# Theory AC15_WO6

theory AC15_WO6
imports HH Cardinal_aux
```(*  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 ==> AC10(n) ==> AC11 ==> AC12 ==> AC15 ==> WO6

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

AC10(succ(n)) ==> AC13(n) ==> AC14 ==> AC15

or

AC1 ==> AC13(1);  AC13(m) ==> AC13(n) ==> AC14 ==> AC15    (m≤n)

So we don't have to prove all implications of both cases.
Moreover we don't need to prove AC13(1) ==> AC1 and AC11 ==> 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"
apply (unfold 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"
apply (unfold 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"
apply (unfold 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 = "%y. 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) ==> AC11                                                       *)
(* ********************************************************************** *)

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

(* ********************************************************************** *)
(* AC11 ==> AC12                                                          *)
(* ********************************************************************** *)

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

(* ********************************************************************** *)
(* AC12 ==> AC15                                                          *)
(* ********************************************************************** *)

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

(* ********************************************************************** *)
(* AC15 ==> 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"
apply (unfold 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 (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) ==> AC13(n-1)  if 2≤n                                       *)
(*                                                                        *)
(* Because of the change to the formal definition of AC10(n) we prove     *)
(* the following obviously equivalent theorem ∈                           *)
(* AC10(n) implies AC13(n) for (1≤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 ==> AC13(1)                                                        *)
(* ********************************************************************** *)

lemma AC1_AC13: "AC1 ==> AC13(1)"
apply (unfold 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) ==> AC13(n) for m ⊆ n                                         *)
(* ********************************************************************** *)

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

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

(* ********************************************************************** *)
(* AC13(n) ==> AC14  if 1 ⊆ n                                            *)
(* ********************************************************************** *)

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

(* ********************************************************************** *)
(* AC14 ==> 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) ==> 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)
done

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

(* ********************************************************************** *)
(* AC11 ==> AC14                                                          *)
(* ********************************************************************** *)

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

end

```