zbMATH — the first resource for mathematics

Semi-intelligible Isar proofs from machine-generated proofs. (English) Zbl 1356.68178
Summary: Sledgehammer is a component of the Isabelle/HOL proof assistant that integrates external automatic theorem provers (ATPs) to discharge interactive proof obligations. As a safeguard against bugs, the proofs found by the external provers are reconstructed in Isabelle. Reconstructing complex arguments involves translating them to Isabelle’s Isar format, supplying suitable justifications for each step. Sledgehammer transforms the proofs by contradiction into direct proofs; it iteratively tests and compresses the output, resulting in simpler and faster proofs; and it supports a wide range of ATPs, including E, LEO-II, Satallax, SPASS, Vampire, veriT, Waldmeister, and Z3.

68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)
03B35 Mechanization of proofs and logical operations
Full Text: DOI
[1] Alama, J.: Escape to Mizar from ATPs. In: Fontaine, P., Schmidt, R., Schulz, S. (eds.) PAAR-2012, EPiC, vol. 21, pp 3-11. EasyChair (2013)
[2] Armand, M., Faure, G., Grégoire, 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 (2011) · Zbl 1350.68223
[3] Autexier, S., Benzmüller, C., Fiedler, A., Horacek, H., Vo, Q.B.: Assertion-level proof representation with under-specification. Electr. Notes Theor. Comput. Sci. 93, 5-23 (2004) · Zbl 1271.68207
[4] Azmy, N., Weidenbach, C.: Computing tiny clause normal forms. In: Bonacina, M.P. (ed.) CADE-24, LNCS, vol. 7898, pp 109-125. Springer (2013) · Zbl 1381.68256
[5] Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp 19-99. Elsevier (2001) · Zbl 0993.03008
[6] Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanovic, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011, LNCS, vol. 6806, pp 171-177. Springer (2011)
[7] Barrett, C., Stump, A., Tinelli, C.: The SMT-LIB standard—Version 2.0. In: Gupta, A., Kroening, D. (eds.) SMT 2010 (2010)
[8] Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II—A cooperative automatic theorem prover for higher-order logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008, LNCS, vol. 5195, pp 162-170. Springer (2008) · Zbl 1165.68446
[9] Besson, F., Fontaine, P., Théry, L.: A flexible proof format for SMT: A proposal. In: Fontaine, P., Stump, A. (eds.) PxTP 2011, pp 15-26 (2011)
[10] Blanchette, J.C.: Automatic proofs and refutations for higher-order logic. Ph.D. thesis, Dept. of Informatics, Technische Universität München (2012)
[11] Blanchette, J.C.: Redirecting proofs by contradiction. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013, EPiC, vol. 14, pp 11-26. EasyChair (2013) · Zbl 1149.68402
[12] Blanchette, JC; Böhme, S; Paulson, LC, Extending sledgehammer with SMT solvers, J. Autom. Reason., 51, 109-128, (2013) · Zbl 1314.68272
[13] Blanchette, J.C., Böhme, S., Popescu, A., Smallbone, N.: Encoding monomorphic and polymorphic types. In: Piterman, N., Smolka, S. (eds.) TACAS 2013, LNCS, vol. 7795, pp 493-507. Springer (2013) · Zbl 1381.68259
[14] Blanchette, J.C., Popescu, A., Wand, D., Weidenbach, C.: More SPASS with Isabelle—Superposition with hard sorts and configurable simplification. In: Beringer, L., Felty, A. (eds.) ITP 2012, LNCS, vol. 7406, pp 345-360. Springer (2012) · Zbl 1360.68742
[15] Böhme, S.: Proving theorems of higher-order logic with SMT solvers. Ph.D. thesis, Dept. of Informatics, Technische Universität München (2012) · Zbl 1246.83140
[16] Böhme, S., Nipkow, T.: Sledgehammer: judgement day. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010, LNCS, vol. 6173, pp 107-121. Springer (2010) · Zbl 1291.68327
[17] Böhme, S., Weber, T.: Fast LCF-style proof reconstruction for Z3. In: Kaufmann, M., Paulson, L. (eds.) ITP 2010, LNCS, vol. 6172, pp 179-194. Springer (2010) · Zbl 1291.68328
[18] 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 (2009) · JFM 60.0020.02
[19] Bradley, AR; Manna, Z, Property-directed incremental invariant generation, Form. Asp. Comput., 20, 379-405, (2008) · Zbl 1149.68402
[20] Brown, C.E.: Satallax: an automatic higher-order prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012, LNCS, vol. 7364, pp 111-117. Springer (2012) · Zbl 1358.68250
[21] Brown, C.E.: Using Satallax to generate proof terms for conjectures in Coq. Presentation at AIPA 2012 (2012) · Zbl 0961.68116
[22] Caminati, M.: Re: [isabelle] sledgehammer, smt and metis. https://lists.cam.ac.uk/pipermail/cl-isabelle-users/2014-November/msg00128.html (2014) · Zbl 1021.68082
[23] Chaieb, A., Nipkow, T.: Verifying and reflecting quantifier elimination for Presburger arithmetic. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005, LNCS, vol. 3835, pp 367-380. Springer (2005) · Zbl 1143.03334
[24] Church, A, A formulation of the simple theory of types, J. Symb. Log., 5, 56-68, (1940) · JFM 66.1192.06
[25] Dahn, BI, Robbins algebras are Boolean: a revision of mccune’s computer-generated solution of robbins problem, J. Algebra, 208, 526-532, (1998) · Zbl 0919.06005
[26] Davis, M.: Obvious logical inferences. In: Hayes, P.J. (ed.) IJCAI ’81, pp 530-531. William Kaufmann (1981) · Zbl 1314.68272
[27] Diekmann, C.: Network security policy verification. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs. http://afp.sf.net/entries/Network_Security_Policy_Verification.shtml (2014)
[28] Fleury, M.: Translation of proofs provided by external provers. Internship report, Technische Universität München. http://perso.eleves.ens-rennes.fr/ mfleur01/documents/Fleury_internship2014.pdf (2014)
[29] Foster, S., Struth, G.: Regular algebras. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs. http://afp.sf.net/entries/Network_Security_Policy_Verification.shtml (2014)
[30] Gentzen, G, Untersuchungen über das logische schließen, Math. Z., 39, 176-210, (1935) · Zbl 0010.14501
[31] Gordon, M.J.C., Melham, T.F. (eds.): Introduction to HOL: A Theorem Proving Environment for Higher Order Logic. Cambridge University Press (1993) · Zbl 0779.68007
[32] Herbrand, J.: Thèses présentées à la faculté des sciences de paris pour obtenir le grade de docteur ès sciences mathématiques. Ph.D. thesis, Science Faculty, Université de Paris (1930)
[33] Hillenbrand, T; Buch, A; Vogt, R; Löchner, B, WALDMEISTER—high-performance equational deduction, J. Autom. Reason., 18, 265-270, (1997)
[34] Hoder, K., Kovacs, L., Voronkov, A.: Vampire usage and demo. Presentation at the Vampire tutorial at CADE-23. http://www.complang.tuwien.ac.at/lkovacs/Cade23_Tutorial_Slides/Session2_Slides.pdf (2011) · Zbl 0139.12303
[35] Huang, X.: Translating machine-generated resolution proofs into ND-proofs at the assertion level. In: Foo, N.Y., Goebel, R. (eds.) PRICAI ’96, LNCS, vol. 1114, pp 399-410. Springer (1996)
[36] Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muṅoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, no. CP-2003-212448 in NASA Technical Reports, pp 56-68 (2003)
[37] Jaśkowski, S, On the rules of suppositions in formal logic, Stud. Logica., 1, 5-32, (1934) · Zbl 0011.09702
[38] Kaliszyk, C., Urban, J.: PRocH: Proof reconstruction for HOL Light. In: Bonacina, M.P. (ed.) CADE-24, LNCS, vol. 7898, pp 267-273. Springer (2013) · Zbl 1381.68271
[39] Kaliszyk, C; Urban, J, Learning-assisted automated reasoning with flyspeck, J. Autom. Reason., 53, 173-213, (2014) · Zbl 1314.68283
[40] Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, IBM Research Symposia Series, pp 85-103. Plenum Press (1972)
[41] Klein, G., Nipkow, T., Paulson, L. (eds.): Archive of Formal Proofs. http://afp.sf.net/
[42] Knuth, D.E., Larrabee, T.L., Roberts, P.M.: Mathematical Writing, Mathematical Association of America (1989)
[43] Korovin, K.: Private communication (2013)
[44] Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: machine learning for Sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013, LNCS, vol. 7998, pp 35-50. Springer (2013) · Zbl 1317.68215
[45] Loveland, D.W.: Automated Theorem Proving: A Logical Basis. North-Holland (1978) · Zbl 0364.68082
[46] Matuszewski, R; Rudnicki, P, Mizar: the first 30 years, Mechanized Mathematics and Its Applications, 4, 3-24, (2005)
[47] Meier, A.: TRAMP: transformation of machine-found proofs into natural deduction proofs at the assertion level (system description). In: McAllester, D. (ed.) CADE-17, LNCS, vol. 1831, pp 460-464. Springer (2000) · Zbl 0963.68537
[48] Meng, J; Paulson, LC, Translating higher-order clauses to first-order clauses, J. Autom. Reason., 40, 35-60, (2008) · Zbl 1203.68188
[49] Meng, J; Paulson, LC, Lightweight relevance filtering for machine-generated resolution problems, J. Appl. Log., 7, 41-57, (2009) · Zbl 1183.68560
[50] Miller, D., Felty, A.: An integration of resolution and natural deduction theorem proving. In: AAAI-86, vol. I: Science, pp 198-202. Morgan Kaufmann (1986) · Zbl 0683.68081
[51] de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008, LNCS, vol. 4963, pp 337-340. Springer (2008)
[52] Nipkow, T, Equational reasoning in isabelle, Sci. Comput. Program., 12, 123-149, (1989) · Zbl 0683.68081
[53] Nipkow, T., Klein, G.: Concrete Semantics—With Isabelle/HOL. Springer (2014) · Zbl 0919.06005
[54] Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer (2002) · Zbl 0994.68131
[55] Pąk, K.: The methods of improving and reorganizing natural deduction proofs. In: MathUI10 (2010) · Zbl 1260.68379
[56] Pąk, K, Methods of lemma extraction in natural deduction proofs, J. Autom. Reason., 50, 217-228, (2013) · Zbl 1260.68379
[57] Pąk, K.: Improving legibility of natural deduction proofs is not trivial. Log. Meth. Comput. Sci. 10(3) (2014) · Zbl 1341.68196
[58] Paulson, L.C.: Isabelle: A Generic Theorem Prover. LNCS, vol. 828. Springer (1994) · Zbl 0825.68059
[59] Paulson, L.C.: Strategic principles in the design of Isabelle. In: CADE-15 Workshop on Strategies in Automated Deduction, pp 11-17 (1998)
[60] Paulson, LC, A generic tableau prover and its integration with isabelle, J. Univ. Comp. Sci., 5, 73-87, (1999) · Zbl 0961.68116
[61] Paulson, L.C.: Three years of experience with Sledgehammer, a practical link between automated and interactive theorem provers. In: Konev, B., Schmidt, R., Schulz, S. (eds.) PAAR-2010, pp 1-10 (2010)
[62] Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: Sutcliffe, G., Schulz, S., Ternovska, E. (eds.) IWIL-2010, EPiC, vol. 2, pp 1-11. EasyChair (2012)
[63] 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 (2007) · Zbl 1144.68364
[64] Pfenning, F.: Analytic and non-analytic proofs. In: Shostak, R.E. (ed.) CADE-7, LNCS, vol. 170, pp 393-413. Springer (1984) · Zbl 0547.03032
[65] Reynolds, A., Tinelli, C, de Moura, L.: Finding conflicting instances of quantified formulas in SMT. In: Claessen, K., Kuncak, V. (eds.) FMCAD 2014, pp 195-202. FMCAD Inc. (2014)
[66] Riazanov, A; Voronkov, A, The design and implementation of vampire, AI Commun., 15, 91-110, (2002) · Zbl 1021.68082
[67] Robinson, AJ, A machine-oriented logic based on the resolution principle, J. ACM, 12, 23-41, (1965) · Zbl 0139.12303
[68] Rudnicki, P, Obvious inferences, J. Autom. Reason., 3, 383-393, (1987) · Zbl 0641.68141
[69] Rudnicki, P., Urban, J.: Escape to ATP for Mizar. In: Fontaine, P., Stump, A. (eds.) PxTP 2011, pp 46-59 (2011) · Zbl 1314.68283
[70] Schulz, S.: System description: E 1.8. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19, LNCS, vol. 8312, pp 735-743. Springer (2013) · Zbl 1407.68442
[71] Smolka, S.J.: Robust, semi-intelligible Isabelle proofs from ATP proofs. B.Sc. thesis, Dept. of Informatics, Technische Universität München (2013)
[72] Smolka, S.J., Blanchette, J.C.: Robust, semi-intelligible Isabelle proofs from ATP proofs. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013, EPiC, vol. 14, pp 117-132. EasyChair (2013)
[73] Steckermeier, A.: Extending Isabelle/HOL with the equality prover Waldmeister. B.Sc. thesis, Dept. of Informatics, Technische Universität München (2014)
[74] Stump, A; Oe, D; Reynolds, A; Hadarean, L; Tinelli, C, SMT proof checking using a logical framework, J. Formal Meth. Sys. Design, 42, 91-118, (2013) · Zbl 1284.68521
[75] Sultana, N., Benzmu̇ller, C.: Understanding LEO-II’s proofs. In: Korovin, K., Schulz, S., Ternovska, E. (eds.) IWIL 2012, EPiC, vol. 22, pp 33-52. EasyChair (2013) · Zbl 1185.68636
[76] Sutcliffe, G.: System description: SystemOnTPTP. In: McAllester, D. (ed.) CADE-17, LNCS, vol. 1831, pp 406-410. Springer (2000) · Zbl 0963.68529
[77] Sutcliffe, G, The TPTP problem library and associated infrastructure—the FOF and CNF parts, v3.5.0, J. Autom. Reason., 43, 337-362, (2009) · Zbl 1185.68636
[78] Sutcliffe, G, The CADE-24 automated theorem proving system competition—CASC-24, AI Commun., 27, 405-416, (2014)
[79] Sutcliffe, G., Zimmer, J., Schulz, S.: TSTP data-exchange formats for automated theorem proving tools. In: Zhang, W., Sorge, V. (eds.) Distributed Constraint Problem Solving and Reasoning in Multi-Agent Systems, Frontiers in Artificial Intelligence and Applications, vol. 112, pp 201-215. IOS Press (2004) · JFM 60.0846.02
[80] Thiemann, R.: Computing N-th roots using the Babylonian method. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs. http://afp.sf.net/entries/Sqrt_Babylonian.shtml (2013)
[81] Urban, J., Sutcliffe, G., Trac, S., Puzis, Y.: Combining Mizar and TPTP semantic presentation and verification tools. In: A. Grabowski, A. Naumowicz (eds.) Computer Reconstruction of the Body of Mathematics, Studies in Logic, Grammar and Rhetoric, vol. 18(31), pp. 121-136. University of Białystok (2009) · Zbl 1185.68636
[82] Wampler-Doty, M.: A complete proof of the Robbins conjecture. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs. http://afp.sf.net/entries/Robbins-Conjecture.shtml (2010)
[83] Weber, T.: Sat-based finite model generation for higher-order logic. Ph.D. thesis, Dept. of Informatics, Technische Universität München (2008)
[84] Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE-22, LNCS, vol. 5663, pp 140-145. Springer (2009)
[85] Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A. (eds.) TPHOLs 1997, LNCS, vol. 1275, pp 307-322. Springer (1997)
[86] Wenzel, M.: Isabelle/Isar—A generic framework for human-readable proof documents. In: In: R. Matuszewski, A. Zalewska (eds.) From Insight to Proof—Festschrift in Honour of Andrzej Trybulec, Studies in Logic, Grammar and Rhetoric, vol. 10(23). University of Białystok (2007)
[87] Wickerson, J., Dodds, M., Parkinson, M.J.: Ribbon proofs for separation logic. In: Felleisen, M., Gardner, P. (eds.) ESOP 2013, LNCS, vol. 7792, pp 189-208. Springer (2013) · Zbl 1381.68065
[88] Zimmer, J., Meier, A., Sutcliffe, G., Zhan, Y.: Integrated proof transformation services. In: Benzmüller, C., Windsteiger, W. (eds.) IJCAR WS 7 (2004) · Zbl 0641.68141
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.