Algebraic coherent confluence and higher globular Kleene algebras

Abstract

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.