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SMTCoq: Mixing Automatic and Interactive Proof Technologies

  • Chantal KellerEmail author
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Part of the Mathematics Education in the Digital Era book series (MEDE, volume 14)

Abstract

SMTCoq is a plugin for the Coq interactive theorem prover to work in conjunction with automated theorem provers based on Boolean Satisfiability (SAT) and Satisfiability Modulo Theories (SMT), in an efficient and expressive way. As such, it allows one to formally establish, in a proof assistant, mathematical results relying on large combinatorial properties that require automatic Boolean reasoning. To this end, the huge SAT proofs coming from such problems can be safely checked in Coq and combined with standard mathematical Coq developments in a generic and modular way, for instance to obtain a formal proof of the Erdős Discrepancy Conjecture. To achieve this objective with the same degree of safety as Coq itself, SMTCoq communicates with SAT and SMT solvers that, in addition to a yes/no answer, can output traces of their internal proof search. The heart of SMTCoq is thus a certified, efficient and modular checker for such traces expressed in a format that can encompass most aspects of SMT reasoning. Preprocessors—that need not be certified—for proof traces coming from the state-of-the-art SMT solvers CVC4 and veriT and SAT solvers zChaff and Glucose are implemented. Coq can thus work in conjunction with widely used provers. From a proof assistant perspective, SMTCoq also provides a mechanism to let Coq users enjoy automation provided by external provers.

Notes

Acknowledgements

This work would be impossible without the help from past, present and future contributors of SMTCoq. Past and present contributors are Mikäl Armand, Clark Barett, Valentin Blot, Burak Ekici, Germain Faure, Benjamin Grégoire, Guy Katz, Tianyi Liang, Alain Mebsout, Andrew Reynolds, Laurent Théry, Cesare Tinelli and Benjamin Werner. Contributors to the certification of the Erdős Discrepancy Conjecture include Maxime Dénès and Pierre-Yves Strub.

The author thanks Véronique Benzaken and Évelyne Contejean for providing good feedback in the choice of the examples.

The author finally thanks the editors for their invitation to contribute to this book, and the reviewers and editors for their valuable feedback.

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Authors and Affiliations

  1. 1.LRI, University of Paris-Sud, CNRS UMR 8623, Université Paris-SaclayOrsay CedexFrance

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