src/HOL/ZF/HOLZF.thy
author wenzelm
Thu May 24 17:25:53 2012 +0200 (2012-05-24)
changeset 47988 e4b69e10b990
parent 46752 e9e7209eb375
child 56073 29e308b56d23
permissions -rw-r--r--
tuned proofs;
     1 (*  Title:      HOL/ZF/HOLZF.thy
     2     Author:     Steven Obua
     3 
     4 Axiomatizes the ZFC universe as an HOL type.  See "Partizan Games in
     5 Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
     6 *)
     7 
     8 theory HOLZF 
     9 imports Main
    10 begin
    11 
    12 typedecl ZF
    13 
    14 axiomatization
    15   Empty :: ZF and
    16   Elem :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" and
    17   Sum :: "ZF \<Rightarrow> ZF" and
    18   Power :: "ZF \<Rightarrow> ZF" and
    19   Repl :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF" and
    20   Inf :: ZF
    21 
    22 definition Upair :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
    23   "Upair a b == Repl (Power (Power Empty)) (% x. if x = Empty then a else b)"
    24 
    25 definition Singleton:: "ZF \<Rightarrow> ZF" where
    26   "Singleton x == Upair x x"
    27 
    28 definition union :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
    29   "union A B == Sum (Upair A B)"
    30 
    31 definition SucNat:: "ZF \<Rightarrow> ZF" where
    32   "SucNat x == union x (Singleton x)"
    33 
    34 definition subset :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" where
    35   "subset A B == ! x. Elem x A \<longrightarrow> Elem x B"
    36 
    37 axiomatization where
    38   Empty: "Not (Elem x Empty)" and
    39   Ext: "(x = y) = (! z. Elem z x = Elem z y)" and
    40   Sum: "Elem z (Sum x) = (? y. Elem z y & Elem y x)" and
    41   Power: "Elem y (Power x) = (subset y x)" and
    42   Repl: "Elem b (Repl A f) = (? a. Elem a A & b = f a)" and
    43   Regularity: "A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. Elem y x \<longrightarrow> Not (Elem y A)))" and
    44   Infinity: "Elem Empty Inf & (! x. Elem x Inf \<longrightarrow> Elem (SucNat x) Inf)"
    45 
    46 definition Sep :: "ZF \<Rightarrow> (ZF \<Rightarrow> bool) \<Rightarrow> ZF" where
    47   "Sep A p == (if (!x. Elem x A \<longrightarrow> Not (p x)) then Empty else 
    48   (let z = (\<some> x. Elem x A & p x) in
    49    let f = % x. (if p x then x else z) in Repl A f))" 
    50 
    51 thm Power[unfolded subset_def]
    52 
    53 theorem Sep: "Elem b (Sep A p) = (Elem b A & p b)"
    54   apply (auto simp add: Sep_def Empty)
    55   apply (auto simp add: Let_def Repl)
    56   apply (rule someI2, auto)+
    57   done
    58 
    59 lemma subset_empty: "subset Empty A"
    60   by (simp add: subset_def Empty)
    61 
    62 theorem Upair: "Elem x (Upair a b) = (x = a | x = b)"
    63   apply (auto simp add: Upair_def Repl)
    64   apply (rule exI[where x=Empty])
    65   apply (simp add: Power subset_empty)
    66   apply (rule exI[where x="Power Empty"])
    67   apply (auto)
    68   apply (auto simp add: Ext Power subset_def Empty)
    69   apply (drule spec[where x=Empty], simp add: Empty)+
    70   done
    71 
    72 lemma Singleton: "Elem x (Singleton y) = (x = y)"
    73   by (simp add: Singleton_def Upair)
    74 
    75 definition Opair :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
    76   "Opair a b == Upair (Upair a a) (Upair a b)"
    77 
    78 lemma Upair_singleton: "(Upair a a = Upair c d) = (a = c & a = d)"
    79   by (auto simp add: Ext[where x="Upair a a"] Upair)
    80 
    81 lemma Upair_fsteq: "(Upair a b = Upair a c) = ((a = b & a = c) | (b = c))"
    82   by (auto simp add: Ext[where x="Upair a b"] Upair)
    83 
    84 lemma Upair_comm: "Upair a b = Upair b a"
    85   by (auto simp add: Ext Upair)
    86 
    87 theorem Opair: "(Opair a b = Opair c d) = (a = c & b = d)"
    88   proof -
    89     have fst: "(Opair a b = Opair c d) \<Longrightarrow> a = c"
    90       apply (simp add: Opair_def)
    91       apply (simp add: Ext[where x="Upair (Upair a a) (Upair a b)"])
    92       apply (drule spec[where x="Upair a a"])
    93       apply (auto simp add: Upair Upair_singleton)
    94       done
    95     show ?thesis
    96       apply (auto)
    97       apply (erule fst)
    98       apply (frule fst)
    99       apply (auto simp add: Opair_def Upair_fsteq)
   100       done
   101   qed
   102 
   103 definition Replacement :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF option) \<Rightarrow> ZF" where
   104   "Replacement A f == Repl (Sep A (% a. f a \<noteq> None)) (the o f)"
   105 
   106 theorem Replacement: "Elem y (Replacement A f) = (? x. Elem x A & f x = Some y)"
   107   by (auto simp add: Replacement_def Repl Sep) 
   108 
   109 definition Fst :: "ZF \<Rightarrow> ZF" where
   110   "Fst q == SOME x. ? y. q = Opair x y"
   111 
   112 definition Snd :: "ZF \<Rightarrow> ZF" where
   113   "Snd q == SOME y. ? x. q = Opair x y"
   114 
   115 theorem Fst: "Fst (Opair x y) = x"
   116   apply (simp add: Fst_def)
   117   apply (rule someI2)
   118   apply (simp_all add: Opair)
   119   done
   120 
   121 theorem Snd: "Snd (Opair x y) = y"
   122   apply (simp add: Snd_def)
   123   apply (rule someI2)
   124   apply (simp_all add: Opair)
   125   done
   126 
   127 definition isOpair :: "ZF \<Rightarrow> bool" where
   128   "isOpair q == ? x y. q = Opair x y"
   129 
   130 lemma isOpair: "isOpair (Opair x y) = True"
   131   by (auto simp add: isOpair_def)
   132 
   133 lemma FstSnd: "isOpair x \<Longrightarrow> Opair (Fst x) (Snd x) = x"
   134   by (auto simp add: isOpair_def Fst Snd)
   135   
   136 definition CartProd :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
   137   "CartProd A B == Sum(Repl A (% a. Repl B (% b. Opair a b)))"
   138 
   139 lemma CartProd: "Elem x (CartProd A B) = (? a b. Elem a A & Elem b B & x = (Opair a b))"
   140   apply (auto simp add: CartProd_def Sum Repl)
   141   apply (rule_tac x="Repl B (Opair a)" in exI)
   142   apply (auto simp add: Repl)
   143   done
   144 
   145 definition explode :: "ZF \<Rightarrow> ZF set" where
   146   "explode z == { x. Elem x z }"
   147 
   148 lemma explode_Empty: "(explode x = {}) = (x = Empty)"
   149   by (auto simp add: explode_def Ext Empty)
   150 
   151 lemma explode_Elem: "(x \<in> explode X) = (Elem x X)"
   152   by (simp add: explode_def)
   153 
   154 lemma Elem_explode_in: "\<lbrakk> Elem a A; explode A \<subseteq> B\<rbrakk> \<Longrightarrow> a \<in> B"
   155   by (auto simp add: explode_def)
   156 
   157 lemma explode_CartProd_eq: "explode (CartProd a b) = (% (x,y). Opair x y) ` ((explode a) \<times> (explode b))"
   158   by (simp add: explode_def set_eq_iff CartProd image_def)
   159 
   160 lemma explode_Repl_eq: "explode (Repl A f) = image f (explode A)"
   161   by (simp add: explode_def Repl image_def)
   162 
   163 definition Domain :: "ZF \<Rightarrow> ZF" where
   164   "Domain f == Replacement f (% p. if isOpair p then Some (Fst p) else None)"
   165 
   166 definition Range :: "ZF \<Rightarrow> ZF" where
   167   "Range f == Replacement f (% p. if isOpair p then Some (Snd p) else None)"
   168 
   169 theorem Domain: "Elem x (Domain f) = (? y. Elem (Opair x y) f)"
   170   apply (auto simp add: Domain_def Replacement)
   171   apply (rule_tac x="Snd x" in exI)
   172   apply (simp add: FstSnd)
   173   apply (rule_tac x="Opair x y" in exI)
   174   apply (simp add: isOpair Fst)
   175   done
   176 
   177 theorem Range: "Elem y (Range f) = (? x. Elem (Opair x y) f)"
   178   apply (auto simp add: Range_def Replacement)
   179   apply (rule_tac x="Fst x" in exI)
   180   apply (simp add: FstSnd)
   181   apply (rule_tac x="Opair x y" in exI)
   182   apply (simp add: isOpair Snd)
   183   done
   184 
   185 theorem union: "Elem x (union A B) = (Elem x A | Elem x B)"
   186   by (auto simp add: union_def Sum Upair)
   187 
   188 definition Field :: "ZF \<Rightarrow> ZF" where
   189   "Field A == union (Domain A) (Range A)"
   190 
   191 definition app :: "ZF \<Rightarrow> ZF => ZF" (infixl "\<acute>" 90) --{*function application*} where
   192   "f \<acute> x == (THE y. Elem (Opair x y) f)"
   193 
   194 definition isFun :: "ZF \<Rightarrow> bool" where
   195   "isFun f == (! x y1 y2. Elem (Opair x y1) f & Elem (Opair x y2) f \<longrightarrow> y1 = y2)"
   196 
   197 definition Lambda :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF" where
   198   "Lambda A f == Repl A (% x. Opair x (f x))"
   199 
   200 lemma Lambda_app: "Elem x A \<Longrightarrow> (Lambda A f)\<acute>x = f x"
   201   by (simp add: app_def Lambda_def Repl Opair)
   202 
   203 lemma isFun_Lambda: "isFun (Lambda A f)"
   204   by (auto simp add: isFun_def Lambda_def Repl Opair)
   205 
   206 lemma domain_Lambda: "Domain (Lambda A f) = A"
   207   apply (auto simp add: Domain_def)
   208   apply (subst Ext)
   209   apply (auto simp add: Replacement)
   210   apply (simp add: Lambda_def Repl)
   211   apply (auto simp add: Fst)
   212   apply (simp add: Lambda_def Repl)
   213   apply (rule_tac x="Opair z (f z)" in exI)
   214   apply (auto simp add: Fst isOpair_def)
   215   done
   216 
   217 lemma Lambda_ext: "(Lambda s f = Lambda t g) = (s = t & (! x. Elem x s \<longrightarrow> f x = g x))"
   218 proof -
   219   have "Lambda s f = Lambda t g \<Longrightarrow> s = t"
   220     apply (subst domain_Lambda[where A = s and f = f, symmetric])
   221     apply (subst domain_Lambda[where A = t and f = g, symmetric])
   222     apply auto
   223     done
   224   then show ?thesis
   225     apply auto
   226     apply (subst Lambda_app[where f=f, symmetric], simp)
   227     apply (subst Lambda_app[where f=g, symmetric], simp)
   228     apply auto
   229     apply (auto simp add: Lambda_def Repl Ext)
   230     apply (auto simp add: Ext[symmetric])
   231     done
   232 qed
   233 
   234 definition PFun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
   235   "PFun A B == Sep (Power (CartProd A B)) isFun"
   236 
   237 definition Fun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF" where
   238   "Fun A B == Sep (PFun A B) (\<lambda> f. Domain f = A)"
   239 
   240 lemma Fun_Range: "Elem f (Fun U V) \<Longrightarrow> subset (Range f) V"
   241   apply (simp add: Fun_def Sep PFun_def Power subset_def CartProd)
   242   apply (auto simp add: Domain Range)
   243   apply (erule_tac x="Opair xa x" in allE)
   244   apply (auto simp add: Opair)
   245   done
   246 
   247 lemma Elem_Elem_PFun: "Elem F (PFun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"
   248   apply (simp add: PFun_def Sep Power subset_def, clarify)
   249   apply (erule_tac x=p in allE)
   250   apply (auto simp add: CartProd isOpair Fst Snd)
   251   done
   252 
   253 lemma Fun_implies_PFun[simp]: "Elem f (Fun U V) \<Longrightarrow> Elem f (PFun U V)"
   254   by (simp add: Fun_def Sep)
   255 
   256 lemma Elem_Elem_Fun: "Elem F (Fun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V" 
   257   by (auto simp add: Elem_Elem_PFun dest: Fun_implies_PFun)
   258 
   259 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"
   260   apply (frule Elem_Elem_PFun[where p=x], simp)
   261   apply (frule Elem_Elem_PFun[where p=y], simp)
   262   apply (subgoal_tac "isFun F")
   263   apply (simp add: isFun_def isOpair_def)  
   264   apply (auto simp add: Fst Snd, blast)
   265   apply (auto simp add: PFun_def Sep)
   266   done
   267 
   268 lemma Fun_total: "\<lbrakk>Elem F (Fun U V); Elem a U\<rbrakk> \<Longrightarrow> \<exists>x. Elem (Opair a x) F"
   269   using [[simp_depth_limit = 2]]
   270   by (auto simp add: Fun_def Sep Domain)
   271 
   272 
   273 lemma unique_fun_value: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> ?! y. Elem (Opair x y) f"
   274   by (auto simp add: Domain isFun_def)
   275 
   276 lemma fun_value_in_range: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> Elem (f\<acute>x) (Range f)"
   277   apply (auto simp add: Range)
   278   apply (drule unique_fun_value)
   279   apply simp
   280   apply (simp add: app_def)
   281   apply (rule exI[where x=x])
   282   apply (auto simp add: the_equality)
   283   done
   284 
   285 lemma fun_range_witness: "\<lbrakk>isFun f; Elem y (Range f)\<rbrakk> \<Longrightarrow> ? x. Elem x (Domain f) & f\<acute>x = y"
   286   apply (auto simp add: Range)
   287   apply (rule_tac x="x" in exI)
   288   apply (auto simp add: app_def the_equality isFun_def Domain)
   289   done
   290 
   291 lemma Elem_Fun_Lambda: "Elem F (Fun U V) \<Longrightarrow> ? f. F = Lambda U f"
   292   apply (rule exI[where x= "% x. (THE y. Elem (Opair x y) F)"])
   293   apply (simp add: Ext Lambda_def Repl Domain)
   294   apply (simp add: Ext[symmetric])
   295   apply auto
   296   apply (frule Elem_Elem_Fun)
   297   apply auto
   298   apply (rule_tac x="Fst z" in exI)
   299   apply (simp add: isOpair_def)
   300   apply (auto simp add: Fst Snd Opair)
   301   apply (rule the1I2)
   302   apply auto
   303   apply (drule Fun_implies_PFun)
   304   apply (drule_tac x="Opair x ya" and y="Opair x yb" in PFun_inj)
   305   apply (auto simp add: Fst Snd)
   306   apply (drule Fun_implies_PFun)
   307   apply (drule_tac x="Opair x y" and y="Opair x ya" in PFun_inj)
   308   apply (auto simp add: Fst Snd)
   309   apply (rule the1I2)
   310   apply (auto simp add: Fun_total)
   311   apply (drule Fun_implies_PFun)
   312   apply (drule_tac x="Opair a x" and y="Opair a y" in PFun_inj)
   313   apply (auto simp add: Fst Snd)
   314   done
   315  
   316 lemma Elem_Lambda_Fun: "Elem (Lambda A f) (Fun U V) = (A = U & (! x. Elem x A \<longrightarrow> Elem (f x) V))"
   317 proof -
   318   have "Elem (Lambda A f) (Fun U V) \<Longrightarrow> A = U"
   319     by (simp add: Fun_def Sep domain_Lambda)
   320   then show ?thesis
   321     apply auto
   322     apply (drule Fun_Range)
   323     apply (subgoal_tac "f x = ((Lambda U f) \<acute> x)")
   324     prefer 2
   325     apply (simp add: Lambda_app)
   326     apply simp
   327     apply (subgoal_tac "Elem (Lambda U f \<acute> x) (Range (Lambda U f))")
   328     apply (simp add: subset_def)
   329     apply (rule fun_value_in_range)
   330     apply (simp_all add: isFun_Lambda domain_Lambda)
   331     apply (simp add: Fun_def Sep PFun_def Power domain_Lambda isFun_Lambda)
   332     apply (auto simp add: subset_def CartProd)
   333     apply (rule_tac x="Fst x" in exI)
   334     apply (auto simp add: Lambda_def Repl Fst)
   335     done
   336 qed    
   337 
   338 
   339 definition is_Elem_of :: "(ZF * ZF) set" where
   340   "is_Elem_of == { (a,b) | a b. Elem a b }"
   341 
   342 lemma cond_wf_Elem:
   343   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"
   344   shows "P a"
   345 proof -
   346   {
   347     fix P
   348     fix U
   349     fix a
   350     assume P_induct: "(\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> (Elem x U \<longrightarrow> P x))"
   351     assume a_in_U: "Elem a U"
   352     have "P a"
   353       proof -
   354         term "P"
   355         term Sep
   356         let ?Z = "Sep U (Not o P)"
   357         have "?Z = Empty \<longrightarrow> P a" by (simp add: Ext Sep Empty a_in_U)     
   358         moreover have "?Z \<noteq> Empty \<longrightarrow> False"
   359           proof 
   360             assume not_empty: "?Z \<noteq> Empty" 
   361             note thereis_x = Regularity[where A="?Z", simplified not_empty, simplified]
   362             then obtain x where x_def: "Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
   363             then have x_induct:"! y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y" by (simp add: Sep)
   364             have "Elem x U \<longrightarrow> P x" 
   365               by (rule impE[OF spec[OF P_induct, where x=x], OF x_induct], assumption)
   366             moreover have "Elem x U & Not(P x)"
   367               apply (insert x_def)
   368               apply (simp add: Sep)
   369               done
   370             ultimately show "False" by auto
   371           qed
   372         ultimately show "P a" by auto
   373       qed
   374   }
   375   with hyps show ?thesis by blast
   376 qed    
   377 
   378 lemma cond2_wf_Elem:
   379   assumes 
   380      special_P: "? U. ! x. Not(Elem x U) \<longrightarrow> (P x)"
   381      and P_induct: "\<forall>x. (\<forall>y. Elem y x \<longrightarrow> P y) \<longrightarrow> P x"
   382   shows
   383      "P a"
   384 proof -
   385   have "? U Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))"
   386   proof -
   387     from special_P obtain U where U:"! x. Not(Elem x U) \<longrightarrow> (P x)" ..
   388     show ?thesis
   389       apply (rule_tac exI[where x=U])
   390       apply (rule exI[where x="P"])
   391       apply (rule ext)
   392       apply (auto simp add: U)
   393       done
   394   qed    
   395   then obtain U where "? Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
   396   then obtain Q where UQ: "P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
   397   show ?thesis
   398     apply (auto simp add: UQ)
   399     apply (rule cond_wf_Elem)
   400     apply (rule P_induct[simplified UQ])
   401     apply simp
   402     done
   403 qed
   404 
   405 primrec nat2Nat :: "nat \<Rightarrow> ZF" where
   406   nat2Nat_0[intro]:  "nat2Nat 0 = Empty"
   407 | nat2Nat_Suc[intro]:  "nat2Nat (Suc n) = SucNat (nat2Nat n)"
   408 
   409 definition Nat2nat :: "ZF \<Rightarrow> nat" where
   410   "Nat2nat == inv nat2Nat"
   411 
   412 lemma Elem_nat2Nat_inf[intro]: "Elem (nat2Nat n) Inf"
   413   apply (induct n)
   414   apply (simp_all add: Infinity)
   415   done
   416 
   417 definition Nat :: ZF
   418  where  "Nat == Sep Inf (\<lambda> N. ? n. nat2Nat n = N)"
   419 
   420 lemma Elem_nat2Nat_Nat[intro]: "Elem (nat2Nat n) Nat"
   421   by (auto simp add: Nat_def Sep)
   422 
   423 lemma Elem_Empty_Nat: "Elem Empty Nat"
   424   by (auto simp add: Nat_def Sep Infinity)
   425 
   426 lemma Elem_SucNat_Nat: "Elem N Nat \<Longrightarrow> Elem (SucNat N) Nat"
   427   by (auto simp add: Nat_def Sep Infinity)
   428   
   429 lemma no_infinite_Elem_down_chain:
   430   "Not (? f. isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N)))"
   431 proof -
   432   {
   433     fix f
   434     assume f:"isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N))"
   435     let ?r = "Range f"
   436     have "?r \<noteq> Empty"
   437       apply (auto simp add: Ext Empty)
   438       apply (rule exI[where x="f\<acute>Empty"])
   439       apply (rule fun_value_in_range)
   440       apply (auto simp add: f Elem_Empty_Nat)
   441       done
   442     then have "? x. Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))"
   443       by (simp add: Regularity)
   444     then obtain x where x: "Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))" ..
   445     then have "? N. Elem N (Domain f) & f\<acute>N = x" 
   446       apply (rule_tac fun_range_witness)
   447       apply (simp_all add: f)
   448       done
   449     then have "? N. Elem N Nat & f\<acute>N = x" 
   450       by (simp add: f)
   451     then obtain N where N: "Elem N Nat & f\<acute>N = x" ..
   452     from N have N': "Elem N Nat" by auto
   453     let ?y = "f\<acute>(SucNat N)"
   454     have Elem_y_r: "Elem ?y ?r"
   455       by (simp_all add: f Elem_SucNat_Nat N fun_value_in_range)
   456     have "Elem ?y (f\<acute>N)" by (auto simp add: f N')
   457     then have "Elem ?y x" by (simp add: N)
   458     with x have "Not (Elem ?y ?r)" by auto
   459     with Elem_y_r have "False" by auto
   460   }
   461   then show ?thesis by auto
   462 qed
   463 
   464 lemma Upair_nonEmpty: "Upair a b \<noteq> Empty"
   465   by (auto simp add: Ext Empty Upair)  
   466 
   467 lemma Singleton_nonEmpty: "Singleton x \<noteq> Empty"
   468   by (auto simp add: Singleton_def Upair_nonEmpty)
   469 
   470 lemma notsym_Elem: "Not(Elem a b & Elem b a)"
   471 proof -
   472   {
   473     fix a b
   474     assume ab: "Elem a b"
   475     assume ba: "Elem b a"
   476     let ?Z = "Upair a b"
   477     have "?Z \<noteq> Empty" by (simp add: Upair_nonEmpty)
   478     then have "? x. Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))"
   479       by (simp add: Regularity)
   480     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))" ..
   481     then have "x = a \<or> x = b" by (simp add: Upair)
   482     moreover have "x = a \<longrightarrow> Not (Elem b ?Z)"
   483       by (auto simp add: x ba)
   484     moreover have "x = b \<longrightarrow> Not (Elem a ?Z)"
   485       by (auto simp add: x ab)
   486     ultimately have "False"
   487       by (auto simp add: Upair)
   488   }    
   489   then show ?thesis by auto
   490 qed
   491 
   492 lemma irreflexiv_Elem: "Not(Elem a a)"
   493   by (simp add: notsym_Elem[of a a, simplified])
   494 
   495 lemma antisym_Elem: "Elem a b \<Longrightarrow> Not (Elem b a)"
   496   apply (insert notsym_Elem[of a b])
   497   apply auto
   498   done
   499 
   500 primrec NatInterval :: "nat \<Rightarrow> nat \<Rightarrow> ZF" where
   501   "NatInterval n 0 = Singleton (nat2Nat n)"
   502 | "NatInterval n (Suc m) = union (NatInterval n m) (Singleton (nat2Nat (n+m+1)))"
   503 
   504 lemma n_Elem_NatInterval[rule_format]: "! q. q <= m \<longrightarrow> Elem (nat2Nat (n+q)) (NatInterval n m)"
   505   apply (induct m)
   506   apply (auto simp add: Singleton union)
   507   apply (case_tac "q <= m")
   508   apply auto
   509   apply (subgoal_tac "q = Suc m")
   510   apply auto
   511   done
   512 
   513 lemma NatInterval_not_Empty: "NatInterval n m \<noteq> Empty"
   514   by (auto intro:   n_Elem_NatInterval[where q = 0, simplified] simp add: Empty Ext)
   515 
   516 lemma increasing_nat2Nat[rule_format]: "0 < n \<longrightarrow> Elem (nat2Nat (n - 1)) (nat2Nat n)"
   517   apply (case_tac "? m. n = Suc m")
   518   apply (auto simp add: SucNat_def union Singleton)
   519   apply (drule spec[where x="n - 1"])
   520   apply arith
   521   done
   522 
   523 lemma represent_NatInterval[rule_format]: "Elem x (NatInterval n m) \<longrightarrow> (? u. n \<le> u & u \<le> n+m & nat2Nat u = x)"
   524   apply (induct m)
   525   apply (auto simp add: Singleton union)
   526   apply (rule_tac x="Suc (n+m)" in exI)
   527   apply auto
   528   done
   529 
   530 lemma inj_nat2Nat: "inj nat2Nat"
   531 proof -
   532   {
   533     fix n m :: nat
   534     assume nm: "nat2Nat n = nat2Nat (n+m)"
   535     assume mg0: "0 < m"
   536     let ?Z = "NatInterval n m"
   537     have "?Z \<noteq> Empty" by (simp add: NatInterval_not_Empty)
   538     then have "? x. (Elem x ?Z) & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" 
   539       by (auto simp add: Regularity)
   540     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
   541     then have "? u. n \<le> u & u \<le> n+m & nat2Nat u = x" 
   542       by (simp add: represent_NatInterval)
   543     then obtain u where u: "n \<le> u & u \<le> n+m & nat2Nat u = x" ..
   544     have "n < u \<longrightarrow> False"
   545     proof 
   546       assume n_less_u: "n < u"
   547       let ?y = "nat2Nat (u - 1)"
   548       have "Elem ?y (nat2Nat u)"
   549         apply (rule increasing_nat2Nat)
   550         apply (insert n_less_u)
   551         apply arith
   552         done
   553       with u have "Elem ?y x" by auto
   554       with x have "Not (Elem ?y ?Z)" by auto
   555       moreover have "Elem ?y ?Z" 
   556         apply (insert n_Elem_NatInterval[where q = "u - n - 1" and n=n and m=m])
   557         apply (insert n_less_u)
   558         apply (insert u)
   559         apply auto
   560         done
   561       ultimately show False by auto
   562     qed
   563     moreover have "u = n \<longrightarrow> False"
   564     proof 
   565       assume "u = n"
   566       with u have "nat2Nat n = x" by auto
   567       then have nm_eq_x: "nat2Nat (n+m) = x" by (simp add: nm)
   568       let ?y = "nat2Nat (n+m - 1)"
   569       have "Elem ?y (nat2Nat (n+m))"
   570         apply (rule increasing_nat2Nat)
   571         apply (insert mg0)
   572         apply arith
   573         done
   574       with nm_eq_x 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 = "m - 1" and n=n and m=m])
   578         apply (insert mg0)
   579         apply auto
   580         done
   581       ultimately show False by auto      
   582     qed
   583     ultimately have "False" using u by arith
   584   }
   585   note lemma_nat2Nat = this
   586   have th:"\<And>x y. \<not> (x < y \<and> (\<forall>(m\<Colon>nat). y \<noteq> x + m))" by presburger
   587   have th': "\<And>x y. \<not> (x \<noteq> y \<and> (\<not> x < y) \<and> (\<forall>(m\<Colon>nat). x \<noteq> y + m))" by presburger
   588   show ?thesis
   589     apply (auto simp add: inj_on_def)
   590     apply (case_tac "x = y")
   591     apply auto
   592     apply (case_tac "x < y")
   593     apply (case_tac "? m. y = x + m & 0 < m")
   594     apply (auto intro: lemma_nat2Nat)
   595     apply (case_tac "y < x")
   596     apply (case_tac "? m. x = y + m & 0 < m")
   597     apply simp
   598     apply simp
   599     using th apply blast
   600     apply (case_tac "? m. x = y + m")
   601     apply (auto intro: lemma_nat2Nat)
   602     apply (drule sym)
   603     using lemma_nat2Nat apply blast
   604     using th' apply blast    
   605     done
   606 qed
   607 
   608 lemma Nat2nat_nat2Nat[simp]: "Nat2nat (nat2Nat n) = n"
   609   by (simp add: Nat2nat_def inv_f_f[OF inj_nat2Nat])
   610 
   611 lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n"
   612   apply (simp add: Nat2nat_def)
   613   apply (rule_tac f_inv_into_f)
   614   apply (auto simp add: image_def Nat_def Sep)
   615   done
   616 
   617 lemma Nat2nat_SucNat: "Elem N Nat \<Longrightarrow> Nat2nat (SucNat N) = Suc (Nat2nat N)"
   618   apply (auto simp add: Nat_def Sep Nat2nat_def)
   619   apply (auto simp add: inv_f_f[OF inj_nat2Nat])
   620   apply (simp only: nat2Nat.simps[symmetric])
   621   apply (simp only: inv_f_f[OF inj_nat2Nat])
   622   done
   623   
   624 
   625 (*lemma Elem_induct: "(\<And>x. \<forall>y. Elem y x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
   626   by (erule wf_induct[OF wf_is_Elem_of, simplified is_Elem_of_def, simplified])*)
   627 
   628 lemma Elem_Opair_exists: "? z. Elem x z & Elem y z & Elem z (Opair x y)"
   629   apply (rule exI[where x="Upair x y"])
   630   by (simp add: Upair Opair_def)
   631 
   632 lemma UNIV_is_not_in_ZF: "UNIV \<noteq> explode R"
   633 proof
   634   let ?Russell = "{ x. Not(Elem x x) }"
   635   have "?Russell = UNIV" by (simp add: irreflexiv_Elem)
   636   moreover assume "UNIV = explode R"
   637   ultimately have russell: "?Russell = explode R" by simp
   638   then show "False"
   639   proof(cases "Elem R R")
   640     case True     
   641     then show ?thesis 
   642       by (insert irreflexiv_Elem, auto)
   643   next
   644     case False
   645     then have "R \<in> ?Russell" by auto
   646     then have "Elem R R" by (simp add: russell explode_def)
   647     with False show ?thesis by auto
   648   qed
   649 qed
   650 
   651 definition SpecialR :: "(ZF * ZF) set" where
   652   "SpecialR \<equiv> { (x, y) . x \<noteq> Empty \<and> y = Empty}"
   653 
   654 lemma "wf SpecialR"
   655   apply (subst wf_def)
   656   apply (auto simp add: SpecialR_def)
   657   done
   658 
   659 definition Ext :: "('a * 'b) set \<Rightarrow> 'b \<Rightarrow> 'a set" where
   660   "Ext R y \<equiv> { x . (x, y) \<in> R }" 
   661 
   662 lemma Ext_Elem: "Ext is_Elem_of = explode"
   663   by (auto simp add: Ext_def is_Elem_of_def explode_def)
   664 
   665 lemma "Ext SpecialR Empty \<noteq> explode z"
   666 proof 
   667   have "Ext SpecialR Empty = UNIV - {Empty}"
   668     by (auto simp add: Ext_def SpecialR_def)
   669   moreover assume "Ext SpecialR Empty = explode z"
   670   ultimately have "UNIV = explode(union z (Singleton Empty)) "
   671     by (auto simp add: explode_def union Singleton)
   672   then show "False" by (simp add: UNIV_is_not_in_ZF)
   673 qed
   674 
   675 definition implode :: "ZF set \<Rightarrow> ZF" where
   676   "implode == inv explode"
   677 
   678 lemma inj_explode: "inj explode"
   679   by (auto simp add: inj_on_def explode_def Ext)
   680 
   681 lemma implode_explode[simp]: "implode (explode x) = x"
   682   by (simp add: implode_def inj_explode)
   683 
   684 definition regular :: "(ZF * ZF) set \<Rightarrow> bool" where
   685   "regular R == ! A. A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. (y, x) \<in> R \<longrightarrow> Not (Elem y A)))"
   686 
   687 definition set_like :: "(ZF * ZF) set \<Rightarrow> bool" where
   688   "set_like R == ! y. Ext R y \<in> range explode"
   689 
   690 definition wfzf :: "(ZF * ZF) set \<Rightarrow> bool" where
   691   "wfzf R == regular R & set_like R"
   692 
   693 lemma regular_Elem: "regular is_Elem_of"
   694   by (simp add: regular_def is_Elem_of_def Regularity)
   695 
   696 lemma set_like_Elem: "set_like is_Elem_of"
   697   by (auto simp add: set_like_def image_def Ext_Elem)
   698 
   699 lemma wfzf_is_Elem_of: "wfzf is_Elem_of"
   700   by (auto simp add: wfzf_def regular_Elem set_like_Elem)
   701 
   702 definition SeqSum :: "(nat \<Rightarrow> ZF) \<Rightarrow> ZF" where
   703   "SeqSum f == Sum (Repl Nat (f o Nat2nat))"
   704 
   705 lemma SeqSum: "Elem x (SeqSum f) = (? n. Elem x (f n))"
   706   apply (auto simp add: SeqSum_def Sum Repl)
   707   apply (rule_tac x = "f n" in exI)
   708   apply auto
   709   done
   710 
   711 definition Ext_ZF :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF" where
   712   "Ext_ZF R s == implode (Ext R s)"
   713 
   714 lemma Elem_implode: "A \<in> range explode \<Longrightarrow> Elem x (implode A) = (x \<in> A)"
   715   apply (auto)
   716   apply (simp_all add: explode_def)
   717   done
   718 
   719 lemma Elem_Ext_ZF: "set_like R \<Longrightarrow> Elem x (Ext_ZF R s) = ((x,s) \<in> R)"
   720   apply (simp add: Ext_ZF_def)
   721   apply (subst Elem_implode)
   722   apply (simp add: set_like_def)
   723   apply (simp add: Ext_def)
   724   done
   725 
   726 primrec Ext_ZF_n :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> nat \<Rightarrow> ZF" where
   727   "Ext_ZF_n R s 0 = Ext_ZF R s"
   728 | "Ext_ZF_n R s (Suc n) = Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"
   729 
   730 definition Ext_ZF_hull :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF" where
   731   "Ext_ZF_hull R s == SeqSum (Ext_ZF_n R s)"
   732 
   733 lemma Elem_Ext_ZF_hull:
   734   assumes set_like_R: "set_like R" 
   735   shows "Elem x (Ext_ZF_hull R S) = (? n. Elem x (Ext_ZF_n R S n))"
   736   by (simp add: Ext_ZF_hull_def SeqSum)
   737   
   738 lemma Elem_Elem_Ext_ZF_hull:
   739   assumes set_like_R: "set_like R" 
   740           and x_hull: "Elem x (Ext_ZF_hull R S)"
   741           and y_R_x: "(y, x) \<in> R"
   742   shows "Elem y (Ext_ZF_hull R S)"
   743 proof -
   744   from Elem_Ext_ZF_hull[OF set_like_R] x_hull 
   745   have "? n. Elem x (Ext_ZF_n R S n)" by auto
   746   then obtain n where n:"Elem x (Ext_ZF_n R S n)" ..
   747   with y_R_x have "Elem y (Ext_ZF_n R S (Suc n))"
   748     apply (auto simp add: Repl Sum)
   749     apply (rule_tac x="Ext_ZF R x" in exI) 
   750     apply (auto simp add: Elem_Ext_ZF[OF set_like_R])
   751     done
   752   with Elem_Ext_ZF_hull[OF set_like_R, where x=y] show ?thesis
   753     by (auto simp del: Ext_ZF_n.simps)
   754 qed
   755 
   756 lemma wfzf_minimal:
   757   assumes hyps: "wfzf R" "C \<noteq> {}"
   758   shows "\<exists>x. x \<in> C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> C)"
   759 proof -
   760   from hyps have "\<exists>S. S \<in> C" by auto
   761   then obtain S where S:"S \<in> C" by auto  
   762   let ?T = "Sep (Ext_ZF_hull R S) (\<lambda> s. s \<in> C)"
   763   from hyps have set_like_R: "set_like R" by (simp add: wfzf_def)
   764   show ?thesis
   765   proof (cases "?T = Empty")
   766     case True
   767     then have "\<forall> z. \<not> (Elem z (Sep (Ext_ZF R S) (\<lambda> s. s \<in> C)))"      
   768       apply (auto simp add: Ext Empty Sep Ext_ZF_hull_def SeqSum)
   769       apply (erule_tac x="z" in allE, auto)
   770       apply (erule_tac x=0 in allE, auto)
   771       done
   772     then show ?thesis 
   773       apply (rule_tac exI[where x=S])
   774       apply (auto simp add: Sep Empty S)
   775       apply (erule_tac x=y in allE)
   776       apply (simp add: set_like_R Elem_Ext_ZF)
   777       done
   778   next
   779     case False
   780     from hyps have regular_R: "regular R" by (simp add: wfzf_def)
   781     from 
   782       regular_R[simplified regular_def, rule_format, OF False, simplified Sep] 
   783       Elem_Elem_Ext_ZF_hull[OF set_like_R]
   784     show ?thesis by blast
   785   qed
   786 qed
   787 
   788 lemma wfzf_implies_wf: "wfzf R \<Longrightarrow> wf R"
   789 proof (subst wf_def, rule allI)
   790   assume wfzf: "wfzf R"
   791   fix P :: "ZF \<Rightarrow> bool"
   792   let ?C = "{x. P x}"
   793   {
   794     assume induct: "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x)"
   795     let ?C = "{x. \<not> (P x)}"
   796     have "?C = {}"
   797     proof (rule ccontr)
   798       assume C: "?C \<noteq> {}"
   799       from
   800         wfzf_minimal[OF wfzf C]
   801       obtain x where x: "x \<in> ?C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> ?C)" ..
   802       then have "P x"
   803         apply (rule_tac induct[rule_format])
   804         apply auto
   805         done
   806       with x show "False" by auto
   807     qed
   808     then have "! x. P x" by auto
   809   }
   810   then show "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (! x. P x)" by blast
   811 qed
   812 
   813 lemma wf_is_Elem_of: "wf is_Elem_of"
   814   by (auto simp add: wfzf_is_Elem_of wfzf_implies_wf)
   815 
   816 lemma in_Ext_RTrans_implies_Elem_Ext_ZF_hull:  
   817   "set_like R \<Longrightarrow> x \<in> (Ext (R^+) s) \<Longrightarrow> Elem x (Ext_ZF_hull R s)"
   818   apply (simp add: Ext_def Elem_Ext_ZF_hull)
   819   apply (erule converse_trancl_induct[where r="R"])
   820   apply (rule exI[where x=0])
   821   apply (simp add: Elem_Ext_ZF)
   822   apply auto
   823   apply (rule_tac x="Suc n" in exI)
   824   apply (simp add: Sum Repl)
   825   apply (rule_tac x="Ext_ZF R z" in exI)
   826   apply (auto simp add: Elem_Ext_ZF)
   827   done
   828 
   829 lemma implodeable_Ext_trancl: "set_like R \<Longrightarrow> set_like (R^+)"
   830   apply (subst set_like_def)
   831   apply (auto simp add: image_def)
   832   apply (rule_tac x="Sep (Ext_ZF_hull R y) (\<lambda> z. z \<in> (Ext (R^+) y))" in exI)
   833   apply (auto simp add: explode_def Sep set_eqI 
   834     in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
   835   done
   836  
   837 lemma Elem_Ext_ZF_hull_implies_in_Ext_RTrans[rule_format]:
   838   "set_like R \<Longrightarrow> ! x. Elem x (Ext_ZF_n R s n) \<longrightarrow> x \<in> (Ext (R^+) s)"
   839   apply (induct_tac n)
   840   apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)
   841   done
   842 
   843 lemma "set_like R \<Longrightarrow> Ext_ZF (R^+) s = Ext_ZF_hull R s"
   844   apply (frule implodeable_Ext_trancl)
   845   apply (auto simp add: Ext)
   846   apply (erule in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
   847   apply (simp add: Elem_Ext_ZF Ext_def)
   848   apply (auto simp add: Elem_Ext_ZF Elem_Ext_ZF_hull)
   849   apply (erule Elem_Ext_ZF_hull_implies_in_Ext_RTrans[simplified Ext_def, simplified], assumption)
   850   done
   851 
   852 lemma wf_implies_regular: "wf R \<Longrightarrow> regular R"
   853 proof (simp add: regular_def, rule allI)
   854   assume wf: "wf R"
   855   fix A
   856   show "A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A))"
   857   proof
   858     assume A: "A \<noteq> Empty"
   859     then have "? x. x \<in> explode A" 
   860       by (auto simp add: explode_def Ext Empty)
   861     then obtain x where x:"x \<in> explode A" ..   
   862     from iffD1[OF wf_eq_minimal wf, rule_format, where Q="explode A", OF x]
   863     obtain z where "z \<in> explode A \<and> (\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> explode A)" by auto    
   864     then show "\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A)"      
   865       apply (rule_tac exI[where x = z])
   866       apply (simp add: explode_def)
   867       done
   868   qed
   869 qed
   870 
   871 lemma wf_eq_wfzf: "(wf R \<and> set_like R) = wfzf R"
   872   apply (auto simp add: wfzf_implies_wf)
   873   apply (auto simp add: wfzf_def wf_implies_regular)
   874   done
   875 
   876 lemma wfzf_trancl: "wfzf R \<Longrightarrow> wfzf (R^+)"
   877   by (auto simp add: wf_eq_wfzf[symmetric] implodeable_Ext_trancl wf_trancl)
   878 
   879 lemma Ext_subset_mono: "R \<subseteq> S \<Longrightarrow> Ext R y \<subseteq> Ext S y"
   880   by (auto simp add: Ext_def)
   881 
   882 lemma set_like_subset: "set_like R \<Longrightarrow> S \<subseteq> R \<Longrightarrow> set_like S"
   883   apply (auto simp add: set_like_def)
   884   apply (erule_tac x=y in allE)
   885   apply (drule_tac y=y in Ext_subset_mono)
   886   apply (auto simp add: image_def)
   887   apply (rule_tac x="Sep x (% z. z \<in> (Ext S y))" in exI) 
   888   apply (auto simp add: explode_def Sep)
   889   done
   890 
   891 lemma wfzf_subset: "wfzf S \<Longrightarrow> R \<subseteq> S \<Longrightarrow> wfzf R"
   892   by (auto intro: set_like_subset wf_subset simp add: wf_eq_wfzf[symmetric])  
   893 
   894 end
   895