Consistency and Completeness of Rewriting in the Calculus of Constructions

Abstract

Adding rewriting to a proof assistant based on the Curry-Howard isomorphism,such as Coq, may greatly improve usability of the tool. Unfortunately adding anarbitrary set of rewrite rules may render the underlying formal systemundecidable and inconsistent. While ways to ensure termination and confluence,and hence decidability of type-checking, have already been studied to someextent, logical consistency has got little attention so far. In this paper weshow that consistency is a consequence of canonicity, which in turn followsfrom the assumption that all functions defined by rewrite rules are complete.We provide a sound and terminating, but necessarily incomplete algorithm toverify this property. The algorithm accepts all definitions that followdependent pattern matching schemes presented by Coquand and studied by McBridein his PhD thesis. It also accepts many definitions by rewriting, containingrules which depart from standard pattern matching.Comment: 20 page