  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://www-sop.inria.fr/marelle/Loic.Pottier/gb-keappa.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
non-equalities 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.
