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(* Title: HOL/ZF/HOLZF.thy
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ID: $Id$
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Author: Steven Obua
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Axiomatizes the ZFC universe as an HOL type.
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See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
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*)
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theory HOLZF
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imports Helper
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begin
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typedecl ZF
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consts
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Empty :: ZF
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Elem :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"
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Sum :: "ZF \<Rightarrow> ZF"
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Power :: "ZF \<Rightarrow> ZF"
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Repl :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF"
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Inf :: ZF
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constdefs
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Upair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"Upair a b == Repl (Power (Power Empty)) (% x. if x = Empty then a else b)"
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Singleton:: "ZF \<Rightarrow> ZF"
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"Singleton x == Upair x x"
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union :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"union A B == Sum (Upair A B)"
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SucNat:: "ZF \<Rightarrow> ZF"
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"SucNat x == union x (Singleton x)"
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subset :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"
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"subset A B == ! x. Elem x A \<longrightarrow> Elem x B"
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finalconsts
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Empty Elem Sum Power Repl Inf
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axioms
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Empty: "Not (Elem x Empty)"
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Ext: "(x = y) = (! z. Elem z x = Elem z y)"
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Sum: "Elem z (Sum x) = (? y. Elem z y & Elem y x)"
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Power: "Elem y (Power x) = (subset y x)"
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Repl: "Elem b (Repl A f) = (? a. Elem a A & b = f a)"
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Regularity: "A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. Elem y x \<longrightarrow> Not (Elem y A)))"
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Infinity: "Elem Empty Inf & (! x. Elem x Inf \<longrightarrow> Elem (SucNat x) Inf)"
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constdefs
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Sep:: "ZF \<Rightarrow> (ZF \<Rightarrow> bool) \<Rightarrow> ZF"
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"Sep A p == (if (!x. Elem x A \<longrightarrow> Not (p x)) then Empty else
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(let z = (\<some> x. Elem x A & p x) in
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let f = % x. (if p x then x else z) in Repl A f))"
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thm Power[unfolded subset_def]
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theorem Sep: "Elem b (Sep A p) = (Elem b A & p b)"
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apply (auto simp add: Sep_def Empty)
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apply (auto simp add: Let_def Repl)
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apply (rule someI2, auto)+
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done
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lemma subset_empty: "subset Empty A"
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by (simp add: subset_def Empty)
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theorem Upair: "Elem x (Upair a b) = (x = a | x = b)"
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apply (auto simp add: Upair_def Repl)
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apply (rule exI[where x=Empty])
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apply (simp add: Power subset_empty)
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apply (rule exI[where x="Power Empty"])
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apply (auto)
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apply (auto simp add: Ext Power subset_def Empty)
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apply (drule spec[where x=Empty], simp add: Empty)+
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done
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lemma Singleton: "Elem x (Singleton y) = (x = y)"
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by (simp add: Singleton_def Upair)
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constdefs
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Opair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"Opair a b == Upair (Upair a a) (Upair a b)"
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lemma Upair_singleton: "(Upair a a = Upair c d) = (a = c & a = d)"
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by (auto simp add: Ext[where x="Upair a a"] Upair)
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lemma Upair_fsteq: "(Upair a b = Upair a c) = ((a = b & a = c) | (b = c))"
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by (auto simp add: Ext[where x="Upair a b"] Upair)
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lemma Upair_comm: "Upair a b = Upair b a"
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by (auto simp add: Ext Upair)
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theorem Opair: "(Opair a b = Opair c d) = (a = c & b = d)"
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proof -
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have fst: "(Opair a b = Opair c d) \<Longrightarrow> a = c"
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apply (simp add: Opair_def)
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apply (simp add: Ext[where x="Upair (Upair a a) (Upair a b)"])
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apply (drule spec[where x="Upair a a"])
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apply (auto simp add: Upair Upair_singleton)
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done
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show ?thesis
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apply (auto)
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apply (erule fst)
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apply (frule fst)
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apply (auto simp add: Opair_def Upair_fsteq)
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done
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qed
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constdefs
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Replacement :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF option) \<Rightarrow> ZF"
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"Replacement A f == Repl (Sep A (% a. f a \<noteq> None)) (the o f)"
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theorem Replacement: "Elem y (Replacement A f) = (? x. Elem x A & f x = Some y)"
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by (auto simp add: Replacement_def Repl Sep)
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constdefs
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Fst :: "ZF \<Rightarrow> ZF"
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"Fst q == SOME x. ? y. q = Opair x y"
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Snd :: "ZF \<Rightarrow> ZF"
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"Snd q == SOME y. ? x. q = Opair x y"
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theorem Fst: "Fst (Opair x y) = x"
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apply (simp add: Fst_def)
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apply (rule someI2)
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apply (simp_all add: Opair)
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done
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theorem Snd: "Snd (Opair x y) = y"
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apply (simp add: Snd_def)
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apply (rule someI2)
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apply (simp_all add: Opair)
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done
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constdefs
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isOpair :: "ZF \<Rightarrow> bool"
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"isOpair q == ? x y. q = Opair x y"
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lemma isOpair: "isOpair (Opair x y) = True"
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by (auto simp add: isOpair_def)
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lemma FstSnd: "isOpair x \<Longrightarrow> Opair (Fst x) (Snd x) = x"
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by (auto simp add: isOpair_def Fst Snd)
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constdefs
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CartProd :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"CartProd A B == Sum(Repl A (% a. Repl B (% b. Opair a b)))"
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lemma CartProd: "Elem x (CartProd A B) = (? a b. Elem a A & Elem b B & x = (Opair a b))"
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apply (auto simp add: CartProd_def Sum Repl)
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apply (rule_tac x="Repl B (Opair a)" in exI)
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apply (auto simp add: Repl)
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done
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constdefs
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explode :: "ZF \<Rightarrow> ZF set"
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"explode z == { x. Elem x z }"
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lemma explode_Empty: "(explode x = {}) = (x = Empty)"
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by (auto simp add: explode_def Ext Empty)
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lemma explode_Elem: "(x \<in> explode X) = (Elem x X)"
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by (simp add: explode_def)
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lemma Elem_explode_in: "\<lbrakk> Elem a A; explode A \<subseteq> B\<rbrakk> \<Longrightarrow> a \<in> B"
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by (auto simp add: explode_def)
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lemma explode_CartProd_eq: "explode (CartProd a b) = (% (x,y). Opair x y) ` ((explode a) \<times> (explode b))"
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by (simp add: explode_def expand_set_eq CartProd image_def)
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lemma explode_Repl_eq: "explode (Repl A f) = image f (explode A)"
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by (simp add: explode_def Repl image_def)
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constdefs
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Domain :: "ZF \<Rightarrow> ZF"
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"Domain f == Replacement f (% p. if isOpair p then Some (Fst p) else None)"
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Range :: "ZF \<Rightarrow> ZF"
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"Range f == Replacement f (% p. if isOpair p then Some (Snd p) else None)"
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theorem Domain: "Elem x (Domain f) = (? y. Elem (Opair x y) f)"
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apply (auto simp add: Domain_def Replacement)
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apply (rule_tac x="Snd x" in exI)
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apply (simp add: FstSnd)
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apply (rule_tac x="Opair x y" in exI)
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apply (simp add: isOpair Fst)
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done
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theorem Range: "Elem y (Range f) = (? x. Elem (Opair x y) f)"
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apply (auto simp add: Range_def Replacement)
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apply (rule_tac x="Fst x" in exI)
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apply (simp add: FstSnd)
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apply (rule_tac x="Opair x y" in exI)
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apply (simp add: isOpair Snd)
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done
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theorem union: "Elem x (union A B) = (Elem x A | Elem x B)"
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by (auto simp add: union_def Sum Upair)
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constdefs
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Field :: "ZF \<Rightarrow> ZF"
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"Field A == union (Domain A) (Range A)"
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constdefs
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"\<acute>" :: "ZF \<Rightarrow> ZF => ZF" (infixl 90) --{*function application*}
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app_def: "f \<acute> x == (THE y. Elem (Opair x y) f)"
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constdefs
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isFun :: "ZF \<Rightarrow> bool"
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"isFun f == (! x y1 y2. Elem (Opair x y1) f & Elem (Opair x y2) f \<longrightarrow> y1 = y2)"
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constdefs
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Lambda :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF"
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"Lambda A f == Repl A (% x. Opair x (f x))"
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lemma Lambda_app: "Elem x A \<Longrightarrow> (Lambda A f)\<acute>x = f x"
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by (simp add: app_def Lambda_def Repl Opair)
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lemma isFun_Lambda: "isFun (Lambda A f)"
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by (auto simp add: isFun_def Lambda_def Repl Opair)
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lemma domain_Lambda: "Domain (Lambda A f) = A"
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apply (auto simp add: Domain_def)
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apply (subst Ext)
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apply (auto simp add: Replacement)
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apply (simp add: Lambda_def Repl)
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apply (auto simp add: Fst)
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apply (simp add: Lambda_def Repl)
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apply (rule_tac x="Opair z (f z)" in exI)
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apply (auto simp add: Fst isOpair_def)
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done
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lemma Lambda_ext: "(Lambda s f = Lambda t g) = (s = t & (! x. Elem x s \<longrightarrow> f x = g x))"
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proof -
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have "Lambda s f = Lambda t g \<Longrightarrow> s = t"
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apply (subst domain_Lambda[where A = s and f = f, symmetric])
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apply (subst domain_Lambda[where A = t and f = g, symmetric])
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apply auto
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done
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then show ?thesis
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apply auto
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apply (subst Lambda_app[where f=f, symmetric], simp)
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apply (subst Lambda_app[where f=g, symmetric], simp)
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apply auto
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apply (auto simp add: Lambda_def Repl Ext)
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apply (auto simp add: Ext[symmetric])
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done
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qed
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constdefs
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PFun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"PFun A B == Sep (Power (CartProd A B)) isFun"
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Fun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
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"Fun A B == Sep (PFun A B) (\<lambda> f. Domain f = A)"
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lemma Fun_Range: "Elem f (Fun U V) \<Longrightarrow> subset (Range f) V"
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apply (simp add: Fun_def Sep PFun_def Power subset_def CartProd)
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apply (auto simp add: Domain Range)
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apply (erule_tac x="Opair xa x" in allE)
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apply (auto simp add: Opair)
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done
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lemma Elem_Elem_PFun: "Elem F (PFun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"
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apply (simp add: PFun_def Sep Power subset_def, clarify)
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apply (erule_tac x=p in allE)
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apply (auto simp add: CartProd isOpair Fst Snd)
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done
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lemma Fun_implies_PFun[simp]: "Elem f (Fun U V) \<Longrightarrow> Elem f (PFun U V)"
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by (simp add: Fun_def Sep)
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lemma Elem_Elem_Fun: "Elem F (Fun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"
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by (auto simp add: Elem_Elem_PFun dest: Fun_implies_PFun)
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lemma PFun_inj: "Elem F (PFun U V) \<Longrightarrow> Elem x F \<Longrightarrow> Elem y F \<Longrightarrow> Fst x = Fst y \<Longrightarrow> Snd x = Snd y"
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apply (frule Elem_Elem_PFun[where p=x], simp)
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apply (frule Elem_Elem_PFun[where p=y], simp)
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apply (subgoal_tac "isFun F")
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apply (simp add: isFun_def isOpair_def)
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apply (auto simp add: Fst Snd, blast)
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apply (auto simp add: PFun_def Sep)
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done
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ML {* simp_depth_limit := 2 *}
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lemma Fun_total: "\<lbrakk>Elem F (Fun U V); Elem a U\<rbrakk> \<Longrightarrow> \<exists>x. Elem (Opair a x) F"
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by (auto simp add: Fun_def Sep Domain)
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ML {* simp_depth_limit := 1000 *}
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lemma unique_fun_value: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> ?! y. Elem (Opair x y) f"
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by (auto simp add: Domain isFun_def)
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lemma fun_value_in_range: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> Elem (f\<acute>x) (Range f)"
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apply (auto simp add: Range)
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apply (drule unique_fun_value)
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apply simp
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apply (simp add: app_def)
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apply (rule exI[where x=x])
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apply (auto simp add: the_equality)
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done
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lemma fun_range_witness: "\<lbrakk>isFun f; Elem y (Range f)\<rbrakk> \<Longrightarrow> ? x. Elem x (Domain f) & f\<acute>x = y"
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apply (auto simp add: Range)
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apply (rule_tac x="x" in exI)
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apply (auto simp add: app_def the_equality isFun_def Domain)
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done
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lemma Elem_Fun_Lambda: "Elem F (Fun U V) \<Longrightarrow> ? f. F = Lambda U f"
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apply (rule exI[where x= "% x. (THE y. Elem (Opair x y) F)"])
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apply (simp add: Ext Lambda_def Repl Domain)
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apply (simp add: Ext[symmetric])
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apply auto
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apply (frule Elem_Elem_Fun)
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apply auto
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apply (rule_tac x="Fst z" in exI)
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apply (simp add: isOpair_def)
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apply (auto simp add: Fst Snd Opair)
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apply (rule theI2')
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apply auto
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apply (drule Fun_implies_PFun)
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apply (drule_tac x="Opair x ya" and y="Opair x yb" in PFun_inj)
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apply (auto simp add: Fst Snd)
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apply (drule Fun_implies_PFun)
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apply (drule_tac x="Opair x y" and y="Opair x ya" in PFun_inj)
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apply (auto simp add: Fst Snd)
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apply (rule theI2')
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apply (auto simp add: Fun_total)
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apply (drule Fun_implies_PFun)
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apply (drule_tac x="Opair a x" and y="Opair a y" in PFun_inj)
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apply (auto simp add: Fst Snd)
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done
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lemma Elem_Lambda_Fun: "Elem (Lambda A f) (Fun U V) = (A = U & (! x. Elem x A \<longrightarrow> Elem (f x) V))"
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proof -
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have "Elem (Lambda A f) (Fun U V) \<Longrightarrow> A = U"
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by (simp add: Fun_def Sep domain_Lambda)
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then show ?thesis
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apply auto
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apply (drule Fun_Range)
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apply (subgoal_tac "f x = ((Lambda U f) \<acute> x)")
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prefer 2
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apply (simp add: Lambda_app)
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apply simp
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apply (subgoal_tac "Elem (Lambda U f \<acute> x) (Range (Lambda U f))")
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apply (simp add: subset_def)
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apply (rule fun_value_in_range)
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apply (simp_all add: isFun_Lambda domain_Lambda)
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apply (simp add: Fun_def Sep PFun_def Power domain_Lambda isFun_Lambda)
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344 |
apply (auto simp add: subset_def CartProd)
|
|
345 |
apply (rule_tac x="Fst x" in exI)
|
|
346 |
apply (auto simp add: Lambda_def Repl Fst)
|
|
347 |
done
|
|
348 |
qed
|
|
349 |
|
|
350 |
|
|
351 |
constdefs
|
|
352 |
is_Elem_of :: "(ZF * ZF) set"
|
|
353 |
"is_Elem_of == { (a,b) | a b. Elem a b }"
|
|
354 |
|
|
355 |
lemma cond_wf_Elem:
|
|
356 |
assumes hyps:"\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> Elem x U \<longrightarrow> P x" "Elem a U"
|
|
357 |
shows "P a"
|
|
358 |
proof -
|
|
359 |
{
|
|
360 |
fix P
|
|
361 |
fix U
|
|
362 |
fix a
|
|
363 |
assume P_induct: "(\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> (Elem x U \<longrightarrow> P x))"
|
|
364 |
assume a_in_U: "Elem a U"
|
|
365 |
have "P a"
|
|
366 |
proof -
|
|
367 |
term "P"
|
|
368 |
term Sep
|
|
369 |
let ?Z = "Sep U (Not o P)"
|
|
370 |
have "?Z = Empty \<longrightarrow> P a" by (simp add: Ext Sep Empty a_in_U)
|
|
371 |
moreover have "?Z \<noteq> Empty \<longrightarrow> False"
|
|
372 |
proof
|
|
373 |
assume not_empty: "?Z \<noteq> Empty"
|
|
374 |
note thereis_x = Regularity[where A="?Z", simplified not_empty, simplified]
|
|
375 |
then obtain x where x_def: "Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
|
|
376 |
then have x_induct:"! y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y" by (simp add: Sep)
|
|
377 |
have "Elem x U \<longrightarrow> P x"
|
|
378 |
by (rule impE[OF spec[OF P_induct, where x=x], OF x_induct], assumption)
|
|
379 |
moreover have "Elem x U & Not(P x)"
|
|
380 |
apply (insert x_def)
|
|
381 |
apply (simp add: Sep)
|
|
382 |
done
|
|
383 |
ultimately show "False" by auto
|
|
384 |
qed
|
|
385 |
ultimately show "P a" by auto
|
|
386 |
qed
|
|
387 |
}
|
|
388 |
with hyps show ?thesis by blast
|
|
389 |
qed
|
|
390 |
|
|
391 |
lemma cond2_wf_Elem:
|
|
392 |
assumes
|
|
393 |
special_P: "? U. ! x. Not(Elem x U) \<longrightarrow> (P x)"
|
|
394 |
and P_induct: "\<forall>x. (\<forall>y. Elem y x \<longrightarrow> P y) \<longrightarrow> P x"
|
|
395 |
shows
|
|
396 |
"P a"
|
|
397 |
proof -
|
|
398 |
have "? U Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))"
|
|
399 |
proof -
|
|
400 |
from special_P obtain U where U:"! x. Not(Elem x U) \<longrightarrow> (P x)" ..
|
|
401 |
show ?thesis
|
|
402 |
apply (rule_tac exI[where x=U])
|
|
403 |
apply (rule exI[where x="P"])
|
|
404 |
apply (rule ext)
|
|
405 |
apply (auto simp add: U)
|
|
406 |
done
|
|
407 |
qed
|
|
408 |
then obtain U where "? Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
|
|
409 |
then obtain Q where UQ: "P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
|
|
410 |
show ?thesis
|
|
411 |
apply (auto simp add: UQ)
|
|
412 |
apply (rule cond_wf_Elem)
|
|
413 |
apply (rule P_induct[simplified UQ])
|
|
414 |
apply simp
|
|
415 |
done
|
|
416 |
qed
|
|
417 |
|
|
418 |
consts
|
|
419 |
nat2Nat :: "nat \<Rightarrow> ZF"
|
|
420 |
|
|
421 |
primrec
|
|
422 |
nat2Nat_0[intro]: "nat2Nat 0 = Empty"
|
|
423 |
nat2Nat_Suc[intro]: "nat2Nat (Suc n) = SucNat (nat2Nat n)"
|
|
424 |
|
|
425 |
constdefs
|
|
426 |
Nat2nat :: "ZF \<Rightarrow> nat"
|
|
427 |
"Nat2nat == inv nat2Nat"
|
|
428 |
|
|
429 |
lemma Elem_nat2Nat_inf[intro]: "Elem (nat2Nat n) Inf"
|
|
430 |
apply (induct n)
|
|
431 |
apply (simp_all add: Infinity)
|
|
432 |
done
|
|
433 |
|
|
434 |
constdefs
|
|
435 |
Nat :: ZF
|
|
436 |
"Nat == Sep Inf (\<lambda> N. ? n. nat2Nat n = N)"
|
|
437 |
|
|
438 |
lemma Elem_nat2Nat_Nat[intro]: "Elem (nat2Nat n) Nat"
|
|
439 |
by (auto simp add: Nat_def Sep)
|
|
440 |
|
|
441 |
lemma Elem_Empty_Nat: "Elem Empty Nat"
|
|
442 |
by (auto simp add: Nat_def Sep Infinity)
|
|
443 |
|
|
444 |
lemma Elem_SucNat_Nat: "Elem N Nat \<Longrightarrow> Elem (SucNat N) Nat"
|
|
445 |
by (auto simp add: Nat_def Sep Infinity)
|
|
446 |
|
|
447 |
lemma no_infinite_Elem_down_chain:
|
|
448 |
"Not (? f. isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N)))"
|
|
449 |
proof -
|
|
450 |
{
|
|
451 |
fix f
|
|
452 |
assume f:"isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N))"
|
|
453 |
let ?r = "Range f"
|
|
454 |
have "?r \<noteq> Empty"
|
|
455 |
apply (auto simp add: Ext Empty)
|
|
456 |
apply (rule exI[where x="f\<acute>Empty"])
|
|
457 |
apply (rule fun_value_in_range)
|
|
458 |
apply (auto simp add: f Elem_Empty_Nat)
|
|
459 |
done
|
|
460 |
then have "? x. Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))"
|
|
461 |
by (simp add: Regularity)
|
|
462 |
then obtain x where x: "Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))" ..
|
|
463 |
then have "? N. Elem N (Domain f) & f\<acute>N = x"
|
|
464 |
apply (rule_tac fun_range_witness)
|
|
465 |
apply (simp_all add: f)
|
|
466 |
done
|
|
467 |
then have "? N. Elem N Nat & f\<acute>N = x"
|
|
468 |
by (simp add: f)
|
|
469 |
then obtain N where N: "Elem N Nat & f\<acute>N = x" ..
|
|
470 |
from N have N': "Elem N Nat" by auto
|
|
471 |
let ?y = "f\<acute>(SucNat N)"
|
|
472 |
have Elem_y_r: "Elem ?y ?r"
|
|
473 |
by (simp_all add: f Elem_SucNat_Nat N fun_value_in_range)
|
|
474 |
have "Elem ?y (f\<acute>N)" by (auto simp add: f N')
|
|
475 |
then have "Elem ?y x" by (simp add: N)
|
|
476 |
with x have "Not (Elem ?y ?r)" by auto
|
|
477 |
with Elem_y_r have "False" by auto
|
|
478 |
}
|
|
479 |
then show ?thesis by auto
|
|
480 |
qed
|
|
481 |
|
|
482 |
lemma Upair_nonEmpty: "Upair a b \<noteq> Empty"
|
|
483 |
by (auto simp add: Ext Empty Upair)
|
|
484 |
|
|
485 |
lemma Singleton_nonEmpty: "Singleton x \<noteq> Empty"
|
|
486 |
by (auto simp add: Singleton_def Upair_nonEmpty)
|
|
487 |
|
|
488 |
lemma antisym_Elem: "Not(Elem a b & Elem b a)"
|
|
489 |
proof -
|
|
490 |
{
|
|
491 |
fix a b
|
|
492 |
assume ab: "Elem a b"
|
|
493 |
assume ba: "Elem b a"
|
|
494 |
let ?Z = "Upair a b"
|
|
495 |
have "?Z \<noteq> Empty" by (simp add: Upair_nonEmpty)
|
|
496 |
then have "? x. Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))"
|
|
497 |
by (simp add: Regularity)
|
|
498 |
then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))" ..
|
|
499 |
then have "x = a \<or> x = b" by (simp add: Upair)
|
|
500 |
moreover have "x = a \<longrightarrow> Not (Elem b ?Z)"
|
|
501 |
by (auto simp add: x ba)
|
|
502 |
moreover have "x = b \<longrightarrow> Not (Elem a ?Z)"
|
|
503 |
by (auto simp add: x ab)
|
|
504 |
ultimately have "False"
|
|
505 |
by (auto simp add: Upair)
|
|
506 |
}
|
|
507 |
then show ?thesis by auto
|
|
508 |
qed
|
|
509 |
|
|
510 |
lemma irreflexiv_Elem: "Not(Elem a a)"
|
|
511 |
by (simp add: antisym_Elem[of a a, simplified])
|
|
512 |
|
|
513 |
lemma antisym_Elem: "Elem a b \<Longrightarrow> Not (Elem b a)"
|
|
514 |
apply (insert antisym_Elem[of a b])
|
|
515 |
apply auto
|
|
516 |
done
|
|
517 |
|
|
518 |
consts
|
|
519 |
NatInterval :: "nat \<Rightarrow> nat \<Rightarrow> ZF"
|
|
520 |
|
|
521 |
primrec
|
|
522 |
"NatInterval n 0 = Singleton (nat2Nat n)"
|
|
523 |
"NatInterval n (Suc m) = union (NatInterval n m) (Singleton (nat2Nat (n+m+1)))"
|
|
524 |
|
|
525 |
lemma n_Elem_NatInterval[rule_format]: "! q. q <= m \<longrightarrow> Elem (nat2Nat (n+q)) (NatInterval n m)"
|
|
526 |
apply (induct m)
|
|
527 |
apply (auto simp add: Singleton union)
|
|
528 |
apply (case_tac "q <= m")
|
|
529 |
apply auto
|
|
530 |
apply (subgoal_tac "q = Suc m")
|
|
531 |
apply auto
|
|
532 |
done
|
|
533 |
|
|
534 |
lemma NatInterval_not_Empty: "NatInterval n m \<noteq> Empty"
|
|
535 |
by (auto intro: n_Elem_NatInterval[where q = 0, simplified] simp add: Empty Ext)
|
|
536 |
|
|
537 |
lemma increasing_nat2Nat[rule_format]: "0 < n \<longrightarrow> Elem (nat2Nat (n - 1)) (nat2Nat n)"
|
|
538 |
apply (case_tac "? m. n = Suc m")
|
|
539 |
apply (auto simp add: SucNat_def union Singleton)
|
|
540 |
apply (drule spec[where x="n - 1"])
|
|
541 |
apply arith
|
|
542 |
done
|
|
543 |
|
|
544 |
lemma represent_NatInterval[rule_format]: "Elem x (NatInterval n m) \<longrightarrow> (? u. n \<le> u & u \<le> n+m & nat2Nat u = x)"
|
|
545 |
apply (induct m)
|
|
546 |
apply (auto simp add: Singleton union)
|
|
547 |
apply (rule_tac x="Suc (n+m)" in exI)
|
|
548 |
apply auto
|
|
549 |
done
|
|
550 |
|
|
551 |
lemma inj_nat2Nat: "inj nat2Nat"
|
|
552 |
proof -
|
|
553 |
{
|
|
554 |
fix n m :: nat
|
|
555 |
assume nm: "nat2Nat n = nat2Nat (n+m)"
|
|
556 |
assume mg0: "0 < m"
|
|
557 |
let ?Z = "NatInterval n m"
|
|
558 |
have "?Z \<noteq> Empty" by (simp add: NatInterval_not_Empty)
|
|
559 |
then have "? x. (Elem x ?Z) & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))"
|
|
560 |
by (auto simp add: Regularity)
|
|
561 |
then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
|
|
562 |
then have "? u. n \<le> u & u \<le> n+m & nat2Nat u = x"
|
|
563 |
by (simp add: represent_NatInterval)
|
|
564 |
then obtain u where u: "n \<le> u & u \<le> n+m & nat2Nat u = x" ..
|
|
565 |
have "n < u \<longrightarrow> False"
|
|
566 |
proof
|
|
567 |
assume n_less_u: "n < u"
|
|
568 |
let ?y = "nat2Nat (u - 1)"
|
|
569 |
have "Elem ?y (nat2Nat u)"
|
|
570 |
apply (rule increasing_nat2Nat)
|
|
571 |
apply (insert n_less_u)
|
|
572 |
apply arith
|
|
573 |
done
|
|
574 |
with u have "Elem ?y x" by auto
|
|
575 |
with x have "Not (Elem ?y ?Z)" by auto
|
|
576 |
moreover have "Elem ?y ?Z"
|
|
577 |
apply (insert n_Elem_NatInterval[where q = "u - n - 1" and n=n and m=m])
|
|
578 |
apply (insert n_less_u)
|
|
579 |
apply (insert u)
|
|
580 |
apply auto
|
|
581 |
done
|
|
582 |
ultimately show False by auto
|
|
583 |
qed
|
|
584 |
moreover have "u = n \<longrightarrow> False"
|
|
585 |
proof
|
|
586 |
assume "u = n"
|
|
587 |
with u have "nat2Nat n = x" by auto
|
|
588 |
then have nm_eq_x: "nat2Nat (n+m) = x" by (simp add: nm)
|
|
589 |
let ?y = "nat2Nat (n+m - 1)"
|
|
590 |
have "Elem ?y (nat2Nat (n+m))"
|
|
591 |
apply (rule increasing_nat2Nat)
|
|
592 |
apply (insert mg0)
|
|
593 |
apply arith
|
|
594 |
done
|
|
595 |
with nm_eq_x have "Elem ?y x" by auto
|
|
596 |
with x have "Not (Elem ?y ?Z)" by auto
|
|
597 |
moreover have "Elem ?y ?Z"
|
|
598 |
apply (insert n_Elem_NatInterval[where q = "m - 1" and n=n and m=m])
|
|
599 |
apply (insert mg0)
|
|
600 |
apply auto
|
|
601 |
done
|
|
602 |
ultimately show False by auto
|
|
603 |
qed
|
|
604 |
ultimately have "False" using u by arith
|
|
605 |
}
|
|
606 |
note lemma_nat2Nat = this
|
|
607 |
show ?thesis
|
|
608 |
apply (auto simp add: inj_on_def)
|
|
609 |
apply (case_tac "x = y")
|
|
610 |
apply auto
|
|
611 |
apply (case_tac "x < y")
|
|
612 |
apply (case_tac "? m. y = x + m & 0 < m")
|
|
613 |
apply (auto intro: lemma_nat2Nat, arith)
|
|
614 |
apply (case_tac "y < x")
|
|
615 |
apply (case_tac "? m. x = y + m & 0 < m")
|
|
616 |
apply auto
|
|
617 |
apply (drule sym)
|
|
618 |
apply (auto intro: lemma_nat2Nat, arith)
|
|
619 |
done
|
|
620 |
qed
|
|
621 |
|
|
622 |
lemma Nat2nat_nat2Nat[simp]: "Nat2nat (nat2Nat n) = n"
|
|
623 |
by (simp add: Nat2nat_def inv_f_f[OF inj_nat2Nat])
|
|
624 |
|
|
625 |
lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n"
|
|
626 |
apply (simp add: Nat2nat_def)
|
|
627 |
apply (rule_tac f_inv_f)
|
|
628 |
apply (auto simp add: image_def Nat_def Sep)
|
|
629 |
done
|
|
630 |
|
|
631 |
lemma Nat2nat_SucNat: "Elem N Nat \<Longrightarrow> Nat2nat (SucNat N) = Suc (Nat2nat N)"
|
|
632 |
apply (auto simp add: Nat_def Sep Nat2nat_def)
|
|
633 |
apply (auto simp add: inv_f_f[OF inj_nat2Nat])
|
|
634 |
apply (simp only: nat2Nat.simps[symmetric])
|
|
635 |
apply (simp only: inv_f_f[OF inj_nat2Nat])
|
|
636 |
done
|
|
637 |
|
|
638 |
|
|
639 |
(*lemma Elem_induct: "(\<And>x. \<forall>y. Elem y x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
|
|
640 |
by (erule wf_induct[OF wf_is_Elem_of, simplified is_Elem_of_def, simplified])*)
|
|
641 |
|
|
642 |
lemma Elem_Opair_exists: "? z. Elem x z & Elem y z & Elem z (Opair x y)"
|
|
643 |
apply (rule exI[where x="Upair x y"])
|
|
644 |
by (simp add: Upair Opair_def)
|
|
645 |
|
|
646 |
lemma UNIV_is_not_in_ZF: "UNIV \<noteq> explode R"
|
|
647 |
proof
|
|
648 |
let ?Russell = "{ x. Not(Elem x x) }"
|
|
649 |
have "?Russell = UNIV" by (simp add: irreflexiv_Elem)
|
|
650 |
moreover assume "UNIV = explode R"
|
|
651 |
ultimately have russell: "?Russell = explode R" by simp
|
|
652 |
then show "False"
|
|
653 |
proof(cases "Elem R R")
|
|
654 |
case True
|
|
655 |
then show ?thesis
|
|
656 |
by (insert irreflexiv_Elem, auto)
|
|
657 |
next
|
|
658 |
case False
|
|
659 |
then have "R \<in> ?Russell" by auto
|
|
660 |
then have "Elem R R" by (simp add: russell explode_def)
|
|
661 |
with False show ?thesis by auto
|
|
662 |
qed
|
|
663 |
qed
|
|
664 |
|
|
665 |
constdefs
|
|
666 |
SpecialR :: "(ZF * ZF) set"
|
|
667 |
"SpecialR \<equiv> { (x, y) . x \<noteq> Empty \<and> y = Empty}"
|
|
668 |
|
|
669 |
lemma "wf SpecialR"
|
|
670 |
apply (subst wf_def)
|
|
671 |
apply (auto simp add: SpecialR_def)
|
|
672 |
done
|
|
673 |
|
|
674 |
constdefs
|
|
675 |
Ext :: "('a * 'b) set \<Rightarrow> 'b \<Rightarrow> 'a set"
|
|
676 |
"Ext R y \<equiv> { x . (x, y) \<in> R }"
|
|
677 |
|
|
678 |
lemma Ext_Elem: "Ext is_Elem_of = explode"
|
|
679 |
by (auto intro: ext simp add: Ext_def is_Elem_of_def explode_def)
|
|
680 |
|
|
681 |
lemma "Ext SpecialR Empty \<noteq> explode z"
|
|
682 |
proof
|
|
683 |
have "Ext SpecialR Empty = UNIV - {Empty}"
|
|
684 |
by (auto simp add: Ext_def SpecialR_def)
|
|
685 |
moreover assume "Ext SpecialR Empty = explode z"
|
|
686 |
ultimately have "UNIV = explode(union z (Singleton Empty)) "
|
|
687 |
by (auto simp add: explode_def union Singleton)
|
|
688 |
then show "False" by (simp add: UNIV_is_not_in_ZF)
|
|
689 |
qed
|
|
690 |
|
|
691 |
constdefs
|
|
692 |
implode :: "ZF set \<Rightarrow> ZF"
|
|
693 |
"implode == inv explode"
|
|
694 |
|
|
695 |
lemma inj_explode: "inj explode"
|
|
696 |
by (auto simp add: inj_on_def explode_def Ext)
|
|
697 |
|
|
698 |
lemma implode_explode[simp]: "implode (explode x) = x"
|
|
699 |
by (simp add: implode_def inj_explode)
|
|
700 |
|
|
701 |
constdefs
|
|
702 |
regular :: "(ZF * ZF) set \<Rightarrow> bool"
|
|
703 |
"regular R == ! A. A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. (y, x) \<in> R \<longrightarrow> Not (Elem y A)))"
|
20565
|
704 |
set_like :: "(ZF * ZF) set \<Rightarrow> bool"
|
|
705 |
"set_like R == ! y. Ext R y \<in> range explode"
|
19203
|
706 |
wfzf :: "(ZF * ZF) set \<Rightarrow> bool"
|
20565
|
707 |
"wfzf R == regular R & set_like R"
|
19203
|
708 |
|
|
709 |
lemma regular_Elem: "regular is_Elem_of"
|
|
710 |
by (simp add: regular_def is_Elem_of_def Regularity)
|
|
711 |
|
20565
|
712 |
lemma set_like_Elem: "set_like is_Elem_of"
|
|
713 |
by (auto simp add: set_like_def image_def Ext_Elem)
|
19203
|
714 |
|
|
715 |
lemma wfzf_is_Elem_of: "wfzf is_Elem_of"
|
20565
|
716 |
by (auto simp add: wfzf_def regular_Elem set_like_Elem)
|
19203
|
717 |
|
|
718 |
constdefs
|
|
719 |
SeqSum :: "(nat \<Rightarrow> ZF) \<Rightarrow> ZF"
|
|
720 |
"SeqSum f == Sum (Repl Nat (f o Nat2nat))"
|
|
721 |
|
|
722 |
lemma SeqSum: "Elem x (SeqSum f) = (? n. Elem x (f n))"
|
|
723 |
apply (auto simp add: SeqSum_def Sum Repl)
|
|
724 |
apply (rule_tac x = "f n" in exI)
|
|
725 |
apply auto
|
|
726 |
done
|
|
727 |
|
|
728 |
constdefs
|
|
729 |
Ext_ZF :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"
|
|
730 |
"Ext_ZF R s == implode (Ext R s)"
|
|
731 |
|
|
732 |
lemma Elem_implode: "A \<in> range explode \<Longrightarrow> Elem x (implode A) = (x \<in> A)"
|
|
733 |
apply (auto)
|
|
734 |
apply (simp_all add: explode_def)
|
|
735 |
done
|
|
736 |
|
20565
|
737 |
lemma Elem_Ext_ZF: "set_like R \<Longrightarrow> Elem x (Ext_ZF R s) = ((x,s) \<in> R)"
|
19203
|
738 |
apply (simp add: Ext_ZF_def)
|
|
739 |
apply (subst Elem_implode)
|
20565
|
740 |
apply (simp add: set_like_def)
|
19203
|
741 |
apply (simp add: Ext_def)
|
|
742 |
done
|
|
743 |
|
|
744 |
consts
|
|
745 |
Ext_ZF_n :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> nat \<Rightarrow> ZF"
|
|
746 |
|
|
747 |
primrec
|
|
748 |
"Ext_ZF_n R s 0 = Ext_ZF R s"
|
|
749 |
"Ext_ZF_n R s (Suc n) = Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"
|
|
750 |
|
|
751 |
constdefs
|
|
752 |
Ext_ZF_hull :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"
|
|
753 |
"Ext_ZF_hull R s == SeqSum (Ext_ZF_n R s)"
|
|
754 |
|
|
755 |
lemma Elem_Ext_ZF_hull:
|
20565
|
756 |
assumes set_like_R: "set_like R"
|
19203
|
757 |
shows "Elem x (Ext_ZF_hull R S) = (? n. Elem x (Ext_ZF_n R S n))"
|
|
758 |
by (simp add: Ext_ZF_hull_def SeqSum)
|
|
759 |
|
|
760 |
lemma Elem_Elem_Ext_ZF_hull:
|
20565
|
761 |
assumes set_like_R: "set_like R"
|
19203
|
762 |
and x_hull: "Elem x (Ext_ZF_hull R S)"
|
|
763 |
and y_R_x: "(y, x) \<in> R"
|
|
764 |
shows "Elem y (Ext_ZF_hull R S)"
|
|
765 |
proof -
|
20565
|
766 |
from Elem_Ext_ZF_hull[OF set_like_R] x_hull
|
19203
|
767 |
have "? n. Elem x (Ext_ZF_n R S n)" by auto
|
|
768 |
then obtain n where n:"Elem x (Ext_ZF_n R S n)" ..
|
|
769 |
with y_R_x have "Elem y (Ext_ZF_n R S (Suc n))"
|
|
770 |
apply (auto simp add: Repl Sum)
|
|
771 |
apply (rule_tac x="Ext_ZF R x" in exI)
|
20565
|
772 |
apply (auto simp add: Elem_Ext_ZF[OF set_like_R])
|
19203
|
773 |
done
|
20565
|
774 |
with Elem_Ext_ZF_hull[OF set_like_R, where x=y] show ?thesis
|
19203
|
775 |
by (auto simp del: Ext_ZF_n.simps)
|
|
776 |
qed
|
|
777 |
|
|
778 |
lemma wfzf_minimal:
|
|
779 |
assumes hyps: "wfzf R" "C \<noteq> {}"
|
|
780 |
shows "\<exists>x. x \<in> C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> C)"
|
|
781 |
proof -
|
|
782 |
from hyps have "\<exists>S. S \<in> C" by auto
|
|
783 |
then obtain S where S:"S \<in> C" by auto
|
|
784 |
let ?T = "Sep (Ext_ZF_hull R S) (\<lambda> s. s \<in> C)"
|
20565
|
785 |
from hyps have set_like_R: "set_like R" by (simp add: wfzf_def)
|
19203
|
786 |
show ?thesis
|
|
787 |
proof (cases "?T = Empty")
|
|
788 |
case True
|
|
789 |
then have "\<forall> z. \<not> (Elem z (Sep (Ext_ZF R S) (\<lambda> s. s \<in> C)))"
|
|
790 |
apply (auto simp add: Ext Empty Sep Ext_ZF_hull_def SeqSum)
|
|
791 |
apply (erule_tac x="z" in allE, auto)
|
|
792 |
apply (erule_tac x=0 in allE, auto)
|
|
793 |
done
|
|
794 |
then show ?thesis
|
|
795 |
apply (rule_tac exI[where x=S])
|
|
796 |
apply (auto simp add: Sep Empty S)
|
|
797 |
apply (erule_tac x=y in allE)
|
20565
|
798 |
apply (simp add: set_like_R Elem_Ext_ZF)
|
19203
|
799 |
done
|
|
800 |
next
|
|
801 |
case False
|
|
802 |
from hyps have regular_R: "regular R" by (simp add: wfzf_def)
|
|
803 |
from
|
|
804 |
regular_R[simplified regular_def, rule_format, OF False, simplified Sep]
|
20565
|
805 |
Elem_Elem_Ext_ZF_hull[OF set_like_R]
|
19203
|
806 |
show ?thesis by blast
|
|
807 |
qed
|
|
808 |
qed
|
|
809 |
|
|
810 |
lemma wfzf_implies_wf: "wfzf R \<Longrightarrow> wf R"
|
|
811 |
proof (subst wf_def, rule allI)
|
|
812 |
assume wfzf: "wfzf R"
|
|
813 |
fix P :: "ZF \<Rightarrow> bool"
|
|
814 |
let ?C = "{x. P x}"
|
|
815 |
{
|
|
816 |
assume induct: "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x)"
|
|
817 |
let ?C = "{x. \<not> (P x)}"
|
|
818 |
have "?C = {}"
|
|
819 |
proof (rule ccontr)
|
|
820 |
assume C: "?C \<noteq> {}"
|
|
821 |
from
|
|
822 |
wfzf_minimal[OF wfzf C]
|
|
823 |
obtain x where x: "x \<in> ?C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> ?C)" ..
|
|
824 |
then have "P x"
|
|
825 |
apply (rule_tac induct[rule_format])
|
|
826 |
apply auto
|
|
827 |
done
|
|
828 |
with x show "False" by auto
|
|
829 |
qed
|
|
830 |
then have "! x. P x" by auto
|
|
831 |
}
|
|
832 |
then show "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (! x. P x)" by blast
|
|
833 |
qed
|
|
834 |
|
|
835 |
lemma wf_is_Elem_of: "wf is_Elem_of"
|
|
836 |
by (auto simp add: wfzf_is_Elem_of wfzf_implies_wf)
|
|
837 |
|
|
838 |
lemma in_Ext_RTrans_implies_Elem_Ext_ZF_hull:
|
20565
|
839 |
"set_like R \<Longrightarrow> x \<in> (Ext (R^+) s) \<Longrightarrow> Elem x (Ext_ZF_hull R s)"
|
19203
|
840 |
apply (simp add: Ext_def Elem_Ext_ZF_hull)
|
|
841 |
apply (erule converse_trancl_induct[where r="R"])
|
|
842 |
apply (rule exI[where x=0])
|
|
843 |
apply (simp add: Elem_Ext_ZF)
|
|
844 |
apply auto
|
|
845 |
apply (rule_tac x="Suc n" in exI)
|
|
846 |
apply (simp add: Sum Repl)
|
|
847 |
apply (rule_tac x="Ext_ZF R z" in exI)
|
|
848 |
apply (auto simp add: Elem_Ext_ZF)
|
|
849 |
done
|
|
850 |
|
20565
|
851 |
lemma implodeable_Ext_trancl: "set_like R \<Longrightarrow> set_like (R^+)"
|
|
852 |
apply (subst set_like_def)
|
19203
|
853 |
apply (auto simp add: image_def)
|
|
854 |
apply (rule_tac x="Sep (Ext_ZF_hull R y) (\<lambda> z. z \<in> (Ext (R^+) y))" in exI)
|
|
855 |
apply (auto simp add: explode_def Sep set_ext
|
|
856 |
in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
|
|
857 |
done
|
|
858 |
|
|
859 |
lemma Elem_Ext_ZF_hull_implies_in_Ext_RTrans[rule_format]:
|
20565
|
860 |
"set_like R \<Longrightarrow> ! x. Elem x (Ext_ZF_n R s n) \<longrightarrow> x \<in> (Ext (R^+) s)"
|
19203
|
861 |
apply (induct_tac n)
|
|
862 |
apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)
|
|
863 |
done
|
|
864 |
|
20565
|
865 |
lemma "set_like R \<Longrightarrow> Ext_ZF (R^+) s = Ext_ZF_hull R s"
|
19203
|
866 |
apply (frule implodeable_Ext_trancl)
|
|
867 |
apply (auto simp add: Ext)
|
|
868 |
apply (erule in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
|
|
869 |
apply (simp add: Elem_Ext_ZF Ext_def)
|
|
870 |
apply (auto simp add: Elem_Ext_ZF Elem_Ext_ZF_hull)
|
|
871 |
apply (erule Elem_Ext_ZF_hull_implies_in_Ext_RTrans[simplified Ext_def, simplified], assumption)
|
|
872 |
done
|
|
873 |
|
|
874 |
lemma wf_implies_regular: "wf R \<Longrightarrow> regular R"
|
|
875 |
proof (simp add: regular_def, rule allI)
|
|
876 |
assume wf: "wf R"
|
|
877 |
fix A
|
|
878 |
show "A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A))"
|
|
879 |
proof
|
|
880 |
assume A: "A \<noteq> Empty"
|
|
881 |
then have "? x. x \<in> explode A"
|
|
882 |
by (auto simp add: explode_def Ext Empty)
|
|
883 |
then obtain x where x:"x \<in> explode A" ..
|
|
884 |
from iffD1[OF wf_eq_minimal wf, rule_format, where Q="explode A", OF x]
|
|
885 |
obtain z where "z \<in> explode A \<and> (\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> explode A)" by auto
|
|
886 |
then show "\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A)"
|
|
887 |
apply (rule_tac exI[where x = z])
|
|
888 |
apply (simp add: explode_def)
|
|
889 |
done
|
|
890 |
qed
|
|
891 |
qed
|
|
892 |
|
20565
|
893 |
lemma wf_eq_wfzf: "(wf R \<and> set_like R) = wfzf R"
|
19203
|
894 |
apply (auto simp add: wfzf_implies_wf)
|
|
895 |
apply (auto simp add: wfzf_def wf_implies_regular)
|
|
896 |
done
|
|
897 |
|
|
898 |
lemma wfzf_trancl: "wfzf R \<Longrightarrow> wfzf (R^+)"
|
|
899 |
by (auto simp add: wf_eq_wfzf[symmetric] implodeable_Ext_trancl wf_trancl)
|
|
900 |
|
|
901 |
lemma Ext_subset_mono: "R \<subseteq> S \<Longrightarrow> Ext R y \<subseteq> Ext S y"
|
|
902 |
by (auto simp add: Ext_def)
|
|
903 |
|
20565
|
904 |
lemma set_like_subset: "set_like R \<Longrightarrow> S \<subseteq> R \<Longrightarrow> set_like S"
|
|
905 |
apply (auto simp add: set_like_def)
|
19203
|
906 |
apply (erule_tac x=y in allE)
|
|
907 |
apply (drule_tac y=y in Ext_subset_mono)
|
|
908 |
apply (auto simp add: image_def)
|
|
909 |
apply (rule_tac x="Sep x (% z. z \<in> (Ext S y))" in exI)
|
|
910 |
apply (auto simp add: explode_def Sep)
|
|
911 |
done
|
|
912 |
|
|
913 |
lemma wfzf_subset: "wfzf S \<Longrightarrow> R \<subseteq> S \<Longrightarrow> wfzf R"
|
20565
|
914 |
by (auto intro: set_like_subset wf_subset simp add: wf_eq_wfzf[symmetric])
|
19203
|
915 |
|
|
916 |
end
|