Quotients dans le CCI

On étudie plusieurs voies pour formaliser les quotients dans le Calcul de Constructions Inductive: les quotients naïfs (en utilisant les travaux de [LW99] et [BB96],onmontre qu´ils conduisent à une incohérence), les quotients décidables, les quotients classiques de la théorie des types (comme ceux étudiés dans [Hof95], et finallement on propose une notion de quotients fonctionnels qui résolvent les problèmes des précédentes notions, et semblent satisfaisants théoriquement et pratiquement. Pour chaque notion introduite au cours du papier, on donne la traduction correspond- ante dans le système Coq [pro99]

Sub-groups of Zn, standard basis, and linear diophantine systems

We caracterize a (affine) sub-group of Zn by a polynomial ideal and standard (Gröbner) basis. We show how to use these standard basis to solve directly many algorithmic problems on (affine) sub-groups and their non-negative parts : * the membership problem * the triviality problem * find the smallest non zero vector * find the minimal non-negative elements * solve linear diohantine systems (Ax = 0, x>=0), (Ax = b, x>=0), Ax <= b on integers, and find smallest solutions Finally we conjecture other properties of these standard basi

Extraction dans le Cci

On présente ici une méthode permettant d'extraire de n'importe quel terme du CCI (toutes sortes confondues) un programme Ocaml acceptable par OCAML, et dont l'évaluation est efficace

Solutions minimales des systemes diophantiens lineaires : bornes et algorithmes

On donne de nouvelles bornes et de nouveaux algorithmes concernant les solutions minimales de systèmes diophantiens linéaires. Nos bornes sont simplement exponentielles en la dimension du système, alors que les bornes connues jusqu'il y a peu étaient doublement exponentielles

Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics

International audienceWe 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

Grobner bases of toric ideals

We study here \grobner\ bases of ideals which define toric varieties. We connect these ideals with the sub-lattices of $Z^d$, then deduce properties on their \grobner\ bases, and give applications of these results. The main contributions of the report are a bound on the degree of the \grobner\ bases, the fact that they contain Minkowski successive minima of a lattice (in particular shortest vector), and the algorithm (derived from Buchberger algorithm), which starts with ideal of polynomials with less variables than usua

Ambiguous Typing

We propose a method to approximate the language of mathematicians by using amgiguities in the notations, in the context of type theory

Proof Certificates for Algebra and their Application to Automatic Geometry Theorem Proving

Post-proceedings of ADG 2008 (Automated Deduction in Geometry)International audienceIntegrating decision procedures in proof assistants in a safe way is a major challenge. In this paper, we describe how, starting from Hilbert's Nullstellensatz theorem, we combine a modified version of Buchberger's algorithm and some reflexive techniques to get an effective procedure that automatically produces formal proofs of theorems in geometry. The method is implemented in the Coq system but, since our specialised version of Buchberger's algorithm outputs explicit proof certificates, it could be easily adapted to other proof assistants

Élimination des quantificateurs sur les réels pour Coq

National audienceOn présente ici l'implémentation en OCaml d'une tactique Coq qui réalise une procédure d'élimination des quantificateurs pour les réels, basée sur la méthode de Hormander