src/ZF/AC/WO6_WO1.thy
author clasohm
Tue Feb 06 12:27:17 1996 +0100 (1996-02-06)
changeset 1478 2b8c2a7547ab
parent 1401 0c439768f45c
child 6068 2d8f3e1f1151
permissions -rw-r--r--
expanded tabs
     1 (*  Title:      ZF/AC/WO6_WO1.thy
     2     ID:         $Id$
     3     Author:     Krzysztof Grabczewski
     4 
     5 The proof of "WO6 ==> WO1".
     6 
     7 From the book "Equivalents of the Axiom of Choice" by Rubin & Rubin,
     8 pages 2-5
     9 *)
    10 
    11 WO6_WO1 = "rel_is_fun" + AC_Equiv + Let +
    12 
    13 consts
    14 (* Auxiliary definitions used in proof *)
    15   NN            :: i => i
    16   uu            :: [i, i, i, i] => i
    17   vv1, ww1, gg1 :: [i, i, i] => i
    18   vv2, ww2, gg2 :: [i, i, i, i] => i
    19 
    20 defs
    21 
    22   NN_def  "NN(y) == {m:nat. EX a. EX f. Ord(a) & domain(f)=a  &  
    23                             (UN b<a. f`b) = y & (ALL b<a. f`b lepoll m)}"
    24 
    25   uu_def  "uu(f, beta, gamma, delta) == (f`beta * f`gamma) Int f`delta"
    26 
    27   (*Definitions for case 1*)
    28 
    29   vv1_def "vv1(f,m,b) ==                                                
    30            let g = LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & 
    31                                    domain(uu(f,b,g,d)) lepoll m));      
    32                d = LEAST d. domain(uu(f,b,g,d)) ~= 0 &                  
    33                            domain(uu(f,b,g,d)) lepoll m         
    34            in  if(f`b ~= 0, domain(uu(f,b,g,d)), 0)"
    35 
    36   ww1_def "ww1(f,m,b) == f`b - vv1(f,m,b)"
    37 
    38   gg1_def "gg1(f,a,m) == lam b:a++a. if (b<a, vv1(f,m,b), ww1(f,m,b--a))"
    39   
    40   (*Definitions for case 2*)
    41 
    42   vv2_def "vv2(f,b,g,s) ==   
    43            if(f`g ~= 0, {uu(f, b, g, LEAST d. uu(f,b,g,d) ~= 0)`s}, 0)"
    44 
    45   ww2_def "ww2(f,b,g,s) == f`g - vv2(f,b,g,s)"
    46 
    47   gg2_def "gg2(f,a,b,s) == lam g:a++a. if (g<a, vv2(f,b,g,s), ww2(f,b,g--a,s))"
    48   
    49 end