Connecting Gröbner bases programs with Coq to do proofs in algebra, geometry and arithmetics

WORKSHOP TO DEFINE DECONTAMINATION AND DECOMMISSIONING NEEDS FOR THE U.S. DEPARTMENT OF ENERGY

Loïc Pottier

Others  :  http://CEUR-WS.org/Vol-418/paper5.pdf
PID  :  3475

来源: CEUR

【 摘 要 】

We describe how we connected three programs that compute Gröbner bases [1] to Coq [11], to do automated proofs on algebraic, geometrical and arithmetical expressions. The result is a set of Coq tactics and a certificate mechanism 1. The programs are: F4 [5], GB [4], and gbcoq [10]. F4 and GB are the fastest (up to our knowledge) available programs that compute Gröbner 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 non-equalities on polynomials with coefficients and indeterminates in R or Z, and are done by reducing to Gröbner computation, via Hilbert’s Nullstellensatz. We adapted also the results of [7], to allow to prove some theorems about modular arithmetics. The connection between Coq and the programs that compute Gröbner 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.