Structured Induction Proofs in Isabelle/Isar

  • Makarius Wenzel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4108)


Isabelle/Isar is a generic framework for human-readable formal proof documents, based on higher-order natural deduction. The Isar proof language provides general principles that may be instantiated to particular object-logics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out well-structured and readable. Our techniques can be applied to non-standard variants of induction as well, such as co-induction and nominal induction. This demonstrates that Isar provides a viable platform for building domain-specific tools that support fully-formal mathematical proof composition.


Automate Reasoning Natural Deduction Induction Rule Major Premise Inductive Case 
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.
    Aspinall, D.: Proof General: A generic tool for proof development. In: European Joint Conferences on Theory and Practice of Software (ETAPS) (2000)Google Scholar
  2. 2.
    Barras, B., et al.: The Coq Proof Assistant Reference Manual, version 8. Inria (2006)Google Scholar
  3. 3.
    Bauer, G., Wenzel, M.: Calculational reasoning revisited — an Isabelle/Isar experience. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Dixon, L., Fleuriot, J.D.: A proof-centric approach to mathematical assistants. Journal of Applied Logic: Special Issue on Mathematics Assistance Systems (to appear)Google Scholar
  5. 5.
    Gentzen, G.: Untersuchungen über das logische Schließen. Math. Zeitschrift (1935)Google Scholar
  6. 6.
    Mizar mathematical library,
  7. 7.
    Nipkow, T.: Structured proofs in Isar/HOL. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHGoogle Scholar
  9. 9.
    Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science, Academic Press, London (1990)Google Scholar
  10. 10.
    Rudnicki, P.: An overview of the MIZAR project. In: 1992 Workshop on Types for Proofs and Programs, Chalmers University of Technology, Bastad (1992)Google Scholar
  11. 11.
    Schroeder-Heister, P.: A natural extension of natural deduction. Journal of Symbolic Logic 49(4) (1984)Google Scholar
  12. 12.
    Trybulec, A.: Some features of the Mizar language. Presented at a workshop in Turin (1993)Google Scholar
  13. 13.
    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, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Urban, C., Tasson, C.: Nominal techniques in Isabelle/HOL. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Wenzel, M.: Isar — a generic interpretative approach to readable formal proof documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    Wenzel, M.: Isabelle/Isar — a versatile environment for human-readable formal proof documents. PhD thesis, Institut für Informatik, TU München (2002),
  17. 17.
    Wenzel, M., Paulson, L.C.: Isabelle/Isar. In: Wiedijk, F. (ed.) The Seventeen Provers of the World. LNCS (LNAI), vol. 3600, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Wiedijk, F.: Comparing mathematical provers. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Wiedijk, F., Wenzel, M.: A comparison of the mathematical proof languages Mizar and Isar. Journal of Automated Reasoning 29(3–4) (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Makarius Wenzel
    • 1
  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany

Personalised recommendations