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Computer-Assisted Mathematics at Work

The Hahn-Banach Theorem in Isabelle/Isar

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Types for Proofs and Programs (TYPES 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1956))

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Abstract

We present a complete formalization of the Hahn-Banach theorem in the simply-typed set-theory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for high-level reasoning based on natural deduction. The _nal result is presented as a readable formal proof document, following usual presentations in mathematical textbooks quite closely. Our case study demonstrates that Isabelle/Isar is capable to support this kind of application of formal logic very well, while being open for an even larger scope.

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References

  1. D. Aspinall. Proof General: A generic tool for proof development. In European Joint Conferences on Theory and Practice of Software (ETAPS), 2000.

    Google Scholar 

  2. D. Aspinall. Protocols for interactive e-Proof. In Theorem Proving in Higher Order Logics (TPHOLs), 2000. Unpublished work-in-progress paper, http://zermelo.dcs.ed.ac.uk/_da/drafts/eproof.ps.gz.

  3. H. P. Barendregt. The quest for correctness. In Images of SMC Research, pages 39–58. Stichting Mathematisch Centrum, Amsterdam, 1996.

    Google Scholar 

  4. H. P. Barendregt, G. Barthe, and M. Ruys. A two level approach towards lean proof-checking. In S. Berardi and M. Coppo, editors, Types for Proofs and Programs (TYPES’95), LNCS 1158, 1995.

    Google Scholar 

  5. G. Bauer. Lesbare Formale Beweise in Isabelle/Isar-Der Satz von Hahn-Banach. Master’s thesis, Technische Universität München, 1999.

    Google Scholar 

  6. G. Bauer. The Hahn-Banach Theorem for real vector spaces. Technische Universität München, 2000. Isabelle/Isar proof document, http://isabelle.in.tum.de/library/HOL/HOL-Real/HahnBanach/document.pdf.

  7. Y. Bertot, G. Dowek, A. Hirschowitz, C. Paulin, and L. Thery, editors. Theorem Proving in Higher Order Logics: TPHOLs’ 99, LNCS 1690, 1999.

    MATH  Google Scholar 

  8. R. Burstall. Teaching people to write proofs: a tool. In CafeOBJ Symposium, Numazu, Japan, April 1998.

    Google Scholar 

  9. P. Callaghan and Z. Luo. Mathematical vernacular in type theory-based proof assistants. Workshop on User Interfaces in Theorem Proving, Eindhoven, 1998.

    Google Scholar 

  10. J. Cederquist, T. Coquand, and S. Negri. The Hahn-Banach theorem in type theory. In G. Sambin and J. Smith, editors, Twenty-Five years of Constructiveype Theory. Oxford University Press, 1998.

    Google Scholar 

  11. A. Church. A formulation of the simple theory of types. Journal of Symbolic Logic, pages 56–68, 1940.

    Google Scholar 

  12. C. Cornes, J. Courant, J.-C. Filli atre, G. Huet, P. Manoury, and C. Muñoz. The Coq Proof Assistant User’s Guide, version 6.1. INRIA-Rocquencourt et CNRS-ENS Lyon, 1996.

    Google Scholar 

  13. Y. Coscoy, G. Kahn, and L. Théry. Extracting text from proofs. In Typed Lambda Calculus and Applications, volume 902 of LNCS. Springer, 1995.

    Google Scholar 

  14. E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. Texts and monographs in computer science. Springer, 1990.

    Google Scholar 

  15. 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 

  16. H. Heuser. Funktionalanalysis: Theorie und Anwendung. Teubner, 1986.

    Google Scholar 

  17. P. Mäenpää and A. Ranta. The type theory and type checker of GF. In Colloquium on Principles, Logics, and Implementations of High-Level Programming Languages. Workshop on Logical Frameworks and Meta-languages, Paris, 1999.

    Google Scholar 

  18. M. Muzalewski. An Outline of PC Mizar. Fondation of Logic, Mathematics and Informatics-Mizar Users Group, 1993. http://www.cs.kun.nl/_freek/mizar/mizarmanual.ps.gz.

  19. L. Narici and E. Beckenstein. The Hahn-Banach Theorem: The life and times. In Topology Atlas. York University, Toronto, Ontario, Canada, 1996. http://at.yorku.ca/topology/preprint.htm and http://at.yorku.ca/p/a/a/a/16.htm.

    Google Scholar 

  20. R. P. Nederpelt, J. H. Geuvers, and R. C. de Vrijer, editors. Selected Papers on Automath, Studies in Logic 133. North Holland, 1994.

    Google Scholar 

  21. B. Nowak and A. Trybulec. Hahn-Banach Theorem. Journal of Formalized Mathematics, 5, 1993. http://mizar.uwb.edu.pl/JFM/Vol5/hahnban.html.

  22. S. Owre, S. Rajan, J. M. Rushby, N. Shankar, and M. Srivas. PVS: combining specification, proof checking, and model checking. In R. Alur and T. A. Henzinger, editors, Computer Aided Verification, LNCS 1102, 1996.

    Chapter  Google Scholar 

  23. L. C. Paulson. Isabelle: the next 700 theorem provers. In P. Odifreddi, editor, Logic and Computer Science. Academic Press, 1990.

    Google Scholar 

  24. L. C. Paulson. Isabelle: A Generic Theorem Prover. LNCS 828. Springer, 1994.

    Book  MATH  Google Scholar 

  25. P. Rudnicki. An overview of the MIZAR project. In 1992 Workshop on Types for Proofs and Programs. Chalmers University of Technology, Bastad, 1992.

    Google Scholar 

  26. D. Syme. Declarative Theorem Proving for Operational Semantics. PhD thesis, University of Cambridge, 1998.

    Google Scholar 

  27. D. Syme. Three tactic theorem proving. In Bertot et al. [7].

    Google Scholar 

  28. A. Trybulec. Some features of the Mizar language. Presented at a workshop in Turin, Italy, 1993.

    Google Scholar 

  29. M. Wenzel. Type classes and overloading in higher-order logic. In 10th International Conference on Theorem Proving in Higher Order Logics, TPHOLs’97, LNCS 1275, 1997.

    Google Scholar 

  30. M. Wenzel. Isar-a generic interpretative approach to readable formal proof documents. In Bertot et al. [7].

    Google Scholar 

  31. M. Wenzel. The Isabelle/Isar Reference manual. Technische Universität München, 2000. Part of the Isabelle documentation, http://isabelle.in.tum.de/doc/isar-ref.pdf.

  32. M. Wenzel. Miscellaneous Isabelle/Isar examples for Higher-Order Logic. Technische Universität München, 2000. Isabelle/Isar proof document, http://isabelle.in.tum.de/library/HOL/Isar_examples/document.pdf.

  33. M. Wenzel. Using Axiomatic Type Classes in Isabelle, 2000. Part of the Isabelle documentation, http://isabelle.in.tum.de/doc/axclass.pdf.

  34. F. Wiedijk. Mizar: An impression. Unpublished paper, 1999. http://www.cs.kun.nl/_freek/mizar/mizarintro.ps.gz.

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Author information

Authors and Affiliations

  1. Technische Universität München, Institut für Informatik, Arcisstra\e 21, 80290, München, Germany

    Gertrud Bauer & Markus Wenzel

Authors
  1. Gertrud Bauer
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  2. Markus Wenzel
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Editor information

Editors and Affiliations

  1. Department of Computing Science, Chalmers University of Technology and University of Göteborg, 41296, Göteborg, Sweden

    Thierry Coquand , Peter Dybjer , Bengt Nordström  & Jan Smith , ,  & 

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Bauer, G., Wenzel, M. (2000). Computer-Assisted Mathematics at Work. In: Coquand, T., Dybjer, P., Nordström, B., Smith, J. (eds) Types for Proofs and Programs. TYPES 1999. Lecture Notes in Computer Science, vol 1956. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44557-9_4

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  • DOI: https://doi.org/10.1007/3-540-44557-9_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41517-6

  • Online ISBN: 978-3-540-44557-9

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