Advertisement

Building theories in Nuprl

  • David A. Basin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 363)

Abstract

This paper provides an account of how mathematical knowledge is represented, reasoned about, and used computationally in a mechanized constructive theorem proving environment. We accomplish this by presenting a particular theory developed in the Nuprl proof development system: finite set theory culminating in Ramsey's theorem. We believe that this development is interesting as a case study in the relationship between constructive mathematics and computer science. Moreover, the aspects we emphasize — the high-level development of definitions and lemmas, the use of tactics to automate reasoning, and the use of type theory as a programming logic — are not restricted in relevance to this particular theory, and indicate the promise of our approach for other branches of constructive mathematics.

Keywords

Automate Deduction Constructive Mathematic Ramsey Number Inductive Case Edge Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    David A. Basin. Building theories in Nuprl. Technical Report 88-932, Cornell University, 1988.Google Scholar
  2. [2]
    David A. Basin. An environment for automated reasoning about partial functions. In 9th International Conference On Automated Deduction, pages 101–110, Argonne, Illinois, 1988.Google Scholar
  3. [3]
    Robert S. Boyer and J. Strother Moore. A Computational Logic. Academic Press, 1979.Google Scholar
  4. [4]
    N. G. de Bruijn. A survey of the project AUTOMATH. In Essays in Combinatory Logic, Lambda Calculus, and Formalism. Academic Press, 1980.Google Scholar
  5. [5]
    W.R. Cleaveland II. Type-Theoretic Models of Concurrency. PhD thesis, Cornell, 1987.Google Scholar
  6. [6]
    R.L. Constable et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice Hall, 1986.Google Scholar
  7. [7]
    R.L. Constable, Knoblock T., and J Bates. Writing programs that construct proofs. Journal of Automated Reasoning, 1(3):285–326, 1985.Google Scholar
  8. [8]
    Thierry Coquand and Gérard Huet. Constructions: A higher order proof system for mechanizing mathematics. In B. Buchberger, editor, EUROCAL '85: European Conference on Computer Algebra, pages 151–184. Springer-Verlag, 1985.Google Scholar
  9. [9]
    Amy Felty and Dale Miller. Specifying theorem provers in a higher-order logic programming language. In 9th International Conference On Automated Deduction, Argonne, Illinois, 1988.Google Scholar
  10. [10]
    A. Ferro et al. Decision procedures for elementary fragments of set theory. In Fifth Conference on Automated Deduction, Les Arcs, France, 1980.Google Scholar
  11. [11]
    Michael J. Gordon, Robin Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanized Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.Google Scholar
  12. [12]
    Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer. Ramsey Theory. John Wiley & Sons, 1980.Google Scholar
  13. [13]
    Douglas J. Howe. Implementing number theory: An experiment with Nuprl. In 8th International Conference On Automated Deduction, Oxford, UK, 1986.Google Scholar
  14. [14]
    Douglas J. Howe. Automating Reasoning in an Implementation of Constructive Type Theory. PhD thesis, Cornell University, 1988.Google Scholar
  15. [15]
    Christoph Kreitz. Constructive automata theory implemented with the Nuprl proof development system. Technical Report 86-779, Cornell University, 1986.Google Scholar
  16. [16]
    Per 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
  17. [17]
    David A. McAllester. ONTIC: A knowledge representation system for mathematics. Technical Report 979, MIT, 1987.Google Scholar
  18. [18]
    Lawrence Paulson. Lessons learned from LCF: a survey of natural deduction proofs. Comp. J., 28(5), 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • David A. Basin
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthaca

Personalised recommendations