-----Description-----  
This task involves performing integer division with remainder on natural numbers. Given two natural numbers, x (the dividend) and y (the divisor), the objective is to determine the quotient and remainder. When y is non-zero, the quotient and remainder should satisfy the condition that the dividend equals the divisor multiplied by the quotient plus the remainder, with the remainder being nonnegative and strictly less than y. In the case where y is zero, the result should indicate that no division is performed by returning x as the quotient and 0 as the remainder.

-----Input-----  
The input consists of two natural numbers:  
• x: A natural number representing the dividend.  
• y: A natural number representing the divisor.

-----Output-----  
The output is a pair of integers (r, q) where:  
• If y ≠ 0, then q is the quotient and r is the remainder such that:  
   - q * Int.ofNat y + r = Int.ofNat x  
   - 0 ≤ r < Int.ofNat y  
   - 0 ≤ q  
• If y = 0, then the output is (Int.ofNat x, 0).

-----Note-----  
The specification regarding the division properties applies only when y is non-zero. When y = 0, the function safely returns (x, 0) in its integer form.
Difficulty: basic

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

-- !benchmark @start solution_aux

-- !benchmark @end solution_aux

-- !benchmark @start precond_aux

-- !benchmark @end precond_aux
@[reducible, simp]
def DivisionFunction_precond (x : Nat) (y : Nat) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond


-- !benchmark @start code_aux
def divMod (x y : Nat) : Int × Int :=
  let q : Int := Int.ofNat (x / y)
  let r : Int := Int.ofNat (x % y)
  (r, q)
-- !benchmark @end code_aux


def DivisionFunction (x : Nat) (y : Nat) (h_precond : DivisionFunction_precond (x) (y)) : Int × Int :=
  -- !benchmark @start code
  if y = 0 then (Int.ofNat x, 0) else divMod x y
  -- !benchmark @end code


-- !benchmark @start postcond_aux

-- !benchmark @end postcond_aux


@[reducible, simp]
def DivisionFunction_postcond (x : Nat) (y : Nat) (result: Int × Int) (h_precond : DivisionFunction_precond (x) (y)) :=
  -- !benchmark @start postcond
  let (r, q) := result;
  (y = 0 → r = Int.ofNat x ∧ q = 0) ∧
  (y ≠ 0 → (q * Int.ofNat y + r = Int.ofNat x) ∧ (0 ≤ r ∧ r < Int.ofNat y) ∧ (0 ≤ q))
  -- !benchmark @end postcond


-- !benchmark @start proof_aux

-- !benchmark @end proof_aux


theorem DivisionFunction_spec_satisfied (x: Nat) (y: Nat) (h_precond : DivisionFunction_precond (x) (y)) :
    DivisionFunction_postcond (x) (y) (DivisionFunction (x) (y) h_precond) h_precond := by
  -- !benchmark @start proof
  unfold DivisionFunction_postcond DivisionFunction
  simp
  split
  {
    simp
    intro h₁
    contradiction
  }
  {

    have h_left : (¬y = 0 →
    (divMod x y).snd * ↑y + (divMod x y).fst = ↑x ∧
      (0 ≤ (divMod x y).fst ∧ (divMod x y).fst < ↑y) ∧ 0 ≤ (divMod x y).snd) := by
      intro h₁
      have h₁_left : (divMod x y).snd * ↑y + (divMod x y).fst = ↑x := by
        unfold divMod
        simp
        have h₁ := Int.ediv_add_emod' x y
        apply h₁
      have h₁_right : (0 ≤ (divMod x y).1 ∧ (divMod x y).1 < ↑y) ∧ 0 ≤ (divMod x y).2 := by
        have h₂_left : (0 ≤ (divMod x y).1 ∧ (divMod x y).1 < ↑y) := by
          have h₃_left : 0 ≤ (divMod x y).1 := by
            unfold divMod
            simp
            rw [←Int.natCast_mod, ←Int.ofNat_eq_natCast]
            have ha := Int.zero_le_ofNat (x % y)
            apply ha
          have h₃_right : (divMod x y).1 < ↑y := by
            unfold divMod
            simp
            rw [←Int.natCast_mod]
            have ha := @Int.natMod_lt x y h₁
            rw [Int.natMod, ←Int.natCast_mod, Int.toNat_ofNat] at ha
            rw [←Mathlib.Tactic.Zify.natCast_lt]
            apply ha
          exact ⟨h₃_left, h₃_right⟩

        have h₂_right : 0 ≤ (divMod x y).2 := by
          unfold divMod
          simp
          rw [←Int.natCast_div]
          have ha := Int.natCast_nonneg (x / y)
          apply ha
        exact ⟨h₂_left, h₂_right⟩
      exact ⟨h₁_left, h₁_right⟩
    have h_right : (y = 0 → (divMod x y).fst = ↑x ∧ (divMod x y).snd = 0) := by
      intro h₀
      have h₁_left : (divMod x y).1 = ↑x := by
        unfold divMod
        simp
        rw [←Int.natCast_mod, h₀, Nat.mod_zero]
      have h₁_right : (divMod x y).snd = 0 := by
        unfold divMod
        simp
        rw [←Int.natCast_div, h₀, Nat.div_zero, ←@Int.cast_id 0]
        rfl
      exact ⟨h₁_left, h₁_right⟩
    constructor
    · simp_all [h_left]
    · simp_all [h_right]
  }
  -- !benchmark @end proof


Tests:
input: x: 10
y: 3
x: 15
y: 5
x: 7
y: 2
x: 0
y: 4
x: 10
y: 0
expected: (1, 3)
(0, 3)
(1, 3)
(0, 0)
(10, 0)
unexpected: (2, 2)
(0, 3)
(1, 4)
(3, 0)
(1, 1)
(0, 4)
(3, 1)
(0, 7)
(1, 2)
(0, 1)
(1, 0)
(2, 0)
(0, 10)
(10, 1)
(5, 5)

Inputs rejected by the precondition: