-----Description----- 
This task requires writing a Lean 4 method that determines whether an array of integers contains at least one pair of consecutive numbers. The method should return true if there is any index where an element, when increased by one, equals the next element in the array. If no such consecutive pair exists, the method should return false.

-----Input-----
The input consists of:
a: An array of integers (the array may be empty or non-empty).

-----Output-----
The output is a Boolean value:
Returns true if there is at least one index where an element plus one equals the following element.
Returns false if the array does not contain any consecutive numbers.

-----Note-----
There are no additional preconditions; the method will function correctly regardless of the array's size.
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 containsConsecutiveNumbers_precond (a : Array Int) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond


-- !benchmark @start code_aux

-- !benchmark @end code_aux


def containsConsecutiveNumbers (a : Array Int) (h_precond : containsConsecutiveNumbers_precond (a)) : Bool :=
  -- !benchmark @start code
  if a.size ≤ 1 then
    false
  else
    let withIndices := a.mapIdx (fun i x => (i, x))
    withIndices.any (fun (i, x) =>
      i < a.size - 1 && x + 1 == a[i+1]!)
  -- !benchmark @end code


-- !benchmark @start postcond_aux

-- !benchmark @end postcond_aux


@[reducible, simp]
def containsConsecutiveNumbers_postcond (a : Array Int) (result: Bool) (h_precond : containsConsecutiveNumbers_precond (a)) :=
  -- !benchmark @start postcond
  (∃ i, i < a.size - 1 ∧ a[i]! + 1 = a[i + 1]!) ↔ result
  -- !benchmark @end postcond


-- !benchmark @start proof_aux

-- !benchmark @end proof_aux


theorem containsConsecutiveNumbers_spec_satisfied (a: Array Int) (h_precond : containsConsecutiveNumbers_precond (a)) :
    containsConsecutiveNumbers_postcond (a) (containsConsecutiveNumbers (a) h_precond) h_precond := by
  -- !benchmark @start proof
  unfold containsConsecutiveNumbers containsConsecutiveNumbers_postcond
  constructor
  · simp_all
    intro i hi hconsec
    have hi' : 1 + i < a.size := by
        rw [Nat.add_comm]
        exact Nat.add_lt_of_lt_sub hi
    have hi'' : i < a.size := by
      have : i < 1 + i := by
        simp [Nat.lt_add_of_pos_left]
      exact Nat.lt_trans this hi'
    constructor
    · exact Nat.lt_of_add_right_lt hi'
    · apply Array.any_iff_exists.mpr
      simp
      exists i
      simp [hi, hi'']
      have : a[i]! = a[i] := by
        exact getElem!_pos a i hi''
      rw [←this]
      exact hconsec
  · simp
    intro ha h
    have h' := Array.any_iff_exists.mp h
    simp at h'
    rcases h' with ⟨i, hi, ⟨hi', hconsec⟩⟩
    have : a[i]! = a[i] := by
      exact getElem!_pos a i hi
    exists i
    rw [this]
    simp_all
  -- !benchmark @end proof


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

Inputs rejected by the precondition: