Final Report Summary - MATHFOR (Formalization of Constructive Mathematics) Correctness of software is a crucial issue and is becoming more and more important. One possible approach (complementary to others) to address this problem is the use of so called "proof-assistants", which are software programs checking the correctness of mathematical arguments. Some impressive projects show the relevance of this approach, e.g. the CompCert project of Leroy (INRIA) which develops in this way a formally verified C compiler.Another potential application of such proof assistants recently emerged in mathematics: to check the correctness of some mathematical arguments that are simply too long, or extremely complex, to be checked manually. Two recent achievement are the formal verification of the Feit-Thompson theorem by Gonthier and his team, and the formal verification of a recent proof of the Kepler conjecture, by Hales. In the past 6 years, a famous mathematician Vladimir Voevodsky, has stressed the importance of having proofs formally checked in his field of research, and suggested some extremely original ideas for improving these proof systems and making them possible to represent the (extremely abstract) mathematics he is using. One crucial issue however was if it was possible to combine these ideas while keeping the connections between reasoning and computations that have been used crucially in the verification of the C compiler, and in the checking of the Feit-Thompson theorem. The main result of this project has been to give a positive answer to this question, suggesting the design of new improved proof assistants.
