Implementation of Bourbaki's Elements of Mathematics in Coq: Part Two, From Natural Numbers to Real Numbers

This paper describes a formalization of the first book of the series ``Elements of Mathematics'' by Nicolas Bourbaki, using the Coq proof assistant. In a first paper published in this journal, we presented the axioms and basic constructions (corresponding to a part of the first two chapters of book I, theory of sets). We discuss here the set of integers (third chapter of book I, theory of set), the sets Z and Q (first chapter of book II, Algebra) and the set of real numbers (Chapter 4 of book III, General topology). We start with a comparison of the Bourbaki approach, the Coq standard library, and the Ssreflect library, then present our implementation