Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics
Abstract
We describe how we connected three programs that compute Groebner bases to Coq, to do automated proofs on algebraic, geometrical and arithmetical expressions. The result is a set of Coq tactics and a certificate mechanism (downloadable at http://wwwsop.inria.fr/marelle/Loic.Pottier/gbkeappa.tgz). The programs are: F4, GB \, and gbcoq. F4 and GB are the fastest (up to our knowledge) available programs that compute Groebner bases. Gbcoq is slow in general but is proved to be correct (in Coq), and we adapted it to our specific problem to be efficient. The automated proofs concern equalities and nonequalities on polynomials with coefficients and indeterminates in R or Z, and are done by reducing to Groebner computation, via Hilbert's Nullstellensatz. We adapted also the results of Harrison, to allow to prove some theorems about modular arithmetics. The connection between Coq and the programs that compute Groebner bases is done using the "external" tactic of Coq that allows to call arbitrary programs accepting xml inputs and outputs. We also produce certificates in order to make the proof scripts independant from the external programs.
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 arXiv:
 arXiv:1007.3615
 Bibcode:
 2010arXiv1007.3615P
 Keywords:

 Computer Science  Symbolic Computation;
 Computer Science  Logic in Computer Science
 EPrint:
 Knowledge Exchange: Automated Provers and Proof Assistants, Doha : Qatar (2008)