Confluence by Critical Pair Analysis

Knuth and Bendix showed that confluence of a terminating first-order rewrite system can be reduced to the joinability of its finitely many critical pairs. We show that this is still true of a rewrite system RT ∪ RNT such that RT is terminating and RNT is a left-linear, rank non-increasing, possibly non-terminating rewrite system. Confluence can then be reduced to the joinability of the critical pairs of RT and to the existence of decreasing diagrams for the critical pairs of RT inside RNT as well as for the rigid parallel critical pairs of RNT.