Section: Scientific Foundations

Libraries of formalized mathematics

It is well known that advanced mathematics can play a crucial role in the design and correctness of sophisticated and sometimes critical software. In some cases, using a proof system is the only option to mechanize the correctness of such programs; this can require the formalization of a wide variety of mathematical theories, and a careful design of these formal libraries for them to be maintainable, combinable and reusable. Furthermore, the ability to formalize advanced contemporary mathematics is still a form of ultimate quality tests for proof systems, and also a way to gain visibility. One of our objectives is to make modern and large pieces of mathematics available as usable formal libraries. Recent examples of complex proofs (Four Color Theorem, Kepler conjecture, classification of finite groups, Fermat theorem) challenge the way the mathematical literature is refereed and published. We think that the development of these formal libraries of mathematics may also change the way certain mathematical result become accepted as theorems. Crafting large bodies of formalized mathematics is a challenging task. These libraries obey similar requirements as software : modularity and usability stem from appropriate data-structures, design patterns and corpus of lemmas. But the appropriate methodology leading to the relevant solutions is often far from obvious, and this is where research has to be done and know-how has to be gained. Up to recently, formal developments were seldom collaborative and rarely benefitted from reusable previous work. The maturity of proof assistants is now sufficient to envision a more modern conception of formal software, as required by large scale verification projects like T. Hales’ proof of the Kepler conjecture or the Feit-Thompson theorem. Several members of the TypiCal team are committed in such big formalization projects, or in more specific but related side projects.