New unified treatment of sequent calculi by Sara Kalvala
combines the old LK and Modal with the new ILL (Int. Linear Logic)
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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
include
<DL>
<DT>Makefile
<DD>compiles the files under Poly/ML or SML of New Jersey. Can also
run all the tests described below.<P>
<DT>ROOT.ML
<DD>loads all source files. Enter an ML image containing FOL and
type: use "ROOT.ML";
</DL>
There are also several subdirectories of examples. To execute the examples on
some directory <Dir>, enter an ML image containing ZF and type
<TT>use "<Dir>/ROOT.ML";</TT>
<DL>
<DT>AC
<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
Gr`abczewski.<P>
<DT>Coind
<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>
<DT>IMP
<DD>subdirectory containing a semantics equivalence proof between
operational and denotational definitions of a simple programming language.
Thanks to Heiko Loetzbeyer & Robert Sandner.<P>
<DT>Resid
<DD>subdirectory containing a proof of the Church-Rosser Theorem. It is
by Ole Rasmussen, following the Coq proof by Gérard Huet.<P>
<DT>ex
<DD>subdirectory containing various examples.
</DL>
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:
<UL>
<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.
</UL>
Useful references on ZF set theory:
<UL>
<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)
</UL>
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