Problem: trial_division_naive

Your task is to implement the "Trial Division (Naive)" Primality Test and verify its correctness in Lean. Preconditions: The input n is a natural number (Nat). Postconditions: Implement the function `check_prime` which returns true if n is prime and false otherwise. (1) Algorithm: You should use a simple trial division loop (or recursion) that checks every integer from 2 up to n - 1. (2) Functional Correctness: You must verify that your result matches the mathematical definition of primality (a number n > 1 divisible only by 1 and itself). Verification Challenges: (1) Loop Invariant: You must maintain an invariant stating that "for all k checked so far (2 <= k < i), k does not divide n". (2) Quantifier Instantiation: When the loop finishes without finding a divisor, you must prove that this implies the universal quantifier forall d. 1 < d < n -> n % d != 0.

import Mathlib

-- Precondition definitions
@[reducible, simp]
def check_prime_precond (n : Nat) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond

-- !benchmark @start auxcode
-- !benchmark @end auxcode

-- Main function definitions
def check_prime (n : Nat) (h_precond : check_prime_precond n) : Bool :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def check_prime_postcond (n : Nat) (result : Bool) (h_precond : check_prime_precond n) : Prop :=
  -- !benchmark @start postcond
  result = true ↔ (n > 1 ∧ ∀ d : Nat, 1 < d ∧ d < n → n % d ≠ 0)
  -- !benchmark @end postcond

-- !benchmark @start lemma
-- !benchmark @end lemma

-- Proof content
theorem check_prime_postcond_satisfied (n : Nat) (h_precond : check_prime_precond n) :
    check_prime_postcond n (check_prime n h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof