# Theory AC18_AC19

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theory AC18_AC19
imports AC_Equiv
begin

    (*  Title:      ZF/AC/AC18_AC19.thy
        ID:         $Id: AC18_AC19.thy,v 1.12 2005/06/17 14:15:10 haftmann Exp $
        Author:     Krzysztof Grabczewski

    The proof of AC1 ==> AC18 ==> AC19 ==> AC1
    *)

    theory AC18_AC19 imports AC_Equiv begin

    constdefs
      uu    :: "i => i"
        "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

      PROP AC18 ==> 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