Labelings for Decreasing Diagrams

TL;DR

This paper advances the automation of decreasing diagrams for confluence proofs in term rewrite systems by developing combined labelings and an incremental approach, with implementation and experimental validation.

Contribution

It introduces a method to lexicographically combine labelings for decreasing diagrams and an incremental confluence proof technique, enhancing automation and applicability.

Findings

Rule labeling benefits from the new approach

Incremental method simplifies confluence proofs

Experimental results validate the effectiveness

Abstract

This article is concerned with automating the decreasing diagrams technique of van Oostrom for establishing confluence of term rewrite systems. We study abstract criteria that allow to lexicographically combine labelings to show local diagrams decreasing. This approach has two immediate benefits. First, it allows to use labelings for linear rewrite systems also for left-linear ones, provided some mild conditions are satisfied. Second, it admits an incremental method for proving confluence which subsumes recent developments in automating decreasing diagrams. The techniques proposed in the article have been implemented and experimental results demonstrate how, e.g., the rule labeling benefits from our contributions.