src/ZF/Order.thy
changeset 435 ca5356bd315a
child 578 efc648d29dd0
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/ZF/Order.thy	Tue Jun 21 17:20:34 1994 +0200
     1.3 @@ -0,0 +1,31 @@
     1.4 +(*  Title: 	ZF/Order.thy
     1.5 +    ID:         $Id$
     1.6 +    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     1.7 +    Copyright   1994  University of Cambridge
     1.8 +
     1.9 +Orders in Zermelo-Fraenkel Set Theory 
    1.10 +*)
    1.11 +
    1.12 +Order = WF + Perm + 
    1.13 +consts
    1.14 +  part_ord        :: "[i,i]=>o"		(*Strict partial ordering*)
    1.15 +  linear, tot_ord :: "[i,i]=>o"		(*Strict total ordering*)
    1.16 +  well_ord        :: "[i,i]=>o"		(*Well-ordering*)
    1.17 +  ord_iso         :: "[i,i,i,i]=>i"	(*Order isomorphisms*)
    1.18 +  pred            :: "[i,i,i]=>i"	(*Set of predecessors*)
    1.19 +
    1.20 +rules
    1.21 +  part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"
    1.22 +
    1.23 +  linear_def   "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"
    1.24 +
    1.25 +  tot_ord_def  "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"
    1.26 +
    1.27 +  well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"
    1.28 +
    1.29 +  ord_iso_def  "ord_iso(A,r,B,s) == \
    1.30 +\                   {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"
    1.31 +
    1.32 +  pred_def     "pred(A,x,r) == {y:A. <y,x>:r}"
    1.33 +
    1.34 +end