Abstract: Reliable Reconstruction of Fine-Grained Proofs in a Proof Assistant

We present a reliable and fast reconstruction of proofs generated by the SMT solver veriT in Isabelle, building upon the work by Barbosa et al. The fine-grained proof format of veriT makes the reconstruction simple and efficient. For typical proof steps, such as arithmetic reasoning and skolemization, our reconstruction is able to avoid expensive search. We compare our procedure with the existing Z3 integration and show similar levels of robustness while solving more problems. Skipping some steps of the very detailed proofs is possible and improves the performance of proof checking.

[1]  Dennis Clark,et al.  The Prime Number Theorem , 2002 .

[2]  Cesare Tinelli,et al.  Finding conflicting instances of quantified formulas in SMT , 2014, 2014 Formal Methods in Computer-Aided Design (FMCAD).

[3]  Lawrence C. Paulson,et al.  An Isabelle/HOL Formalisation of Green’s Theorem , 2016, Journal of Automated Reasoning.

[4]  Pascal Fontaine,et al.  SMT Solvers for Rodin , 2012, ABZ.

[5]  K. Rustan M. Leino,et al.  Trigger Selection Strategies to Stabilize Program Verifiers , 2016, CAV.

[6]  Cezary Kaliszyk,et al.  MaSh: Machine Learning for Sledgehammer , 2013, ITP.

[7]  Sascha Böhme,et al.  Fast LCF-Style Proof Reconstruction for Z3 , 2010, ITP.

[8]  Clark W. Barrett,et al.  The SMT-LIB Standard Version 2.0 , 2010 .

[9]  Uwe Waldmann,et al.  Formalization of Bachmair and Ganzinger's Ordered Resolution Prover , 2018, Arch. Formal Proofs.

[10]  Benjamin Grégoire,et al.  A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses , 2011, CPP.

[11]  Kenneth L. McMillan,et al.  Interpolants from Z3 proofs , 2011, 2011 Formal Methods in Computer-Aided Design (FMCAD).

[12]  Pascal Fontaine,et al.  Scalable Fine-Grained Proofs for Formula Processing , 2017, CADE.

[13]  Cesare Tinelli,et al.  Extending SMT Solvers to Higher-Order Logic , 2019, CADE.

[14]  Michael J. C. Gordon,et al.  Edinburgh LCF: A mechanised logic of computation , 1979 .

[15]  Heiko Becker,et al.  Formalization of Knuth-Bendix Orders for Lambda-Free Higher-Order Terms , 2016, Arch. Formal Proofs.

[16]  Stephan Schulz,et al.  E - a brainiac theorem prover , 2002, AI Commun..

[17]  Cesare Tinelli,et al.  SMT proof checking using a logical framework , 2013, Formal Methods Syst. Des..

[18]  Pascal Fontaine,et al.  Revisiting Enumerative Instantiation , 2018, TACAS.

[19]  Manuel Eberl Elementary Facts About the Distribution of Primes , 2019, Arch. Formal Proofs.

[20]  René Thiemann,et al.  An Incremental Simplex Algorithm with Unsatisfiable Core Generation , 2018, Arch. Formal Proofs.

[21]  Tobias Nipkow,et al.  Sledgehammer: Judgement Day , 2010, IJCAR.

[22]  M. Fleury,et al.  Reconstructing veriT Proofs in Isabelle/HOL , 2019, PxTP.

[23]  Cesare Tinelli,et al.  SMTCoq: A Plug-In for Integrating SMT Solvers into Coq , 2017, CAV.

[24]  Clark W. Barrett,et al.  Cooperating Theorem Provers: A Case Study Combining HOL-Light and CVC Lite , 2006, Electron. Notes Theor. Comput. Sci..

[25]  Christopher L. Conway,et al.  Cvc4 , 2011, CAV.

[26]  Nikolaj Bjørner,et al.  Proofs and Refutations, and Z3 , 2008, LPAR Workshops.

[27]  Lawrence C. Paulson,et al.  Lightweight relevance filtering for machine-generated resolution problems , 2009, J. Appl. Log..

[28]  Sascha Böhme,et al.  Semi-intelligible Isar Proofs from Machine-Generated Proofs , 2015, Journal of Automated Reasoning.

[29]  Haniel Barbosa,et al.  Efficient Instantiation Techniques in SMT (Work In Progress) , 2016, PAAR@IJCAR.

[30]  Guillaume Burel,et al.  Expressing theories in the λΠ-calculus modulo theory and in the Dedukti system , 2016 .

[31]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[32]  Pascal Fontaine,et al.  veriT: An Open, Trustable and Efficient SMT-Solver , 2009, CADE.

[33]  Pascal Fontaine,et al.  Congruence Closure with Free Variables , 2017, TACAS.

[34]  Bruno Dutertre,et al.  Integrating Simplex with DPLL(T ) , 2006 .

[35]  Sascha Böhme,et al.  Proving Theorems of Higher-Order Logic with SMT Solvers , 2012 .

[36]  Lawrence C. Paulson,et al.  Extending Sledgehammer with SMT Solvers , 2011, Journal of Automated Reasoning.

[37]  Cesare Tinelli,et al.  Satisfiability Modulo Theories , 2021, Handbook of Satisfiability.