## Section: New Results

### Proving tools

#### Connecting an SMT prover and Coq

Participants : Michaël Armand, Germain Faure [project-team Typical] , Benjamin Grégoire, Chantal Keller [project-team Typical] , Laurent Théry.

We completed our work on integrating SAT technology inside Coq. This work has been described in a publication at the conference ITP10 in Edinburgh [7] . Furthermore this serves as a basis for the integration of SMT technology. We are now capable of replaying traces produced by the SMT prover VeriT that deal with congruence closure. This works is supported by the ANR Decert project.

#### Proofs certificates for theorems in geometry

Participants : Benjamin Grégoire, Loïc Pottier, Laurent Théry.

We completed our work of previous years on Gröbner bases for geometric theorems by publishing a paper [6] .

#### Geometric Algebras and Automatic Theorem Proving

Participants : Laurent Fuchs [Université de Poitiers] , Laurent Théry.

We extended our formalisation of geometric algebras with some notions of bracket algebras. This lets us derive a reflexive tactic that is capable of proving elementary problems in incidence geometry fully automatically. This work was presented at the conference ADG'2010 in Munich. This work is also supported by the ANR Galapagos project.

#### A tactic on polynomial equalities: nsatz

Participant : Loïc Pottier.

We re-wrote and finished the implementation of the tactic “nsatz”, which implements Hilbert’s Nullstellensatz: it proves equations between polynomials from similar hypotheses. It extends the “ring” tactic. “nsatz” is implemented using the “type classes” of the Coq system, and works on integral domains, with specializations on Z, Q and R. This is available in the distributed version of the Coq system (8.3). We plan to extend this work by providing certificates in Coq for Gröbner bases, and other useful computational objects in computer algebra of this kind (dimension, invariants, etc).

#### D-Modules

Participant : Loïc Pottier.

We studied normalization of non-commutative polynomials ad exponentials in the Weyl algebra, and found a method of normalization by evaluation which reduces to the commutative case, which is suitable for an easy implementation and proof in Coq. Extension to non-commutative Gröbner bases is planned.