An effective proof of the well-foundedness of the multiset path ordering

Article

Abstract

The main contribution of this paper is an effective proof of the well-foundedness of MPO, as a term of the Calculus of Inductive Constructions. This proof is direct, short and simple. It is a sequence of nested inductions and it only relies on the fact that the multiset order, restricted to a well-founded part of the base set, is well-founded. In particular, it does not require to establish preliminarily the transitivity of the relation, although we prove it as an additional property. The terms we consider are not supposed to be ground nor the signature to be finite. All the proofs have been carried out in the Coq proof-assistant.

Keywords

Termination Well-foundedness Multiset path order Constructive logic 

References

  1. 1.
    Baader F., Nipkow T. (1998) Term Rewriting and All That. Cambridge University Press, New YorkMATHGoogle Scholar
  2. 2.
    CoLoR: a Coq Library on Rewriting and termination. http://color.loria.frGoogle Scholar
  3. 3.
    Coquand T., Huet G. (1985) Constructions : A Higher Order Proof System for Mechanizing Mathematics. EUROCAL 85. LNCS, vol. 203. Springer, Berlin Heidelberg New YorkGoogle Scholar
  4. 4.
    Coupet-Grimal, S., Delobel, W.: A Constructive Axiomatization of the Recursive Path Ordering (Submitted, 2006)Google Scholar
  5. 5.
    Dawson, E., Goré, R.: A general theorem on termination of rewriting. In: Computer Science Logic, CSL’04, no. 3210 in LNCS, pp. 100–114. Springer, Berlin Heidelberg New York (2004)Google Scholar
  6. 6.
    Dershowitz N. (1982) Orderings for term rewriting systems. Theor. Comput. Sci. 3(17):279–301MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ferreira, M., Zantema, H.: Well-foundedness of term orderings. In: 4th International Workshop on Conditional Term Rewriting Systems (CTRS’94), no. 968 in LNCS, pp. 106–123. Springer, Berlin Heidelberg New York (1995)Google Scholar
  8. 8.
    Giménez, E.: Un calcul de constructions infinies et son application à la vérification de systèmes communicants. Thèse d’université, Ecole Normale Supérieure de Lyon (1996)Google Scholar
  9. 9.
    Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and types. Cambridge Tracts in Theoretical Computer Science, vol. 7 (1988)Google Scholar
  10. 10.
    Huet, G., Lankford, D.: On the uniform halting problem for term rewriting systems. Technical Report 283, IRIA (1978)Google Scholar
  11. 11.
    Jouannaud, J.P., Rubio, A.: The higher-order recursive path ordering. In: Proceedings of the 14th annual IEEE Symposium on Logic in Computer Science (LICS’99), pp. 402–411. Trento, Italy (1999)Google Scholar
  12. 12.
    Jouannaud, J.P., Rubio, A.: Higher-Order Recursive Path Orderings a la carte. Technical Report. http://www.lix.polytechnique.fr/Labo/Jean-Pierre.Jouannaud/biblio.html (2003)Google Scholar
  13. 13.
    Koprowski, A.: Well-foundedness of the Higher-Order Recursive Path Ordering in Coq. Master thesis, Free University of Amsterdam (The Netherlands) and Warsaw University (Poland) (2004)Google Scholar
  14. 14.
    Kruskal J. (1960) Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Trans. AMS 95, 210–225MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Leclerc, F.: Termination proof of term rewriting system with the multiset path ordering. A complete development in the system Coq. In: TLCA, pp. 312–327 (1995)Google Scholar
  16. 16.
    Lescanne P. (1982) Some properties of decomposition ordering, a simplification ordering to prove termination of rewriting systems. R.A.I.R.O. Theor. Inform. 14(4):331–347MathSciNetGoogle Scholar
  17. 17.
    Nash-Williams C.S.J.A. (1963) On well-quasi-ordering finite trees. Proc. Camb. Philos. Soc. 59(4):833–835MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Nipkow, T.: An Inductive Proof of the Well-foundedness of the Multiset Order. Due to Wilfried Buchholz. Tech. rep. http://www4.informatik.tu-muenchen.de/~nipkow/misc/index.html (1998)Google Scholar
  19. 19.
    Paulin-Mohring, C.: Définitions inductives en théorie des types d’ordre supérieur. Habilitation à diriger les recherches, Université Claude Bernard Lyon I (1996)Google Scholar
  20. 20.
    van Raamsdonk, F.: On termination of higher-order rewriting. In: Proceedings of the 12th International Conference on Rewriting Techniques and Applications (RTA’01), pp. 261–275. Utrecht, The Netherlands (2001)Google Scholar
  21. 21.
    Seisenberger, M.: Kruskal’s tree theorem in a constructive theory of inductive definitions. In: Schuster, P., Berger, U., Osswald, H. (eds.) Reuniting the Antipodes—Constructive and Nonstandard Views of the Continuum, Symposion in San Servolo/Venice, Italy, May 17–22, 1999 (Synthese Library 306, Kluwer, Dordrecht, 2001)Google Scholar
  22. 22.
    Team, T.C.D.: The Coq Proof Assistant Reference Manual—Version 8.0. Tech. rep., LogiCal Project-INRIA (2004)Google Scholar
  23. 23.
    The Coq Proof Assistant. http://www.coq.inria.frGoogle Scholar
  24. 24.
    Veldman W. (2004) An intuitionistic proof of kruskal’s theorem. Arch. Math. Log. 43(2):215–264MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Centre de Mathématiques et d’InformatiqueMarseille Cedex 13France

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