 oranges"" 19 = 8
-- #test implementation ""0 apples and 1 oranges"" 3 = 2
-- #test implementation ""2 apples and 3 oranges"" 100 = 95
-- #test implementation ""100 apples and 1 oranges"" 120 = 19","theorem correctness
(s: String) (n : Nat)
: problem_spec implementation s n
:=","by
unfold problem_spec
let result := implementation s n
use result
simp [result]
sorry",,,
def pluck(numbers: List[Int]) -> List[Int],"Given an array representing a branch of a tree that has non-negative integer nodes
your task is to pluck one of the nodes and return it.
The plucked node should be the node with the smallest even value.
If multiple nodes with the same smallest even value are found return the node that has smallest index.

The plucked node should be returned in a list, [ smallest_value, its index ],
If there are no even values or the given array is empty, return [].
","[{""input"": [4, 2, 3], ""expected_output"": [2, 1]}, {""input"": [1, 2, 3], ""expected_output"": [2, 1]}, {""input"": [], ""expected_output"": []}, {""input"": [5, 0, 3, 0, 4, 2], ""expected_output"": [0, 1]}]","def pluck(numbers: List[Int]) -> List[Int]
""""""Given an array representing a branch of a tree that has non-negative integer nodes
your task is to pluck one of the nodes and return it.
The plucked node should be the node with the smallest even value.
If multiple nodes with the same smallest even value are found return the node that has smallest index.

The plucked node should be returned in a list, [ smallest_value, its index ],
If there are no even values or the given array is empty, return [].
""""""","def problem_spec
-- function signature
(implementation: List Nat → List Nat)
-- inputs
(numbers: List Nat) :=
-- spec
let spec (result: List Nat) :=
(result.length = 0 ↔ ∀ i, i < numbers.length → numbers[i]! % 2 = 1) ∧
(result.length = 2 ↔ ∃ i, i < numbers.length ∧
  numbers[i]! % 2 = 0 ∧
  result[0]! = numbers[i]! ∧
  result[1]! = i ∧
  (∀ j, j < numbers.length → j < i → (numbers[j]! % 2 = 1 ∨ numbers[i]! < numbers[j]!)) ∧
  (∀ k, k < numbers.length → numbers[k]! % 2 = 0 → numbers[i]! ≤ numbers[k]!));
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(impl: List Nat → List Nat)
-- inputs
(x: List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Nat) : List Nat :=,sorry,"-- #test implementation [4, 2, 3] = [2, 1]
-- #test implementation [1, 2, 3] = [2, 1]
-- #test implementation [] = []
-- #test implementation [5, 0, 3, 0, 4, 2] = [0, 1]
--","theorem correctness
(numbers: List Nat)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def search(numbers: List[int]) -> int,"You are given a non-empty list of positive integers. Return the greatest integer that is greater than
zero, and has a frequency greater than or equal to the value of the integer itself.
The frequency of an integer is the number of times it appears in the list.
If no such a value exist, return -1.
","[{""input"": [4, 1, 2, 2, 3, 1], ""expected_output"": 2}, {""input"": [1, 2, 2, 3, 3, 3, 4, 4, 4], ""expected_output"": 3}, {""input"": [5, 5, 4, 4, 4], ""expected_output"": -1}]","def search(numbers: List[int]) -> int
""""""You are given a non-empty list of positive integers. Return the greatest integer that is greater than
zero, and has a frequency greater than or equal to the value of the integer itself.
The frequency of an integer is the number of times it appears in the list.
If no such a value exist, return -1.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(numbers: List Int) :=
-- spec
let spec (result: Int) :=
0 < numbers.length ∧ numbers.all (fun n => 0 < n) →
(result ≠ -1 ↔ ∃ i : Nat, i < numbers.length ∧
  numbers[i]! = result ∧ numbers[i]! > 0 ∧
  numbers[i]! ≤ (numbers.filter (fun x => x = numbers[i]!)).length ∧
  (¬∃ j : Nat, j < numbers.length ∧
  numbers[i]! < numbers[j]! ∧ numbers[j]! ≤ numbers.count numbers[j]!));
-- program termination
∃ result, implementation numbers = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(x: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (numbers: List Int) : (Int) :=,sorry,"-- #test implementation [4, 1, 2, 2, 3, 1] = 2
-- #test implementation [1, 2, 2, 3, 3, 4, 4, 4] = 3
-- #test implementation [5, 5, 4, 4, 4] = -1","theorem correctness
(numbers: List Int)
: problem_spec implementation numbers
:=","by
unfold problem_spec
let result := implementation numbers
use result
simp [result]
sorry",,,
def strange_sort_list(lst: List[int]) -> List[int],"Given list of integers, return list in strange order.
Strange sorting is when you start with the minimum value,
then the maximum of the remaining integers, then the minimum and so on.
","[{""input"": [1, 2, 3, 4], ""expected_output"": [1, 4, 2, 3]}, {""input"": [5, 5, 5, 5], ""expected_output"": [5, 5, 5, 5]}, {""input"": [], ""expected_output"": []}]","def strange_sort_list(lst: List[int]) -> List[int]
""""""Given list of integers, return list in strange order.
Strange sorting is when you start with the minimum value,
then the maximum of the remaining integers, then the minimum and so on.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int)
-- inputs
(lst: List Int) :=
-- spec
let spec (result: List Int) :=
  let sorted_lst := lst.mergeSort;
  (List.Perm lst result)
  ∧ (forall i, (0 <= i ∧ i < lst.length ∧ i % 2 = 0) → result[i]! = sorted_lst[i / 2]!)
  ∧ (forall i, (0 <= i ∧ i < lst.length ∧ i % 2 = 1) → result[i]! = sorted_lst[lst.length - (i-1)/2 - 1]!)
-- program termination
∃ result, implementation lst = result ∧ spec result","def generated_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int): List Int :=,sorry,"-- #test implementation [1, 2, 3, 4] = [1, 4, 2, 3]
-- #test implementation [5, 6, 7, 8, 9] = [5, 9, 6, 8, 7]
-- #test implementation [1, 2, 3, 4, 5] = [1, 5, 2, 4, 3]
-- #test implementation [5, 6, 7, 8, 9, 1] = [1, 9, 5, 8, 6, 7]
-- #test implementation [5, 5, 5, 5] = [5, 5, 5, 5]
-- #test implementation [] = []
-- #test implementation [1,2,3,4,5,6,7,8] = [1, 8, 2, 7, 3, 6, 4, 5]
-- #test implementation [0,2,2,2,5,5,-5,-5] = [-5, 5, -5, 5, 0, 2, 2, 2]
-- #test implementation [111111] = [111111]","theorem correctness
(lst: List Int)
: problem_spec implementation lst
:=","by
sorry",,,
"def triangle_area(a: float, b: float, c: float) -> float","Given the lengths of the three sides of a triangle. Return the area of the triangle rounded to 2 decimal points
if the three sides form a valid triangle. Otherwise return -1. Three sides make a valid triangle when the sum of
any two sides is greater than the third side.
","[{""input"": ""(3, 4, 5)"", ""expected_output"": 6}, {""input"": ""(1, 2, 10)"", ""expected_output"": -1}]","def triangle_area(a: float, b: float, c: float) -> float
""""""Given the lengths of the three sides of a triangle. Return the area of the triangle rounded to 2 decimal points
if the three sides form a valid triangle. Otherwise return -1. Three sides make a valid triangle when the sum of
any two sides is greater than the third side.
""""""","def problem_spec
-- function signature
(implementation: Rat → Rat → Rat → Rat)
-- inputs
(a: Rat) (b: Rat) (c: Rat) :=
-- spec
let spec (result : Rat) :=
  let is_valid_triangle :=
    (a + b > c) ∧  (a + c > b) ∧ (b + c > a);
  let s :=
    (a + b + c) / 2;
  if is_valid_triangle then
    |result^2 - (s * (s-a) * (s-b) * (s-c))| ≤ ((1: Rat)/10000)
  else
    result = -1
-- program termination
∃ result, implementation a b c = result ∧
spec result","def generated_spec
-- function signature
(impl: Rat → Rat → Rat → Rat)
-- inputs
(a: Rat) (b: Rat) (c: Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b c, problem_spec impl a b c) ↔
(∀ a b c, generated_spec impl a b c) :=",sorry,def implementation (a: Rat) (b: Rat) (c: Rat): Rat :=,sorry,"-- #test implementation 3 4 5 = 6.00
-- #test implementation 1 2 10 = -1
-- #test implementation 4 8 5 = 8.18
-- #test implementation 2 2 2 = 1.73
-- #test implementation 1 2 3 = -1
-- #test implementation 10 5 7 = 16.25
-- #test implementation 2 6 3 = -1
-- #test implementation 1 1 1 =  0.43
-- #test implementation 2 2 10 = -1","theorem correctness
(a: Rat) (b: Rat) (c: Rat)
: problem_spec implementation a b c
:=","by
sorry",,,
"def will_it_fly(q: List[int], w: int) -> bool","Write a function that returns True if the object q will fly, and False otherwise.
The object q will fly if it's balanced (it is a palindromic list) and the sum of its elements is
less than or equal the maximum possible weight w.
","[{""input"": ""([1, 2], 5)"", ""expected_output"": false}, {""input"": ""([3, 2, 3], 1)"", ""expected_output"": false}, {""input"": ""([3, 2, 3], 9)"", ""expected_output"": true}, {""input"": ""([3], 5)"", ""expected_output"": true}]","def will_it_fly(q: List[int], w: int) -> bool
""""""Write a function that returns True if the object q will fly, and False otherwise.
The object q will fly if it's balanced (it is a palindromic list) and the sum of its elements is
less than or equal the maximum possible weight w.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int → Bool)
-- inputs
(q: List Int) (w: Int) :=
-- spec
let spec (result : Bool) :=
  (result → (List.Palindrome q)) ∧
  (result → (List.sum q ≤ w)) ∧
  (¬(List.Palindrome q) → ¬ result) ∧
  (¬(List.sum q ≤ w) → ¬ result)
-- program termination
∃ result, implementation q w = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Int → Bool)
-- inputs
(q: List Int) (w: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ q w, problem_spec impl q w) ↔
(∀ q w, generated_spec impl q w) :=",sorry,def implementation (q: List Int) (w: Int) : Bool :=,(List.Palindrome q) ∧ (List.sum q ≤ w),"-- #test implementation [3, 2, 3] 9 = True
-- #test implementation [1, 2] 5 = False
-- #test implementation [3] 5 = True
-- #test implementation [3, 2, 3] 1 = False
-- #test implementation [1, 2, 3] 6 = False
-- #test implementation [5] 5 = True","theorem correctness
(q: List Int) (w: Int)
: problem_spec implementation q w
:=","by
unfold problem_spec
let result := implementation q w
use result
simp [result]
simp [implementation]
apply And.intro
intro h_is_palindrome h_q_sum
assumption
apply And.intro
intro h_palindrome
simp [h_palindrome]
intro h_q_sum h_palindrome
assumption",,,
def smallest_change(arr: List[int]) -> int,"Given an array arr of integers, find the minimum number of elements that
need to be changed to make the array palindromic. A palindromic array is an array that
is read the same backwards and forwards. In one change, you can change one element to any other element.
","[{""input"": [1, 2, 3, 5, 4, 7, 9, 6], ""expected_output"": 4}, {""input"": [1, 2, 3, 4, 3, 2, 2], ""expected_output"": 1}, {""input"": [1, 2, 3, 2, 1], ""expected_output"": 0}]","def smallest_change(arr: List[int]) -> int
""""""Given an array arr of integers, find the minimum number of elements that
need to be changed to make the array palindromic. A palindromic array is an array that
is read the same backwards and forwards. In one change, you can change one element to any other element.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(arr: List Int) :=
-- spec
let spec (result : Int) :=
  let swaps_done (arr1: List Int) (arr2: List Int) :=
    ((List.finRange (arr1.length)).filter (fun idx => arr1[idx]? ≠ arr2[idx]?)).length/2
  ∀ palin_perm, (List.Perm arr palin_perm) ∧ (List.Palindrome palin_perm) →
    result ≤ (swaps_done arr palin_perm)
-- program termination
∃ result, implementation arr = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : Int :=,((List.finRange (arr.length/2)).filter (fun i => arr[i]? != arr[arr.length - 1 - i]?)).length,"-- #test implementation [1,2,3,5,4,7,9,6] = 4
-- #test implementation [1, 2, 3, 4, 3, 2, 2] = 1
-- #test implementation [1, 4, 2] = 1
-- #test implementation [1, 4, 4, 2] = 1
-- #test implementation [1, 2, 3, 2, 1] = 0
-- #test implementation [3, 1, 1, 3] = 0
-- #test implementation [1] = 0
-- #test implementation [0, 1] = 1","theorem correctness
(arr: List Int)
: problem_spec implementation arr
:=","by
sorry",,,
"def total_match(lst1: List[str], lst2: List[str]) -> List[str]","Write a function that accepts two lists of strings and returns the list that has
total number of chars in the all strings of the list less than the other list.
If the two lists have the same number of chars, return the first list.
","[{""input"": ""([], [])"", ""expected_output"": []}, {""input"": ""(['hi', 'admin'], ['hI', 'Hi'])"", ""expected_output"": [""hI"", ""Hi""]}, {""input"": ""(['hi', 'admin'], ['hi', 'hi', 'admin', 'project'])"", ""expected_output"": [""hi"", ""admin""]}, {""input"": ""(['hi', 'admin'], ['hI', 'hi', 'hi'])"", ""expected_output"": [""hI"", ""hi"", ""hi""]}, {""input"": ""(['4'], ['1', '2', '3', '4', '5'])"", ""expected_output"": [""4""]}]","def total_match(lst1: List[str], lst2: List[str]) -> List[str]
""""""Write a function that accepts two lists of strings and returns the list that has
total number of chars in the all strings of the list less than the other list.
If the two lists have the same number of chars, return the first list.
""""""","def problem_spec
-- function signature
(implementation: List String → List String → List String)
-- inputs
(lst1: List String) (lst2: List String) :=
let sum_chars (xs: List String) : Int :=
  xs.foldl (λ acc a => acc + a.length) 0;
-- spec
let spec (result : List String) :=
  ((result = lst1) ∨ (result = lst2))
  ∧
  (sum_chars result ≤ sum_chars lst1)
  ∧
  (sum_chars result ≤ sum_chars lst2)
  ∧
  ((sum_chars lst1 = sum_chars lst2) → (result = lst1))
-- program termination
∃ result, implementation lst1 lst2 = result ∧
spec result","def generated_spec
-- function signature
(impl: List String → List String → List String)
-- inputs
(lst1: List String) (lst2: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst1 lst2, problem_spec impl lst1 lst2) ↔
(∀ lst1 lst2, generated_spec impl lst1 lst2) :=",sorry,def implementation (lst1: List String) (lst2: List String) : List String :=,sorry,"-- #test implementation [] [] = []
-- #test implementation [""hi"", ""admin""] [""hi"", ""hi""] = [""hi"", ""hi""]
-- #test implementation [""hi"", ""admin""] [""hi"", ""hi"", ""admin"", ""project""] = [""hi"", ""admin""]
-- #test implementation [""4""] [""1"", ""2"", ""3"", ""4"", ""5""] = [""4""]
-- #test implementation [""hi"", ""admin""] [""hI"", ""Hi""] = [""hI"", ""Hi""]
-- #test implementation [""hi"", ""admin""] [""hI"", ""hi"", ""hi""] = [""hI"", ""hi"", ""hi""]
-- #test implementation [""hi"", ""admin""] [""hI"", ""hi"", ""hii""] = [""hi"", ""admin""]
-- #test implementation [] [""this""] = []
-- #test implementation [""this""] [] == []","theorem correctness
(lst1: List String) (lst2: List String)
: problem_spec implementation lst1 lst2
:=","by
sorry",,,
def is_multiply_prime(a: int) -> bool,"Write a function that returns true if the given number is the multiplication of 3 prime numbers
and false otherwise. Knowing that (a) is less then 100.
","[{""input"": 30, ""expected_output"": true}]","def is_multiply_prime(a: int) -> bool
""""""Write a function that returns true if the given number is the multiplication of 3 prime numbers
and false otherwise. Knowing that (a) is less then 100.
""""""","def problem_spec
-- function signature
(implementation: Int → Bool)
-- inputs
(a: Int) :=
-- spec
let spec (result: Bool) :=
  (a < 100) →
    result ↔ exists a' b c, (Nat.Prime a') ∧ (Nat.Prime b) ∧ (Nat.Prime c) ∧ (a == a'*b*c)
-- program termination
∃ result, implementation a = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → Bool)
-- inputs
(a: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a, problem_spec impl a) ↔
(∀ a, generated_spec impl a) :=",sorry,def implementation (a: Int) : Bool :=,sorry,"-- #test implementation 5 = False
-- #test implementation 30 = True
-- #test implementation 8 = True
-- #test implementation 10 = False
-- #test implementation 125 = True
-- #test implementation (3 * 5 * 7) = True
-- #test implementation (3 * 6 * 7) = False
-- #test implementation (9 * 9 * 9) = False
-- #test implementation (11 * 9 * 9) = False
-- #test implementation (11*13*7) = True","theorem correctness
(a: Int)
: problem_spec implementation a
:=","by
sorry",,,
"def is_simple_power(x: int, n: int) -> bool","Your task is to write a function that returns true if a number x is a simple
power of n and false in other cases. x is a simple power of n if n**int=x
","[{""input"": ""(1, 4)"", ""expected_output"": true}, {""input"": ""(2, 2)"", ""expected_output"": true}, {""input"": ""(8, 2)"", ""expected_output"": true}, {""input"": ""(3, 2)"", ""expected_output"": false}, {""input"": ""(3, 1)"", ""expected_output"": false}, {""input"": ""(5, 3)"", ""expected_output"": false}]","def is_simple_power(x: int, n: int) -> bool
""""""Your task is to write a function that returns true if a number x is a simple
power of n and false in other cases. x is a simple power of n if n**int=x
""""""","def problem_spec
-- function signature
(implementation: Int → Int → Bool)
-- inputs
(x: Int) (n: Int) :=
-- spec
let spec (result: Bool) :=
  result ↔ exists k: Nat, x = n^k
-- program termination
∃ result, implementation x n = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → Int → Bool)
-- inputs
(x: Int) (n: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x n, problem_spec impl x n) ↔
(∀ x n, generated_spec impl x n) :=",sorry,def implementation (x: Int) (n: Int) : Bool :=,sorry,"-- #test implementation 16 2 = True
-- #test implementation 143214 16 = False
-- #test implementation 4 2 = True
-- #test implementation 9 3 = True
-- #test implementation 16 4 = True
-- #test implementation 24 2 = False
-- #test implementation 128 4 = False
-- #test implementation 12 6 = False
-- #test implementation 1 1 = True
-- #test implementation 1 12 = True","theorem correctness
(x: Int) (n: Int)
: problem_spec implementation x n
:=","by
sorry",,,
def iscube(a: int) -> bool,"Write a function that takes an integer a and returns True if this integer is a cube of some integer number.
Note: you may assume the input is always valid.
","[{""input"": 1, ""expected_output"": true}, {""input"": 2, ""expected_output"": false}, {""input"": -1, ""expected_output"": true}, {""input"": 64, ""expected_output"": true}, {""input"": 0, ""expected_output"": true}, {""input"": 180, ""expected_output"": false}]","def iscube(a: int) -> bool
""""""Write a function that takes an integer a and returns True if this integer is a cube of some integer number.
Note: you may assume the input is always valid.
""""""","def problem_spec
-- function signature
(implementation: Int → Bool)
-- inputs
(a: Int) :=
-- spec
let spec (result: Bool) :=
  result ↔ exists n: Int, a = n^3
-- program termination
∃ result, implementation a = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → Bool)
-- inputs
(a: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a, problem_spec impl a) ↔
(∀ a, generated_spec impl a) :=",sorry,def implementation (a: Int) : Bool :=,sorry,"-- #test implementation 1 = True
-- #test implementation 2 = False
-- #test implementation -1 = True
-- #test implementation 64 = True
-- #test implementation 180 = False
-- #test implementation 1000 = True
-- #test implementation 0 = True
-- #test implementation 1729 = False","theorem correctness
(a: Int)
: problem_spec implementation a
:=","by
sorry",,,
def hex_key(num: string) -> int,"You have been tasked to write a function that receives
a hexadecimal number as a string and counts the number of hexadecimal
digits that are primes (prime number, or a prime, is a natural number
greater than 1 that is not a product of two smaller natural numbers).
Hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
Prime numbers are 2, 3, 5, 7, 11, 13, 17,...
So you have to determine a number of the following digits: 2, 3, 5, 7,
B (=decimal 11), D (=decimal 13).
Note: you may assume the input is always correct or empty string,
and symbols A,B,C,D,E,F are always uppercase.
","[{""input"": ""AB"", ""expected_output"": 1}, {""input"": ""1077E"", ""expected_output"": 2}, {""input"": ""ABED1A33"", ""expected_output"": 4}, {""input"": ""123456789ABCDEF0"", ""expected_output"": 6}, {""input"": ""2020"", ""expected_output"": 2}]","def hex_key(num: string) -> int
""""""You have been tasked to write a function that receives
a hexadecimal number as a string and counts the number of hexadecimal
digits that are primes (prime number, or a prime, is a natural number
greater than 1 that is not a product of two smaller natural numbers).
Hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
Prime numbers are 2, 3, 5, 7, 11, 13, 17,...
So you have to determine a number of the following digits: 2, 3, 5, 7,
B (=decimal 11), D (=decimal 13).
Note: you may assume the input is always correct or empty string,
and symbols A,B,C,D,E,F are always uppercase.
""""""","def problem_spec
-- function signature
(implementation: String → Int)
-- inputs
(num: String) :=
-- spec
let spec (result: Int) :=
  let num_val (ch : Char) :=
    if ch.isDigit then
      (ch.toNat - '0'.toNat)
    else if ch.isUpper then
      ((ch.toNat - 'A'.toNat) + 10)
    else 0;
  0 < num.length →
  (
    let char_val := num_val num.toList[0]!;
    (Nat.Prime char_val →
      (1 < num.length → result = char_val + implementation (num.drop 1)) ∧
      (1 = num.length → result = char_val)) ∧
    (¬Nat.Prime char_val →
      (1 < num.length → result = implementation (num.drop 1)) ∧
      (1 = num.length → result = 0))
  )
-- program termination
∃ result, implementation num = result ∧
spec result","def generated_spec
-- function signature
(impl: String → Int)
-- inputs
(num: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ num, problem_spec impl num) ↔
(∀ num, generated_spec impl num) :=",sorry,def implementation (num: String) : Int :=,"let IsPrimeHexDigit (c: Char): Int :=
    if c = '2' ∨ c = '3' ∨ c = '5' ∨ c = '7' ∨ c = 'B' ∨ c = 'D' then 1 else 0;
  num.foldl (λ acc a => acc + (IsPrimeHexDigit a)) 0","-- #test implementation ""AB"" = 1
-- #test implementation ""1077E"" = 2
-- #test implementation ""ABED1A33"" = 4
-- #test implementation ""2020"" = 2
-- #test implementation ""123456789ABCDEF0"" = 6
-- #test implementation ""112233445566778899AABBCCDDEEFF00"" = 12
-- #test implementation """" = 0","theorem correctness
(num: String)
: problem_spec implementation num
:=","by
sorry",,,
def decimal_to_binary(decimal: nat) -> string,"You will be given a number in decimal form and your task is to convert it to
binary format. The function should return a string, with each character representing a binary
number. Each character in the string will be '0' or '1'.

There will be an extra couple of characters 'db' at the beginning and at the end of the string.
The extra characters are there to help with the format.
","[{""input"": 15, ""expected_output"": ""db1111db""}, {""input"": 32, ""expected_output"": ""db100000db""}]","def decimal_to_binary(decimal: nat) -> string
""""""You will be given a number in decimal form and your task is to convert it to
binary format. The function should return a string, with each character representing a binary
number. Each character in the string will be '0' or '1'.

There will be an extra couple of characters 'db' at the beginning and at the end of the string.
The extra characters are there to help with the format.
""""""","def problem_spec
-- function signature
(implementation: Nat → String)
-- inputs
(decimal: Nat) :=
-- spec
let spec (result: String) :=
  4 < result.length ∧
  result.drop (result.length - 2) = ""db"" ∧
  result.take 2 = ""db"" ∧
  let resultTrimmed := (result.toList.drop 2).dropLast.dropLast.map (fun c => c.toNat - '0'.toNat)
  decimal = Nat.ofDigits 2 resultTrimmed.reverse
-- program termination
∃ result, implementation decimal = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → String)
-- inputs
(decimal: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ decimal, problem_spec impl decimal) ↔
(∀ decimal, generated_spec impl decimal) :=",sorry,def implementation (decimal: Nat) : String :=,"""db"" ++ (Nat.toDigits 2 decimal).asString ++ ""db""","-- #test implementation 0 = ""db0db""
-- #test implementation 32 = ""db100000db""
-- #test implementation 103 = ""db1100111db""
-- #test implementation 15 = ""db1111db""","theorem correctness
(decimal: Nat)
: problem_spec implementation decimal
:=","by
sorry",,,
def is_happy(s: str) -> bool,"You are given a string s.
Your task is to check if the string is happy or not.
A string is happy if its length is at least 3 and every 3 consecutive letters are distinct
","[{""input"": ""a"", ""output"": false}, {""input"": ""aa"", ""output"": false}, {""input"": ""abcd"", ""output"": true}, {""input"": ""aabb"", ""output"": false}, {""input"": ""adb"", ""output"": true}]","def is_happy(s: str) -> bool
""""""You are given a string s.
Your task is to check if the string is happy or not.
A string is happy if its length is at least 3 and every 3 consecutive letters are distinct
""""""","def problem_spec
-- function signature
(implementation: String → Bool)
-- inputs
(s: String) :=
-- spec
let spec (result : Bool) :=
  result ↔
  (3 ≤ s.length) ∧
  ¬ (∃ i j, i < j ∧ j < s.length ∧ j - i ≤ 2 ∧ s.data.get! i = s.data.get! j)
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Bool :=,sorry,"-- #test implementation ""a"" = False
-- #test implementation ""aa"" = False
-- #test implementation ""abcd"" = True
-- #test implementation ""aabb"" = False
-- #test implementation ""adb"" = True","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
def numerical_letter_grade(grades: list[float]) -> list[str],"It is the last week of the semester and the teacher has to give the grades
to students. The teacher has been making her own algorithm for grading.
The only problem is, she has lost the code she used for grading.
She has given you a list of GPAs for some students and you have to write
a function that can output a list of letter grades using the following table:
         GPA       |    Letter grade
          4.0                A+
        > 3.7                A
        > 3.3                A-
        > 3.0                B+
        > 2.7                B
        > 2.3                B-
        > 2.0                C+
        > 1.7                C
        > 1.3                C-
        > 1.0                D+
        > 0.7                D
        > 0.0                D-
          0.0                E
Note: I have included a hypothesis that Float is hashable, not sure if this will mess up proving attempts but we can modify it if so. Reviewer: please think if there's a better way.
Example:
grade_equation([4.0, 3, 1.7, 2, 3.5]) ==> ['A+', 'B', 'C-', 'C', 'A-']
","[{""input"": [4.0, 3, 1.7, 2, 3.5], ""output"": [""A+"", ""B"", ""C-"", ""C"", ""A-""]}]","def numerical_letter_grade(grades: list[float]) -> list[str]
""""""It is the last week of the semester and the teacher has to give the grades
to students. The teacher has been making her own algorithm for grading.
The only problem is, she has lost the code she used for grading.
She has given you a list of GPAs for some students and you have to write
a function that can output a list of letter grades using the following table:
         GPA       |    Letter grade
          4.0                A+
        > 3.7                A
        > 3.3                A-
        > 3.0                B+
        > 2.7                B
        > 2.3                B-
        > 2.0                C+
        > 1.7                C
        > 1.3                C-
        > 1.0                D+
        > 0.7                D
        > 0.0                D-
          0.0                E
Note: I have included a hypothesis that Float is hashable, not sure if this will mess up proving attempts but we can modify it if so. Reviewer: please think if there's a better way.
Example:
grade_equation([4.0, 3, 1.7, 2, 3.5]) ==> ['A+', 'B', 'C-', 'C', 'A-']
""""""","def problem_spec
-- function signature
(implementation: List Float → List String)
-- inputs
(grades: List Float) :=
-- spec
let grade_dict : List (Float × String) :=
  [
    (4.0, ""A+""),
    (3.7, ""A""),
    (3.3, ""A-""),
    (3.0, ""B+""),
    (2.7, ""B""),
    (2.3, ""B-""),
    (2.0, ""C+""),
    (1.7, ""C""),
    (1.3, ""C-""),
    (1.0, ""D+""),
    (0.7, ""D""),
    (0.0, ""D-"")
  ]
let spec (result : List String) :=
  grades.all (fun grade => 0.0 ≤ grade ∧ grade ≤ 4.0) ∧
  result.length = grades.length ∧
  ∀ i, i < grades.length →
    let number_grade := grades[i]!
    let number_grade_keys := grade_dict.map (fun (g, _) => g)
    if 0.0 < number_grade then
      ∃ i : Nat, i < number_grade_keys.length ∧
        number_grade_keys[i]! ≤ number_grade ∧
        (∀ k' : Nat, k' < number_grade_keys.length → number_grade_keys[k']! ≤ number_grade → number_grade_keys[k']! ≤ number_grade_keys[i]!) ∧
        result[i]! = (grade_dict[i]!).snd
    else
      result[i]! = ""E""
-- program termination
∃ result,
  implementation grades = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Float → List String)
-- inputs
(grades: List Float) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ grades, problem_spec impl grades) ↔
(∀ grades, generated_spec impl grades) :=",sorry,def implementation (grades: List Float) : List String :=,sorry,"-- #test implementation [4.0, 3, 1.7, 2, 3.5] = ['A+', 'B', 'C-', 'C', 'A-']","theorem correctness
(grades: List Float)
: problem_spec implementation grades
:=","by
sorry",,,
def prime_length(s: str) -> bool,"Write a function that takes a string and returns True if the string
length is a prime number or False otherwise
","[{""input"": ""Hello"", ""output"": true}, {""input"": ""abcdcba"", ""output"": true}, {""input"": ""kittens"", ""output"": true}, {""input"": ""orange"", ""output"": false}]","def prime_length(s: str) -> bool
""""""Write a function that takes a string and returns True if the string
length is a prime number or False otherwise
""""""","def problem_spec
-- function signature
(implementation: String → Bool)
-- inputs
(s: String) :=
-- spec
let spec (result : Bool) :=
let is_prime (n: Nat) : Prop :=
  ¬ (∃ k, 2 ≤ k ∧ k < n ∧ n % k = 0);
  result ↔ is_prime s.length
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Bool :=,Nat.Prime s.length,"-- #test implementation ""Hello"" = True
-- #test implementation ""abcdcba"" = True
-- #test implementation ""kittens"" = True
-- #test implementation ""orange"" = False","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
unfold problem_spec
let result := implementation s
use result
simp [result]
simp [implementation]
apply Iff.intro
intro h_is_prime
simp [Nat.prime_def] at h_is_prime
intro x h_2_le_x h_x_lt_s_len
have h_p' := h_is_prime.2 x
by_contra h_s_len_mod_x
have h_x_dvd_s_len: x ∣ s.length := by
  apply Nat.dvd_of_mod_eq_zero
  exact h_s_len_mod_x
simp [h_x_dvd_s_len] at h_p'
have h_x_eq_s_len_false: ¬ (x = s.length) := by
  linarith
simp [h_x_eq_s_len_false] at h_p'
have h_x_ne_1: ¬ (x = 1) := by
  linarith
simp [h_x_ne_1] at h_p'
sorry",,,
def starts_one_ends(n: int) -> int,"Given a positive integer n, return the count of the numbers of n-digit
positive integers that start or end with 1.
Note: For reviewer, I believe this is the most straightforward spec, and I am relying on Set cardianlity not being computable in general. The point of this problem is really to privide a formula.
Note: But I guess a program that goes through each number and adds 1 will be the same as a program that computes in O(1) under this view.
","[{""input"": 1, ""output"": 1}, {""input"": 2, ""output"": 18}]","def starts_one_ends(n: int) -> int
""""""Given a positive integer n, return the count of the numbers of n-digit
positive integers that start or end with 1.
Note: For reviewer, I believe this is the most straightforward spec, and I am relying on Set cardianlity not being computable in general. The point of this problem is really to privide a formula.
Note: But I guess a program that goes through each number and adds 1 will be the same as a program that computes in O(1) under this view.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result : Nat) :=
  0 < n →
  result = {k : ℕ | 10 ^ (n - 1) ≤ k ∧ k < 10 ^ n ∧ (k.repr.front = '1' ∨ k.repr.back = '1')}.ncard
-- program termination
∃ result,
  implementation n = result ∧
  spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 1 = 1
-- #test implementation 2 = 18","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
sorry",,,
def solve(n: int) -> str,"Given a positive integer N, return the total sum of its digits in binary.
","[{""input"": 1000, ""output"": ""1""}, {""input"": 150, ""output"": ""110""}, {""input"": 147, ""output"": ""1100""}]","def solve(n: int) -> str
""""""Given a positive integer N, return the total sum of its digits in binary.
""""""","def problem_spec
-- function signature
(implementation: Nat → String)
-- inputs
(n: Nat) :=
-- spec
let spec (result : String) :=
  0 < n →
  result.all (fun c => c = '0' ∨ c = '1') →
  Nat.ofDigits 2 (result.data.map (fun c => if c = '0' then 0 else 1)).reverse = (Nat.digits 10 n).sum
-- program termination
∃ result,
  implementation n = result ∧
  spec result","def generated_spec
-- function signature
(impl: Nat → String)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : String :=,sorry,"-- #test implementation 1000 = ""1""
-- #test implementation 150 = ""110""
-- #test implementation 147 = ""1100""","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
sorry",,,
def solve(n: list[int]) -> int,"Given a non-empty list of integers lst, add the even elements that are at odd indices.
","[{""input"": [4, 2, 6, 7], ""output"": 2}]","def solve(n: list[int]) -> int
""""""Given a non-empty list of integers lst, add the even elements that are at odd indices.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(lst: List Int) :=
-- spec
let spec (result : Int) :=
  lst.length = 0 → result = 0 ∧
  lst.length > 0 →
    if lst.length > 1 then
      result = (if Even lst[1]! then lst[1]! else 0) + implementation (lst.drop 2)
    else
      result = (if Even lst[1]! then lst[1]! else 0)
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int) : Int :=,sorry,"-- #test implementation [4, 2, 6, 7] = 2","theorem correctness
(lst: List Int)
: problem_spec implementation lst
:=","by
sorry",,,
def anti_shuffle(s : str) -> str,"""""""
Write a function that takes a string and returns an ordered version of it.
Ordered version of string, is a string where all words (separated by space)
are replaced by a new word where all the characters arranged in
ascending order based on ascii value.
Note: You should keep the order of words and blank spaces in the sentence.
","[{""input"": ""Hi"", ""output"": ""Hi""}, {""input"": ""hello"", ""output"": ""ehllo""}, {""input"": ""Hello World!!!"", ""output"": ""Hello !!!Wdlor""}]","def anti_shuffle(s : str) -> str
""""""""""""
Write a function that takes a string and returns an ordered version of it.
Ordered version of string, is a string where all words (separated by space)
are replaced by a new word where all the characters arranged in
ascending order based on ascii value.
Note: You should keep the order of words and blank spaces in the sentence.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(s: String) :=
-- spec
let spec (result : String) :=
  result.length = s.length ∧
  let words := result.split (fun c => c = ' ');
  let s_words := s.split (fun c => c = ' ');
  s_words.length = words.length ∧
  ∀ i, i < words.length →
    words[i]!.length = s_words[i]!.length ∧
    ((∀ j, j < words[i]!.length →
      (words[i]!.data[j]! ∈ s_words[i]!.data ∧
      s_words[i]!.data[j]! ∈ words[i]!.data ∧
      words[i]!.data.count (words[i]!.data[j]!) = s_words[i]!.data.count (s_words[i]!.data[j]!))) ∧
    List.Sorted Nat.le (words[i]!.data.map (fun c => c.val.toNat)))
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""Hi"" = ""Hi""
-- #test implementation ""hello"" = ""ehllo""
-- #test implementation ""Hello World!!!"" = ""Hello !!!Wdlor""","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
"def get_coords_sorted(lst : list(list(int)), x : int) -> list((int, int))","You are given a 2 dimensional data, as a nested lists,
which is similar to matrix, however, unlike matrices,
each row may contain a different number of columns.
Given lst, and integer x, find integers x in the list,
and return list of tuples, [(x1, y1), (x2, y2) ...] such that
each tuple is a coordinate - (row, columns), starting with 0.
Sort coordinates initially by rows in ascending order.
Also, sort coordinates of the row by columns in descending order.
","[{""input"": [[[1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 1, 6], [1, 2, 3, 4, 5, 1]], 1], ""output"": [""(0"", ""0)"", ""(1"", ""4)"", ""(1"", ""0)"", ""(2"", ""5)"", ""(2"", ""0)""]}, {""input"": [[], 1], ""output"": []}, {""input"": [[[], [1], [1, 2, 3]], 3], ""output"": [""(2"", ""2)""]}]","def get_coords_sorted(lst : list(list(int)), x : int) -> list((int, int))
""""""You are given a 2 dimensional data, as a nested lists,
which is similar to matrix, however, unlike matrices,
each row may contain a different number of columns.
Given lst, and integer x, find integers x in the list,
and return list of tuples, [(x1, y1), (x2, y2) ...] such that
each tuple is a coordinate - (row, columns), starting with 0.
Sort coordinates initially by rows in ascending order.
Also, sort coordinates of the row by columns in descending order.
""""""","def problem_spec
-- function signature
(implementation: List (List Int) → Int → List (Nat × Nat))
-- inputs
(lst: List (List Int))
(x: Int) :=
-- spec
let spec (result : List (Nat × Nat)) :=
  (∀ i, i < result.length →
    let (row, col) := result[i]!
    row < lst.length ∧
    col < lst[row]!.length ∧
    (lst[row]!)[col]! = x) ∧
  (∀ᵉ (i < lst.length) (j < lst[i]!.length),
    (lst[i]!)[j]! = x → (i, j) ∈ result) ∧
  (result.map (fun (r, c) => r)).Sorted Nat.le ∧
  (∀ i < result.length,
    let (row, col) := result[i]!
    ((result.filter (fun (r, c) => r = row)).map (fun (r, c) => c)).Sorted (fun a b => a ≥ b))
-- program termination
∃ result,
  implementation lst x = result ∧
  spec result","def generated_spec
-- function signature
(impl: List (List Int) → Int → List (Nat × Nat))
-- inputs
(lst: List (List Int))
(x: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst x, problem_spec impl lst x) ↔
(∀ lst x, generated_spec impl lst x) :=",sorry,def implementation (lst: List (List Int)) (x: Int) : List (Nat × Nat) :=,sorry,"-- #test implementation ""Hi"" = ""Hi""
-- #test implementation ""hello"" = ""ehllo""
-- #test implementation ""Hello World!!!"" = ""Hello !!!Wdlor""","theorem correctness
(lst: List (List Int))
(x: Int)
: problem_spec implementation lst x
:=","by
sorry",,,
def sort_array(lst : list(int)) -> list(int),"""""""
Given an array of non-negative integers, return a copy of the given array after sorting,
you will sort the given array in ascending order if the sum( first index value, last index value) is odd,
or sort it in descending order if the sum( first index value, last index value) is even.
Note(George): I have elected to ignore the copy part.
","[{""input"": [], ""output"": []}, {""input"": [5], ""output"": [5]}, {""input"": [2, 4, 3, 0, 1, 5], ""output"": [0, 1, 2, 3, 4, 5]}, {""input"": [2, 4, 3, 0, 1, 5, 6], ""output"": [6, 5, 4, 3, 2, 1, 0]}]","def sort_array(lst : list(int)) -> list(int)
""""""""""""
Given an array of non-negative integers, return a copy of the given array after sorting,
you will sort the given array in ascending order if the sum( first index value, last index value) is odd,
or sort it in descending order if the sum( first index value, last index value) is even.
Note(George): I have elected to ignore the copy part.
""""""","def problem_spec
-- function signature
(implementation: List Nat → List Nat)
-- inputs
(lst: List Nat) :=
-- spec
let spec (result : List Nat) :=
  lst.length > 0 →
  result.length = lst.length ∧
  (∀ i, i < result.length →
    result[i]! ∈ lst ∧
    lst[i]! ∈ result ∧
    result.count (lst[i]!) = lst.count (lst[i]!)) ∧
  (lst.head! + lst.getLast!) ≡ 1 [MOD 2] →
    result.Sorted Nat.le ∧
  (lst.head! + lst.getLast!) ≡ 0 [MOD 2] →
    result.Sorted (fun a b => a ≥ b)
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Nat → List Nat)
-- inputs
(lst: List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Nat) : List Nat :=,sorry,"-- #test implementation [] = []
-- #test implementation [5] = [5]
-- #test implementation [2, 4, 3, 0, 1, 5] = [0, 1, 2, 3, 4, 5]
-- #test implementation [2, 4, 3, 0, 1, 5, 6] = [6, 5, 4, 3, 2, 1, 0]","theorem correctness
(lst: List Nat)
: problem_spec implementation lst
:=","by
sorry",,,
def encrypt(str : str) -> str,"Create a function encrypt that takes a string as an argument and
returns a string encrypted with the alphabet being rotated.
The alphabet should be rotated in a manner such that the letters
shift down by two multiplied to two places.
","[{""input"": ""hi"", ""output"": ""lm""}, {""input"": ""asdfghjkl"", ""output"": ""ewhjklnop""}, {""input"": ""gf"", ""output"": ""kj""}, {""input"": ""et"", ""output"": ""ix""}]","def encrypt(str : str) -> str
""""""Create a function encrypt that takes a string as an argument and
returns a string encrypted with the alphabet being rotated.
The alphabet should be rotated in a manner such that the letters
shift down by two multiplied to two places.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(str: String) :=
-- spec
let spec (result : String) :=
  result.data.all (fun c => c.isLower) →
  result.length = str.length ∧
  (∀ i, i < str.length →
    let c := str.data[i]!
    let c' := result.data[i]!
    ((c'.toNat - 97) + 2 * 2) % 26 = (c.toNat - 97))
-- program termination
∃ result,
  implementation str = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(str: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ str, problem_spec impl str) ↔
(∀ str, generated_spec impl str) :=",sorry,def implementation (str: String) : String :=,sorry,"-- #test implementation ""hi"" = ""lm""
-- #test implementation ""asdfghjkl"" = ""ewhjklnop""
-- #test implementation ""gf"" = ""kj""
-- #test implementation ""et"" = ""ix""","theorem correctness
(str: String)
: problem_spec implementation str
:=","by
sorry",,,
def next_smallest(lst: List[int]) -> Optional[int],"You are given a list of integers.
Write a function next_smallest() that returns the 2nd smallest element of the list.
Return None if there is no such element.
TODO(George): Remove this when being reviewed
The spec is defined as: if result is none there is no second smallest element, which
exists in a finite list iff there are at least two distinct elements in the list.
If result is some x, then x is the second smallest element of the list, the spec
obtains the sublist of elements smaller than the result, and checks that this
sublist does not contain two distinct elements (they are all the same).
","[{""input"": [1, 2, 3, 4, 5], ""output"": 2}, {""input"": [5, 1, 4, 3, 2], ""output"": 2}, {""input"": [], ""output"": ""None""}, {""input"": [1, 1], ""output"": ""None""}]","def next_smallest(lst: List[int]) -> Optional[int]
""""""You are given a list of integers.
Write a function next_smallest() that returns the 2nd smallest element of the list.
Return None if there is no such element.
TODO(George): Remove this when being reviewed
The spec is defined as: if result is none there is no second smallest element, which
exists in a finite list iff there are at least two distinct elements in the list.
If result is some x, then x is the second smallest element of the list, the spec
obtains the sublist of elements smaller than the result, and checks that this
sublist does not contain two distinct elements (they are all the same).
""""""","def problem_spec
-- function signature
(implementation: List Int → Option Int)
-- inputs
(lst: List Int) :=
-- spec
let spec (result : Option Int) :=
  match result with
  | none => ¬ (∃ i j, i < lst.length ∧ j < lst.length ∧ i ≠ j ∧ lst.get! i < lst.get! j)
  | some result =>
    let smaller_els := lst.filter (· < result);
    0 < smaller_els.length ∧
    smaller_els.all (λ x => x = smaller_els.get! 0);
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Int → Option Int)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int) : Option Int :=,sorry,"-- #test implementation [1, 2, 3, 4, 5] = some 2
-- #test implementation [5, 1, 4, 3, 2] = some 2
-- #test implementation [] = none
-- #test implementation [1, 1] = none","theorem correctness
(lst: List Int)
: problem_spec implementation lst
:=","by
sorry",,,
def is_bored(s: str) -> int,"You'll be given a string of words, and your task is to count the number
of boredoms. A boredom is a sentence that starts with the word ""I"".
Sentences are delimited by '.', '?' or '!'.
","[{""input"": ""Hello world"", ""expected_output"": 0}, {""input"": ""The sky is blue. The sun is shining. I love this weather"", ""expected_output"": 1}]","def is_bored(s: str) -> int
""""""You'll be given a string of words, and your task is to count the number
of boredoms. A boredom is a sentence that starts with the word ""I"".
Sentences are delimited by '.', '?' or '!'.
""""""","def problem_spec
-- function signature
(implementation: String → Nat)
-- inputs
(s: String) :=
-- spec
let spec (result : Nat) :=
  let is_sentence_is_boredom (s: String) : Bool :=
    (s.startsWith ""I "" ∨ s.startsWith "" I"") ∧ '.' ∉ s.data ∧ '?' ∉ s.data ∧ '!' ∉ s.data
  match s.data.findIdx? (λ c => c = '.' ∨ c = '?' ∨ c = '!') with
  | some i =>
    let j := i + 1;
    let substring := s.drop j;
    result = (if is_sentence_is_boredom substring then 1 else 0) + implementation substring
  | none =>
    result = if is_sentence_is_boredom s then 1 else 0
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Nat)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Nat :=,sorry,"-- #test implementation ""Hello world"" = 0
-- #test implementation ""The sky is blue. The sun is shining. I love this weather"" = 1","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
"def any_int(a: float, b: float, c: float) -> bool","Create a function that takes 3 numbers.
Returns true if one of the numbers is equal to the sum of the other two, and all numbers are integers.
Returns false in any other cases.
","[{""input"": [5, 2, 7], ""expected_output"": true}, {""input"": [3, 2, 2], ""expected_output"": false}, {""input"": [3, -2, 1], ""expected_output"": true}, {""input"": [3.6, -2.2, 2], ""expected_output"": false}]","def any_int(a: float, b: float, c: float) -> bool
""""""Create a function that takes 3 numbers.
Returns true if one of the numbers is equal to the sum of the other two, and all numbers are integers.
Returns false in any other cases.
""""""","def problem_spec
-- function signature
(implementation: Rat → Rat → Rat → Bool)
-- inputs
(a: Rat) (b: Rat) (c: Rat) :=
-- spec
let spec (result : Bool) :=
  let nums := [a, b, c];
  result = true ↔ ∃ i j k : ℕ, {i, j, k} ⊆ ({0, 1, 2} : Set ℕ) ∧ i ≠ j ∧ j ≠ k ∧ k ≠ i ∧ (nums[i]! + nums[j]! = nums[k]!) ∧ a.den = 1 ∧ a.den = b.den ∧ a.den = c.den
-- program termination
∃ result,
  implementation a b c = result ∧
  spec result","def generated_spec
-- function signature
(impl: Rat → Rat → Rat → Bool)
-- inputs
(a: Rat) (b: Rat) (c: Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b c, problem_spec impl a b c) ↔
(∀ a b c, generated_spec impl a b c) :=",sorry,def implementation (a: Rat) (b: Rat) (c: Rat) : Bool :=,sorry,"-- #test implementation 5 2 7 = true
-- #test implementation 3 2 2 = false
-- #test implementation 3 (-2) 1 = true
-- #test implementation 3.6 (-2.2) 2 = false","theorem correctness
(a: Rat) (b: Rat) (c: Rat)
: problem_spec implementation a b c
:=","by
sorry",,,
def encode(s : str) -> str,"Write a function that takes a message, and encodes in such a
way that it swaps case of all letters, replaces all vowels in
the message with the letter that appears 2 places ahead of that
vowel in the english alphabet.
Assume only letters.
","[{""input"": ""test"", ""expected_output"": ""TGST""}, {""input"": ""This is a message"", ""expected_output"": ""tHKS KS C MGSSCGG""}]","def encode(s : str) -> str
""""""Write a function that takes a message, and encodes in such a
way that it swaps case of all letters, replaces all vowels in
the message with the letter that appears 2 places ahead of that
vowel in the english alphabet.
Assume only letters.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(s: String) :=
-- spec
let spec (result : String) :=
  s.data.all (λ c => c.isAlpha) →
  result.length = s.length ∧
  (∀ i, i < s.length →
    let c := s.data.get! i;
    let c' := result.data.get! i;
    match c with
    | 'a' | 'e' | 'i' | 'o' | 'u' | 'A' | 'E' | 'I' | 'O' | 'U' =>
      c.isUpper → c'.val = c.toLower.val + 2 ∧
      c.isLower → c'.val = c.toUpper.val + 2
    | _ =>
      c.isUpper → c' = c.toLower ∧
      c.isLower → c' = c.toUpper)
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""test"" = ""TGST""
-- #test implementation ""This is a message"" = ""tHKS KS C MGSSCGG""","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
def sum_largest_prime(lst : list[int]) -> int,"You are given a list of integers.
You need to find the largest prime value and return the sum of its digits.
Note(George): Modified to use List of nats because all examples are nats.
","[{""input"": [0, 3, 2, 1, 3, 5, 7, 4, 5, 5, 5, 2, 181, 32, 4, 32, 3, 2, 32, 324, 4, 3], ""expected_output"": 10}, {""input"": [1, 0, 1, 8, 2, 4597, 2, 1, 3, 40, 1, 2, 1, 2, 4, 2, 5, 1], ""expected_output"": 25}, {""input"": [1, 3, 1, 32, 5107, 34, 83278, 109, 163, 23, 2323, 32, 30, 1, 9, 3], ""expected_output"": 13}, {""input"": [0, 724, 32, 71, 99, 32, 6, 0, 5, 91, 83, 0, 5, 6], ""expected_output"": 11}, {""input"": [0, 81, 12, 3, 1, 21], ""expected_output"": 3}, {""input"": [0, 8, 1, 2, 1, 7], ""expected_output"": 7}]","def sum_largest_prime(lst : list[int]) -> int
""""""You are given a list of integers.
You need to find the largest prime value and return the sum of its digits.
Note(George): Modified to use List of nats because all examples are nats.
""""""","def problem_spec
-- function signature
(implementation: List Nat → Nat)
-- inputs
(lst: List Nat) :=
-- spec
let spec (result : Nat) :=
  lst.any (fun num => Nat.Prime num) →
    result > 0 ∧ ∃ i, i < lst.length ∧ Prime (lst.get! i) ∧
    (∀ j, j < lst.length ∧ Prime (lst.get! j) → lst.get! i ≤ lst.get! j) ∧
    result = (Nat.digits 10 (lst.get! i)).sum
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Nat → Nat)
-- inputs
(lst: List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Nat) : Nat :=,sorry,"-- #test implementation [0,3,2,1,3,5,7,4,5,5,5,2,181,32,4,32,3,2,32,324,4,3] = 10
-- #test implementation [1,0,1,8,2,4597,2,1,3,40,1,2,1,2,4,2,5,1] = 25
-- #test implementation [1,3,1,32,5107,34,83278,109,163,23,2323,32,30,1,9,3] = 13
-- #test implementation [0,724,32,71,99,32,6,0,5,91,83,0,5,6] = 11
-- #test implementation [0,81,12,3,1,21] = 3
-- #test implementation [0,8,1,2,1,7] = 7","theorem correctness
(lst: List Nat)
: problem_spec implementation lst
:=","by
sorry",,,
"def check_dict_case(s : dict[str, str]) -> bool","Given a dictionary, return True if all keys are strings in lower
case or all keys are strings in upper case, else return False.
The function should return False is the given dictionary is empty.
Note(George): Modified the problem to use strings only for both keys and values.
","[{""input"": {""a"": ""apple"", ""b"": ""banana""}, ""expected_output"": true}, {""input"": {""a"": ""apple"", ""A"": ""banana"", ""B"": ""banana""}, ""expected_output"": false}, {""input"": {""a"": ""apple"", ""b"": ""banana""}, ""expected_output"": false}, {""input"": {""Name"": ""John"", ""Age"": ""36"", ""City"": ""Houston""}, ""expected_output"": false}, {""input"": {""STATE"": ""NC"", ""ZIP"": ""12345""}, ""expected_output"": true}]","def check_dict_case(s : dict[str, str]) -> bool
""""""Given a dictionary, return True if all keys are strings in lower
case or all keys are strings in upper case, else return False.
The function should return False is the given dictionary is empty.
Note(George): Modified the problem to use strings only for both keys and values.
""""""","def problem_spec
-- function signature
(implementation: Std.HashMap String String → Bool)
-- inputs
(D: Std.HashMap String String) :=
-- spec
let spec (result : Bool) :=
  match D.isEmpty with
  | true => result = false
  | false =>
    let keys := D.keys
    let all_lower := keys.all (fun s => s.toLower = s)
    let all_upper := keys.all (fun s => s.toUpper = s)
    result = true ↔ all_lower || all_upper
-- program termination
∃ result,
  implementation D = result ∧
  spec result","def generated_spec
-- function signature
(impl: Std.HashMap String String → Bool)
-- inputs
(D: Std.HashMap String String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ D, problem_spec impl D) ↔
(∀ D, generated_spec impl D) :=",sorry,def implementation (D: Std.HashMap String String) : Bool :=,sorry,"-- #test implementation (Std.HashMap.ofList [(""a"", ""apple""), (""b"", ""banana"")]) = true
-- #test implementation (Std.HashMap.ofList [(""a"", ""apple""), (""A"", ""banana""), (""B"", ""banana"")]) = false
-- #test implementation (Std.HashMap.ofList [(""a"", ""apple""), (""b"", ""banana""), (""a"", ""apple"")]) = false
-- #test implementation (Std.HashMap.ofList [(""Name"", ""John""), (""Age"", ""36""), (""City"", ""Houston"")]) = false
-- #test implementation (Std.HashMap.ofList [(""STATE"", ""NC""), (""ZIP"", ""12345"")]) = true","theorem correctness
(D: Std.HashMap String String)
: problem_spec implementation D
:=","by
sorry",,,
def count_up_to(n : int) -> list[int],"Implement a function that takes an non-negative integer and returns an array of the first n
integers that are prime numbers and less than n.
","[{""input"": 5, ""expected_output"": [2, 3]}, {""input"": 11, ""expected_output"": [2, 3, 5, 7]}, {""input"": 0, ""expected_output"": []}, {""input"": 20, ""expected_output"": [2, 3, 5, 7, 11, 13, 17, 19]}, {""input"": 1, ""expected_output"": []}, {""input"": 18, ""expected_output"": [2, 3, 5, 7, 11, 13, 17]}]","def count_up_to(n : int) -> list[int]
""""""Implement a function that takes an non-negative integer and returns an array of the first n
integers that are prime numbers and less than n.
""""""","def problem_spec
-- function signature
(implementation: Nat → List Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result : List Nat) :=
  match n with
  | 0 => result = []
  | n => n > 0 → (∀ i, i < result.length → (Nat.Prime (result.get! i)) ∧ (result.get! i) < n) ∧
         (∀ i : Nat, i < n → Nat.Prime i → i ∈ result)
-- program termination
∃ result,
  implementation n = result ∧
  spec result","def generated_spec
-- function signature
(impl: Nat → List Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : List Nat :=,sorry,"-- #test implementation 5 = [2,3]
-- #test implementation 11 = [2,3,5,7]
-- #test implementation 0 = []
-- #test implementation 20 = [2,3,5,7,11,13,17,19]
-- #test implementation 1 = []
-- #test implementation 18 = [2,3,5,7,11,13,17]","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
sorry",,,
"def multiply(a : Int, b : Int) -> Int","Complete the function that takes two integers and returns
the product of their unit digits.
Assume the input is always valid.
-- Note(George): I'm finding it hard to not leak the implementation here, so I opted to make the spec more convoluted.
","[{""input"": ""148, 412"", ""expected_output"": 16}, {""input"": ""19, 28"", ""expected_output"": 72}, {""input"": ""2020, 1851"", ""expected_output"": 0}, {""input"": ""14, -15"", ""expected_output"": 20}]","def multiply(a : Int, b : Int) -> Int
""""""Complete the function that takes two integers and returns
the product of their unit digits.
Assume the input is always valid.
-- Note(George): I'm finding it hard to not leak the implementation here, so I opted to make the spec more convoluted.
""""""","def problem_spec
-- function signature
(implementation: Int → Int → Int)
-- inputs
(a b: Int) :=
-- spec
let spec (result : Int) :=
  |result| ≤ 81 ∧
  result % 10 = (a * b) % 10 ∧
  ((b%10) ≠ 0 → (result % (b%10) = 0 ∧ (result/ (b%10)) % 100 = (a%10))) ∧
  ((a%10) ≠ 0 → (result % (a%10) = 0 ∧ (result/ (a%10)) % 100 = (b%10))) ∧
  ((a%10 = 0) ∨ (b%10 = 0) → result = 0)
-- program termination
∃ result,
  implementation a b = result ∧
  spec result","def generated_spec
-- function signature
(impl: Int → Int → Int)
-- inputs
(a b: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b, problem_spec impl a b) ↔
(∀ a b, generated_spec impl a b) :=",sorry,def implementation (a b: Int) : Int :=,sorry,"-- #test implementation 148 412 = 16
-- #test implementation 19 28 = 72
-- #test implementation 2020 1851 = 0
-- #test implementation 14 -15 = 20","theorem correctness
(a b: Int)
: problem_spec implementation a b
:=","by
sorry",,,
def count_upper(s : String) -> Int,"Given a string s, count the number of uppercase vowels in even indices.
-- Note(George): I also feel like this one is hard to not leak, I tried a trick about keeping implementation for a recursive call in the spec. Let me know if this doesn't work..
","[{""input"": ""aBCdEf"", ""expected_output"": 1}, {""input"": ""abcdefg"", ""expected_output"": 0}, {""input"": ""dBBE"", ""expected_output"": 0}]","def count_upper(s : String) -> Int
""""""Given a string s, count the number of uppercase vowels in even indices.
-- Note(George): I also feel like this one is hard to not leak, I tried a trick about keeping implementation for a recursive call in the spec. Let me know if this doesn't work..
""""""","def problem_spec
-- function signature
(implementation: String → Int)
-- inputs
(s: String) :=
-- spec
let uppercase_vowels := ['A', 'E', 'I', 'O', 'U']
let spec (result : Int) :=
  let chars := s.toList
  (result = 0 ↔ ∀ i, i < chars.length → chars[i]! ∉ uppercase_vowels) ∧
  (1 < chars.length →
    (result - 1 = implementation (chars.drop 2).toString ↔ chars[0]! ∈ uppercase_vowels) ∨
    (result = implementation (chars.drop 2).toString ↔ chars[0]! ∉ uppercase_vowels)
  )
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Int)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Int :=,sorry,"-- #test implementation ""aBCdEf"" = 1
-- #test implementation ""abcdefg"" = 0
-- #test implementation ""dBBE"" = 0","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
def closest_integer(s : String) -> Option Int,"Create a function that takes a value (string) representing a number
and returns the closest integer to it. If the number is equidistant
from two integers, round it away from zero.
","[{""input"": ""10"", ""expected_output"": 10}, {""input"": ""15.3"", ""expected_output"": 15}]","def closest_integer(s : String) -> Option Int
""""""Create a function that takes a value (string) representing a number
and returns the closest integer to it. If the number is equidistant
from two integers, round it away from zero.
""""""","def problem_spec
-- function signature
(implementation: String → Option Int)
-- inputs
(s: String) :=
-- spec
let numeric_characters := [""-"", ""0"", ""1"", ""2"", ""3"", ""4"", ""5"", ""6"", ""7"", ""8"", ""9"", "".""]
let is_valid_string :=
  s.length > 0 ∧
  s.count (""."".get! 0) ≤ 1 ∧
  s.count (""-"".get! 0) ≤ 1 ∧
  s.data.all (fun c => numeric_characters.contains (String.mk [c])) ∧
  (s.count (""-"".get! 0) = 1 → s.data.head! = '-')

let spec (result : Option Int) := match result with
  | some result =>
    is_valid_string ∧
    let parts := s.split (fun c => c = '.')
    (parts.length = 1 → result = s.toInt!) ∧
    (parts.length = 2 →
      let integer_part := parts.get! 0
      let is_negative := s.data.head! = '-'
      |((integer_part.toInt! - result) : ℚ)| ≤ 0.5 ∧
      (is_negative → |((integer_part.toInt! - result) : ℚ)| = 0.5 → integer_part.toInt? < result) ∧
      (¬is_negative → |((integer_part.toInt! - result) : ℚ)| = 0.5 → integer_part.toInt? > result)
    )
  | none => ¬ is_valid_string
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Option Int)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Option Int :=,sorry,"-- #test implementation ""10"" = some 10
-- #test implementation ""15.3"" = some 15","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
def make_a_pile(n: int) -> List[int],"Given a positive integer n, you have to make a pile of n levels of stones.
The first level has n stones.
The number of stones in the next level is:
  - the next odd number if n is odd.
  - the next even number if n is even.
Return the number of stones in each level in a list, where element at index
i represents the number of stones in the level (i+1).
","[{""input"": 3, ""expected_output"": [3, 5, 7]}]","def make_a_pile(n: int) -> List[int]
""""""Given a positive integer n, you have to make a pile of n levels of stones.
The first level has n stones.
The number of stones in the next level is:
  - the next odd number if n is odd.
  - the next even number if n is even.
Return the number of stones in each level in a list, where element at index
i represents the number of stones in the level (i+1).
""""""","def problem_spec
-- function signature
(implementation: Int → List Int)
-- inputs
(n: Int) :=
-- spec
let spec (result: List Int) :=
  result.length = n ∧
  (forall i: Nat, (1 <= i ∧ i < n) → (result[i]! = result[i-1]! + 2)) ∧
  result[0]! = n
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → List Int)
-- inputs
(n: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Int) : List Int :=,sorry,"-- #test implementation 3 = [3, 5, 7]
-- #test implementation 4 = [4,6,8,10]
-- #test implementation 5 = [5, 7, 9, 11, 13]
-- #test implementation 6 = [6, 8, 10, 12, 14, 16]
-- #test implementation 8 = [8, 10, 12, 14, 16, 18, 20, 22]","theorem correctness
(n: Int)
: problem_spec implementation n
:=","by
sorry",,,
def words_string(s: string) -> List[string],"You will be given a string of words separated by commas or spaces. Your task is
to split the string into words and return an array of the words.
","[{""input"": ""Hi, my name is John"", ""expected_output"": [""Hi"", ""my"", ""name"", ""is"", ""John""]}, {""input"": ""One, two, three, four, five, six"", ""expected_output"": [""One"", ""two"", ""three"", ""four"", ""five"", ""six""]}]","def words_string(s: string) -> List[string]
""""""You will be given a string of words separated by commas or spaces. Your task is
to split the string into words and return an array of the words.
""""""","def problem_spec
-- function signature
(implementation: String → List String)
-- inputs
(s: String) :=
-- spec
let spec (result: List String) :=
  let chars := s.toList;
  let first := s.takeWhile (fun c => c ≠ ',' ∧ c ≠ ' ');
  (result = [] ↔ (∀ x ∈ chars, x = ' ' ∨ x = ',') ∨ s = """") ∧
  (result ≠ [] ↔ result = [first] ++ (implementation (s.drop (first.length + 1))))

-- program termination
∃ result, implementation s = result ∧
spec result","def generated_spec
-- function signature
(impl: String → List String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : List String :=,"List.map (fun s => (s.toList.filter (fun c => c != ',')).asString) ((s.splitOn).filter (fun s => s != """" ∧ s != "",""))","-- #test implementation ""Hi, my name is John"" = [""Hi"", ""my"", ""name"", ""is"", ""John""]
-- #test implementation ""One, two, three, four, five, six"" = [""One"", ""two"", ""three"", ""four"", ""five"", ""six""]
-- #test implementation ""Hi, my name"" = [""Hi"", ""my"", ""name""]
-- #test implementation ""One,, two, three, four, five, six,"" = [""One"", ""two"", ""three"", ""four"", ""five"", ""six""]
-- #test implementation """" = []
-- #test implementation ""ahmed     , gamal"" = [""ahmed"", ""gamal""]","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
"def choose_num(x: int, y: int) -> int","This function takes two positive numbers x and y and returns the
biggest even integer number that is in the range [x, y] inclusive. If
there's no such number, then the function should return -1.
","[{""input"": ""(12, 15)"", ""expected_output"": 14}, {""input"": ""(13, 12)"", ""expected_output"": -1}]","def choose_num(x: int, y: int) -> int
""""""This function takes two positive numbers x and y and returns the
biggest even integer number that is in the range [x, y] inclusive. If
there's no such number, then the function should return -1.
""""""","def problem_spec
-- function signature
(implementation: Int → Int → Int)
-- inputs
(x: Int) (y: Int) :=
-- spec
let spec (result: Int) :=
  (result = -1 ∨ (x ≤ result ∧ result ≤ y ∧ Even result)) ∧
  (result = -1 ∨ (forall i: Int, (x ≤ i ∧ i ≤ y ∧ Even i) → result ≥ i)) ∧
  (result = -1 ↔ (x > y ∨ (x == y ∧ Odd x ∧ Odd y)))
-- program termination
∃ result, implementation x y = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → Int → Int)
-- inputs
(x: Int) (y: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x y, problem_spec impl x y) ↔
(∀ x y, generated_spec impl x y) :=",sorry,def implementation (x: Int) (y: Int) : Int :=,sorry,"-- #test implementation 12 15 = 14
-- #test implementation 13 12 = -1
-- #test implementation 33 12354 = 12354
-- #test implementation 5234 5233 = -1
-- #test implementation 6 29 = 28
-- #test implementation 27 10 = (-1)
-- #test implementation 7 7 = -1
-- #test implementation 546 546 = 546","theorem correctness
(x: Int) (y: Int)
: problem_spec implementation x y
:=","by
sorry",,,
"def rounded_avg(n: nat, m: nat) -> Option[string]","You are given two positive integers n and m, and your task is to compute the
average of the integers from n through m (including n and m).
Round the answer to the nearest integer and convert that to binary.
If n is greater than m, return none.
","[{""input"": ""(1, 5)"", ""expected_output"": ""0b11""}, {""input"": ""(7, 5)"", ""expected_output"": ""None""}, {""input"": ""(10, 20)"", ""expected_output"": ""0b1111""}, {""input"": ""(20, 33)"", ""expected_output"": ""0b11010""}]","def rounded_avg(n: nat, m: nat) -> Option[string]
""""""You are given two positive integers n and m, and your task is to compute the
average of the integers from n through m (including n and m).
Round the answer to the nearest integer and convert that to binary.
If n is greater than m, return none.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat → Option String)
-- inputs
(n: Nat) (m: Nat) :=
-- spec
let spec (result: Option String) :=
  (n > m ↔ result.isNone) ∧
  (n ≤ m ↔ result.isSome) ∧
  (n ≤ m →
    (result.isSome ∧
    let val := Option.getD result """";
    let xs := List.Ico n (m+1);
    let avg := xs.sum / xs.length;
    (val.take 2 = ""0b"") ∧
    (Nat.ofDigits 2 ((val.drop 2).toList.map (fun c => c.toNat - '0'.toNat)).reverse = avg)))
-- program termination
∃ result, implementation n m = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → Option String)
-- inputs
(n: Nat) (m: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n m, problem_spec impl n m) ↔
(∀ n m, generated_spec impl n m) :=",sorry,def implementation (n: Nat) (m: Nat) : Option String :=,"if n > m then
  none
else
  let xs := List.Ico n (m+1);
  let avg := xs.sum / xs.length;
  some (""0b"" ++ (Nat.toDigits 2 avg).asString)","-- #test implementation 1 5 = some ""0b11""
-- #test implementation 7 13 = some ""0b1010""
-- #test implementation 964 977 = some ""0b1111001010""
-- #test implementation 996 997 = some ""0b1111100100""
-- #test implementation 185 546 = some ""0b101101110""
-- #test implementation 362 496 = some ""0b110101101""
-- #test implementation 350 902 = some ""0b1001110010""
-- #test implementation 197 233 = some ""0b11010111""
-- #test implementation 7 5 = none
-- #test implementation 5 1 = none
-- #test implementation 5 5 = some ""0b101""","theorem correctness
(n: Nat) (m: Nat)
: problem_spec implementation n m
:=","by
sorry",,,
def unique_digits(x: List[nat]) -> List[nat],"Given a list of positive integers x. return a sorted list of all
elements that hasn't any even digit.

Note: Returned list should be sorted in increasing order.
","[{""input"": [15, 33, 1422, 1], ""expected_output"": [1, 15, 33]}, {""input"": [152, 323, 1422, 10], ""expected_output"": []}]","def unique_digits(x: List[nat]) -> List[nat]
""""""Given a list of positive integers x. return a sorted list of all
elements that hasn't any even digit.

Note: Returned list should be sorted in increasing order.
""""""","def problem_spec
-- function signature
(implementation: List Nat → List Nat)
-- inputs
(x: List Nat) :=
-- spec
let spec (result: List Nat) :=
  let has_even_digits(i: Nat): Bool :=
    (List.filter (fun d => Even d) (Nat.digits 10 i)).length > 0;
  (List.Sorted Nat.le result) ∧
  (forall i, i ∈ result ↔ (i ∈ x ∧ !(has_even_digits i)))
-- program termination
∃ result, implementation x = result ∧
spec result","def generated_spec
-- function signature
(impl: List Nat → List Nat)
-- inputs
(x: List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=",sorry,def implementation (x: List Nat) : List Nat :=,sorry,"-- #test implementation [15, 33, 1422, 1] = [1, 15, 33]
-- #test implementation [152, 323, 1422, 10] = []
-- #test implementation [12345, 2033, 111, 151] = [111, 151]
-- #test implementation [135, 103, 31] = [31, 135]","theorem correctness
(x: List Nat)
: problem_spec implementation x
:=","by
sorry",,,
def by_length(arr: List[int]) -> List[string],"Given an array of integers, sort the integers that are between 1 and 9 inclusive,
reverse the resulting array, and then replace each digit by its corresponding name from
""One"", ""Two"", ""Three"", ""Four"", ""Five"", ""Six"", ""Seven"", ""Eight"", ""Nine"".
","[{""input"": [2, 1, 1, 4, 5, 8, 2, 3], ""expected_output"": [""Eight"", ""Five"", ""Four"", ""Three"", ""Two"", ""Two"", ""One"", ""One""]}, {""input"": [], ""expected_output"": []}, {""input"": [1, -1, 55], ""expected_output"": [""One""]}]","def by_length(arr: List[int]) -> List[string]
""""""Given an array of integers, sort the integers that are between 1 and 9 inclusive,
reverse the resulting array, and then replace each digit by its corresponding name from
""One"", ""Two"", ""Three"", ""Four"", ""Five"", ""Six"", ""Seven"", ""Eight"", ""Nine"".
""""""","def problem_spec
-- function signature
(implementation: List Int → List String)
-- inputs
(arr: List Int) :=
-- spec
let spec (result: List String) :=
  let digits: List String := [""One"", ""Two"", ""Three"", ""Four"", ""Five"", ""Six"", ""Seven"", ""Eight"", ""Nine""];
  (forall s: String, (s ∈ result → s ∈ digits)) ∧
  (arr.length ≥ result.length) ∧
  (forall x: Nat, ((x: Int) ∈ arr ∧ 1 ≤ x ∧ x ≤ 9) → (digits[x-1]! ∈ result)) ∧
  (List.Sorted Int.le (List.map (fun (s: String) => (List.indexOf s digits) + 1) result).reverse)
-- program termination
∃ result, implementation arr = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → List String)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : List String :=,sorry,"-- #test implementation [2, 1, 1, 4, 5, 8, 2, 3] = [""Eight"", ""Five"", ""Four"", ""Three"", ""Two"", ""Two"", ""One"", ""One""]
-- #test implementation [] = []
-- #test implementation [1, -1 , 55] = [""One""]
-- #test implementation [1, -1, 3, 2] = [""Three"", ""Two"", ""One""]
-- #test implementation [9, 4, 8] = [""Nine"", ""Eight"", ""Four""]","theorem correctness
(arr: List Int)
: problem_spec implementation arr
:=","by
sorry",,,
def f(n: int) -> List[int],"Implement the function f that takes n as a parameter,
and returns a list of size n, such that the value of the element at index i is the factorial of i if i is even
or the sum of numbers from 1 to i otherwise.
i starts from 1.
the factorial of i is the multiplication of the numbers from 1 to i (1 * 2 * ... * i).
","[{""input"": 5, ""expected_output"": [1, 2, 6, 24, 15]}]","def f(n: int) -> List[int]
""""""Implement the function f that takes n as a parameter,
and returns a list of size n, such that the value of the element at index i is the factorial of i if i is even
or the sum of numbers from 1 to i otherwise.
i starts from 1.
the factorial of i is the multiplication of the numbers from 1 to i (1 * 2 * ... * i).
""""""","def problem_spec
-- function signature
(implementation: Int → List Int)
-- inputs
(n: Int) :=
-- spec
let spec (result: List Int) :=
  (result.length = n) ∧
  (forall i: Nat, (1 ≤ i ∧ i ≤ n ∧ Even i) → (result[i-1]! = Nat.factorial i)) ∧
  (forall i: Nat, (1 ≤ i ∧ i ≤ n ∧ Odd i) → (result[i-1]! = (List.range (i+1)).sum))
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Int → List Int)
-- inputs
(n: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Int) : List Int :=,sorry,"-- #test implementation 5 = [1, 2, 6, 24, 15]
-- #test implementation 7 = [1, 2, 6, 24, 15, 720, 28]
-- #test implementation 1 = [1]
-- #test implementation 3 = [1, 2, 6]","theorem correctness
(n: Int)
: problem_spec implementation n
:=","by
sorry",,,
"def even_odd_palindrome(n: nat) -> (nat, nat)","Given a positive integer n, return a tuple that has the number of even and odd
integer palindromes that fall within the range(1, n), inclusive.
","[{""input"": 3, ""expected_output"": ""(1, 2)""}, {""input"": 12, ""expected_output"": ""(4, 6)""}]","def even_odd_palindrome(n: nat) -> (nat, nat)
""""""Given a positive integer n, return a tuple that has the number of even and odd
integer palindromes that fall within the range(1, n), inclusive.
""""""","def problem_spec
-- function signature
(implementation: Nat → Nat × Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat × Nat) :=
  let is_palindrome (k: Nat): Prop :=
    List.Palindrome (Nat.digits 10 k);
  let even_palindrome (k: Nat): Prop :=
    (Even k) ∧ (is_palindrome k);
  let odd_palindrome (k: Nat): Prop :=
    (Odd k) ∧ (is_palindrome k);
  n > 0 →
  (1 < n →
    let impl_n_minus_1 := implementation (n - 1);
    ((even_palindrome n) → result.1 = 1 + impl_n_minus_1.1) ∧
    ((odd_palindrome n) → result.2 = 1 + impl_n_minus_1.2) ∧
    (¬ (odd_palindrome n) → result.2 = impl_n_minus_1.2) ∧
    (¬ (even_palindrome n) → result.1 = impl_n_minus_1.1))
  ∧
  (n = 1 → (result.1 = 0) ∧ (result.2 = 1));
-- program termination
∃ result, implementation n = result ∧
spec result","def generated_spec
-- function signature
(impl: Nat → Nat × Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat × Nat :=,sorry,"-- #test implementation 123 = (8, 13)
-- #test implementation 12 = (4, 6)
-- #test implementation 3 = (1, 2)
-- #test implementation 63 = (6, 8)
-- #test implementation 25 = (5, 6)
-- #test implementation 19 = (4, 6)
-- #test implementation 9 = (4, 5)","theorem correctness
(n: Nat)
: problem_spec implementation n
:=","by
sorry",,,
def count_nums(arr: List[int]) -> int,"Write a function count_nums which takes an array of integers and returns
the number of elements which has a sum of digits > 0.
If a number is negative, then its first signed digit will be negative:
e.g. -123 has signed digits -1, 2, and 3.
","[{""input"": [], ""expected_output"": 0}, {""input"": [-1, 11, -11], ""expected_output"": 1}, {""input"": [1, 1, 2], ""expected_output"": 3}]","def count_nums(arr: List[int]) -> int
""""""Write a function count_nums which takes an array of integers and returns
the number of elements which has a sum of digits > 0.
If a number is negative, then its first signed digit will be negative:
e.g. -123 has signed digits -1, 2, and 3.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(arr: List Int) :=

-- spec
let spec (result: Int) :=
  let dig_sum (x: Int): Int :=
    let digs := x.natAbs.digits 10;
    if x >= 0 then
      (List.map (fun t => (t: Int)) digs).sum
    else
      (List.map (fun t => (t: Int)) (digs.drop 1)).sum - (digs[0]! : Int);
  (arr = [] → result = 0) ∧
  (arr ≠ [] → 0 < (dig_sum arr[0]!) → result = 1 + implementation (arr.drop 1)) ∧
  (arr ≠ [] → (dig_sum arr[0]!) ≤ 0 → result = implementation (arr.drop 1));
-- program termination
∃ result, implementation arr = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : Int :=,"let signed_digits(x: Int): List Int :=
    let x': Nat := Int.natAbs x;
    let xs: List Nat := Nat.digits 10 x';
    if x >= 0 then
      xs
    else
      (-Int.ofNat (xs[0]!)) :: xs.tail;
  ((List.map (fun (x: Int) => (signed_digits x).sum) arr).filter (fun (x: Int) => x > 0)).length","-- #test implementation [] = 0
-- #test implementation [-1, -2, 0] = 0
-- #test implementation [1, 1, 2, -2, 3, 4, 5] = 6
-- #test implementation [1, 6, 9, -6, 0, 1, 5] = 5
-- #test implementation [1, 100, 98, -7, 1, -1] = 4
-- #test implementation [12, 23, 34, -45, -56, 0] = 5
-- #test implementation [-0, 1^0] = 1
-- #test implementation [1] = 1","theorem correctness
(arr: List Int)
: problem_spec implementation arr
:=","by
sorry",,,
def move_one_ball(arr: List[int]) -> bool,"We have an array 'arr' of N integers arr[1], arr[2], ..., arr[N].The
numbers in the array will be randomly ordered. Your task is to determine if
it is possible to get an array sorted in non-decreasing order by performing
the following operation on the given array:
    You are allowed to perform right shift operation any number of times.

One right shift operation means shifting all elements of the array by one
position in the right direction. The last element of the array will be moved to
the starting position in the array i.e. 0th index.

If it is possible to obtain the sorted array by performing the above operation
then return True else return False.
If the given array is empty then return True.

Note: The given list is guaranteed to have unique elements.
","[{""input"": [3, 4, 5, 1, 2], ""expected_output"": true}, {""input"": [3, 5, 4, 1, 2], ""expected_output"": false}]","def move_one_ball(arr: List[int]) -> bool
""""""We have an array 'arr' of N integers arr[1], arr[2], ..., arr[N].The
numbers in the array will be randomly ordered. Your task is to determine if
it is possible to get an array sorted in non-decreasing order by performing
the following operation on the given array:
    You are allowed to perform right shift operation any number of times.

One right shift operation means shifting all elements of the array by one
position in the right direction. The last element of the array will be moved to
the starting position in the array i.e. 0th index.

If it is possible to obtain the sorted array by performing the above operation
then return True else return False.
If the given array is empty then return True.

Note: The given list is guaranteed to have unique elements.
""""""","def problem_spec
-- function signature
(implementation: List Int → Bool)
-- inputs
(arr: List Int) :=
let is_shifted (xs: List Int) (ys: List Int) (i: Nat) :=
  (xs.length = ys.length) ∧
  (0 <= i) ∧
  (i < xs.length) ∧
  (forall j, (0 <= j ∧ j < ys.length) → (ys[j]! = xs[(j-i) % xs.length]!))
-- spec
let spec (result: Bool) :=
  ((arr = []) → (result = True)) ∧
  result ↔ (exists i, exists arr_shifted, (is_shifted arr arr_shifted i) ∧ (List.Sorted Int.le arr_shifted))
-- program termination
∃ result, implementation arr = result ∧
spec result","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : Bool :=,sorry,"-- #test implementation [3, 4, 5, 1, 2] = True
-- #test implementation [3, 5, 10, 1, 2] = True
-- #test implementation [4, 3, 1, 2] = False
-- #test implementation [3, 5, 4, 1, 2] = False
-- #test implementation [] = True","theorem correctness
(arr: List Int)
: problem_spec implementation arr
:=","by
sorry",,,
"def exchange(lst1: list[int], lst2: list[int]) -> str","In this problem, you will implement a function that takes two lists of numbers,
and determines whether it is possible to perform an exchange of elements
between them to make lst1 a list of only even numbers.
There is no limit on the number of exchanged elements between lst1 and lst2.
If it is possible to exchange elements between the lst1 and lst2 to make
all the elements of lst1 to be even, return ""YES"".
Otherwise, return ""NO"". It is assumed that the input lists will be non-empty.
","[{""input"": ""([1, 2, 3, 4], [1, 2, 3, 4])"", ""expected_output"": ""YES""}, {""input"": ""([1, 2, 3, 4], [1, 5, 3, 4])"", ""expected_output"": ""NO""}]","def exchange(lst1: list[int], lst2: list[int]) -> str
""""""In this problem, you will implement a function that takes two lists of numbers,
and determines whether it is possible to perform an exchange of elements
between them to make lst1 a list of only even numbers.
There is no limit on the number of exchanged elements between lst1 and lst2.
If it is possible to exchange elements between the lst1 and lst2 to make
all the elements of lst1 to be even, return ""YES"".
Otherwise, return ""NO"". It is assumed that the input lists will be non-empty.
""""""","def problem_spec
-- function signature
(implementation: List Int → List Int → String)
-- inputs
(lst1: List Int)
(lst2: List Int) :=
-- spec
let spec (result : String) :=
  lst1 ≠ [] → lst2 ≠ [] →
  let bool_result := ∃ exchange: List (Nat × Nat),
    let lst1_idxs := exchange.map (fun (a, b) => a)
    let lst2_idxs := exchange.map (fun (a, b) => b)
    lst1_idxs.all (fun i => i < lst1.length) ∧
    lst2_idxs.all (fun i => i < lst2.length) ∧
    lst1_idxs.Nodup ∧
    lst2_idxs.Nodup ∧
    ∀ i, i < lst1.length →
      (i ∉ lst1_idxs → Even (lst1.get! i)) ∧
      (i ∈ lst1_idxs →
        -- find the (a, b) in exchange where a = i
        let i_idx := (lst1_idxs.indexesOf i).head!
        Even (lst2.get! (lst2_idxs.get! i_idx)))
  (bool_result → result = ""YES"") ∧
  (result = ""NO"" → ¬ bool_result) ∧
  (result ≠ ""YES"" ∧ result ≠ ""NO"" → False)
-- program termination
∃ result,
  implementation lst1 lst2 = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Int → List Int → String)
-- inputs
(lst1: List Int)
(lst2: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst1 lst2, problem_spec impl lst1 lst2) ↔
(∀ lst1 lst2, generated_spec impl lst1 lst2) :=",sorry,def implementation (lst1: List Int) (lst2: List Int) : String :=,sorry,"-- #test implementation ([1, 2, 3, 4], [1, 2, 3, 4]) = ""YES""
-- #test implementation ([1, 2, 3, 4], [1, 5, 3, 4]) = ""NO""","theorem correctness
(lst1: List Int)
(lst2: List Int)
: problem_spec implementation lst1 lst2
:=","by
sorry",,,
"def histogram(s : str) -> Dict[str, int]","Given a string representing a space separated lowercase letters, return a dictionary
of the letter with the most repetition and containing the corresponding count.
If several letters have the same occurrence, return all of them.
-- Note(George): I believe the equality extensionality for HashMaps makes this spec true.
","[{""input"": ""a b c"", ""expected_output"": {""a"": 1, ""b"": 1, ""c"": 1}}, {""input"": ""a b b a"", ""expected_output"": {""a"": 2, ""b"": 2}}, {""input"": ""a b c a b"", ""expected_output"": {""a"": 2, ""b"": 2}}, {""input"": ""b b b b a"", ""expected_output"": {""b"": 4}}, {""input"": """", ""expected_output"": {}}]","def histogram(s : str) -> Dict[str, int]
""""""Given a string representing a space separated lowercase letters, return a dictionary
of the letter with the most repetition and containing the corresponding count.
If several letters have the same occurrence, return all of them.
-- Note(George): I believe the equality extensionality for HashMaps makes this spec true.
""""""","def problem_spec
-- function signature
(implementation: String → Std.HashMap Char Nat)
-- inputs
(s: String) :=
-- spec
let spec (result : Std.HashMap Char Nat) :=
    let chars := s.splitOn "" ""
    chars.all (fun c => c.length = 1) ∧ s.all (fun c => c.isLower ∨ c = ' ') →
    ∀ key ∈ result.keys,
      (key.isLower ∧
      key ∈ s.data ∧
      result.get! key = s.count key) ∧
    (∀ char ∈ s.data, char.isLower →
      ((∃ char2 ∈ s.data, char2.isLower ∧ char2 ≠ char ∧
      s.count char < s.count char2) ↔ char ∉ result.keys))
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Std.HashMap Char Nat)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : Std.HashMap Char Nat :=,sorry,"-- #test implementation 'a b c' = {'a': 1, 'b': 1, 'c': 1}
-- #test implementation 'a b b a' = {'a': 2, 'b': 2}
-- #test implementation 'a b c a b' = {'a': 2, 'b': 2}
-- #test implementation 'b b b b a' = {'b': 4}
-- #test implementation '' = {}","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
"def reverse_delete(s : str, c : str) -> (str, bool)","We are given two strings s and c, you have to deleted all the characters in s that are equal to any character in c
then check if the result string is palindrome.
A string is called palindrome if it reads the same backward as forward.
You should return a tuple containing the result string and True/False for the check.
-- Note: We assume the deletions preserve the order of the remaining characters.
","[{""input"": [""abcde"", ""ae""], ""expected_output"": ""(\""bcd\"", False)""}, {""input"": [""abcdef"", ""b""], ""expected_output"": ""(\""acdef\"", False)""}, {""input"": [""abcdedcba"", ""ab""], ""expected_output"": ""('cdedc', True)""}]","def reverse_delete(s : str, c : str) -> (str, bool)
""""""We are given two strings s and c, you have to deleted all the characters in s that are equal to any character in c
then check if the result string is palindrome.
A string is called palindrome if it reads the same backward as forward.
You should return a tuple containing the result string and True/False for the check.
-- Note: We assume the deletions preserve the order of the remaining characters.
""""""","def problem_spec
-- function signature
(implementation: String → String → (String × Bool))
-- inputs
(s: String)
(c: String) :=
-- spec
let spec (result : String × Bool) :=
  let (result_str, result_bool) := result
  result_bool = (List.Palindrome result_str.data) ∧
  (c.data.length = 0 → result_str = s) ∧
  (c.data.length > 0 →
    result_str =
    (implementation
      (String.join ((s.data.filter (fun x => x ≠ c.data.head!)).map (fun c => String.mk [c])))
      (c.drop 1)).fst)

-- program termination
∃ result,
  implementation s c = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String → (String × Bool))
-- inputs
(s: String)
(c: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s c, problem_spec impl s c) ↔
(∀ s c, generated_spec impl s c) :=",sorry,def implementation (s: String) (c: String) : String × Bool :=,sorry,"-- #test implementation ""abcde"" ""ae"" = (""bcd"", False)
-- #test implementation ""abcdef"" ""b"" = (""acdef"", False)
-- #test implementation ""abcdedcba"" ""ab"" = (""cdedc"", True)","theorem correctness
(s c: String)
: problem_spec implementation s c
:=","by
sorry",,,
def odd_count(lst : list[str]) -> list[str],"Given a list of strings, where each string consists of only digits, return a list.
Each element i of the output should be ""the number of odd elements in the
string i of the input."" where all the i's should be replaced by the number
of odd digits in the i'th string of the input.
Note(George): Found it hard to not leak the implementation, so I opted for a recursive statement.
","[{""input"": [""1234567""], ""expected_output"": [""the number of odd elements 4n the str4ng 4 of the 4nput.""]}, {""input"": [""3"", ""11111111""], ""expected_output"": [""the number of odd elements 1n the str1ng 1 of the 1nput."", ""the number of odd elements 8n the str8ng 8 of the 8nput.""]}]","def odd_count(lst : list[str]) -> list[str]
""""""Given a list of strings, where each string consists of only digits, return a list.
Each element i of the output should be ""the number of odd elements in the
string i of the input."" where all the i's should be replaced by the number
of odd digits in the i'th string of the input.
Note(George): Found it hard to not leak the implementation, so I opted for a recursive statement.
""""""","def problem_spec
-- function signature
(implementation: List String → List String)
-- inputs
(lst: List String) :=
-- spec
let spec (result : List String) :=
  lst.all (fun s => s.data.all (fun c => c.isDigit)) →
  (result.length = 0 ↔ lst.length = 0) ∧
  (result.length > 0 →
  let num_odd_digits := (lst.head!.data.filter (fun c => c.isDigit ∧ c.toNat % 2 = 1)).length
  result.head! = ""the number of odd elements "" ++ num_odd_digits.repr ++ ""n the str"" ++ num_odd_digits.repr ++ ""ng "" ++ num_odd_digits.repr ++ "" of the "" ++ num_odd_digits.repr ++ ""nput.""
  ∧ result.tail! = implementation lst.tail!)
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List String → List String)
-- inputs
(lst: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List String) : List String :=,sorry,"-- #test implementation ['1234567'] = [""the number of odd elements 4n the str4ng 4 of the 4nput.""]
-- #test implementation ['3',""11111111""] = [""the number of odd elements 1n the str1ng 1 of the 1nput."",
--  ""the number of odd elements 8n the str8ng 8 of the 8nput.""]","theorem correctness
(lst: List String)
: problem_spec implementation lst
:=","by
sorry",,,
def minSubArraySum(nums : list[int]) -> int,"Given an array of integers nums, find the minimum sum of any non-empty sub-array
of nums.
","[{""input"": [2, 3, 4, 1, 2, 4], ""expected_output"": 1}, {""input"": [-1, -2, -3], ""expected_output"": -6}]","def minSubArraySum(nums : list[int]) -> int
""""""Given an array of integers nums, find the minimum sum of any non-empty sub-array
of nums.
""""""","def problem_spec
-- function signature
(implementation: List Int → Int)
-- inputs
(nums: List Int) :=
-- spec
let spec (result : Int) :=
  (∀ subarray ∈ nums.sublists,
    subarray.length > 0 →
    result ≤ subarray.sum) ∧
  (∃ subarray ∈ nums.sublists,
    subarray.length > 0 ∧
    result = subarray.sum)
-- program termination
∃ result,
  implementation nums = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(nums: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ nums, problem_spec impl nums) ↔
(∀ nums, generated_spec impl nums) :=",sorry,def implementation (nums: List Int) : Int :=,sorry,"-- #test implementation [2, 3, 4, 1, 2, 4] = 1
-- #test implementation [-1, -2, -3] = -6","theorem correctness
(nums: List Int)
: problem_spec implementation nums
:=","by
sorry",,,
"def max_fill_count(grid : list[list[int]], capacity : int) -> int","You are given a rectangular grid of wells. Each row represents a single well,
and each 1 in a row represents a single unit of water.
Each well has a corresponding bucket that can be used to extract water from it,
and all buckets have the same capacity.
Your task is to use the buckets to empty the wells.
Output the number of times you need to lower the buckets.
Constraints:
    * all wells have the same length
    * 1 <= grid.length <= 10^2
    * 1 <= grid[:,1].length <= 10^2
    * grid[i][j] -> 0 | 1
    * 1 <= capacity <= 10
","[{""input"": ""([[0,0,1,0], [0,1,0,0], [1,1,1,1]], 1)"", ""expected_output"": 6}, {""input"": ""([[0,0,1,1], [0,0,0,0], [1,1,1,1], [0,1,1,1]], 2)"", ""expected_output"": 5}, {""input"": ""([[0,0,0], [0,0,0]], 5)"", ""expected_output"": 0}]","def max_fill_count(grid : list[list[int]], capacity : int) -> int
""""""You are given a rectangular grid of wells. Each row represents a single well,
and each 1 in a row represents a single unit of water.
Each well has a corresponding bucket that can be used to extract water from it,
and all buckets have the same capacity.
Your task is to use the buckets to empty the wells.
Output the number of times you need to lower the buckets.
Constraints:
    * all wells have the same length
    * 1 <= grid.length <= 10^2
    * 1 <= grid[:,1].length <= 10^2
    * grid[i][j] -> 0 | 1
    * 1 <= capacity <= 10
""""""","def problem_spec
-- function signature
(implementation: List (List Nat) → Nat → Nat)
-- inputs
(grid: List (List Nat))
(capacity: Nat) :=
-- spec
let spec (result : Nat) :=
  (grid.all (fun row => row.all (fun cell => cell = 0 ∨ cell = 1))) →
  (∃ len : Nat, grid.all (fun row => row.length = len)) →
  (result = 0 ↔ grid.length = 0) ∧
  (grid.length > 0 →
    let well_water_count := grid.head!.sum;
    result - (well_water_count / capacity) - (if well_water_count % capacity > 0 then 1 else 0) = implementation grid.tail! capacity)
-- program termination
∃ result,
  implementation grid capacity = result ∧
  spec result","def generated_spec
-- function signature
(impl: List (List Nat) → Nat → Nat)
-- inputs
(grid: List (List Nat))
(capacity: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ grid capacity, problem_spec impl grid capacity) ↔
(∀ grid capacity, generated_spec impl grid capacity) :=",sorry,def implementation (grid: List (List Nat)) (capacity: Nat) : Nat :=,sorry,"-- #test implementation [[0,0,1,0], [0,1,0,0], [1,1,1,1]] 1 = 6
-- #test implementation [[0,0,1,1], [0,0,0,0], [1,1,1,1], [0,1,1,1]] 2 = 5
-- #test implementation [[0,0,0], [0,0,0]] 5 = 0","theorem correctness
(grid: List (List Nat))
(capacity: Nat)
: problem_spec implementation grid capacity
:=","by
sorry",,,
"def max_fill_count(grid : list[list[int]], capacity : int) -> int","Please write a function that sorts an array of non-negative integers according to
number of ones in their binary representation in ascending order.
For similar number of ones, sort based on decimal value.
","[{""input"": [1, 5, 2, 3, 4], ""expected_output"": [1, 2, 3, 4, 5]}, {""input"": [1, 0, 2, 3, 4], ""expected_output"": [0, 1, 2, 3, 4]}]","def max_fill_count(grid : list[list[int]], capacity : int) -> int
""""""Please write a function that sorts an array of non-negative integers according to
number of ones in their binary representation in ascending order.
For similar number of ones, sort based on decimal value.
""""""","def problem_spec
-- function signature
(implementation: List Nat → List Nat)
-- inputs
(lst: List Nat) :=
-- spec
let spec (result : List Nat) :=
  ∀ x : Nat, lst.count x = result.count x ∧
  result.length = lst.length ∧
  (∀ i j : Nat, i < j → j < result.length →
    Nat.digits 2 (result.get! i) < Nat.digits 2 (result.get! j) ∨
    (Nat.digits 2 (result.get! i) = Nat.digits 2 (result.get! j) ∧ result.get! i < result.get! j))
-- program termination
∃ result,
  implementation lst = result ∧
  spec result","def generated_spec
-- function signature
(impl: List Nat → List Nat)
-- inputs
(lst: List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Nat) : List Nat :=,sorry,"-- #test implementation [1, 5, 2, 3, 4] = [1, 2, 3, 4, 5]
-- #test implementation [1, 0, 2, 3, 4] = [0, 1, 2, 3, 4]","theorem correctness
(lst: List Nat)
: problem_spec implementation lst
:=","by
sorry",,,
"def select_words(s : str, n : int) -> list[str]","Given a string s and a natural number n, you have been tasked to implement
a function that returns a list of all words from string s that contain exactly
n consonants, in order these words appear in the string s.
If the string s is empty then the function should return an empty list.
Note: you may assume the input string contains only letters and spaces.
","[{""input"": ""(\""Mary had a little lamb\"", 4)"", ""expected_output"": [""little""]}, {""input"": ""(\""Mary had a little lamb\"", 3)"", ""expected_output"": [""Mary"", ""lamb""]}, {""input"": ""(\""simple white space\"", 2)"", ""expected_output"": []}, {""input"": ""(\""Hello world\"", 4)"", ""expected_output"": [""world""]}, {""input"": ""(\""Uncle sam\"", 3)"", ""expected_output"": [""Uncle""]}]","def select_words(s : str, n : int) -> list[str]
""""""Given a string s and a natural number n, you have been tasked to implement
a function that returns a list of all words from string s that contain exactly
n consonants, in order these words appear in the string s.
If the string s is empty then the function should return an empty list.
Note: you may assume the input string contains only letters and spaces.
""""""","def problem_spec
-- function signature
(implementation: String → Nat → List String)
-- inputs
(s: String)
(n: Nat) :=
-- spec
let spec (result : List String) :=
  let is_consonant (c: Char) :=
    c ∉ ['a', 'e', 'i', 'o', 'u'] ∧
    c ∉ ['A', 'E', 'I', 'O', 'U'] ∧
    c.isAlpha
  s.all (fun c => c = ' ' ∨ c.isAlpha) →
  let words := s.splitOn "" ""
  (result = [] ↔ (s.length = 0 ∨ words.all (fun word => (word.data.filter (fun c => is_consonant c)).length ≠ n))) ∧
  (result.length > 0 →
    let first_word := result[0]!
    first_word ∈ words ∧
    (first_word.data.filter (fun c => is_consonant c)).length = n ∧
    let first_word_idx := words.indexOf first_word
    (∀ i, i < first_word_idx →
      (words[i]!.data.filter (fun c => is_consonant c)).length ≠ n) ∧
    result.tail! =
      implementation ((words.drop (first_word_idx + 1)).foldl (fun acc word => acc ++ "" "" ++ word) """") n
  )

-- program termination
∃ result,
  implementation s n = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → Nat → List String)
-- inputs
(s: String)
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s n, problem_spec impl s n) ↔
(∀ s n, generated_spec impl s n) :=",sorry,def implementation (s: String) (n: Nat) : List String :=,sorry,"-- #test implementation ""Mary had a little lamb"" 4 = [""little""]
-- #test implementation ""Mary had a little lamb"" 3 = [""Mary"", ""lamb""]
-- #test implementation ""simple white space"" 2 = []
-- #test implementation ""Hello world"" 4 = [""world""]
-- #test implementation ""Uncle sam"" 3 = [""Uncle""]","theorem correctness
(s: String)
(n: Nat)
: problem_spec implementation s n
:=","by
sorry",,,
def get_closest_vowel(s : str) -> str,"You are given a word. Your task is to find the closest vowel that stands between
two consonants from the right side of the word (case sensitive).

Vowels in the beginning and ending doesn't count. Return empty string if you didn't
find any vowel met the above condition.

You may assume that the given string contains English letter only.
Note: The ""closest"" is interpreted as the closest to the end of the word, not the closest to the consonants.
","[{""input"": ""yogurt"", ""expected_output"": ""u""}, {""input"": ""FULL"", ""expected_output"": ""U""}, {""input"": ""quick"", ""expected_output"": ""i""}, {""input"": ""ab"", ""expected_output"": """"}]","def get_closest_vowel(s : str) -> str
""""""You are given a word. Your task is to find the closest vowel that stands between
two consonants from the right side of the word (case sensitive).

Vowels in the beginning and ending doesn't count. Return empty string if you didn't
find any vowel met the above condition.

You may assume that the given string contains English letter only.
Note: The ""closest"" is interpreted as the closest to the end of the word, not the closest to the consonants.
""""""","def problem_spec
-- function signature
(implementation: String → String)
-- inputs
(s: String) :=
-- spec
let spec (result : String) :=
  s.data.all (fun c => c.isAlpha) →
  let is_consonant (c: Char) :=
    c ∉ ['a', 'e', 'i', 'o', 'u'] ∧
    c ∉ ['A', 'E', 'I', 'O', 'U'] ∧
    c.isAlpha
  (result = """" → ¬ ∃ (i j k : Nat), i < j ∧ j < k ∧ k < s.length ∧ is_consonant s.data[i]! ∧ ¬ is_consonant s.data[j]! ∧ is_consonant s.data[k]!) ∧
  (result ≠ """" →
    result.length = 1 ∧
    result.data[0]! ∈ s.data ∧
    ¬ is_consonant result.data[0]! ∧
    ∃ (i j k : Nat),
      i < j ∧ j < k ∧ k < s.length ∧
      is_consonant s.data[i]! ∧ ¬ is_consonant s.data[j]! ∧ is_consonant s.data[k]! ∧
      result.data[0]! = s.data[j]! ∧
      (∀ (i' j' k' : Nat),
        i' < j' ∧ j' < k' ∧ k' < s.length ∧ is_consonant s.data[i']! ∧ ¬ is_consonant s.data[j']! ∧ is_consonant s.data[k']! →
        j' ≤ j)
  )
-- program termination
∃ result,
  implementation s = result ∧
  spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""yogurt"" = ""u""
-- #test implementation ""FULL"" = ""U""
-- #test implementation ""quick"" = ""i""
-- #test implementation ""ab"" = """"","theorem correctness
(s: String)
: problem_spec implementation s
:=","by
sorry",,,
def match_parens(l : list[str]) -> str,"You are given a list of two strings, both strings consist of open
parentheses '(' or close parentheses ')' only.
Your job is to check if it is possible to concatenate the two strings in
some order, that the resulting string will be good.
A string S is considered to be good if and only if all parentheses in S
are balanced. For example: the string '(())()' is good, while the string
'())' is not.
Return 'Yes' if there's a way to make a good string, and return 'No' otherwise.
","[{""input"": [""()("", "")""], ""expected_output"": ""Yes""}, {""input"": ["")"", "")""], ""expected_output"": ""No""}]","def match_parens(l : list[str]) -> str
""""""You are given a list of two strings, both strings consist of open
parentheses '(' or close parentheses ')' only.
Your job is to check if it is possible to concatenate the two strings in
some order, that the resulting string will be good.
A string S is considered to be good if and only if all parentheses in S
are balanced. For example: the string '(())()' is good, while the string
'())' is not.
Return 'Yes' if there's a way to make a good string, and return 'No' otherwise.
""""""","def problem_spec
-- function signature
(implementation: List String → String)
-- inputs
(l: List String) :=
-- spec
let spec (result : String) :=
  l.length = 2 →
  l[0]!.all (fun c => c = '(' ∨ c = ')') →
  l[1]!.all (fun c => c = '(' ∨ c = ')') →
  let res := (balanced_paren_non_computable (l[0]! ++ l[1]!) '(' ')' ∨
            balanced_paren_non_computable (l[1]! ++ l[0]!) '(' ')')
  (res → result = ""Yes"") ∧
  (¬ res → result = ""No"")
-- program termination
∃ result,
  implementation l = result ∧
  spec result","def generated_spec
-- function signature
(impl: List String → String)
-- inputs
(l: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ l, problem_spec impl l) ↔
(∀ l, generated_spec impl l) :=",sorry,def implementation (l: List String) : String :=,sorry,"-- #test implementation ['()(', ')'] = ""Yes""
-- #test implementation [')', ')'] = ""No""","theorem correctness
(l: List String)
: problem_spec implementation l
:=","by
sorry",,,
"def maximum(arr: List[int], k: int) -> List[int]","Given an array arr of integers and a positive integer k, return a sorted list of length
k with the maximum k numbers in arr.
Note:
    1. The length of the array will be in the range of [1, 1000].
    2. The elements in the array will be in the range of [-1000, 1000].
    3. 0 <= k <= len(arr)
","[{""input"": [[2, 4, 3, 1], 3], ""expected_output"": [2, 3, 4]}, {""input"": [[2, 4, 3, 1], 0], ""expected_output"": []}]","def maximum(arr: List[int], k: int) -> List[int]
""""""Given an array arr of integers and a positive integer k, return a sorted list of length
k with the maximum k numbers in arr.
Note:
    1. The length of the array will be in the range of [1, 1000].
    2. The elements in the array will be in the range of [-1000, 1000].
    3. 0 <= k <= len(arr)
""""""","def problem_spec
-- function signature
(impl: List Int → Int → List Int)
-- inputs
(arr: List Int)
(k: Int) :=
-- spec
let spec (result: List Int) :=
    1 ≤ arr.length → arr.length ≤ 1000 → arr.all (fun x => -1000 ≤ x ∧ x ≤ 1000) → 0 ≤ k → k ≤ arr.length →
    result.length = k ∧
    result.Sorted (· ≤ ·) ∧
    ∀ x ∈ result, x ∈ arr ∧

    let result_reversed := result.reverse; -- reverse to get last element
    match result_reversed with
    | [] => k = 0
    | max :: remaining_reversed =>
      arr.max? = some max ∧
      impl (arr.erase max) (k-1) = (remaining_reversed.reverse)
-- program terminates
∃ result, impl arr k = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Int → List Int)
-- inputs
(arr: List Int)
(k: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr k, problem_spec impl arr k) ↔
(∀ arr k, generated_spec impl arr k) :=",sorry,def implementation (arr: List Int) (k: Int) : List Int :=,sorry,"-- #test implementation [2, 4, 3, 1] 3 = [2, 3, 4]
-- #test implementation [2, 4, 3, 1] 0 = []","theorem correctness
(arr: List Int)
(k: Int)
: problem_spec implementation arr k  :=",sorry,,,
def solution(lst: List[int]) -> int,"Given a non-empty list of integers, return the sum of all of the odd elements that
are in even positions.
","[{""input"": [5, 8, 7, 1], ""expected_output"": 12}, {""input"": [3, 3, 3, 3, 3], ""expected_output"": 9}, {""input"": [30, 13, 24, 321], ""expected_output"": 0}]","def solution(lst: List[int]) -> int
""""""Given a non-empty list of integers, return the sum of all of the odd elements that
are in even positions.
""""""","def problem_spec
-- function signature
(impl: List Int → Int)
-- inputs
(lst: List Int) :=
-- spec
let spec (result : Int) :=
lst ≠ [] → ∀ i,  i < lst.length ∧ i % 2 = 0 →
  (lst.length = 1 → impl lst = 0) ∧
  (i + 1 < lst.length →
    (lst[i + 1]! % 2 = 1 →
    impl (lst.drop i) = lst[i + 1]! + (if i + 2 < lst.length then impl (lst.drop (i+2)) else 0)) ∧
    (lst[i + 1]! % 2 = 0 →
    impl (lst.drop i) = if i + 2 < lst.length then impl (lst.drop (i+2)) else 0)
  )
-- program terminates
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int) : Int :=,sorry,"-- #test implementation ([5, 8, 7, 1]: List Int) = 12
-- #test implementation ([3, 3, 3, 3, 3]: List Int) = 9
-- #test implementation ([30, 13, 24, 321]: List Int) = 0","theorem correctness
(lst: List Int)
: problem_spec implementation lst :=",sorry,,,
"def add_elements(arr: List[int], k: int) -> int","Given a non-empty array of integers arr and an integer k, return
the sum of the elements with at most two digits from the first k elements of arr.

Constraints:
    1. 1 <= len(arr) <= 100
    2. 1 <= k <= len(arr)
","[{""input"": [[111, 21, 3, 4000, 5, 6, 7, 8, 9], 4], ""expected_output"": 24}]","def add_elements(arr: List[int], k: int) -> int
""""""Given a non-empty array of integers arr and an integer k, return
the sum of the elements with at most two digits from the first k elements of arr.

Constraints:
    1. 1 <= len(arr) <= 100
    2. 1 <= k <= len(arr)
""""""","def problem_spec
-- function signature
(impl: List Int → Nat → Int)
-- inputs
(arr: List Int)
(k: Nat) :=
-- spec
let spec (result: Int) :=
  1 ≤ arr.length → arr.length ≤ 100 → 1 ≤ k → k ≤ arr.length →
  ((∀ i, 0 ≤ i ∧ i < k → ¬(arr[i]! ≤ 99 ∧ -99 ≤ arr[i]!)) → result = 0) ∧
  ∃ i, i < k
    ∧ arr[i]! ≤ 99 ∧ -99 ≤ arr[i]!
    ∧ result = arr[i]! + (if i = 0 then 0 else impl arr i)
    ∧ ∀ i', i < i' ∧ i' < k → ¬(arr[i']! ≤ 99 ∧ -99 ≤ arr[i']!)
-- program termination
∃ result, impl arr k = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Nat → Int)
-- inputs
(arr: List Int)
(k: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr k, problem_spec impl arr k) ↔
(∀ arr k, generated_spec impl arr k) :=",sorry,def implementation (arr: List Int) (k: Nat) : Int :=,sorry,"-- #test implementation ([111, 21, 3, 4000, 5, 6, 7, 8, 9]: List Int) 4 = 24","theorem correctness
(arr: List Int)
(k: Nat)
: problem_spec implementation arr k :=",sorry,,,
def get_odd_collatz (n: int) -> List[int],"Given a positive integer n, return a sorted list that has the odd numbers in collatz sequence.

The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined
as follows: start with any positive integer n. Then each term is obtained from the
previous term as follows: if the previous term is even, the next term is one half of
the previous term. If the previous term is odd, the next term is 3 times the previous
term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

Note:
    1. Collatz(1) is [1].
    2. returned list sorted in increasing order.
","[{""input"": 5, ""expected_output"": [1, 5]}]","def get_odd_collatz (n: int) -> List[int]
""""""Given a positive integer n, return a sorted list that has the odd numbers in collatz sequence.

The Collatz conjecture is a conjecture in mathematics that concerns a sequence defined
as follows: start with any positive integer n. Then each term is obtained from the
previous term as follows: if the previous term is even, the next term is one half of
the previous term. If the previous term is odd, the next term is 3 times the previous
term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.

Note:
    1. Collatz(1) is [1].
    2. returned list sorted in increasing order.
""""""","def problem_spec
-- function signature
(impl: Nat → List Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: List Nat) :=
n > 0 →
result.Sorted (· < ·) ∧
∀ m, m ∈ result ↔ Odd m ∧ collatz_reachable n m -- m is reachable from starting point n
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → List Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : List Nat :=,sorry,"-- #test implementation 5 = [1, 5]","theorem correctness
(n: Nat)
: problem_spec implementation n :=",sorry,"/--
name: collatz_reachable
use: |
  Helper to check if a natural number m is reachable in the Collatz sequence starting at n.
problems:
  - 123
-/
def collatz_reachable (n m : Nat) : Prop :=
  ∃ k, Nat.iterate (fun x => if x % 2 = 0 then x / 2 else x * 3 + 1) k n = m",,
def valid_date(date: str) -> Bool,"You have to write a function which validates a given date string and
returns True if the date is valid otherwise False.
The date is valid if all of the following rules are satisfied:
1. The date string is not empty.
2. The number of days is not less than 1 or higher than 31 days for months 1,3,5,7,8,10,12. And the number of days is not less than 1 or higher than 30 days for months 4,6,9,11. And, the number of days is not less than 1 or higher than 29 for the month 2.
3. The months should not be less than 1 or higher than 12.
4. The date should be in the format: mm-dd-yyyy
","[{""input"": ""03-11-2000"", ""expected_output"": true}, {""input"": ""15-01-2012"", ""expected_output"": false}, {""input"": ""04-0-2040"", ""expected_output"": false}, {""input"": ""06-04-2020"", ""expected_output"": true}, {""input"": ""06/04/2020"", ""expected_output"": false}]","def valid_date(date: str) -> Bool
""""""You have to write a function which validates a given date string and
returns True if the date is valid otherwise False.
The date is valid if all of the following rules are satisfied:
1. The date string is not empty.
2. The number of days is not less than 1 or higher than 31 days for months 1,3,5,7,8,10,12. And the number of days is not less than 1 or higher than 30 days for months 4,6,9,11. And, the number of days is not less than 1 or higher than 29 for the month 2.
3. The months should not be less than 1 or higher than 12.
4. The date should be in the format: mm-dd-yyyy
""""""","def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(date: String) :=
-- spec
let spec (result: Bool) :=
  result = true ↔
    ∃ m1 m2 sep1 d1 d2 sep2 y1 y2 y3 y4 : Char,
    date = String.mk [m1, m2, sep1, d1, d2, sep2, y1, y2, y3, y4] ∧
    sep1 = '-' ∧ sep2 = '-' ∧
    [m1, m2, d1, d2, y1, y2, y3, y4].all Char.isDigit ∧
    let month := (String.mk [m1, m2]).toNat!;
    let day := (String.mk [d1, d2]).toNat!;
    1 ≤ month ∧ month ≤ 12 ∧
    (month ∈ ({4, 6, 9, 11}: List Nat) → 1 ≤ day ∧ day ≤ 30) ∧
    (month ∈ ({1, 3, 5, 7, 8, 10, 12}: List Nat) → 1 ≤ day ∧ day ≤ 31) ∧
    (month ∈ ({2}: List Nat) → 1 ≤ day ∧ day ≤ 29)
-- program terminates
∃ result, impl date = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(date: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ date, problem_spec impl date) ↔
(∀ date, generated_spec impl date) :=",sorry,def implementation (date: String) : Bool :=,sorry,"-- #test implementation ""03-11-2000"" = true
-- #test implementation ""15-01-2012"" = false
-- #test implementation ""04-0-2040"" = false
-- #test implementation ""06-04-2020"" = true
-- #test implementation ""06/04/2020"" = false","theorem correctness
(date: String)
: problem_spec implementation date :=",sorry,,,
"def minPath(grid, k)","Given a grid with N rows and N columns (N >= 2) and a positive integer k,
each cell of the grid contains a value. Every integer in the range [1, N * N]
inclusive appears exactly once on the cells of the grid.

You have to find the minimum path of length k in the grid. You can start
from any cell, and in each step you can move to any of the neighbor cells,
in other words, you can go to cells which share an edge with you current
cell.
Please note that a path of length k means visiting exactly k cells (not
necessarily distinct).
You CANNOT go off the grid.
A path A (of length k) is considered less than a path B (of length k) if
after making the ordered lists of the values on the cells that A and B go
through (let's call them lst_A and lst_B), lst_A is lexicographically less
than lst_B, in other words, there exist an integer index i (1 <= i <= k)
such that lst_A[i] < lst_B[i] and for any j (1 <= j < i) we have
lst_A[j] = lst_B[j].
It is guaranteed that the answer is unique.
Return an ordered list of the values on the cells that the minimum path go through.
","[{""input"": [[[1, 2, 3], [4, 5, 6], [7, 8, 9]], 3], ""expected_output"": [1, 2, 3]}, {""input"": [[5, 9, 3], [4, 1, 6], [7, 8, 2], 1], ""expected_output"": [1]}]","def minPath(grid, k)
""""""Given a grid with N rows and N columns (N >= 2) and a positive integer k,
each cell of the grid contains a value. Every integer in the range [1, N * N]
inclusive appears exactly once on the cells of the grid.

You have to find the minimum path of length k in the grid. You can start
from any cell, and in each step you can move to any of the neighbor cells,
in other words, you can go to cells which share an edge with you current
cell.
Please note that a path of length k means visiting exactly k cells (not
necessarily distinct).
You CANNOT go off the grid.
A path A (of length k) is considered less than a path B (of length k) if
after making the ordered lists of the values on the cells that A and B go
through (let's call them lst_A and lst_B), lst_A is lexicographically less
than lst_B, in other words, there exist an integer index i (1 <= i <= k)
such that lst_A[i] < lst_B[i] and for any j (1 <= j < i) we have
lst_A[j] = lst_B[j].
It is guaranteed that the answer is unique.
Return an ordered list of the values on the cells that the minimum path go through.
""""""","def problem_spec
-- function signature
(impl: List (List Nat) → Nat → List Nat)
-- inputs
(grid: List (List Nat))
(k: Nat) :=
-- spec
let lexographically_less (a b: List Nat) : Prop :=
  a.length = b.length ∧ a.length = k ∧
  (∃ i, i < k ∧ a.get! i < b.get! i ∧
  (∀ j, j < i → a.get! j = b.get! j));
let rec is_valid_path (k': Nat) (path: List Nat) (grid: List (List Nat)) : Prop :=
  let n := grid.length;
  path.length = k' →
  (∃ i j,
    (i < n ∧ j < n ∧ path.get! 0 = (grid.get! i).get! j) ∧
    (1 < path.length →
      ( ∃ i' j', i' < n ∧ j' < n ∧
        (path.get! 1 = (grid.get! i').get! j') ∧
        ((abs ((i: Int) - (i': Int)) = 1 ∧ j = j') ∨
         (abs ((j: Int) - (j': Int)) = 1 ∧ i = i'))) ∧
      (is_valid_path (k' - 1) (path.drop 1) grid))
  );
let spec (result: List Nat) :=
  let n := grid.length;
  (∀ i, i < n → (grid.get! i).length = n) →
  (∀ i j, i < n → j < n ↔ ((grid.get! i).get! j) ∈ [1, n^2]) →
  is_valid_path k result grid ∧ (∀ path, is_valid_path k path grid → lexographically_less result path);
-- program terminates
∃ result, impl grid k = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List (List Nat) → Nat → List Nat)
-- inputs
(grid: List (List Nat))
(k: Nat): Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ grid k, problem_spec impl grid k) ↔
(∀ grid k, generated_spec impl grid k) :=",sorry,def implementation (grid: List (List Nat)) (k: Nat) : List Nat :=,sorry,"-- #test implementation [[1,2,3], [4,5,6], [7,8,9]] 3 = [1,2,3]
-- #test implementation [[5,9,3], [4,1,6], [7,8,2]] 1 = [1]","theorem correctness
(grid: List (List Nat))
(k: Nat)
: problem_spec implementation grid k :=",sorry,,,
def is_sorted(lst: List[int]) -> Bool,"Given a list of numbers, return whether or not they are sorted
in ascending order. If list has more than 1 duplicate of the same
number, return False. Assume no negative numbers and only integers.
","[{""input"": [5], ""expected_output"": true}, {""input"": [1, 2, 3, 4, 5], ""expected_output"": true}, {""input"": [1, 3, 2, 4, 5], ""expected_output"": false}, {""input"": [1, 2, 3, 4, 5, 6], ""expected_outupt"": true}, {""input"": [1, 2, 3, 4, 5, 6, 7], ""expected_output"": true}, {""input"": [1, 3, 2, 4, 5, 6, 7], ""expected_output"": false}, {""input"": [1, 2, 2, 3, 3, 4], ""expected_output"": true}, {""input"": [1, 2, 2, 2, 3, 4], ""expected_output"": false}]","def is_sorted(lst: List[int]) -> Bool
""""""Given a list of numbers, return whether or not they are sorted
in ascending order. If list has more than 1 duplicate of the same
number, return False. Assume no negative numbers and only integers.
""""""","def problem_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(lst: List Int) :=
-- spec
let sorted_ascending := lst.Sorted (· ≤ ·);
let ms := Multiset.ofList lst;
let multiple_duplicates := ∃ i, i ∈ lst ∧ 2 < ms.count i;
let spec (res: Bool) :=
  res → sorted_ascending ∧
  res → ¬multiple_duplicates ∧
  multiple_duplicates → ¬res ∧
  ¬sorted_ascending → ¬res;
-- program terminates
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Bool)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int) : Bool :=,sorry,"-- #test implementation [5] = true
-- #test implementation [1, 2, 3, 4, 5] = true
-- #test implementation [1, 3, 2, 4, 5] = false
-- #test implementation [1, 2, 3, 4, 5, 6] = true
-- #test implementation [1, 2, 3, 4, 5, 6, 7] = true
-- #test implementation [1, 3, 2, 4, 5, 6, 7] = false
-- #test implementation [1, 2, 2, 3, 3, 4] = true
-- #test implementation [1, 2, 2, 2, 3, 4] = false","theorem correctness
(lst: List Int)
: problem_spec implementation lst :=",sorry,,,
"def intersection(interval1: Tuple[Int, Int], interval2: Tuple[Int, Int]) -> str","You are given two intervals,
where each interval is a pair of integers. For example, interval = (start, end) = (1, 2).
The given intervals are closed which means that the interval (start, end)
includes both start and end.
For each given interval, it is assumed that its start is less or equal its end.
Your task is to determine whether the length of intersection of these two
intervals is a prime number.
Example, the intersection of the intervals (1, 3), (2, 4) is (2, 3)
which its length is 1, which not a prime number.
If the length of the intersection is a prime number, return ""YES"",
otherwise, return ""NO"".
If the two intervals don't intersect, return ""NO"".
","[{""input"": [""(1"", ""2)"", ""(2"", ""3)""], ""expected_output"": ""NO""}, {""input"": [""(-1"", ""1)"", ""(0"", ""4)""], ""expected_output"": ""NO""}, {""input"": [""(-3"", ""-1)"", ""(-5"", ""5)""], ""expected_output"": ""YES""}]","def intersection(interval1: Tuple[Int, Int], interval2: Tuple[Int, Int]) -> str
""""""You are given two intervals,
where each interval is a pair of integers. For example, interval = (start, end) = (1, 2).
The given intervals are closed which means that the interval (start, end)
includes both start and end.
For each given interval, it is assumed that its start is less or equal its end.
Your task is to determine whether the length of intersection of these two
intervals is a prime number.
Example, the intersection of the intervals (1, 3), (2, 4) is (2, 3)
which its length is 1, which not a prime number.
If the length of the intersection is a prime number, return ""YES"",
otherwise, return ""NO"".
If the two intervals don't intersect, return ""NO"".
""""""","def problem_spec
-- function signature
(impl: Int × Int → Int × Int → String)
-- inputs
(interval1: Int × Int)
(interval2: Int × Int) :=
-- spec
let spec (result: String) :=
let (s1, e1) := interval1;
let (s2, e2) := interval2;
s1 ≤ e1 → s2 ≤ e2 →
let intersectionStart := max s1 s2;
let intersectionEnd := min e1 e2;
let hasIntersection := intersectionStart ≤ intersectionEnd;
let isPrime := Nat.Prime (intersectionEnd - intersectionStart).toNat;
(result = ""YES"" ↔ hasIntersection ∧ isPrime) ∧
(result = ""NO"" ↔ ¬hasIntersection ∨ ¬isPrime) ∧
(result = ""YES"" ∨ result = ""NO"")
-- program terminates
∃ result, impl interval1 interval2 = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Int × Int → Int × Int → String)
-- inputs
(interval1: Int × Int)
(interval2: Int × Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ interval1 interval2, problem_spec impl interval1 interval2) ↔
(∀ interval1 interval2, generated_spec impl interval1 interval2) :=",sorry,def implementation (interval1: Int × Int) (interval2: Int × Int) : String :=,sorry,"-- #test implementation (1, 2) (2, 3) = ""NO""
-- #test implementation (-1, 1) (0, 4) = ""NO""
-- #test implementation (-3, -1) (-5, 5) = ""YES""","theorem correctness
(interval1: Int × Int)
(interval2: Int × Int)
: problem_spec implementation interval1 interval2 :=",sorry,,,
def prod_signs(arr: List[int]) -> Optional[int],"You are given an array arr of integers and you need to return
sum of magnitudes of integers multiplied by product of all signs
of each number in the array, represented by 1, -1 or 0.
Note: return None for empty arr.
","[{""input"": [1, 2, 2, -4], ""expected_output"": -9}, {""input"": [0, 1], ""expected_output"": 0}, {""input"": [], ""expected_output"": ""None""}]","def prod_signs(arr: List[int]) -> Optional[int]
""""""You are given an array arr of integers and you need to return
sum of magnitudes of integers multiplied by product of all signs
of each number in the array, represented by 1, -1 or 0.
Note: return None for empty arr.
""""""","def problem_spec
-- function signature
(impl: List Int → Option Int)
-- inputs
(arr: List Int) :=
-- spec
let spec (result: Option Int) :=
  match result with
  | none => arr = []
  | some result =>
  let magnitude_sum := (arr.map (fun x => Int.ofNat x.natAbs)).sum;
    let neg_count_odd := (arr.filter (fun x => x < 0)).length % 2 = 1;
    let has_zero := 0 ∈ arr;
    (result < 0 ↔ (neg_count_odd ∧ ¬has_zero)
      ∧ result = magnitude_sum * -1) ∧
    (0 < result ↔ (¬neg_count_odd ∧ ¬has_zero)
      ∧ result = magnitude_sum) ∧
    (result = 0 ↔ has_zero)
-- program terminates
∃ result, impl arr = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Option Int)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : Option Int :=,sorry,"-- #test implementation ([1, 2, 2, -4]: List Int) = (-9: Int)
-- #test implementation ([0, 1]: List Int) = (0: Int)
-- #test implementation ([]: List Int) = none","theorem correctness
(arr: List Int)
: problem_spec implementation arr :=",sorry,,,
def split_words(txt),"Given a string of words, return a list of words split on whitespace, if no whitespaces exists in the text you
should split on commas ',' if no commas exists you should return the number of lower-case letters with odd order in the
alphabet, ord('a') = 0, ord('b') = 1, ... ord('z') = 25
","[{""input"": ""Hello world!"", ""expected_output"": [""Hello"", ""world!""]}, {""input"": ""Hello,world!"", ""expected_output"": [""Hello"", ""world!""]}, {""input"": ""abcdef"", ""expected_output"": 3}]","def split_words(txt)
""""""Given a string of words, return a list of words split on whitespace, if no whitespaces exists in the text you
should split on commas ',' if no commas exists you should return the number of lower-case letters with odd order in the
alphabet, ord('a') = 0, ord('b') = 1, ... ord('z') = 25
""""""","def problem_spec
-- function signature
-- return a tuple of Option (List String) and Option Nat
(impl: String → Option (List String) × Option Nat)
-- inputs
(text: String) :=
-- spec
let spec (result: Option (List String) × Option Nat) :=
  -- both cannot be None
  let words := result.fst;
  let ord := result.snd;
  0 < text.length →
  ¬ (words = none ∧ ord = none) ∧
    (words = none ↔ ∀ ch, ch ∈ text.toList →  (ch = ',' ∨ ch = ' ')) ∧
    (∀ num, ord = some num → (text.get! 0).toNat = num) ∧
    (∀ lst, words = some lst → ∀ i, i < lst.length →
      let str := lst.get! i;
      text.containsSubstr str) ∧
    (∀ lst, words = some lst →
      let first := text.takeWhile (fun c => c ≠ ',' ∧ c ≠ ' ');
      let nextImpl := impl (text.drop (first.length + 1));
      let nextWords := nextImpl.fst;
      (∃ nextLst, nextWords = some nextLst ∧
        lst = [first] ++ nextLst))
-- program terminates
∃ result, impl text = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Option (List String) × Option Nat)
-- inputs
(text: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ text, problem_spec impl text) ↔
(∀ text, generated_spec impl text) :=",sorry,def implementation (text: String) : Option (List String) × Option Nat :=,sorry,"-- #test implementation ""Hello world!"" = (some [""Hello"", ""world!""], none)
-- #test implementation ""Hello,world!"" = (some [""Hello"", ""world!""], none)
-- #test implementation ""abcdef"" = (none, some 3)","theorem correctness
(text: String)
: problem_spec implementation text :=",sorry,,,
def tri(n: int) -> List[int],"Everyone knows Fibonacci sequence, it was studied deeply by mathematicians in
the last couple centuries. However, what people don't know is Tribonacci sequence.
Tribonacci sequence is defined by the recurrence:
tri(1) = 3
tri(n) = 1 + n / 2, if n is even.
tri(n) =  tri(n - 1) + tri(n - 2) + tri(n + 1), if n is odd.
For example:
tri(2) = 1 + (2 / 2) = 2
tri(4) = 3
tri(3) = tri(2) + tri(1) + tri(4)
       = 2 + 3 + 3 = 8
You are given a non-negative integer number n, you have to a return a list of the
first n + 1 numbers of the Tribonacci sequence.
","[{""input"": 3, ""expected_output"": [1, 3, 2, 8]}]","def tri(n: int) -> List[int]
""""""Everyone knows Fibonacci sequence, it was studied deeply by mathematicians in
the last couple centuries. However, what people don't know is Tribonacci sequence.
Tribonacci sequence is defined by the recurrence:
tri(1) = 3
tri(n) = 1 + n / 2, if n is even.
tri(n) =  tri(n - 1) + tri(n - 2) + tri(n + 1), if n is odd.
For example:
tri(2) = 1 + (2 / 2) = 2
tri(4) = 3
tri(3) = tri(2) + tri(1) + tri(4)
       = 2 + 3 + 3 = 8
You are given a non-negative integer number n, you have to a return a list of the
first n + 1 numbers of the Tribonacci sequence.
""""""","def problem_spec
-- function signature
(impl: Nat → List Int)
-- inputs
(n: Nat) :=
-- spec
let spec (result: List Int) :=
  0 < result.length ∧
  result.length = n ∧
  let i := result.length-1;
  (i = 0 → result[0]! = 1) ∧ -- base case
  (i = 1 → result[1]! = 3) ∧
  (2 ≤ i ∧ i % 2 = 0 → result[i]! = 1 + i / 2) ∧
  (2 ≤ i ∧ i % 2 = 1 → result[i]! = result[i-2]! + result[i-1]! + (1 + (i+1) / 2)) ∧
  if i = 0 then true else result.take i = impl (i-1)
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → List Int)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : List Int:=,sorry,"-- #test implementation 3 = [1, 3, 2, 8]","theorem correctness
(n: Nat)
: problem_spec implementation n :=",sorry,,,
def digits(n: int) -> int,"Given a positive integer n, return the product of the odd digits.
Return 0 if all digits are even.
","[{""input"": 1, ""expected_output"": 1}, {""input"": 4, ""expected_output"": 0}, {""input"": 235, ""expected_output"": 15}]","def digits(n: int) -> int
""""""Given a positive integer n, return the product of the odd digits.
Return 0 if all digits are even.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
  0 < n →
  (n < 10 → (n % 2 = 1 → result = n) ∧ (n % 2 = 0 → result = 0)) ∧
  (10 ≤ n →
    let digit := n % 10;
    let rest := n / 10;
    (digit % 2 = 1 →
      if (Nat.toDigits 10 rest).all (fun x => Even (x.toNat - '0'.toNat))
        then impl rest = 0 ∧ result = digit
      else
        result = impl rest * digit)
    ∧
    (digit % 2 = 0 →
      result = impl rest))
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,"-- #test implementation 1 = 1
-- #test implementation 4 = 0
-- #test implementation 235 = 15","theorem correctness
(n: Nat)
: problem_spec implementation n :=",sorry,,,
def is_nested(string: str) -> Bool,"Create a function that takes a string as input which contains only parentheses.
The function should return True if and only if there is a valid subsequence of parentheses
where at least one parenthesis in the subsequence is nested.
","[{""input"": ""(())"", ""expected_output"": true}, {""input"": ""()))))))((((()"", ""expected_output"": false}, {""input"": ""()()"", ""expected_output"": false}, {""input"": ""()"", ""expected_output"": false}, {""input"": ""(()())"", ""expected_output"": true}, {""input"": ""(())(("", ""expected_output"": true}]","def is_nested(string: str) -> Bool
""""""Create a function that takes a string as input which contains only parentheses.
The function should return True if and only if there is a valid subsequence of parentheses
where at least one parenthesis in the subsequence is nested.
""""""","def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(string: String) :=
-- spec
let spec (result: Bool) :=
string.toList.all (fun x => x = '(' ∨ x = ')') →
result = true ↔
  ∃ x : String,
    is_subsequence x.toList string.toList ∧
    balanced_paren_non_computable x '(' ')' ∧
    2 ≤ count_max_paren_depth x
-- program termination
∃ result, impl string = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(lst: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ string, problem_spec impl string) ↔
(∀ string, generated_spec impl string) :=",sorry,def implementation (lst: String) : Bool :=,sorry,"-- #test implementation ""(())"" = true
-- #test implementation ""()))))))((((()"" = false
-- #test implementation ""()()"" = false
-- #test implementation ""()"" = false
-- #test implementation ""(()())"" = true
-- #test implementation ""(())(("" = true","theorem correctness
(string: String)
: problem_spec implementation string :=",sorry,"/--
name: string_is_paren_balanced_helper
use: |
  Helper function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced_helper
(paren_string: String) (num_open: Int): Bool
:=
-- Recursively check if the string is balanced
if paren_string.isEmpty then
  num_open = 0
else
  let c := paren_string.get! 0
  if c == '(' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open + 1)
  else if c == ')' then
    string_is_paren_balanced_helper (paren_string.drop 1) (num_open - 1)
  else
    string_is_paren_balanced_helper (paren_string.drop 1) num_open
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: string_is_paren_balanced
use: |
  Function to check if a string is balanced with respect to parentheses.
problems:
  - 1
  - 6
  - 132
sample_problems:
  - 0
-/
def string_is_paren_balanced
(paren_string: String): Bool
:=
string_is_paren_balanced_helper paren_string 0

/--
name: count_max_paren_depth_helper
use: |
  Helper to count the maximum depth of parentheses in a string.
problems:
  - 6
  - 132
-/
def count_max_paren_depth_helper
(paren_string: String) (num_open: Int) (max_depth: Nat): Nat :=
-- Recursively count the maximum depth of parentheses
if paren_string.isEmpty then
  max_depth
else
  let c := paren_string.get! 0
  if c == '(' then
    let new_num_open := num_open + 1
    count_max_paren_depth_helper (paren_string.drop 1) (new_num_open) (max_depth.max new_num_open.toNat)
  else if c == ')' then
    count_max_paren_depth_helper (paren_string.drop 1) (num_open - 1) max_depth
  else
    count_max_paren_depth_helper (paren_string.drop 1) num_open max_depth
termination_by paren_string.length
decreasing_by
  all_goals
  {
    rename_i h_non_empty_string
    rw [String.drop_eq, String.length]
    simp
    rw [String.isEmpty_iff] at h_non_empty_string
    by_cases h_paren_nil : paren_string.length ≤ 0
    rw [Nat.le_zero_eq] at h_paren_nil
    rw [←string_eq_iff_data_eq] at h_non_empty_string
    have h_temp : """".data = [] := by simp
    rw [h_temp] at h_non_empty_string
    rw [String.length] at h_paren_nil
    rw [List.length_eq_zero] at h_paren_nil
    contradiction
    have h_temp : paren_string.length > 0 := by linarith
    assumption
  }

/--
name: count_max_paren_depth
use: |
  Function to count the maximum depth of parentheses in a string.
problems:
  - 6
  - 132
-/
def count_max_paren_depth
(paren_string: String): Nat :=
count_max_paren_depth_helper paren_string 0 0

/--
name: is_subsequence
use: |
  Helper to check if List Char xs is a subsequence of List Char ys.
problems:
  - 132
-/
def is_subsequence (xs ys : List Char) : Bool :=
  match xs, ys with
  | [], _ => true
  | _, [] => false
  | x::xs', y::ys' =>
      if x = y then is_subsequence xs' ys' else is_subsequence xs ys'",,
def sum_squares(lst: List[float]) -> int,"You are given a list of numbers.
You need to return the sum of squared numbers in the given list,
round each element in the list to the upper int(Ceiling) first.
","[{""input"": [1, 2, 3], ""expected_output"": 14}, {""input"": [1, 4, 9], ""expected_output"": 98}, {""input"": [1, 3, 5, 7], ""expected_output"": 84}, {""input"": [1.4, 4.2, 0], ""expected_output"": 29}, {""input"": [-2.4, 1, 1], ""expected_output"": 6}]","def sum_squares(lst: List[float]) -> int
""""""You are given a list of numbers.
You need to return the sum of squared numbers in the given list,
round each element in the list to the upper int(Ceiling) first.
""""""","def problem_spec
-- function signature
(impl: List Rat → Int)
-- inputs
(lst: List Rat) :=
-- spec
let spec (result: Int) :=
  (lst = [] → result = 0) ∧
  (lst != [] → 0 ≤ result - lst[0]!.ceil^2 ∧ (impl (lst.drop 1) = (result - lst[0]!.ceil^2)))
-- program termination
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Rat → Int)
-- inputs
(lst: List Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Rat) : Int :=,sorry,"-- #test implementation [1, 2, 3] = 14
-- #test implementation [1, 4, 9] = 98
-- #test implementation [1, 3, 5, 7] = 84
-- #test implementation [1.4, 4.2, 0] = 29
-- #test implementation [-2.4, 1, 1] = 6","theorem correctness
(lst: List Rat)
: problem_spec implementation lst :=",sorry,,,
def check_if_last_char_is_a_letter(txt: str) -> Bool,"Create a function that returns True if the last character
of a given string is an alphabetical character and is not
a part of a word, and False otherwise.
Note: ""word"" is a group of characters separated by space.
","[{""input"": ""apple pie"", ""expected_output"": false}, {""input"": ""apple pi e"", ""expected_output"": true}, {""input"": ""apple pi e "", ""expected_output"": false}, {""input"": """", ""expected_output"": false}]","def check_if_last_char_is_a_letter(txt: str) -> Bool
""""""Create a function that returns True if the last character
of a given string is an alphabetical character and is not
a part of a word, and False otherwise.
Note: ""word"" is a group of characters separated by space.
""""""","def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(txt: String) :=
-- spec
let spec (result: Bool) :=
  let words := txt.splitOn "" "";
  match words with
  | [] => result = False
  | [last_word] => (result ↔ last_word.length = 1 ∧ (let diff := (last_word.get 0).toLower.toNat - 'a'.toNat; 0 ≤ diff ∧ diff ≤ 25))
  | head::tail => result ↔ (let tail_txt := String.join tail; impl tail_txt);
-- program terminates
∃ result, impl txt = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(txt: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ txt, problem_spec impl txt) ↔
(∀ txt, generated_spec impl txt) :=",sorry,def implementation (txt: String) : Bool :=,sorry,"-- #test implementation ""apple pie"" = false
-- #test implementation ""apple pi e"" = true
-- #test implementation ""apple pi e "" = false
-- #test implementation """" = false","theorem correctness
(txt: String)
: problem_spec implementation txt :=",sorry,,,
def can_arrange(arr: List[int]) -> int,"Create a function which returns the largest index of an element which
is not greater than or equal to the element immediately preceding it. If
no such element exists then return -1. The given array will not contain
duplicate values.
","[{""input"": [1, 2, 4, 3, 5], ""expected_output"": 3}, {""input"": [1, 2, 3], ""expected_output"": -1}]","def can_arrange(arr: List[int]) -> int
""""""Create a function which returns the largest index of an element which
is not greater than or equal to the element immediately preceding it. If
no such element exists then return -1. The given array will not contain
duplicate values.
""""""","def problem_spec
-- function signature
(impl: List Int → Int)
-- inputs
(arr: List Int) :=
-- spec
let spec (result: Int) :=
  ¬arr.any (fun x => 1 < arr.count x) →
  (arr.length = 0 ∨ arr.length = 1 → result = -1) ∧
  (1 < arr.length →
    let last := arr.length-1;
    let i := if arr[last]! < arr[last-1]! then Int.ofNat last else -1;
    result = max (impl (arr.take last)) i);
-- program termination
∃ result, impl arr = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(arr: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ arr, problem_spec impl arr) ↔
(∀ arr, generated_spec impl arr) :=",sorry,def implementation (arr: List Int) : Int :=,sorry,"-- #test implementation [1, 2, 4, 3, 5] = 3
-- #test implementation [1, 2, 3] = -1","theorem correctness
(arr: List Int)
: problem_spec implementation arr :=",sorry,,,
"def largest_smallest_integers(lst: List[int]) -> Tuple[ Optional[Int], Optional[Int] ]","Create a function that returns a tuple (a, b), where 'a' is
the largest of negative integers, and 'b' is the smallest
of positive integers in a list.
If there is no negative or positive integers, return them as None.
","[{""input"": [2, 4, 1, 3, 5, 7], ""expected_output"": ""(None, 1)""}, {""input"": [], ""expected_output"": ""(None, None)""}, {""input"": [0], ""expected_output"": ""(None, None)""}]","def largest_smallest_integers(lst: List[int]) -> Tuple[ Optional[Int], Optional[Int] ]
""""""Create a function that returns a tuple (a, b), where 'a' is
the largest of negative integers, and 'b' is the smallest
of positive integers in a list.
If there is no negative or positive integers, return them as None.
""""""","def problem_spec
-- function signature
(impl: List Int → Option Int × Option Int)
-- inputs
(lst: List Int) :=
-- spec
let spec (result: Option Int × Option Int) :=
  let (a, b) := result;
  (match a with
  | none => ¬(∃ i, i ∈ lst ∧ i < 0)
  | some a => a < 0 ∧ a ∈ lst ∧ ∀ i, i ∈ lst ∧ i < 0 → i ≤ a) ∧
  (match b with
  | none => ¬(∃ i, i ∈ lst ∧ 0 < i)
  | some b => 0 < b ∧ b ∈ lst ∧ ∀ i, i ∈ lst ∧ 0 < i → b ≤ i)
-- program termination
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Option Int × Option Int)
-- inputs
(lst: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List Int) : (Option Int × Option Int) :=,sorry,"-- #test implementation [2, 4, 1, 3, 5, 7] = (none, some 1)
-- #test implementation [] = (none, none)
-- #test implementation [0] = (none, none)","theorem correctness
(lst: List Int)
: problem_spec implementation lst :=",sorry,,,
def is_equal_to_sum_even(n: int) -> Bool,"Evaluate whether the given number n can be written as the sum of exactly 4 positive even numbers
","[{""input"": 4, ""expected_output"": false}, {""input"": 6, ""expected_output"": false}, {""input"": 8, ""expected_output"": true}]","def is_equal_to_sum_even(n: int) -> Bool
""""""Evaluate whether the given number n can be written as the sum of exactly 4 positive even numbers
""""""","def problem_spec
-- function signature
(impl: Int → Bool)
-- inputs
(n: Int) :=
-- spec
let spec (result: Bool) :=
  let sum_exists := ∃ a b c d : Nat,
    Even a ∧
    Even b ∧
    Even c ∧
    Even d ∧
    (a + b + c + d = n);
  result = true ↔ sum_exists
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Int → Bool)
-- inputs
(n: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Int) : Bool :=,sorry,"-- #test implementation 4 = false
-- #test implementation 6 = false
-- #test implementation 8 = true","theorem correctness
(n: Int)
: problem_spec implementation n :=",sorry,,,
def special_factorial(n: int) -> int,"The Brazilian factorial is defined as:
brazilian_factorial(n) = n! * (n-1)! * (n-2)! * ... * 1!
where n > 0. Please write a function that computes the Brazilian factorial.
","[{""input"": 4, ""expected_output"": 288}]","def special_factorial(n: int) -> int
""""""The Brazilian factorial is defined as:
brazilian_factorial(n) = n! * (n-1)! * (n-2)! * ... * 1!
where n > 0. Please write a function that computes the Brazilian factorial.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
let factorial := Nat.factorial n;
(0 < n → result / factorial = impl (n - 1)) ∧
(n = 0 → result = 1);
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,-- #test implementation 4 = 288,"theorem correctness
(n: Nat)
: problem_spec implementation n :=",sorry,,,
def fix_spaces(text: str) -> str,"Given a string text, replace all spaces in it with underscores,
and if a string has more than 2 consecutive spaces,
then replace all consecutive spaces with -
","[{""input"": ""Example"", ""expected_output"": ""Example""}, {""input"": ""Example 1"", ""expected_output"": ""Example_1""}, {""input"": "" Example 2"", ""expected_output"": ""_Example_2""}, {""input"": "" Example   3"", ""expected_output"": ""_Example-3""}]","def fix_spaces(text: str) -> str
""""""Given a string text, replace all spaces in it with underscores,
and if a string has more than 2 consecutive spaces,
then replace all consecutive spaces with -
""""""","def problem_spec
-- function signature
(impl: String → String)
-- inputs
(text: String) :=
-- spec
let spec (result: String) :=
  (result = """" → text = """") ∧
  (result ≠ """" → (
    (∃ pref s, text = pref ++ s
      ∧ pref.length = 1
      ∧ pref ≠ "" ""
      ∧ result = pref ++ impl s)
    ∨
    (∃ pref s : String, text = pref ++ s ∧ pref ≠ """" ∧ (∀ ch, ch ∈ pref.toList → ch = ' ')
      ∧ let k := pref.length;
        (k ≤ 2 → result = (String.replicate k '_') ++ (impl (text.drop k)))
      ∧ (2 < k → result = ""-"" ++ (impl (text.drop k)))) )
  )
-- program termination
∃ result, impl text = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(text: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ text, problem_spec impl text) ↔
(∀ text, generated_spec impl text) :=",sorry,def implementation (text: String) : String :=,sorry,"-- #test implementation ""Example"" = ""Example""
-- #test implementation ""Example 1"" = ""Example_1""
-- #test implementation "" Example 2"" = ""_Example_2""
-- #test implementation "" Example   3"" = ""_Example-3""","theorem correctness
(text: String)
: problem_spec implementation text :=",sorry,,,
def file_name_check(file_name: str) -> str,"Create a function which takes a string representing a file's name, and returns
'Yes' if the the file's name is valid, and returns 'No' otherwise.
A file's name is considered to be valid if and only if all the following conditions
are met:
- There should not be more than three digits ('0'-'9') in the file's name.
- The file's name contains exactly one dot '.'
- The substring before the dot should not be empty, and it starts with a letter from
the latin alphapet ('a'-'z' and 'A'-'Z').
- The substring after the dot should be one of these: ['txt', 'exe', 'dll']
","[{""input"": ""example.txt"", ""expected_output"": ""Yes""}, {""input"": ""1example.dll"", ""expected_output"": ""No""}]","def file_name_check(file_name: str) -> str
""""""Create a function which takes a string representing a file's name, and returns
'Yes' if the the file's name is valid, and returns 'No' otherwise.
A file's name is considered to be valid if and only if all the following conditions
are met:
- There should not be more than three digits ('0'-'9') in the file's name.
- The file's name contains exactly one dot '.'
- The substring before the dot should not be empty, and it starts with a letter from
the latin alphapet ('a'-'z' and 'A'-'Z').
- The substring after the dot should be one of these: ['txt', 'exe', 'dll']
""""""","def problem_spec
-- function signature
(impl: String → String)
-- inputs
(file_name : String) :=
-- spec
let spec (result: String) :=
let valid := (file_name.toList.filter Char.isDigit).length ≤ 3 ∧
  (file_name.toList.filter (· = '.')).length = 1 ∧
  ∃ before after : String,
    file_name = before ++ ""."" ++ after ∧
    before != """" ∧
    Char.isAlpha (before.get! 0) ∧
    (after = ""txt"" ∨ after = ""exe"" ∨ after = ""dll"")
(result = ""Yes"" ↔ valid) ∧
(result = ""No"" ↔ ¬valid)

-- program termination
∃ result, impl file_name = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(file_name: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ file_name, problem_spec impl file_name) ↔
(∀ file_name, generated_spec impl file_name) :=",sorry,def implementation (file_name : String) : String :=,sorry,"-- #test implementation ""example.txt"" = ""Yes""
-- #test implementation ""1example.dll"" = ""No""","theorem correctness
(file_name : String)
: problem_spec implementation file_name :=",sorry,,,
def sum_squares(lst: List[int]) -> int,"This function will take a list of integers. For all entries in the list, the function shall square the integer entry if its index is a
multiple of 3 and will cube the integer entry if its index is a multiple of 4 and not a multiple of 3. The function will not
change the entries in the list whose indexes are not a multiple of 3 or 4. The function shall then return the sum of all entries.
","[{""input"": [1, 2, 3], ""expected_output"": 6}, {""input"": [], ""expected_output"": 0}, {""input"": [-1, -5, 2, -1, -5], ""expected_output"": -126}]","def sum_squares(lst: List[int]) -> int
""""""This function will take a list of integers. For all entries in the list, the function shall square the integer entry if its index is a
multiple of 3 and will cube the integer entry if its index is a multiple of 4 and not a multiple of 3. The function will not
change the entries in the list whose indexes are not a multiple of 3 or 4. The function shall then return the sum of all entries.
""""""","def problem_spec
-- function signature
(impl: List Int → Int)
-- inputs
(lst : List Int) :=
-- spec
let spec (result : Int) :=
let last := lst.length-1;
(lst = [] → result = 0) ∧
(lst ≠ [] ∧ last % 3 = 0 → result = lst[last]! ^ 2 + impl (lst.take last)) ∧
(lst ≠ [] ∧ last % 4 = 0 ∧ last % 3 != 0 → result = lst[last]! ^ 3 + impl (lst.take last)) ∧
(lst ≠ [] ∧ last % 3 != 0 ∧ last % 4 != 0 → result = lst[last]! + impl (lst.take last))
-- program termination
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(lst : List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst : List Int) : Int :=,sorry,"-- #test implementation [1, 2, 3] = 6
-- #test implementation [] = 0
-- #test implementation [-1, -5, 2, -1, -5] = -126","theorem correctness
(lst : List Int)
: problem_spec implementation lst :=",sorry,,,
def words_in_sentence(sentence: str) -> str,"You are given a string representing a sentence,
the sentence contains some words separated by a space,
and you have to return a string that contains the words from the original sentence,
whose lengths are prime numbers,
the order of the words in the new string should be the same as the original one.

Constraints:
* 1 <= len(sentence) <= 100
* sentence contains only letters
","[{""input"": ""This is a test"", ""expected_output"": ""is""}, {""input"": ""lets go for swimming"", ""expected_output"": ""go for""}]","def words_in_sentence(sentence: str) -> str
""""""You are given a string representing a sentence,
the sentence contains some words separated by a space,
and you have to return a string that contains the words from the original sentence,
whose lengths are prime numbers,
the order of the words in the new string should be the same as the original one.

Constraints:
* 1 <= len(sentence) <= 100
* sentence contains only letters
""""""","def problem_spec
-- function signature
(impl: String → String)
-- inputs
(sentence: String) :=
-- spec
let spec (result: String) :=
let words := sentence.splitOn;
let result_words := result.splitOn;
  1 ≤ sentence.length → sentence.length ≤ 100 →
  sentence.all (fun x => Char.isAlpha x) →
  result_words.length ≤ words.length ∧
  (∀ word ∈ result_words, word ∈ words ∧ Nat.Prime word.length) ∧
  match result_words with
  | [] => ∀ word ∈ words, ¬(Nat.Prime word.length)
  | head :: tail => if Nat.Prime head.length ∧ head = words[0]! then String.join tail = impl (String.join (words.drop 1))
    else result = impl (String.join (words.drop 1))
-- program termination
∃ result, impl sentence = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(sentence: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ sentence, problem_spec impl sentence) ↔
(∀ sentence, generated_spec impl sentence) :=",sorry,def implementation (sentence : String) : String :=,sorry,"-- #test implementation ""This is a test"" = ""is""
-- #test implementation ""lets go for swimming"" = ""go for""","theorem correctness
(sentence : String)
: problem_spec implementation sentence :=",sorry,,,
"def simplify(x: str, n: str) -> Bool","Your task is to implement a function that will simplify the expression
x * n. The function returns True if x * n evaluates to a whole number and False
otherwise. Both x and n, are string representation of a fraction, and have the following format,
<numerator>/<denominator> where both numerator and denominator are positive whole numbers.

You can assume that x, and n are valid fractions, and do not have zero as denominator.
","[{""input"": [""1/5"", ""5/1""], ""expected_output"": true}, {""input"": [""1/6"", ""2/1""], ""expected_output"": false}, {""input"": [""7/10"", ""10/2""], ""expected_output"": false}]","def simplify(x: str, n: str) -> Bool
""""""Your task is to implement a function that will simplify the expression
x * n. The function returns True if x * n evaluates to a whole number and False
otherwise. Both x and n, are string representation of a fraction, and have the following format,
<numerator>/<denominator> where both numerator and denominator are positive whole numbers.

You can assume that x, and n are valid fractions, and do not have zero as denominator.
""""""","def problem_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(x: String) (n: String) :=
-- spec
let spec (result: Bool) :=
let fx := x.splitOn ""/"";
let fn := n.splitOn ""/"";
fx.length = 2 → fn.length = 2 →
fx.all String.isNat → fn.all String.isNat →
let p1 := fx[0]!.toNat!;
let q1 := fx[1]!.toNat!;
let p2 := fn[0]!.toNat!;
let q2 := fn[1]!.toNat!;
q1 ≠ 0 → q2 ≠ 0 →
(result ↔ (∃ k, k * p1 * p2 = q1 * q2))
-- program termination
∃ result, impl x n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(x: String) (n: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ x n, problem_spec impl x n) ↔
(∀ x n, generated_spec impl x n) :=",sorry,def implementation (x: String) (n: String) : Bool :=,sorry,"-- #test implementation ""1/5"" ""5/1"" = True
-- #test implementation ""1/6"" ""2/1"" = False
-- #test implementation ""7/10"" ""10/2"" = False","theorem correctness
(x: String) (n: String)
: problem_spec implementation x n :=",sorry,,,
def order_by_points(nums: List[int]) -> List[int],"Write a function which sorts the given list of integers
in ascending order according to the sum of their digits.
Note: if there are several items with similar sum of their digits,
order them based on their index in original list.
","[{""input"": [1, 11, -1, -11, -12], ""expected_output"": [-1, -11, 1, -12, 11]}, {""input"": [], ""expected_output"": []}]","def order_by_points(nums: List[int]) -> List[int]
""""""Write a function which sorts the given list of integers
in ascending order according to the sum of their digits.
Note: if there are several items with similar sum of their digits,
order them based on their index in original list.
""""""","def problem_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(nums: List Int) :=
-- spec
let spec (result: List Int) :=
List.Perm nums result ∧
match result with
| [] => nums = []
| head::tail =>
  let head_sum := digit_sum head;
  (∀ num ∈ nums,
    let sum := digit_sum num;
    sum > head_sum ∨
   (sum = head_sum ∧ nums.indexOf num ≥ nums.indexOf head))
  ∧ impl (nums.erase head) = tail
-- program termination
∃ result, impl nums = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → List Int)
-- inputs
(nums: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ nums, problem_spec impl nums) ↔
(∀ nums, generated_spec impl nums) :=",sorry,def implementation (nums: List Int) : List Int :=,sorry,"-- #test implementation [1, 11, -1, -11, -12] = [-1, -11, 1, -12, 11]
-- #test implementation [] = []","theorem correctness
(nums: List Int)
: problem_spec implementation nums :=",sorry,"/--
name: digit_sum
use: |
  Helper to sum the digits of a number. If the number is negative, the
  negative sign is treated as part of the first digit.
problems:
  - 145
-/
def digit_sum (n : Int) : Int :=
  let ds := (toString n.natAbs).toList.map fun c => c.toNat - Char.toNat '0'
  match ds with
  | [] => 0
  | d :: ds' =>
    let tail := ds'.foldl (· + ·) 0
    if n < 0 then Int.ofNat tail - Int.ofNat d
    else Int.ofNat (d + tail)",,
def specialFilter(nums: List[int]) -> int,"Write a function that takes an array of numbers as input and returns
the number of elements in the array that are greater than 10 and both
first and last digits of a number are odd (1, 3, 5, 7, 9).
","[{""input"": [15, -73, 14, -15], ""expected_output"": 1}, {""input"": [33, -2, -3, 45, 21, 109], ""expected_output"": 2}]","def specialFilter(nums: List[int]) -> int
""""""Write a function that takes an array of numbers as input and returns
the number of elements in the array that are greater than 10 and both
first and last digits of a number are odd (1, 3, 5, 7, 9).
""""""","def problem_spec
-- function signature
(impl: List Int → Int)
-- inputs
(nums: List Int) :=
-- spec
let spec (result: Int) :=
match nums with
| [] => result = 0
| head::tail =>
  let first_digit := (toString head.natAbs).toList[0]!.toNat - Char.toNat '0';
  let last_digit := head % 10;
  let valid := head > 10 ∧ Odd first_digit ∧ Odd last_digit
  if valid then result = 1 + impl tail else result = impl tail
-- program termination
∃ result, impl nums = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Int → Int)
-- inputs
(nums: List Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ nums, problem_spec impl nums) ↔
(∀ nums, generated_spec impl nums) :=",sorry,def implementation (nums: List Int) : Int :=,sorry,"-- #test implementation [15, -73, 14, -15] = 1
-- #test implementation [33, -2, -3, 45, 21, 109] = 2","theorem correctness
(nums: List Int)
: problem_spec implementation nums :=",sorry,,,
def get_max_triples(n: int) -> int,"You are given a positive integer n. You have to create an integer array a of length n.
For each i (1 ≤ i ≤ n), the value of a[i] = i * i - i + 1.
Return the number of triples (a[i], a[j], a[k]) of a where i < j < k,
and a[i] + a[j] + a[k] is a multiple of 3.
","[{""input"": 5, ""expected_output"": 1}]","def get_max_triples(n: int) -> int
""""""You are given a positive integer n. You have to create an integer array a of length n.
For each i (1 ≤ i ≤ n), the value of a[i] = i * i - i + 1.
Return the number of triples (a[i], a[j], a[k]) of a where i < j < k,
and a[i] + a[j] + a[k] is a multiple of 3.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
∃ (S : Finset (Nat × Nat × Nat)), S.card = result ∧
  ∀ (triple: Nat × Nat × Nat),
    let (i, j, k) := triple;
    let a_i := i * i - i + 1;
    let a_j := j * j - j + 1;
    let a_k := k * k - k + 1;
    (1 ≤ i ∧ i < j ∧ j < k ∧ k ≤ n ∧
    (a_i + a_j + a_k) % 3 = 0)
    ↔ triple ∈ S
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(nums: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=",sorry,def implementation (n: Nat) : Nat :=,sorry,-- #test implementation 5 = 1,"theorem correctness
(n: Nat)
: problem_spec implementation n :=",sorry,,,
"def bf(planet1: str, planet2: str) -> List[str]","There are eight planets in our solar system: the closest to the Sun
is Mercury, the next one is Venus, then Earth, Mars, Jupiter, Saturn,
Uranus, Neptune.
Write a function that takes two planet names as strings planet1 and planet2.
The function should return a tuple containing all planets whose orbits are
located between the orbit of planet1 and the orbit of planet2, sorted by
the proximity to the sun.
The function should return an empty tuple if planet1 or planet2
are not correct planet names.
","[{""input"": ""(\""Jupiter\"", \""Neptune\"")"", ""expected_output"": ""(\""Saturn\"", \""Uranus\"")""}, {""input"": ""(\""Earth\"", \""Mercury\"")"", ""expected_output"": ""(\""Venus\"")""}, {""input"": ""(\""Mercury\"", \""Uranus\"")"", ""expected_output"": ""(\""Venus\"", \""Earth\"", \""Mars\"", \""Jupiter\"", \""Saturn\"")""}]","def bf(planet1: str, planet2: str) -> List[str]
""""""There are eight planets in our solar system: the closest to the Sun
is Mercury, the next one is Venus, then Earth, Mars, Jupiter, Saturn,
Uranus, Neptune.
Write a function that takes two planet names as strings planet1 and planet2.
The function should return a tuple containing all planets whose orbits are
located between the orbit of planet1 and the orbit of planet2, sorted by
the proximity to the sun.
The function should return an empty tuple if planet1 or planet2
are not correct planet names.
""""""","def problem_spec
-- function signature
(impl: String → String → List String)
-- inputs
(planet1: String)
(planet2: String) :=
-- spec
let spec (result: List String) :=
let planets := [""Mercury"", ""Venus"", ""Earth"", ""Mars"", ""Jupiter"", ""Saturn"", ""Uranus"", ""Neptune""];
if planet1 ∉ planets ∨ planet2 ∉ planets then
  result = []
else
  let index1 := planets.indexOf planet1;
  let index2 := planets.indexOf planet2;
  let minIdx := if index1 < index2 then index1 else index2;
  let maxIdx := if index1 < index2 then index2 else index1;
  (∀ str ∈ result, str ∈ planets) ∧
  (∀ planet ∈ planets, planet ∈ result ↔
    planets.indexOf planet < maxIdx ∧
    minIdx < planets.indexOf planet) ∧
  result.Sorted (fun a b => planets.indexOf a < planets.indexOf b)
-- program termination
∃ result, impl planet1 planet2 = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String → List String)
-- inputs
(planet1: String)
(planet2: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ planet1 planet2, problem_spec impl planet1 planet2) ↔
(∀ planet1 planet2, generated_spec impl planet1 planet2) :=",sorry,def implementation (planet1: String) (planet2: String) : List String :=,sorry,"-- #test implementation ""Jupiter"" ""Neptune"" = [""Saturn"", ""Uranus""]
-- #test implementation ""Earth"" ""Mercury"" = [""Venus""]
-- #test implementation ""Mercury"" ""Uranus"" = [""Venus"", ""Earth"", ""Mars"", ""Jupiter"", ""Saturn""]","theorem correctness
(planet1: String)
(planet2: String)
: problem_spec implementation planet1 planet2 :=",sorry,,,
def sorted_list_sum(lst: List[str]) -> List[str],"Write a function that accepts a list of strings as a parameter,
deletes the strings that have odd lengths from it,
and returns the resulted list with a sorted order,
The list is always a list of strings and never an array of numbers,
and it may contain duplicates.
The order of the list should be ascending by length of each word, and you
should return the list sorted by that rule.
If two words have the same length, sort the list alphabetically.
The function should return a list of strings in sorted order.
You may assume that all words will have the same length.
","[{""input"": [""aa"", ""a"", ""aaa""], ""output"": [""aa""]}, {""input"": [""ab"", ""a"", ""aaa"", ""cd""], ""output"": [""ab"", ""cd""]}]","def sorted_list_sum(lst: List[str]) -> List[str]
""""""Write a function that accepts a list of strings as a parameter,
deletes the strings that have odd lengths from it,
and returns the resulted list with a sorted order,
The list is always a list of strings and never an array of numbers,
and it may contain duplicates.
The order of the list should be ascending by length of each word, and you
should return the list sorted by that rule.
If two words have the same length, sort the list alphabetically.
The function should return a list of strings in sorted order.
You may assume that all words will have the same length.
""""""","def problem_spec
-- function signature
(impl: List String → List String)
-- inputs
(lst: List String) :=
-- spec
let spec (result: List String) :=
match result with
| [] => ∀ str ∈ lst, Odd str.length
| head::tail =>
  Even head.length ∧
  (∀ str ∈ lst,
    Odd str.length ∨
    str.length > head.length ∨
    str.length = head.length ∧ str ≥ head)
  ∧ impl (lst.erase head) = tail
-- program termination
∃ result, impl lst = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List String → List String)
-- inputs
(lst: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ lst, problem_spec impl lst) ↔
(∀ lst, generated_spec impl lst) :=",sorry,def implementation (lst: List String) : List String :=,sorry,"-- #test implementation [""aa"", ""a"", ""aaa""] = [""aa""]
-- #test implementation [""ab"", ""a"", ""aaa"", ""cd""] = [""ab"", ""cd""]","theorem correctness
(lst: List String)
: problem_spec implementation lst :=",sorry,,,
"def x_or_y(int n, int x, int y) -> int","A simple program which should return the value of x if n is
a prime number and should return the value of y otherwise.
","[{""input"": [7, 34, 12], ""expected_output"": 34}, {""input"": [15, 8, 5], ""expected_output"": 5}]","def x_or_y(int n, int x, int y) -> int
""""""A simple program which should return the value of x if n is
a prime number and should return the value of y otherwise.
""""""","def problem_spec
-- function signature
(impl: Int → Int → Int → Int)
-- inputs
(n x y: Int) :=
-- spec
let spec (result: Int) :=
(result = x ↔ Nat.Prime n.toNat) ∧
(result = y ↔ (¬ Nat.Prime n.toNat ∨ n ≤ 1))
-- program terminates
∃ result, impl n x y = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Int → Int → Int → Int)
-- inputs
(n x y: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ n x y, problem_spec impl n x y) ↔
(∀ n x y, generated_spec impl n x y) :=",sorry,def implementation (n x y: Int) : Int :=,sorry,"-- #test implementation 7 34 12 = 34
-- #test implementation 15 8 5 = 5","theorem correctness
(n x y: Int)
: problem_spec implementation n x y :=",sorry,,,
def double_the_difference(numbers: List[float]) -> Int,"Given a list of numbers, return the sum of squares of the numbers
in the list that are odd. Ignore numbers that are negative or not integers.
","[{""input"": [1, 3, 2, 0], ""expected_output"": 10}, {""input"": [""-1. -2"", 0], ""expected_output"": 0}, {""input"": [9, -2], ""expected_output"": 81}, {""input"": [0], ""expected_output"": 0}]","def double_the_difference(numbers: List[float]) -> Int
""""""Given a list of numbers, return the sum of squares of the numbers
in the list that are odd. Ignore numbers that are negative or not integers.
""""""","def problem_spec
-- function signature
(impl: List Rat → Int)
-- inputs
(numbers: List Rat) :=
let isEven (n : Rat) := n % 2 = 0;
let isNegative (n : Rat) := n < 0;
let isNotInteger (n : Rat) := ¬ n.isInt;
-- spec
let spec (result: Int) :=
0 < numbers.length →
0 ≤ result ∧
if numbers.length = 1
then result = if (isEven numbers[0]! ∨ isNegative numbers[0]! ∨ isNotInteger numbers[0]!) then (0 : Int) else numbers[0]!.floor ^ 2
else result = if (isEven numbers[0]! ∨ isNegative numbers[0]! ∨ isNotInteger numbers[0]!) then (0 : Int) else numbers[0]!.floor ^ 2 + impl numbers.tail
-- program terminates
∃ result, impl numbers = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Rat → Int)
-- inputs
(numbers: List Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ numbers, problem_spec impl numbers) ↔
(∀ numbers, generated_spec impl numbers) :=",sorry,def implementation (numbers: List Rat) : Int :=,sorry,"-- #test implementation ([1, 3, 2, 0]: List Rat) = (10: Int)
-- #test implementation ([-1, -2, 0]: List Int) = (0: Int)
-- #test implementation ([9, -2]: List Int) = 81
-- #test implementation ([0]: List Int) = 0","theorem correctness
(numbers: List Rat)
: problem_spec implementation numbers :=",sorry,,,
"def compare(scores: List float, guesses: List float) -> List [float]","I think we all remember that feeling when the result of some long-awaited
event is finally known. The feelings and thoughts you have at that moment are
definitely worth noting down and comparing.
Your task is to determine if a person correctly guessed the results of a number of matches.
You are given two arrays of scores and guesses of equal length, where each index shows a match.
Return an array of the same length denoting how far off each guess was. If they have guessed correctly,
the value is 0, and if not, the value is the absolute difference between the guess and the score.

Note: to reviewer, the reason for not using |.| to get the absolute value is to avoid leaking the implementation.
","[{""input"": [[1, 2, 3, 4, 5, 1], [1, 2, 3, 4, 2, -2]], ""expected_output"": [0, 0, 0, 0, 3, 3]}, {""input"": [[0, 5, 0, 0, 0, 4], [4, 1, 1, 0, 0, -2]], ""expected_output"": [4, 4, 1, 0, 0, 6]}]","def compare(scores: List float, guesses: List float) -> List [float]
""""""I think we all remember that feeling when the result of some long-awaited
event is finally known. The feelings and thoughts you have at that moment are
definitely worth noting down and comparing.
Your task is to determine if a person correctly guessed the results of a number of matches.
You are given two arrays of scores and guesses of equal length, where each index shows a match.
Return an array of the same length denoting how far off each guess was. If they have guessed correctly,
the value is 0, and if not, the value is the absolute difference between the guess and the score.

Note: to reviewer, the reason for not using |.| to get the absolute value is to avoid leaking the implementation.
""""""","def problem_spec
-- function signature
(impl: List Rat → List Rat → List Rat)
-- inputs
(scores guesses: List Rat) :=
-- spec
let spec (result: List Rat) :=
  result.length = scores.length ∧
  scores.length = guesses.length ∧
  ∀ i, i < scores.length →
  if scores[i]! > guesses[i]! then result[i]! + guesses[i]! = scores[i]!
  else result[i]! + scores[i]! = guesses[i]!
-- program terminates
∃ result, impl scores guesses = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List Rat → List Rat → List Rat)
-- inputs
(scores guesses: List Rat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ scores guesses, problem_spec impl scores guesses) ↔
(∀ scores guesses, generated_spec impl scores guesses) :=",sorry,def implementation (scores guesses: List Rat) : List Rat :=,sorry,"-- #test implementation [1,2,3,4,5,1] [1,2,3,4,2,-2] = [0,0,0,0,3,3]
-- #test implementation [0,5,0,0,0,4] [4,1,1,0,0,-2] = [4,4,1,0,0,6]","theorem correctness
(scores guesses: List Rat)
: problem_spec implementation scores guesses :=",sorry,,,
"def Strongest_Extension(class_name: String, extensions: List[String]) -> String","You will be given the name of a class (a string) and a list of extensions.
The extensions are to be used to load additional classes to the class. The
strength of the extension is as follows: Let CAP be the number of the uppercase
letters in the extension's name, and let SM be the number of lowercase letters
in the extension's name, the strength is given by the fraction CAP - SM.
You should find the strongest extension and return a string in this
format: ClassName.StrongestExtensionName.
If there are two or more extensions with the same strength, you should
choose the one that comes first in the list.
For example, if you are given ""Slices"" as the class and a list of the
extensions: ['SErviNGSliCes', 'Cheese', 'StuFfed'] then you should
return 'Slices.SErviNGSliCes' since 'SErviNGSliCes' is the strongest extension
(its strength is -1).
","[{""input"": [""my_class"", [""AA"", ""Be"", ""CC""]], ""expected_output"": ""my_class.AA""}]","def Strongest_Extension(class_name: String, extensions: List[String]) -> String
""""""You will be given the name of a class (a string) and a list of extensions.
The extensions are to be used to load additional classes to the class. The
strength of the extension is as follows: Let CAP be the number of the uppercase
letters in the extension's name, and let SM be the number of lowercase letters
in the extension's name, the strength is given by the fraction CAP - SM.
You should find the strongest extension and return a string in this
format: ClassName.StrongestExtensionName.
If there are two or more extensions with the same strength, you should
choose the one that comes first in the list.
For example, if you are given ""Slices"" as the class and a list of the
extensions: ['SErviNGSliCes', 'Cheese', 'StuFfed'] then you should
return 'Slices.SErviNGSliCes' since 'SErviNGSliCes' is the strongest extension
(its strength is -1).
""""""","def problem_spec
-- function signature
(impl: String → List String → String)
-- inputs
(class_name: String)
(extensions: List String) :=
let strength (extension: String) :=
  let cap := (extension.toList.filter (fun c => c.isUpper)).length;
  let sm := (extension.toList.filter (fun c => c.isLower)).length;
  cap - sm;
-- spec
let spec (result: String) :=
let last_pos := result.revPosOf '.';
0 < extensions.length ∧ extensions.all (fun x => 0 < x.length) ∧ 0 < class_name.length →
0 < result.length ∧
last_pos.isSome ∧
let class_name' := result.take (last_pos.get!).byteIdx;
let extension_name := result.drop ((last_pos.get!).byteIdx + 1);
class_name' = class_name ∧
extension_name ∈ extensions ∧
let strength_of_extensions := extensions.map (fun ext => strength ext);
∃ j : Nat, j < extensions.length ∧
  extensions[j]! = extension_name ∧
  (∀ i : Nat, i < j → strength_of_extensions[i]! < strength_of_extensions[j]!)
  ∧ strength_of_extensions[j]! = strength_of_extensions.max?.get!;
-- program terminates
∃ result, impl class_name extensions = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → List String → String)
-- inputs
(class_name: String)
(extensions: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ class_name extensions, problem_spec impl class_name extensions) ↔
(∀ class_name extensions, generated_spec impl class_name extensions) :=",sorry,def implementation (class_name: String) (extensions: List String) : String :=,sorry,"-- #test implementation 'my_class', ['AA', 'Be', 'CC'] = 'my_class.AA'","theorem correctness
(class_name: String)
(extensions: List String)
: problem_spec implementation class_name extensions :=",sorry,,,
"def cycpattern_check(String a, String b) -> Bool","You are given 2 words. You need to return True if the second word or any of its rotations is a substring in the first word, else False
","[{""input"": [""abcd"", ""abd""], ""expected_output"": false}, {""input"": [""hello"", ""ell""], ""expected_output"": true}, {""input"": [""whassup"", ""psus""], ""expected_output"": false}, {""input"": [""abab"", ""baa""], ""expected_output"": true}, {""input"": [""efef"", ""eeff""], ""expected_output"": false}, {""input"": [""himenss"", ""simen""], ""expected_output"": true}]","def cycpattern_check(String a, String b) -> Bool
""""""You are given 2 words. You need to return True if the second word or any of its rotations is a substring in the first word, else False
""""""","def problem_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(a b: String) :=
-- spec
let spec (result: Bool) :=
(b.length = 0 → result) ∧
(0 < b.length →
result ↔ ((b.length ≤ a.length) ∧
  (∃ i : Nat, i < b.length ∧
  let b_rotation := b.drop i ++ b.take i;
  a.containsSubstr b_rotation)));
-- program terminates
∃ result, impl a b = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String → Bool)
-- inputs
(a b: String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b, problem_spec impl a b) ↔
(∀ a b, generated_spec impl a b) :=",sorry,def implementation (a b: String) : Bool :=,sorry,"-- #test implementation ""abcd"" ""abd"" = False
-- #test implementation ""hello"" ""ell"" = True
-- #test implementation ""whassup"" ""psus"" = False
-- #test implementation ""abab"" ""baa"" = True
-- #test implementation ""efef"" ""eeff"" = False
-- #test implementation ""himenss"" ""simen"" = True","theorem correctness
(a b: String)
: problem_spec implementation a b :=",sorry,,,
"def even_odd_count(num: int) -> Tuple[int, int]","Given an integer. return a tuple that has the number of even and odd digits respectively.
","[{""input"": -12, ""expected_output"": [1, 1]}, {""input"": 123, ""expected_output"": [1, 2]}]","def even_odd_count(num: int) -> Tuple[int, int]
""""""Given an integer. return a tuple that has the number of even and odd digits respectively.
""""""","def problem_spec
-- function signature
(impl: Int → Int × Int)
-- inputs
(num: Int) :=
-- spec
let spec (result: Int × Int) :=
  let (even_count, odd_count) := result;
  let numAbs := |num|.toNat;
  let numBy10 := numAbs/10;
  let (even_count', odd_count') := impl numBy10;
  (result = impl numAbs) ∧
  (0 ≤ num → (Even num ↔ 1 + even_count' = even_count) ∧ (Odd num ↔ even_count' = even_count)) ∧
  (0 ≤ num → (Odd num ↔ 1 + odd_count' = odd_count) ∧ (Even num ↔ odd_count' = odd_count));
-- program terminates
∃ result, impl num = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Int → Int × Int)
-- inputs
(num: Int) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ num, problem_spec impl num) ↔
(∀ num, generated_spec impl num) :=",sorry,def implementation (num: Int) : Int × Int :=,sorry,"-- #test implementation -12 = (1, 1)
-- #test implementation 123 = (1, 2)","theorem correctness
(num: Int)
: problem_spec implementation num :=",sorry,,,
def int_to_mini_roman(num: Nat) -> String,"Given a positive integer, obtain its roman numeral equivalent as a string,
and return it in lowercase.
Restrictions: 1 <= num <= 1000
","[{""input"": 19, ""expected_output"": ""xix""}, {""input"": 152, ""expected_output"": ""clii""}, {""input"": 426, ""expected_output"": ""cdxxvi""}]","def int_to_mini_roman(num: Nat) -> String
""""""Given a positive integer, obtain its roman numeral equivalent as a string,
and return it in lowercase.
Restrictions: 1 <= num <= 1000
""""""","def problem_spec
-- function signature
(impl: Nat → String)
-- inputs
(num: Nat) :=
-- spec
let spec (result: String) :=
1 ≤ num ∧ num ≤ 1000 ∧ (result.data.all (fun c => c.isLower)) →
isValidRoman result ∧ romanToDecimal result = num
-- program terminates
∃ result, impl num = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → String)
-- inputs
(num: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ num, problem_spec impl num) ↔
(∀ num, generated_spec impl num) :=",sorry,def implementation (num: Nat) : String :=,sorry,"-- #test implementation 19 = ""xix""
-- #test implementation 152 = ""clii""
-- #test implementation 426 = ""cdxxvi""","theorem correctness
(num: Nat)
: problem_spec implementation num :=",sorry,"/--
name: romanCharToValue
use: |
  Map from valid characters and their values
problems:
  - 156
sample_problems: []
-/
def romanCharToValue : Char → Nat
| 'i' => 1
| 'v' => 5
| 'x' => 10
| 'l' => 50
| 'c' => 100
| 'd' => 500
| 'm' => 1000
| _   => 0

/--
name: validSubtractivePairs
use: |
  Legal subtractive pairs: first char can precede second for subtraction (in roman numerals)
problems:
  - 156
sample_problems: []
-/
def validSubtractivePairs : List (Char × Char) :=
  [('i', 'v'), ('i', 'x'), ('x', 'l'), ('x', 'c'), ('c', 'd'), ('c', 'm')]

/--
name: maxRepetitions
use: |
  Max allowed repetitions for each character (in roman numerals)
problems:
  - 156
sample_problems: []
-/
def maxRepetitions : Char → Nat
| 'i' | 'x' | 'c' | 'm' => 3
| 'v' | 'l' | 'd'       => 1
| _                    => 0

/--
name: countRepetitions
use: |
  Helper to count consecutive repetitions (in roman numerals)
problems:
  - 156
sample_problems: []
-/
def countRepetitions : List Char → Char → Nat → Nat
| [], _, n => n
| (h :: t), c, n => if h = c then countRepetitions t c (n + 1) else n

/--
name: validRepetition
use: |
  Helper to validate proper use of repetitions (in roman numerals)
problems:
  - 156
sample_problems: []
-/
partial def validRepetition : List Char → Bool
| [] => true
| c :: rest =>
  let max := maxRepetitions c
  let count := countRepetitions rest c 1
  count ≤ max ∧ validRepetition (rest.drop (count - 1))

/--
name: validSubtractiveOrder
use: |
  Helper to validate legal subtractive combinations (in roman numerals)
problems:
  - 156
sample_problems: []
-/
def validSubtractiveOrder : List Char → Bool
| [] | [_] => true
| c1 :: c2 :: rest =>
  match romanCharToValue c1, romanCharToValue c2 with
  | v1, v2 =>
    if v1 < v2 then
      -- check if c1 and c2 form a legal subtractive pair
      (c1, c2) ∈ validSubtractivePairs ∧ validSubtractiveOrder rest
    else if v1 = 0 ∨ v2 = 0 then
      false
    else
      validSubtractiveOrder (c2 :: rest)

/--
name: isValidRoman
use: |
  Function to check if a string is a roman numeral
problems:
  - 156
sample_problems: []
-/
def isValidRoman (s : String) : Bool :=
  s.data.all (λ c => romanCharToValue c ≠ 0) ∧
  validRepetition s.data ∧
  validSubtractiveOrder s.data

/--
name: romanToDecimalAux
use: |
  Helper to convert list of roman characters to decimal
problems:
  - 156
sample_problems: []
-/
def romanToDecimalAux : List Char → Nat
| [] => 0
| c1 :: c2 :: rest =>
    let val1 := romanCharToValue c1
    let val2 := romanCharToValue c2
    if val1 < val2 then
      -- subtractive notation
      (val2 - val1) + romanToDecimalAux rest
    else
      val1 + romanToDecimalAux (c2 :: rest)
| [c] => romanCharToValue c

/--
name: romanToDecimalAux
use: |
  Function to convert a valid lowercase Roman numeral string to Nat
problems:
  - 156
sample_problems: []
-/
def romanToDecimal (s : String) : Nat :=
  romanToDecimalAux s.data",,
"def right_angle_triangle(a: Nat, b: Nat, c: Nat) -> Bool","Given the lengths of the three sides of a triangle. Return True if the three
sides form a right-angled triangle, False otherwise.
A right-angled triangle is a triangle in which one angle is right angle or
90 degree.
","[{""input"": [3, 4, 5], ""expected_output"": true}, {""input"": [1, 2, 3], ""expected_output"": false}]","def right_angle_triangle(a: Nat, b: Nat, c: Nat) -> Bool
""""""Given the lengths of the three sides of a triangle. Return True if the three
sides form a right-angled triangle, False otherwise.
A right-angled triangle is a triangle in which one angle is right angle or
90 degree.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat → Nat → Bool)
-- inputs
(a b c: Nat) :=
-- spec
let spec (result: Bool) :=
result ↔
  0 < a ∧ 0 < b ∧ 0 < c ∧
  ((a * a + b * b = c * c) ∨
  (a * a + c * c = b * b) ∨
  (b * b + c * c = a * a))
-- program terminates
∃ result, impl a b c = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → Nat → Bool)
-- inputs
(a b c: Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b c, problem_spec impl a b c) ↔
(∀ a b c, generated_spec impl a b c) :=",sorry,def implementation (a b c: Nat) : Bool :=,sorry,"-- #test implementation ([1, 2, 2, -4]: List Int) = (-9: Int)
-- #test implementation ([0, 1]: List Int) = (0: Int)
-- #test implementation ([]: List Int) = none","theorem correctness
(a b c: Nat)
: problem_spec implementation a b c :=",sorry,,,
def find_max(words: List String) -> String,"Write a function that accepts a list of strings.
The list contains different words. Return the word with maximum number
of unique characters. If multiple strings have maximum number of unique
characters, return the one which comes first in lexicographical order.
","[{""input"": [""name"", ""of"", ""string""], ""expected_output"": ""string""}, {""input"": [""name"", ""enam"", ""game""], ""expected_output"": ""enam""}, {""input"": [""aaaaaaa"", ""bb"", ""cc""], ""expected_output"": ""aaaaaaa""}]","def find_max(words: List String) -> String
""""""Write a function that accepts a list of strings.
The list contains different words. Return the word with maximum number
of unique characters. If multiple strings have maximum number of unique
characters, return the one which comes first in lexicographical order.
""""""","def problem_spec
-- function signature
(impl: List String → String)
-- inputs
(words: List String) :=
let unique_chars (string: String) :=
  let string_idx := {i: Nat | i < string.length}.toFinset;
  let characters := string_idx.image (fun i => string.toList.get! i);
  characters.card;
-- spec
let spec (result: String) :=
(result = """" ↔ words.length = 0) ∧
(words.length != 0 → result ∈ words ∧
let unique_chars_list := words.map unique_chars;
let max_unique_chars := unique_chars_list.max?.get!;
unique_chars result = max_unique_chars ∧
∀ i : Nat, i < words.length →
  unique_chars_list[i]! = max_unique_chars →
  result ≤ words[i]!);
-- program terminates
∃ result, impl words = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List String → String)
-- inputs
(words: List String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ words, problem_spec impl words) ↔
(∀ words, generated_spec impl words) :=",sorry,def implementation (words: List String) : String :=,sorry,"-- #test implementation [""name"", ""of"", ""string""]= ""string""
-- #test implementation [""name"", ""enam"", ""game""] = ""enam""
-- #test implementation [""aaaaaaa"", ""bb"" ,""cc""] = ""aaaaaaa""","theorem correctness
(words: List String)
: problem_spec implementation words :=",sorry,,,
"def eat(number: Nat, need: Nat, remaining: Nat) -> List Nat","You're a hungry rabbit, and you already have eaten a certain number of carrots,
but now you need to eat more carrots to complete the day's meals.
you should return an array of [ total number of eaten carrots after your meals,
                                the number of carrots left after your meals ]
if there are not enough remaining carrots, you will eat all remaining carrots, but will still be hungry.

Variables:
  @number : integer
      the number of carrots that you have eaten.
  @need : integer
      the number of carrots that you need to eat.
  @remaining : integer
      the number of remaining carrots thet exist in stock

Constrain:
* 0 <= number <= 1000
* 0 <= need <= 1000
* 0 <= remaining <= 1000
","[{""input"": [5, 6, 10], ""expected_output"": [11, 4]}, {""input"": [4, 8, 9], ""expected_output"": [12, 1]}, {""input"": [1, 10, 10], ""expected_output"": [11, 0]}, {""input"": [2, 11, 5], ""expected_output"": [7, 0]}]","def eat(number: Nat, need: Nat, remaining: Nat) -> List Nat
""""""You're a hungry rabbit, and you already have eaten a certain number of carrots,
but now you need to eat more carrots to complete the day's meals.
you should return an array of [ total number of eaten carrots after your meals,
                                the number of carrots left after your meals ]
if there are not enough remaining carrots, you will eat all remaining carrots, but will still be hungry.

Variables:
  @number : integer
      the number of carrots that you have eaten.
  @need : integer
      the number of carrots that you need to eat.
  @remaining : integer
      the number of remaining carrots thet exist in stock

Constrain:
* 0 <= number <= 1000
* 0 <= need <= 1000
* 0 <= remaining <= 1000
""""""","def problem_spec
-- function signature
(impl: Nat → Nat → Nat → List Nat)
-- inputs
(number need remaining: Nat) :=
-- spec
let spec (result: List Nat) :=
number ≤ 1000 → need ≤ 1000 → remaining ≤ 1000 →
result.length = 2 ∧
(need ≤ remaining → result[0]! - need = number ∧
need = remaining - result[1]!) ∧
(remaining < need → result[0]! - remaining = number ∧
result[1]! = 0);
-- program terminates
∃ result, impl number need remaining = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → Nat → List Nat)
-- inputs
(a b c : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b c, problem_spec impl a b c) ↔
(∀ a b c, generated_spec impl a b c) :=",sorry,def implementation (a b c: Nat) : List Nat :=,sorry,"-- #test implementation 5 6 10 = [11, 4]
-- #test implementation 4 8 9 = [12, 1]
-- #test implementation 1 10 10 = [11, 0]
-- #test implementation 2 11 5 = [7, 0]","theorem correctness
(a b c: Nat)
: problem_spec implementation a b c :=",sorry,,,
"def do_algebra(operator: List String, operand: List Nat) -> Int","Given two lists operator, and operand. The first list has basic algebra operations, and
the second list is a list of integers. Use the two given lists to build the algebric
expression and return the evaluation of this expression.

The basic algebra operations:
Addition ( + )
Subtraction ( - )
Multiplication ( * )
Floor division ( // )
Exponentiation ( ** )

Note:
    The length of operator list is equal to the length of operand list minus one.
    Operand is a list of of non-negative integers.
    Operator list has at least one operator, and operand list has at least two operands.
","[{""input"": [[""+"", ""*"", ""-""], [2, 3, 4, 5]], ""expected_output"": 9}]","def do_algebra(operator: List String, operand: List Nat) -> Int
""""""Given two lists operator, and operand. The first list has basic algebra operations, and
the second list is a list of integers. Use the two given lists to build the algebric
expression and return the evaluation of this expression.

The basic algebra operations:
Addition ( + )
Subtraction ( - )
Multiplication ( * )
Floor division ( // )
Exponentiation ( ** )

Note:
    The length of operator list is equal to the length of operand list minus one.
    Operand is a list of of non-negative integers.
    Operator list has at least one operator, and operand list has at least two operands.
""""""","def problem_spec
-- function signature
(impl: List String → List Nat → Int)
-- inputs
(operator : List String)
(operand : List Nat) :=
-- spec
let spec (result: Int) :=
operator.length = operand.length - 1 ∧ 0 < operator.length ∧ operand.all (fun n => 0 ≤ n) →
let inline_tokens : List String := mergeAlternately operand operator
evalArith_precedence inline_tokens result
-- program terminates
∃ result, impl operator operand = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: List String → List Nat → Int)
-- inputs
(operator : List String)
(operand : List Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ operator operand, problem_spec impl operator operand) ↔
(∀ operator operand, generated_spec impl operator operand) :=",sorry,def implementation (operator: List String) (operand : List Nat) : Int :=,sorry,"-- #test implementation ['+', '*', '-'] [2,3,4,5] = 9","theorem correctness
(operator : List String) (operand : List Nat)
: problem_spec implementation operator operand :=",sorry,"/--
name: mergeAlternately
use: |
  Helper Methods to mergeAlternately a list of strings
problems:
  - 160
sample_problems: []
-/
def mergeAlternately : List Nat → List String → List String
| [], []       => []
| [], y :: ys  => y :: mergeAlternately [] ys
| x :: xs, []  => x.repr :: mergeAlternately xs []
| x :: xs, y :: ys => x.repr :: y :: mergeAlternately xs ys

/--
name: applyOp
use: |
  Helper method to apply operations on two integers
problems:
  - 160
sample_problems: []
-/
def applyOp (x y : Int) : String → Option Int
  | ""+""  => some (x + y)
  | ""-""  => some (x - y)
  | ""*""  => some (x * y)
  | ""//"" => if y == 0 then none else some (x / y)
  | ""**"" =>
    if x < 0 then none
    else some (Int.ofNat ((Int.toNat x) ^ (Int.toNat y)))
  | _     => none

/--
name: evalArith_pass
use: |
  Noncomputable relational spec for a single step evaluation (any op, ignoring precedence).
problems:
  - 160
sample_problems: []
-/
inductive evalArith_pass : List String → Int → Prop
| num {s : String} {n : Nat} (h : s.toNat! = n) :
    evalArith_pass [s] (Int.ofNat n)
| binOp {ts1 ts2 : List String} {op : String} {r1 r2 r : Int}
    (h1 : evalArith_pass ts1 r1)
    (h2 : evalArith_pass ts2 r2)
    (hop : applyOp r1 r2 op = some r) :
    evalArith_pass (ts1 ++ op :: ts2) r

/--
name: evalArith_exp
use: |
  Relational spec for exponentiation (highest precedence).
problems:
  - 160
sample_problems: []
-/
inductive evalArith_exp : List String → Int → Prop
| of_pass {ts r} (h : evalArith_pass ts r) : evalArith_exp ts r
| step {ts1 ts2 r1 r2 r} (h1 : evalArith_exp ts1 r1) (h2 : evalArith_exp ts2 r2)
    (hop : applyOp r1 r2 ""**"" = some r) :
    evalArith_exp (ts1 ++ ""**"" :: ts2) r

/--
name: evalArith_mul
use: |
  Relational spec for multiplication/division (middle precedence).
problems:
  - 160
sample_problems: []
-/
inductive evalArith_mul : List String → Int → Prop
| of_exp {ts r} (h : evalArith_exp ts r) : evalArith_mul ts r
| step {ts1 ts2 r1 r2 r} (h1 : evalArith_mul ts1 r1) (h2 : evalArith_mul ts2 r2)
    (hop : applyOp r1 r2 ""*"" = some r ∨ applyOp r1 r2 ""//"" = some r) :
    evalArith_mul (ts1 ++ ""*"" :: ts2) r

/--
name: evalArith_mul
use: |
  Relational spec for addition/subtraction (lowest precedence).
problems:
  - 160
sample_problems: []
-/
inductive evalArith_add : List String → Int → Prop
| of_mul {ts r} (h : evalArith_mul ts r) : evalArith_add ts r
| step {ts1 ts2 r1 r2 r} (h1 : evalArith_add ts1 r1) (h2 : evalArith_add ts2 r2)
    (hop : applyOp r1 r2 ""+"" = some r ∨ applyOp r1 r2 ""-"" = some r) :
    evalArith_add (ts1 ++ ""+"" :: ts2) r

/--
name: evalArith_mul
use: |
  Main function to evaluate an expression
problems:
  - 160
sample_problems: []
-/
def evalArith_precedence (ts : List String) (r : Int) : Prop :=
  evalArith_add ts r",,
def solve(string : String) -> String,"You are given a string s.
if s[i] is a letter, reverse its case from lower to upper or vise versa,
otherwise keep it as it is.
If the string contains no letters, reverse the string.
The function should return the resulted string.
","[{""input"": ""1234"", ""expected_output"": ""4321""}, {""input"": ""ab"", ""expected_output"": ""AB""}, {""input"": ""#a@C"", ""expected_output"": ""#A@c""}]","def solve(string : String) -> String
""""""You are given a string s.
if s[i] is a letter, reverse its case from lower to upper or vise versa,
otherwise keep it as it is.
If the string contains no letters, reverse the string.
The function should return the resulted string.
""""""","def problem_spec
-- function signature
(impl: String → String)
-- inputs
(string : String) :=
-- spec
let spec (result: String) :=
result.length = string.length ∧
let hasNoAlphabet := string.all (λ c => not (c.isAlpha));
(hasNoAlphabet →
  result.toList = string.toList.reverse) ∧
(hasNoAlphabet = false →
  ∀ i, i < string.length →
  let c := string.get! ⟨i⟩;
  (c.isAlpha → ((c.isLower → c.toUpper = result.get! ⟨i⟩) ∨
              (c.isUpper → c.toLower = result.get! ⟨i⟩))) ∧
  (¬ c.isAlpha → c = result.get! ⟨i⟩))
-- program terminates
∃ result, impl string = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: String → String)
-- inputs
(s : String) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ s, problem_spec impl s) ↔
(∀ s, generated_spec impl s) :=",sorry,def implementation (s: String) : String :=,sorry,"-- #test implementation ""1234"" = ""4321""
-- #test implementation ""ab"" = ""AB""
-- #test implementation ""#a@C"" = ""#A@c""","theorem correctness
(s: String)
: problem_spec implementation s :=",sorry,,,
"def generate_integers(a : Nat, b : Nat) -> List Nat","Given two positive integers a and b, return the even digits between a
and b, in ascending order.
","[{""input"": [2, 8], ""expected_output"": [2, 4, 6, 8]}, {""input"": [8, 2], ""expected_output"": [2, 4, 6, 8]}, {""input"": [10, 14], ""expected_output"": [10, 12, 14]}]","def generate_integers(a : Nat, b : Nat) -> List Nat
""""""Given two positive integers a and b, return the even digits between a
and b, in ascending order.
""""""","def problem_spec
-- function signature
(impl: Nat → Nat → List Nat)
-- inputs
(a b : Nat) :=
let isAscendingBy2 (r : List Nat) :=
∀ i, i < r.length - 1 → r[i+1]! - r[i]! = 2
-- spec
let spec (result: List Nat) :=
result.all (fun n => n % 2 = 0) ∧ isAscendingBy2 result ∧
1 < result.length ∧
let min_a_b := min a b;
let max_a_b := max a b;
if min_a_b = max_a_b ∧ (min_a_b % 2 = 1)
then result = []
else ((result[0]! = if 2 ∣ min_a_b then min_a_b else (min_a_b + 1)) ∧
(result[result.length-1]! = if 2 ∣ max_a_b then max_a_b else max_a_b - 1))
-- program terminates
∃ result, impl a b = result ∧
-- return value satisfies spec
spec result","def generated_spec
-- function signature
(impl: Nat → Nat → List Nat)
-- inputs
(a b : Nat) : Prop :=","theorem spec_isomorphism:
∀ impl,
(∀ a b, problem_spec impl a b) ↔
(∀ a b, generated_spec impl a b) :=",sorry,def implementation (a b: Nat) : List Nat :=,sorry,"-- #test implementation 2 8 = [2, 4, 6, 8]
-- #test implementation 8 2 = [2, 4, 6, 8]
-- #test implementation 10 14 = [10, 12, 14]","theorem correctness
(a b: Nat)
: problem_spec implementation a b :=",sorry,,,