Squier's coherence theorem and its generalisations provide a categorical interpretation of contracting homotopies via confluent and terminating rewriting. In particular, this approach relates standardisation to coherence results in the context of higher dimensional rewriting systems. On the other hand, modal Kleene algebras (MKAs) have provided a description of properties of abstract rewriting systems, and classic (one-dimensional) consistency results have been formalised in this setting. In this talk, I will recall the notion of higher Kleene algebra, introduced as an extension of MKAs, and which provide a formal setting for reasoning about (higher dimensional) coherence proofs for abstract rewriting systems. In this setting, I will describe and sketch a proof of the Church-Rosser theorem with higher-dimensional witnesses, and briefly explain how Squier's coherence theorem is formalised.