src/ZF/Order.thy
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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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  mono_map        :: [i,i,i,i]=>i       (*Order-preserving maps*)
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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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  ord_iso_map     :: [i,i,i,i]=>i       (*Construction for linearity theorem*)
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defs
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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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  mono_map_def "mono_map(A,r,B,s) == 
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                   {f: A->B. ALL x:A. ALL y:A. <x,y>:r --> <f`x,f`y>:s}"
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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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  ord_iso_map_def
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     "ord_iso_map(A,r,B,s) == 
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       UN x:A. UN y:B. UN f: ord_iso(pred(A,x,r), r, pred(B,y,s), s).   
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            {<x,y>}"
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end