1478
|
1 |
(* Title: ZF/AC/AC18_AC19.thy
|
1123
|
2 |
ID: $Id$
|
1478
|
3 |
Author: Krzysztof Grabczewski
|
1123
|
4 |
|
12776
|
5 |
The proof of AC1 ==> AC18 ==> AC19 ==> AC1
|
1123
|
6 |
*)
|
|
7 |
|
12776
|
8 |
theory AC18_AC19 = AC_Equiv:
|
|
9 |
|
|
10 |
constdefs
|
|
11 |
uu :: "i => i"
|
|
12 |
"uu(a) == {c Un {0}. c \<in> a}"
|
|
13 |
|
|
14 |
|
|
15 |
(* ********************************************************************** *)
|
|
16 |
(* AC1 ==> AC18 *)
|
|
17 |
(* ********************************************************************** *)
|
|
18 |
|
|
19 |
lemma PROD_subsets:
|
|
20 |
"[| f \<in> (\<Pi>b \<in> {P(a). a \<in> A}. b); \<forall>a \<in> A. P(a)<=Q(a) |]
|
|
21 |
==> (\<lambda>a \<in> A. f`P(a)) \<in> (\<Pi>a \<in> A. Q(a))"
|
|
22 |
by (rule lam_type, drule apply_type, auto)
|
|
23 |
|
|
24 |
lemma lemma_AC18:
|
|
25 |
"[| \<forall>A. 0 \<notin> A --> (\<exists>f. f \<in> (\<Pi>X \<in> A. X)); A \<noteq> 0 |]
|
|
26 |
==> (\<Inter>a \<in> A. \<Union>b \<in> B(a). X(a, b)) \<subseteq>
|
|
27 |
(\<Union>f \<in> \<Pi>a \<in> A. B(a). \<Inter>a \<in> A. X(a, f`a))"
|
|
28 |
apply (rule subsetI)
|
|
29 |
apply (erule_tac x = "{{b \<in> B (a) . x \<in> X (a,b) }. a \<in> A}" in allE)
|
|
30 |
apply (erule impE, fast)
|
|
31 |
apply (erule exE)
|
|
32 |
apply (rule UN_I)
|
|
33 |
apply (fast elim!: PROD_subsets)
|
|
34 |
apply (simp, fast elim!: not_emptyE dest: apply_type [OF _ RepFunI])
|
|
35 |
done
|
|
36 |
|
|
37 |
lemma AC1_AC18: "AC1 ==> AC18"
|
|
38 |
apply (unfold AC1_def AC18_def)
|
|
39 |
apply (fast elim!: lemma_AC18 apply_type intro!: equalityI INT_I UN_I)
|
|
40 |
done
|
|
41 |
|
|
42 |
(* ********************************************************************** *)
|
|
43 |
(* AC18 ==> AC19 *)
|
|
44 |
(* ********************************************************************** *)
|
|
45 |
|
|
46 |
text{*Hard to express because of the need for meta-quantifiers in AC18*}
|
|
47 |
lemma "AC18 ==> AC19"
|
|
48 |
proof -
|
|
49 |
assume ac18 [unfolded AC18_def, norm_hhf]: AC18
|
|
50 |
show AC19
|
|
51 |
apply (unfold AC18_def AC19_def)
|
|
52 |
apply (intro allI impI)
|
|
53 |
apply (rule ac18 [of _ "%x. x", THEN mp], blast)
|
|
54 |
done
|
|
55 |
qed
|
1123
|
56 |
|
12776
|
57 |
(* ********************************************************************** *)
|
|
58 |
(* AC19 ==> AC1 *)
|
|
59 |
(* ********************************************************************** *)
|
|
60 |
|
|
61 |
lemma RepRep_conj:
|
|
62 |
"[| A \<noteq> 0; 0 \<notin> A |] ==> {uu(a). a \<in> A} \<noteq> 0 & 0 \<notin> {uu(a). a \<in> A}"
|
|
63 |
apply (unfold uu_def, auto)
|
|
64 |
apply (blast dest!: sym [THEN RepFun_eq_0_iff [THEN iffD1]])
|
|
65 |
done
|
|
66 |
|
|
67 |
lemma lemma1_1: "[|c \<in> a; x = c Un {0}; x \<notin> a |] ==> x - {0} \<in> a"
|
|
68 |
apply clarify
|
|
69 |
apply (rule subst_elem , (assumption))
|
|
70 |
apply (fast elim: notE subst_elem)
|
|
71 |
done
|
|
72 |
|
|
73 |
lemma lemma1_2:
|
|
74 |
"[| f`(uu(a)) \<notin> a; f \<in> (\<Pi>B \<in> {uu(a). a \<in> A}. B); a \<in> A |]
|
|
75 |
==> f`(uu(a))-{0} \<in> a"
|
|
76 |
apply (unfold uu_def, fast elim!: lemma1_1 dest!: apply_type)
|
|
77 |
done
|
1123
|
78 |
|
12776
|
79 |
lemma lemma1: "\<exists>f. f \<in> (\<Pi>B \<in> {uu(a). a \<in> A}. B) ==> \<exists>f. f \<in> (\<Pi>B \<in> A. B)"
|
|
80 |
apply (erule exE)
|
|
81 |
apply (rule_tac x = "\<lambda>a\<in>A. if (f` (uu(a)) \<in> a, f` (uu(a)), f` (uu(a))-{0})"
|
|
82 |
in exI)
|
|
83 |
apply (rule lam_type)
|
|
84 |
apply (simp add: lemma1_2)
|
|
85 |
done
|
|
86 |
|
|
87 |
lemma lemma2_1: "a\<noteq>0 ==> 0 \<in> (\<Union>b \<in> uu(a). b)"
|
|
88 |
by (unfold uu_def, auto)
|
|
89 |
|
|
90 |
lemma lemma2: "[| A\<noteq>0; 0\<notin>A |] ==> (\<Inter>x \<in> {uu(a). a \<in> A}. \<Union>b \<in> x. b) \<noteq> 0"
|
|
91 |
apply (erule not_emptyE)
|
|
92 |
apply (rule_tac a = "0" in not_emptyI)
|
|
93 |
apply (fast intro!: lemma2_1)
|
|
94 |
done
|
|
95 |
|
|
96 |
lemma AC19_AC1: "AC19 ==> AC1"
|
|
97 |
apply (unfold AC19_def AC1_def, clarify)
|
|
98 |
apply (case_tac "A=0", force)
|
|
99 |
apply (erule_tac x = "{uu (a) . a \<in> A}" in allE)
|
|
100 |
apply (erule impE)
|
|
101 |
apply (erule RepRep_conj , (assumption))
|
|
102 |
apply (rule lemma1)
|
|
103 |
apply (drule lemma2 , (assumption))
|
|
104 |
apply auto
|
|
105 |
done
|
1123
|
106 |
|
1203
|
107 |
end
|