Advertisement

Journal of Automated Reasoning

, Volume 29, Issue 3–4, pp 189–224 | Cite as

A Compendium of Continuous Lattices in MIZAR

  • Grzegorz Bancerek
  • Piotr Rudnicki
Article

Abstract

This paper reports on the MIZAR formalization of the theory of continuous lattices as presented in Gierz et al.: A Compendium of Continuous Lattices, 1980. By a MIZAR formalization we mean a formulation of theorems, definitions, and proofs written in the MIZAR language whose correctness is verified by the MIZAR processor. This effort was originally motivated by the question of whether or not the MIZAR system was sufficiently developed for the task of expressing advanced mathematics. The current state of the formalization – 57 MIZAR articles written by 16 authors – indicates that in principle the MIZAR system has successfully met the challenge. To our knowledge it is the most sizable effort aimed at mechanically checking some substantial and relatively recent field of advanced mathematics. However, it does not mean that doing mathematics in MIZAR is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of the Gierz et al. compendium has (i) prompted many improvements of the MIZAR proof checking system, (ii) caused numerous revisions of the the MIZAR data base, and (iii) contributed to the “to do” list of further changes to the MIZAR system.

MIZAR QED project formalization of mathematics set theory proof checking theory of continuous lattices mathematical knowledge management 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bancerek, G.: Tarski's classes and ranks, Formalized Mathematics 1(3) (1990), 563–567. MML: CLASSES1.Google Scholar
  2. 2.
    Bancerek, G.: König's theorem, Formalized Mathematics 1(3) (1990), 589–593. MML: CARD_3.Google Scholar
  3. 3.
    Bancerek, G.: Complete lattices, Formalized Mathematics 2(5) (1991), 719–725. MML: LATTICE3.Google Scholar
  4. 4.
    Bancerek, G.: Bounds in posets and relational substructures, Formalized Mathematics 6(1) (1997), 81–91. MML: YELLOW_0.Google Scholar
  5. 5.
    Bancerek, G.: Directed sets, nets, ideals, filters, and maps, Formalized Mathematics 6(1) (1997), 93–107. MML: WAYBEL_0.Google Scholar
  6. 6.
    Bancerek, G.: The “way-below” relation, FormalizedMathematics 6(1) (1997), 169–176. MML: WAYBEL_3.Google Scholar
  7. 7.
    Bancerek, G.: Duality in relation structures, Formalized Mathematics 6(2) (1997), 227–232. MML: YELLOW_7.Google Scholar
  8. 8.
    Bancerek, G.: Prime ideals and filters, Formalized Mathematics 6(2) (1997), 241–247. MMLGoogle Scholar
  9. 9.
    Bancerek, G.: Bases and refinements of topologies, Formalized Mathematics 7(1) (1998), 35–43. MML: YELLOW_9.Google Scholar
  10. 10.
    Bancerek, G.: The Lawson topology, Formalized Mathematics 7(2) (1998), 163–168. MML: WAYBEL19.Google Scholar
  11. 11.
    Bancerek, G.: Lawson topology in continuous lattices, Formalized Mathematics 7(2) (1998), 177–184. MML: WAYBEL21.Google Scholar
  12. 12.
    Bancerek, G.: Development of the theory of continuous lattices in MIZAR, in M. Kerber and M. Kohlhase (eds), Symbolic Computation and Automated Reasoning, A. K. Peters, 2001.Google Scholar
  13. 13.
    Bancerek, G.: Continuous lattices of maps between T0 spaces, Formalized Mathematics 9(1) (2001), 111–117. MML: WAYBEL26.Google Scholar
  14. 14.
    Bancerek, G.: Categorial background for duality theory, Formalized Mathematics 9(4) (2001), 755–765. MML: YELLOW21.Google Scholar
  15. 15.
    Bancerek, G.: Duality based on the Galois connection. Part I, Formalized Mathematics 9(4) (2001), 767–778. MML: WAYBEL34.Google Scholar
  16. 16.
    Bancerek, G., Endou, N. and Shidama, Y.: Lim-inf convergence and its compactness, Mechanized Mathematics and Its Applications 2(1) (2002), 29–35.Google Scholar
  17. 17.
    Byli´nski, C.: Some basic properties of sets, Formalized Mathematics 1(1) (1990), 47–53. MML: ZFMISC_1.Google Scholar
  18. 18.
    Byli´nski, C.: Introduction to categories and functors, Formalized Mathematics 1(2) (1990), 409–420. MML: CAT_1.Google Scholar
  19. 19.
    Byli´nski, C.: Category ens, Formalized Mathematics 2(4) (1991), 527–533. MML: ENS_1.Google Scholar
  20. 20.
    Byli´nski, C.: Galois connections, Formalized Mathematics 6(1) (1997), 131–143. MML: WAYBEL_1.Google Scholar
  21. 21.
    Byli´nski, C. and Rudnicki, P.: The Scott topology. Part II, Formalized Mathematics 6(3) (1997), 441–446. MML: WAYBEL14.Google Scholar
  22. 22.
    Fleuriot, J. and Paulson, L. C.: A combination of nonstandard analysis and geometry theorem proving, with application to Newton's Principia, in C. Kirchner and H. Kirchner (eds), 15th International Conf. on Automated Deduction: CADE-15, LNAI 1421, Springer, 1998, pp. 3–16.Google Scholar
  23. 23.
    Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S.: A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980.Google Scholar
  24. 24.
    Grabowski, A. and Milewski, R.: Boolean posets, posets under inclusion and products of relational structures, Formalized Mathematics 6(1) (1997), 117–121. MML: YELLOW_1.Google Scholar
  25. 25.
    Gryko, J.: Injective spaces, Formalized Mathematics 7(1) (1998), 57–62. MML: WAYBEL18.Google Scholar
  26. 26.
    Ja´skowski, S.: On the Rules of Supposition in Formal Logic, Studia Logica,Warsaw University, 1934. Reprinted in S. McCall, Polish Logic in 1920–1939, Clarendon Press, Oxford.Google Scholar
  27. 27.
    Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge, 1982.Google Scholar
  28. 28.
    van Benthem Jutting, L. S.: Checking Landau's “Grundlagen” in the Automath system, Ph.D. thesis, The Eindhoven, 1977.Google Scholar
  29. 29.
    Kelley, J. L.: General Topology, Van Nostrand, New York, 1955.Google Scholar
  30. 30.
    Korniłowicz, A.: Cartesian products of relations and relational structures, Formalized Mathematics 6(1) (1997), 145–152. MML: YELLOW_3.Google Scholar
  31. 31.
    Korniłowicz, A.: On the topological properties of meet-continuous lattices, Formalized Mathematics 6(2) (1997), 269–277. MML: WAYBEL_9.Google Scholar
  32. 32.
    Landau, E. G. H.: Grundlagen der Analysis, Akademische Verlag, Leipzig, 1930.Google Scholar
  33. 33.
    Library Committee of the Association of Mizar Users. Preliminaries to Structures, JFM, Addenda. MML: STRUCT_0.Google Scholar
  34. 34.
    Library Committee of the Association of Mizar Users. Preliminaries to Arithmetic, JFM, Addenda. MML: ARYTM.Google Scholar
  35. 35.
    MIZAR Manuals. http://mizar.org/project/bibliography.html.Google Scholar
  36. 36.
    Madras, B.: Product of family of universal algebras, Formalized Mathematics 4(1) (1993), 103–108. MML: PRALG_1.Google Scholar
  37. 37.
    Milewski, R.: Algebraic lattices, Formalized Mathematics 6(2) (1997), 249–254. MML: WAYBEL_8.Google Scholar
  38. 38.
    Nederpelt, R. P., Geuvers, J. H. and de Vrijer, R. C.: Selected Papers on Automath, North-Holland, Amsterdam, 1994.Google Scholar
  39. 39.
    Nipkow, T.: Winskel is (almost) right: Towards a mechanized semantics textbook, Formal Aspects of Computing 10 (1998), 171–186.Google Scholar
  40. 40.
    Padlewska, B. and Darmochwał, A.: Topological spaces and continuous functions, Formalized Mathematics 1(1) (1990), 223–230. MML: PRE_TOPC.Google Scholar
  41. 41.
    Paulson, L. C. and Grabczewski, K.: Mechanizing set theory: Cardinal arithmetic and the axiom of choice, J. Automated Reasoning 17 (1996), 291–323.Google Scholar
  42. 42.
    Rasiowa, H. and Sikorski, R.: The Mathematics of Metamathematics, PWN, Warszawa, 1968.Google Scholar
  43. 43.
    Rudnicki, P.: Kernel projections and quotient lattices, Formalized Mathematics 7(2) (1998), 169–175. MML: WAYBEL20.Google Scholar
  44. 44.
    Rudnicki, P., Schwarzweller, Ch. and Trybulec, A.: Commutative algebra in the Mizar system, J. Symbolic Comput. 32 (2001), 143–169.Google Scholar
  45. 45.
    Rudnicki, P. and Trybulec, A.: On equivalents of well-foundedness, J. Automated Reasoning 23(3–4) (1999), 197–234.Google Scholar
  46. 46.
    Rudnicki, P. and Trybulec, A.: Mathematical knowledge management in MIZAR, 1st Int.Workshop on MKM, Sept. 24–26, 2001. http://www.risc.uni-linz.ac.at/institute/conferences/MKM2001.Google Scholar
  47. 47.
    Shibakov, A Yu. and Trybulec, A.: The Cantor set, Formalized Mathematics 5(2) (1996), 233–236. MML: CANTOR_1.Google Scholar
  48. 48.
    Trybulec, A.: Tarski Grothendieck set theory, Formalized Mathematics 1(1) (1990), 9–11. MML: TARSKI.Google Scholar
  49. 49.
    Trybulec, A.: Built-in concepts, Formalized Mathematics 1(1) (1990), 13–15. MML: AXIOMS.Google Scholar
  50. 50.
    Trybulec, A.: Categories without uniqueness of cod and dom, Formalized Mathematics 5(2) (1996), 259–267. MML: ALTCAT_1.Google Scholar
  51. 51.
    Trybulec, A.: Functors for alternative categories, Formalized Mathematics 5(4) (1996), 595–608. MML: FUNCTOR0.Google Scholar
  52. 52.
    Trybulec, A.: Moore–Smith convergence, Formalized Mathematics 6(2) (1997), 213–225.Google Scholar
  53. 53.
    Trybulec A.: Scott topology, Formalized Mathematics 6(2) (1997), 311–319. MML: WAYBEL11.Google Scholar
  54. 54.
    Trybulec, W. A.: Partially ordered sets, Formalized Mathematics 1(2) (1990), 313–319. MML: ORDERS_1.Google Scholar
  55. 55.
    Trybulec, Z. and Świ¸eczkowska, H.: Boolean properties of sets, Formalized Mathematics 1(1) (1990), 17–23. MML: BOOLE.Google Scholar
  56. 56.
    Wiedijk, F.: Mizar: An impression. http://www.cs.kun.nl/~freek/notes.Google Scholar
  57. 57.
    Wiedijk, F.: Estimating the cost of a standard library for a mathematical proof checker. http: //www.cs.kun.nl/~freek/notes.Google Scholar
  58. 58.
    Wiedijk, F.: The de Bruijn factor. http://www.cs.kun.nl/~freek/notes.Google Scholar
  59. 59.
    Woronowicz, E. and Zalewska, A.: Properties of binary relations, Formalized Mathematics 1(1) (1990), 85–89. MML: RELAT_2.Google Scholar
  60. 60.
    Żukowski, S.: Introduction to lattice theory, Formalized Mathematics 1(1) (1990), 215–222. MML: LATTICES.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Grzegorz Bancerek
  • Piotr Rudnicki

There are no affiliations available

Personalised recommendations