(* 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\<noteq>0 \<Longrightarrow> B \<lesssim> A*B"
unfolding lepoll_def
apply (erule not_emptyE)
apply (rule_tac x = "\<lambda>z \<in> B. \<langle>x,z\<rangle>" in exI)
apply (fast intro!: snd_conv lam_injective)
done
lemma cons_times_nat_not_Finite:
"0\<notin>A \<Longrightarrow> \<forall>B \<in> {cons(0,x*nat). x \<in> A}. \<not>Finite(B)"
apply clarify
apply (rule nat_not_Finite [THEN notE] )
apply (subgoal_tac "x \<noteq> 0")
apply (blast intro: lepoll_Sigma [THEN lepoll_Finite])+
done
lemma lemma1: "\<lbrakk>\<Union>(C)=A; a \<in> A\<rbrakk> \<Longrightarrow> \<exists>B \<in> C. a \<in> B \<and> B \<subseteq> A"
by fast
lemma lemma2:
"\<lbrakk>pairwise_disjoint(A); B \<in> A; C \<in> A; a \<in> B; a \<in> C\<rbrakk> \<Longrightarrow> B=C"
by (unfold pairwise_disjoint_def, blast)
lemma lemma3:
"\<forall>B \<in> {cons(0, x*nat). x \<in> A}. pairwise_disjoint(f`B) \<and>
sets_of_size_between(f`B, 2, n) \<and> \<Union>(f`B)=B
\<Longrightarrow> \<forall>B \<in> A. \<exists>! u. u \<in> f`cons(0, B*nat) \<and> u \<subseteq> cons(0, B*nat) \<and>
0 \<in> u \<and> 2 \<lesssim> u \<and> u \<lesssim> 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: "\<lbrakk>A \<lesssim> i; Ord(i)\<rbrakk> \<Longrightarrow> {P(a). a \<in> A} \<lesssim> i"
unfolding lepoll_def
apply (erule exE)
apply (rule_tac x = "\<lambda>x \<in> RepFun(A,P). \<mu> j. \<exists>a\<in>A. x=P(a) \<and> f`a=j"
in exI)
apply (rule_tac d = "\<lambda>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:
"\<lbrakk>B \<in> A; 2 \<lesssim> u(B)\<rbrakk> \<Longrightarrow> (\<lambda>x \<in> A. {fst(x). x \<in> u(x)-{0}})`B \<noteq> 0"
apply simp
apply (fast dest: lepoll_Diff_sing
elim: lepoll_trans [THEN succ_lepoll_natE] ssubst
intro!: lepoll_refl)
done
lemma lemma5_2:
"\<lbrakk>B \<in> A; u(B) \<subseteq> cons(0, B*nat)\<rbrakk>
\<Longrightarrow> (\<lambda>x \<in> A. {fst(x). x \<in> u(x)-{0}})`B \<subseteq> B"
apply auto
done
lemma lemma5_3:
"\<lbrakk>n \<in> nat; B \<in> A; 0 \<in> u(B); u(B) \<lesssim> succ(n)\<rbrakk>
\<Longrightarrow> (\<lambda>x \<in> A. {fst(x). x \<in> u(x)-{0}})`B \<lesssim> n"
apply simp
apply (fast elim!: Diff_lepoll [THEN lemma4 [OF _ nat_into_Ord]])
done
lemma ex_fun_AC13_AC15:
"\<lbrakk>\<forall>B \<in> {cons(0, x*nat). x \<in> A}.
pairwise_disjoint(f`B) \<and>
sets_of_size_between(f`B, 2, succ(n)) \<and> \<Union>(f`B)=B;
n \<in> nat\<rbrakk>
\<Longrightarrow> \<exists>f. \<forall>B \<in> A. f`B \<noteq> 0 \<and> f`B \<subseteq> B \<and> f`B \<lesssim> 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: "\<lbrakk>n \<in> nat; 1\<le>n; AC10(n)\<rbrakk> \<Longrightarrow> AC11"
by (unfold AC10_def AC11_def, blast)
(* ********************************************************************** *)
(* AC11 \<Longrightarrow> AC12 *)
(* ********************************************************************** *)
theorem AC11_AC12: "AC11 \<Longrightarrow> AC12"
by (unfold AC10_def AC11_def AC11_def AC12_def, blast)
(* ********************************************************************** *)
(* AC12 \<Longrightarrow> AC15 *)
(* ********************************************************************** *)
theorem AC12_AC15: "AC12 \<Longrightarrow> 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) \<Longrightarrow> (\<Union>a<x. F(a)) = (\<Union>a \<in> x. F(a))"
by (fast intro!: ltI dest!: ltD)
lemma AC15_WO6_aux1:
"\<forall>x \<in> Pow(A)-{0}. f`x\<noteq>0 \<and> f`x \<subseteq> x \<and> f`x \<lesssim> m
\<Longrightarrow> (\<Union>i<\<mu> 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:
"\<forall>x \<in> Pow(A)-{0}. f`x\<noteq>0 \<and> f`x \<subseteq> x \<and> f`x \<lesssim> m
\<Longrightarrow> \<forall>x < (\<mu> x. HH(f,A,x)={A}). HH(f,A,x) \<lesssim> 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 \<Longrightarrow> 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 = "\<mu> i. HH (f,A,i) ={A}" in exI)
apply (rule_tac x = "\<lambda>j \<in> (\<mu> 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: "\<lbrakk>n \<in> nat; 1\<le>n; AC10(n)\<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> 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 = "\<lambda>x \<in> 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: "\<lbrakk>m\<le>n; AC13(m)\<rbrakk> \<Longrightarrow> 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: "\<lbrakk>n \<in> nat; 1\<le>n; AC13(n)\<rbrakk> \<Longrightarrow> AC14"
by (unfold AC13_def AC14_def, auto)
(* ********************************************************************** *)
(* AC14 \<Longrightarrow> AC15 *)
(* ********************************************************************** *)
theorem AC14_AC15: "AC14 \<Longrightarrow> 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: "\<lbrakk>A\<noteq>0; A \<lesssim> 1\<rbrakk> \<Longrightarrow> \<exists>a. A={a}"
by (fast elim!: not_emptyE lepoll_1_is_sing)
lemma AC13_AC1_lemma:
"\<forall>B \<in> A. f(B)\<noteq>0 \<and> f(B)<=B \<and> f(B) \<lesssim> 1
\<Longrightarrow> (\<lambda>x \<in> A. THE y. f(x)={y}) \<in> (\<Prod>X \<in> 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) \<Longrightarrow> AC1"
unfolding AC13_def AC1_def
apply (fast elim!: AC13_AC1_lemma)
done
(* ********************************************************************** *)
(* AC11 \<Longrightarrow> AC14 *)
(* ********************************************************************** *)
theorem AC11_AC14: "AC11 \<Longrightarrow> AC14"
unfolding AC11_def AC14_def
apply (fast intro!: AC10_AC13)
done
end