(* Title: ZF/AC/AC18_AC19.thy ID: $Id: AC18_AC19.thy,v 1.13 2007/10/07 19:19:33 wenzelm Exp $ Author: Krzysztof Grabczewski The proof of AC1 ==> AC18 ==> AC19 ==> AC1 *) theory AC18_AC19 imports AC_Equiv begin definition uu :: "i => i" where "uu(a) == {c Un {0}. c ∈ a}" (* ********************************************************************** *) (* AC1 ==> AC18 *) (* ********************************************************************** *) lemma PROD_subsets: "[| f ∈ (Π b ∈ {P(a). a ∈ A}. b); ∀a ∈ A. P(a)<=Q(a) |] ==> (λa ∈ A. f`P(a)) ∈ (Π a ∈ A. Q(a))" by (rule lam_type, drule apply_type, auto) lemma lemma_AC18: "[| ∀A. 0 ∉ A --> (∃f. f ∈ (Π X ∈ A. X)); A ≠ 0 |] ==> (\<Inter>a ∈ A. \<Union>b ∈ B(a). X(a, b)) ⊆ (\<Union>f ∈ Π a ∈ A. B(a). \<Inter>a ∈ A. X(a, f`a))" apply (rule subsetI) apply (erule_tac x = "{{b ∈ B (a) . x ∈ X (a,b) }. a ∈ A}" in allE) apply (erule impE, fast) apply (erule exE) apply (rule UN_I) apply (fast elim!: PROD_subsets) apply (simp, fast elim!: not_emptyE dest: apply_type [OF _ RepFunI]) done lemma AC1_AC18: "AC1 ==> PROP AC18" apply (unfold AC1_def) apply (rule AC18.intro) apply (fast elim!: lemma_AC18 apply_type intro!: equalityI INT_I UN_I) done (* ********************************************************************** *) (* AC18 ==> AC19 *) (* ********************************************************************** *) theorem (in AC18) AC19 apply (unfold AC19_def) apply (intro allI impI) apply (rule AC18 [of _ "%x. x", THEN mp], blast) done (* ********************************************************************** *) (* AC19 ==> AC1 *) (* ********************************************************************** *) lemma RepRep_conj: "[| A ≠ 0; 0 ∉ A |] ==> {uu(a). a ∈ A} ≠ 0 & 0 ∉ {uu(a). a ∈ A}" apply (unfold uu_def, auto) apply (blast dest!: sym [THEN RepFun_eq_0_iff [THEN iffD1]]) done lemma lemma1_1: "[|c ∈ a; x = c Un {0}; x ∉ a |] ==> x - {0} ∈ a" apply clarify apply (rule subst_elem, assumption) apply (fast elim: notE subst_elem) done lemma lemma1_2: "[| f`(uu(a)) ∉ a; f ∈ (Π B ∈ {uu(a). a ∈ A}. B); a ∈ A |] ==> f`(uu(a))-{0} ∈ a" apply (unfold uu_def, fast elim!: lemma1_1 dest!: apply_type) done lemma lemma1: "∃f. f ∈ (Π B ∈ {uu(a). a ∈ A}. B) ==> ∃f. f ∈ (Π B ∈ A. B)" apply (erule exE) apply (rule_tac x = "λa∈A. if (f` (uu(a)) ∈ a, f` (uu(a)), f` (uu(a))-{0})" in exI) apply (rule lam_type) apply (simp add: lemma1_2) done lemma lemma2_1: "a≠0 ==> 0 ∈ (\<Union>b ∈ uu(a). b)" by (unfold uu_def, auto) lemma lemma2: "[| A≠0; 0∉A |] ==> (\<Inter>x ∈ {uu(a). a ∈ A}. \<Union>b ∈ x. b) ≠ 0" apply (erule not_emptyE) apply (rule_tac a = 0 in not_emptyI) apply (fast intro!: lemma2_1) done lemma AC19_AC1: "AC19 ==> AC1" apply (unfold AC19_def AC1_def, clarify) apply (case_tac "A=0", force) apply (erule_tac x = "{uu (a) . a ∈ A}" in allE) apply (erule impE) apply (erule RepRep_conj, assumption) apply (rule lemma1) apply (drule lemma2, assumption, auto) done end
lemma PROD_subsets:
[| f ∈ (Πb∈{P(a) . a ∈ A}. b); ∀a∈A. P(a) ⊆ Q(a) |]
==> (λa∈A. f ` P(a)) ∈ (Πa∈A. Q(a))
lemma lemma_AC18:
[| ∀A. 0 ∉ A --> (∃f. f ∈ (ΠX∈A. X)); A ≠ 0 |]
==> (\<Inter>a∈A. \<Union>b∈B(a). X(a, b)) ⊆
(\<Union>f∈Πa∈A. B(a). \<Inter>a∈A. X(a, f ` a))
lemma AC1_AC18:
AC1 ==> PROP AC18
theorem
AC19
lemma RepRep_conj:
[| A ≠ 0; 0 ∉ A |] ==> {uu(a) . a ∈ A} ≠ 0 ∧ 0 ∉ {uu(a) . a ∈ A}
lemma lemma1_1:
[| c ∈ a; x = c ∪ {0}; x ∉ a |] ==> x - {0} ∈ a
lemma lemma1_2:
[| f ` uu(a) ∉ a; f ∈ (ΠB∈{uu(a) . a ∈ A}. B); a ∈ A |] ==> f ` uu(a) - {0} ∈ a
lemma lemma1:
∃f. f ∈ (ΠB∈{uu(a) . a ∈ A}. B) ==> ∃f. f ∈ (ΠB∈A. B)
lemma lemma2_1:
a ≠ 0 ==> 0 ∈ (\<Union>b∈uu(a). b)
lemma lemma2:
[| A ≠ 0; 0 ∉ A |] ==> (\<Inter>x∈{uu(a) . a ∈ A}. \<Union>b∈x. b) ≠ 0
lemma AC19_AC1:
AC19 ==> AC1