--- a/src/ZF/AC/WO6_WO1.thy Thu Apr 13 14:15:36 1995 +0200
+++ b/src/ZF/AC/WO6_WO1.thy Thu Apr 13 14:18:22 1995 +0200
@@ -6,16 +6,23 @@
From the book "Equivalents of the Axiom of Choice" by Rubin & Rubin,
pages 2-5
+
+
+ vv1_def "vv1(f,m,b) == if(f`b ~= 0, \
+\ domain(uu(f, b, thing(f,b,g,d), \
+\ LEAST d. domain(uu(f, b, thing(f,b,g,d), d)) ~= 0 & \
+\ domain(uu(f,b, thing(f,b,g,d), d)) lepoll m)), 0)"
+
*)
-WO6_WO1 = "rel_is_fun" + AC_Equiv +
+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 :: "[i, i, i] => i"
- vv2, ww2 :: "[i, i, i, i] => i"
+ 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
@@ -23,23 +30,27 @@
\ (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))"
- vv1_def "vv1(f,b,m) == if(f`b ~= 0, \
-\ domain(uu(f,b, \
-\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
-\ domain(uu(f,b,g,d)) lesspoll m)), \
-\ LEAST d. domain(uu(f,b, \
-\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
-\ domain(uu(f,b,g,d)) lesspoll m)), d)) ~= 0 & \
-\ domain(uu(f,b, \
-\ LEAST g. (EX d. Ord(d) & (domain(uu(f,b,g,d)) ~= 0 & \
-\ domain(uu(f,b,g,d)) lesspoll m)), d)) lesspoll m)), 0)"
-
- ww1_def "ww1(f,b,m) == f`b - vv1(f,b,m)"
+ (*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)"
+\ 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