## Cano, Guillaume and Cohen, Cyril and Dénès, Maxime and Mörtberg, Anders and Siles, Vincent - Formalized linear algebra over Elementary Divisor Rings in Coq

lmcs:1639 - Logical Methods in Computer Science, June 22, 2016, Volume 12, Issue 2
Formalized linear algebra over Elementary Divisor Rings in Coq

Authors: Cano, Guillaume and Cohen, Cyril and Dénès, Maxime and Mörtberg, Anders and Siles, Vincent

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend Bézout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility.

Source : oai:arXiv.org:1601.07472
DOI : 10.2168/LMCS-12(2:7)2016
Volume: Volume 12, Issue 2
Published on: June 22, 2016
Submitted on: November 11, 2014
Keywords: Computer Science - Logic in Computer Science,Mathematics - Rings and Algebras,I.2.3,F.4.1