The Optimal Fixed Point Combinator

  • Arthur Charguéraud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6172)


In this paper, we develop a general theory of fixed point combinators, in higher-order logic equipped with Hilbert’s epsilon operator. This combinator allows for a direct and effective formalization of corecursive values, recursive and corecursive functions, as well as functions mixing recursion and corecursion. It supports higher-order recursion, nested recursion, and offers a proper treatment of partial functions in the sense that domains need not be hardwired in the definition of functionals. Our work, which has been entirely implemented in Coq, unifies and generalizes existing results on contraction conditions and complete ordered families of equivalences, and relies on the theory of optimal fixed points for the treatment of partial functions. It provides a practical way to formalize circular definitions in higher-order logic.


Equivalence Relation Point Theorem Contraction Condition Partial Function Recursive Function 
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  1. 1.
    Balaa, A., Bertot, Y.: Fix-point equations for well-founded recursion in type theory. In: Aagaard, M., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 1–16. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Barthe, G., Forest, J., Pichardie, D., Rusu, V.: Defining and reasoning about recursive functions: A practical tool for the Coq proof assistant. In: Hagiya, M., Wadler, P. (eds.) FLOPS 2006. LNCS, vol. 3945, pp. 114–129. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Barthe, G., Frade, M.J., Giménez, E., Pinto, L., Uustalu, T.: Type-based termination of recursive definitions. Mathematical Structures in Computer Science 14(1), 97–141 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertot, Y.: Filters on coinductive streams, an application to eratosthenes’ sieve. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 102–115. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Bertot, Y., Komendantskaya, E.: Inductive and Coinductive Components of Corecursive Functions in Coq. In: Proceedings of CMCS’08, April 2008. ENTCS, vol. 203, pp. 25–47 (2008)Google Scholar
  6. 6.
    Bertot, Y., Komendantsky, V.: Fixed point semantics and partial recursion in Coq. In: Antoy, S., Albert, E. (eds.) PPDP, pp. 89–96. ACM, New York (2008)CrossRefGoogle Scholar
  7. 7.
    Bove, A., Capretta, V.: Nested general recursion and partiality in type theory. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 121–135. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Dubois, C., Donzeau-Gouge, V.: A step towards the mechanization of partial functions: domains as inductive predicates. In: CADE-15 Workshop on mechanization of partial functions (1998)Google Scholar
  9. 9.
    Di Gianantonio, P., Miculan, M.: A unifying approach to recursive and co-recursive definitions. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 148–161. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Di Gianantonio, P., Miculan, M.: Unifying recursive and co-recursive definitions in sheaf categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 136–150. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Harrison, J.: Inductive definitions: Automation and application. In: Schubert, E.T., Alves-Foss, J., Windley, P. (eds.) TPHOLs 1995. LNCS, vol. 971, pp. 200–213. Springer, Heidelberg (1995)Google Scholar
  12. 12.
    Krauss, A.: Partial and nested recursive function definitions in higher-order logic. Journal of Automated Reasoning (to appear, December 2009)Google Scholar
  13. 13.
    Krstić, S.: Inductive fixpoints in higher order logic (February 2004)Google Scholar
  14. 14.
    Krstić, S., Matthews, J.: Inductive invariants for nested recursion. In: Basin, D.A., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 253–269. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Letouzey, P.: Programmation fonctionnelle certifiée: L’extraction de programmes dans l’assistant Coq (June 1, 2007)Google Scholar
  16. 16.
    Manna, Z., Shamir, A.: The theoretical aspects of the optimal FixedPoint. SIAM Journal on Computing 5(3), 414–426 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Matthews, J.: Recursive function definition over coinductive types. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 73–90. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Nordström, B.: Terminating general recursion. BIT 28(3), 605–619 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Slind, K.: Reasoning about Terminating Functional Programs. PhD thesis, Institut für Informatik, Technische Universität München (1999)Google Scholar
  20. 20.
    Sozeau, M.: Subset coercions in coq. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 237–252. Springer, Heidelberg (2007)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Arthur Charguéraud
    • 1
  1. 1.INRIA 

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