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<HTML><HEAD><TITLE>ZF/README</TITLE></HEAD><BODY>
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<H2>ZF: Zermelo-Fraenkel Set Theory</H2>
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This directory contains the ML sources of the Isabelle system for
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ZF Set Theory, based on FOL.<p>
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There are several subdirectories of examples:
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<DL>
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<DT>AC
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<DD>subdirectory containing proofs from the book "Equivalents of the Axiom
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of Choice, II" by H. Rubin and J.E. Rubin, 1985. Thanks to Krzysztof
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Gr`abczewski.<P>
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<DT>Coind
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<DD>subdirectory containing a large example of proof by co-induction. It
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is by Jacob Frost following a paper by Robin Milner and Mads Tofte.<P>
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<DT>IMP
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<DD>subdirectory containing a semantics equivalence proof between
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operational and denotational definitions of a simple programming language.
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Thanks to Heiko Loetzbeyer & Robert Sandner.<P>
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<DT>Resid
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<DD>subdirectory containing a proof of the Church-Rosser Theorem. It is
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by Ole Rasmussen, following the Coq proof by Gérard Huet.<P>
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<DT>ex
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<DD>subdirectory containing various examples.
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</DL>
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Isabelle/ZF formalizes the greater part of elementary set theory,
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including relations, functions, injections, surjections, ordinals and
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cardinals. Results proved include Cantor's Theorem, the Recursion
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Theorem, the Schroeder-Bernstein Theorem, and (assuming AC) the
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Wellordering Theorem.<P>
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Isabelle/ZF also provides theories of lists, trees, etc., for
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formalizing computational notions. It supports inductive definitions
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of infinite-branching trees for any cardinality of branching.<P>
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Useful references for Isabelle/ZF:
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<UL>
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<LI> Lawrence C. Paulson,<BR>
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Set theory for verification: I. From foundations to functions.<BR>
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J. Automated Reasoning 11 (1993), 353-389.
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<LI> Lawrence C. Paulson,<BR>
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Set theory for verification: II. Induction and recursion.<BR>
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Report 312, Computer Lab (1993).<BR>
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<LI> Lawrence C. Paulson,<BR>
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A fixedpoint approach to implementing (co)inductive definitions. <BR>
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In: A. Bundy (editor),<BR>
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CADE-12: 12th International Conference on Automated Deduction,<BR>
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(Springer LNAI 814, 1994), 148-161.
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</UL>
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Useful references on ZF set theory:
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<UL>
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<LI> Paul R. Halmos, Naive Set Theory (Van Nostrand, 1960)
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<LI> Patrick Suppes, Axiomatic Set Theory (Dover, 1972)
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<LI> Keith J. Devlin,<BR>
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Fundamentals of Contemporary Set Theory (Springer, 1979)
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<LI> Kenneth Kunen<BR>
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Set Theory: An Introduction to Independence Proofs<BR>
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(North-Holland, 1980)
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</UL>
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</BODY></HTML>
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