We extend the formalisation of confluence results in Kleene algebras to a
formalisation of coherent proofs by confluence. To this end, we introduce the
structure of modal higher-dimensional globular Kleene algebra, a
higher-dimensional generalisation of modal and concurrent Kleene algebra. We
give a calculation of a coherent Church-Rosser theorem and Newman's lemma in
higher-dimensional Kleene algebras. We interpret these results in the context
of higher-dimensional rewriting systems described by polygraphs.Comment: Pre-print (second version