Daria Walukiewicz-Chrzaszcz ; Jacek Chrzaszcz - Consistency and Completeness of Rewriting in the Calculus of Constructions

lmcs:1141 - Logical Methods in Computer Science, September 15, 2008, Volume 4, Issue 3 - https://doi.org/10.2168/LMCS-4(3:8)2008
Consistency and Completeness of Rewriting in the Calculus of Constructions

Authors: Daria Walukiewicz-Chrzaszcz ; Jacek Chrzaszcz

    Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules are complete. We provide a sound and terminating, but necessarily incomplete algorithm to verify this property. The algorithm accepts all definitions that follow dependent pattern matching schemes presented by Coquand and studied by McBride in his PhD thesis. It also accepts many definitions by rewriting, containing rules which depart from standard pattern matching.

    Volume: Volume 4, Issue 3
    Published on: September 15, 2008
    Accepted on: June 25, 2015
    Submitted on: February 1, 2007
    Keywords: Computer Science - Logic in Computer Science,Computer Science - Symbolic Computation,F.4.1,F.4.2

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