Reductions in Higher-Order Rewriting and Their Equivalence

TL;DR

This paper extends proof terms to higher-order rewriting, allowing nested composition, and introduces equivalence notions and a standardization procedure to analyze and compare higher-order reductions.

Contribution

It proposes a new notion of higher-order proof terms called rewrites that support nested composition and establishes their equivalence and standardization methods.

Findings

Permutation and projection equivalences coincide.

A standardization procedure computes canonical rewrite representatives.

Supports nested composition in higher-order proof terms.

Abstract

Proof terms are syntactic expressions that represent computations in term rewriting. They were introduced by Meseguer and exploited by van Oostrom and de Vrijer to study equivalence of reductions in (left-linear) first-order term rewriting systems. We study the problem of extending the notion of proof term to higher-order rewriting, which generalizes the first-order setting by allowing terms with binders and higher-order substitution. In previous works that devise proof terms for higher-order rewriting, such as Bruggink's, it has been noted that the challenge lies in reconciling composition of proof terms and higher-order substitution (\b{eta}-equivalence). This led Bruggink to reject "nested" composition, other than at the outermost level. In this paper, we propose a notion of higher-order proof term we dub rewrites that supports nested composition. We then define two notions of equivalence on rewrites, namely permutation equivalence and projection equivalence, and show that they coincide. We also propose a standardization procedure, that computes a canonical representative of the permutation equivalence class of a rewrite.