## VerifyThis 2016: Strassen's Matrix Multiplication

This is a complete solution to VerifyThis 2016 challenge 1. The details are described in the paper The Matrix Reproved.

A self-contained solution to task 1 can be found in file naive.mlw Other files solve task 2 and 3:

- sum_extended.mlw prove a few extra lemmas on top of what the standard library already gives
- matrices.mlw contains the theories of matrices, matrix arithmetic, and block product. Except for providing the requested program, it solves task 2.
- matrices_ring_simp.mlw is a support file for proof by reflection of matrix algebraic equations
- strassen.mlw is the implementation of Strassen's Algorithm for task 3. It also contains the program associated to task 2 (assoc_proof).

To replay proofs: be sure you have all the three SMT solvers Alt-Ergo, CVC4 and Z3.
Run `why3 replay -L . FILE` to replay the session associated to `FILE.mlw`.

Challenge text:

Challenge 1: Matrix Multiplication Consider the following pseudocode algorithm, which is naive implementation of matrix multiplication. For simplicity we assume that the matrices are square. int[][] matrixMultiply(int[][] A, int[][] B) { int n = A.length; // initialise C int[][] C = new int[n][n]; for (int i = 0; i < n; i++) { for (int k = 0; k < n; k++) { for (int j = 0; j < n; j++) { C[i][j] += A[i][k] * B[k][j]; } } } return C; } Tasks. 1. Provide a specification to describe the behaviour of this algorithm, and prove that it correctly implements its specification. 2. Show that matrix multiplication is associative, i.e., the order in which matrices are multiplied can be disregarded: A(BC) = (AB)C. To show this, you should write a program that performs the two different computations, and then prove that the result of the two computations is always the same. 3. [Optional, if time permits] In the literature, there exist many proposals for more efficient matrix multiplication algorithms. Strassenâ€™s algorithm was one of the first. The key idea of the algorithm is to use a recursive algorithm that reduces the number of multiplications on submatrices (from 8 to 7), see https://en.wikipedia.org/wiki/Strassen_algorithm for an explanation. A relatively clean Java implementation (and Python and C++) can be found here: https://martin-thoma.com/strassen-algorithm-in-python-java-cpp/. Prove that the naive algorithm above has the same behaviour as Strassenâ€™s algorithm. Proving it for a restricted case, like a 2x2 matrix should be straightforward, the challenge is to prove it for arbitrary matrices with size 2^n.

**Authors:** Martin Clochard / Léon Gondelman / Mário Pereira

**Topics:** Matrices / Proofs by reflection

**Tools:** Why3

**References:** VerifyThis @ ETAPS 2016

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This example is split into several files.