-----Description-----  
The task is to determine the position of a target integer in a given array. The goal is to return the index corresponding to the first occurrence of the target value. If the target is not present in the array, the result should indicate that by returning the size of the array. This description focuses entirely on understanding the problem without specifying any particular implementation method.

-----Input-----  
The input consists of:  
• a: An array of integers.  
• e: An integer representing the target to search for in the array.

-----Output-----  
The output is a natural number (Nat) which is:  
• The index of the first occurrence of the target integer if found.  
• The size of the array if the target integer is not present.  

-----Note-----  
There are no strict preconditions on the input; the method should work correctly for any array of integers. The specification ensures that the returned index is always valid: it is either within the array bounds with a matching element or equals the array’s size if the element is absent.
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 LinearSearch_precond (a : Array Int) (e : Int) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond


-- !benchmark @start code_aux

-- !benchmark @end code_aux


def LinearSearch (a : Array Int) (e : Int) (h_precond : LinearSearch_precond (a) (e)) : Nat :=
  -- !benchmark @start code
  let rec loop (n : Nat) : Nat :=
    if n < a.size then
      if a[n]! = e then n
      else loop (n + 1)
    else n
  loop 0
  -- !benchmark @end code

-- !benchmark @start postcond_aux

-- !benchmark @end postcond_aux


@[reducible, simp]
def LinearSearch_postcond (a : Array Int) (e : Int) (result: Nat) (h_precond : LinearSearch_precond (a) (e)) :=
  -- !benchmark @start postcond
  result ≤ a.size ∧ (result = a.size ∨ a[result]! = e) ∧ (∀ i, i < result → a[i]! ≠ e)
  -- !benchmark @end postcond


-- !benchmark @start proof_aux

-- !benchmark @end proof_aux


theorem LinearSearch_spec_satisfied (a: Array Int) (e: Int) (h_precond : LinearSearch_precond (a) (e)) :
    LinearSearch_postcond (a) (e) (LinearSearch (a) (e) h_precond) h_precond := by
  -- !benchmark @start proof
  unfold LinearSearch_postcond LinearSearch
  apply And.intro
  . let aux (x : Nat) : (0 ≤ x) → (x ≤ a.size) → LinearSearch.loop a e x ≤ a.size := by
      intro hx₀ hx₁
      let nx := a.size - x
      have hn₁ : nx = a.size - x := by rfl
      have hn₂ : x = a.size - nx := by
        rw [hn₁, Nat.sub_sub_self]
        apply hx₁
      rw [hn₂]
      induction nx with
      | zero =>
        unfold LinearSearch.loop
        simp
      | succ n₁ ih =>
        by_cases hp : (a.size ≤ n₁)
        . rw [Nat.sub_eq_zero_of_le hp] at ih
          have h_tmp : a.size ≤ n₁ + 1 := Nat.le_add_right_of_le hp
          rw [Nat.sub_eq_zero_of_le h_tmp]
          exact ih
        . have hq : n₁ < a.size := Nat.not_le.mp hp
          unfold LinearSearch.loop
          simp
          split_ifs
          . simp
          . rw [Nat.sub_add_eq, Nat.sub_add_cancel]
            exact ih
            apply Nat.zero_lt_sub_of_lt
            exact hq
          simp
    have h₀ : 0 ≤ a.size := by simp
    have h_triv : 0 ≤ 0 := by simp
    exact aux 0 h_triv h₀
  . apply And.intro
    . let aux (x : Nat) : (x ≥ 0) → (x ≤ a.size) → LinearSearch.loop a e x = a.size ∨ a[LinearSearch.loop a e x]! = e := by
        intro hx₀ hx₁
        let nx := a.size - x
        have hn₁ : nx = a.size - x := by rfl
        have hn₂ : x = a.size - nx := by
          rw [hn₁, Nat.sub_sub_self]
          apply hx₁
        rw [hn₂]
        induction nx with
        | zero =>
          unfold LinearSearch.loop
          simp
        | succ n₁ ih =>
          -- Ohh boy...
          by_cases hp : (a.size ≤ n₁)
          . rw [Nat.sub_eq_zero_of_le hp] at ih
            have h_tmp : a.size ≤ n₁ + 1 := Nat.le_add_right_of_le hp
            rw [Nat.sub_eq_zero_of_le h_tmp]
            exact ih
          . have hq : n₁ < a.size := Nat.not_le.mp hp
            apply Or.elim ih -- Didn't find elem, so we're gonna also return a.size...
            . intro ih₁
              unfold LinearSearch.loop
              split_ifs
              . rename_i h₁ h₂
                rw [h₂]
                simp
              . rename_i ha₁ ha₂
                rw [Nat.sub_add_eq, Nat.sub_add_cancel]
                rw [ih₁]
                simp
                apply Nat.zero_lt_sub_of_lt
                exact hq
              . rename_i h₁
                have ha₁ := Nat.not_lt.mp h₁
                have ha₂ := Nat.sub_le a.size (n₁ + 1)
                have ha := Nat.eq_iff_le_and_ge.mpr ⟨ha₁, ha₂⟩
                rw [←ha]
                simp
            . intro ih₂
              unfold LinearSearch.loop
              split_ifs
              . rename_i h₁ h₂
                rw [h₂]
                simp
              . rename_i ha₁ ha₂
                rw [Nat.sub_add_eq, Nat.sub_add_cancel]
                rw [ih₂]
                simp
                apply Nat.zero_lt_sub_of_lt
                exact hq
              . rename_i h₁
                have ha₁ := Nat.not_lt.mp h₁
                have ha₂ := Nat.sub_le a.size (n₁ + 1)
                have ha := Nat.eq_iff_le_and_ge.mpr ⟨ha₁, ha₂⟩
                rw [←ha]
                simp
      have h₀ : 0 ≤ 0 := by simp
      have h₁ : 0 ≤ a.size := by simp
      exact aux 0 h₀ h₁
    . let aux (x : Nat) : (0 ≤ x) → (x ≤ a.size) → (∀ i, x ≤ i → i < LinearSearch.loop a e x → a[i]! ≠ e) := by
        intro hx₀ hx₁ i
        let nx := a.size - x
        have hn₁ : nx = a.size - x := by rfl
        have hn₂ : x = a.size - nx := by
          rw [hn₁, Nat.sub_sub_self]
          apply hx₁
        rw [hn₂]
        induction nx with
        | zero =>
          -- There's no such i
          unfold LinearSearch.loop
          simp
          intro hxi hi
          have h_contr := Nat.lt_of_le_of_lt hxi hi
          have h : a.size ≤ a.size := by simp
          have h : a.size - a.size < a.size - a.size := Nat.sub_lt_sub_right h h_contr
          simp at h
        | succ n ih =>
          intro hxi
          unfold LinearSearch.loop
          simp
          split_ifs
          . rename_i h₁ h₂
            intro h_contr
            have h := Nat.lt_of_le_of_lt hxi h_contr
            simp at h
          . rename_i h₁ h₂
            by_cases hp : (a.size ≤ n)
            . rw [Nat.sub_eq_zero_iff_le.mpr hp] at ih
              intro hi
              have hp₁ : a.size ≤ n + 1 := by
                have h₁' : n ≤ n + 1 := by simp
                exact Nat.le_trans hp h₁'
              rw [Nat.sub_eq_zero_iff_le.mpr hp₁] at hxi
              rw [Nat.sub_eq_zero_iff_le.mpr hp₁] at hi
              rw [Nat.sub_eq_zero_iff_le.mpr hp₁] at h₂
              have ih₁ := ih hxi
              simp at hi
              unfold LinearSearch.loop at ih₁
              split_ifs at ih₁
              . rename_i h₁'
                simp at ih₁
                exact ih₁ hi
              . rename_i h₁'
                contradiction
            . have hq : n < a.size := Nat.not_le.mp hp
              have hq' : 1 ≤ a.size - n := by
                have h : 0 < a.size - n := by
                  exact Nat.sub_pos_of_lt hq
                exact Nat.one_le_iff_ne_zero.mpr (Nat.ne_zero_of_lt h)
              rw [Nat.sub_add_eq, Nat.sub_add_cancel hq']
              intro hi
              by_cases h_bounds : (a.size - n ≤ i)
              . exact ih h_bounds hi
              . have h_bounds' : ( i + 1 < a.size - n + 1) := (@Nat.add_lt_add_iff_right 1 i (a.size - n)).mpr (Nat.not_le.mp h_bounds)
                have h := Nat.le_of_lt_add_one h_bounds'
                apply Nat.le_sub_of_add_le at h
                rw [← Nat.sub_add_eq] at h
                have hi_fixed := Nat.eq_iff_le_and_ge.mpr ⟨hxi, h⟩
                rw [hi_fixed] at h₂
                exact h₂
          . intro h_contr
            have h := Nat.lt_of_le_of_lt hxi h_contr
            simp at h
      have h₀ : 0 ≤ a.size := by simp
      have h_triv : 0 ≤ 0 := by simp
      intro i
      have h_tmp : 0 ≤ i := Nat.zero_le i
      exact aux 0 h_triv h₀ i h_tmp


  -- !benchmark @end proof


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

Inputs rejected by the precondition: