## 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.

Our previous work on integrating SAT technology has been used as a basis to obtain SMT automation within Coq. We are now capable of replaying traces produced by the SMT prover VeriT that deal with conjunctive normal forms, congruence closures, and linear arithmetic. We are actively working on adding quantified formulae. This work is supported by the ANR Decert project. A prelimary version [10] of this work has been presented at the workshop PSATTT'11, a full version [9] at the conference CPP11. The generic exchange proof format [13] for SMT has been presented at the workshop PXTP'11.

#### Geometric Algebras and Automatic Theorem Proving

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

We have completed our work on Grassman-Cayley algebras. This has been published in the post-proceedings of the ADG'10 conference. We are now working on the natural continuation of this work: Clifford's algebras. We have very encouraging premilary results.

#### Taylor models in Coq

Participants : Erik Martin-Dorel [project-team Arénaire] , Ioana Paşca [project-team Arénaire] , Micaela Mayero [Université Paris XIII] , Laurence Rideau, Laurent Théry.

Taylor models are a very effective way to approximate real functions with polynomials. We have started a formalisation of these models in the Coq prover. In a first step, we have concentrated our efforts in having a computational version of these models within Coq using native computations, certified floating point and interval arithmetics. Since our first evaluations show that they behave well computationally, we are now working on completing this work with the corresponding correctness proofs. This work is supported by the ANR Tamadi.

#### Tactics on polynomial equalities: nsatz

Participant : Loïc Pottier.

We started describing in the Coq programming language an efficient algorithm to compute Gröbner bases, similar to the one written in ocaml for the nsatz tactic. We hope to prove it correct and to use it for proofs by reflexion in commutative algebra.

#### D-Modules

Participant : Loïc Pottier.

We studied normalization of non-commutative polynomials ad exponentials in the Weyl algebra. The normal forms we found are similar with the one described found by Blasiak and Flajolet for graph models.