Automated Confluence Proof by Decreasing Diagrams based on Rule-Labelling

Abstract

Decreasing diagrams technique (van Oostrom, 1994) is a technique that
can be widely applied to prove confluence of rewrite systems.  To
directly apply the decreasing diagrams technique to prove confluence
of rewrite systems, rule-labelling heuristic has been proposed by van
Oostrom (2008).  We show how constraints for ensuring confluence of
term rewriting systems constructed based on the rule-labelling
heuristic are encoded as linear arithmetic constraints suitable for
solving the satisfiability of them by external SMT solvers.  We point
out an additional constraint omitted in (van Oostrom, 2008) that is
needed to guarantee the soundness of confluence proofs based on the
rule-labelling heuristic extended to deal with non-right-linear rules.
We also present several extensions of the rule-labelling heuristic by
which the applicability of the technique is enlarged.