Confluence of Orthogonal Term Rewriting Systems in the Prototype Verification System

Formalizing the dependency pair criterion for innermost termination

“…This library is a development that already contains the basic elements of abstract reduction systems and TRS, such as reducibility, confluence and noetherianity regarding a given relation, notions of subterms and replacement, etc. Furthermore, this theory embraces several elaborate formalizations regarding such systems, such as confluence of abstract reduction systems (see [11]), the Critical Pair Theorem (see [7]) and orthogonal TRSs and their confluence (see [8]).…”

“…This work presents a complete formalization of the DP criterion for innermost reduction. The formalization extends the PVS library for TRSs (named also TRS) that encompasses the basic notions of rewriting as well as some elaborate results (e.g., [7], [8]). This library includes specifications of terms, positions, substitutions, abstract reduction relations, and term rewriting systems which are adequate for the development of formalizations that remain close to article and textbook proofs, as the one presented in this paper.…”

Unificação, confluência e tipos com interseção para sistemas de reescrita nominal

“…With these syntactic restrictions confluence of orthogonal rewriting theories is guaranteed [Rosen, 1973]. This result has been formalised for TRSs in a couple of proof assistants [Rocha-Oliveira et al, 2016, Thiemann, 2013.…”

“…In this section, from now on, the PVS theory TRS [Avelar et al, 2014, Galdino andAyala-Rincón, 2009] will be used to illustrate the PVS rules and some data structure. This theory includes the subtheory orthogonality developed in a previous work and presented in [Rocha-Oliveira et al, 2016], that is a formalisation of the well know theorem of confluence of orthogonal systems in the context of TRSs. As mentioned in the introduction, a similar result for Nominal Rewriting Systems is discussed in this thesis.…”

“…To do so, it would be necessary to provide the proper definitions of nominal rewriting using equivariant matching and closed nominal rewriting, with nominal equivariance-free matching. In the context of TRSs, confluence of orthogonal systems and the Critical Pair Criterion are formalised in PVS, as shown in [Galdino and Ayala-Rincón, 2010] and [Rocha-Oliveira et al, 2016].…”