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The Not So Simple Proof-Irrelevant Model of CC

  • Alexandre Miquel
  • Benjamin Werner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2646)

Abstract

It is well-known that the Calculus of Constructions (CC) bears a simple set-theoretical model in which proof-terms are mapped onto a single object—a property which is known as proof-irrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model raises several unexpected difficulties related to the interpretation of the impredicative level, especially for the soundness property which is surprisingly difficult to be given a correct proof in this simple framework. We propose a way to tackle these difficulties, thus giving a (more) detailed elementary consistency proof of CC without going back to a translation to F ω . We also discuss some possible alternatives and possible extensions of our construction.

Keywords

Type Theory Interpretation Function Predicative Level Typing Derivation Dependent Product 
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.

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References

  1. 1.
    P. Aczel. On relating type theories and set theories, in Types for Proofs and Programs, edited by Altenkirch, Naraschewski and Reus, Proceedings of Types’ 98, LNCS 1657 (1999).CrossRefGoogle Scholar
  2. 2.
    T. Altenkirch. Extensional Equality in Intensional Type Theory. In Proceedings of the fourteenth Annual IEEE Symposium on Logic in Computer Science, IEEE, 1999.Google Scholar
  3. 3.
    H. Barendregt. Lambda Calculi with Types. Technical Report 91-19, Catholic University Nijmegen, 1991. In Handbook of Logic in Computer Science, Vol II, Elsevier, 1992.Google Scholar
  4. 4.
    F. Barbanera, S. Berardi. Proof-irrelevance out of Excluded Middle and Choice in the Calculus of Constructions. Journal of Functional Programming, vol. 6(3), 519–525, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Barras and B. Werner. Coq in Coq. Manuscript.Google Scholar
  6. 6.
    G. Barthe. Domain-free pure type systems. Journal of Functional Programming, vol. 10(5), p. 412–452, 2000.CrossRefMathSciNetGoogle Scholar
  7. 7.
    L. Chicli, L. Pottier and C. Simpson. Quotient Types in Coq. Presentation at the TYPES’01 Workshop, Nijmegen, 2001.Google Scholar
  8. 8.
    Th. Coquand. Une Théorie des Constructions. Thèse de Doctorat, Université Paris 7, janvier 1985.Google Scholar
  9. 9.
    Th. Coquand. Metamathematical Investigations of a Calculus of Constructions. In P. Oddifredi (editor), Logic and Computer Science. Academic Press, 1990. Rapport de recherche INRIA 1088.Google Scholar
  10. 10.
    Th. Coquand and G. Huet. The Calculus of Constructions. Information and Computation, 76(2/3), 1988.Google Scholar
  11. 11.
    H. Geuvers. Logics and Type Systems. PhD Thesis, Katholieke Universiteit Nijmegen, 1993.Google Scholar
  12. 12.
    J.-Y. Girard. Interprétation fonctionnelle et élimination des coupures de l’arithmétique d’ordre supérieur, Thèse d’État, Université Paris 7, 1972.Google Scholar
  13. 13.
    J.-Y. Girard. Translation and appendices Y. Lafont and P. Taylor. Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1989.Google Scholar
  14. 14.
    J.-L. Krivine. Théorie des ensembles. Cassini, Paris, 1998.zbMATHGoogle Scholar
  15. 15.
    P. Martin-Löf. Intuitionistic Type Theory. Studies in Proof Theory, Bibliopolis, 1984.Google Scholar
  16. 16.
    J. McKinna and R. Pollack. Pure Type Systems formalized, in TLCA’93, M. Bezem and J. F. Groote Eds, LNCS 664, Springer-Verlag, Berlin, 1993.Google Scholar
  17. 17.
    P.-A. Melliès and B. Werner. A Generic Normalization Proof for Pure Type Systems. In TYPES’96, E. Gimenez and C. Paulin-Mohring Eds, LNCS 1512, Springer-Verlag, Berlin, 1998.Google Scholar
  18. 18.
    A. Miquel. A Model for Impredicative Type Systems with Universes, Intersection Types and Subtyping. Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS’00), 2000.Google Scholar
  19. 19.
    A. Miquel. Le calcul des constructions implicite: syntaxe et sémantique. Thèse de doctorat, Université Paris 7, 2001.Google Scholar
  20. 20.
    J. Reynolds. Polymorphism is not Set-Theoretic, Semantics of Data Types G. Kahn, D. B. MacQueen and G. Plotkin Eds. LNCS 173, pp. 145–156, Springer-Verlag, Berlin, 1984.Google Scholar
  21. 21.
    T. Streicher. Semantics of Type Theory. Progress in Theoretical Computer Science. Birkhaeuser Verlag, Basel 1991.Google Scholar
  22. 22.
    B. Werner. Sets in Types, Types in Sets. In, M. Abadi and T. Itoh (Eds), Theoretical Aspects of Computer Science, TACS’97, LNCS 1281, Springer-Verlag, 1997.CrossRefGoogle Scholar
  23. 23.
    B. Werner. Une Théorie des Constructions Inductives. Thèse de Doctorat, Université Paris 7, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alexandre Miquel
    • 1
  • Benjamin Werner
    • 2
  1. 1.Laboratoire de Recherche en InformatiqueUniversité Paris-SudORSAY cedexFrance
  2. 2.INRIA-RocquencourtLE CHESNAY cedexFrance

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