Assia Mahboubi

A machine-checked proof of the odd order theorem

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.

An introduction to small scale reflection in Coq

This tutorial presents the SSReflect extension to the Coq system. This extension consists of an extension to the Coq language of script, and of a set of libraries, originating from the formal proof of the Four Color theorem. This tutorial proposes a guided tour in some of the basic libraries distributed in the SSReflect package. It focuses on the application of the small scale reflection methodology to the formalization of finite objects in intuitionistic type theory.

A modular formalisation of finite group theory

In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a long-term effort to formalise the Feit-Thompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most compositional way.

Computer-checked mathematics: a formal proof of the odd order theorem

The Odd Order Theorem is a landmark result in finite group theory, due to W. Feit and J. G. Thompson [1], which states that every finite group of odd order is solvable. It is famous for its crucial role in the classification of finite simple groups, for the novel methods introduced by its original proof but also for the striking contrast between the simplicity of its statement and the unusual length and complexity of its proof. After a six year collaborative effort, we managed to formalize and machine-check a complete proof of this theorem [2] using the Coq proof assistant [3]. The resulting collection of libraries of formalized mathematics covers a wide variety of topics, mostly in algebra, as this proof relies on a sophisticated combination of local analysis and character theory. In this tutorial we comment on the role played by the different features of the proof assistant, from the meta-theory of its underlying logic to the implementation of its various components. We will also discuss some issues raised by the translation of mathematical textbooks into formal libraries and the perspectives it opens on the use of a computer to do mathematics.

The rooster and the butterflies

This paper describes a machine-checked proof of the Jordan-Holder theorem for finite groups. This purpose of this description is to discuss the representation of the elementary concepts of finite group theory inside type theory. The design choices underlying these representations were crucial to the successful formalization of a complete proof of the Odd Order Theorem with the Coq system.