From Formal Proofs to Mathematical Proofs: A Safe, Incremental Way for Building in First-order Decision Procedures

Abstract

Abstract. 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.

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

“…Currently, one sometimes needs to introduce auxiliary functions or explicitly apply type casting functions to address this problem. As already mentioned, the use of rewriting or (certified) decision procedures in the equivalence on types solves many of these problems (Blanqui 2005;Blanqui et al 2008;Strub 2010a). This approach is adopted in Coq modulo theories (CoqMT), which is a new extension of Coq where type equivalence can be extended by decision procedures such as linear arithmetic (Strub 2010b).…”

“…We could perhaps benefit from a recent library of lemmas and tactics to reason on heterogeneous equalities (Hur 2009). But our preferred route will be to use an extension of Coq incorporating rewriting or decision procedures in the equivalence of types (Blanqui 2005;Blanqui et al 2008;Strub 2010a). Some recent experiments show that, with CoqMT (Strub 2010b), all casts can be removed † .…”

LF P

“…Systems supporting the Poincaré Principle [4], or Deduction Modulo [9], where derivation is separated from verification, can be directly incorporated in LFP . Similarly, we can abstractly subsume the system presented in [6], which addresses a specific instance of our problem: how to outsource the computation of a decision procedure in Type Theory in a sound and principled way via an abstract conversion rule.…”

Coq without Type Casts: A Complete Proof of Coq Modulo Theory

“…There are many ways to fix the problem: by adding rewrite rules to increase the computing ability [7]; by building in full extensionality to the price of loosing decidability [19,17]; by building in a restricted form of extensionality taking advantage of equations present in the typing context [8]; and by building in an even weaker form of extensionality restricted to an object level first-order theory [20]. The difficulty is indeed to find the right trade-off between the expressivity of the type system, the decidability of type-checking, and the efficiency of the implementation.…”