Problem: polymul_karatsuba

Your task is to implement the Karatsuba algorithm for fast polynomial multiplication and verify its correctness in Dafny.  Preconditions: Inputs a and b are seq<int> representing polynomials. Structure: The length of a and b are equal and must be a power of 2. Size: The total length is at most 1000. Bounds: Input coefficients are within $[-1,000,000, 1,000,000]$. Requirements: Implement poly_multiply that returns the product of the two polynomials. (1) Algorithm Logic: + Base Case: If $N$ is small, perform scalar multiplication. + Split: Divide $A(x)$ and $B(x)$ into halves $A_0, A_1$ and $B_0, B_1$. + Recurse: Compute $z_0 = A_0 B_0$, $z_2 = A_1 B_1$, and $z_1 = (A_0+A_1)(B_0+B_1)$. + Combine: Result $= z_0 + (z_1 - z_0 - z_2)x^m + z_2 x^{2m}$. (2) Functional Correctness: Verify that the result mathematically matches the standard convolution definition (same as the Naive version). Verification Challenges: (1) Algebraic Equivalence: You must prove that the Karatsuba formula equals the standard expansion. This requires a ghost proof establishing $(A_0+A_1)(B_0+B_1) - A_0B_0 - A_1B_1 = A_0B_1 + A_1B_0$.

// <preamble>
// Helper to convert inputs. In Dafny, we use int directly, so this is identity.
function to_int(s: seq<int>): seq<int> {
    s
}

// Helper: Enforce bounds on coefficients to prevent overflow.
predicate coeffs_bounded(s: seq<int>) {
    forall i :: 0 <= i < |s| ==> -1_000_000 <= s[i] <= 1_000_000
}

// Recursive definition of the convolution sum
function spec_convolution_sum(a: seq<int>, b: seq<int>, k: int, current_i: int): int
    decreases if current_i < 0 then 0 else current_i + 1
{
    if current_i < 0 then
        0
    else
        // Calculate the term for the current index i
        var term := if current_i < |a| && (k - current_i) < |b| && (k - current_i) >= 0 then
            a[current_i] * b[k - current_i]
        else
            0;
        // Recursive step
        term + spec_convolution_sum(a, b, k, current_i - 1)
}

// Wrapper for the full convolution coefficient at index k
function spec_poly_mul_coeff(a: seq<int>, b: seq<int>, k: int): int {
    spec_convolution_sum(a, b, k, k)
}
// </preamble>

// <helpers>

// </helpers>

// <proofs>

// </proofs>

// <spec>
method poly_multiply(a: seq<int>, b: seq<int>) returns (res: seq<int>)
    requires |a| > 0
    requires |b| > 0
    // Constraints from Description
    requires |a| == |b|
    requires exists k :: 0 <= k <= 10 && |a| == (1 << k) // Enforce Power of 2 (up to 1024)
    requires |a| + |b| <= 1000
    // Safety Precondition: Coefficients are bounded
    requires coeffs_bounded(a)
    requires coeffs_bounded(b)
    
    ensures |res| == |a| + |b| - 1
    // Functional Correctness
    ensures forall k :: 0 <= k < |res| ==> res[k] == spec_poly_mul_coeff(a, b, k)
// </spec>
// <code>
{
    // Implement and verify the Karatsuba polynomial multiplication here.
}
// </code>

method Main() {}