Problem: polymul_karatsuba

Your task is to implement the Karatsuba Multiplication Algorithm for polynomials and verify its functional correctness in Lean. Given two polynomials A and B represented as coefficient lists, compute their product. Preconditions: 1. Lengths: A and B have equal lengths, which are powers of two (e.g., 2, 4, 8...). 2. Constraints: The sum of lengths is at most 1000. Coefficients are integers between -1,000,000 and 1,000,000. Postconditions: Implement `karatsuba_mul` using the divide-and-conquer approach. (1) Algorithm: - Base Case: Small N (e.g., N=1), use simple multiplication. - Recursive Step: Split A and B into halves (A0, A1, B0, B1). Compute z0 = A0*B0, z2 = A1*B1, and z1 = (A0+A1)*(B0+B1). - Combine: result = z0 + (z1 - z0 - z2)x^m + z2 x^2m. (2) Output: Returns a list of size |A| + |B| - 1. (3) Functional Correctness: Verify that the result matches the standard mathematical convolution of A and B.  Verification Challenges: (1) Algebraic Equivalence: You must prove that the Karatsuba formula mathematically equals the standard polynomial multiplication formula. This typically requires a helper lemma showing (A0+A1)(B0+B1) - A0B0 - A1B1 = A0B1 + A1B0. (2) Termination: Prove that the recursion on list length terminates.

import Mathlib

-- Precondition definitions
@[reducible, simp]
def karatsuba_mul_precond (a b : List Int) : Prop :=
  -- !benchmark @start precond
  a.length > 0 ∧
  b.length > 0 ∧
  a.length + b.length ≤ 1000 ∧
  (∃ k : Nat, 2 ^ k = a.length) ∧
  (∃ k : Nat, 2 ^ k = b.length) ∧
  a.length = b.length ∧
  (∀ c, c ∈ a → -1000000 ≤ c ∧ c ≤ 1000000) ∧
  (∀ c, c ∈ b → -1000000 ≤ c ∧ c ≤ 1000000)
  -- !benchmark @end precond

def spec_poly_mul_coeff (a b : List Int) (k : Nat) : Int :=
  -- Omitted definition for brevity, same as naive
  0 -- placeholder

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

-- Main function definitions
def karatsuba_mul (a b : List Int) (h_precond : karatsuba_mul_precond a b) : List Int :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def karatsuba_mul_postcond (a b : List Int) (result : List Int) (h_precond : karatsuba_mul_precond a b) : Prop :=
  -- !benchmark @start postcond
   result.length = a.length + b.length - 1 ∧
   ∀ k : Nat, k < result.length →
     result.getD k 0 = (List.range (k + 1)).foldl (fun acc i =>
         if i < a.length ∧ k - i < b.length then
           acc + a.getD i 0 * b.getD (k - i) 0
         else acc) 0
  -- !benchmark @end postcond

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

-- Proof content
theorem karatsuba_mul_postcond_satisfied (a b : List Int) (h_precond : karatsuba_mul_precond a b) :
    karatsuba_mul_postcond a b (karatsuba_mul a b h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof