src/ZF/Order.thy

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(*  Title:      ZF/Order.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Orders in Zermelo-Fraenkel Set Theory 
*)

Order = WF + Perm + 
consts
  part_ord        :: [i,i]=>o           (*Strict partial ordering*)
  linear, tot_ord :: [i,i]=>o           (*Strict total ordering*)
  well_ord        :: [i,i]=>o           (*Well-ordering*)
  mono_map        :: [i,i,i,i]=>i       (*Order-preserving maps*)
  ord_iso         :: [i,i,i,i]=>i       (*Order isomorphisms*)
  pred            :: [i,i,i]=>i		(*Set of predecessors*)
  ord_iso_map     :: [i,i,i,i]=>i       (*Construction for linearity theorem*)

defs
  part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"

  linear_def   "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"

  tot_ord_def  "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"

  well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"

  mono_map_def "mono_map(A,r,B,s) == 
                   {f: A->B. ALL x:A. ALL y:A. <x,y>:r --> <f`x,f`y>:s}"

  ord_iso_def  "ord_iso(A,r,B,s) == 
                   {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"

  pred_def     "pred(A,x,r) == {y:A. <y,x>:r}"

  ord_iso_map_def
     "ord_iso_map(A,r,B,s) == 
       UN x:A. UN y:B. UN f: ord_iso(pred(A,x,r), r, pred(B,y,s), s).   
            {<x,y>}"

constdefs

  first :: [i, i, i] => o
    "first(u, X, R) == u:X & (ALL v:X. v~=u --> <u,v> : R)"

end