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Experiments with ZF set theory in HOL and Isabelle

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Higher Order Logic Theorem Proving and Its Applications (TPHOLs 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 971))

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Abstract

Most general purpose proof assistants support versions of typed higher order logic. Experience has shown that these logics are capable of representing most of the mathematical models needed in Computer Science. However, perhaps there exist applications where ZF-style set theory is more natural, or even necessary. Examples may include Scott's classical inverse-limit construction of a model of the untyped λ-calculus (D∞) and the semantics of parts of the Z specification notation. This paper compares the representation and use of ZF set theory within both HOL and Isabelle. The main case study is the construction of D ∞. The advantages and disadvantages of higher-order set theory versus first-order set theory are explored experimentally. This study also provides a comparison of the proof infrastructure of HOL and Isabelle.

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References

  1. S. Agerholm. Formalising a model of the λ-calculus in HOL-ST. Technical Report 354, University of Cambridge Computer Laboratory, November 1994.

    Google Scholar 

  2. S. Agerholm. A HOL Basis for Reasoning about Functional Programs. PhD thesis, BRICS, Department of Computer Science, University of Aarhus, December 1994. Available as Technical Report RS-94-44.

    Google Scholar 

  3. S. Agerholm. A comparison of HOL-ST and Isabelle/ZF. Technical Report 369, University of Cambridge Computer Laboratory, 1995.

    Google Scholar 

  4. D. Cantone, A. Ferro, and E. Omodeo, editors. Computable Set Theory, volume 1. Clarendon Press, Oxford, 1989.

    Google Scholar 

  5. R. L. Constable et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, 1986.

    Google Scholar 

  6. F. Corella. Mechanizing set theory. Technical Report 232, University of Cambridge Computer Laboratory, 1991.

    Google Scholar 

  7. G. Dowek, A. Felty, H. Herbelin, G. Huet, C. Murthy, C. Parent, C. Paulin-Mohring, and B. Werner. The Coq proof assistant user's guide — version 5.8. Technical Report 154, INRIA-Rocquencourt, 1993.

    Google Scholar 

  8. Roman Matuszewski (ed). Formalized Mathematics. Université Catholique de Louvain, 1990-. Subscription is $10 per issue or $50 per year (including postage). Subscriptions and orders should be addressed to: Fondation Philippe le Hodey, MIZAR, Av.F.Roosevelt 35, 1050 Brussels, Belgium (fax: +32 (2) 640.89.68).

    Google Scholar 

  9. W. M. Farmer, J. D. Guttman, and F. Javier Thayer. IMPS: An interactive mathematical proof system. Journal of Automated Reasoning, 11(2):213–248, 1993.

    Google Scholar 

  10. S. Finn and M. P. Fourman. L2 — The LAMBDA Logic. Abstract Hardware Limited, September 1993. In LAMBDA 4.3 Reference Manuals.

    Google Scholar 

  11. M. J. C. Gordon. Merging HOL with set theory: preliminary experiments. Technical Report 353, University of Cambridge Computer Laboratory, 1994.

    Google Scholar 

  12. M. J. C. Gordon and T. F. Melham, editors. Introduction to HOL: A Theorem-proving Environment for Higher-Order Logic. Cambridge University Press, 1993.

    Google Scholar 

  13. F. K. Hanna, N. Daeche, and M. Longley. Veritas+: a specification language based on type theory. In M. Leeser and G. Brown, editors, Hardware specification, verification and synthesis: mathematical aspects, volume 408 of Lecture Notes in Computer Science, pages 358–379. Springer-Verlag, 1989.

    Google Scholar 

  14. C. B. Jones. Systematic Software Development using VDM. Prentice Hall International, 1990.

    Google Scholar 

  15. L. Lamport. TLA+. Available on the World Wide Web at the URL: http://www.research.digital.com/SRC/tla/tla.html.

    Google Scholar 

  16. Z. Luo and R. Pollack. LEGO proof development system: User's manual. Technical Report ECS-LFCS-92-211, University of Edinburgh, LFCS, Computer Science Department, University of Edinburgh, The King's Buildings, Edinburgh, EH9 3JZ, May 1992.

    Google Scholar 

  17. L. Magnusson and B. Nordström. The ALF proof editor and its proof engine. In Types for Proofs and Programs: International Workshop TYPES '93, number 806 in Lecture Notes in Computer Science, pages 213–237. Springer-Verlag, 1994.

    Google Scholar 

  18. D. A. McAllester. ONTIC: A Knowledge Representation System for Mathematics. MIT Press, 1989.

    Google Scholar 

  19. P. M. Melliar-Smith and John Rushby. The enhanced HDM system for specification and verification. In Proc. Verkshop III, volume 10 of ACM Software Engineering Notes, pages 41–43. Springer-Verlag, 1985.

    Google Scholar 

  20. L. C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge Tracts in Theoretical Computing 2, Cambridge University Press, 1987.

    Google Scholar 

  21. L. C. Paulson. Set theory for verification: I. From foundations to functions. Journal of Automated Reasoning, 11(3):353–389, 1993.

    Google Scholar 

  22. L. C. Paulson. Set theory for verification: II. Induction and Recursion. Technical Report 312, University of Cambridge Computer Laboratory, 1993.

    Google Scholar 

  23. L. C. Paulson. Isabelle: A Generic Theorem Prover, volume 828 of Lecture Notes in Computer Science. Springer-Verlag, 1994.

    Google Scholar 

  24. K. D. Petersen. Graph model of lambda in higher order logic. In J. J. Joyce and C. H. Seger, editors, Proceedings of the 6th International Workshop on Higher Order Logic Theorem Proving and its Applications, volume 780 of Lecture Notes in Computer Science. Springer-Verlag, 1994.

    Google Scholar 

  25. G. Plotkin. Domains. Course notes, Department of Computer Science, University of Edinburgh, 1983.

    Google Scholar 

  26. PVS World Wide Web page. http://www.csl.sri.com/pvs/overview.html.

    Google Scholar 

  27. Piotr Rudnicki. An Overview of the MIZAR Project. Unpublished; but available by anonymous FTP from menaik.cs.ualberta.ca in the directory pub/Mizar/Mizar_Over.tar.Z, 1992.

    Google Scholar 

  28. M. Saaltink. Z and EVES. Technical Report TR-91-5449-02, Odyssey Research Associates, 265 Carling Avenue, Suite 506, Ottawa, Ontario K1S 2E1, Canada, October 1991.

    Google Scholar 

  29. M. Smyth and G. D. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11, 1982.

    Google Scholar 

  30. J. M. Spivey. The Z Notation: A Reference Manual. Prentice Hall International Series in Computer Science, 2nd edition, 1992.

    Google Scholar 

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Authors and Affiliations

  1. Computer Laboratory, University of Cambridge, New Museums Site, Pembroke Street, CB2 3QG, Cambridge, UK

    Sten Agerholm & Mike Gordon

Authors
  1. Sten Agerholm
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  2. Mike Gordon
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Editor information

E. Thomas Schubert Philip J. Windley James Alves-Foss

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© 1995 Springer-Verlag Berlin Heidelberg

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Agerholm, S., Gordon, M. (1995). Experiments with ZF set theory in HOL and Isabelle. In: Thomas Schubert, E., Windley, P.J., Alves-Foss, J. (eds) Higher Order Logic Theorem Proving and Its Applications. TPHOLs 1995. Lecture Notes in Computer Science, vol 971. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60275-5_55

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  • DOI: https://doi.org/10.1007/3-540-60275-5_55

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  • Print ISBN: 978-3-540-60275-0

  • Online ISBN: 978-3-540-44784-9

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