Abstract. This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.

Abstract. This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.

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This paper provides a gentle introduction to the art of programming type inference with the mechanism of Canonical Structures. Programmable type inference has been one of the key ingredients for the successful formalization of the Odd Order Theorem using the Coq proof assistant. The paper concludes comparing the language of Canonical Structures to the one of Type Classes and Unification Hints.

Une formalisation modulaire des groupes finis Résumé : Ce rapport présente une formalisation de théorie des groupesélémentaires réalisée dans le système Coq. Ce travail est la premièreétape d'un projet d'envergure, qui a pour but de construire une preuve formelle du théorème de Feit-Thompson. Comme nos développements formels ultérieurs reposeront de façon cruciale sur cette base, nous avons consacré un soin particulierà la modularité de ces fondations.