We report on the formalization of a sufficient condition for confluence of first-order left-linear rewrite systems within the proof assistant Isabelle/HOL. This criterion, originally proposed by Okui (1998), is based on simultaneous critical pairs, which finitely represent peaks consisting of a multi-step and a normal step. It properly subsumes the formalized result on development-closed critical pairs.

Several critical pair criteria are known that guarantee confluence of left-linear term rewrite systems. The correctness of most of these have been formalized in a proof assistant. An important exception has been the development closedness criterion of van Oostrom. Its proof requires a high level of understanding about overlapping redexes and descendants as well as several intermediate results related to these concepts. We present a formalization in the proof assistant Isabelle/HOL. The result has been integrated into the certifier CeTA.

Abstract Proof terms are a useful concept for comparing computations in term rewriting. We analyze proof terms with composition, with an eye towards automation. We revisit permutation equivalence and projection equivalence, two key notions presented in the literature. We report on the integration of proof terms with composition into ProTeM, a tool for manipulating proof terms.