  This article is concerned with automating the decreasing diagrams technique
of van Oostrom for establishing confluence of term rewrite systems. We study
abstract criteria that allow to lexicographically combine labelings to show
local diagrams decreasing. This approach has two immediate benefits. First, it
allows to use labelings for linear rewrite systems also for left-linear ones,
provided some mild conditions are satisfied. Second, it admits an incremental
method for proving confluence which subsumes recent developments in automating
decreasing diagrams. The techniques proposed in the article have been
implemented and experimental results demonstrate how, e.g., the rule labeling
benefits from our contributions.
