A two-level approach towards lean proof-checking

  • Gilles BartheEmail author
  • Mark Ruys
  • Henk Barendregt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1158)


We present a simple and effective methodology for equational reasoning in proof checkers. The method is based on a two-level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive and oracle types. The potential of our two-level approach is illustrated by some examples developed in Lego.


Type Theory Equational Theory Computer Algebra System Universal Algebra Equational Problem 
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  1. 1.
    A. Bailey. Representing algebra in Lego, M.Sc. thesis, University of Edinburgh, October 1993.Google Scholar
  2. 2.
    C. Ballarins, K. Homann and J. Calmet. Theorems and algorithms: an interface between Maple and Isabelle, in the proceedings of ISSAC'95.Google Scholar
  3. 3.
    H.P. Barendregt. Typed λ-calculi, Handbook of logic in computer science, Abramsky and al eds, OUP 1992.Google Scholar
  4. 4.
    G. Barthe. Towards a mathematical vernacular, manuscript, presented at the HISC workshop, Amsterdam, March 1994.Google Scholar
  5. 5.
    G. Barthe. Formalising mathematics in type theory: fundamentals and case studies, manuscript, June 1994, submitted for publication.Google Scholar
  6. 6.
    G. Barthe and H. Elbers. Towards lean proof checking, to appear in the proceedings of DISCO'96, Lecture Notes in Computer Science, Springer-Verlag, 1996. An extended version will appear as a CWI technical report.Google Scholar
  7. 7.
    G. Barthe and H. Geuvers. Congruence types, to appear in the proceedings of CSL'95, 1995.Google Scholar
  8. 8.
    G. Barthe, M. Ruys and H. Barendregt. A two-level approach towards lean proofchecking, to appear as a CWI technical report, 1996.Google Scholar
  9. 9.
    V. Breazu-Tannen. Combining algebra and higher-order types, in the proceedings of LICS'88, pp 82–90, IEEE, 1988.Google Scholar
  10. 10.
    P. Cohn. Universal algebra, Mathematics and its Applications, Vol. 6, D. Reidel, 1981.Google Scholar
  11. 11.
    R. Constable. Metalevel Programming in Constructive Type Theory, Logic and Algebra of Specification, F. Bauer and al eds, NATO Asi Series, 1994.Google Scholar
  12. 12.
    R. Constable and al. Implementing mathematics with the NuPrl proof development system, Prentice Hall, 1986.Google Scholar
  13. 13.
    G. Dowek and al. The Coq proof assistant user's guide Technical Report, INRIA, November 1993.Google Scholar
  14. 14.
    H. Elbers. A machine-assisted construction of the real numbers, M.Sc. thesis, University of Nijmegen, September 1993.Google Scholar
  15. 15.
    J. Harrison and L. Théry. Extending the HOL theorem prover with a computer algebra system to reason about the reals, in proceedings of HOL'93, LNCS, 1993.Google Scholar
  16. 16.
    D. Howe. Automating reasoning in an implementation of constructive type theory, Ph.D. thesis, Cornell University, 1988.Google Scholar
  17. 17.
    P. Jackson. Exploring abstract algebra in constructive type theory, in the proceedings of CADE-12, LNAI 814, June 1994.Google Scholar
  18. 18.
    J.W. Klop. Term-rewriting systems, in Handbook of logic in computer science (volume 2), Abramsky and al eds, OUP 1992.Google Scholar
  19. 19.
    Z. Luo. Computation and reasoning: a type theory for computer science, OUP, 1994.Google Scholar
  20. 20.
    Z. Luo and R. Pollack. LEGO proof development system: user's manual, Technical Report, University of Edinburgh, May 1992.Google Scholar
  21. 21.
    L. Magnusson and B. Nordström. The Alf proof editor and its proof engine, in the proceedings of Types for Proofs and Programs, LNCS 806, May 1993.Google Scholar
  22. 22.
    P. Martin-Löf. An intuitionistic theory of types, Bibliopolis, 1984.Google Scholar
  23. 23.
    R. Nederpelt and al. Selected papers on AUTOMATH, North-Holland, 1994.Google Scholar
  24. 24.
    B. Nordström, K. Petersson and J. Smith. Programming in Martin-Löf 's type theory, OUP, 1990.Google Scholar
  25. 25.
    M.P.J. Ruys. Ph.D. thesis, University of Nijmegen, forthcoming (1996).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsUniversity of NijmegenThe Netherlands

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