author lcp Thu, 13 Apr 1995 14:18:22 +0200 changeset 1042 04ef9b8ef1af parent 1041 6664d0b54d0f child 1043 ffa68eb2730b
Defined vv1 using let. Introduced gg1, gg2.
 src/ZF/AC/WO6_WO1.thy file | annotate | diff | comparison | revisions
```--- 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```