Set theory for verification. I: from foundations to functions
Journal of Automated Reasoning
Term rewriting and all that
Set Theory, Higher Order Logic or Both?
TPHOLs '96 Proceedings of the 9th International Conference on Theorem Proving in Higher Order Logics
Organizing Numerical Theories Using Axiomatic Type Classes
Journal of Automated Reasoning
Defining functions on equivalence classes
ACM Transactions on Computational Logic (TOCL)
Isabelle/HOL: a proof assistant for higher-order logic
Isabelle/HOL: a proof assistant for higher-order logic
Proving bounds for real linear programs in Isabelle/HOL
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
TYPES'04 Proceedings of the 2004 international conference on Types for Proofs and Programs
Imperative Functional Programming with Isabelle/HOL
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
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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.