src/ZF/Order.thy
author lcp
Tue Jun 21 17:20:34 1994 +0200 (1994-06-21)
changeset 435 ca5356bd315a
child 578 efc648d29dd0
permissions -rw-r--r--
Addition of cardinals and order types, various tidying
     1 (*  Title: 	ZF/Order.thy
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Orders in Zermelo-Fraenkel Set Theory 
     7 *)
     8 
     9 Order = WF + Perm + 
    10 consts
    11   part_ord        :: "[i,i]=>o"		(*Strict partial ordering*)
    12   linear, tot_ord :: "[i,i]=>o"		(*Strict total ordering*)
    13   well_ord        :: "[i,i]=>o"		(*Well-ordering*)
    14   ord_iso         :: "[i,i,i,i]=>i"	(*Order isomorphisms*)
    15   pred            :: "[i,i,i]=>i"	(*Set of predecessors*)
    16 
    17 rules
    18   part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"
    19 
    20   linear_def   "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"
    21 
    22   tot_ord_def  "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"
    23 
    24   well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"
    25 
    26   ord_iso_def  "ord_iso(A,r,B,s) == \
    27 \                   {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"
    28 
    29   pred_def     "pred(A,x,r) == {y:A. <y,x>:r}"
    30 
    31 end