Abstract
We present a new methodology for exchanging unsatisfiability proofs between an untrusted SMT solver and a sceptical proof assistant with computation capabilities like Coq. We advocate modular SMT proofs that separate boolean reasoning and theory reasoning; and structure the communication between theories using Nelson-Oppen combination scheme. We present the design and implementation of a Coq reflexive verifier that is modular and allows for fine-tuned theory-specific verifiers. The current verifier is able to verify proofs for quantifier-free formulae mixing linear arithmetic and uninterpreted functions. Our proof generation scheme benefits from the efficiency of state-of-the-art SMT solvers while being independent from a specific SMT solver proof format. Our only requirement for the SMT solver is the ability to extract unsat cores and generate boolean models. In practice, unsat cores are relatively small and their proof is obtained with a modest overhead by our proof-producing prover. We present experiments assessing the feasibility of the approach for benchmarks obtained from the SMT competition.
This work was partly funded by the ANR DeCert, FNRAE ASCERT and Région Bretagne CertLogS projects.
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References
Armand, M., Faure, G., Gregoire, B., Keller, C., Théry, L., Werner, B.: A Modular Integration of SAT/SMT Solvers to Coq Through Proof Witnesses. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 135–150. Springer, Heidelberg (2011)
Armand, M., Grégoire, B., Spiwack, A., Théry, L.: Extending Coq with Imperative Features and its Application to SAT Verification. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 83–98. Springer, Heidelberg (2010)
Barret, C., Stump, A., Tinelli, C.: The SMT-LIB standard: Version 2.0 (2010)
Barrett, C.W., Tinelli, C.: CVC3. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 298–302. Springer, Heidelberg (2007)
Besson, F.: Fast Reflexive Arithmetic Tactics the Linear Case and Beyond. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 48–62. Springer, Heidelberg (2007)
Böhme, S., Weber, T.: Fast LCF-Style Proof Reconstruction for Z3. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 179–194. Springer, Heidelberg (2010)
Boulton, R.J.: Efficiency in a Fully-Expansive Theorem Prover. PhD thesis, University of Cambridge Computer Laboratory, Technical Report 337 (1994)
Bouton, T., de Oliveira, D.C.B., Déharbe, D., Fontaine, P.: veriT: An Open, Trustable and Efficient SMT-Solver. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 151–156. Springer, Heidelberg (2009)
Contejean, E., Corbineau, P.: Reflecting Proofs in First-Order Logic with Equality. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 7–22. Springer, Heidelberg (2005)
de Moura, L.M., Bjørner, N.: Proofs and Refutations, and Z3. In: LPAR 2008 Workshops: KEAPPA. CEUR-WS.org, vol. 418 (2008)
de Moura, L., Bjørner, N.S.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)
de Moura, L.M., Rueß, H., Shankar, N.: Justifying equality. ENTCS 125(3), 69–85 (2005)
de Moura, L., Rueß, H., Sorea, M.: Lazy Theorem Proving for Bounded Model Checking Over Infinite Domains. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 438–455. Springer, Heidelberg (2002)
Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52(3), 365–473 (2005)
Fontaine, P., Marion, J.-Y., Merz, S., Nieto, L.P., Tiu, A.F.: Expressiveness + Automation + Soundness: Towards Combining SMT Solvers and Interactive Proof Assistants. In: Hermanns, H. (ed.) TACAS 2006. LNCS, vol. 3920, pp. 167–181. Springer, Heidelberg (2006)
Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: ICFP 2002, pp. 235–246. ACM (2002)
Grégoire, B., Mahboubi, A.: Proving Equalities in a Commutative Ring Done Right in Coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 98–113. Springer, Heidelberg (2005)
Hurd, J.: Integrating Gandalf and HOL. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 311–322. Springer, Heidelberg (1999)
Leroy, X.: Formal certification of a compiler back-end or: programming a compiler with a proof assistant. In: POPL 2006, pp. 42–54. ACM (2006)
McLaughlin, S., Barrett, C., Ge, Y.: Cooperating theorem provers: A case study combining HOL-Light and CVC Lite. ENTCS 144(2), 43–51 (2006)
Necula, G.C.: Compiling with Proofs. PhD thesis, CMU (1998)
Necula, G.C., Lee, P.: Proof Generation in the Touchstone Theorem Prover. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 25–44. Springer, Heidelberg (2000)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1, 245–257 (1979)
Nieuwenhuis, R., Oliveras, A.: Proof-Producing Congruence Closure. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 453–468. Springer, Heidelberg (2005)
Paulson, L.C., Susanto, K.W.: Source-Level Proof Reconstruction for Interactive Theorem Proving. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 232–245. Springer, Heidelberg (2007)
Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. In: SC, pp. 4–13 (1991)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley (1998)
Stengle, G.: A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen 207(2), 87–97 (1973)
Weber, T., Amjad, H.: Efficiently checking propositional refutations in HOL theorem provers. J. Applied Logic 7(1), 26–40 (2009)
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Besson, F., Cornilleau, PE., Pichardie, D. (2011). Modular SMT Proofs for Fast Reflexive Checking Inside Coq. In: Jouannaud, JP., Shao, Z. (eds) Certified Programs and Proofs. CPP 2011. Lecture Notes in Computer Science, vol 7086. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25379-9_13
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