Certified Equational Reasoning via Ordered Completion

On the one hand, equational reasoning is a fundamental part of automated theorem proving with ordered completion as a key technique. On the other hand, the complexity of corresponding, often highly optimized, automated reasoning tools makes implementations inherently error-prone. As a remedy, we provide a formally verified certifier for ordered completion based techniques. This certifier is code generated from an accompanying Isabelle/HOL formalization of ordered rewriting and ordered completion incorporating an advanced ground joinability criterion. It allows us to rigorously validate generated proof certificates from several domains: ordered completion, satisfiability in equational logic, and confluence of conditional term rewriting.

[1]  Geoff Sutcliffe.  The TPTP Problem Library and Associated Infrastructure , 2009, Journal of Automated Reasoning.

[2]  Thomas Hillenbrand,et al.  On using ground joinable equations in equational theorem proving , 2003, J. Symb. Comput..

[3]  Christian Sternagel,et al.  Certifying Confluence of Almost Orthogonal CTRSs via Exact Tree Automata Completion , 2016, FSCD.

[4]  Christian Sternagel,et al.  Certified Ordered Completion , 2018, ArXiv.

[5]  Haruhiko Sato,et al.  Encoding Dependency Pair Techniques and Control Strategies for Maximal Completion , 2015, CADE.

[6]  Ryan Wisnesky,et al.  Algebraic data integration* , 2015, Journal of Functional Programming.

[7]  Georg Moser,et al.  MædMax: A Maximal Ordered Completion Tool , 2018, IJCAR.

[8]  Paliath Narendran,et al.  On Ground-Confluence of Term Rewriting Systems , 1990, Inf. Comput..

[9]  Aart Middeldorp,et al.  Conditional Confluence (System Description) , 2014, RTA-TLCA.

[10]  Dagmar Knorr.  Akademisches Schreiben lehren und lernen – Spektren einer prozessorientierten Schreibdidaktik , 2019 .

[11]  Larry Wos,et al.  Short Single Axioms for Boolean Algebra , 2002, Journal of Automated Reasoning.

[12]  Tobias Nipkow,et al.  Ordered Rewriting and Confluence , 1990, CADE.

[13]  Nao Hirokawa,et al.  Infinite Runs in Abstract Completion , 2017, FSCD.

[14]  René Thiemann,et al.  Formalizing Knuth-Bendix Orders and Knuth-Bendix Completion , 2013, RTA.

[15]  René Thiemann,et al.  The Certification Problem Format , 2014, UITP.

[16]  Thomas Sternagel,et al.  Recording Completion for Certificates in Equational Reasoning , 2015, CPP.

[17]  Tobias Nipkow,et al.  Code Generation via Higher-Order Rewrite Systems , 2010, FLOPS.

[18]  Lawrence Charles Paulson,et al.  Isabelle/HOL: A Proof Assistant for Higher-Order Logic , 2002 .

[19]  Sarah Winkler,et al.  A Ground Joinability Criterion for Ordered Completion∗ , 2017 .

[20]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[21]  William McCune,et al.  Solution of the Robbins Problem , 1997, Journal of Automated Reasoning.

[22]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[23]  Heiko Becker,et al.  A Transfinite Knuth-Bendix Order for Lambda-Free Higher-Order Terms , 2017, CADE.