Inductive Types in the Calculus of Algebraic Constructions

  • Frédéric Blanqui
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2701)


In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that CIC as a whole can be seen as a CAC, and that it can be extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.


Induction Hypothesis Type Theory Predicate Symbol Proof Assistant Inductive Type 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Abel. Termination checking with types. Technical Report 0201, Ludwig Maximilians Universität, München, Germany, 2002.Google Scholar
  2. 2.
    F. Barbanera, M. Fernández, and H. Geuvers. Modularity of strong normalization and confluence in the algebraic-λ-cube. In Proceedings of the 9th IEEE Symposium on Logic in Computer Science, 1994.Google Scholar
  3. 3.
    H. Barendregt. Lambda calculi with types. In S. Abramski, D. Gabbay, and T. Maibaum, editors, Handbook of logic in computer science, volume 2. Oxford University Press, 1992.Google Scholar
  4. 4.
    F. Blanqui. Definitions by rewriting in the Calculus of Constructions (extended abstract). In Proceedings of the 16th IEEE Symposium on Logic in Computer Science, 2001.Google Scholar
  5. 5.
    F. Blanqui. Théorie des Types et Récriture. PhD thesis, Université Paris XI, Orsay, France, 2001. Available in english as “Type Theory and Rewriting”.Google Scholar
  6. 6.
    F. Blanqui. Definitions by rewriting in the Calculus of Constructions, 2003. Journal submission, 68 pages.Google Scholar
  7. 7.
    F. Blanqui. A short and flexible strong normalization proof for the Calculus of Algebraic Constructions with curried rewriting, 2003. Draft.Google Scholar
  8. 8.
    T. Coquand. Pattern matching with dependent types. In Proceedings of the International Workshop on Types for Proofs and Programs, 1992.
  9. 9.
    T. Coquand and G. Huet. The Calculus of Constructions. Information and Computation, 76(2–3):95–120, 1988.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    T. Coquand and C. Paulin-Mohring. Inductively defined types. In Proceedings of the International Conference on Computer Logic, Lecture Notes in Computer Science 417, 1988.Google Scholar
  11. 11.
    N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 6. North-Holland, 1990.Google Scholar
  12. 12.
    P. Dybjer. A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic, 65(2):525–549, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1988.Google Scholar
  14. 14.
    R. Harper and J. Mitchell. Parametricity and variants of Girard’s J operator. Information Processing Letters, 70:1–5, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J.-P. Jouannaud and M. Okada. Executable higher-order algebraic specification languages. In Proceedings of the 6th IEEE Symposium on Logic in Computer Science, 1991.Google Scholar
  16. 16.
    J.W. Klop, V. van Oostrom, and F. van Raamsdonk. Combinatory reduction systems: introduction and survey. Theoretical Computer Science, 121:279–308, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Matthes. Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. PhD thesis, Ludwig Maximilians Universität, München, Germany, 1998.zbMATHGoogle Scholar
  18. 18.
    C. McBride. Dependently typed functional programs and their proofs. PhD thesis, University of Edinburgh, United Kingdom, 1999.Google Scholar
  19. 19.
    N. P. Mendler. Inductive Definition in Type Theory. PhD thesis, Cornell University, United States, 1987.Google Scholar
  20. 20.
    C. Paulin-Mohring. Personal communication, 2001.Google Scholar
  21. 21.
    R. Pollack. Dependently typed records in type theory. Formal Aspects of Computing, 13(3–5):341–363, 2002.Google Scholar
  22. 22.
    Coq Development Team. The Coq Proof Assistant Reference Manual — Version 7.3. INRIA Rocquencourt, France, 2002. Scholar
  23. 23.
    B. Werner. Une Théorie des Constructions Inductives. PhD thesis, Université Paris VII, France, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Frédéric Blanqui
    • 1
  1. 1.Laboratoire d’Informatique de l’École PolytechniquePalaiseau CedexFrance

Personalised recommendations