Ordinals in HOL: Transfinite Arithmetic up to (and Beyond) ω1

  • Michael Norrish
  • Brian Huffman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7998)


We describe a comprehensive HOL mechanisation of the theory of ordinal numbers, focusing on the basic arithmetic operations. Mechanised results include the existence of fixpoints such as ε 0, the existence of normal forms, and the validation of algorithms used in the ACL2 theorem-proving system.


Ordinal Number HOL4 Theory Transitivity Rule Mechanise Proof Algebraic Type 
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.
    Castéran, P., Contejean, E.: On ordinal notations,
  2. 2.
    Gordon, M.J.C., Reynolds, J., Hunt, Jr., W.A., Kaufmann, M.: An integration of HOL and ACL2. In: Proceedings of Formal Methods in Computer-Aided Design (FMCAD), pp. 153–160. IEEE Computer Society (2006)Google Scholar
  3. 3.
    Harrison, J.: The HOL wellorder library. HOL88 documentation (May 1992)Google Scholar
  4. 4.
    Huffman, B.: Countable ordinals. Archive of Formal Proofs, Formal proof development (November 2005),
  5. 5.
    Kaufmann, M., Slind, K.: Proof pearl: Wellfounded induction on the ordinals up to ε 0. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 294–301. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Manolios, P., Vroon, D.: Ordinal arithmetic: Algorithms and mechanization. Journal of Automated Reasoning 34(4), 387–423 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Traytel, D., Popescu, A., Blanchette, J.C.: Foundational, compositional (co)datatypes for higher-order logic: Category theory applied to theorem proving. In: Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, pp. 596–605. IEEE (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Norrish
    • 1
    • 2
  • Brian Huffman
    • 3
  1. 1.Canberra Research Lab.NICTAAustralia
  2. 2.Australian National UniversityAustralia
  3. 3.Galois, Inc.USA

Personalised recommendations