Proof search with set variable instantiation in the Calculus of Constructions

  • Amy Felty
Session 8B
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1104)


We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higher-order logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of first-order logic that allows existential quantification over set variables. The method finds maximal solutions for this special class of higher-order variables. This class of variables can also be identified in CC. The existence of a correspondence between higher-order logic and higher-order type theories such as CC is well-known. CC can be viewed as an extension of higher-order logic where the basic terms of the language, the simply-typed λ-terms, are replaced with terms containing dependent types. We adapt Bledsoe's procedure to the corresponding class of variables in CC and extend it to handle terms with dependent types.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. C. Bailin and D. Barker-Plummer. \(\mathcal{Z}\)-match: An inference rule for incrementally elaborating set instantiation. Journal of Automated Reasoning, 11(3):391–428, Dec. 1993.CrossRefGoogle Scholar
  2. 2.
    W. W. Bledsoe. A maximal method for set variables in automatic theorem proving. Machine Intelligence, 9:53–100, 1979.Google Scholar
  3. 3.
    A. Church. A formulation of the simple theory of types. Journal of Symbolic Logic, 5:56–68, 1940.Google Scholar
  4. 4.
    R. L. Constable et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, 1986.Google Scholar
  5. 5.
    T. Coquand and G. Huet. The calculus of constructions. Information and Computation, 76(2/3):95–120, February/March 1988.CrossRefGoogle Scholar
  6. 6.
    C. Cornes, J. Courant, J.-C. Filliâtre, G. Huet, P. Manoury, C. Paulin-Mohring, C. Muñoz, C. Murthy, C. Parent, A. Saïbi, and B. Werner. The Coq Proof Assistant reference manual. Technical report, INRIA, 1995.Google Scholar
  7. 7.
    G. Dowek. Démonstration Automatique dans le Calcul des Constructions. PhD thesis, L'Université Paris VII, Dec. 1991.Google Scholar
  8. 8.
    G. Dowek. A complete proof synthesis method for the cube of type systems. Journal of Logic and Computation, 3(3):287–315, 1993.Google Scholar
  9. 9.
    A. Felty. Encoding the calculus of constructions in a higher-order logic. In Eighth Annual Symposium on Logic in Computer Science, pages 233–244, June 1993.Google Scholar
  10. 10.
    M. J. C. Gordon and T. F. Melham. Introduction to HOL—A Theorem Proving Environment for Higher Order Logic. Cambridge University Press, 1993.Google Scholar
  11. 11.
    R. Harper, F. Honsell, and G. Plotkin. A framework for defining logics. Journal of the ACM, 40(1):143–184, Jan. 1993.CrossRefGoogle Scholar
  12. 12.
    W. A. Howard. The formulae-as-type notion of construction, 1969. In To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus, and Formalism, pages 479–490. Academic Press, 1980.Google Scholar
  13. 13.
    G. Huet. A uniform approach to type theory. In G. Huet, editor, Logical Foundations of Functional Programming. Addison Wesley, 1990.Google Scholar
  14. 14.
    P. Martin-Löf. Intuitionistic Type Theory. Studies in Proof Theory Lecture Notes. BIBLIOPOLES, Napoli, 1984.Google Scholar
  15. 15.
    D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov. Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic, 51:125–157, 1991.CrossRefGoogle Scholar
  16. 16.
    L. C. Paulson. Isabelle: A Generic Theorem Prover, volume 828 of Lecture Note in Computer Science. Springer-Verlag, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Amy Felty
    • 1
  1. 1.Bell LaboratoriesLucent TechnologiesMurray HillUSA

Personalised recommendations