Connecting Gr"obner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics

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.

💡 Research Summary

The paper presents a practical integration of three Gröbner‑basis computation systems—F4, GB, and gbcoq—with the Coq proof assistant, enabling fully automated reasoning about algebraic equalities, inequalities, geometric relations, and arithmetic statements. The authors observe that while Coq provides a trustworthy logical kernel, it lacks efficient decision procedures for polynomial reasoning, and existing external computer‑algebra systems, although fast, cannot be trusted without additional verification. To bridge this gap, they exploit Coq’s “external” tactic, which allows a Coq proof script to invoke an arbitrary program that communicates via XML.

The workflow proceeds in four stages. First, a Coq goal involving polynomials over ℝ or ℤ is transformed, using Hilbert’s Nullstellensatz for equalities and a variant for inequalities, into a finite set of polynomial equations whose ideal membership decides the original statement. For modular arithmetic the authors adapt Harrison’s technique, reducing statements to polynomial equations over finite fields ℤ/pℤ. Second, the transformed set is serialized as an XML document and handed to one of the three external engines. F4 and GB are state‑of‑the‑art matrix‑based algorithms that can handle large systems in seconds; gbcoq is a Coq‑implemented, formally verified Gröbner engine that is slower but provides a correctness guarantee by construction. Third, the chosen engine computes a Gröbner basis and returns it, again in XML. Fourth, Coq parses the result and checks whether the constant 1 belongs to the generated ideal; if so, the original goal is discharged automatically.

A central contribution is the “certificate” mechanism. The XML output not only contains the Gröbner basis but also a complete log of the transformation steps, the chosen monomial ordering, and the mapping between the original goal and the generated ideal. When a proof script is later replayed, Coq can ignore the external program entirely and verify the certificate internally, thus achieving proof‑script independence from any external executable. This approach satisfies both efficiency (by using fast external engines during proof development) and trustworthiness (by checking certificates during proof checking).

The authors evaluate their system on a suite of benchmarks drawn from algebra, geometry, and number theory. For most benchmarks, F4 or GB produce a Gröbner basis in sub‑second time, while gbcoq takes from a few seconds up to a minute on the same inputs. Certificate verification, however, consistently completes within a few seconds for all three engines. The experiments demonstrate that the hybrid strategy—fast external computation for proof search, followed by internal certificate validation—offers the best trade‑off between speed and reliability.

In conclusion, the paper delivers a robust, extensible framework that brings high‑performance Gröbner‑basis computation into the trusted environment of Coq. By coupling external algebraic engines with a formally verified fallback (gbcoq) and a rigorous certificate infrastructure, the authors enable automated proofs of non‑trivial polynomial statements while preserving the foundational guarantees of the Coq kernel. Future work is suggested in extending the approach to richer algebraic structures (e.g., rational functions, algebraic curves) and improving the user experience through higher‑level tactics and IDE integration.