Revisiting Cut-Elimination: One Difficult Proof Is Really a Proof

  • Christian Urban
  • Bozhi Zhu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5117)


Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term- rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation, which implements proof transformation that permute cuts over other inference rules. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the informal proof given by the first author in his PhD-thesis. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.


Inference Rule Classical Logic Strong Normalisation Variable Convention Main Formula 
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.
    Aydemir, B., Charguéraud, A., Pierce, B.C., Pollack, R., Weirich, S.: Engineering Formal Metatheory. In: Proc. of the 35rd Symposium on Principles of Programming Languages (POPL), pp. 3–15. ACM, New York (2008)Google Scholar
  2. 2.
    Barbanera, F., Berardi, S.: A Symmetric Lambda Calculus for “Classical” Program Extraction. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 495–515. Springer, Heidelberg (1994)Google Scholar
  3. 3.
    Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, vol. 103. North-Holland, Amsterdam (1981)zbMATHGoogle Scholar
  4. 4.
    Brauner, P., Houtmann, C., Kirchner, C.: Principles of Superdeduction. In: Proc. of the 22nd Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 41–50 (2007)Google Scholar
  5. 5.
    Gentzen, G.: Untersuchungen über das logische Schließen I and II. Mathematische Zeitschrift 39, 176–210, 405–431 (1935)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Harper, R., Pfenning, F.: On Equivalence and Canonical Forms in the LF Type Theory. ACM Transactions on Computational Logic 6(1), 61–101 (2005)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)zbMATHGoogle Scholar
  8. 8.
    Kleene, S.C.: Disjunction and Existence Under Implication in Elementary Intuitionistic Formalisms. Journal of Symbolic Logic 27(1), 11–18 (1962)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Pfenning, F.: Structural Cut Elimination. Information and Computation 157(1–2), 84–141 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pitts, A.: Alpha-Structural Recursion and Induction. Journal of the ACM 53, 459–506 (2006)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Prawitz, D.: Ideas and Results of Proof Theory. In: Proceedings of the 2nd Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, vol. 63, pp. 235–307. North-Holland, Amsterdam (1971)Google Scholar
  12. 12.
    Schürmann, C., Sarnat, J.: Towards a Judgemental Reconstruction of Logical Relation Proofs. In: Proc. of the 23rd IEEE Symposium on Logic in Computer Science (LICS) (to appear, 2008)Google Scholar
  13. 13.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science, vol. 43. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  14. 14.
    Urban, C.: Classical Logic and Computation. PhD thesis, Cambridge University (October 2000)Google Scholar
  15. 15.
    Urban, C., Berghofer, S.: A Recursion Combinator for Nominal Datatypes Implemented in Isabelle/HOL. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 498–512. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Urban, C., Berghofer, S., Norrish, M.: Barendregt’s Variable Convention in Rule Inductions. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 35–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Urban, C., Bierman, G.: Strong Normalisation of Cut-Elimination in Classical Logic. Fundamenta Informaticae 45(1–2), 123–155 (2001)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Urban, C., Cheney, J., Berghofer, S.: Mechanizing the Metatheory of LF. In: Proc. of the 23rd IEEE Symposium on Logic in Computer Science (LICS). Technical report (to appear, 2008),
  19. 19.
    Urban, C., Tasson, C.: Nominal Techniques in Isabelle/HOL. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 38–53. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    van Bakel, S., Lengrand, S., Lescanne, P.: The Language X: Circuits, Computations and Classical Logic. In: Coppo, M., Lodi, E., Pinna, G.M. (eds.) ICTCS 2005. LNCS, vol. 3701, pp. 81–96. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Yoshida, N., Berger, M., Honda, K.: Strong Normalisation in the π-Calculus. In: Proc. of the 16th IEEE Symposium on Logic in Computer Science (LICS), pp. 311–322 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christian Urban
    • 1
  • Bozhi Zhu
    • 2
  1. 1.Technical University of Munich 
  2. 2.North China Electric Power University 

Personalised recommendations