(*  Title:      ZF/AC/AC15_WO6.thy
    Author:     Krzysztof Grabczewski

The proofs needed to state that AC10, ..., AC15 are equivalent to the rest.
We need the following:

WO1 \<Longrightarrow> AC10(n) \<Longrightarrow> AC11 \<Longrightarrow> AC12 \<Longrightarrow> AC15 \<Longrightarrow> WO6

In order to add the formulations AC13 and AC14 we need:

AC10(succ(n)) \<Longrightarrow> AC13(n) \<Longrightarrow> AC14 \<Longrightarrow> AC15

or

AC1 \<Longrightarrow> AC13(1);  AC13(m) \<Longrightarrow> AC13(n) \<Longrightarrow> AC14 \<Longrightarrow> AC15    (m\<le>n)

So we don't have to prove all implications of both cases.
Moreover we don't need to prove AC13(1) \<Longrightarrow> AC1 and AC11 \<Longrightarrow> AC14 as
Rubin & Rubin do.
*)

theory AC15_WO6
imports HH Cardinal_aux
begin


(* ********************************************************************** *)
(* Lemmas used in the proofs in which the conclusion is AC13, AC14        *)
(* or AC15                                                                *)
(*  - cons_times_nat_not_Finite                                           *)
(*  - ex_fun_AC13_AC15                                                    *)
(* ********************************************************************** *)

lemma lepoll_Sigma: "A≠0 ==> B < A*B"
  unfolding lepoll_def
apply (erule not_emptyE)
apply (rule_tac x = "λz ∈ B. ⟨x,z⟩" in exI)
apply (fast intro!: snd_conv lam_injective)
done

lemma cons_times_nat_not_Finite:
     "0∉A ==> ∀B ∈ {cons(0,x*nat). x ∈ A}. ¬Finite(B)"
apply clarify 
apply (rule nat_not_Finite [THEN notE] )
apply (subgoal_tac "x ≠ 0")
 apply (blast intro: lepoll_Sigma [THEN lepoll_Finite])+
done

lemma lemma1: "[∪(C)=A; a ∈ A] ==> ∃B ∈ C. a ∈ B ∧ B ⊆ A"
by fast

lemma lemma2: 
        "[pairwise_disjoint(A); B ∈ A; C ∈ A; a ∈ B; a ∈ C] ==> B=C"
by (unfold pairwise_disjoint_def, blast) 

lemma lemma3: 
     "∀B ∈ {cons(0, x*nat). x ∈ A}. pairwise_disjoint(f`B) ∧
             sets_of_size_between(f`B, 2, n) ∧ ∪(f`B)=B
     ==> ∀B ∈ A. ∃! u. u ∈ f`cons(0, B*nat) ∧ u ⊆ cons(0, B*nat) ∧
             0 ∈ u ∧ 2 < u ∧ u < n"
  unfolding sets_of_size_between_def
apply (rule ballI)
apply (erule_tac x="cons(0, B*nat)" in ballE)
 apply (blast dest: lemma1 intro!: lemma2, blast)
done

lemma lemma4: "[A < i; Ord(i)] ==> {P(a). a ∈ A} < i"
  unfolding lepoll_def
apply (erule exE)
apply (rule_tac x = "λx ∈ RepFun(A,P). μ j. ∃a∈A. x=P(a) ∧ f`a=j" 
       in exI)
apply (rule_tac d = "λy. P (converse (f) `y) " in lam_injective)
apply (erule RepFunE)
apply (frule inj_is_fun [THEN apply_type], assumption)
apply (fast intro: LeastI2 elim!: Ord_in_Ord inj_is_fun [THEN apply_type])
apply (erule RepFunE)
apply (rule LeastI2)
  apply fast
 apply (fast elim!: Ord_in_Ord inj_is_fun [THEN apply_type])
apply (fast elim: sym left_inverse [THEN ssubst])
done

lemma lemma5_1:
     "[B ∈ A; 2 < u(B)] ==> (λx ∈ A. {fst(x). x ∈ u(x)-{0}})`B ≠ 0"
apply simp
apply (fast dest: lepoll_Diff_sing 
            elim: lepoll_trans [THEN succ_lepoll_natE] ssubst
            intro!: lepoll_refl)
done

lemma lemma5_2:
     "[B ∈ A; u(B) ⊆ cons(0, B*nat)]
      ==> (λx ∈ A. {fst(x). x ∈ u(x)-{0}})`B ⊆ B"
apply auto 
done

lemma lemma5_3:
     "[n ∈ nat; B ∈ A; 0 ∈ u(B); u(B) < succ(n)]
      ==> (λx ∈ A. {fst(x). x ∈ u(x)-{0}})`B < n"
apply simp
apply (fast elim!: Diff_lepoll [THEN lemma4 [OF _ nat_into_Ord]])
done

lemma ex_fun_AC13_AC15:
     "[∀B ∈ {cons(0, x*nat). x ∈ A}.
                pairwise_disjoint(f`B) ∧
                sets_of_size_between(f`B, 2, succ(n)) ∧ ∪(f`B)=B;
         n ∈ nat]
      ==> ∃f. ∀B ∈ A. f`B ≠ 0 ∧ f`B ⊆ B ∧ f`B < n"
by (fast del: subsetI notI
         dest!: lemma3 theI intro!: lemma5_1 lemma5_2 lemma5_3)


(* ********************************************************************** *)
(* The target proofs                                                      *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* AC10(n) \<Longrightarrow> AC11                                                       *)
(* ********************************************************************** *)

theorem AC10_AC11: "[n ∈ nat; 1≤n; AC10(n)] ==> AC11"
by (unfold AC10_def AC11_def, blast)

(* ********************************************************************** *)
(* AC11 \<Longrightarrow> AC12                                                          *)
(* ********************************************************************** *)

theorem AC11_AC12: "AC11 ==> AC12"
by (unfold AC10_def AC11_def AC11_def AC12_def, blast)

(* ********************************************************************** *)
(* AC12 \<Longrightarrow> AC15                                                          *)
(* ********************************************************************** *)

theorem AC12_AC15: "AC12 ==> AC15"
  unfolding AC12_def AC15_def
apply (blast del: ballI 
             intro!: cons_times_nat_not_Finite ex_fun_AC13_AC15)
done

(* ********************************************************************** *)
(* AC15 \<Longrightarrow> WO6                                                           *)
(* ********************************************************************** *)

lemma OUN_eq_UN: "Ord(x) ==> (∪a<x. F(a)) = (∪a ∈ x. F(a))"
by (fast intro!: ltI dest!: ltD)

lemma AC15_WO6_aux1:
     "∀x ∈ Pow(A)-{0}. f`x≠0 ∧ f`x ⊆ x ∧ f`x < m
      ==> (∪i<μ x. HH(f,A,x)={A}. HH(f,A,i)) = A"
apply (simp add: Ord_Least [THEN OUN_eq_UN])
apply (rule equalityI)
apply (fast dest!: less_Least_subset_x)
apply (blast del: subsetI 
           intro!: f_subsets_imp_UN_HH_eq_x [THEN Diff_eq_0_iff [THEN iffD1]])
done

lemma AC15_WO6_aux2:
     "∀x ∈ Pow(A)-{0}. f`x≠0 ∧ f`x ⊆ x ∧ f`x < m
      ==> ∀x < (μ x. HH(f,A,x)={A}). HH(f,A,x) < m"
apply (rule oallI)
apply (drule ltD [THEN less_Least_subset_x])
apply (frule HH_subset_imp_eq)
apply (erule ssubst)
apply (blast dest!: HH_subset_x_imp_subset_Diff_UN [THEN not_emptyI2])
        (*but can't use del: DiffE despite the obvious conflict*)
done

theorem AC15_WO6: "AC15 ==> WO6"
  unfolding AC15_def WO6_def
apply (rule allI)
apply (erule_tac x = "Pow (A) -{0}" in allE)
apply (erule impE, fast)
apply (elim bexE conjE exE)
apply (rule bexI)
 apply (rule conjI, assumption)
 apply (rule_tac x = "μ i. HH (f,A,i) ={A}" in exI)
 apply (rule_tac x = "λj ∈ (μ i. HH (f,A,i) ={A}) . HH (f,A,j) " in exI)
 apply (simp_all add: ltD)
apply (fast intro!: Ord_Least lam_type [THEN domain_of_fun]
            elim!: less_Least_subset_x AC15_WO6_aux1 AC15_WO6_aux2) 
done


(* ********************************************************************** *)
(* The proof needed in the first case, not in the second                  *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* AC10(n) \<Longrightarrow> AC13(n-1)  if 2\<le>n                                       *)
(*                                                                        *)
(* Because of the change to the formal definition of AC10(n) we prove     *)
(* the following obviously equivalent theorem \<in>                           *)
(* AC10(n) implies AC13(n) for (1\<le>n)                                   *)
(* ********************************************************************** *)

theorem AC10_AC13: "[n ∈ nat; 1≤n; AC10(n)] ==> AC13(n)"
apply (unfold AC10_def AC13_def, safe)
apply (erule allE) 
apply (erule impE [OF _ cons_times_nat_not_Finite], assumption) 
apply (fast elim!: impE [OF _ cons_times_nat_not_Finite] 
            dest!: ex_fun_AC13_AC15)
done

(* ********************************************************************** *)
(* The proofs needed in the second case, not in the first                 *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* AC1 \<Longrightarrow> AC13(1)                                                        *)
(* ********************************************************************** *)

lemma AC1_AC13: "AC1 ==> AC13(1)"
  unfolding AC1_def AC13_def
apply (rule allI)
apply (erule allE)
apply (rule impI)
apply (drule mp, assumption) 
apply (elim exE)
apply (rule_tac x = "λx ∈ A. {f`x}" in exI)
apply (simp add: singleton_eqpoll_1 [THEN eqpoll_imp_lepoll])
done

(* ********************************************************************** *)
(* AC13(m) \<Longrightarrow> AC13(n) for m \<subseteq> n                                         *)
(* ********************************************************************** *)

lemma AC13_mono: "[m≤n; AC13(m)] ==> AC13(n)"
  unfolding AC13_def
apply (drule le_imp_lepoll)
apply (fast elim!: lepoll_trans)
done

(* ********************************************************************** *)
(* The proofs necessary for both cases                                    *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* AC13(n) \<Longrightarrow> AC14  if 1 \<subseteq> n                                            *)
(* ********************************************************************** *)

theorem AC13_AC14: "[n ∈ nat; 1≤n; AC13(n)] ==> AC14"
by (unfold AC13_def AC14_def, auto)

(* ********************************************************************** *)
(* AC14 \<Longrightarrow> AC15                                                          *)
(* ********************************************************************** *)

theorem AC14_AC15: "AC14 ==> AC15"
by (unfold AC13_def AC14_def AC15_def, fast)

(* ********************************************************************** *)
(* The redundant proofs; however cited by Rubin & Rubin                   *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* AC13(1) \<Longrightarrow> AC1                                                        *)
(* ********************************************************************** *)

lemma lemma_aux: "[A≠0; A < 1] ==> ∃a. A={a}"
by (fast elim!: not_emptyE lepoll_1_is_sing)

lemma AC13_AC1_lemma:
     "∀B ∈ A. f(B)≠0 ∧ f(B)<=B ∧ f(B) < 1
      ==> (λx ∈ A. THE y. f(x)={y}) ∈ (∏X ∈ A. X)"
apply (rule lam_type)
apply (drule bspec, assumption)
apply (elim conjE)
apply (erule lemma_aux [THEN exE], assumption)
apply (simp add: the_equality)
done

theorem AC13_AC1: "AC13(1) ==> AC1"
  unfolding AC13_def AC1_def
apply (fast elim!: AC13_AC1_lemma)
done

(* ********************************************************************** *)
(* AC11 \<Longrightarrow> AC14                                                          *)
(* ********************************************************************** *)

theorem AC11_AC14: "AC11 ==> AC14"
  unfolding AC11_def AC14_def
apply (fast intro!: AC10_AC13)
done

end