Programmation fonctionnelle certifiée : L'extraction de programmes dans l'assistant Coq. (Certified functional programming : Program extraction within Coq proof assistant)

This work concerns the generation of programs which are certified to be correct byconstruction. These programs are obtained by extracting relevant information from construc-tive proofs made with the Coq proof assistant.Such a translation, named “extraction”, of constructive proofs into functional programsis not new, and corresponds to an isomorphism known as Curry-Howard’s. An extractiontool has been part of Coq assistant for a long time. But this old extraction tool sufferedfrom several limitations : in particular, some Coq proofs were refused by it, whereas someothers led to incorrect programs.In order to overcome these limitations, we built a completely new extraction tool for Coq,including both a new theory and a new implementation. Concerning theory, we developednew correctness proofs for this extraction mechanism. These new proofs are both complexand original. Concerning implementation, we focused on the generation of efficient andrealistic code, which can be integrated in large-scale software developments, using modulesand interfaces.Finally, we also present several case studies illustrating the capabilities of our new extrac-tion. For example, we describe the certification of a modular library of finite set structures,and the production of programs about real exact arithmetic, starting from a formalizationof constructive real analysis. These examples show the progress already achieved, even if thesituation is not perfect yet, in particular in the last study.