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(* Title: ZF/Order.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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Orders in Zermelo-Fraenkel Set Theory
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*)
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Order = WF + Perm +
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consts
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part_ord :: "[i,i]=>o" (*Strict partial ordering*)
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linear, tot_ord :: "[i,i]=>o" (*Strict total ordering*)
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well_ord :: "[i,i]=>o" (*Well-ordering*)
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ord_iso :: "[i,i,i,i]=>i" (*Order isomorphisms*)
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pred :: "[i,i,i]=>i" (*Set of predecessors*)
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rules
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part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"
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linear_def "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"
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tot_ord_def "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"
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well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"
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ord_iso_def "ord_iso(A,r,B,s) == \
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\ {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"
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pred_def "pred(A,x,r) == {y:A. <y,x>:r}"
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end
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