Theory HOLZF

(*  Title:      HOL/ZF/HOLZF.thy
    Author:     Steven Obua

Axiomatizes the ZFC universe as an HOL type.  See "Partizan Games in
Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
*)

theory HOLZF 
imports Main
begin

typedecl ZF

axiomatization
  Empty :: ZF and
  Elem :: "ZF ⇒ ZF ⇒ bool" and
  Sum :: "ZF ⇒ ZF" and
  Power :: "ZF ⇒ ZF" and
  Repl :: "ZF ⇒ (ZF ⇒ ZF) ⇒ ZF" and
  Inf :: ZF

definition Upair :: "ZF ⇒ ZF ⇒ ZF" where
  "Upair a b == Repl (Power (Power Empty)) (% x. if x = Empty then a else b)"

definition Singleton:: "ZF ⇒ ZF" where
  "Singleton x == Upair x x"

definition union :: "ZF ⇒ ZF ⇒ ZF" where
  "union A B == Sum (Upair A B)"

definition SucNat:: "ZF ⇒ ZF" where
  "SucNat x == union x (Singleton x)"

definition subset :: "ZF ⇒ ZF ⇒ bool" where
  "subset A B ≡ ∀x. Elem x A ⟶ Elem x B"

axiomatization where
  Empty: "Not (Elem x Empty)" and
  Ext: "(x = y) = (∀z. Elem z x = Elem z y)" and
  Sum: "Elem z (Sum x) = (∃y. Elem z y ∧ Elem y x)" and
  Power: "Elem y (Power x) = (subset y x)" and
  Repl: "Elem b (Repl A f) = (∃a. Elem a A ∧ b = f a)" and
  Regularity: "A ≠ Empty ⟶ (∃x. Elem x A ∧ (∀y. Elem y x ⟶ Not (Elem y A)))" and
  Infinity: "Elem Empty Inf ∧ (∀x. Elem x Inf ⟶ Elem (SucNat x) Inf)"

definition Sep :: "ZF ⇒ (ZF ⇒ bool) ⇒ ZF" where
  "Sep A p == (if (∀x. Elem x A ⟶ Not (p x)) then Empty else 
  (let z = (ϵ x. Elem x A & p x) in
   let f = λx. (if p x then x else z) in Repl A f))" 

thm Power[unfolded subset_def]

theorem Sep: "Elem b (Sep A p) = (Elem b A ∧ p b)"
  apply (auto simp add: Sep_def Empty)
  apply (auto simp add: Let_def Repl)
  apply (rule someI2, auto)+
  done

lemma subset_empty: "subset Empty A"
  by (simp add: subset_def Empty)

theorem Upair: "Elem x (Upair a b) = (x = a ∨ x = b)"
  apply (auto simp add: Upair_def Repl)
  apply (rule exI[where x=Empty])
  apply (simp add: Power subset_empty)
  apply (rule exI[where x="Power Empty"])
  apply (auto)
  apply (auto simp add: Ext Power subset_def Empty)
  apply (drule spec[where x=Empty], simp add: Empty)+
  done

lemma Singleton: "Elem x (Singleton y) = (x = y)"
  by (simp add: Singleton_def Upair)

definition Opair :: "ZF ⇒ ZF ⇒ ZF" where
  "Opair a b == Upair (Upair a a) (Upair a b)"

lemma Upair_singleton: "(Upair a a = Upair c d) = (a = c & a = d)"
  by (auto simp add: Ext[where x="Upair a a"] Upair)

lemma Upair_fsteq: "(Upair a b = Upair a c) = ((a = b & a = c) | (b = c))"
  by (auto simp add: Ext[where x="Upair a b"] Upair)

lemma Upair_comm: "Upair a b = Upair b a"
  by (auto simp add: Ext Upair)

theorem Opair: "(Opair a b = Opair c d) = (a = c & b = d)"
  proof -
    have fst: "(Opair a b = Opair c d) ⟹ a = c"
      apply (simp add: Opair_def)
      apply (simp add: Ext[where x="Upair (Upair a a) (Upair a b)"])
      apply (drule spec[where x="Upair a a"])
      apply (auto simp add: Upair Upair_singleton)
      done
    show ?thesis
      apply (auto)
      apply (erule fst)
      apply (frule fst)
      apply (auto simp add: Opair_def Upair_fsteq)
      done
  qed

definition Replacement :: "ZF ⇒ (ZF ⇒ ZF option) ⇒ ZF" where
  "Replacement A f == Repl (Sep A (% a. f a ≠ None)) (the o f)"

theorem Replacement: "Elem y (Replacement A f) = (∃x. Elem x A ∧ f x = Some y)"
  by (auto simp add: Replacement_def Repl Sep) 

definition Fst :: "ZF ⇒ ZF" where
  "Fst q == SOME x. ∃y. q = Opair x y"

definition Snd :: "ZF ⇒ ZF" where
  "Snd q == SOME y. ∃x. q = Opair x y"

theorem Fst: "Fst (Opair x y) = x"
  apply (simp add: Fst_def)
  apply (rule someI2)
  apply (simp_all add: Opair)
  done

theorem Snd: "Snd (Opair x y) = y"
  apply (simp add: Snd_def)
  apply (rule someI2)
  apply (simp_all add: Opair)
  done

definition isOpair :: "ZF ⇒ bool" where
  "isOpair q == ∃x y. q = Opair x y"

lemma isOpair: "isOpair (Opair x y) = True"
  by (auto simp add: isOpair_def)

lemma FstSnd: "isOpair x ⟹ Opair (Fst x) (Snd x) = x"
  by (auto simp add: isOpair_def Fst Snd)
  
definition CartProd :: "ZF ⇒ ZF ⇒ ZF" where
  "CartProd A B == Sum(Repl A (% a. Repl B (% b. Opair a b)))"

lemma CartProd: "Elem x (CartProd A B) = (∃a b. Elem a A ∧ Elem b B ∧ x = (Opair a b))"
  apply (auto simp add: CartProd_def Sum Repl)
  apply (rule_tac x="Repl B (Opair a)" in exI)
  apply (auto simp add: Repl)
  done

definition explode :: "ZF ⇒ ZF set" where
  "explode z == { x. Elem x z }"

lemma explode_Empty: "(explode x = {}) = (x = Empty)"
  by (auto simp add: explode_def Ext Empty)

lemma explode_Elem: "(x ∈ explode X) = (Elem x X)"
  by (simp add: explode_def)

lemma Elem_explode_in: "⟦ Elem a A; explode A ⊆ B⟧ ⟹ a ∈ B"
  by (auto simp add: explode_def)

lemma explode_CartProd_eq: "explode (CartProd a b) = (% (x,y). Opair x y) ` ((explode a) × (explode b))"
  by (simp add: explode_def set_eq_iff CartProd image_def)

lemma explode_Repl_eq: "explode (Repl A f) = image f (explode A)"
  by (simp add: explode_def Repl image_def)

definition Domain :: "ZF ⇒ ZF" where
  "Domain f == Replacement f (% p. if isOpair p then Some (Fst p) else None)"

definition Range :: "ZF ⇒ ZF" where
  "Range f == Replacement f (% p. if isOpair p then Some (Snd p) else None)"

theorem Domain: "Elem x (Domain f) = (∃y. Elem (Opair x y) f)"
  apply (auto simp add: Domain_def Replacement)
  apply (rule_tac x="Snd xa" in exI)
  apply (simp add: FstSnd)
  apply (rule_tac x="Opair x y" in exI)
  apply (simp add: isOpair Fst)
  done

theorem Range: "Elem y (Range f) = (∃x. Elem (Opair x y) f)"
  apply (auto simp add: Range_def Replacement)
  apply (rule_tac x="Fst x" in exI)
  apply (simp add: FstSnd)
  apply (rule_tac x="Opair x y" in exI)
  apply (simp add: isOpair Snd)
  done

theorem union: "Elem x (union A B) = (Elem x A | Elem x B)"
  by (auto simp add: union_def Sum Upair)

definition Field :: "ZF ⇒ ZF" where
  "Field A == union (Domain A) (Range A)"

definition app :: "ZF ⇒ ZF => ZF" (infixl ‹´› 90) ― ‹function application› where
  "f ´ x == (THE y. Elem (Opair x y) f)"

definition isFun :: "ZF ⇒ bool" where
  "isFun f == (∀x y1 y2. Elem (Opair x y1) f & Elem (Opair x y2) f ⟶ y1 = y2)"

definition Lambda :: "ZF ⇒ (ZF ⇒ ZF) ⇒ ZF" where
  "Lambda A f == Repl A (% x. Opair x (f x))"

lemma Lambda_app: "Elem x A ⟹ (Lambda A f)´x = f x"
  by (simp add: app_def Lambda_def Repl Opair)

lemma isFun_Lambda: "isFun (Lambda A f)"
  by (auto simp add: isFun_def Lambda_def Repl Opair)

lemma domain_Lambda: "Domain (Lambda A f) = A"
  apply (auto simp add: Domain_def)
  apply (subst Ext)
  apply (auto simp add: Replacement)
  apply (simp add: Lambda_def Repl)
  apply (auto simp add: Fst)
  apply (simp add: Lambda_def Repl)
  apply (rule_tac x="Opair z (f z)" in exI)
  apply (auto simp add: Fst isOpair_def)
  done

lemma Lambda_ext: "(Lambda s f = Lambda t g) = (s = t ∧ (∀x. Elem x s ⟶ f x = g x))"
proof -
  have "Lambda s f = Lambda t g ⟹ s = t"
    apply (subst domain_Lambda[where A = s and f = f, symmetric])
    apply (subst domain_Lambda[where A = t and f = g, symmetric])
    apply auto
    done
  then show ?thesis
    apply auto
    apply (subst Lambda_app[where f=f, symmetric], simp)
    apply (subst Lambda_app[where f=g, symmetric], simp)
    apply auto
    apply (auto simp add: Lambda_def Repl Ext)
    apply (auto simp add: Ext[symmetric])
    done
qed

definition PFun :: "ZF ⇒ ZF ⇒ ZF" where
  "PFun A B == Sep (Power (CartProd A B)) isFun"

definition Fun :: "ZF ⇒ ZF ⇒ ZF" where
  "Fun A B == Sep (PFun A B) (λ f. Domain f = A)"

lemma Fun_Range: "Elem f (Fun U V) ⟹ subset (Range f) V"
  apply (simp add: Fun_def Sep PFun_def Power subset_def CartProd)
  apply (auto simp add: Domain Range)
  apply (erule_tac x="Opair xa x" in allE)
  apply (auto simp add: Opair)
  done

lemma Elem_Elem_PFun: "Elem F (PFun U V) ⟹ Elem p F ⟹ isOpair p & Elem (Fst p) U & Elem (Snd p) V"
  apply (simp add: PFun_def Sep Power subset_def, clarify)
  apply (erule_tac x=p in allE)
  apply (auto simp add: CartProd isOpair Fst Snd)
  done

lemma Fun_implies_PFun[simp]: "Elem f (Fun U V) ⟹ Elem f (PFun U V)"
  by (simp add: Fun_def Sep)

lemma Elem_Elem_Fun: "Elem F (Fun U V) ⟹ Elem p F ⟹ isOpair p & Elem (Fst p) U & Elem (Snd p) V" 
  by (auto simp add: Elem_Elem_PFun dest: Fun_implies_PFun)

lemma PFun_inj: "Elem F (PFun U V) ⟹ Elem x F ⟹ Elem y F ⟹ Fst x = Fst y ⟹ Snd x = Snd y"
  apply (frule Elem_Elem_PFun[where p=x], simp)
  apply (frule Elem_Elem_PFun[where p=y], simp)
  apply (subgoal_tac "isFun F")
  apply (simp add: isFun_def isOpair_def)  
  apply (auto simp add: Fst Snd)
  apply (auto simp add: PFun_def Sep)
  done

lemma Fun_total: "⟦Elem F (Fun U V); Elem a U⟧ ⟹ ∃x. Elem (Opair a x) F"
  using [[simp_depth_limit = 2]]
  by (auto simp add: Fun_def Sep Domain)


lemma unique_fun_value: "⟦isFun f; Elem x (Domain f)⟧ ⟹ ∃!y. Elem (Opair x y) f"
  by (auto simp add: Domain isFun_def)

lemma fun_value_in_range: "⟦isFun f; Elem x (Domain f)⟧ ⟹ Elem (f´x) (Range f)"
  apply (auto simp add: Range)
  apply (drule unique_fun_value)
  apply simp
  apply (simp add: app_def)
  apply (rule exI[where x=x])
  apply (auto simp add: the_equality)
  done

lemma fun_range_witness: "⟦isFun f; Elem y (Range f)⟧ ⟹ ∃x. Elem x (Domain f) & f´x = y"
  apply (auto simp add: Range)
  apply (rule_tac x="x" in exI)
  apply (auto simp add: app_def the_equality isFun_def Domain)
  done

lemma Elem_Fun_Lambda: "Elem F (Fun U V) ⟹ ∃f. F = Lambda U f"
  apply (rule exI[where x= "% x. (THE y. Elem (Opair x y) F)"])
  apply (simp add: Ext Lambda_def Repl Domain)
  apply (simp add: Ext[symmetric])
  apply auto
  apply (frule Elem_Elem_Fun)
  apply auto
  apply (rule_tac x="Fst z" in exI)
  apply (simp add: isOpair_def)
  apply (auto simp add: Fst Snd Opair)
  apply (rule the1I2)
  apply auto
  apply (drule Fun_implies_PFun)
  apply (drule_tac x="Opair x ya" and y="Opair x yb" in PFun_inj)
  apply (auto simp add: Fst Snd)
  apply (drule Fun_implies_PFun)
  apply (drule_tac x="Opair x y" and y="Opair x ya" in PFun_inj)
  apply (auto simp add: Fst Snd)
  apply (rule the1I2)
  apply (auto simp add: Fun_total)
  apply (drule Fun_implies_PFun)
  apply (drule_tac x="Opair a x" and y="Opair a y" in PFun_inj)
  apply (auto simp add: Fst Snd)
  done
 
lemma Elem_Lambda_Fun: "Elem (Lambda A f) (Fun U V) = (A = U ∧ (∀x. Elem x A ⟶ Elem (f x) V))"
proof -
  have "Elem (Lambda A f) (Fun U V) ⟹ A = U"
    by (simp add: Fun_def Sep domain_Lambda)
  then show ?thesis
    apply auto
    apply (drule Fun_Range)
    apply (subgoal_tac "f x = ((Lambda U f) ´ x)")
    prefer 2
    apply (simp add: Lambda_app)
    apply simp
    apply (subgoal_tac "Elem (Lambda U f ´ x) (Range (Lambda U f))")
    apply (simp add: subset_def)
    apply (rule fun_value_in_range)
    apply (simp_all add: isFun_Lambda domain_Lambda)
    apply (simp add: Fun_def Sep PFun_def Power domain_Lambda isFun_Lambda)
    apply (auto simp add: subset_def CartProd)
    apply (rule_tac x="Fst x" in exI)
    apply (auto simp add: Lambda_def Repl Fst)
    done
qed    


definition is_Elem_of :: "(ZF * ZF) set" where
  "is_Elem_of == { (a,b) | a b. Elem a b }"

lemma cond_wf_Elem:
  assumes hyps:"∀x. (∀y. Elem y x ⟶ Elem y U ⟶ P y) ⟶ Elem x U ⟶ P x" "Elem a U"
  shows "P a"
proof -
  {
    fix P
    fix U
    fix a
    assume P_induct: "(∀x. (∀y. Elem y x ⟶ Elem y U ⟶ P y) ⟶ (Elem x U ⟶ P x))"
    assume a_in_U: "Elem a U"
    have "P a"
      proof -
        term "P"
        term Sep
        let ?Z = "Sep U (Not o P)"
        have "?Z = Empty ⟶ P a" by (simp add: Ext Sep Empty a_in_U)     
        moreover have "?Z ≠ Empty ⟶ False"
          proof 
            assume not_empty: "?Z ≠ Empty" 
            note thereis_x = Regularity[where A="?Z", simplified not_empty, simplified]
            then obtain x where x_def: "Elem x ?Z ∧ (∀y. Elem y x ⟶ Not (Elem y ?Z))" ..
            then have x_induct:"∀y. Elem y x ⟶ Elem y U ⟶ P y" by (simp add: Sep)
            have "Elem x U ⟶ P x" 
              by (rule impE[OF spec[OF P_induct, where x=x], OF x_induct], assumption)
            moreover have "Elem x U & Not(P x)"
              apply (insert x_def)
              apply (simp add: Sep)
              done
            ultimately show "False" by auto
          qed
        ultimately show "P a" by auto
      qed
  }
  with hyps show ?thesis by blast
qed    

lemma cond2_wf_Elem:
  assumes 
     special_P: "∃U. ∀x. Not(Elem x U) ⟶ (P x)"
     and P_induct: "∀x. (∀y. Elem y x ⟶ P y) ⟶ P x"
  shows
     "P a"
proof -
  have "∃U Q. P = (λ x. (Elem x U ⟶ Q x))"
  proof -
    from special_P obtain U where U: "∀x. Not(Elem x U) ⟶ (P x)" ..
    show ?thesis
      apply (rule_tac exI[where x=U])
      apply (rule exI[where x="P"])
      apply (rule ext)
      apply (auto simp add: U)
      done
  qed    
  then obtain U where "∃Q. P = (λ x. (Elem x U ⟶ Q x))" ..
  then obtain Q where UQ: "P = (λ x. (Elem x U ⟶ Q x))" ..
  show ?thesis
    apply (auto simp add: UQ)
    apply (rule cond_wf_Elem)
    apply (rule P_induct[simplified UQ])
    apply simp
    done
qed

primrec nat2Nat :: "nat ⇒ ZF" where
  nat2Nat_0[intro]:  "nat2Nat 0 = Empty"
| nat2Nat_Suc[intro]:  "nat2Nat (Suc n) = SucNat (nat2Nat n)"

definition Nat2nat :: "ZF ⇒ nat" where
  "Nat2nat == inv nat2Nat"

lemma Elem_nat2Nat_inf[intro]: "Elem (nat2Nat n) Inf"
  apply (induct n)
  apply (simp_all add: Infinity)
  done

definition Nat :: ZF
 where  "Nat == Sep Inf (λN. ∃n. nat2Nat n = N)"

lemma Elem_nat2Nat_Nat[intro]: "Elem (nat2Nat n) Nat"
  by (auto simp add: Nat_def Sep)

lemma Elem_Empty_Nat: "Elem Empty Nat"
  by (auto simp add: Nat_def Sep Infinity)

lemma Elem_SucNat_Nat: "Elem N Nat ⟹ Elem (SucNat N) Nat"
  by (auto simp add: Nat_def Sep Infinity)
  
lemma no_infinite_Elem_down_chain:
  "Not (∃f. isFun f ∧ Domain f = Nat ∧ (∀N. Elem N Nat ⟶ Elem (f´(SucNat N)) (f´N)))"
proof -
  {
    fix f
    assume f: "isFun f ∧ Domain f = Nat ∧ (∀N. Elem N Nat ⟶ Elem (f´(SucNat N)) (f´N))"
    let ?r = "Range f"
    have "?r ≠ Empty"
      apply (auto simp add: Ext Empty)
      apply (rule exI[where x="f´Empty"])
      apply (rule fun_value_in_range)
      apply (auto simp add: f Elem_Empty_Nat)
      done
    then have "∃x. Elem x ?r ∧ (∀y. Elem y x ⟶ Not(Elem y ?r))"
      by (simp add: Regularity)
    then obtain x where x: "Elem x ?r ∧ (∀y. Elem y x ⟶ Not(Elem y ?r))" ..
    then have "∃N. Elem N (Domain f) & f´N = x" 
      apply (rule_tac fun_range_witness)
      apply (simp_all add: f)
      done
    then have "∃N. Elem N Nat & f´N = x" 
      by (simp add: f)
    then obtain N where N: "Elem N Nat & f´N = x" ..
    from N have N': "Elem N Nat" by auto
    let ?y = "f´(SucNat N)"
    have Elem_y_r: "Elem ?y ?r"
      by (simp_all add: f Elem_SucNat_Nat N fun_value_in_range)
    have "Elem ?y (f´N)" by (auto simp add: f N')
    then have "Elem ?y x" by (simp add: N)
    with x have "Not (Elem ?y ?r)" by auto
    with Elem_y_r have "False" by auto
  }
  then show ?thesis by auto
qed

lemma Upair_nonEmpty: "Upair a b ≠ Empty"
  by (auto simp add: Ext Empty Upair)  

lemma Singleton_nonEmpty: "Singleton x ≠ Empty"
  by (auto simp add: Singleton_def Upair_nonEmpty)

lemma notsym_Elem: "Not(Elem a b & Elem b a)"
proof -
  {
    fix a b
    assume ab: "Elem a b"
    assume ba: "Elem b a"
    let ?Z = "Upair a b"
    have "?Z ≠ Empty" by (simp add: Upair_nonEmpty)
    then have "∃x. Elem x ?Z ∧ (∀y. Elem y x ⟶ Not(Elem y ?Z))"
      by (simp add: Regularity)
    then obtain x where x:"Elem x ?Z ∧ (∀y. Elem y x ⟶ Not(Elem y ?Z))" ..
    then have "x = a ∨ x = b" by (simp add: Upair)
    moreover have "x = a ⟶ Not (Elem b ?Z)"
      by (auto simp add: x ba)
    moreover have "x = b ⟶ Not (Elem a ?Z)"
      by (auto simp add: x ab)
    ultimately have "False"
      by (auto simp add: Upair)
  }    
  then show ?thesis by auto
qed

lemma irreflexiv_Elem: "Not(Elem a a)"
  by (simp add: notsym_Elem[of a a, simplified])

lemma antisym_Elem: "Elem a b ⟹ Not (Elem b a)"
  apply (insert notsym_Elem[of a b])
  apply auto
  done

primrec NatInterval :: "nat ⇒ nat ⇒ ZF" where
  "NatInterval n 0 = Singleton (nat2Nat n)"
| "NatInterval n (Suc m) = union (NatInterval n m) (Singleton (nat2Nat (n+m+1)))"

lemma n_Elem_NatInterval[rule_format]: "∀q. q ≤ m ⟶ Elem (nat2Nat (n+q)) (NatInterval n m)"
  apply (induct m)
  apply (auto simp add: Singleton union)
  apply (case_tac "q <= m")
  apply auto
  apply (subgoal_tac "q = Suc m")
  apply auto
  done

lemma NatInterval_not_Empty: "NatInterval n m ≠ Empty"
  by (auto intro:   n_Elem_NatInterval[where q = 0, simplified] simp add: Empty Ext)

lemma increasing_nat2Nat[rule_format]: "0 < n ⟶ Elem (nat2Nat (n - 1)) (nat2Nat n)"
  apply (case_tac "∃m. n = Suc m")
  apply (auto simp add: SucNat_def union Singleton)
  apply (drule spec[where x="n - 1"])
  apply arith
  done

lemma represent_NatInterval[rule_format]: "Elem x (NatInterval n m) ⟶ (∃u. n ≤ u ∧ u ≤ n+m ∧ nat2Nat u = x)"
  apply (induct m)
  apply (auto simp add: Singleton union)
  apply (rule_tac x="Suc (n+m)" in exI)
  apply auto
  done

lemma inj_nat2Nat: "inj nat2Nat"
proof -
  {
    fix n m :: nat
    assume nm: "nat2Nat n = nat2Nat (n+m)"
    assume mg0: "0 < m"
    let ?Z = "NatInterval n m"
    have "?Z ≠ Empty" by (simp add: NatInterval_not_Empty)
    then have "∃x. (Elem x ?Z) ∧ (∀y. Elem y x ⟶ Not (Elem y ?Z))" 
      by (auto simp add: Regularity)
    then obtain x where x:"Elem x ?Z ∧ (∀y. Elem y x ⟶ Not (Elem y ?Z))" ..
    then have "∃u. n ≤ u & u ≤ n+m & nat2Nat u = x" 
      by (simp add: represent_NatInterval)
    then obtain u where u: "n ≤ u & u ≤ n+m ∧ nat2Nat u = x" ..
    have "n < u ⟶ False"
    proof 
      assume n_less_u: "n < u"
      let ?y = "nat2Nat (u - 1)"
      have "Elem ?y (nat2Nat u)"
        apply (rule increasing_nat2Nat)
        apply (insert n_less_u)
        apply arith
        done
      with u have "Elem ?y x" by auto
      with x have "Not (Elem ?y ?Z)" by auto
      moreover have "Elem ?y ?Z" 
        apply (insert n_Elem_NatInterval[where q = "u - n - 1" and n=n and m=m])
        apply (insert n_less_u)
        apply (insert u)
        apply auto
        done
      ultimately show False by auto
    qed
    moreover have "u = n ⟶ False"
    proof 
      assume "u = n"
      with u have "nat2Nat n = x" by auto
      then have nm_eq_x: "nat2Nat (n+m) = x" by (simp add: nm)
      let ?y = "nat2Nat (n+m - 1)"
      have "Elem ?y (nat2Nat (n+m))"
        apply (rule increasing_nat2Nat)
        apply (insert mg0)
        apply arith
        done
      with nm_eq_x have "Elem ?y x" by auto
      with x have "Not (Elem ?y ?Z)" by auto
      moreover have "Elem ?y ?Z" 
        apply (insert n_Elem_NatInterval[where q = "m - 1" and n=n and m=m])
        apply (insert mg0)
        apply auto
        done
      ultimately show False by auto      
    qed
    ultimately have "False" using u by arith
  }
  note lemma_nat2Nat = this
  have th:"⋀x y. ¬ (x < y ∧ (∀(m::nat). y ≠ x + m))" by presburger
  have th': "⋀x y. ¬ (x ≠ y ∧ (¬ x < y) ∧ (∀(m::nat). x ≠ y + m))" by presburger
  show ?thesis
    apply (auto simp add: inj_on_def)
    apply (case_tac "x = y")
    apply auto
    apply (case_tac "x < y")
    apply (case_tac "∃m. y = x + m & 0 < m")
    apply (auto intro: lemma_nat2Nat)
    apply (case_tac "y < x")
    apply (case_tac "∃m. x = y + m & 0 < m")
    apply simp
    apply simp
    using th apply blast
    apply (case_tac "∃m. x = y + m")
    apply (auto intro: lemma_nat2Nat)
    apply (drule sym)
    using lemma_nat2Nat apply blast
    using th' apply blast    
    done
qed

lemma Nat2nat_nat2Nat[simp]: "Nat2nat (nat2Nat n) = n"
  by (simp add: Nat2nat_def inv_f_f[OF inj_nat2Nat])

lemma nat2Nat_Nat2nat[simp]: "Elem n Nat ⟹ nat2Nat (Nat2nat n) = n"
  apply (simp add: Nat2nat_def)
  apply (rule_tac f_inv_into_f)
  apply (auto simp add: image_def Nat_def Sep)
  done

lemma Nat2nat_SucNat: "Elem N Nat ⟹ Nat2nat (SucNat N) = Suc (Nat2nat N)"
  apply (auto simp add: Nat_def Sep Nat2nat_def)
  apply (auto simp add: inv_f_f[OF inj_nat2Nat])
  apply (simp only: nat2Nat.simps[symmetric])
  apply (simp only: inv_f_f[OF inj_nat2Nat])
  done
  

(*lemma Elem_induct: "(⋀x. ∀y. Elem y x ⟶ P y ⟹ P x) ⟹ P a"
  by (erule wf_induct[OF wf_is_Elem_of, simplified is_Elem_of_def, simplified])*)

lemma Elem_Opair_exists: "∃z. Elem x z & Elem y z & Elem z (Opair x y)"
  apply (rule exI[where x="Upair x y"])
  by (simp add: Upair Opair_def)

lemma UNIV_is_not_in_ZF: "UNIV ≠ explode R"
proof
  let ?Russell = "{ x. Not(Elem x x) }"
  have "?Russell = UNIV" by (simp add: irreflexiv_Elem)
  moreover assume "UNIV = explode R"
  ultimately have russell: "?Russell = explode R" by simp
  then show "False"
  proof(cases "Elem R R")
    case True     
    then show ?thesis 
      by (insert irreflexiv_Elem, auto)
  next
    case False
    then have "R ∈ ?Russell" by auto
    then have "Elem R R" by (simp add: russell explode_def)
    with False show ?thesis by auto
  qed
qed

definition SpecialR :: "(ZF * ZF) set" where
  "SpecialR ≡ { (x, y) . x ≠ Empty ∧ y = Empty}"

lemma "wf SpecialR"
  apply (subst wf_def)
  apply (auto simp add: SpecialR_def)
  done

definition Ext :: "('a * 'b) set ⇒ 'b ⇒ 'a set" where
  "Ext R y ≡ { x . (x, y) ∈ R }" 

lemma Ext_Elem: "Ext is_Elem_of = explode"
  by (auto simp add: Ext_def is_Elem_of_def explode_def)

lemma "Ext SpecialR Empty ≠ explode z"
proof 
  have "Ext SpecialR Empty = UNIV - {Empty}"
    by (auto simp add: Ext_def SpecialR_def)
  moreover assume "Ext SpecialR Empty = explode z"
  ultimately have "UNIV = explode(union z (Singleton Empty)) "
    by (auto simp add: explode_def union Singleton)
  then show "False" by (simp add: UNIV_is_not_in_ZF)
qed

definition implode :: "ZF set ⇒ ZF" where
  "implode == inv explode"

lemma inj_explode: "inj explode"
  by (auto simp add: inj_on_def explode_def Ext)

lemma implode_explode[simp]: "implode (explode x) = x"
  by (simp add: implode_def inj_explode)

definition regular :: "(ZF * ZF) set ⇒ bool" where
  "regular R == ∀A. A ≠ Empty ⟶ (∃x. Elem x A ∧ (∀y. (y, x) ∈ R ⟶ Not (Elem y A)))"

definition set_like :: "(ZF * ZF) set ⇒ bool" where
  "set_like R == ∀y. Ext R y ∈ range explode"

definition wfzf :: "(ZF * ZF) set ⇒ bool" where
  "wfzf R == regular R ∧ set_like R"

lemma regular_Elem: "regular is_Elem_of"
  by (simp add: regular_def is_Elem_of_def Regularity)

lemma set_like_Elem: "set_like is_Elem_of"
  by (auto simp add: set_like_def image_def Ext_Elem)

lemma wfzf_is_Elem_of: "wfzf is_Elem_of"
  by (auto simp add: wfzf_def regular_Elem set_like_Elem)

definition SeqSum :: "(nat ⇒ ZF) ⇒ ZF" where
  "SeqSum f == Sum (Repl Nat (f o Nat2nat))"

lemma SeqSum: "Elem x (SeqSum f) = (∃n. Elem x (f n))"
  apply (auto simp add: SeqSum_def Sum Repl)
  apply (rule_tac x = "f n" in exI)
  apply auto
  done

definition Ext_ZF :: "(ZF * ZF) set ⇒ ZF ⇒ ZF" where
  "Ext_ZF R s == implode (Ext R s)"

lemma Elem_implode: "A ∈ range explode ⟹ Elem x (implode A) = (x ∈ A)"
  apply (auto)
  apply (simp_all add: explode_def)
  done

lemma Elem_Ext_ZF: "set_like R ⟹ Elem x (Ext_ZF R s) = ((x,s) ∈ R)"
  apply (simp add: Ext_ZF_def)
  apply (subst Elem_implode)
  apply (simp add: set_like_def)
  apply (simp add: Ext_def)
  done

primrec Ext_ZF_n :: "(ZF * ZF) set ⇒ ZF ⇒ nat ⇒ ZF" where
  "Ext_ZF_n R s 0 = Ext_ZF R s"
| "Ext_ZF_n R s (Suc n) = Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"

definition Ext_ZF_hull :: "(ZF * ZF) set ⇒ ZF ⇒ ZF" where
  "Ext_ZF_hull R s == SeqSum (Ext_ZF_n R s)"

lemma Elem_Ext_ZF_hull:
  assumes set_like_R: "set_like R" 
  shows "Elem x (Ext_ZF_hull R S) = (∃n. Elem x (Ext_ZF_n R S n))"
  by (simp add: Ext_ZF_hull_def SeqSum)
  
lemma Elem_Elem_Ext_ZF_hull:
  assumes set_like_R: "set_like R" 
          and x_hull: "Elem x (Ext_ZF_hull R S)"
          and y_R_x: "(y, x) ∈ R"
  shows "Elem y (Ext_ZF_hull R S)"
proof -
  from Elem_Ext_ZF_hull[OF set_like_R] x_hull 
  have "∃n. Elem x (Ext_ZF_n R S n)" by auto
  then obtain n where n:"Elem x (Ext_ZF_n R S n)" ..
  with y_R_x have "Elem y (Ext_ZF_n R S (Suc n))"
    apply (auto simp add: Repl Sum)
    apply (rule_tac x="Ext_ZF R x" in exI) 
    apply (auto simp add: Elem_Ext_ZF[OF set_like_R])
    done
  with Elem_Ext_ZF_hull[OF set_like_R, where x=y] show ?thesis
    by (auto simp del: Ext_ZF_n.simps)
qed

lemma wfzf_minimal:
  assumes hyps: "wfzf R" "C ≠ {}"
  shows "∃x. x ∈ C ∧ (∀y. (y, x) ∈ R ⟶ y ∉ C)"
proof -
  from hyps have "∃S. S ∈ C" by auto
  then obtain S where S:"S ∈ C" by auto  
  let ?T = "Sep (Ext_ZF_hull R S) (λ s. s ∈ C)"
  from hyps have set_like_R: "set_like R" by (simp add: wfzf_def)
  show ?thesis
  proof (cases "?T = Empty")
    case True
    then have "∀ z. ¬ (Elem z (Sep (Ext_ZF R S) (λ s. s ∈ C)))"      
      apply (auto simp add: Ext Empty Sep Ext_ZF_hull_def SeqSum)
      apply (erule_tac x="z" in allE, auto)
      apply (erule_tac x=0 in allE, auto)
      done
    then show ?thesis 
      apply (rule_tac exI[where x=S])
      apply (auto simp add: Sep Empty S)
      apply (erule_tac x=y in allE)
      apply (simp add: set_like_R Elem_Ext_ZF)
      done
  next
    case False
    from hyps have regular_R: "regular R" by (simp add: wfzf_def)
    from 
      regular_R[simplified regular_def, rule_format, OF False, simplified Sep] 
      Elem_Elem_Ext_ZF_hull[OF set_like_R]
    show ?thesis by blast
  qed
qed

lemma wfzf_implies_wf: "wfzf R ⟹ wf R"
proof (subst wf_def, rule allI)
  assume wfzf: "wfzf R"
  fix P :: "ZF ⇒ bool"
  let ?C = "{x. P x}"
  {
    assume induct: "(∀x. (∀y. (y, x) ∈ R ⟶ P y) ⟶ P x)"
    let ?C = "{x. ¬ (P x)}"
    have "?C = {}"
    proof (rule ccontr)
      assume C: "?C ≠ {}"
      from
        wfzf_minimal[OF wfzf C]
      obtain x where x: "x ∈ ?C ∧ (∀y. (y, x) ∈ R ⟶ y ∉ ?C)" ..
      then have "P x"
        apply (rule_tac induct[rule_format])
        apply auto
        done
      with x show "False" by auto
    qed
    then have "∀x. P x" by auto
  }
  then show "(∀x. (∀y. (y, x) ∈ R ⟶ P y) ⟶ P x) ⟶ (∀x. P x)" by blast
qed

lemma wf_is_Elem_of: "wf is_Elem_of"
  by (auto simp add: wfzf_is_Elem_of wfzf_implies_wf)

lemma in_Ext_RTrans_implies_Elem_Ext_ZF_hull:  
  "set_like R ⟹ x ∈ (Ext (R⇧+) s) ⟹ Elem x (Ext_ZF_hull R s)"
  apply (simp add: Ext_def Elem_Ext_ZF_hull)
  apply (erule converse_trancl_induct[where r="R"])
  apply (rule exI[where x=0])
  apply (simp add: Elem_Ext_ZF)
  apply auto
  apply (rule_tac x="Suc n" in exI)
  apply (simp add: Sum Repl)
  apply (rule_tac x="Ext_ZF R z" in exI)
  apply (auto simp add: Elem_Ext_ZF)
  done

lemma implodeable_Ext_trancl: "set_like R ⟹ set_like (R⇧+)"
  apply (subst set_like_def)
  apply (auto simp add: image_def)
  apply (rule_tac x="Sep (Ext_ZF_hull R y) (λ z. z ∈ (Ext (R⇧+) y))" in exI)
  apply (auto simp add: explode_def Sep set_eqI 
    in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
  done
 
lemma Elem_Ext_ZF_hull_implies_in_Ext_RTrans[rule_format]:
  "set_like R ⟹ ∀x. Elem x (Ext_ZF_n R s n) ⟶ x ∈ (Ext (R⇧+) s)"
  apply (induct_tac n)
  apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)
  done

lemma "set_like R ⟹ Ext_ZF (R⇧+) s = Ext_ZF_hull R s"
  apply (frule implodeable_Ext_trancl)
  apply (auto simp add: Ext)
  apply (erule in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
  apply (simp add: Elem_Ext_ZF Ext_def)
  apply (auto simp add: Elem_Ext_ZF Elem_Ext_ZF_hull)
  apply (erule Elem_Ext_ZF_hull_implies_in_Ext_RTrans[simplified Ext_def, simplified], assumption)
  done

lemma wf_implies_regular: "wf R ⟹ regular R"
proof (simp add: regular_def, rule allI)
  assume wf: "wf R"
  fix A
  show "A ≠ Empty ⟶ (∃x. Elem x A ∧ (∀y. (y, x) ∈ R ⟶ ¬ Elem y A))"
  proof
    assume A: "A ≠ Empty"
    then have "∃x. x ∈ explode A" 
      by (auto simp add: explode_def Ext Empty)
    then obtain x where x:"x ∈ explode A" ..   
    from iffD1[OF wf_eq_minimal wf, rule_format, where Q="explode A", OF x]
    obtain z where "z ∈ explode A ∧ (∀y. (y, z) ∈ R ⟶ y ∉ explode A)" by auto    
    then show "∃x. Elem x A ∧ (∀y. (y, x) ∈ R ⟶ ¬ Elem y A)"      
      apply (rule_tac exI[where x = z])
      apply (simp add: explode_def)
      done
  qed
qed

lemma wf_eq_wfzf: "(wf R ∧ set_like R) = wfzf R"
  apply (auto simp add: wfzf_implies_wf)
  apply (auto simp add: wfzf_def wf_implies_regular)
  done

lemma wfzf_trancl: "wfzf R ⟹ wfzf (R⇧+)"
  by (auto simp add: wf_eq_wfzf[symmetric] implodeable_Ext_trancl wf_trancl)

lemma Ext_subset_mono: "R ⊆ S ⟹ Ext R y ⊆ Ext S y"
  by (auto simp add: Ext_def)

lemma set_like_subset: "set_like R ⟹ S ⊆ R ⟹ set_like S"
  apply (auto simp add: set_like_def)
  apply (erule_tac x=y in allE)
  apply (drule_tac y=y in Ext_subset_mono)
  apply (auto simp add: image_def)
  apply (rule_tac x="Sep x (% z. z ∈ (Ext S y))" in exI) 
  apply (auto simp add: explode_def Sep)
  done

lemma wfzf_subset: "wfzf S ⟹ R ⊆ S ⟹ wfzf R"
  by (auto intro: set_like_subset wf_subset simp add: wf_eq_wfzf[symmetric])  

end