In this paper we present a further simplification of the proof of Toyama's result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms.
We then show that Toyama's modularity result scales up to rewriting modulo equations under natural assumptions met by all relations introduced in the litterature.