Algebraic coherent confluence and higher globular Kleene algebras

Abstract

We extend the formalisation of confluence results in Kleene algebras to aformalisation of coherent confluence proofs. For this, we introduce thestructure of higher globular Kleene algebra, a higher-dimensionalgeneralisation of modal and concurrent Kleene algebra. We calculate a coherentChurch-Rosser theorem and a coherent Newman's lemma in higher Kleene algebrasby equational reasoning. We instantiate these results in the context of higherrewriting systems modelled by polygraphs