De Bruijn’s weak diamond property revisited

In this paper we revisit an unpublished but influential technical report from 1978 by N.G. de Bruijn, written in the framework of the Automath project. This report describes a technique for proving confluence of abstract reduction systems, called the weak diamond property. It paved the way for the powerful technique developed by Van Oostrom to prove confluence of abstract reduction systems, called decreasing diagrams. We first revisit in detail De Bruijn’s old proof, providing a few corrections and hints for understanding. We find that this original criterion and proof technique are still worthwhile. Next, we establish that De Bruijn’s confluence criterion can be used to derive the decreasing diagrams theorem (the reverse was already known). We also provide a short proof of decreasing diagrams in the spirit of De Bruijn. We finally address the issue of completeness of this method.