The Implicit Calculus of Constructions Extending Pure Type Systems with an Intersection Type Binder and Subtyping

  • Alexandre Miquel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)

Abstract

In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection type binder—called the implicit dependent product. Unlike the usual approach of Type Assignment Systems, the implicit product can be used at every place in the universe hierarchy. We study syntactical properties of this calculus such as the βη-subject reduction property, and we show that the implicit product induces a rich subtyping relation over the type system in a natural way. We also illustrate the specificities of this calculus by revisiting the impredicative encodings of the Calculus of Constructions, and we show that their translation into the implicit calculus helps to reflect the computational meaning of the underlying terms in a more accurate way.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. Altenkirch. Constructions, Inductive types and Strong Normalization. PhD thesis, University of Edinburgh, 1993.Google Scholar
  2. 2.
    Henk Barendregt. Introduction to generalized type systems. Technical Report 90-8, University of Nijmegen, Department of Informatics, May 1990.Google Scholar
  3. 3.
    B. Barras, S. Boutin, C. Cornes, J. Courant, J.C. Filliâtre, E. Giménez, H. Herbelin, G. Huet, C. Muñoz, C. Murthy, C. Parent, C. Paulin, A. Saïbi, and B. Werner. The Coq Proof Assistant Reference Manual — Version V6.1. Technical Report 0203, INRIA, August 1997.Google Scholar
  4. 4.
    J. H. Geuvers and M. J. Nederhof. A modular proof of strong normalization for the calculus of constructions. In Journal of Functional Programming, volume 1,2(1991), pages 155–189, 1991.MATHMathSciNetGoogle Scholar
  5. 5.
    P. Giannini, F. Honsell, and S. Ronchi della Rocca. Type inference: some results, some problems. In Fundamenta Informaticæ, volume 19(1,2), pages 87–126, 1993.MATHGoogle Scholar
  6. 6.
    M. Hagiya and Y. Toda. On implicit arguments. Technical Report 95-1, Department of Information Science, Faculty of Science, University of Tokyo, 1995.Google Scholar
  7. 7.
    Paul B. Jackson. The Nuprl proof development system, version 4.1 reference manual and user’s guide. Technical report, Cornell University, 1994.Google Scholar
  8. 8.
    D. Leivant. Polymorphic type inference. In Proceedings of the 10th ACM Symposium on Principles of Programming Languages, pages 88–98, 1983.Google Scholar
  9. 9.
    Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, 1994.Google Scholar
  10. 10.
    Zhaohui Luo and Randy Pollack. Lego proof development system: User’s manual. Technical Report 92-228, LFCS, 1992.Google Scholar
  11. 11.
    Lena Magnusson. Introduction to ALF—an interactive proof editor. In Uffe H. Engberg, Kim G. Larsen, and Peter D. Mosses, editors, Proceedings of the 6th Nordic Workshop on Programming Theory (Aarhus, Denmark, 17–19 October, 1994), number NS-94-6 in Notes Series, page 269, Department of Computer Science, University of Aarhus, December 1994. BRICS. vi+483.Google Scholar
  12. 12.
    Alexandre Miquel. Arguments implicites dans le calcul des constructions: étude d’un formalisme á la Curry. Master’s thesis, Université Denis-Diderot Paris 7, octobre 1998.Google Scholar
  13. 13.
    Alexandre Miquel. A model for impredicative type systems with universes, intersection types and subtyping. In Proceedings of the 15 th Annual IEEE Symposium on Logic in Computer Science (LICS’00), 2000.Google Scholar
  14. 14.
    R. Pollack. Implicit syntax. In Gérard Huet and Gordon Plotkin, editors, Proceedings of the First Workshop on Logical Frameworks (Antibes), may 1990.Google Scholar
  15. 15.
    A. Saïbi. Algébre Constructive en Théorie des Types, Outils génériques pour la modélisation et la démonstration, Application à la théorie des Catégories. PhD thesis, Université Paris VI, 1998.Google Scholar
  16. 16.
    S. van Bakel, L. Liquori, R. Ronchi della Rocca, and P. Urzyczyn. Comparing Cubes. In A. Nerode and Yu. V. Matiyasevich, editors, Proceedings of LFCS’ 94. Third International Symposium on Logical Foundations of Computer Science, St. Petersburg, Russia, volume 813 of Lecture Notes in Computer Science, pages 353–365. Springer-Verlag, 1994.Google Scholar
  17. 17.
    J. B. Wells. Typability and type checking in system F are equivalent and undecidable. In Annals of Pure and Applied Logic, volume 98(1–3), pages 111–156, 1999.CrossRefMathSciNetGoogle Scholar
  18. 18.
    B. Werner. Une théorie des Constructions Inductives. PhD thesis, Université Paris VII, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Alexandre Miquel
    • 1
  1. 1.INRIA Rocquencourt - Projet LogiCalLe Chesnay cedexFrance

Personalised recommendations