Abstract

 
 
 A theory for Abstract Reduction Systems (ARS) in the proof assistant PVS (Prototype Verification System) called ars is described. Adequate specifications of basic definitions and notions of the theory of ARSs such as reduction, confluence and normal form are given and well-known results formalized. The formalizations include non trivial results of the theory of ARSs such as the correctness of the principle of Noetherian Induction, Newman’s Lemma and its generalizations, and Commutation Lemmas, among others. Although term rewriting proving technologies have been provided in several specification languages and proof assistants, to our knowledge, before the development presented in this paper there was no complete formalization of an abstract reduction theory in PVS. This makes relevant the presented ars specification as the basis of a PVStheory called trs for the general treatment of Term Rewriting Systems.
 
 
Highlights
Concepts and properties related with Abstract Reduction Systems (ARS) and Term Rewriting Systems (TRS) have been specified in several proof assistants, e.g., Rewriting Rule Laboratory (RRL) [9], ACL2 [18], Coq [8], Isabelle [14], BoyerMoore [19], Otter [4] among others
These rewriting based hardware specifications are synthesized to commercial reconfigurable hardware by applying the system FELIX [10] and their correctness is verified over the proof assistant PVS after translating the rewriting specification to a corresponding logic theory with the system SAEPTUM [2]
On the one hand ars is built over the PVS theory for binary relations being the closures specified in terms of “iteration” of the binary relations
Summary
Concepts and properties related with Abstract Reduction Systems (ARS) and Term Rewriting Systems (TRS) have been specified in several proof assistants, e.g., RRL [9], ACL2 [18], Coq [8], Isabelle [14], BoyerMoore [19], Otter [4] among others. The last mentioned step should be improved by making available a full theory of rewriting methods in PVS, that to our knowledge before our full PVS development for TRS reported in [5] was not available in this proof assistant With this motivation, this paper introduces a PVS theory called ars for dealing with properties of ARSs. Basic ARS notions are adequately specified in such a way that non elementary proof techniques such as Noetherian induction are straightforwardly applicable. Well-known results that are considered proof benchmarks such as Newman’s, Yokouchi’s and commutation Lemma are verified [6] These specifications are built over PVS theories for sets and relations. The files of this theory are available at www.mat.unb.br/∼ayala/TCgroup
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