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(* Title: ZF/Zorn.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1994 University of Cambridge
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Based upon the article
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Abrial & Laffitte,
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Towards the Mechanization of the Proofs of Some
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Classical Theorems of Set Theory.
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Union_in_Pow is proved in ZF.ML
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*)
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Zorn = OrderArith + AC + "inductive" +
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consts
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Subset_rel :: "i=>i"
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increasing :: "i=>i"
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chain, maxchain :: "i=>i"
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super :: "[i,i]=>i"
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rules
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Subset_rel_def "Subset_rel(A) == {z: A*A . EX x y. z=<x,y> & x<=y & x~=y}"
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increasing_def "increasing(A) == {f: Pow(A)->Pow(A). ALL x. x<=A --> x<=f`x}"
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485
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516
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chain_def "chain(A) == {F: Pow(A). ALL X:F. ALL Y:F. X<=Y | Y<=X}"
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super_def "super(A,c) == {d: chain(A). c<=d & c~=d}"
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maxchain_def "maxchain(A) == {c: chain(A). super(A,c)=0}"
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(** We could make the inductive definition conditional on next: increasing(S)
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but instead we make this a side-condition of an introduction rule. Thus
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the induction rule lets us assume that condition! Many inductive proofs
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are therefore unconditional.
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**)
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consts
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"TFin" :: "[i,i]=>i"
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inductive
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domains "TFin(S,next)" <= "Pow(S)"
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intrs
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nextI "[| x : TFin(S,next); next: increasing(S) \
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\ |] ==> next`x : TFin(S,next)"
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Pow_UnionI "Y : Pow(TFin(S,next)) ==> Union(Y) : TFin(S,next)"
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monos "[Pow_mono]"
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con_defs "[increasing_def]"
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type_intrs "[CollectD1 RS apply_funtype, Union_in_Pow]"
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end
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