-----Description----- 
This task requires writing a Lean 4 method that verifies if every odd index in an array of integers holds an odd number. In other words, for each index in the array that is odd, the number located at that index must also be odd. The method should return true if this condition is satisfied for every odd index; otherwise, it should return false.

-----Input-----
The input consists of:
a: An array of integers.

-----Output-----
The output is a Boolean value:
Returns true if, for every odd index in the array, the corresponding element is odd.
Returns false if there is at least one odd index where the corresponding element is not odd.

-----Note-----
There are no preconditions; the method will work for any array of integers.
Difficulty: basic

Lean 4 solution:
-- !benchmark @start import type=solution

-- !benchmark @end import

-- !benchmark @start solution_aux
def isOdd (n : Int) : Bool :=
  n % 2 == 1
-- !benchmark @end solution_aux

-- !benchmark @start precond_aux

-- !benchmark @end precond_aux

@[reducible, simp]
def isOddAtIndexOdd_precond (a : Array Int) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond


-- !benchmark @start code_aux

-- !benchmark @end code_aux


def isOddAtIndexOdd (a : Array Int) (h_precond : isOddAtIndexOdd_precond (a)) : Bool :=
  -- !benchmark @start code
  -- First create pairs of (index, value) for all elements in the array
  let indexedArray := a.mapIdx fun i x => (i, x)

  -- Check if all elements at odd indices are odd numbers
  indexedArray.all fun (i, x) => !(isOdd i) || isOdd x
  -- !benchmark @end code


-- !benchmark @start postcond_aux

-- !benchmark @end postcond_aux


@[reducible, simp]
def isOddAtIndexOdd_postcond (a : Array Int) (result: Bool) (h_precond : isOddAtIndexOdd_precond (a)) :=
  -- !benchmark @start postcond
  result ↔ (∀ i, (hi : i < a.size) → isOdd i → isOdd (a[i]))
  -- !benchmark @end postcond


-- !benchmark @start proof_aux

-- !benchmark @end proof_aux


theorem isOddAtIndexOdd_spec_satisfied (a: Array Int) (h_precond : isOddAtIndexOdd_precond (a)) :
    isOddAtIndexOdd_postcond (a) (isOddAtIndexOdd (a) h_precond) h_precond := by
  -- !benchmark @start proof
  unfold isOddAtIndexOdd isOddAtIndexOdd_postcond
  simp_all
  constructor
  · intro h
    intro i hi h_odd
    -- Since the function returns true, all elements in satisfy the predicate
    have h_all_odd : (a.mapIdx (fun j x => (j, x))).all (fun (i, x) => !(isOdd i) || isOdd x) = true := by
      simp_all
    -- Apply the property of Array.all
    rw [Array.all_iff_forall] at h_all_odd
    simp_all
    have h_sat_i : !(isOdd i) || isOdd a[i] := by
      simp
      apply h_all_odd i hi
    simp [h_odd] at h_sat_i
    exact h_sat_i
  · intro h
    apply Array.all_iff_forall.mpr
    intro i hi
    simp
    intro hi'
    have h_sat : isOdd i → isOdd a[i] := by
      apply h i hi'
    rw [Decidable.imp_iff_not_or] at h_sat
    simp at h_sat
    exact h_sat
  -- !benchmark @end proof


Tests:
input: a: #[1, 2, 3, 4, 5]
a: #[1, 3, 5, 7, 9]
a: #[2, 4, 6, 8, 10]
a: #[]
a: #[7]
a: #[0, 1, 0, 1]
a: #[0, 2, 4, 6]
expected: False
True
False
True
True
True
False
unexpected: True
False
True
False
False
False
True

Inputs rejected by the precondition: