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Mizar Light for HOL Light

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Theorem Proving in Higher Order Logics (TPHOLs 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2152))

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Abstract

There are two different approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system). Most provers are procedural. However declarative proofs are much closer in style to informal mathematical reasoning than procedural ones.

There have been attempts to put declarative interfaces on top of procedural proof assistants, like John Harrison’s Mizar mode for HOL and Markus Wenzel’s Isar mode for Isabelle. However in those cases the declarative assistant is a layer on top of the procedural basis, having a separate syntax and a different ‘feel’ from the underlying system.

This paper shows that the procedural and the declarative ways of proving are related and can be integrated seamlessly. It presents an implementation of the Mizar proof language on top of HOL that consists of only 41 lines of ML. This shows how close the procedural and declarative styles of proving really are.

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References

  1. Henk Barendregt. The impact of the lambda calculus. Bulletin of Symbolic Logic, 3(2), 1997.

    Google Scholar 

  2. Bruno Barras, e.a. The Coq Proof Assistant Reference Manual, 2000. ftp:http://ftp.inria.fr/INRIA/coq/V6.3.1/doc/Reference-Manual-all.ps.gz.

  3. Gertrud Bauer. Lesbare formale Beweise in Isabelle/Isar — der Satz von Hahn-Banach. Master’s thesis, TU München, November 1999. http://www.in.tum.de/~bauerg/HahnBanach-DA.pdf.

  4. Gertrud Bauer. The Hahn-Banach Theorem for real vector spaces. Part of the Isabelle99-2 distribution, http://isabelle.in.tum.de/library/HOL/HOL-Real/library/HOL/HOL-Real/HahnBanach/document.pdf, February 2001.

  5. Gertrud Bauer and Markus Wenzel. Computer-assisted mathematics at work — the Hahn-Banach theorem in Isabelle/Isar. In Thierry Coquand, Peter Dybjer, Bengt Nordström, and Jan Smith, editors, Types for Proofs and Programs: TYPES’99, volume 1956 of LNCS, 2000.

    Chapter  Google Scholar 

  6. Jan Cederquist, Thierry Coquand, and Sara Negri. The Hahn-Banach Theorem in Type Theory. In G. Sambin and J. Smith, editors, Twenty-Five years of Constructive Type Theory, Oxford, 1998. Oxford University Press.

    Google Scholar 

  7. M.J.C. Gordon and T.F. Melham, editors. Introduction to HOL. Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  8. M.J.C. Gordon, R. Milner, and C.P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of LNCS. Springer Verlag, Berlin, Heidelberg, New York, 1979.

    Google Scholar 

  9. John Harrison. A Mizar Mode for HOL. In Proceedings of the 9th International Conference on Theorem Proving in Higher Order Logics, TPHOLs’ 96, volume 1125 of LNCS, pages 203–220. Springer, 1996.

    Google Scholar 

  10. John Harrison. The HOL Light manual (1.1), 2000. http://www.cl.cam.ac.uk/users/jrh/hol-light/manual-1.1.ps.gz.

  11. David A. McAllester. Ontic: A Knowledge Representation System for Mathematics. The MIT Press Series in Artificial Intelligence. MIT Press, 1989.

    Google Scholar 

  12. M. Muzalewski. An Outline of PC Mizar. Fondation Philippe le Hodey, Brussels, 1993. http://www.cs.kun.nl/~freek/mizar/mizarmanual.ps.gz.

    Google Scholar 

  13. R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

    MATH  Google Scholar 

  14. L.C. Paulson. The Isabelle Reference Manual, 2000. http://www.cl.cam.ac.uk.Research/HVG/Isabelle/dist/Isabelle99-1/doc/ref.pdf.

  15. Don Syme. Three Tactic Theorem Proving. In Theorem Proving in Higher Order Logics, TPHOLs’ 99, volume 1690 of LNCS, pages 203–220. Springer, 1999.

    Chapter  Google Scholar 

  16. M. Wenzel. The Isabelle/Isar Reference Manual. TU München, München, 1999. http://isabelle.in.tum.de/doc/isar-ref.pdf.

    Google Scholar 

  17. F. Wiedijk. Mizar: An impression. http://www.cs.kun.nl/~freek/mizar/mizarintro.ps.gz, 1999.

  18. Vincent Zammit. On the Implementation of an Extensible Declarative Proof Language. In Theorem Proving in Higher Order Logics, TPHOLs’ 99, volume 1690 of LNCS, pages 185–202. Springer, 1999.

    Chapter  Google Scholar 

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Wiedijk, F. (2001). Mizar Light for HOL Light. In: Boulton, R.J., Jackson, P.B. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2001. Lecture Notes in Computer Science, vol 2152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44755-5_26

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  • DOI: https://doi.org/10.1007/3-540-44755-5_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42525-0

  • Online ISBN: 978-3-540-44755-9

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