Inductive Consequences in the Calculus of Constructions

  • Daria Walukiewicz-Chrząszcz
  • Jacek Chrząszcz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6172)


Extending the calculus of constructions with rewriting would greatly improve the efficiency of proof assistants such as Coq. In this paper we address the issue of the logical power of such an extension.

In our previous work we proposed a procedure to check completeness of user-defined rewrite systems. In many cases this procedure demonstrates that only a basic subset of the rules is sufficient for completeness. Now we investigate the question whether the remaining rules are inductive consequences of the basic subset.

We show that the answer is positive for most practical rewrite systems. It is negative for some systems whose critical pair diagrams require rewriting under a lambda. However the positive answer can be recovered when the notion of inductive consequences is modified by allowing a certain form of functional extensionality.


Inductive Consequence Type Theory Additional Rule Critical Pair Proof Assistant 
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  1. 1.
    Abel, A., Coquand, T., Pagano, M.: A modular type-checking algorithm for type theory with singleton types and proof irrelevance. In: Curien, P.-L. (ed.) TLCA 2009. LNCS, vol. 5608, pp. 5–19. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Altenkirch, T., McBride, C., Swierstra, W.: Observational equality, now? In: PLPV 2007, pp. 57–68. ACM, New York (2007)CrossRefGoogle Scholar
  3. 3.
    Barbanera, F., Fernández, M., Geuvers, H.: Modularity of strong normalization in the algebraic-λ-cube. Journal of Functional Programming 7(6), 613–660 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blanqui, F.: Definitions by rewriting in the Calculus of Constructions. Mathematical Structures in Computer Science 15(1), 37–92 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blanqui, F., Jouannaud, J.P., Okada, M.: The calculus of algebraic constructions. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 301–316. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Blanqui, F., Jouannaud, J.P., Strub, P.Y.: Building decision procedures in the calculus of inductive constructions. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 328–342. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Chrząszcz, J.: Modules in Coq are and will be correct. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 130–146. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Chrząszcz, J.: Modules in Type Theory with Generative Definitions. PhD thesis, Warsaw University and University Paris-Sud (2004)Google Scholar
  9. 9.
    The Coq proof assistant,
  10. 10.
    Ehrig, H., Mahr, B.: Fundamentals of Algebraic Specification 1. Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science, vol. 6. Springer, Heidelberg (1985)zbMATHGoogle Scholar
  11. 11.
    Hofmann, M., Streicher, T.: The groupoid model refutes uniqueness of identity proofs. In: LICS 1994, pp. 208–212. IEEE Computer Society, Los Alamitos (1994)Google Scholar
  12. 12.
    Letouzey, P.: A new extraction for Coq. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 200–219. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Oury, N.: Extensionality in the calculus of constructions. In: Hurd, J., Melham, T.F. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 278–293. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Paulin-Mohring, C.: Inductive definitions in the system Coq: Rules and properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 328–345. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  15. 15.
    Terese (ed.): Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  16. 16.
    Walukiewicz-Chrząszcz, D.: Termination of rewriting in the calculus of constructions. Journal of Functional Programming 13(2), 339–414 (2003)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Walukiewicz-Chrząszcz, D.: Termination of Rewriting in the Calculus of Constructions. PhD thesis, Warsaw University and University Paris-Sud (2003)Google Scholar
  18. 18.
    Walukiewicz-Chrząszcz, D., Chrząszcz, J.: Inductive consequences in the calculus of constructions,
  19. 19.
    Walukiewicz-Chrząszcz, D., Chrząszcz, J.: Consistency and completeness of rewriting in the calculus of constructions. Logical Methods of Computer Science 4(3) (2008)Google Scholar
  20. 20.
    Werner, B.: On the strength of proof-irrelevant type theories. Logical Methods in Computer Science 4(3) (2008)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Daria Walukiewicz-Chrząszcz
    • 1
  • Jacek Chrząszcz
    • 1
  1. 1.Institute of InformaticsWarsaw UniversityWarsawPoland

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