ZF: Zermelo-Fraenkel Set Theory
This directory contains the ML sources of the Isabelle system for
ZF Set Theory, based on FOL.
There are several subdirectories of examples:
- AC
- subdirectory containing proofs from the book "Equivalents of the Axiom
of Choice, II" by H. Rubin and J.E. Rubin, 1985. Thanks to Krzysztof
Gr`abczewski.
- Coind
- subdirectory containing a large example of proof by co-induction. It
is by Jacob Frost following a paper by Robin Milner and Mads Tofte.
- IMP
- subdirectory containing a semantics equivalence proof between
operational and denotational definitions of a simple programming language.
Thanks to Heiko Loetzbeyer & Robert Sandner.
- Resid
- subdirectory containing a proof of the Church-Rosser Theorem. It is
by Ole Rasmussen, following the Coq proof by G�ard Huet.
- ex
- subdirectory containing various examples.
Isabelle/ZF formalizes the greater part of elementary set theory,
including relations, functions, injections, surjections, ordinals and
cardinals. Results proved include Cantor's Theorem, the Recursion
Theorem, the Schroeder-Bernstein Theorem, and (assuming AC) the
Wellordering Theorem.
Isabelle/ZF also provides theories of lists, trees, etc., for
formalizing computational notions. It supports inductive definitions
of infinite-branching trees for any cardinality of branching.
Useful references for Isabelle/ZF:
- Lawrence C. Paulson,
Set theory for verification: I. From foundations to functions.
J. Automated Reasoning 11 (1993), 353-389.
- Lawrence C. Paulson,
Set theory for verification: II. Induction and recursion.
Report 312, Computer Lab (1993).
- Lawrence C. Paulson,
A fixedpoint approach to implementing (co)inductive definitions.
In: A. Bundy (editor),
CADE-12: 12th International Conference on Automated Deduction,
(Springer LNAI 814, 1994), 148-161.
Useful references on ZF set theory:
- Paul R. Halmos, Naive Set Theory (Van Nostrand, 1960)
- Patrick Suppes, Axiomatic Set Theory (Dover, 1972)
- Keith J. Devlin,
Fundamentals of Contemporary Set Theory (Springer, 1979)
- Kenneth Kunen
Set Theory: An Introduction to Independence Proofs
(North-Holland, 1980)