On abstract normalisation beyond neededness

TL;DR

This paper investigates the normalization of multistep strategies in abstract rewrite systems, focusing on necessary sets of redexes, and applies the results to PPC and lambda-calculus with parallel-or.

Contribution

It proves a normalization theorem for multistep strategies reducing necessary and never-gripping sets of redexes in ARS, extending to non-sequential systems.

Findings

Normalisation theorem for ARS with multistep strategies

Application to PPC calculus of patterns

Application to lambda-calculus with parallel-or

Abstract

We study normalisation of multistep strategies, strategies that reduce a set of redexes at a time, focussing on the notion of necessary sets, those which contain at least one redex that cannot be avoided in order to reach a normal form. This is particularly appealing in the setting of non-sequential rewrite systems, in which terms that are not in normal form may not have any needed redex. We first prove a normalisation theorem for abstract rewrite systems (ARS), a general rewriting framework encompassing many rewriting systems developed by P-A.Melli\`es in his PhD thesis. The theorem states that multistep strategies reducing so called necessary and never-gripping sets of redexes at a time are normalising in any ARS. Gripping refers to an abstract property reflecting the behavior of higher-order substitution. We then apply this result to the particular case of PPC, a calculus of patterns and to the lambda-calculus with parallel-or.