Section: Software and Platforms

CoqMT

Participants : Qian Wang [correspondant] , Jean-Pierre Jouannaud.

The proof-assistant Coq is based on a complex type theory, which resulted from various extensions of the Calculus of Constructions studied independently from each other. With the collaboration of Bruno Barras, we decided to address the challenge of proving the real type theory underlying Coq, and even, indeed, of its recent extension CoqMT developed in Formes by Pierre-Yves Strub. To this end, we have studied formally the theory CoqMTU, which extends the pure Calculus of Constructions by inductive types, a predicative hierarchy of universes, and a decidable theory T for some first-order inductive types. Recently, we were able to announce the complete certification of CoqMTU in Coq augmented with appropriate intuitionistic set-theoretic axioms in order to fight Gödel's incompleteness theorem˜[16] . As a consequence, Coq and CoqMTU are the first proof assistants, of which consistency (relative to intuitionistic set theory IZF augmented with the afore-mentioned axioms) is formally entirely proved (in Coq). While previous formal proofs for Coq and other proof assistants all assumed strong normalization, the present one proves strong normalization thanks to the new notion of strongly-normalizing model introduced by Bruno Barras. While consistency is done already, decidability of type-checking in CoqMTU remains to be done. This is a straightforward consequence for Coq, but a non-trivial task for CoqMTU because of the interaction between inductive types and the first-order theory T. It should however be done by the summer of 2014. We consider this work as a major scientific achievement of the team.