From formal proofs to mathematical proofs: a safe, incremental way for   building in first-order decision procedures
  We investigate here a new version of the Calculus of Inductive Constructions (CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary first-order decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated first-order decision procedures that can be taken from the shelves provided they deliver a proof certificate. The soundness of the whole system becomes an incremental property following from the soundness of the certificate checkers and that of the kernel. A detailed example shows that the resulting style of proofs becomes closer to that of the working mathematician. 
Fr\'ed\'eric Blanqui (INRIA Lorraine - LORIA), Jean-Pierre Jouannaud   (LIX, INRIA Saclay Ile de France), Pierre-Yves Strub ...