Implementing constructive real analysis (preliminary report)

  • Jawahar Chirimar
  • Douglas J. Howe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 613)

Abstract

In this paper we present the results of an investigation into the use of the Nuprl proof development system to implement higher constructive mathematics. As a first step in exploring the issues involved, we have developed a basis for formalizing substantial parts of real analysis. More specifically, we have: developed type-theoretic representations of concepts from Bishop's treatment of constructive mathematics that allow reasonably direct formalizations; used Nuprl's facility for sound extension of its inference system to implement automated reasoners for analysis; and tested these ideas in a formalization of rational and real arithmetic and of a proof of the completeness theorem for the reals (every Cauchy sequence converges).

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References

  1. [1]
    E. Bishop. Foundations of Constructive Analysis. McGraw-Hill, New York, 1967.Google Scholar
  2. [2]
    R. L. Constable, et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Englewood Cliffs, New Jersey, 1986.Google Scholar
  3. [3]
    T. Coquand and G. Huet. The calculus of constructions. Information and Computation, 76:95–120, 1988.CrossRefGoogle Scholar
  4. [4]
    S. Feferman. A language and axioms for explicit mathematics. In Dold, A. and B. Eckmann, editor, Algebra and Logic, volume 450 of Lecture Notes in Mathematics, pages 87–139. Springer-Verlag, 1975.Google Scholar
  5. [5]
    M. J. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF: A Mechanized Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.Google Scholar
  6. [6]
    S. Hayashi and H. Nakano. PX: A Computational Logic. Foundations of Computing. MIT Press, Cambridge, MA, 1988.Google Scholar
  7. [7]
    D. J. Howe. Computational metatheory in Nuprl. CADE-9, pages 238–257, May 1988.Google Scholar
  8. [8]
    C. Jones. Completing the rationals and metric spaces in LEGO. In Proceedings of the Second B.R.A. Workshop on Logical Frameworks, Edinburgh, UK, May 1991. (To appear.).Google Scholar
  9. [9]
    L. S. Jutting. Checking Landau's “Grundlagen” in the AUTOMATH system. PhD thesis, Eindhoven University, 1977.Google Scholar
  10. [10]
    P. Martin-Löf. Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153–175, Amsterdam, 1982. North Holland.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Jawahar Chirimar
    • 1
  • Douglas J. Howe
    • 2
  1. 1.Department of Computer and Information ScienceUniversity of PennsylvaniaUSA
  2. 2.Department of Computer ScienceCornell UniversityUSA

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