Skip to main content

Semantics of Mizar as an Isabelle Object Logic

Abstract

We formally define the foundations of the Mizar system as an object logic in the Isabelle logical framework. For this, we propose adequate mechanisms to represent the various components of Mizar. We express Mizar types in a uniform way, provide a common type intersection operation, allow reasoning about type inhabitation, and develop a type inference mechanism. We provide Mizar-like definition mechanisms which require the same proof obligations and provide same derived properties. Structures and set comprehension operators can be defined as definitional extensions. Re-formalized proofs from various parts of the Mizar Library show the practical usability of the specified foundations.

References

  1. Abrial, J.: Modeling in Event-B—System and Software Engineering. Cambridge University Press, Cambridge (2010)

    MATH  Book  Google Scholar 

  2. Adams, M.: Proof auditing formalised mathematics. J. Formaliz. Reason. 9(1), 3–32 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Agerholm, S., Gordon, M.J.C.: Experiments with ZF set theory in HOL and Isabelle. In: Schubert, E.T., Windley, P.J., Alves-Foss, J. (eds.) Higher Order Logic Theorem Proving and Its Applications, Volume 971 of LNCS, pp. 32–45. Springer, Berlin (1995)

    MATH  Chapter  Google Scholar 

  4. Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reason. 52(2), 191–213 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  5. Asperti, A., Bancerek, G., Trybulec, A. (eds.): Mathematical Knowledge Management (MKM 2004), Volume 3119 of LNCS. Springer, Berlin (2004)

    Google Scholar 

  6. Bancerek, G.: Tarski’s classes and ranks. Formaliz. Math. 1(3), 563–567 (1990)

    Google Scholar 

  7. Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pąk, K.: The role of the Mizar Mathematical Library for interactive proof development in Mizar. J. Autom. Reason. (2017). https://doi.org/10.1007/s10817-017-9440-6

    MATH  Article  Google Scholar 

  8. Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pąk, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) Intelligent Computer Mathematics—International Conference, CICM 2015, Volume 9150 of LNCS, pp. 261–279. Springer, Berlin (2015)

    MATH  Google Scholar 

  9. Bancerek, G., Rudnicki, P.: A compendium of continuous lattices in MIZAR. J. Autom. Reason. 29(3–4), 189–224 (2002)

    MATH  Article  Google Scholar 

  10. Bancerek, G., Rudnicki, P.: Information retrieval in MML. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) Mathematical Knowledge Management, MKM 2003, Volume 2594 of LNCS, pp. 119–132. Springer, Berlin (2003)

    Google Scholar 

  11. Bancerek, G., Urban, J.: Integrated semantic browsing of the Mizar Mathematical Library for authoring Mizar articles. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) Mathematical Knowledge Management (MKM 2004), Volume 3119 of LNCS, pp. 44–57. Springer, Berlin (2004)

    Google Scholar 

  12. Barras, B., Tankink, C., Tassi, E.: Asynchronous processing of Coq documents: from the kernel up to the user interface. In: Urban, C., Zhang, X. (eds.) Interactive Theorem Proving, ITP 2015, Volume 9236 of LNCS, pp. 51–66. Springer, Berlin (2015)

    MATH  Google Scholar 

  13. Blanchette, J.C., Greenaway, D., Kaliszyk, C., Kühlwein, D., Urban, J.: A learning-based fact selector for Isabelle/HOL. J. Autom. Reason. 57(3), 219–244 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  14. Blanchette, J.C., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving, ITP 2010, Volume 6172 of LNCS, pp. 131–146. Springer, Berlin (2010)

    Google Scholar 

  15. Brown, C.E.: The Egal Manual (2014)

  16. Brown, C.E., Urban, J.: Extracting higher-order goals from the Mizar Mathematical Library. In: Kohlhase, M., Johansson, M., Miller, B.R., de Moura, L., Tompa, F.W. (eds.) Intelligent Computer Mathematics (CICM 2016), Volume 9791 of LNCS, pp. 99–114. Springer, Berlin (2016)

    Google Scholar 

  17. Byliński, C.: Introduction to categories and functors. Formaliz. Math. 1(2), 409–420 (1990)

    Google Scholar 

  18. Corbineau, P.: A declarative language for the Coq proof assistant. In: Miculan, M., Scagnetto, I., Honsell, F. (eds.) Types for Proofs and Programs, International Conference, TYPES 2007, Volume 4941 of LNCS, pp. 69–84. Springer, Berlin (2007)

    Google Scholar 

  19. Dahn, I.: Interpretation of a Mizar-like logic in first-order logic. In: Caferra, R., Salzer, G. (eds.) First-Order Theorem Proving (FTP 1998), Volume 1761 of LNCS, pp. 137–151. Springer, Berlin (1998)

    Google Scholar 

  20. Dahn, I., Wernhard, C.: First order proof problems extracted from an article in the Mizar Mathematical Library. In: Bonacina, M.P., Furbach, U. (eds.) First-Order Theorem Proving (FTP 1997), RISC-Linz Report Series No. 97–50, pp. 58–62. Johannes Kepler Universität, Linz (1997)

  21. Davis, M.: Obvious logical inferences. In: Hayes, P.J. (ed.) International Joint Conference on Artificial Intelligence (IJCAI 1981), pp. 530–531. William Kaufmann, Burlington (1981)

    Google Scholar 

  22. de Moura, L.M., Kong, S., Avigad, J., van Doorn, F., von Raumer, J.: The Lean theorem prover (system description). In: Felty, A.P., Middeldorp, A. (eds.) Conference on Automated Deduction, CADE 2015, Volume 9195 of LNCS, pp. 378–388. Springer, Berlin (2015)

    Google Scholar 

  23. Dunchev, C., Coen, C.S., Tassi, E.: Implementing HOL in an higher order logic programming language. In: Dowek, G., Licata, D.R., Alves, S. (eds.) Logical Frameworks and Meta-Languages Theory and Practice, LFMTP 2016, pp. 4:1–4:10. ACM, Albion (2016)

    Google Scholar 

  24. Elgot, C.C., Robinson, A.: Random-access stored-program machines, an approach to programming languages. J. ACM 11(4), 365–399 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  25. Felty, A.P., Gunter, E.L., Hannan, J., Miller, D., Nadathur, G., Scedrov, A.: Lambda-Prolog: an extended logic programming language. In: Lusk, E.L., Overbeek, R.A. (eds.) International Conference on Automated Deduction, CADE, Volume 310 of LNCS, pp. 754–755. Springer, Berlin (1988)

    Chapter  Google Scholar 

  26. Fitch, F.B.: Symbolic Logic. An Introduction. The Ronald Press Company, New York (1952)

    MATH  Google Scholar 

  27. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formaliz. Reason. 3(2), 153–245 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  29. Hähnle, R., Kerber, M., Weidenbach, C.: Common syntax of the DFGSchwerpunktprogramm deduction. Technical Report TR 10/96, Fakultät für Informatik, Universität Karlsruhe, Karlsruhe, Germany (1996)

  30. Harrison, J.: A Mizar mode for HOL. In: von Wright, J., Grundy, J., Harrison, J. (eds.) Theorem Proving in Higher Order Logics: TPHOLs 1996, Volume 1125 of LNCS, pp. 203–220. Springer, Berlin (1996)

    Chapter  Google Scholar 

  31. Hilbert, D.: Foundations of Geometry. Open Court, Illinois (1971)

    MATH  Google Scholar 

  32. Iancu, M., Kohlhase, M., Rabe, F., Urban, J.: The Mizar Mathematical Library in OMDoc: translation and applications. J. Autom. Reason. 50(2), 191–202 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  33. Jaśkowski, S.: On the rules of suppositions. Studia Logica 1, 32 (1934)

    MATH  Google Scholar 

  34. Kaliszyk, C., Pąk, K.: Isabelle formalization of set theoretic structures and set comprehensions. In: Blamer, J., Kutsia, T., Simos, D. (eds.) Mathematical Aspects of Computer and Information Sciences, MACIS 2017, Volume 10693 of LNCS. Springer, Berlin (2017)

    Google Scholar 

  35. Kaliszyk, C., Pąk, K.: Presentation and manipulation of Mizar properties in an Isabelle object logic. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds.) Intelligent Computer Mathematics - CICM 2017, Volume 10383 of LNCS, pp. 193–207. Springer, Berlin (2017)

    Google Scholar 

  36. Kaliszyk, C., Pąk, K.: Progress in the independent certification of Mizar Mathematical Library in Isabelle. In: Ganzha, M., Maciaszek, L.A., Paprzycki, M. (eds.) Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, pp. 227–236 (2017)

  37. Kaliszyk, C., Pąk, K., Urban, J.: Towards a Mizar environment for Isabelle: Foundations and language. In: Avigad, J., Chlipala, A. (eds.) Proceedings of 5th Conference on Certified Programs and Proofs (CPP 2016), pp. 58–65. ACM (2016)

  38. Kaliszyk, C., Urban, J.: MizAR 40 for Mizar 40. J. Autom. Reason. 55(3), 245–256 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  39. Kaliszyk, C., Wiedijk, F.: Merging procedural and declarative proof. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds.) Types for Proofs and Programs, International Conference, TYPES 2008, Volume 5497 of LNCS, pp. 203–219. Springer, Berlin (2008)

  40. Kobayashi, N. (ed.): Proceedings Eighth Workshop on Intersection Types and Related Systems, ITRS 2016, Volume 242 of EPTCS (2017)

  41. Korniłowicz, A.: Flexary connectives in Mizar. Comput. Lang. Syst. Struct. 44, 238–250 (2015)

    MATH  Google Scholar 

  42. Korniłowicz, A., Schwarzweller, C.: Computers and algorithms in Mizar. Mech. Math. Appl. 4(1), 43–50 (2005)

    Google Scholar 

  43. Krauss, A., Schropp, A.: A mechanized translation from higher-order logic to set theory. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving (ITP 2010), Volume 6172 of LNCS, pp. 323–338. Springer, Berlin (2010)

    Google Scholar 

  44. Kuncar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. In: Urban, C., Zhang, X. (eds.) Interactive Theorem Proving - 6th International Conference, ITP 2015, Volume 9236 of LNCS, pp. 234–252. Springer, Berlin (2015)

  45. Kuncar, O., Popescu, A.: Safety and conservativity of definitions in HOL and Isabelle/HOL. PACMPL 2((POPL)), 24:1–24:26 (2018)

    Google Scholar 

  46. Kunčar, O.: Reconstruction of the Mizar type system in the HOL light system. In: Pavlu, J., Safrankova, J. (eds.) WDS Proceedings of Contributed Papers: Part I—Mathematics and Computer Sciences, pp. 7–12. Matfyzpress (2010)

  47. Lee, G., Rudnicki, P.: Alternative aggregates in Mizar. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) Proceedings of 6th International Conference on Mathematical Knowledge Management (MKM 2007), Volume 4573 of LNCS, pp. 327–341. Springer (2007)

  48. Megill, N.D.: Metamath: A Computer Language for Pure Mathematics. Lulu Press, Morrisville (2007)

    Google Scholar 

  49. Merz, S.: Mechanizing TLA in Isabelle. In: Rodošek, R. (ed.) Workshop on Verification in New Orientations, pp. 54–74. University of Maribor, Maribor (1995)

    Google Scholar 

  50. Nakamura, Y., Trybulec, A.: A mathematical model of CPU. Formaliz. Math. 3(2), 151–160 (1992)

    Google Scholar 

  51. Naraschewski, W., Wenzel, M.: Object-oriented verification based on record subtyping in higher-order logic. In: Grundy, J., Newey, M.C. (eds) Theorem Proving in Higher Order Logics, 11th International Conference, TPHOLs’98, volume 1479 of LNCS, pp. 349–366. Springer, Berlin (1998)

  52. Naumowicz, A.: Enhanced processing of adjectives in Mizar. In: Grabowski, A., Naumowicz, A. (eds.) Computer Reconstruction of the Body of Mathematics, Volume 18(31) of Studies in Logic, Grammar and Rhetoric, pp. 89–101. University of Białystok, Białystok (2009)

    Google Scholar 

  53. Naumowicz, A.: Automating boolean set operations in Mizar proof checking with the aid of an external SAT solver. J. Autom. Reason. 55(3), 285–294 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  54. Naumowicz, A., Byliński, C.: Improving Mizar texts with properties and requirements. In: Asperti et al. [5], pp. 290–301

  55. Naumowicz, A., Korniłowicz, A.: A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Theorem Proving in Higher Order Logics, TPHOLS, Volume 5674 of LNCS, pp. 67–72. Springer, Berlin (2009)

    MATH  Google Scholar 

  56. Naumowicz, A., Piliszek, R.: Accessing the Mizar library with a weakly strict Mizar parser. In: Kohlhase, M., Johansson, M., Miller, B.R., de Moura, L., Tompa, F.W. (eds.) Intelligent Computer Mathematics, CICM 2016, Volume 9791 of LNCS, pp. 77–82. Springer, Berlin (2016)

    Google Scholar 

  57. Obua, S.: Partizan games in Isabelle/HOLZF. In: Barkaoui, K., Cavalcanti, A., Cerone, A. (eds.) Theoretical Aspects of Computing—ICTAC 2006, Volume 4281 of LNCS, pp. 272–286. Springer, Berlin (2006)

    Chapter  Google Scholar 

  58. Obua, S., Fleuriot, J.D., Scott, P., Aspinall, D.: ProofPeer: collaborative theorem proving. CoRR. arXiv:1404.6186 (2014)

  59. Obua, S., Fleuriot, J.D., Scott, P., Aspinall, D.: Type Inference for ZFH. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) Intelligent Computer Mathematics—International Conference, CICM, Volume 9150 of LNCS, pp. 87–101. Springer (2015)

  60. Ono, K.: On a practical way of describing formal deductions. Nagoya Math. J. 21, 115–121 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  61. Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science (1990), pp. 361–386 (1990)

  62. Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Autom. Reason. 11(3), 353–389 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  63. Pąk, K.: Topological manifolds. Formaliz. Math. 22(2), 179–186 (2014)

    MATH  Article  Google Scholar 

  64. Rabe, F.: A logical framework combining model and proof theory. Math. Struct. Comput. Sci. 23(5), 945–1001 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  65. Rudnicki, P.: Obvious inferences. J. Autom. Reason. 3(4), 383–393 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  66. Schürmann, C.: The Twelf proof assistant. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Theorem Proving in Higher Order Logics, 22nd International Conference, TPHOLs 2009, Volume 5674 of LNCS, pp. 79–83. Springer, Berlin (2009)

  67. Syme, D.: Three tactic theorem proving. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin-Mohring, C., Théry, L. (eds.) Theorem Proving in Higher Order Logics, TPHOLs 1999, Volume 1690 of LNCS, pp. 203–220. Springer, Berlin (1999)

    Google Scholar 

  68. Trybulec, A., Korniłowicz, A., Naumowicz, A., Kuperberg, K.T.: Formal mathematics for mathematicians—special issue. J. Autom. Reason. 50(2), 119–121 (2013)

    MathSciNet  Article  Google Scholar 

  69. Urban, J.: MPTP—motivation, implementation, first experiments. J. Autom. Reason. 33(3–4), 319–339 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  70. Urban, J.: XML-izing Mizar: making semantic processing and presentation of MML easy. In: Kohlhase, M. (ed.) Mathematical Knowledge Management (MKM 2005), Volume 3863 of LNCS, pp. 346–360. Springer, Berlin (2005)

    Google Scholar 

  71. Urban, J.: MizarMode—an integrated proof assistance tool for the Mizar way of formalizing mathematics. J. Appl. Logic 4(4), 414–427 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  72. Urban, J.: MoMM—fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. Artif. Intell. Tools 15(1), 109–130 (2006)

    Article  Google Scholar 

  73. Urban, J.: MPTP 0.2: design, implementation, and initial experiments. J. Autom. Reason. 37(1–2), 21–43 (2006)

    MATH  Google Scholar 

  74. Urban, J., Bancerek, G.: Presenting and explaining Mizar. Electr. Notes Theor. Comput. Sci. 174(2), 63–74 (2007)

    MATH  Article  Google Scholar 

  75. Urban, J., Hoder, K., Voronkov, A.: Evaluation of automated theorem proving on the Mizar Mathematical Library. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) International Congress on Mathematical Software (ICMS 2010), Volume 6327 of LNCS, pp. 155–166. Springer, Berlin (2010)

    Google Scholar 

  76. Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reason. 50(2), 229–241 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  77. Urban, J., Sutcliffe, G.: ATP-based cross-verification of Mizar proofs: method, systems, and first experiments. Math. Comput. Sci. 2(2), 231–251 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  78. Weidenbach, C., Afshordel, B., Brahm, U., Cohrs, C., Engel, T., Keen, E., Theobalt, C., Topić, D.: System description: SPASS version 1.0.0. In: Automated Deduction - CADE-16, volume 1632 of LNCS, pp. 378–382. Springer (1999). https://doi.org/10.1007/3-540-48660-7_34

  79. Wenzel, M.: Isar—a generic interpretative approach to readable formal proof documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs 1999, Volume 1690 of LNCS, pp. 167–184. Springer (1999)

  80. Wenzel, M.: Asynchronous user interaction and tool integration in Isabelle/PIDE. In: Klein, G., Gamboa, R. (eds.) Interactive Theorem Proving, ITP 2014, Volume 8558 of LNCS, pp. 515–530. Springer, Berlin (2014)

    Google Scholar 

  81. Wenzel, M.: The Isabelle/Isar reference manual (2017)

  82. Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed, O.A., Muñoz, C.A., Tahar, S. (eds.) Theorem Proving in Higher Order Logics, 21st International Conference, TPHOLs 2008, Volume 5170 of LNCS, pp. 33–38. Springer (2008)

  83. Wenzel, M., Wiedijk, F.: A comparison of Mizar and Isar. J. Autom. Reason. 29(3–4), 389–411 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  84. Wiedijk, F.: CHECKER—notes on the basic inference step in Mizar. http://www.cs.kun.nl/~freek/mizar/by.dvi (2000). Accessed 25 Aug 2018

  85. Wiedijk, F.: Mizar light for HOL light. In: Boulton, R.J., Jackson, P.B. (eds.) Theorem Proving in Higher Order Logics, TPHOLs 2001, Volume 2152 of LNCS, pp. 378–394. Springer, Berlin (2001)

    Google Scholar 

  86. Wiedijk, F.: A synthesis of the procedural and declarative styles of interactive theorem proving. Log. Methods Comput. Sci. 8(1:30), 1–26 (2012)

  87. Zhan, B.: Formalization of the fundamental group in untyped set theory using auto2. In: Ayala-Rincón, M., Muñoz, C.A. (eds.) Interactive Theorem Proving—ITP 2017, Volume 10499 of LNCS, pp. 514–530. Springer, Berlin (2017)

    Google Scholar 

Download references

Acknowledgements

Open access funding provided by University of Innsbruck and Medical University of Innsbruck. We would like to thank the anonymous reviewers, as well as Josef Urban, Chad Brown, and Julian Parsert for their comments on the previous versions of this paper. This work has been supported by the European Research Council (ERC) Grant No. 714034 SMART, OeAD Scientific & Technological Cooperation with Poland grant, and the PolishNational Science Center granted by decision noDEC-2015/19/D/ST6/01473.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Cezary Kaliszyk.

Additional information

The paper has been financed by the resources of the Polish National Science Center granted by decision no DEC-2015/19/D/ST6/01473 and the ERC starting Grant No. 714034 SMART.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kaliszyk, C., Pąk, K. Semantics of Mizar as an Isabelle Object Logic. J Autom Reasoning 63, 557–595 (2019). https://doi.org/10.1007/s10817-018-9479-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-018-9479-z