## Section: New Results

### Type theory and formalization of mathematics

#### Group theory

Participants : Georges Gonthier [Microsoft Research] , Assia Mahboubi [project-team Typical] , Laurence Rideau, Laurent Théry, Sidi Ould Biha.

We participate in the collaborative research agreement “Mathematical Components” with Microsoft Research. This project aims at evaluating the applicability of a new approach to mathematical proofs called “small-scale reflection”, especially in the domain of finite group theory [3] .

This year, we have consolidated the Phd work of Sidi Ould Biha. The algebraic structures for linear algebra are now part of the main development line of the "Mathematical Components" libraries. In conjunction with some basic notions of representation theory, we have now all the pre-requisite elements for formalising the character theory that is needed for the Feit-Thompson theorem. In particular, we have included special support for finite aspects, for instance finite dimension vector spaces. This work is also supported by the Formath European project.

#### Proofs in geometry

Participants : Tuan Minh Pham, Yves Bertot.

We completed our work on developing a proving tool that integrates the capabilities of a proof system lik Coq, a proo management interface like Pcoq, and a tool for dynamic geometry manipulation and visualization like GeoGebra. This work was presented at the UITP conference.

We integrated the previous work of F. Guilhot on the formalization of high-school geometry with the work of J. Narboux on the area method for automatic proof in geometry. This also involved removing many of the axioms present in the initial work of Guilhot, where axioms were often used for definitional purposes.

Last we completed our work on describing orientation in geometry proofs.

#### Towards constructive algebraic topology

Participants : Yves Bertot, Laurence Rideau.

As part of our collaboration in the Formath European project, we gave a one week course of ssreflect at the university of La Rioja in Logroño, Spain, and we participated in the formalization of “incidence simplicial matrices” in ssreflect. We started working on an article describing this work.

#### On Bernstein coefficients

Participants : Assia Mahboubi [project-team Typical] , Yves Bertot.

As a contribution to our long term objective of developing a formally verified implementation of cylindrical algebraic decomposition, we studied the proof that the number of alternation in a polynomial's Bernstein coefficients gives an upper bound of the number roots for this polynomial in the corresponding interval. An article describing this work has been submitted for publication and is already available as a pre-print [14] .

#### Computing with polynomials and matrices

Participants : Maxime Dénès, Stefania Dumbrava [Bremen university] , Laurent Théry.

The libraries of the project "Mathematical Components" propose a rather complete formalisation of polynomials and matrices. Unfortunately, these objects cannot be used directly for computing. In her internship, Stefania Dumbrava has been working on providing some computational contents to these objects. In particular she has investigated how persistent arrays could be effectively used for this purpose.

#### Co-recursion and real numbers

Participants : Yves Bertot, Nicolas Julien, Ioana Paşca.

The work we did on the formal verification of programs that combine Newton's method and rounding has been summarized in an article that is submitted for publication and is already available as a pre-print [15] .

#### Regularity of interval matrices

Participants : Yves Bertot, Guillaume Cano, Ioana Paşca.

We have formalized a collection of criteria for the regularity of matrices with interval coefficients taken from the work of Rex and Rohn. This work leads to a publication in a conference [12] and to a chapter in Ioana Pasca's thesis [5] .

The formalization relies on a theorem of mathematics whose proof has yet to be completed: the Perron-Frobenius theorem. The formal verification of this theorem is under way, it implies adding new concepts to the libraries, among which complex numbers, general topology, compacts, etc.

#### Type-based termination

Participants : Benjamin Grégoire, Jorge Luis Sacchini.

We extended the Calculus of Inductive Constructions with a type-based mechanism for ensuring termination of recursive functions. In [11] we published a preliminary version where only natural numbers are considered. We are currently working on the full version with inductive types which will be part of Jorge Luis Sacchini's Phd thesis.

#### Native compilation of terms with primitive structures

Participants : Benjamin Grégoire, Maxime Dénès.

We integrated the native compiler of the Ocaml language into a scheme for the efficient reduction of terms in the calculus of inductive constructions. On some examples, efficiency gains can reach a tenfold increase in speed. We expect this to have a strong impact on the capability to perform proofs by reflection involving heavy computation in the Coq system.