src/ZF/Order.thy
author clasohm
Sat Dec 09 13:36:11 1995 +0100 (1995-12-09)
changeset 1401 0c439768f45c
parent 1155 928a16e02f9f
child 1478 2b8c2a7547ab
permissions -rw-r--r--
removed quotes from consts and syntax sections
     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   mono_map        :: [i,i,i,i]=>i	(*Order-preserving maps*)
    15   ord_iso         :: [i,i,i,i]=>i	(*Order isomorphisms*)
    16   pred            :: [i,i,i]=>i	(*Set of predecessors*)
    17   ord_iso_map     :: [i,i,i,i]=>i	(*Construction for linearity theorem*)
    18 
    19 defs
    20   part_ord_def "part_ord(A,r) == irrefl(A,r) & trans[A](r)"
    21 
    22   linear_def   "linear(A,r) == (ALL x:A. ALL y:A. <x,y>:r | x=y | <y,x>:r)"
    23 
    24   tot_ord_def  "tot_ord(A,r) == part_ord(A,r) & linear(A,r)"
    25 
    26   well_ord_def "well_ord(A,r) == tot_ord(A,r) & wf[A](r)"
    27 
    28   mono_map_def "mono_map(A,r,B,s) == 
    29                    {f: A->B. ALL x:A. ALL y:A. <x,y>:r --> <f`x,f`y>:s}"
    30 
    31   ord_iso_def  "ord_iso(A,r,B,s) == 
    32                    {f: bij(A,B). ALL x:A. ALL y:A. <x,y>:r <-> <f`x,f`y>:s}"
    33 
    34   pred_def     "pred(A,x,r) == {y:A. <y,x>:r}"
    35 
    36   ord_iso_map_def
    37      "ord_iso_map(A,r,B,s) == 
    38        UN x:A. UN y:B. UN f: ord_iso(pred(A,x,r), r, pred(B,y,s), s).   
    39             {<x,y>}"
    40 
    41 end