src/HOL/ZF/HOLZF.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 24124 4399175e3014
child 24784 102e0e732495
permissions -rw-r--r--
moved Finite_Set before Datatype
     1 (*  Title:      HOL/ZF/HOLZF.thy
     2     ID:         $Id$
     3     Author:     Steven Obua
     4 
     5     Axiomatizes the ZFC universe as an HOL type.
     6     See "Partizan Games in Isabelle/HOLZF", available from http://www4.in.tum.de/~obua/partizan
     7 *)
     8 
     9 theory HOLZF 
    10 imports Helper
    11 begin
    12 
    13 typedecl ZF
    14 
    15 axiomatization
    16   Empty :: ZF and
    17   Elem :: "ZF \<Rightarrow> ZF \<Rightarrow> bool" and
    18   Sum :: "ZF \<Rightarrow> ZF" and
    19   Power :: "ZF \<Rightarrow> ZF" and
    20   Repl :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF" and
    21   Inf :: ZF
    22 
    23 constdefs
    24   Upair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
    25   "Upair a b == Repl (Power (Power Empty)) (% x. if x = Empty then a else b)"
    26   Singleton:: "ZF \<Rightarrow> ZF"
    27   "Singleton x == Upair x x"
    28   union :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
    29   "union A B == Sum (Upair A B)"
    30   SucNat:: "ZF \<Rightarrow> ZF"
    31   "SucNat x == union x (Singleton x)"
    32   subset :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"
    33   "subset A B == ! x. Elem x A \<longrightarrow> Elem x B"
    34 
    35 axioms
    36   Empty: "Not (Elem x Empty)"
    37   Ext: "(x = y) = (! z. Elem z x = Elem z y)"
    38   Sum: "Elem z (Sum x) = (? y. Elem z y & Elem y x)"
    39   Power: "Elem y (Power x) = (subset y x)"
    40   Repl: "Elem b (Repl A f) = (? a. Elem a A & b = f a)"
    41   Regularity: "A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. Elem y x \<longrightarrow> Not (Elem y A)))"
    42   Infinity: "Elem Empty Inf & (! x. Elem x Inf \<longrightarrow> Elem (SucNat x) Inf)"
    43 
    44 constdefs
    45   Sep:: "ZF \<Rightarrow> (ZF \<Rightarrow> bool) \<Rightarrow> ZF"
    46   "Sep A p == (if (!x. Elem x A \<longrightarrow> Not (p x)) then Empty else 
    47   (let z = (\<some> x. Elem x A & p x) in
    48    let f = % x. (if p x then x else z) in Repl A f))" 
    49 
    50 thm Power[unfolded subset_def]
    51 
    52 theorem Sep: "Elem b (Sep A p) = (Elem b A & p b)"
    53   apply (auto simp add: Sep_def Empty)
    54   apply (auto simp add: Let_def Repl)
    55   apply (rule someI2, auto)+
    56   done
    57 
    58 lemma subset_empty: "subset Empty A"
    59   by (simp add: subset_def Empty)
    60 
    61 theorem Upair: "Elem x (Upair a b) = (x = a | x = b)"
    62   apply (auto simp add: Upair_def Repl)
    63   apply (rule exI[where x=Empty])
    64   apply (simp add: Power subset_empty)
    65   apply (rule exI[where x="Power Empty"])
    66   apply (auto)
    67   apply (auto simp add: Ext Power subset_def Empty)
    68   apply (drule spec[where x=Empty], simp add: Empty)+
    69   done
    70 
    71 lemma Singleton: "Elem x (Singleton y) = (x = y)"
    72   by (simp add: Singleton_def Upair)
    73 
    74 constdefs 
    75   Opair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
    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 constdefs 
   104   Replacement :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF option) \<Rightarrow> ZF"
   105   "Replacement A f == Repl (Sep A (% a. f a \<noteq> None)) (the o f)"
   106 
   107 theorem Replacement: "Elem y (Replacement A f) = (? x. Elem x A & f x = Some y)"
   108   by (auto simp add: Replacement_def Repl Sep) 
   109 
   110 constdefs
   111   Fst :: "ZF \<Rightarrow> ZF"
   112   "Fst q == SOME x. ? y. q = Opair x y"
   113   Snd :: "ZF \<Rightarrow> ZF"
   114   "Snd q == SOME y. ? x. q = Opair x y"
   115 
   116 theorem Fst: "Fst (Opair x y) = x"
   117   apply (simp add: Fst_def)
   118   apply (rule someI2)
   119   apply (simp_all add: Opair)
   120   done
   121 
   122 theorem Snd: "Snd (Opair x y) = y"
   123   apply (simp add: Snd_def)
   124   apply (rule someI2)
   125   apply (simp_all add: Opair)
   126   done
   127 
   128 constdefs 
   129   isOpair :: "ZF \<Rightarrow> bool"
   130   "isOpair q == ? x y. q = Opair x y"
   131 
   132 lemma isOpair: "isOpair (Opair x y) = True"
   133   by (auto simp add: isOpair_def)
   134 
   135 lemma FstSnd: "isOpair x \<Longrightarrow> Opair (Fst x) (Snd x) = x"
   136   by (auto simp add: isOpair_def Fst Snd)
   137   
   138 constdefs
   139   CartProd :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
   140   "CartProd A B == Sum(Repl A (% a. Repl B (% b. Opair a b)))"
   141 
   142 lemma CartProd: "Elem x (CartProd A B) = (? a b. Elem a A & Elem b B & x = (Opair a b))"
   143   apply (auto simp add: CartProd_def Sum Repl)
   144   apply (rule_tac x="Repl B (Opair a)" in exI)
   145   apply (auto simp add: Repl)
   146   done
   147 
   148 constdefs
   149   explode :: "ZF \<Rightarrow> ZF set"
   150   "explode z == { x. Elem x z }"
   151 
   152 lemma explode_Empty: "(explode x = {}) = (x = Empty)"
   153   by (auto simp add: explode_def Ext Empty)
   154 
   155 lemma explode_Elem: "(x \<in> explode X) = (Elem x X)"
   156   by (simp add: explode_def)
   157 
   158 lemma Elem_explode_in: "\<lbrakk> Elem a A; explode A \<subseteq> B\<rbrakk> \<Longrightarrow> a \<in> B"
   159   by (auto simp add: explode_def)
   160 
   161 lemma explode_CartProd_eq: "explode (CartProd a b) = (% (x,y). Opair x y) ` ((explode a) \<times> (explode b))"
   162   by (simp add: explode_def expand_set_eq CartProd image_def)
   163 
   164 lemma explode_Repl_eq: "explode (Repl A f) = image f (explode A)"
   165   by (simp add: explode_def Repl image_def)
   166 
   167 constdefs
   168   Domain :: "ZF \<Rightarrow> ZF"
   169   "Domain f == Replacement f (% p. if isOpair p then Some (Fst p) else None)"
   170   Range :: "ZF \<Rightarrow> ZF"
   171   "Range f == Replacement f (% p. if isOpair p then Some (Snd p) else None)"
   172 
   173 theorem Domain: "Elem x (Domain f) = (? y. Elem (Opair x y) f)"
   174   apply (auto simp add: Domain_def Replacement)
   175   apply (rule_tac x="Snd x" in exI)
   176   apply (simp add: FstSnd)
   177   apply (rule_tac x="Opair x y" in exI)
   178   apply (simp add: isOpair Fst)
   179   done
   180 
   181 theorem Range: "Elem y (Range f) = (? x. Elem (Opair x y) f)"
   182   apply (auto simp add: Range_def Replacement)
   183   apply (rule_tac x="Fst x" in exI)
   184   apply (simp add: FstSnd)
   185   apply (rule_tac x="Opair x y" in exI)
   186   apply (simp add: isOpair Snd)
   187   done
   188 
   189 theorem union: "Elem x (union A B) = (Elem x A | Elem x B)"
   190   by (auto simp add: union_def Sum Upair)
   191 
   192 constdefs
   193   Field :: "ZF \<Rightarrow> ZF"
   194   "Field A == union (Domain A) (Range A)"
   195 
   196 constdefs
   197   "\<acute>"         :: "ZF \<Rightarrow> ZF => ZF"    (infixl 90) --{*function application*} 
   198   app_def:  "f \<acute> x == (THE y. Elem (Opair x y) f)"
   199 
   200 constdefs
   201   isFun :: "ZF \<Rightarrow> bool"
   202   "isFun f == (! x y1 y2. Elem (Opair x y1) f & Elem (Opair x y2) f \<longrightarrow> y1 = y2)"
   203 
   204 constdefs
   205   Lambda :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF"
   206   "Lambda A f == Repl A (% x. Opair x (f x))"
   207 
   208 lemma Lambda_app: "Elem x A \<Longrightarrow> (Lambda A f)\<acute>x = f x"
   209   by (simp add: app_def Lambda_def Repl Opair)
   210 
   211 lemma isFun_Lambda: "isFun (Lambda A f)"
   212   by (auto simp add: isFun_def Lambda_def Repl Opair)
   213 
   214 lemma domain_Lambda: "Domain (Lambda A f) = A"
   215   apply (auto simp add: Domain_def)
   216   apply (subst Ext)
   217   apply (auto simp add: Replacement)
   218   apply (simp add: Lambda_def Repl)
   219   apply (auto simp add: Fst)
   220   apply (simp add: Lambda_def Repl)
   221   apply (rule_tac x="Opair z (f z)" in exI)
   222   apply (auto simp add: Fst isOpair_def)
   223   done
   224 
   225 lemma Lambda_ext: "(Lambda s f = Lambda t g) = (s = t & (! x. Elem x s \<longrightarrow> f x = g x))"
   226 proof -
   227   have "Lambda s f = Lambda t g \<Longrightarrow> s = t"
   228     apply (subst domain_Lambda[where A = s and f = f, symmetric])
   229     apply (subst domain_Lambda[where A = t and f = g, symmetric])
   230     apply auto
   231     done
   232   then show ?thesis
   233     apply auto
   234     apply (subst Lambda_app[where f=f, symmetric], simp)
   235     apply (subst Lambda_app[where f=g, symmetric], simp)
   236     apply auto
   237     apply (auto simp add: Lambda_def Repl Ext)
   238     apply (auto simp add: Ext[symmetric])
   239     done
   240 qed
   241 
   242 constdefs 
   243   PFun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
   244   "PFun A B == Sep (Power (CartProd A B)) isFun"
   245   Fun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"
   246   "Fun A B == Sep (PFun A B) (\<lambda> f. Domain f = A)"
   247 
   248 lemma Fun_Range: "Elem f (Fun U V) \<Longrightarrow> subset (Range f) V"
   249   apply (simp add: Fun_def Sep PFun_def Power subset_def CartProd)
   250   apply (auto simp add: Domain Range)
   251   apply (erule_tac x="Opair xa x" in allE)
   252   apply (auto simp add: Opair)
   253   done
   254 
   255 lemma Elem_Elem_PFun: "Elem F (PFun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"
   256   apply (simp add: PFun_def Sep Power subset_def, clarify)
   257   apply (erule_tac x=p in allE)
   258   apply (auto simp add: CartProd isOpair Fst Snd)
   259   done
   260 
   261 lemma Fun_implies_PFun[simp]: "Elem f (Fun U V) \<Longrightarrow> Elem f (PFun U V)"
   262   by (simp add: Fun_def Sep)
   263 
   264 lemma Elem_Elem_Fun: "Elem F (Fun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V" 
   265   by (auto simp add: Elem_Elem_PFun dest: Fun_implies_PFun)
   266 
   267 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"
   268   apply (frule Elem_Elem_PFun[where p=x], simp)
   269   apply (frule Elem_Elem_PFun[where p=y], simp)
   270   apply (subgoal_tac "isFun F")
   271   apply (simp add: isFun_def isOpair_def)  
   272   apply (auto simp add: Fst Snd, blast)
   273   apply (auto simp add: PFun_def Sep)
   274   done
   275 
   276 lemma Fun_total: "\<lbrakk>Elem F (Fun U V); Elem a U\<rbrakk> \<Longrightarrow> \<exists>x. Elem (Opair a x) F"
   277   using [[simp_depth_limit = 2]]
   278   by (auto simp add: Fun_def Sep Domain)
   279 
   280 
   281 lemma unique_fun_value: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> ?! y. Elem (Opair x y) f"
   282   by (auto simp add: Domain isFun_def)
   283 
   284 lemma fun_value_in_range: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> Elem (f\<acute>x) (Range f)"
   285   apply (auto simp add: Range)
   286   apply (drule unique_fun_value)
   287   apply simp
   288   apply (simp add: app_def)
   289   apply (rule exI[where x=x])
   290   apply (auto simp add: the_equality)
   291   done
   292 
   293 lemma fun_range_witness: "\<lbrakk>isFun f; Elem y (Range f)\<rbrakk> \<Longrightarrow> ? x. Elem x (Domain f) & f\<acute>x = y"
   294   apply (auto simp add: Range)
   295   apply (rule_tac x="x" in exI)
   296   apply (auto simp add: app_def the_equality isFun_def Domain)
   297   done
   298 
   299 lemma Elem_Fun_Lambda: "Elem F (Fun U V) \<Longrightarrow> ? f. F = Lambda U f"
   300   apply (rule exI[where x= "% x. (THE y. Elem (Opair x y) F)"])
   301   apply (simp add: Ext Lambda_def Repl Domain)
   302   apply (simp add: Ext[symmetric])
   303   apply auto
   304   apply (frule Elem_Elem_Fun)
   305   apply auto
   306   apply (rule_tac x="Fst z" in exI)
   307   apply (simp add: isOpair_def)
   308   apply (auto simp add: Fst Snd Opair)
   309   apply (rule theI2')
   310   apply auto
   311   apply (drule Fun_implies_PFun)
   312   apply (drule_tac x="Opair x ya" and y="Opair x yb" in PFun_inj)
   313   apply (auto simp add: Fst Snd)
   314   apply (drule Fun_implies_PFun)
   315   apply (drule_tac x="Opair x y" and y="Opair x ya" in PFun_inj)
   316   apply (auto simp add: Fst Snd)
   317   apply (rule theI2')
   318   apply (auto simp add: Fun_total)
   319   apply (drule Fun_implies_PFun)
   320   apply (drule_tac x="Opair a x" and y="Opair a y" in PFun_inj)
   321   apply (auto simp add: Fst Snd)
   322   done
   323  
   324 lemma Elem_Lambda_Fun: "Elem (Lambda A f) (Fun U V) = (A = U & (! x. Elem x A \<longrightarrow> Elem (f x) V))"
   325 proof -
   326   have "Elem (Lambda A f) (Fun U V) \<Longrightarrow> A = U"
   327     by (simp add: Fun_def Sep domain_Lambda)
   328   then show ?thesis
   329     apply auto
   330     apply (drule Fun_Range)
   331     apply (subgoal_tac "f x = ((Lambda U f) \<acute> x)")
   332     prefer 2
   333     apply (simp add: Lambda_app)
   334     apply simp
   335     apply (subgoal_tac "Elem (Lambda U f \<acute> x) (Range (Lambda U f))")
   336     apply (simp add: subset_def)
   337     apply (rule fun_value_in_range)
   338     apply (simp_all add: isFun_Lambda domain_Lambda)
   339     apply (simp add: Fun_def Sep PFun_def Power domain_Lambda isFun_Lambda)
   340     apply (auto simp add: subset_def CartProd)
   341     apply (rule_tac x="Fst x" in exI)
   342     apply (auto simp add: Lambda_def Repl Fst)
   343     done
   344 qed    
   345 
   346 
   347 constdefs
   348   is_Elem_of :: "(ZF * ZF) set"
   349   "is_Elem_of == { (a,b) | a b. Elem a b }"
   350 
   351 lemma cond_wf_Elem:
   352   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"
   353   shows "P a"
   354 proof -
   355   {
   356     fix P
   357     fix U
   358     fix a
   359     assume P_induct: "(\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> (Elem x U \<longrightarrow> P x))"
   360     assume a_in_U: "Elem a U"
   361     have "P a"
   362       proof -
   363 	term "P"
   364 	term Sep
   365 	let ?Z = "Sep U (Not o P)"
   366 	have "?Z = Empty \<longrightarrow> P a" by (simp add: Ext Sep Empty a_in_U)	
   367 	moreover have "?Z \<noteq> Empty \<longrightarrow> False"
   368 	  proof 
   369 	    assume not_empty: "?Z \<noteq> Empty" 
   370 	    note thereis_x = Regularity[where A="?Z", simplified not_empty, simplified]
   371 	    then obtain x where x_def: "Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
   372             then have x_induct:"! y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y" by (simp add: Sep)
   373 	    have "Elem x U \<longrightarrow> P x" 
   374 	      by (rule impE[OF spec[OF P_induct, where x=x], OF x_induct], assumption)
   375 	    moreover have "Elem x U & Not(P x)"
   376 	      apply (insert x_def)
   377 	      apply (simp add: Sep)
   378 	      done
   379 	    ultimately show "False" by auto
   380 	  qed
   381 	ultimately show "P a" by auto
   382       qed
   383   }
   384   with hyps show ?thesis by blast
   385 qed    
   386 
   387 lemma cond2_wf_Elem:
   388   assumes 
   389      special_P: "? U. ! x. Not(Elem x U) \<longrightarrow> (P x)"
   390      and P_induct: "\<forall>x. (\<forall>y. Elem y x \<longrightarrow> P y) \<longrightarrow> P x"
   391   shows
   392      "P a"
   393 proof -
   394   have "? U Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))"
   395   proof -
   396     from special_P obtain U where U:"! x. Not(Elem x U) \<longrightarrow> (P x)" ..
   397     show ?thesis
   398       apply (rule_tac exI[where x=U])
   399       apply (rule exI[where x="P"])
   400       apply (rule ext)
   401       apply (auto simp add: U)
   402       done
   403   qed    
   404   then obtain U where "? Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
   405   then obtain Q where UQ: "P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..
   406   show ?thesis
   407     apply (auto simp add: UQ)
   408     apply (rule cond_wf_Elem)
   409     apply (rule P_induct[simplified UQ])
   410     apply simp
   411     done
   412 qed
   413 
   414 consts
   415   nat2Nat :: "nat \<Rightarrow> ZF"
   416 
   417 primrec
   418 nat2Nat_0[intro]:  "nat2Nat 0 = Empty"
   419 nat2Nat_Suc[intro]:  "nat2Nat (Suc n) = SucNat (nat2Nat n)"
   420 
   421 constdefs
   422   Nat2nat :: "ZF \<Rightarrow> nat"
   423   "Nat2nat == inv nat2Nat"
   424 
   425 lemma Elem_nat2Nat_inf[intro]: "Elem (nat2Nat n) Inf"
   426   apply (induct n)
   427   apply (simp_all add: Infinity)
   428   done
   429 
   430 constdefs
   431   Nat :: ZF
   432   "Nat == Sep Inf (\<lambda> N. ? n. nat2Nat n = N)"
   433 
   434 lemma Elem_nat2Nat_Nat[intro]: "Elem (nat2Nat n) Nat"
   435   by (auto simp add: Nat_def Sep)
   436 
   437 lemma Elem_Empty_Nat: "Elem Empty Nat"
   438   by (auto simp add: Nat_def Sep Infinity)
   439 
   440 lemma Elem_SucNat_Nat: "Elem N Nat \<Longrightarrow> Elem (SucNat N) Nat"
   441   by (auto simp add: Nat_def Sep Infinity)
   442   
   443 lemma no_infinite_Elem_down_chain:
   444   "Not (? f. isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N)))"
   445 proof -
   446   {
   447     fix f
   448     assume f:"isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N))"
   449     let ?r = "Range f"
   450     have "?r \<noteq> Empty"
   451       apply (auto simp add: Ext Empty)
   452       apply (rule exI[where x="f\<acute>Empty"])
   453       apply (rule fun_value_in_range)
   454       apply (auto simp add: f Elem_Empty_Nat)
   455       done
   456     then have "? x. Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))"
   457       by (simp add: Regularity)
   458     then obtain x where x: "Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))" ..
   459     then have "? N. Elem N (Domain f) & f\<acute>N = x" 
   460       apply (rule_tac fun_range_witness)
   461       apply (simp_all add: f)
   462       done
   463     then have "? N. Elem N Nat & f\<acute>N = x" 
   464       by (simp add: f)
   465     then obtain N where N: "Elem N Nat & f\<acute>N = x" ..
   466     from N have N': "Elem N Nat" by auto
   467     let ?y = "f\<acute>(SucNat N)"
   468     have Elem_y_r: "Elem ?y ?r"
   469       by (simp_all add: f Elem_SucNat_Nat N fun_value_in_range)
   470     have "Elem ?y (f\<acute>N)" by (auto simp add: f N')
   471     then have "Elem ?y x" by (simp add: N)
   472     with x have "Not (Elem ?y ?r)" by auto
   473     with Elem_y_r have "False" by auto
   474   }
   475   then show ?thesis by auto
   476 qed
   477 
   478 lemma Upair_nonEmpty: "Upair a b \<noteq> Empty"
   479   by (auto simp add: Ext Empty Upair)  
   480 
   481 lemma Singleton_nonEmpty: "Singleton x \<noteq> Empty"
   482   by (auto simp add: Singleton_def Upair_nonEmpty)
   483 
   484 lemma antisym_Elem: "Not(Elem a b & Elem b a)"
   485 proof -
   486   {
   487     fix a b
   488     assume ab: "Elem a b"
   489     assume ba: "Elem b a"
   490     let ?Z = "Upair a b"
   491     have "?Z \<noteq> Empty" by (simp add: Upair_nonEmpty)
   492     then have "? x. Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))"
   493       by (simp add: Regularity)
   494     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))" ..
   495     then have "x = a \<or> x = b" by (simp add: Upair)
   496     moreover have "x = a \<longrightarrow> Not (Elem b ?Z)"
   497       by (auto simp add: x ba)
   498     moreover have "x = b \<longrightarrow> Not (Elem a ?Z)"
   499       by (auto simp add: x ab)
   500     ultimately have "False"
   501       by (auto simp add: Upair)
   502   }    
   503   then show ?thesis by auto
   504 qed
   505 
   506 lemma irreflexiv_Elem: "Not(Elem a a)"
   507   by (simp add: antisym_Elem[of a a, simplified])
   508 
   509 lemma antisym_Elem: "Elem a b \<Longrightarrow> Not (Elem b a)"
   510   apply (insert antisym_Elem[of a b])
   511   apply auto
   512   done
   513 
   514 consts
   515   NatInterval :: "nat \<Rightarrow> nat \<Rightarrow> ZF"
   516 
   517 primrec
   518   "NatInterval n 0 = Singleton (nat2Nat n)"
   519   "NatInterval n (Suc m) = union (NatInterval n m) (Singleton (nat2Nat (n+m+1)))"
   520 
   521 lemma n_Elem_NatInterval[rule_format]: "! q. q <= m \<longrightarrow> Elem (nat2Nat (n+q)) (NatInterval n m)"
   522   apply (induct m)
   523   apply (auto simp add: Singleton union)
   524   apply (case_tac "q <= m")
   525   apply auto
   526   apply (subgoal_tac "q = Suc m")
   527   apply auto
   528   done
   529 
   530 lemma NatInterval_not_Empty: "NatInterval n m \<noteq> Empty"
   531   by (auto intro:   n_Elem_NatInterval[where q = 0, simplified] simp add: Empty Ext)
   532 
   533 lemma increasing_nat2Nat[rule_format]: "0 < n \<longrightarrow> Elem (nat2Nat (n - 1)) (nat2Nat n)"
   534   apply (case_tac "? m. n = Suc m")
   535   apply (auto simp add: SucNat_def union Singleton)
   536   apply (drule spec[where x="n - 1"])
   537   apply arith
   538   done
   539 
   540 lemma represent_NatInterval[rule_format]: "Elem x (NatInterval n m) \<longrightarrow> (? u. n \<le> u & u \<le> n+m & nat2Nat u = x)"
   541   apply (induct m)
   542   apply (auto simp add: Singleton union)
   543   apply (rule_tac x="Suc (n+m)" in exI)
   544   apply auto
   545   done
   546 
   547 lemma inj_nat2Nat: "inj nat2Nat"
   548 proof -
   549   {
   550     fix n m :: nat
   551     assume nm: "nat2Nat n = nat2Nat (n+m)"
   552     assume mg0: "0 < m"
   553     let ?Z = "NatInterval n m"
   554     have "?Z \<noteq> Empty" by (simp add: NatInterval_not_Empty)
   555     then have "? x. (Elem x ?Z) & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" 
   556       by (auto simp add: Regularity)
   557     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..
   558     then have "? u. n \<le> u & u \<le> n+m & nat2Nat u = x" 
   559       by (simp add: represent_NatInterval)
   560     then obtain u where u: "n \<le> u & u \<le> n+m & nat2Nat u = x" ..
   561     have "n < u \<longrightarrow> False"
   562     proof 
   563       assume n_less_u: "n < u"
   564       let ?y = "nat2Nat (u - 1)"
   565       have "Elem ?y (nat2Nat u)"
   566 	apply (rule increasing_nat2Nat)
   567 	apply (insert n_less_u)
   568 	apply arith
   569 	done
   570       with u have "Elem ?y x" by auto
   571       with x have "Not (Elem ?y ?Z)" by auto
   572       moreover have "Elem ?y ?Z" 
   573 	apply (insert n_Elem_NatInterval[where q = "u - n - 1" and n=n and m=m])
   574 	apply (insert n_less_u)
   575 	apply (insert u)
   576 	apply auto
   577 	done
   578       ultimately show False by auto
   579     qed
   580     moreover have "u = n \<longrightarrow> False"
   581     proof 
   582       assume "u = n"
   583       with u have "nat2Nat n = x" by auto
   584       then have nm_eq_x: "nat2Nat (n+m) = x" by (simp add: nm)
   585       let ?y = "nat2Nat (n+m - 1)"
   586       have "Elem ?y (nat2Nat (n+m))"
   587 	apply (rule increasing_nat2Nat)
   588 	apply (insert mg0)
   589 	apply arith
   590 	done
   591       with nm_eq_x have "Elem ?y x" by auto
   592       with x have "Not (Elem ?y ?Z)" by auto
   593       moreover have "Elem ?y ?Z" 
   594 	apply (insert n_Elem_NatInterval[where q = "m - 1" and n=n and m=m])
   595 	apply (insert mg0)
   596 	apply auto
   597 	done
   598       ultimately show False by auto      
   599     qed
   600     ultimately have "False" using u by arith
   601   }
   602   note lemma_nat2Nat = this
   603   have th:"\<And>x y. \<not> (x < y \<and> (\<forall>(m\<Colon>nat). y \<noteq> x + m))" by presburger
   604   have th': "\<And>x y. \<not> (x \<noteq> y \<and> (\<not> x < y) \<and> (\<forall>(m\<Colon>nat). x \<noteq> y + m))" by presburger
   605   show ?thesis
   606     apply (auto simp add: inj_on_def)
   607     apply (case_tac "x = y")
   608     apply auto
   609     apply (case_tac "x < y")
   610     apply (case_tac "? m. y = x + m & 0 < m")
   611     apply (auto intro: lemma_nat2Nat)
   612     apply (case_tac "y < x")
   613     apply (case_tac "? m. x = y + m & 0 < m")
   614     apply simp
   615     apply simp
   616     using th apply blast
   617     apply (case_tac "? m. x = y + m")
   618     apply (auto intro: lemma_nat2Nat)
   619     apply (drule sym)
   620     using lemma_nat2Nat apply blast
   621     using th' apply blast    
   622     done
   623 qed
   624 
   625 lemma Nat2nat_nat2Nat[simp]: "Nat2nat (nat2Nat n) = n"
   626   by (simp add: Nat2nat_def inv_f_f[OF inj_nat2Nat])
   627 
   628 lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n"
   629   apply (simp add: Nat2nat_def)
   630   apply (rule_tac f_inv_f)
   631   apply (auto simp add: image_def Nat_def Sep)
   632   done
   633 
   634 lemma Nat2nat_SucNat: "Elem N Nat \<Longrightarrow> Nat2nat (SucNat N) = Suc (Nat2nat N)"
   635   apply (auto simp add: Nat_def Sep Nat2nat_def)
   636   apply (auto simp add: inv_f_f[OF inj_nat2Nat])
   637   apply (simp only: nat2Nat.simps[symmetric])
   638   apply (simp only: inv_f_f[OF inj_nat2Nat])
   639   done
   640   
   641 
   642 (*lemma Elem_induct: "(\<And>x. \<forall>y. Elem y x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
   643   by (erule wf_induct[OF wf_is_Elem_of, simplified is_Elem_of_def, simplified])*)
   644 
   645 lemma Elem_Opair_exists: "? z. Elem x z & Elem y z & Elem z (Opair x y)"
   646   apply (rule exI[where x="Upair x y"])
   647   by (simp add: Upair Opair_def)
   648 
   649 lemma UNIV_is_not_in_ZF: "UNIV \<noteq> explode R"
   650 proof
   651   let ?Russell = "{ x. Not(Elem x x) }"
   652   have "?Russell = UNIV" by (simp add: irreflexiv_Elem)
   653   moreover assume "UNIV = explode R"
   654   ultimately have russell: "?Russell = explode R" by simp
   655   then show "False"
   656   proof(cases "Elem R R")
   657     case True     
   658     then show ?thesis 
   659       by (insert irreflexiv_Elem, auto)
   660   next
   661     case False
   662     then have "R \<in> ?Russell" by auto
   663     then have "Elem R R" by (simp add: russell explode_def)
   664     with False show ?thesis by auto
   665   qed
   666 qed
   667 
   668 constdefs 
   669   SpecialR :: "(ZF * ZF) set"
   670   "SpecialR \<equiv> { (x, y) . x \<noteq> Empty \<and> y = Empty}"
   671 
   672 lemma "wf SpecialR"
   673   apply (subst wf_def)
   674   apply (auto simp add: SpecialR_def)
   675   done
   676 
   677 constdefs
   678   Ext :: "('a * 'b) set \<Rightarrow> 'b \<Rightarrow> 'a set"
   679   "Ext R y \<equiv> { x . (x, y) \<in> R }" 
   680 
   681 lemma Ext_Elem: "Ext is_Elem_of = explode"
   682   by (auto intro: ext simp add: Ext_def is_Elem_of_def explode_def)
   683 
   684 lemma "Ext SpecialR Empty \<noteq> explode z"
   685 proof 
   686   have "Ext SpecialR Empty = UNIV - {Empty}"
   687     by (auto simp add: Ext_def SpecialR_def)
   688   moreover assume "Ext SpecialR Empty = explode z"
   689   ultimately have "UNIV = explode(union z (Singleton Empty)) "
   690     by (auto simp add: explode_def union Singleton)
   691   then show "False" by (simp add: UNIV_is_not_in_ZF)
   692 qed
   693 
   694 constdefs 
   695   implode :: "ZF set \<Rightarrow> ZF"
   696   "implode == inv explode"
   697 
   698 lemma inj_explode: "inj explode"
   699   by (auto simp add: inj_on_def explode_def Ext)
   700 
   701 lemma implode_explode[simp]: "implode (explode x) = x"
   702   by (simp add: implode_def inj_explode)
   703 
   704 constdefs
   705   regular :: "(ZF * ZF) set \<Rightarrow> bool"
   706   "regular R == ! A. A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. (y, x) \<in> R \<longrightarrow> Not (Elem y A)))"
   707   set_like :: "(ZF * ZF) set \<Rightarrow> bool"
   708   "set_like R == ! y. Ext R y \<in> range explode"
   709   wfzf :: "(ZF * ZF) set \<Rightarrow> bool"
   710   "wfzf R == regular R & set_like R"
   711 
   712 lemma regular_Elem: "regular is_Elem_of"
   713   by (simp add: regular_def is_Elem_of_def Regularity)
   714 
   715 lemma set_like_Elem: "set_like is_Elem_of"
   716   by (auto simp add: set_like_def image_def Ext_Elem)
   717 
   718 lemma wfzf_is_Elem_of: "wfzf is_Elem_of"
   719   by (auto simp add: wfzf_def regular_Elem set_like_Elem)
   720 
   721 constdefs
   722   SeqSum :: "(nat \<Rightarrow> ZF) \<Rightarrow> ZF"
   723   "SeqSum f == Sum (Repl Nat (f o Nat2nat))"
   724 
   725 lemma SeqSum: "Elem x (SeqSum f) = (? n. Elem x (f n))"
   726   apply (auto simp add: SeqSum_def Sum Repl)
   727   apply (rule_tac x = "f n" in exI)
   728   apply auto
   729   done
   730 
   731 constdefs
   732   Ext_ZF :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"
   733   "Ext_ZF R s == implode (Ext R s)"
   734 
   735 lemma Elem_implode: "A \<in> range explode \<Longrightarrow> Elem x (implode A) = (x \<in> A)"
   736   apply (auto)
   737   apply (simp_all add: explode_def)
   738   done
   739 
   740 lemma Elem_Ext_ZF: "set_like R \<Longrightarrow> Elem x (Ext_ZF R s) = ((x,s) \<in> R)"
   741   apply (simp add: Ext_ZF_def)
   742   apply (subst Elem_implode)
   743   apply (simp add: set_like_def)
   744   apply (simp add: Ext_def)
   745   done
   746 
   747 consts
   748   Ext_ZF_n :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> nat \<Rightarrow> ZF"
   749 
   750 primrec
   751   "Ext_ZF_n R s 0 = Ext_ZF R s"
   752   "Ext_ZF_n R s (Suc n) = Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"
   753 
   754 constdefs
   755   Ext_ZF_hull :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"
   756   "Ext_ZF_hull R s == SeqSum (Ext_ZF_n R s)"
   757 
   758 lemma Elem_Ext_ZF_hull:
   759   assumes set_like_R: "set_like R" 
   760   shows "Elem x (Ext_ZF_hull R S) = (? n. Elem x (Ext_ZF_n R S n))"
   761   by (simp add: Ext_ZF_hull_def SeqSum)
   762   
   763 lemma Elem_Elem_Ext_ZF_hull:
   764   assumes set_like_R: "set_like R" 
   765           and x_hull: "Elem x (Ext_ZF_hull R S)"
   766           and y_R_x: "(y, x) \<in> R"
   767   shows "Elem y (Ext_ZF_hull R S)"
   768 proof -
   769   from Elem_Ext_ZF_hull[OF set_like_R] x_hull 
   770   have "? n. Elem x (Ext_ZF_n R S n)" by auto
   771   then obtain n where n:"Elem x (Ext_ZF_n R S n)" ..
   772   with y_R_x have "Elem y (Ext_ZF_n R S (Suc n))"
   773     apply (auto simp add: Repl Sum)
   774     apply (rule_tac x="Ext_ZF R x" in exI) 
   775     apply (auto simp add: Elem_Ext_ZF[OF set_like_R])
   776     done
   777   with Elem_Ext_ZF_hull[OF set_like_R, where x=y] show ?thesis
   778     by (auto simp del: Ext_ZF_n.simps)
   779 qed
   780 
   781 lemma wfzf_minimal:
   782   assumes hyps: "wfzf R" "C \<noteq> {}"
   783   shows "\<exists>x. x \<in> C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> C)"
   784 proof -
   785   from hyps have "\<exists>S. S \<in> C" by auto
   786   then obtain S where S:"S \<in> C" by auto  
   787   let ?T = "Sep (Ext_ZF_hull R S) (\<lambda> s. s \<in> C)"
   788   from hyps have set_like_R: "set_like R" by (simp add: wfzf_def)
   789   show ?thesis
   790   proof (cases "?T = Empty")
   791     case True
   792     then have "\<forall> z. \<not> (Elem z (Sep (Ext_ZF R S) (\<lambda> s. s \<in> C)))"      
   793       apply (auto simp add: Ext Empty Sep Ext_ZF_hull_def SeqSum)
   794       apply (erule_tac x="z" in allE, auto)
   795       apply (erule_tac x=0 in allE, auto)
   796       done
   797     then show ?thesis 
   798       apply (rule_tac exI[where x=S])
   799       apply (auto simp add: Sep Empty S)
   800       apply (erule_tac x=y in allE)
   801       apply (simp add: set_like_R Elem_Ext_ZF)
   802       done
   803   next
   804     case False
   805     from hyps have regular_R: "regular R" by (simp add: wfzf_def)
   806     from 
   807       regular_R[simplified regular_def, rule_format, OF False, simplified Sep] 
   808       Elem_Elem_Ext_ZF_hull[OF set_like_R]
   809     show ?thesis by blast
   810   qed
   811 qed
   812 
   813 lemma wfzf_implies_wf: "wfzf R \<Longrightarrow> wf R"
   814 proof (subst wf_def, rule allI)
   815   assume wfzf: "wfzf R"
   816   fix P :: "ZF \<Rightarrow> bool"
   817   let ?C = "{x. P x}"
   818   {
   819     assume induct: "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x)"
   820     let ?C = "{x. \<not> (P x)}"
   821     have "?C = {}"
   822     proof (rule ccontr)
   823       assume C: "?C \<noteq> {}"
   824       from
   825 	wfzf_minimal[OF wfzf C]
   826       obtain x where x: "x \<in> ?C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> ?C)" ..
   827       then have "P x"
   828 	apply (rule_tac induct[rule_format])
   829 	apply auto
   830 	done
   831       with x show "False" by auto
   832     qed
   833     then have "! x. P x" by auto
   834   }
   835   then show "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (! x. P x)" by blast
   836 qed
   837 
   838 lemma wf_is_Elem_of: "wf is_Elem_of"
   839   by (auto simp add: wfzf_is_Elem_of wfzf_implies_wf)
   840 
   841 lemma in_Ext_RTrans_implies_Elem_Ext_ZF_hull:  
   842   "set_like R \<Longrightarrow> x \<in> (Ext (R^+) s) \<Longrightarrow> Elem x (Ext_ZF_hull R s)"
   843   apply (simp add: Ext_def Elem_Ext_ZF_hull)
   844   apply (erule converse_trancl_induct[where r="R"])
   845   apply (rule exI[where x=0])
   846   apply (simp add: Elem_Ext_ZF)
   847   apply auto
   848   apply (rule_tac x="Suc n" in exI)
   849   apply (simp add: Sum Repl)
   850   apply (rule_tac x="Ext_ZF R z" in exI)
   851   apply (auto simp add: Elem_Ext_ZF)
   852   done
   853 
   854 lemma implodeable_Ext_trancl: "set_like R \<Longrightarrow> set_like (R^+)"
   855   apply (subst set_like_def)
   856   apply (auto simp add: image_def)
   857   apply (rule_tac x="Sep (Ext_ZF_hull R y) (\<lambda> z. z \<in> (Ext (R^+) y))" in exI)
   858   apply (auto simp add: explode_def Sep set_ext 
   859     in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
   860   done
   861  
   862 lemma Elem_Ext_ZF_hull_implies_in_Ext_RTrans[rule_format]:
   863   "set_like R \<Longrightarrow> ! x. Elem x (Ext_ZF_n R s n) \<longrightarrow> x \<in> (Ext (R^+) s)"
   864   apply (induct_tac n)
   865   apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)
   866   done
   867 
   868 lemma "set_like R \<Longrightarrow> Ext_ZF (R^+) s = Ext_ZF_hull R s"
   869   apply (frule implodeable_Ext_trancl)
   870   apply (auto simp add: Ext)
   871   apply (erule in_Ext_RTrans_implies_Elem_Ext_ZF_hull)
   872   apply (simp add: Elem_Ext_ZF Ext_def)
   873   apply (auto simp add: Elem_Ext_ZF Elem_Ext_ZF_hull)
   874   apply (erule Elem_Ext_ZF_hull_implies_in_Ext_RTrans[simplified Ext_def, simplified], assumption)
   875   done
   876 
   877 lemma wf_implies_regular: "wf R \<Longrightarrow> regular R"
   878 proof (simp add: regular_def, rule allI)
   879   assume wf: "wf R"
   880   fix A
   881   show "A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A))"
   882   proof
   883     assume A: "A \<noteq> Empty"
   884     then have "? x. x \<in> explode A" 
   885       by (auto simp add: explode_def Ext Empty)
   886     then obtain x where x:"x \<in> explode A" ..   
   887     from iffD1[OF wf_eq_minimal wf, rule_format, where Q="explode A", OF x]
   888     obtain z where "z \<in> explode A \<and> (\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> explode A)" by auto    
   889     then show "\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A)"      
   890       apply (rule_tac exI[where x = z])
   891       apply (simp add: explode_def)
   892       done
   893   qed
   894 qed
   895 
   896 lemma wf_eq_wfzf: "(wf R \<and> set_like R) = wfzf R"
   897   apply (auto simp add: wfzf_implies_wf)
   898   apply (auto simp add: wfzf_def wf_implies_regular)
   899   done
   900 
   901 lemma wfzf_trancl: "wfzf R \<Longrightarrow> wfzf (R^+)"
   902   by (auto simp add: wf_eq_wfzf[symmetric] implodeable_Ext_trancl wf_trancl)
   903 
   904 lemma Ext_subset_mono: "R \<subseteq> S \<Longrightarrow> Ext R y \<subseteq> Ext S y"
   905   by (auto simp add: Ext_def)
   906 
   907 lemma set_like_subset: "set_like R \<Longrightarrow> S \<subseteq> R \<Longrightarrow> set_like S"
   908   apply (auto simp add: set_like_def)
   909   apply (erule_tac x=y in allE)
   910   apply (drule_tac y=y in Ext_subset_mono)
   911   apply (auto simp add: image_def)
   912   apply (rule_tac x="Sep x (% z. z \<in> (Ext S y))" in exI) 
   913   apply (auto simp add: explode_def Sep)
   914   done
   915 
   916 lemma wfzf_subset: "wfzf S \<Longrightarrow> R \<subseteq> S \<Longrightarrow> wfzf R"
   917   by (auto intro: set_like_subset wf_subset simp add: wf_eq_wfzf[symmetric])  
   918 
   919 end