Composants mathématiques pour la théorie des groupes

Sidi Ould Biha 1
1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Formal proof systems have evolved considerably in recent years. Success stories like formal proofs of the four color theorem or the prime numbers theorem have shown that formal proof systems have reached a level of maturity that enables them to tackle non-trivial mathematical problems. Nevertheless, the use of formal proof systems in mathematics is very limited. One of the reasons that explains this situation is the lack of formal proof libraries. This thesis focuses on the development of mathematical components for finite groups theory. It is part of a project that aims to formalize the Feit-Thompson theorem on the classification of finite groups. The main goal of this work is to apply software engineering techniques to facilitate the reuse and organization of large scale formal mathematics developments like the formalization of the Feit-Thompson theorem. This thesis presents the first formalization of the Cayley-Hamilton theorem on polynomials and matrices. It presents also developments on the representation theory of finite groups which is a necessary component for the formalization of the Feit-Thompson theorem. In particular, it presents a formalization of the theory of modules over a field or an algebra and a formalization of the Maschke theorem. These developments have been done in the Coq system and with the SSReflect extension.
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Sidi Ould Biha. Composants mathématiques pour la théorie des groupes. Génie logiciel [cs.SE]. Université Nice Sophia Antipolis, 2010. Français. ⟨tel-00493524⟩

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