Consistency and Completeness of Rewriting in the Calculus of Constructions

TL;DR

This paper investigates the conditions under which adding rewriting rules to the Calculus of Constructions maintains logical consistency, proposing an algorithm to verify the completeness of rewrite functions.

Contribution

It establishes that consistency follows from canonicity, which depends on the completeness of rewrite functions, and introduces a sound, terminating algorithm to verify this property.

Findings

The algorithm accepts definitions following dependent pattern matching.

It also accepts many non-standard rewriting rules.

Consistency is linked to the canonicity of rewrite functions.

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 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.