Termination of rewrite relations on $\lambda$-terms based on Girard's notion of reducibility

TL;DR

This paper extends Girard's reducibility notion to establish termination of diverse rewrite relations on λ-terms, offering a powerful criterion for higher-order rewriting systems.

Contribution

It introduces a generalized termination proof method based on computability closure applicable to various higher-order rewriting frameworks.

Findings

Proves termination of rewriting modulo equational theories

Extends Girard's reducibility to higher-order rewriting

Provides a unified termination criterion for multiple systems

Abstract

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems.