## Section: New Results

### Type theory and formalization of mathematics

#### Foundational aspects of mechanized proofs

Participants : José Grimm, Loïc Pottier.

We attempt to prove all theorems in the “Theory of Sets” of
Bourbaki. The first chapter descripes Formal Mathematics, and we show
that it can be interpreted in the Coq language, thanks to a bunch of
axioms introduced by Carlos Simpson (CNRS, Nice), modulo some
modifications. This work that was started in 2009, when J. Grimm was
in the Apics project-team. A new formulation of this work using `ssreflect` has proved more efficient than the initial formulation
relying on standard Coq.

The second chapter of Bourbaki covers the theory of sets, *per
se*. It defines ordered pairs, correspondences, unions,
intersections and products of a family of sets, as well as equivalence
relations. The work of formalizing this chapter comprises 15000 lines
of Coq script and is described in a technical report and a paper for
the journal of formal reasoning published in 2010.

The third chapter of Bourbaki covers the theory of ordered sets, well-ordered sets, equipotent sets, cardinals, natural integers, and infinite sets; its implementation in Coq is described in [21] . This chapter is longer (22000 lines of code), and there are more exercises (18000 lines of code for about half of the exercises currently implemented).

We also looked at the *univalent foundation* proposed by
V. Voevodsky to provide a new model for equality in type theory and
simplified the proof that he proposed to derive extensionality from
the univalence axiom.

#### Group theory (Character theory)

Participants : Georges Gonthier [Microsoft Research] , Laurence Rideau, Laurent Théry.

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 [4] .

This year, we have initiated the formalisation of the second book of the proof of Feit-Thompson's theorem. The basic properties of character theories are now covered. This lets us formalised the first 4 chapters of the second book, “Character theory for the Odd Order Theorem” by Peterfalvi.

#### Proofs in geometry

Participants : Tuan Minh Pham, Yves Bertot.

The work on elementary (synthetic) geometry has been completed. A publication on the topic has also been presented at a conference [19] . This work was also the main content of Tuan Minh Pham's thesis which was defended in November [5] .

#### Towards constructive algebraic topology

Participants : Laurence Rideau, Maxime Dénès, Yves Bertot.

We have participated in the formalization of a complete chain of
computation from an image (as a bitmap) to the corresponding Betti numbers
and homology groups.
In particular, we improved the formalization of
“incidence simplicial matrices” in `ssreflect` . This work was
described in conference article [17] .

#### Computing with polynomials and matrices

Participants : Maxime Dénès, Yves Bertot.

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.

We have continued our study of executable algorithms to compute with matrices and polynomials inside Coq. In collaboration with other members of the European project Formath, we have looked at implementation of Strassen-Winograd and Karatsuba for fast matrix multiplication and other algorithms for various kinds of matrix normal forms: Smith normal form, Frobenius, and Jordan normal forms. This work is described in an article that has been submitted for publication.

#### Regularity of interval matrices

Participants : Guillaume Cano, Yves Bertot.

As part of our work on the regularity of interval matrices, we still needed to formalize the Perron-Frobenius theorem. This year we concentrated on an important lemma for this formalization, the Bolzano-Weierstrass theorem, which requires a usable formalization of general topology, in particular the concept of compact.

#### Type-based termination

Participants : Jorge Luis Sacchini, Benjamin Grégoire.

The work on this topic has been completed and is described in Jorge-Luis Sacchini's Ph.D thesis, which was defended in June 2011 [6] .

#### Native compilation of terms with primitive structures

Participants : Mathieu Boespflug [McGill University, Canada] , Maxime Dénès, Benjamin Grégoire.

We kept working on the integration of the native compiler of the Ocaml language into a scheme for the efficient reduction of terms in the calculus of inductive constructions. This work is described in a publication at the conference CPP11 in Taiwan [14] .