Layer Systems for Confluence—Formalized

Abstract

Toyama's theorem states that the union of two confluent term rewrite systems with disjoint signatures is again confluent. This is a fundamental result in term rewriting, and several proofs appear in the literature. The underlying proof technique has been adapted to prove further results like persistence of confluence (if a many-sorted term rewrite system is confluent, then the underlying unsorted system is confluent) or the preservation of confluence by currying. In this paper we present a formalization of modularity and related results in Isabelle/HOL. The formalization is based on layer systems, which cover modularity, persistence, currying (and more) in a single framework. The persistence result has been integrated into the certifier CeTA and the confluence tool CSI, allowing us to check confluence proofs based on persistent decomposition, of which modularity is a special case.

A verified ground confluence tool for linear variable-separated rewrite systems in Isabelle/HOL

“…The proof of the if direction in [20] relies on the fact that confluence is preserved under signature extension. Signature extension is a special case of modularity (Toyama [23]), a celebrated result in term rewriting which has recently been formalized in the context of the more general layer framework [12]. Because we deal with LV-TRSs whose left-hand sides may be variables we cannot reuse the formalization and hence we opted for a simple direct proof.…”