(* 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 = (\<some> x. Elem x A & p x) in
let f = \<lambda>x. (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) (\ 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\e>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 = (\ 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 = (\ x. (Elem x U \ Q x))" ..
then obtain Q where UQ: "P = (\ 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 (\N. \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: "(\<And>x. \<forall>y. Elem y x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> 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) (\ 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) (\ 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\<^sup>+) 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\<^sup>+)"
apply (subst set_like_def)
apply (auto simp add: image_def)
apply (rule_tac x="Sep (Ext_ZF_hull R y) (\ z. z \ (Ext (R\<^sup>+) 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\<^sup>+) s)"
apply (induct_tac n)
apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)
done
lemma "set_like R \ Ext_ZF (R\<^sup>+) 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\<^sup>+)"
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
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