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<H2>ZF: Zermelo-Fraenkel Set Theory</H2>

This directory contains the Standard ML sources of the Isabelle system for
ZF Set Theory.  It is loaded upon FOL, not Pure Isabelle.  Important files

<DD>compiles the files under Poly/ML or SML of New Jersey.  Can also
run all the tests described below.<P>

<DD>loads all source files.  Enter an ML image containing FOL and
type: use "ROOT.ML";

There are also several subdirectories of examples.  To execute the examples on
some directory &lt;Dir&gt;, enter an ML image containing ZF and type
<TT>use "&lt;Dir&gt;/ROOT.ML";</TT>

<DD>subdirectory containing proofs from the book "Equivalents of the Axiom
of Choice, II" by H. Rubin and J.E. Rubin, 1985.  Thanks to Krzysztof

<DD>subdirectory containing a large example of proof by co-induction.  It
is by Jacob Frost following a paper by Robin Milner and Mads Tofte.<P>

<DD>subdirectory containing a semantics equivalence proof between
operational and denotational definitions of a simple programming language.
Thanks to Heiko Loetzbeyer & Robert Sandner.<P>

<DD>subdirectory containing a proof of the Church-Rosser Theorem.  It is
by Ole Rasmussen, following the Coq proof by Gérard Huet.<P>

<DD>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.<P>

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.<P>

Useful references for Isabelle/ZF:

<LI>	Lawrence C. Paulson,<BR>
	Set theory for verification: I. From foundations to functions.<BR>
	J. Automated Reasoning 11 (1993), 353-389.

<LI>	Lawrence C. Paulson,<BR>
	Set theory for verification: II. Induction and recursion.<BR>
	Report 312, Computer Lab (1993).<BR>

<LI>	Lawrence C. Paulson,<BR>
	A fixedpoint approach to implementing (co)inductive definitions. <BR> 
	In: A. Bundy (editor),<BR>
	CADE-12: 12th International Conference on Automated Deduction,<BR>
	(Springer LNAI 814, 1994), 148-161.

Useful references on ZF set theory:

<LI>	Paul R. Halmos, Naive Set Theory (Van Nostrand, 1960)

<LI>	Patrick Suppes, Axiomatic Set Theory (Dover, 1972)

<LI>	Keith J. Devlin,<BR>
	Fundamentals of Contemporary Set Theory (Springer, 1979)

<LI>	Kenneth Kunen<BR>
	Set Theory: An Introduction to Independence Proofs<BR>
	(North-Holland, 1980)