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.
It is well known that (ground) confluence is a decidable property of ground term rewrite systems, and that this extends to larger classes. Here we present a formally verified ground confluence checker for linear, variable-separated rewrite systems. To this end, we formalize procedures for ground tree transducers and so-called RR n relations. The ground confluence checker is an important milestone on the way to formalizing the decidability of the first-order theory of ground rewriting for linear, variable-separated rewrite systems. It forms the basis for a formalized confluence checker for leftlinear, right-ground systems.
We introduce layer systems for proving generalizations of the modularity of
confluence for first-order rewrite systems. Layer systems specify how terms can
be divided into layers. We establish structural conditions on those systems
that imply confluence. Our abstract framework covers known results like
modularity, many-sorted persistence, layer-preservation and currying. We
present a counterexample to an extension of persistence to order-sorted
rewriting and derive new sufficient conditions for the extension to hold. All
our proofs are constructive
CSI is a strong automated confluence prover for rewrite systems which has been in development since 2010. In this paper we report on recent extensions that make CSI more powerful, secure, and useful. These extensions include improved confluence criteria but also support for uniqueness of normal forms. Most of the implemented techniques produce machine-readable proof output that can be independently verified by an external tool, thus increasing the trust in CSI. We also report on CSIˆho, a tool built on the same framework and similar ideas as CSI that automatically checks confluence of higher-order rewrite systems.
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