(* Title: ZF/AC/WO6_WO1.thy
ID: $Id$
Author: Krzysztof Grabczewski
The proof of "WO6 ==> WO1".
From the book "Equivalents of the Axiom of Choice" by Rubin & Rubin,
pages 2-5
*)
WO6_WO1 = "rel_is_fun" + AC_Equiv + Let +
consts
(* Auxiliary definitions used in proof *)
NN :: i => i
uu :: [i, i, i, i] => i
vv1, ww1, gg1 :: [i, i, i] => i
vv2, ww2, gg2 :: [i, i, i, i] => i
defs
NN_def "NN(y) == {m:nat. EX a. EX f. Ord(a) & domain(f)=a &
(UN b<a. f`b) = y & (ALL b<a. f`b lepoll m)}"
uu_def "uu(f, beta, gamma, delta) == (f`beta * f`gamma) Int f`delta"
(*Definitions for case 1*)
vv1_def "vv1(f,m,b) ==
let g = LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 &
domain(uu(f,b,g,d)) lepoll m));
d = LEAST d. domain(uu(f,b,g,d)) ~= 0 &
domain(uu(f,b,g,d)) lepoll m
in if(f`b ~= 0, domain(uu(f,b,g,d)), 0)"
ww1_def "ww1(f,m,b) == f`b - vv1(f,m,b)"
gg1_def "gg1(f,a,m) == lam b:a++a. if (b<a, vv1(f,m,b), ww1(f,m,b--a))"
(*Definitions for case 2*)
vv2_def "vv2(f,b,g,s) ==
if(f`g ~= 0, {uu(f, b, g, LEAST d. uu(f,b,g,d) ~= 0)`s}, 0)"
ww2_def "ww2(f,b,g,s) == f`g - vv2(f,b,g,s)"
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))"
end