Theory AC7_AC9

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theory AC7_AC9
imports AC_Equiv
begin

(*  Title:      ZF/AC/AC7_AC9.thy
    ID:         $Id: AC7_AC9.thy,v 1.3 2005/06/17 14:15:10 haftmann Exp $
    Author:     Krzysztof Grabczewski

The proofs needed to state that AC7, AC8 and AC9 are equivalent to the previous
instances of AC.
*)

theory AC7_AC9 imports AC_Equiv begin

(* ********************************************************************** *)
(* Lemmas used in the proofs AC7 ==> AC6 and AC9 ==> AC1                  *)
(*  - Sigma_fun_space_not0                                                *)
(*  - Sigma_fun_space_eqpoll                                              *)
(* ********************************************************************** *)

lemma Sigma_fun_space_not0: "[| 0∉A; B ∈ A |] ==> (nat->Union(A)) * B ≠ 0"
by (blast dest!: Sigma_empty_iff [THEN iffD1] Union_empty_iff [THEN iffD1])

lemma inj_lemma: 
        "C ∈ A ==> (λg ∈ (nat->Union(A))*C.   
                (λn ∈ nat. if(n=0, snd(g), fst(g)`(n #- 1))))   
                ∈ inj((nat->Union(A))*C, (nat->Union(A)) ) "
apply (unfold inj_def)
apply (rule CollectI)
apply (fast intro!: lam_type if_type apply_type fst_type snd_type, auto) 
apply (rule fun_extension, assumption+)
apply (drule lam_eqE [OF _ nat_succI], assumption, simp) 
apply (drule lam_eqE [OF _ nat_0I], simp) 
done

lemma Sigma_fun_space_eqpoll:
     "[| C ∈ A; 0∉A |] ==> (nat->Union(A)) * C ≈ (nat->Union(A))"
apply (rule eqpollI)
apply (simp add: lepoll_def)
apply (fast intro!: inj_lemma)
apply (fast elim!: prod_lepoll_self not_sym [THEN not_emptyE] subst_elem 
            elim: swap)
done


(* ********************************************************************** *)
(* AC6 ==> AC7                                                            *)
(* ********************************************************************** *)

lemma AC6_AC7: "AC6 ==> AC7"
by (unfold AC6_def AC7_def, blast)

(* ********************************************************************** *)
(* AC7 ==> AC6, Rubin & Rubin p. 12, Theorem 2.8                          *)
(* The case of the empty family of sets added in order to complete        *)
(* the proof.                                                             *)
(* ********************************************************************** *)

lemma lemma1_1: "y ∈ (Π B ∈ A. Y*B) ==> (λB ∈ A. snd(y`B)) ∈ (Π B ∈ A. B)"
by (fast intro!: lam_type snd_type apply_type)

lemma lemma1_2:
     "y ∈ (Π B ∈ {Y*C. C ∈ A}. B) ==> (λB ∈ A. y`(Y*B)) ∈ (Π B ∈ A. Y*B)"
apply (fast intro!: lam_type apply_type)
done

lemma AC7_AC6_lemma1:
     "(Π B ∈ {(nat->Union(A))*C. C ∈ A}. B) ≠ 0 ==> (Π B ∈ A. B) ≠ 0"
by (fast intro!: equals0I lemma1_1 lemma1_2)

lemma AC7_AC6_lemma2: "0 ∉ A ==> 0 ∉ {(nat -> Union(A)) * C. C ∈ A}"
by (blast dest: Sigma_fun_space_not0)

lemma AC7_AC6: "AC7 ==> AC6"
apply (unfold AC6_def AC7_def)
apply (rule allI)
apply (rule impI)
apply (case_tac "A=0", simp)
apply (rule AC7_AC6_lemma1)
apply (erule allE) 
apply (blast del: notI
             intro!: AC7_AC6_lemma2 intro: eqpoll_sym eqpoll_trans 
                    Sigma_fun_space_eqpoll)
done


(* ********************************************************************** *)
(* AC1 ==> AC8                                                            *)
(* ********************************************************************** *)

lemma AC1_AC8_lemma1: 
        "∀B ∈ A. ∃B1 B2. B=<B1,B2> & B1 ≈ B2   
        ==> 0 ∉ { bij(fst(B),snd(B)). B ∈ A }"
apply (unfold eqpoll_def, auto)
done

lemma AC1_AC8_lemma2:
     "[| f ∈ (Π X ∈ RepFun(A,p). X); D ∈ A |] ==> (λx ∈ A. f`p(x))`D ∈ p(D)" 
apply (simp, fast elim!: apply_type)
done

lemma AC1_AC8: "AC1 ==> AC8"
apply (unfold AC1_def AC8_def)
apply (fast dest: AC1_AC8_lemma1 AC1_AC8_lemma2)
done


(* ********************************************************************** *)
(* AC8 ==> AC9                                                            *)
(*  - this proof replaces the following two from Rubin & Rubin:           *)
(*    AC8 ==> AC1 and AC1 ==> AC9                                         *)
(* ********************************************************************** *)

lemma AC8_AC9_lemma:
     "∀B1 ∈ A. ∀B2 ∈ A. B1 ≈ B2 
      ==> ∀B ∈ A*A. ∃B1 B2. B=<B1,B2> & B1 ≈ B2"
by fast

lemma AC8_AC9: "AC8 ==> AC9"
apply (unfold AC8_def AC9_def)
apply (intro allI impI)
apply (erule allE)
apply (erule impE, erule AC8_AC9_lemma, force) 
done


(* ********************************************************************** *)
(* AC9 ==> AC1                                                            *)
(* The idea of this proof comes from "Equivalents of the Axiom of Choice" *)
(* by Rubin & Rubin. But (x * y) is not necessarily equipollent to        *)
(* (x * y) Un {0} when y is a set of total functions acting from nat to   *)
(* Union(A) -- therefore we have used the set (y * nat) instead of y.     *)
(* ********************************************************************** *)

lemma snd_lepoll_SigmaI: "b ∈ B ==> X \<lesssim> B × X"
by (blast intro: lepoll_trans prod_lepoll_self eqpoll_imp_lepoll 
                 prod_commute_eqpoll) 

lemma nat_lepoll_lemma:
     "[|0 ∉ A; B ∈ A|] ==> nat \<lesssim> ((nat -> Union(A)) × B) × nat"
by (blast dest: Sigma_fun_space_not0 intro: snd_lepoll_SigmaI)

lemma AC9_AC1_lemma1:
     "[| 0∉A;  A≠0;   
         C = {((nat->Union(A))*B)*nat. B ∈ A}  Un  
             {cons(0,((nat->Union(A))*B)*nat). B ∈ A};  
         B1 ∈ C;  B2 ∈ C |]   
      ==> B1 ≈ B2"
by (blast intro!: nat_lepoll_lemma Sigma_fun_space_eqpoll
                     nat_cons_eqpoll [THEN eqpoll_trans] 
                     prod_eqpoll_cong [OF _ eqpoll_refl]
             intro: eqpoll_trans eqpoll_sym )

lemma AC9_AC1_lemma2:
     "∀B1 ∈ {(F*B)*N. B ∈ A} Un {cons(0,(F*B)*N). B ∈ A}.   
      ∀B2 ∈ {(F*B)*N. B ∈ A} Un {cons(0,(F*B)*N). B ∈ A}.   
        f`<B1,B2> ∈ bij(B1, B2)   
      ==> (λB ∈ A. snd(fst((f`<cons(0,(F*B)*N),(F*B)*N>)`0))) ∈ (Π X ∈ A. X)"
apply (intro lam_type snd_type fst_type)
apply (rule apply_type [OF _ consI1]) 
apply (fast intro!: fun_weaken_type bij_is_fun)
done

lemma AC9_AC1: "AC9 ==> AC1"
apply (unfold AC1_def AC9_def)
apply (intro allI impI)
apply (erule allE)
apply (case_tac "A~=0")
apply (blast dest: AC9_AC1_lemma1 AC9_AC1_lemma2, force) 
done

end

lemma Sigma_fun_space_not0:

  [| 0  A; BA |] ==> (nat -> \<Union>A) × B  0

lemma inj_lemma:

  CA
  ==> (λg∈(nat -> \<Union>A) × C.
          λnnat. if n = 0 then snd(g) else fst(g) ` (n #- 1)) ∈
      inj((nat -> \<Union>A) × C, nat -> \<Union>A)

lemma Sigma_fun_space_eqpoll:

  [| CA; 0  A |] ==> (nat -> \<Union>A) × C  nat -> \<Union>A

lemma AC6_AC7:

  AC6 ==> AC7

lemma lemma1_1:

  y ∈ (ΠBA. Y × B) ==> (λBA. snd(y ` B)) ∈ (ΠBA. B)

lemma lemma1_2:

  y ∈ (ΠB∈{Y × C . CA}. B) ==> (λBA. y ` (Y × B)) ∈ (ΠBA. Y × B)

lemma AC7_AC6_lemma1:

  B∈{(nat -> \<Union>A) × C . CA}. B)  0 ==> (ΠBA. B)  0

lemma AC7_AC6_lemma2:

  0  A ==> 0  {(nat -> \<Union>A) × C . CA}

lemma AC7_AC6:

  AC7 ==> AC6

lemma AC1_AC8_lemma1:

  BA. ∃B1 B2. B = ⟨B1, B2⟩ ∧ B1  B2 ==> 0  {bij(fst(B), snd(B)) . BA}

lemma AC1_AC8_lemma2:

  [| f ∈ (ΠX∈RepFun(A, p). X); DA |] ==> (λxA. f ` p(x)) ` Dp(D)

lemma AC1_AC8:

  AC1 ==> AC8

lemma AC8_AC9_lemma:

  B1A. ∀B2A. B1  B2 ==> ∀BA × A. ∃B1 B2. B = ⟨B1, B2⟩ ∧ B1  B2

lemma AC8_AC9:

  AC8 ==> AC9

lemma snd_lepoll_SigmaI:

  bB ==> X lepoll B × X

lemma nat_lepoll_lemma:

  [| 0  A; BA |] ==> nat lepoll ((nat -> \<Union>A) × B) × nat

lemma AC9_AC1_lemma1:

  [| 0  A; A  0;
     C = {((nat -> \<Union>A) × B) × nat . BA} ∪
         {cons(0, ((nat -> \<Union>A) × B) × nat) . BA};
     B1.0C; B2.0C |]
  ==> B1.0  B2.0

lemma AC9_AC1_lemma2:

  B1∈{(F × B) × N . BA} ∪ {cons(0, (F × B) × N) . BA}.
     ∀B2∈{(F × B) × N . BA} ∪ {cons(0, (F × B) × N) . BA}.
        f ` ⟨B1, B2⟩ ∈ bij(B1, B2)
  ==> (λBA. snd(fst(f ` ⟨cons(0, (F × B) × N), (F × B) × N⟩ ` 0))) ∈ (ΠXA. X)

lemma AC9_AC1:

  AC9 ==> AC1