# Theory AC18_AC19

theory AC18_AC19
imports AC_Equiv
```(*  Title:      ZF/AC/AC18_AC19.thy
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 ∪ {0}. c ∈ a}"

(* ********************************************************************** *)
(* AC1 ==> AC18                                                           *)
(* ********************************************************************** *)

lemma PROD_subsets:
"[| f ∈ (∏b ∈ {P(a). a ∈ A}. b);  ∀a ∈ A. P(a)<=Q(a) |]
==> (λa ∈ A. f`P(a)) ∈ (∏a ∈ A. Q(a))"
by (rule lam_type, drule apply_type, auto)

lemma lemma_AC18:
"[| ∀A. 0 ∉ A ⟶ (∃f. f ∈ (∏X ∈ A. X)); A ≠ 0 |]
==> (⋂a ∈ A. ⋃b ∈ B(a). X(a, b)) ⊆
(⋃f ∈ ∏a ∈ A. B(a). ⋂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 ∪ {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 ∈ (∏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 ∈ (∏B ∈ {uu(a). a ∈ A}. B) ==> ∃f. f ∈ (∏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)
done

lemma lemma2_1: "a≠0 ==> 0 ∈ (⋃b ∈ uu(a). b)"
by (unfold uu_def, auto)

lemma lemma2: "[| A≠0; 0∉A |] ==> (⋂x ∈ {uu(a). a ∈ A}. ⋃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
```