Partizan Games (PGs) were invented by John H. Conway and are described in his book On Numbers and Games. We formalize PGs in Higher Order Logic extended with ZF axioms (HOLZF) using Isabelle, a mechanical proof assistant. We show that PGs can be defined as the unique fixpoint of a function that arises naturally from Conway’s original definition. While the construction of PGs in HOLZF relies heavily on the ZF axioms, operations on PGs are defined on a game type that hides its set theoretic origins. A polymorphic type of sets that are not bigger than ZF sets facilitates this. We formalize the induction principle that Conway uses throughout his proofs about games, and prove its correctness. For these purposes we examine how the notions of well-foundedness in HOL and ZF are related in HOLZF. Finally, games (modulo equality) are added to Isabelle’s numeric types by showing that they are an instance of the axiomatic type class of partially ordered abelian groups.


Proper Class Induction Principle Game Type Polymorphic Type Modulo Equality 
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  1. 1.
    Conway, J.H.: On Numbers And Games, 2nd edn. A K Peters Ltd. (2001)Google Scholar
  2. 2.
    Mamane, L.E.: Surreal Numbers in Coq. In: Filliâtre, J.-C., Paulin-Mohring, C., Werner, B. (eds.) TYPES 2004. LNCS, vol. 3839, pp. 170–185. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Gordon, M.J.C.: Set Theory, Higher Order Logic or Both. In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 190–201. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Agerholm, S.: Formalising a Model of the λ-Calculus in HOL-ST. Technical Report 354, University of Cambridge Computer Laboratory (1994)Google Scholar
  5. 5.
    Agerholm, S., Gordon, M.J.C.: Experiments with ZF Set Theory in HOL and Isabelle. Technical Report RS-95-37, BRICS (1995)Google Scholar
  6. 6.
    Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Automated Reasoning 11, 353–389 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Paulson, L.C.: Set theory for verification: II. Induction and Recursion. J. Automated Reasoning 15, 167–215 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  9. 9.
    Baader, F., Nipkow, T.: Term Rewriting and All That, Cambridge U.P (1998)Google Scholar
  10. 10.
    Jech, T.: Set Theory, 3rd rev. edn. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  11. 11.
    Paulson, L.C.: Organizing Numerical Theories Using Axiomatic Type Classes. Journal of Automated Reasoning 33(1), 29–49 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Paulson, L.C.: Defining Functions on Equivalence Classes. In: ACM Transactions on Computational Logic (in press)Google Scholar
  13. 13.
    Obua, S.: Proving bounds for real linear programs in isabelle/HOL. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 227–244. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Obua, S.: Partizan Games in Isabelle/HOLZF,

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Steven Obua
    • 1
  1. 1.Technische Universität MünchenGarchingGermany

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