Theory HH

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theory HH
imports Hartog
begin

(*  Title:      ZF/AC/HH.thy
    ID:         $Id: HH.thy,v 1.14 2007/10/07 19:29:42 wenzelm Exp $
    Author:     Krzysztof Grabczewski

Some properties of the recursive definition of HH used in the proofs of
  AC17 ==> AC1
  AC1 ==> WO2
  AC15 ==> WO6
*)

theory HH imports AC_Equiv Hartog begin

definition
  HH :: "[i, i, i] => i"  where
    "HH(f,x,a) == transrec(a, %b r. let z = x - (\<Union>c ∈ b. r`c)
                                    in  if f`z ∈ Pow(z)-{0} then f`z else {x})"

subsection{*Lemmas useful in each of the three proofs*}

lemma HH_def_satisfies_eq:
     "HH(f,x,a) = (let z = x - (\<Union>b ∈ a. HH(f,x,b))   
                   in  if f`z ∈ Pow(z)-{0} then f`z else {x})"
by (rule HH_def [THEN def_transrec, THEN trans], simp)

lemma HH_values: "HH(f,x,a) ∈ Pow(x)-{0} | HH(f,x,a)={x}"
apply (rule HH_def_satisfies_eq [THEN ssubst])
apply (simp add: Let_def Diff_subset [THEN PowI], fast)
done

lemma subset_imp_Diff_eq:
     "B ⊆ A ==> X-(\<Union>a ∈ A. P(a)) = X-(\<Union>a ∈ A-B. P(a))-(\<Union>b ∈ B. P(b))"
by fast

lemma Ord_DiffE: "[| c ∈ a-b; b<a |] ==> c=b | b<c & c<a"
apply (erule ltE)
apply (drule Ord_linear [of _ c])
apply (fast elim: Ord_in_Ord)
apply (fast intro!: ltI intro: Ord_in_Ord)
done

lemma Diff_UN_eq_self: "(!!y. y∈A ==> P(y) = {x}) ==> x - (\<Union>y ∈ A. P(y)) = x" 
by (simp, fast elim!: mem_irrefl)

lemma HH_eq: "x - (\<Union>b ∈ a. HH(f,x,b)) = x - (\<Union>b ∈ a1. HH(f,x,b))   
              ==> HH(f,x,a) = HH(f,x,a1)"
apply (subst HH_def_satisfies_eq [of _ _ a1]) 
apply (rule HH_def_satisfies_eq [THEN trans], simp) 
done

lemma HH_is_x_gt_too: "[| HH(f,x,b)={x}; b<a |] ==> HH(f,x,a)={x}"
apply (rule_tac P = "b<a" in impE)
prefer 2 apply assumption+
apply (erule lt_Ord2 [THEN trans_induct])
apply (rule impI)
apply (rule HH_eq [THEN trans])
prefer 2 apply assumption+
apply (rule leI [THEN le_imp_subset, THEN subset_imp_Diff_eq, THEN ssubst], 
       assumption)
apply (rule_tac t = "%z. z-?X" in subst_context)
apply (rule Diff_UN_eq_self)
apply (drule Ord_DiffE, assumption) 
apply (fast elim: ltE, auto) 
done

lemma HH_subset_x_lt_too:
     "[| HH(f,x,a) ∈ Pow(x)-{0}; b<a |] ==> HH(f,x,b) ∈ Pow(x)-{0}"
apply (rule HH_values [THEN disjE], assumption)
apply (drule HH_is_x_gt_too, assumption)
apply (drule subst, assumption)
apply (fast elim!: mem_irrefl)
done

lemma HH_subset_x_imp_subset_Diff_UN:
    "HH(f,x,a) ∈ Pow(x)-{0} ==> HH(f,x,a) ∈ Pow(x - (\<Union>b ∈ a. HH(f,x,b)))-{0}"
apply (drule HH_def_satisfies_eq [THEN subst])
apply (rule HH_def_satisfies_eq [THEN ssubst])
apply (simp add: Let_def Diff_subset [THEN PowI])
apply (drule split_if [THEN iffD1])
apply (fast elim!: mem_irrefl)
done

lemma HH_eq_arg_lt:
     "[| HH(f,x,v)=HH(f,x,w); HH(f,x,v) ∈ Pow(x)-{0}; v ∈ w |] ==> P"
apply (frule_tac P = "%y. y ∈ Pow (x) -{0}" in subst, assumption)
apply (drule_tac a = w in HH_subset_x_imp_subset_Diff_UN)
apply (drule subst_elem, assumption)
apply (fast intro!: singleton_iff [THEN iffD2] equals0I)
done

lemma HH_eq_imp_arg_eq:
  "[| HH(f,x,v)=HH(f,x,w); HH(f,x,w) ∈ Pow(x)-{0}; Ord(v); Ord(w) |] ==> v=w"
apply (rule_tac j = w in Ord_linear_lt)
apply (simp_all (no_asm_simp))
 apply (drule subst_elem, assumption) 
 apply (blast dest: ltD HH_eq_arg_lt)
apply (blast dest: HH_eq_arg_lt [OF sym] ltD) 
done

lemma HH_subset_x_imp_lepoll: 
     "[| HH(f, x, i) ∈ Pow(x)-{0}; Ord(i) |] ==> i lepoll Pow(x)-{0}"
apply (unfold lepoll_def inj_def)
apply (rule_tac x = "λj ∈ i. HH (f, x, j) " in exI)
apply (simp (no_asm_simp))
apply (fast del: DiffE
            elim!: HH_eq_imp_arg_eq Ord_in_Ord HH_subset_x_lt_too 
            intro!: lam_type ballI ltI intro: bexI)
done

lemma HH_Hartog_is_x: "HH(f, x, Hartog(Pow(x)-{0})) = {x}"
apply (rule HH_values [THEN disjE])
prefer 2 apply assumption 
apply (fast del: DiffE
            intro!: Ord_Hartog 
            dest!: HH_subset_x_imp_lepoll 
            elim!: Hartog_lepoll_selfE)
done

lemma HH_Least_eq_x: "HH(f, x, LEAST i. HH(f, x, i) = {x}) = {x}"
by (fast intro!: Ord_Hartog HH_Hartog_is_x LeastI)

lemma less_Least_subset_x:
     "a ∈ (LEAST i. HH(f,x,i)={x}) ==> HH(f,x,a) ∈ Pow(x)-{0}"
apply (rule HH_values [THEN disjE], assumption)
apply (rule less_LeastE)
apply (erule_tac [2] ltI [OF _ Ord_Least], assumption)
done

subsection{*Lemmas used in the proofs of AC1 ==> WO2 and AC17 ==> AC1*}

lemma lam_Least_HH_inj_Pow: 
        "(λa ∈ (LEAST i. HH(f,x,i)={x}). HH(f,x,a))   
         ∈ inj(LEAST i. HH(f,x,i)={x}, Pow(x)-{0})"
apply (unfold inj_def, simp)
apply (fast intro!: lam_type dest: less_Least_subset_x 
            elim!: HH_eq_imp_arg_eq Ord_Least [THEN Ord_in_Ord])
done

lemma lam_Least_HH_inj:
     "∀a ∈ (LEAST i. HH(f,x,i)={x}). ∃z ∈ x. HH(f,x,a) = {z}   
      ==> (λa ∈ (LEAST i. HH(f,x,i)={x}). HH(f,x,a))   
          ∈ inj(LEAST i. HH(f,x,i)={x}, {{y}. y ∈ x})"
by (rule lam_Least_HH_inj_Pow [THEN inj_strengthen_type], simp)

lemma lam_surj_sing: 
        "[| x - (\<Union>a ∈ A. F(a)) = 0;  ∀a ∈ A. ∃z ∈ x. F(a) = {z} |]   
         ==> (λa ∈ A. F(a)) ∈ surj(A, {{y}. y ∈ x})"
apply (simp add: surj_def lam_type Diff_eq_0_iff)
apply (blast elim: equalityE) 
done

lemma not_emptyI2: "y ∈ Pow(x)-{0} ==> x ≠ 0"
by auto

lemma f_subset_imp_HH_subset:
     "f`(x - (\<Union>j ∈ i. HH(f,x,j))) ∈ Pow(x - (\<Union>j ∈ i. HH(f,x,j)))-{0}   
      ==> HH(f, x, i) ∈ Pow(x) - {0}"
apply (rule HH_def_satisfies_eq [THEN ssubst])
apply (simp add: Let_def Diff_subset [THEN PowI] not_emptyI2 [THEN if_P], fast)
done

lemma f_subsets_imp_UN_HH_eq_x:
     "∀z ∈ Pow(x)-{0}. f`z ∈ Pow(z)-{0}
      ==> x - (\<Union>j ∈ (LEAST i. HH(f,x,i)={x}). HH(f,x,j)) = 0"
apply (case_tac "?P ∈ {0}", fast)
apply (drule Diff_subset [THEN PowI, THEN DiffI])
apply (drule bspec, assumption) 
apply (drule f_subset_imp_HH_subset) 
apply (blast dest!: subst_elem [OF _ HH_Least_eq_x [symmetric]] 
             elim!: mem_irrefl)
done

lemma HH_values2: "HH(f,x,i) = f`(x - (\<Union>j ∈ i. HH(f,x,j))) | HH(f,x,i)={x}"
apply (rule HH_def_satisfies_eq [THEN ssubst])
apply (simp add: Let_def Diff_subset [THEN PowI])
done

lemma HH_subset_imp_eq:
     "HH(f,x,i): Pow(x)-{0} ==> HH(f,x,i)=f`(x - (\<Union>j ∈ i. HH(f,x,j)))"
apply (rule HH_values2 [THEN disjE], assumption)
apply (fast elim!: equalityE mem_irrefl dest!: singleton_subsetD)
done

lemma f_sing_imp_HH_sing:
     "[| f ∈ (Pow(x)-{0}) -> {{z}. z ∈ x};   
         a ∈ (LEAST i. HH(f,x,i)={x}) |] ==> ∃z ∈ x. HH(f,x,a) = {z}"
apply (drule less_Least_subset_x)
apply (frule HH_subset_imp_eq)
apply (drule apply_type)
apply (rule Diff_subset [THEN PowI, THEN DiffI])
apply (fast dest!: HH_subset_x_imp_subset_Diff_UN [THEN not_emptyI2], force) 
done

lemma f_sing_lam_bij: 
     "[| x - (\<Union>j ∈ (LEAST i. HH(f,x,i)={x}). HH(f,x,j)) = 0;   
         f ∈ (Pow(x)-{0}) -> {{z}. z ∈ x} |]   
      ==> (λa ∈ (LEAST i. HH(f,x,i)={x}). HH(f,x,a))   
          ∈ bij(LEAST i. HH(f,x,i)={x}, {{y}. y ∈ x})"
apply (unfold bij_def)
apply (fast intro!: lam_Least_HH_inj lam_surj_sing f_sing_imp_HH_sing)
done

lemma lam_singI:
     "f ∈ (Π X ∈ Pow(x)-{0}. F(X))   
      ==> (λX ∈ Pow(x)-{0}. {f`X}) ∈ (Π X ∈ Pow(x)-{0}. {{z}. z ∈ F(X)})"
by (fast del: DiffI DiffE
            intro!: lam_type singleton_eq_iff [THEN iffD2] dest: apply_type)

(*FIXME: both uses have the form ...[THEN bij_converse_bij], so 
  simplification is needed!*)
lemmas bij_Least_HH_x =  
    comp_bij [OF f_sing_lam_bij [OF _ lam_singI] 
              lam_sing_bij [THEN bij_converse_bij], standard]


subsection{*The proof of AC1 ==> WO2*}

(*Establishing the existence of a bijection, namely
converse
 (converse(λx∈x. {x}) O
  Lambda
   (LEAST i. HH(λX∈Pow(x) - {0}. {f ` X}, x, i) = {x},
    HH(λX∈Pow(x) - {0}. {f ` X}, x)))
Perhaps it could be simplified. *)

lemma bijection:
     "f ∈ (Π X ∈ Pow(x) - {0}. X) 
      ==> ∃g. g ∈ bij(x, LEAST i. HH(λX ∈ Pow(x)-{0}. {f`X}, x, i) = {x})"
apply (rule exI) 
apply (rule bij_Least_HH_x [THEN bij_converse_bij])
apply (rule f_subsets_imp_UN_HH_eq_x)
apply (intro ballI apply_type) 
apply (fast intro: lam_type apply_type del: DiffE, assumption) 
apply (fast intro: Pi_weaken_type)
done

lemma AC1_WO2: "AC1 ==> WO2"
apply (unfold AC1_def WO2_def eqpoll_def)
apply (intro allI) 
apply (drule_tac x = "Pow(A) - {0}" in spec) 
apply (blast dest: bijection)
done

end


Lemmas useful in each of the three proofs

lemma HH_def_satisfies_eq:

  HH(f, x, a) =
  (let z = x - (\<Union>ba. HH(f, x, b))
   in if f ` z ∈ Pow(z) - {0} then f ` z else {x})

lemma HH_values:

  HH(f, x, a) ∈ Pow(x) - {0} ∨ HH(f, x, a) = {x}

lemma subset_imp_Diff_eq:

  BA
  ==> X - (\<Union>aA. P(a)) = X - (\<Union>aA - B. P(a)) - (\<Union>bB. P(b))

lemma Ord_DiffE:

  [| ca - b; b < a |] ==> c = bb < cc < a

lemma Diff_UN_eq_self:

  (!!y. yA ==> P(y) = {x}) ==> x - (\<Union>yA. P(y)) = x

lemma HH_eq:

  x - (\<Union>ba. HH(f, x, b)) = x - (\<Union>ba1.0. HH(f, x, b))
  ==> HH(f, x, a) = HH(f, x, a1.0)

lemma HH_is_x_gt_too:

  [| HH(f, x, b) = {x}; b < a |] ==> HH(f, x, a) = {x}

lemma HH_subset_x_lt_too:

  [| HH(f, x, a) ∈ Pow(x) - {0}; b < a |] ==> HH(f, x, b) ∈ Pow(x) - {0}

lemma HH_subset_x_imp_subset_Diff_UN:

  HH(f, x, a) ∈ Pow(x) - {0}
  ==> HH(f, x, a) ∈ Pow(x - (\<Union>ba. HH(f, x, b))) - {0}

lemma HH_eq_arg_lt:

  [| HH(f, x, v) = HH(f, x, w); HH(f, x, v) ∈ Pow(x) - {0}; vw |] ==> P

lemma HH_eq_imp_arg_eq:

  [| HH(f, x, v) = HH(f, x, w); HH(f, x, w) ∈ Pow(x) - {0}; Ord(v); Ord(w) |]
  ==> v = w

lemma HH_subset_x_imp_lepoll:

  [| HH(f, x, i) ∈ Pow(x) - {0}; Ord(i) |] ==> i lepoll Pow(x) - {0}

lemma HH_Hartog_is_x:

  HH(f, x, Hartog(Pow(x) - {0})) = {x}

lemma HH_Least_eq_x:

  HH(f, x, μi. HH(f, x, i) = {x}) = {x}

lemma less_Least_subset_x:

  a ∈ (μi. HH(f, x, i) = {x}) ==> HH(f, x, a) ∈ Pow(x) - {0}

Lemmas used in the proofs of AC1 ==> WO2 and AC17 ==> AC1

lemma lam_Least_HH_inj_Pow:

  aμi. HH(f, x, i) = {x}. HH(f, x, a)) ∈
  inj(μi. HH(f, x, i) = {x}, Pow(x) - {0})

lemma lam_Least_HH_inj:

  aμi. HH(f, x, i) = {x}. ∃zx. HH(f, x, a) = {z}
  ==> (λaμi. HH(f, x, i) = {x}. HH(f, x, a)) ∈
      inj(μi. HH(f, x, i) = {x}, {{y} . yx})

lemma lam_surj_sing:

  [| x - (\<Union>aA. F(a)) = 0; ∀aA. ∃zx. F(a) = {z} |]
  ==> (λaA. F(a)) ∈ surj(A, {{y} . yx})

lemma not_emptyI2:

  y ∈ Pow(x) - {0} ==> x  0

lemma f_subset_imp_HH_subset:

  f ` (x - (\<Union>ji. HH(f, x, j))) ∈ Pow(x - (\<Union>ji. HH(f, x, j))) - {0}
  ==> HH(f, x, i) ∈ Pow(x) - {0}

lemma f_subsets_imp_UN_HH_eq_x:

  z∈Pow(x) - {0}. f ` z ∈ Pow(z) - {0}
  ==> x - (\<Union>jμi. HH(f, x, i) = {x}. HH(f, x, j)) = 0

lemma HH_values2:

  HH(f, x, i) = f ` (x - (\<Union>ji. HH(f, x, j))) ∨ HH(f, x, i) = {x}

lemma HH_subset_imp_eq:

  HH(f, x, i) ∈ Pow(x) - {0}
  ==> HH(f, x, i) = f ` (x - (\<Union>ji. HH(f, x, j)))

lemma f_sing_imp_HH_sing:

  [| f ∈ Pow(x) - {0} -> {{z} . zx}; a ∈ (μi. HH(f, x, i) = {x}) |]
  ==> ∃zx. HH(f, x, a) = {z}

lemma f_sing_lam_bij:

  [| x - (\<Union>jμi. HH(f, x, i) = {x}. HH(f, x, j)) = 0;
     f ∈ Pow(x) - {0} -> {{z} . zx} |]
  ==> (λaμi. HH(f, x, i) = {x}. HH(f, x, a)) ∈
      bij(μi. HH(f, x, i) = {x}, {{y} . yx})

lemma lam_singI:

  f ∈ (ΠX∈Pow(x) - {0}. F(X))
  ==> (λX∈Pow(x) - {0}. {f ` X}) ∈ (ΠX∈Pow(x) - {0}. {{z} . zF(X)})

lemma bij_Least_HH_x:

  [| C - (\<Union>jμi. HHX∈Pow(C) - {0}. {f ` X}, C, i) = {C}.
             HHX∈Pow(C) - {0}. {f ` X}, C, j)) =
     0;
     f ∈ Pow(C) - {0} -> C |]
  ==> converse(λxC. {x}) Oaμi. HHX∈Pow(C) - {0}. {f ` X}, C, i) = {C}.
          HHX∈Pow(C) - {0}. {f ` X}, C, a)) ∈
      bij(μi. HHX∈Pow(C) - {0}. {f ` X}, C, i) = {C}, C)

The proof of AC1 ==> WO2

lemma bijection:

  f ∈ (ΠX∈Pow(x) - {0}. X)
  ==> ∃g. gbij(x, μi. HHX∈Pow(x) - {0}. {f ` X}, x, i) = {x})

lemma AC1_WO2:

  AC1 ==> WO2