A formal proof of Sasaki-Murao algorithm

The goal of this note is to give a presentation of a formal proof of the Sasaki-Murao algorithm [SM82]. This is an elegant algorithm for computing the determinant of a square matrix over an arbitrary commutative ring in polynomial time. Usual presentations of this algorithm are quite complex, and rely on some Sylvester identities [AL04]. We believe that the proof we shall present, which was obtained by formalizing this algorithm in Type Theory (more precisely in the SSReflect [GM10] extension to Coq [Tea10]) is simpler. It does not rely on Sylvester identities and indeed gives a proof of some of them as corollaries. It provides also a good example of how one can use a library of formalized mathematical results to prove formally a computer algebra program. The whole formalization can be found at [MS12].