import Imports.AllImports
import Std
-- start_def problem_details
/--
function_signature: "def exchange(lst1: list[int], lst2: list[int]) -> str"
docstring: |
    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.
test_cases:
  - input: ([1, 2, 3, 4], [1, 2, 3, 4])
    expected_output: "YES"
  - input: ([1, 2, 3, 4], [1, 5, 3, 4])
    expected_output: "NO"
-/
-- end_def problem_details

-- start_def problem_spec
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
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: List Int → List Int → String)
-- inputs
(lst1: List Int)
(lst2: List Int) : Prop :=
--end_def generated_spec
--start_def generated_spec_body
sorry
--end_def generated_spec_body

-- start_def spec_isomorphism
theorem spec_isomorphism:
∀ impl,
(∀ lst1 lst2, problem_spec impl lst1 lst2) ↔
(∀ lst1 lst2, generated_spec impl lst1 lst2) :=
-- end_def spec_isomorphism
-- start_def spec_isomorphism_proof
sorry
-- end_def spec_isomorphism_proof


-- start_def implementation_signature
def implementation (lst1: List Int) (lst2: List Int) : String :=
-- end_def implementation_signature
-- start_def implementation
sorry
-- end_def implementation

-- start_def test_cases
-- #test implementation ([1, 2, 3, 4], [1, 2, 3, 4]) = "YES"
-- #test implementation ([1, 2, 3, 4], [1, 5, 3, 4]) = "NO"
-- end_def test_cases


-- start_def correctness_definition
theorem correctness
(lst1: List Int)
(lst2: List Int)
: problem_spec implementation lst1 lst2
:=
-- end_def correctness_definition
-- start_def correctness_proof
by
sorry
-- end_def correctness_proof
