Problem: integer_exponential

Your task is to implement Integer Exponentiation and verify its correctness in Lean. Given a base b and a non-negative exponent e, compute b to the power of e.  Preconditions: The input base b and exponent e are natural numbers (Nat). Postconditions: Implement the function to compute the exponential. (1) Functional Correctness: You must verify that your implementation returns exactly b ^ e (as strictly defined by Lean's Nat.pow). (2) Termination: You must prove that your recursive implementation terminates. Verification Challenges: (1) Accumulation Invariant: You must prove the invariant (or recursion hypothesis) that ensures the computation correctly builds up to the final power b^e.

import Mathlib

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

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

-- Main function definitions
def exponentiation (b e : Nat) (h_precond : exponentiation_precond b e) : Nat :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def exponentiation_postcond (b e result : Nat) (h_precond : exponentiation_precond b e) : Prop :=
  -- !benchmark @start postcond
  result = b ^ e
  -- !benchmark @end postcond

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

-- Proof content
theorem exponentiation_postcond_satisfied (b e : Nat)
    (h_precond : exponentiation_precond b e) :
    exponentiation_postcond b e (exponentiation b e h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof