src/HOL/ZF/HOLZF.thy
 author haftmann Sun Sep 21 16:56:11 2014 +0200 (2014-09-21) changeset 58410 6d46ad54a2ab parent 57492 74bf65a1910a child 61076 bdc1e2f0a86a permissions -rw-r--r--
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
```     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 xa" 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)
```
```   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
```