Quellcode-Bibliothek AC16_WO4.thy Sprache: Isabelle

(*  Title:      ZF/AC/AC16_WO4.thy
    Author:     Krzysztof Grabczewski

The proof of AC16(n, k) \<Longrightarrow> WO4(n-k)

Tidied (using locales) by LCP
*)

theory AC16_WO4
imports AC16_lemmas
begin

(* ********************************************************************** *)
(* The case of finite set                                                 *)
(* ********************************************************************** *)

lemma lemma1:
     "\Finite(A); 0 nat\
      \<Longrightarrow> \<exists>a f. Ord(a) \<and> domain(f) = a \<and>
                (\<Union>b<a. f`b) = A \<and> (\<forall>b<a. f`b \<lesssim> m)"
  unfolding Finite_def
apply (erule bexE)
apply (drule eqpoll_sym [THEN eqpoll_def [THEN def_imp_iff, THEN iffD1]])
apply (erule exE)
apply (rule_tac x = n in exI)
apply (rule_tac x = "\i \ n. {f`i}" in exI)
apply (simp add: ltD bij_def surj_def)
apply (fast intro!: ltI nat_into_Ord lam_funtype [THEN domain_of_fun]
               singleton_eqpoll_1 [THEN eqpoll_imp_lepoll, THEN lepoll_trans]
                    nat_1_lepoll_iff [THEN iffD2]
          elim!: apply_type ltE)
done

(* ********************************************************************** *)
(* The case of infinite set                                               *)
(* ********************************************************************** *)

(* well_ord(x,r) \<Longrightarrow> well_ord({{y,z}. y \<in> x}, Something(x,z))  **)
lemmas well_ord_paired = paired_bij [THEN bij_is_inj, THEN well_ord_rvimage]

lemma lepoll_trans1: "\A \ B; \ A \ C\ \ \ B \ C"
by (blast intro: lepoll_trans)

(* ********************************************************************** *)
(* There exists a well ordered set y such that ...                        *)
(* ********************************************************************** *)

lemmas lepoll_paired = paired_eqpoll [THEN eqpoll_sym, THEN eqpoll_imp_lepoll]

lemma lemma2: "\y R. well_ord(y,R) \ x \ y = 0 \ \y \ z \ \Finite(y)"
apply (rule_tac x = "{{a,x}. a \ nat \ Hartog (z) }" in exI)
apply (rule well_ord_Un [OF Ord_nat [THEN well_ord_Memrel]
                         Ord_Hartog [THEN well_ord_Memrel], THEN exE])
apply (blast intro!: Ord_Hartog well_ord_Memrel well_ord_paired
                      lepoll_trans1 [OF _ not_Hartog_lepoll_self]
                      lepoll_trans [OF subset_imp_lepoll lepoll_paired]
       elim!: nat_not_Finite [THEN notE]
       elim: mem_asym
       dest!: Un_upper1 [THEN subset_imp_lepoll, THEN lepoll_Finite]
              lepoll_paired [THEN lepoll_Finite])
done

lemma infinite_Un: "\Finite(B) \ \Finite(A \ B)"
by (blast intro: subset_Finite)

(* ********************************************************************** *)
(* There is a v \<in> s(u) such that k \<lesssim> x \<inter> y (in our case succ(k))    *)
(* The idea of the proof is the following \<in>                               *)
(* Suppose not, i.e. every element of s(u) has exactly k-1 elements of y   *)
(* Thence y is less than or equipollent to {v \<in> Pow(x). v \<approx> n#-k}      *)
(*   We have obtained this result in two steps \<in>                          *)
(*   1. y is less than or equipollent to {v \<in> s(u). a \<subseteq> v}                  *)
(*      where a is certain k-2 element subset of y                        *)
(*   2. {v \<in> s(u). a \<subseteq> v} is less than or equipollent                       *)
(*      to {v \<in> Pow(x). v \<approx> n-k}                                       *)
(* ********************************************************************** *)

(*Proof simplified by LCP*)
lemma succ_not_lepoll_lemma:
     "\\(\x \ A. f`x=y); f \ inj(A, B); y \ B\
      \<Longrightarrow> (\<lambda>a \<in> succ(A). if(a=A, y, f`a)) \<in> inj(succ(A), B)"
apply (rule_tac d = "\z. if (z=y, A, converse (f) `z) " in lam_injective)
apply (force simp add: inj_is_fun [THEN apply_type])
(*this preliminary simplification prevents looping somehow*)
apply (simp (no_asm_simp))
apply force
done

lemma succ_not_lepoll_imp_eqpoll: "\\A \ B; A \ B\ \ succ(A) \ B"
  unfolding lepoll_def eqpoll_def bij_def surj_def
apply (fast elim!: succ_not_lepoll_lemma inj_is_fun)
done

(* ********************************************************************** *)
(* There is a k-2 element subset of y                                     *)
(* ********************************************************************** *)

lemmas ordertype_eqpoll =
       ordermap_bij [THEN exI [THEN eqpoll_def [THEN def_imp_iff, THEN iffD2]]]

lemma cons_cons_subset:
     "\a \ y; b \ y-a; u \ x\ \ cons(b, cons(u, a)) \ Pow(x \ y)"
by fast

lemma cons_cons_eqpoll:
     "\a \ k; a \ y; b \ y-a; u \ x; x \ y = 0\
      \<Longrightarrow> cons(b, cons(u, a)) \<approx> succ(succ(k))"
by (fast intro!: cons_eqpoll_succ)

lemma set_eq_cons:
     "\succ(k) \ A; k \ B; B \ A; a \ A-B; k \ nat\ \ A = cons(a, B)"
apply (rule equalityI)
prefer 2 apply fast
apply (rule Diff_eq_0_iff [THEN iffD1])
apply (rule equals0I)
apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll])
apply (drule eqpoll_sym [THEN cons_eqpoll_succ], fast)
apply (drule cons_eqpoll_succ, fast)
apply (fast elim!: lepoll_trans [THEN lepoll_trans, THEN succ_lepoll_natE,
         OF eqpoll_sym [THEN eqpoll_imp_lepoll] subset_imp_lepoll])
done

lemma cons_eqE: "\cons(x,a) = cons(y,a); x \ a\ \ x = y "
by (fast elim!: equalityE)

lemma eq_imp_Int_eq: "A = B \ A \ C = B \ C"
by blast

(* ********************************************************************** *)
(* some arithmetic                                                        *)
(* ********************************************************************** *)

lemma eqpoll_sum_imp_Diff_lepoll_lemma [rule_format]:
     "\k \ nat; m \ nat\
      \<Longrightarrow> \<forall>A B. A \<approx> k #+ m \<and> k \<lesssim> B \<and> B \<subseteq> A \<longrightarrow> A-B \<lesssim> m"
apply (induct_tac "k")
apply (simp add: add_0)
apply (blast intro: eqpoll_imp_lepoll lepoll_trans
                    Diff_subset [THEN subset_imp_lepoll])
apply (intro allI impI)
apply (rule succ_lepoll_imp_not_empty [THEN not_emptyE], fast)
apply (erule_tac x = "A - {xa}" in allE)
apply (erule_tac x = "B - {xa}" in allE)
apply (erule impE)
apply (simp add: add_succ)
apply (fast intro!: Diff_sing_eqpoll lepoll_Diff_sing)
apply (subgoal_tac "A - {xa} - (B - {xa}) = A - B", simp)
apply blast
done

lemma eqpoll_sum_imp_Diff_lepoll:
     "\A \ succ(k #+ m); B \ A; succ(k) \ B; k \ nat; m \ nat\
      \<Longrightarrow> A-B \<lesssim> m"
apply (simp only: add_succ [symmetric])
apply (blast intro: eqpoll_sum_imp_Diff_lepoll_lemma)
done

(* ********************************************************************** *)
(* similar properties for \<approx>                                          *)
(* ********************************************************************** *)

lemma eqpoll_sum_imp_Diff_eqpoll_lemma [rule_format]:
     "\k \ nat; m \ nat\
      \<Longrightarrow> \<forall>A B. A \<approx> k #+ m \<and> k \<approx> B \<and> B \<subseteq> A \<longrightarrow> A-B \<approx> m"
apply (induct_tac "k")
apply (force dest!: eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_0_is_0])
apply (intro allI impI)
apply (rule succ_lepoll_imp_not_empty [THEN not_emptyE])
apply (fast elim!: eqpoll_imp_lepoll)
apply (erule_tac x = "A - {xa}" in allE)
apply (erule_tac x = "B - {xa}" in allE)
apply (erule impE)
apply (force intro: eqpoll_sym intro!: Diff_sing_eqpoll)
apply (subgoal_tac "A - {xa} - (B - {xa}) = A - B", simp)
apply blast
done

lemma eqpoll_sum_imp_Diff_eqpoll:
     "\A \ succ(k #+ m); B \ A; succ(k) \ B; k \ nat; m \ nat\
      \<Longrightarrow> A-B \<approx> m"
apply (simp only: add_succ [symmetric])
apply (blast intro: eqpoll_sum_imp_Diff_eqpoll_lemma)
done

(* ********************************************************************** *)
(* LL can be well ordered                                                 *)
(* ********************************************************************** *)

lemma subsets_lepoll_0_eq_unit: "{x \ Pow(X). x \ 0} = {0}"
by (fast dest!: lepoll_0_is_0 intro!: lepoll_refl)

lemma subsets_lepoll_succ:
     "n \ nat \ {z \ Pow(y). z \ succ(n)} =
                  {z \<in> Pow(y). z \<lesssim> n} \<union> {z \<in> Pow(y). z \<approx> succ(n)}"
by (blast intro: leI le_imp_lepoll nat_into_Ord
                    lepoll_trans eqpoll_imp_lepoll
          dest!: lepoll_succ_disj)

lemma Int_empty:
     "n \ nat \ {z \ Pow(y). z \ n} \ {z \ Pow(y). z \ succ(n)} = 0"
by (blast intro: eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans]
                 succ_lepoll_natE)

locale AC16 =
  fixes x and y and k and l and m and t_n and R and MM and LL and GG and s
  defines k_def:     "k \ succ(l)"
      and MM_def:    "MM \ {v \ t_n. succ(k) \ v \ y}"
      and LL_def:    "LL \ {v \ y. v \ MM}"
      and GG_def:    "GG \ \v \ LL. (THE w. w \ MM \ v \ w) - v"
      and s_def:     "s(u) \ {v \ t_n. u \ v \ k \ v \ y}"
  assumes all_ex:    "\z \ {z \ Pow(x \ y) . z \ succ(k)}.
                       \<exists>! w. w \<in> t_n \<and> z \<subseteq> w "
    and disjoint[iff]:  "x \ y = 0"
    and "includes":  "t_n \ {v \ Pow(x \ y). v \ succ(k #+ m)}"
    and WO_R[iff]:      "well_ord(y,R)"
    and lnat[iff]:      "l \ nat"
    and mnat[iff]:      "m \ nat"
    and mpos[iff]:      "0
    and Infinite[iff]:  "\ Finite(y)"
    and noLepoll:  "\ y \ {v \ Pow(x). v \ m}"
begin

lemma knat [iff]: "k \ nat"
by (simp add: k_def)

(* ********************************************************************** *)
(*   1. y is less than or equipollent to {v \<in> s(u). a \<subseteq> v}                *)
(*      where a is certain k-2 element subset of y                        *)
(* ********************************************************************** *)

lemma Diff_Finite_eqpoll: "\l \ a; a \ y\ \ y - a \ y"
apply (insert WO_R Infinite lnat)
apply (rule eqpoll_trans)
apply (rule Diff_lesspoll_eqpoll_Card)
apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])
apply (blast intro: lesspoll_trans1
            intro!: Card_cardinal
                    Card_cardinal [THEN Card_is_Ord, THEN nat_le_infinite_Ord,
                                   THEN le_imp_lepoll]
            dest: well_ord_cardinal_eqpoll
                   eqpoll_sym  eqpoll_imp_lepoll
                   n_lesspoll_nat [THEN lesspoll_trans2]
                   well_ord_cardinal_eqpoll [THEN eqpoll_sym,
                          THEN eqpoll_imp_lepoll, THEN lepoll_infinite])+
done

lemma s_subset: "s(u) \ t_n"
by (unfold s_def, blast)

lemma sI:
      "\w \ t_n; cons(b,cons(u,a)) \ w; a \ y; b \ y-a; l \ a\
       \<Longrightarrow> w \<in> s(u)"
  unfolding s_def succ_def k_def
apply (blast intro!: eqpoll_imp_lepoll [THEN cons_lepoll_cong]
             intro: subset_imp_lepoll lepoll_trans)
done

lemma in_s_imp_u_in: "v \ s(u) \ u \ v"
by (unfold s_def, blast)

lemma ex1_superset_a:
     "\l \ a; a \ y; b \ y - a; u \ x\
      \<Longrightarrow> \<exists>! c. c \<in> s(u) \<and> a \<subseteq> c \<and> b \<in> c"
apply (rule all_ex [simplified k_def, THEN ballE])
apply (erule ex1E)
apply (rule_tac a = w in ex1I, blast intro: sI)
apply (blast dest: s_subset [THEN subsetD] in_s_imp_u_in)
apply (blast del: PowI
             intro!: cons_cons_subset eqpoll_sym [THEN cons_cons_eqpoll])
done

lemma the_eq_cons:
     "\\v \ s(u). succ(l) \ v \ y;
         l \<approx> a;  a \<subseteq> y;  b \<in> y - a;  u \<in> x\<rbrakk>
      \<Longrightarrow> (THE c. c \<in> s(u) \<and> a \<subseteq> c \<and> b \<in> c) \<inter> y = cons(b, a)"
apply (frule ex1_superset_a [THEN theI], assumption+)
apply (rule set_eq_cons)
apply (fast+)
done

lemma y_lepoll_subset_s:
     "\\v \ s(u). succ(l) \ v \ y;
         l \<approx> a;  a \<subseteq> y;  u \<in> x\<rbrakk>
      \<Longrightarrow> y \<lesssim> {v \<in> s(u). a \<subseteq> v}"
apply (rule Diff_Finite_eqpoll [THEN eqpoll_sym, THEN eqpoll_imp_lepoll,
                                THEN lepoll_trans],  fast+)
apply (rule_tac  f3 = "\b \ y-a. THE c. c \ s (u) \ a \ c \ b \ c"
        in exI [THEN lepoll_def [THEN def_imp_iff, THEN iffD2]])
apply (simp add: inj_def)
apply (rule conjI)
apply (rule lam_type)
apply (frule ex1_superset_a [THEN theI], fast+, clarify)
apply (rule cons_eqE [of _ a])
apply (drule_tac A = "THE c. P (c)" and C = y for P in eq_imp_Int_eq)
apply (simp_all add: the_eq_cons)
done

(* ********************************************************************** *)
(* back to the second part                                                *)
(* ********************************************************************** *)

(*relies on the disjointness of x, y*)
lemma x_imp_not_y [dest]: "a \ x \ a \ y"
by (blast dest:  disjoint [THEN equalityD1, THEN subsetD, OF IntI])

lemma w_Int_eq_w_Diff:
     "w \ x \ y \ w \ (x - {u}) = w - cons(u, w \ y)"
by blast

lemma w_Int_eqpoll_m:
     "\w \ {v \ s(u). a \ v};
         l \<approx> a;  u \<in> x;
         \<forall>v \<in> s(u). succ(l) \<approx> v \<inter> y\<rbrakk>
      \<Longrightarrow> w \<inter> (x - {u}) \<approx> m"
apply (erule CollectE)
apply (subst w_Int_eq_w_Diff)
apply (fast dest!: s_subset [THEN subsetD]
                   "includes" [simplified k_def, THEN subsetD])
apply (blast dest: s_subset [THEN subsetD]
                   "includes" [simplified k_def, THEN subsetD]
             dest: eqpoll_sym [THEN cons_eqpoll_succ, THEN eqpoll_sym]
                   in_s_imp_u_in
            intro!: eqpoll_sum_imp_Diff_eqpoll)
done

(* ********************************************************************** *)
(*   2. {v \<in> s(u). a \<subseteq> v} is less than or equipollent                       *)
(*      to {v \<in> Pow(x). v \<approx> n-k}                                       *)
(* ********************************************************************** *)

lemma eqpoll_m_not_empty: "a \ m \ a \ 0"
apply (insert mpos)
apply (fast elim!: zero_lt_natE dest!: eqpoll_succ_imp_not_empty)
done

lemma cons_cons_in:
     "\z \ xa \ (x - {u}); l \ a; a \ y; u \ x\
      \<Longrightarrow> \<exists>! w. w \<in> t_n \<and> cons(z, cons(u, a)) \<subseteq> w"
apply (rule all_ex [THEN bspec])
  unfolding k_def
apply (fast intro!: cons_eqpoll_succ elim: eqpoll_sym)
done

lemma subset_s_lepoll_w:
     "\\v \ s(u). succ(l) \ v \ y; a \ y; l \ a; u \ x\
      \<Longrightarrow> {v \<in> s(u). a \<subseteq> v} \<lesssim> {v \<in> Pow(x). v \<approx> m}"
apply (rule_tac f3 = "\w \ {v \ s (u) . a \ v}. w \ (x - {u})"
       in exI [THEN lepoll_def [THEN def_imp_iff, THEN iffD2]])
apply (simp add: inj_def)
apply (intro conjI lam_type CollectI)
  apply fast
apply (blast intro: w_Int_eqpoll_m)
apply (intro ballI impI)
(** LEVEL 8 **)
apply (rule w_Int_eqpoll_m [THEN eqpoll_m_not_empty, THEN not_emptyE])
apply (blast, assumption+)
apply (drule equalityD1 [THEN subsetD], assumption)
apply (frule cons_cons_in, assumption+)
apply (blast dest: ex1_two_eq intro: s_subset [THEN subsetD] in_s_imp_u_in)+
done

(* ********************************************************************** *)
(* well_ord_subsets_lepoll_n                                              *)
(* ********************************************************************** *)

lemma well_ord_subsets_eqpoll_n:
     "n \ nat \ \S. well_ord({z \ Pow(y) . z \ succ(n)}, S)"
apply (rule WO_R [THEN well_ord_infinite_subsets_eqpoll_X,
                  THEN eqpoll_def [THEN def_imp_iff, THEN iffD1], THEN exE])
apply (fast intro: bij_is_inj [THEN well_ord_rvimage])+
done

lemma well_ord_subsets_lepoll_n:
     "n \ nat \ \R. well_ord({z \ Pow(y). z \ n}, R)"
apply (induct_tac "n")
apply (force intro!: well_ord_unit simp add: subsets_lepoll_0_eq_unit)
apply (erule exE)
apply (rule well_ord_subsets_eqpoll_n [THEN exE], assumption)
apply (simp add: subsets_lepoll_succ)
apply (drule well_ord_radd, assumption)
apply (erule Int_empty [THEN disj_Un_eqpoll_sum,
                  THEN eqpoll_def [THEN def_imp_iff, THEN iffD1], THEN exE])
apply (fast elim!: bij_is_inj [THEN well_ord_rvimage])
done

lemma LL_subset: "LL \ {z \ Pow(y). z \ succ(k #+ m)}"
  unfolding LL_def MM_def
apply (insert "includes")
apply (blast intro: subset_imp_lepoll eqpoll_imp_lepoll lepoll_trans)
done

lemma well_ord_LL: "\S. well_ord(LL,S)"
apply (rule well_ord_subsets_lepoll_n [THEN exE, of "succ(k#+m)"])
apply simp
apply (blast intro: well_ord_subset [OF _ LL_subset])
done

(* ********************************************************************** *)
(* every element of LL is a contained in exactly one element of MM        *)
(* ********************************************************************** *)

lemma unique_superset_in_MM:
     "v \ LL \ \! w. w \ MM \ v \ w"
apply (unfold MM_def LL_def, safe, fast)
apply (rule lepoll_imp_eqpoll_subset [THEN exE], assumption)
apply (rule_tac x = x in all_ex [THEN ballE])
apply (blast intro: eqpoll_sym)+
done

(* ********************************************************************** *)
(* The function GG satisfies the conditions of WO4                        *)
(* ********************************************************************** *)

(* ********************************************************************** *)
(* The union of appropriate values is the whole x                         *)
(* ********************************************************************** *)

lemma Int_in_LL: "w \ MM \ w \ y \ LL"
by (unfold LL_def, fast)

lemma in_LL_eq_Int:
     "v \ LL \ v = (THE x. x \ MM \ v \ x) \ y"
apply (unfold LL_def, clarify)
apply (subst unique_superset_in_MM [THEN the_equality2])
apply (auto simp add: Int_in_LL)
done

lemma unique_superset1: "a \ LL \ (THE x. x \ MM \ a \ x) \ MM"
by (erule unique_superset_in_MM [THEN theI, THEN conjunct1])

lemma the_in_MM_subset:
     "v \ LL \ (THE x. x \ MM \ v \ x) \ x \ y"
apply (drule unique_superset1)
  unfolding MM_def
apply (fast dest!: unique_superset1 "includes" [THEN subsetD])
done

lemma GG_subset: "v \ LL \ GG ` v \ x"
  unfolding GG_def
apply (frule the_in_MM_subset)
apply (frule in_LL_eq_Int)
apply (force elim: equalityE)
done

lemma nat_lepoll_ordertype: "nat \ ordertype(y, R)"
apply (rule nat_le_infinite_Ord [THEN le_imp_lepoll])
apply (rule Ord_ordertype [OF WO_R])
apply (rule ordertype_eqpoll [THEN eqpoll_imp_lepoll, THEN lepoll_infinite])
apply (rule WO_R)
apply (rule Infinite)
done

lemma ex_subset_eqpoll_n: "n \ nat \ \z. z \ y \ n \ z"
apply (erule nat_lepoll_imp_ex_eqpoll_n)
apply (rule lepoll_trans [OF nat_lepoll_ordertype])
apply (rule ordertype_eqpoll [THEN eqpoll_sym, THEN eqpoll_imp_lepoll])
apply (rule WO_R)
done

lemma exists_proper_in_s: "u \ x \ \v \ s(u). succ(k) \ v \ y"
apply (rule ccontr)
apply (subgoal_tac "\v \ s (u) . k \ v \ y")
prefer 2 apply (simp add: s_def, blast intro: succ_not_lepoll_imp_eqpoll)
  unfolding k_def
apply (insert all_ex "includes" lnat)
apply (rule ex_subset_eqpoll_n [THEN exE], assumption)
apply (rule noLepoll [THEN notE])
apply (blast intro: lepoll_trans [OF y_lepoll_subset_s subset_s_lepoll_w])
done

lemma exists_in_MM: "u \ x \ \w \ MM. u \ w"
apply (erule exists_proper_in_s [THEN bexE])
apply (unfold MM_def s_def, fast)
done

lemma exists_in_LL: "u \ x \ \w \ LL. u \ GG`w"
apply (rule exists_in_MM [THEN bexE], assumption)
apply (rule bexI)
apply (erule_tac [2] Int_in_LL)
  unfolding GG_def
apply (simp add: Int_in_LL)
apply (subst unique_superset_in_MM [THEN the_equality2])
apply (fast elim!: Int_in_LL)+
done

lemma OUN_eq_x: "well_ord(LL,S) \
      (\<Union>b<ordertype(LL,S). GG ` (converse(ordermap(LL,S)) ` b)) = x"
apply (rule equalityI)
apply (rule subsetI)
apply (erule OUN_E)
apply (rule GG_subset [THEN subsetD])
prefer 2 apply assumption
apply (blast intro: ltD  ordermap_bij [THEN bij_converse_bij, THEN bij_is_fun,
                                       THEN apply_type])
apply (rule subsetI)
apply (erule exists_in_LL [THEN bexE])
apply (force intro: ltI [OF _ Ord_ordertype]
                    ordermap_type [THEN apply_type]
             simp add: ordermap_bij [THEN bij_is_inj, THEN left_inverse])
done

(* ********************************************************************** *)
(* Every element of the family is less than or equipollent to n-k (m)     *)
(* ********************************************************************** *)

lemma in_MM_eqpoll_n: "w \ MM \ w \ succ(k #+ m)"
  unfolding MM_def
apply (fast dest: "includes" [THEN subsetD])
done

lemma in_LL_eqpoll_n: "w \ LL \ succ(k) \ w"
by (unfold LL_def MM_def, fast)

lemma in_LL: "w \ LL \ w \ (THE x. x \ MM \ w \ x)"
by (erule subset_trans [OF in_LL_eq_Int [THEN equalityD1] Int_lower1])

lemma all_in_lepoll_m:
      "well_ord(LL,S) \
       \<forall>b<ordertype(LL,S). GG ` (converse(ordermap(LL,S)) ` b) \<lesssim> m"
  unfolding GG_def
apply (rule oallI)
apply (simp add: ltD ordermap_bij [THEN bij_converse_bij, THEN bij_is_fun, THEN apply_type])
apply (insert "includes")
apply (rule eqpoll_sum_imp_Diff_lepoll)
apply (blast del: subsetI
             dest!: ltD
             intro!: eqpoll_sum_imp_Diff_lepoll in_LL_eqpoll_n
             intro: in_LL   unique_superset1 [THEN in_MM_eqpoll_n]
                    ordermap_bij [THEN bij_converse_bij, THEN bij_is_fun,
                                  THEN apply_type])+
done

lemma "conclusion":
     "\a f. Ord(a) \ domain(f) = a \ (\b (\b m)"
apply (rule well_ord_LL [THEN exE])
apply (rename_tac S)
apply (rule_tac x = "ordertype (LL,S)" in exI)
apply (rule_tac x = "\b \ ordertype(LL,S).
                      GG ` (converse (ordermap (LL,S)) ` b)" in exI)
apply (simp add: ltD)
apply (blast intro: lam_funtype [THEN domain_of_fun]
                    Ord_ordertype  OUN_eq_x  all_in_lepoll_m [THEN ospec])
done

end

(* ********************************************************************** *)
(* The main theorem AC16(n, k) \<Longrightarrow> WO4(n-k)                               *)
(* ********************************************************************** *)

theorem AC16_WO4:
     "\AC_Equiv.AC16(k #+ m, k); 0 < k; 0 < m; k \ nat; m \ nat\ \ WO4(m)"
  unfolding AC_Equiv.AC16_def WO4_def
apply (rule allI)
apply (case_tac "Finite (A)")
apply (rule lemma1, assumption+)
apply (cut_tac lemma2)
apply (elim exE conjE)
apply (erule_tac x = "A \ y" in allE)
apply (frule infinite_Un, drule mp, assumption)
apply (erule zero_lt_natE, assumption, clarify)
apply (blast intro: AC16.conclusion [OF AC16.intro])
done

end

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