wneuper/isa: src/HOL/ZF/HOLZF.thy@c28279b29ff1

     1 (*  Title:      HOL/ZF/HOLZF.thy

     2     Author:     Steven Obua

     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 *)

     8 theory HOLZF

     9 imports Helper

    10 begin

    12 typedecl ZF

    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

    22 constdefs

    23   Upair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

    24   "Upair a b == Repl (Power (Power Empty)) (% x. if x = Empty then a else b)"

    25   Singleton:: "ZF \<Rightarrow> ZF"

    26   "Singleton x == Upair x x"

    27   union :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

    28   "union A B == Sum (Upair A B)"

    29   SucNat:: "ZF \<Rightarrow> ZF"

    30   "SucNat x == union x (Singleton x)"

    31   subset :: "ZF \<Rightarrow> ZF \<Rightarrow> bool"

    32   "subset A B == ! x. Elem x A \<longrightarrow> Elem x B"

    34 axioms

    35   Empty: "Not (Elem x Empty)"

    36   Ext: "(x = y) = (! z. Elem z x = Elem z y)"

    37   Sum: "Elem z (Sum x) = (? y. Elem z y & Elem y x)"

    38   Power: "Elem y (Power x) = (subset y x)"

    39   Repl: "Elem b (Repl A f) = (? a. Elem a A & b = f a)"

    40   Regularity: "A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. Elem y x \<longrightarrow> Not (Elem y A)))"

    41   Infinity: "Elem Empty Inf & (! x. Elem x Inf \<longrightarrow> Elem (SucNat x) Inf)"

    43 constdefs

    44   Sep:: "ZF \<Rightarrow> (ZF \<Rightarrow> bool) \<Rightarrow> ZF"

    45   "Sep A p == (if (!x. Elem x A \<longrightarrow> Not (p x)) then Empty else

    46   (let z = (\<some> x. Elem x A & p x) in

    47    let f = % x. (if p x then x else z) in Repl A f))"

    49 thm Power[unfolded subset_def]

    51 theorem Sep: "Elem b (Sep A p) = (Elem b A & p b)"

    52   apply (auto simp add: Sep_def Empty)

    53   apply (auto simp add: Let_def Repl)

    54   apply (rule someI2, auto)+

    55   done

    57 lemma subset_empty: "subset Empty A"

    58   by (simp add: subset_def Empty)

    60 theorem Upair: "Elem x (Upair a b) = (x = a | x = b)"

    61   apply (auto simp add: Upair_def Repl)

    62   apply (rule exI[where x=Empty])

    63   apply (simp add: Power subset_empty)

    64   apply (rule exI[where x="Power Empty"])

    65   apply (auto)

    66   apply (auto simp add: Ext Power subset_def Empty)

    67   apply (drule spec[where x=Empty], simp add: Empty)+

    68   done

    70 lemma Singleton: "Elem x (Singleton y) = (x = y)"

    71   by (simp add: Singleton_def Upair)

    73 constdefs

    74   Opair:: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

    75   "Opair a b == Upair (Upair a a) (Upair a b)"

    77 lemma Upair_singleton: "(Upair a a = Upair c d) = (a = c & a = d)"

    78   by (auto simp add: Ext[where x="Upair a a"] Upair)

    80 lemma Upair_fsteq: "(Upair a b = Upair a c) = ((a = b & a = c) | (b = c))"

    81   by (auto simp add: Ext[where x="Upair a b"] Upair)

    83 lemma Upair_comm: "Upair a b = Upair b a"

    84   by (auto simp add: Ext Upair)

    86 theorem Opair: "(Opair a b = Opair c d) = (a = c & b = d)"

    87   proof -

    88     have fst: "(Opair a b = Opair c d) \<Longrightarrow> a = c"

    89       apply (simp add: Opair_def)

    90       apply (simp add: Ext[where x="Upair (Upair a a) (Upair a b)"])

    91       apply (drule spec[where x="Upair a a"])

    92       apply (auto simp add: Upair Upair_singleton)

    93       done

    94     show ?thesis

    95       apply (auto)

    96       apply (erule fst)

    97       apply (frule fst)

    98       apply (auto simp add: Opair_def Upair_fsteq)

    99       done

   100   qed

   102 constdefs

   103   Replacement :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF option) \<Rightarrow> ZF"

   104   "Replacement A f == Repl (Sep A (% a. f a \<noteq> None)) (the o f)"

   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)

   109 constdefs

   110   Fst :: "ZF \<Rightarrow> ZF"

   111   "Fst q == SOME x. ? y. q = Opair x y"

   112   Snd :: "ZF \<Rightarrow> ZF"

   113   "Snd q == SOME y. ? x. q = Opair x y"

   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

   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

   127 constdefs

   128   isOpair :: "ZF \<Rightarrow> bool"

   129   "isOpair q == ? x y. q = Opair x y"

   131 lemma isOpair: "isOpair (Opair x y) = True"

   132   by (auto simp add: isOpair_def)

   134 lemma FstSnd: "isOpair x \<Longrightarrow> Opair (Fst x) (Snd x) = x"

   135   by (auto simp add: isOpair_def Fst Snd)

   137 constdefs

   138   CartProd :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

   139   "CartProd A B == Sum(Repl A (% a. Repl B (% b. Opair a b)))"

   141 lemma CartProd: "Elem x (CartProd A B) = (? a b. Elem a A & Elem b B & x = (Opair a b))"

   142   apply (auto simp add: CartProd_def Sum Repl)

   143   apply (rule_tac x="Repl B (Opair a)" in exI)

   144   apply (auto simp add: Repl)

   145   done

   147 constdefs

   148   explode :: "ZF \<Rightarrow> ZF set"

   149   "explode z == { x. Elem x z }"

   151 lemma explode_Empty: "(explode x = {}) = (x = Empty)"

   152   by (auto simp add: explode_def Ext Empty)

   154 lemma explode_Elem: "(x \<in> explode X) = (Elem x X)"

   155   by (simp add: explode_def)

   157 lemma Elem_explode_in: "\<lbrakk> Elem a A; explode A \<subseteq> B\<rbrakk> \<Longrightarrow> a \<in> B"

   158   by (auto simp add: explode_def)

   160 lemma explode_CartProd_eq: "explode (CartProd a b) = (% (x,y). Opair x y) ` ((explode a) \<times> (explode b))"

   161   by (simp add: explode_def expand_set_eq CartProd image_def)

   163 lemma explode_Repl_eq: "explode (Repl A f) = image f (explode A)"

   164   by (simp add: explode_def Repl image_def)

   166 constdefs

   167   Domain :: "ZF \<Rightarrow> ZF"

   168   "Domain f == Replacement f (% p. if isOpair p then Some (Fst p) else None)"

   169   Range :: "ZF \<Rightarrow> ZF"

   170   "Range f == Replacement f (% p. if isOpair p then Some (Snd p) else None)"

   172 theorem Domain: "Elem x (Domain f) = (? y. Elem (Opair x y) f)"

   173   apply (auto simp add: Domain_def Replacement)

   174   apply (rule_tac x="Snd x" in exI)

   175   apply (simp add: FstSnd)

   176   apply (rule_tac x="Opair x y" in exI)

   177   apply (simp add: isOpair Fst)

   178   done

   180 theorem Range: "Elem y (Range f) = (? x. Elem (Opair x y) f)"

   181   apply (auto simp add: Range_def Replacement)

   182   apply (rule_tac x="Fst x" in exI)

   183   apply (simp add: FstSnd)

   184   apply (rule_tac x="Opair x y" in exI)

   185   apply (simp add: isOpair Snd)

   186   done

   188 theorem union: "Elem x (union A B) = (Elem x A | Elem x B)"

   189   by (auto simp add: union_def Sum Upair)

   191 constdefs

   192   Field :: "ZF \<Rightarrow> ZF"

   193   "Field A == union (Domain A) (Range A)"

   195 constdefs

   196   app :: "ZF \<Rightarrow> ZF => ZF"    (infixl "\<acute>" 90) --{*function application*}

   197   "f \<acute> x == (THE y. Elem (Opair x y) f)"

   199 constdefs

   200   isFun :: "ZF \<Rightarrow> bool"

   201   "isFun f == (! x y1 y2. Elem (Opair x y1) f & Elem (Opair x y2) f \<longrightarrow> y1 = y2)"

   203 constdefs

   204   Lambda :: "ZF \<Rightarrow> (ZF \<Rightarrow> ZF) \<Rightarrow> ZF"

   205   "Lambda A f == Repl A (% x. Opair x (f x))"

   207 lemma Lambda_app: "Elem x A \<Longrightarrow> (Lambda A f)\<acute>x = f x"

   208   by (simp add: app_def Lambda_def Repl Opair)

   210 lemma isFun_Lambda: "isFun (Lambda A f)"

   211   by (auto simp add: isFun_def Lambda_def Repl Opair)

   213 lemma domain_Lambda: "Domain (Lambda A f) = A"

   214   apply (auto simp add: Domain_def)

   215   apply (subst Ext)

   216   apply (auto simp add: Replacement)

   217   apply (simp add: Lambda_def Repl)

   218   apply (auto simp add: Fst)

   219   apply (simp add: Lambda_def Repl)

   220   apply (rule_tac x="Opair z (f z)" in exI)

   221   apply (auto simp add: Fst isOpair_def)

   222   done

   224 lemma Lambda_ext: "(Lambda s f = Lambda t g) = (s = t & (! x. Elem x s \<longrightarrow> f x = g x))"

   225 proof -

   226   have "Lambda s f = Lambda t g \<Longrightarrow> s = t"

   227     apply (subst domain_Lambda[where A = s and f = f, symmetric])

   228     apply (subst domain_Lambda[where A = t and f = g, symmetric])

   229     apply auto

   230     done

   231   then show ?thesis

   232     apply auto

   233     apply (subst Lambda_app[where f=f, symmetric], simp)

   234     apply (subst Lambda_app[where f=g, symmetric], simp)

   235     apply auto

   236     apply (auto simp add: Lambda_def Repl Ext)

   237     apply (auto simp add: Ext[symmetric])

   238     done

   239 qed

   241 constdefs

   242   PFun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

   243   "PFun A B == Sep (Power (CartProd A B)) isFun"

   244   Fun :: "ZF \<Rightarrow> ZF \<Rightarrow> ZF"

   245   "Fun A B == Sep (PFun A B) (\<lambda> f. Domain f = A)"

   247 lemma Fun_Range: "Elem f (Fun U V) \<Longrightarrow> subset (Range f) V"

   248   apply (simp add: Fun_def Sep PFun_def Power subset_def CartProd)

   249   apply (auto simp add: Domain Range)

   250   apply (erule_tac x="Opair xa x" in allE)

   251   apply (auto simp add: Opair)

   252   done

   254 lemma Elem_Elem_PFun: "Elem F (PFun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"

   255   apply (simp add: PFun_def Sep Power subset_def, clarify)

   256   apply (erule_tac x=p in allE)

   257   apply (auto simp add: CartProd isOpair Fst Snd)

   258   done

   260 lemma Fun_implies_PFun[simp]: "Elem f (Fun U V) \<Longrightarrow> Elem f (PFun U V)"

   261   by (simp add: Fun_def Sep)

   263 lemma Elem_Elem_Fun: "Elem F (Fun U V) \<Longrightarrow> Elem p F \<Longrightarrow> isOpair p & Elem (Fst p) U & Elem (Snd p) V"

   264   by (auto simp add: Elem_Elem_PFun dest: Fun_implies_PFun)

   266 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"

   267   apply (frule Elem_Elem_PFun[where p=x], simp)

   268   apply (frule Elem_Elem_PFun[where p=y], simp)

   269   apply (subgoal_tac "isFun F")

   270   apply (simp add: isFun_def isOpair_def)

   271   apply (auto simp add: Fst Snd, blast)

   272   apply (auto simp add: PFun_def Sep)

   273   done

   275 lemma Fun_total: "\<lbrakk>Elem F (Fun U V); Elem a U\<rbrakk> \<Longrightarrow> \<exists>x. Elem (Opair a x) F"

   276   using [[simp_depth_limit = 2]]

   277   by (auto simp add: Fun_def Sep Domain)

   280 lemma unique_fun_value: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> ?! y. Elem (Opair x y) f"

   281   by (auto simp add: Domain isFun_def)

   283 lemma fun_value_in_range: "\<lbrakk>isFun f; Elem x (Domain f)\<rbrakk> \<Longrightarrow> Elem (f\<acute>x) (Range f)"

   284   apply (auto simp add: Range)

   285   apply (drule unique_fun_value)

   286   apply simp

   287   apply (simp add: app_def)

   288   apply (rule exI[where x=x])

   289   apply (auto simp add: the_equality)

   290   done

   292 lemma fun_range_witness: "\<lbrakk>isFun f; Elem y (Range f)\<rbrakk> \<Longrightarrow> ? x. Elem x (Domain f) & f\<acute>x = y"

   293   apply (auto simp add: Range)

   294   apply (rule_tac x="x" in exI)

   295   apply (auto simp add: app_def the_equality isFun_def Domain)

   296   done

   298 lemma Elem_Fun_Lambda: "Elem F (Fun U V) \<Longrightarrow> ? f. F = Lambda U f"

   299   apply (rule exI[where x= "% x. (THE y. Elem (Opair x y) F)"])

   300   apply (simp add: Ext Lambda_def Repl Domain)

   301   apply (simp add: Ext[symmetric])

   302   apply auto

   303   apply (frule Elem_Elem_Fun)

   304   apply auto

   305   apply (rule_tac x="Fst z" in exI)

   306   apply (simp add: isOpair_def)

   307   apply (auto simp add: Fst Snd Opair)

   308   apply (rule theI2')

   309   apply auto

   310   apply (drule Fun_implies_PFun)

   311   apply (drule_tac x="Opair x ya" and y="Opair x yb" in PFun_inj)

   312   apply (auto simp add: Fst Snd)

   313   apply (drule Fun_implies_PFun)

   314   apply (drule_tac x="Opair x y" and y="Opair x ya" in PFun_inj)

   315   apply (auto simp add: Fst Snd)

   316   apply (rule theI2')

   317   apply (auto simp add: Fun_total)

   318   apply (drule Fun_implies_PFun)

   319   apply (drule_tac x="Opair a x" and y="Opair a y" in PFun_inj)

   320   apply (auto simp add: Fst Snd)

   321   done

   323 lemma Elem_Lambda_Fun: "Elem (Lambda A f) (Fun U V) = (A = U & (! x. Elem x A \<longrightarrow> Elem (f x) V))"

   324 proof -

   325   have "Elem (Lambda A f) (Fun U V) \<Longrightarrow> A = U"

   326     by (simp add: Fun_def Sep domain_Lambda)

   327   then show ?thesis

   328     apply auto

   329     apply (drule Fun_Range)

   330     apply (subgoal_tac "f x = ((Lambda U f) \<acute> x)")

   331     prefer 2

   332     apply (simp add: Lambda_app)

   333     apply simp

   334     apply (subgoal_tac "Elem (Lambda U f \<acute> x) (Range (Lambda U f))")

   335     apply (simp add: subset_def)

   336     apply (rule fun_value_in_range)

   337     apply (simp_all add: isFun_Lambda domain_Lambda)

   338     apply (simp add: Fun_def Sep PFun_def Power domain_Lambda isFun_Lambda)

   339     apply (auto simp add: subset_def CartProd)

   340     apply (rule_tac x="Fst x" in exI)

   341     apply (auto simp add: Lambda_def Repl Fst)

   342     done

   343 qed

   346 constdefs

   347   is_Elem_of :: "(ZF * ZF) set"

   348   "is_Elem_of == { (a,b) | a b. Elem a b }"

   350 lemma cond_wf_Elem:

   351   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"

   352   shows "P a"

   353 proof -

   354   {

   355     fix P

   356     fix U

   357     fix a

   358     assume P_induct: "(\<forall>x. (\<forall>y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y) \<longrightarrow> (Elem x U \<longrightarrow> P x))"

   359     assume a_in_U: "Elem a U"

   360     have "P a"

   361       proof -

   362         term "P"

   363         term Sep

   364         let ?Z = "Sep U (Not o P)"

   365         have "?Z = Empty \<longrightarrow> P a" by (simp add: Ext Sep Empty a_in_U)

   366         moreover have "?Z \<noteq> Empty \<longrightarrow> False"

   367           proof

   368             assume not_empty: "?Z \<noteq> Empty"

   369             note thereis_x = Regularity[where A="?Z", simplified not_empty, simplified]

   370             then obtain x where x_def: "Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..

   371             then have x_induct:"! y. Elem y x \<longrightarrow> Elem y U \<longrightarrow> P y" by (simp add: Sep)

   372             have "Elem x U \<longrightarrow> P x"

   373               by (rule impE[OF spec[OF P_induct, where x=x], OF x_induct], assumption)

   374             moreover have "Elem x U & Not(P x)"

   375               apply (insert x_def)

   376               apply (simp add: Sep)

   377               done

   378             ultimately show "False" by auto

   379           qed

   380         ultimately show "P a" by auto

   381       qed

   382   }

   383   with hyps show ?thesis by blast

   384 qed

   386 lemma cond2_wf_Elem:

   387   assumes

   388      special_P: "? U. ! x. Not(Elem x U) \<longrightarrow> (P x)"

   389      and P_induct: "\<forall>x. (\<forall>y. Elem y x \<longrightarrow> P y) \<longrightarrow> P x"

   390   shows

   391      "P a"

   392 proof -

   393   have "? U Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))"

   394   proof -

   395     from special_P obtain U where U:"! x. Not(Elem x U) \<longrightarrow> (P x)" ..

   396     show ?thesis

   397       apply (rule_tac exI[where x=U])

   398       apply (rule exI[where x="P"])

   399       apply (rule ext)

   400       apply (auto simp add: U)

   401       done

   402   qed

   403   then obtain U where "? Q. P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..

   404   then obtain Q where UQ: "P = (\<lambda> x. (Elem x U \<longrightarrow> Q x))" ..

   405   show ?thesis

   406     apply (auto simp add: UQ)

   407     apply (rule cond_wf_Elem)

   408     apply (rule P_induct[simplified UQ])

   409     apply simp

   410     done

   411 qed

   413 consts

   414   nat2Nat :: "nat \<Rightarrow> ZF"

   416 primrec

   417 nat2Nat_0[intro]:  "nat2Nat 0 = Empty"

   418 nat2Nat_Suc[intro]:  "nat2Nat (Suc n) = SucNat (nat2Nat n)"

   420 constdefs

   421   Nat2nat :: "ZF \<Rightarrow> nat"

   422   "Nat2nat == inv nat2Nat"

   424 lemma Elem_nat2Nat_inf[intro]: "Elem (nat2Nat n) Inf"

   425   apply (induct n)

   426   apply (simp_all add: Infinity)

   427   done

   429 constdefs

   430   Nat :: ZF

   431   "Nat == Sep Inf (\<lambda> N. ? n. nat2Nat n = N)"

   433 lemma Elem_nat2Nat_Nat[intro]: "Elem (nat2Nat n) Nat"

   434   by (auto simp add: Nat_def Sep)

   436 lemma Elem_Empty_Nat: "Elem Empty Nat"

   437   by (auto simp add: Nat_def Sep Infinity)

   439 lemma Elem_SucNat_Nat: "Elem N Nat \<Longrightarrow> Elem (SucNat N) Nat"

   440   by (auto simp add: Nat_def Sep Infinity)

   442 lemma no_infinite_Elem_down_chain:

   443   "Not (? f. isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N)))"

   444 proof -

   445   {

   446     fix f

   447     assume f:"isFun f & Domain f = Nat & (! N. Elem N Nat \<longrightarrow> Elem (f\<acute>(SucNat N)) (f\<acute>N))"

   448     let ?r = "Range f"

   449     have "?r \<noteq> Empty"

   450       apply (auto simp add: Ext Empty)

   451       apply (rule exI[where x="f\<acute>Empty"])

   452       apply (rule fun_value_in_range)

   453       apply (auto simp add: f Elem_Empty_Nat)

   454       done

   455     then have "? x. Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))"

   456       by (simp add: Regularity)

   457     then obtain x where x: "Elem x ?r & (! y. Elem y x \<longrightarrow> Not(Elem y ?r))" ..

   458     then have "? N. Elem N (Domain f) & f\<acute>N = x"

   459       apply (rule_tac fun_range_witness)

   460       apply (simp_all add: f)

   461       done

   462     then have "? N. Elem N Nat & f\<acute>N = x"

   463       by (simp add: f)

   464     then obtain N where N: "Elem N Nat & f\<acute>N = x" ..

   465     from N have N': "Elem N Nat" by auto

   466     let ?y = "f\<acute>(SucNat N)"

   467     have Elem_y_r: "Elem ?y ?r"

   468       by (simp_all add: f Elem_SucNat_Nat N fun_value_in_range)

   469     have "Elem ?y (f\<acute>N)" by (auto simp add: f N')

   470     then have "Elem ?y x" by (simp add: N)

   471     with x have "Not (Elem ?y ?r)" by auto

   472     with Elem_y_r have "False" by auto

   473   }

   474   then show ?thesis by auto

   475 qed

   477 lemma Upair_nonEmpty: "Upair a b \<noteq> Empty"

   478   by (auto simp add: Ext Empty Upair)

   480 lemma Singleton_nonEmpty: "Singleton x \<noteq> Empty"

   481   by (auto simp add: Singleton_def Upair_nonEmpty)

   483 lemma notsym_Elem: "Not(Elem a b & Elem b a)"

   484 proof -

   485   {

   486     fix a b

   487     assume ab: "Elem a b"

   488     assume ba: "Elem b a"

   489     let ?Z = "Upair a b"

   490     have "?Z \<noteq> Empty" by (simp add: Upair_nonEmpty)

   491     then have "? x. Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))"

   492       by (simp add: Regularity)

   493     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not(Elem y ?Z))" ..

   494     then have "x = a \<or> x = b" by (simp add: Upair)

   495     moreover have "x = a \<longrightarrow> Not (Elem b ?Z)"

   496       by (auto simp add: x ba)

   497     moreover have "x = b \<longrightarrow> Not (Elem a ?Z)"

   498       by (auto simp add: x ab)

   499     ultimately have "False"

   500       by (auto simp add: Upair)

   501   }

   502   then show ?thesis by auto

   503 qed

   505 lemma irreflexiv_Elem: "Not(Elem a a)"

   506   by (simp add: notsym_Elem[of a a, simplified])

   508 lemma antisym_Elem: "Elem a b \<Longrightarrow> Not (Elem b a)"

   509   apply (insert notsym_Elem[of a b])

   510   apply auto

   511   done

   513 consts

   514   NatInterval :: "nat \<Rightarrow> nat \<Rightarrow> ZF"

   516 primrec

   517   "NatInterval n 0 = Singleton (nat2Nat n)"

   518   "NatInterval n (Suc m) = union (NatInterval n m) (Singleton (nat2Nat (n+m+1)))"

   520 lemma n_Elem_NatInterval[rule_format]: "! q. q <= m \<longrightarrow> Elem (nat2Nat (n+q)) (NatInterval n m)"

   521   apply (induct m)

   522   apply (auto simp add: Singleton union)

   523   apply (case_tac "q <= m")

   524   apply auto

   525   apply (subgoal_tac "q = Suc m")

   526   apply auto

   527   done

   529 lemma NatInterval_not_Empty: "NatInterval n m \<noteq> Empty"

   530   by (auto intro:   n_Elem_NatInterval[where q = 0, simplified] simp add: Empty Ext)

   532 lemma increasing_nat2Nat[rule_format]: "0 < n \<longrightarrow> Elem (nat2Nat (n - 1)) (nat2Nat n)"

   533   apply (case_tac "? m. n = Suc m")

   534   apply (auto simp add: SucNat_def union Singleton)

   535   apply (drule spec[where x="n - 1"])

   536   apply arith

   537   done

   539 lemma represent_NatInterval[rule_format]: "Elem x (NatInterval n m) \<longrightarrow> (? u. n \<le> u & u \<le> n+m & nat2Nat u = x)"

   540   apply (induct m)

   541   apply (auto simp add: Singleton union)

   542   apply (rule_tac x="Suc (n+m)" in exI)

   543   apply auto

   544   done

   546 lemma inj_nat2Nat: "inj nat2Nat"

   547 proof -

   548   {

   549     fix n m :: nat

   550     assume nm: "nat2Nat n = nat2Nat (n+m)"

   551     assume mg0: "0 < m"

   552     let ?Z = "NatInterval n m"

   553     have "?Z \<noteq> Empty" by (simp add: NatInterval_not_Empty)

   554     then have "? x. (Elem x ?Z) & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))"

   555       by (auto simp add: Regularity)

   556     then obtain x where x:"Elem x ?Z & (! y. Elem y x \<longrightarrow> Not (Elem y ?Z))" ..

   557     then have "? u. n \<le> u & u \<le> n+m & nat2Nat u = x"

   558       by (simp add: represent_NatInterval)

   559     then obtain u where u: "n \<le> u & u \<le> n+m & nat2Nat u = x" ..

   560     have "n < u \<longrightarrow> False"

   561     proof

   562       assume n_less_u: "n < u"

   563       let ?y = "nat2Nat (u - 1)"

   564       have "Elem ?y (nat2Nat u)"

   565         apply (rule increasing_nat2Nat)

   566         apply (insert n_less_u)

   567         apply arith

   568         done

   569       with u have "Elem ?y x" by auto

   570       with x have "Not (Elem ?y ?Z)" by auto

   571       moreover have "Elem ?y ?Z"

   572         apply (insert n_Elem_NatInterval[where q = "u - n - 1" and n=n and m=m])

   573         apply (insert n_less_u)

   574         apply (insert u)

   575         apply auto

   576         done

   577       ultimately show False by auto

   578     qed

   579     moreover have "u = n \<longrightarrow> False"

   580     proof

   581       assume "u = n"

   582       with u have "nat2Nat n = x" by auto

   583       then have nm_eq_x: "nat2Nat (n+m) = x" by (simp add: nm)

   584       let ?y = "nat2Nat (n+m - 1)"

   585       have "Elem ?y (nat2Nat (n+m))"

   586         apply (rule increasing_nat2Nat)

   587         apply (insert mg0)

   588         apply arith

   589         done

   590       with nm_eq_x have "Elem ?y x" by auto

   591       with x have "Not (Elem ?y ?Z)" by auto

   592       moreover have "Elem ?y ?Z"

   593         apply (insert n_Elem_NatInterval[where q = "m - 1" and n=n and m=m])

   594         apply (insert mg0)

   595         apply auto

   596         done

   597       ultimately show False by auto

   598     qed

   599     ultimately have "False" using u by arith

   600   }

   601   note lemma_nat2Nat = this

   602   have th:"\<And>x y. \<not> (x < y \<and> (\<forall>(m\<Colon>nat). y \<noteq> x + m))" by presburger

   603   have th': "\<And>x y. \<not> (x \<noteq> y \<and> (\<not> x < y) \<and> (\<forall>(m\<Colon>nat). x \<noteq> y + m))" by presburger

   604   show ?thesis

   605     apply (auto simp add: inj_on_def)

   606     apply (case_tac "x = y")

   607     apply auto

   608     apply (case_tac "x < y")

   609     apply (case_tac "? m. y = x + m & 0 < m")

   610     apply (auto intro: lemma_nat2Nat)

   611     apply (case_tac "y < x")

   612     apply (case_tac "? m. x = y + m & 0 < m")

   613     apply simp

   614     apply simp

   615     using th apply blast

   616     apply (case_tac "? m. x = y + m")

   617     apply (auto intro: lemma_nat2Nat)

   618     apply (drule sym)

   619     using lemma_nat2Nat apply blast

   620     using th' apply blast

   621     done

   622 qed

   624 lemma Nat2nat_nat2Nat[simp]: "Nat2nat (nat2Nat n) = n"

   625   by (simp add: Nat2nat_def inv_f_f[OF inj_nat2Nat])

   627 lemma nat2Nat_Nat2nat[simp]: "Elem n Nat \<Longrightarrow> nat2Nat (Nat2nat n) = n"

   628   apply (simp add: Nat2nat_def)

   629   apply (rule_tac f_inv_onto_f)

   630   apply (auto simp add: image_def Nat_def Sep)

   631   done

   633 lemma Nat2nat_SucNat: "Elem N Nat \<Longrightarrow> Nat2nat (SucNat N) = Suc (Nat2nat N)"

   634   apply (auto simp add: Nat_def Sep Nat2nat_def)

   635   apply (auto simp add: inv_f_f[OF inj_nat2Nat])

   636   apply (simp only: nat2Nat.simps[symmetric])

   637   apply (simp only: inv_f_f[OF inj_nat2Nat])

   638   done

   641 (*lemma Elem_induct: "(\<And>x. \<forall>y. Elem y x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"

   642   by (erule wf_induct[OF wf_is_Elem_of, simplified is_Elem_of_def, simplified])*)

   644 lemma Elem_Opair_exists: "? z. Elem x z & Elem y z & Elem z (Opair x y)"

   645   apply (rule exI[where x="Upair x y"])

   646   by (simp add: Upair Opair_def)

   648 lemma UNIV_is_not_in_ZF: "UNIV \<noteq> explode R"

   649 proof

   650   let ?Russell = "{ x. Not(Elem x x) }"

   651   have "?Russell = UNIV" by (simp add: irreflexiv_Elem)

   652   moreover assume "UNIV = explode R"

   653   ultimately have russell: "?Russell = explode R" by simp

   654   then show "False"

   655   proof(cases "Elem R R")

   656     case True

   657     then show ?thesis

   658       by (insert irreflexiv_Elem, auto)

   659   next

   660     case False

   661     then have "R \<in> ?Russell" by auto

   662     then have "Elem R R" by (simp add: russell explode_def)

   663     with False show ?thesis by auto

   664   qed

   665 qed

   667 constdefs

   668   SpecialR :: "(ZF * ZF) set"

   669   "SpecialR \<equiv> { (x, y) . x \<noteq> Empty \<and> y = Empty}"

   671 lemma "wf SpecialR"

   672   apply (subst wf_def)

   673   apply (auto simp add: SpecialR_def)

   674   done

   676 constdefs

   677   Ext :: "('a * 'b) set \<Rightarrow> 'b \<Rightarrow> 'a set"

   678   "Ext R y \<equiv> { x . (x, y) \<in> R }"

   680 lemma Ext_Elem: "Ext is_Elem_of = explode"

   681   by (auto intro: ext simp add: Ext_def is_Elem_of_def explode_def)

   683 lemma "Ext SpecialR Empty \<noteq> explode z"

   684 proof

   685   have "Ext SpecialR Empty = UNIV - {Empty}"

   686     by (auto simp add: Ext_def SpecialR_def)

   687   moreover assume "Ext SpecialR Empty = explode z"

   688   ultimately have "UNIV = explode(union z (Singleton Empty)) "

   689     by (auto simp add: explode_def union Singleton)

   690   then show "False" by (simp add: UNIV_is_not_in_ZF)

   691 qed

   693 constdefs

   694   implode :: "ZF set \<Rightarrow> ZF"

   695   "implode == inv explode"

   697 lemma inj_explode: "inj explode"

   698   by (auto simp add: inj_on_def explode_def Ext)

   700 lemma implode_explode[simp]: "implode (explode x) = x"

   701   by (simp add: implode_def inj_explode)

   703 constdefs

   704   regular :: "(ZF * ZF) set \<Rightarrow> bool"

   705   "regular R == ! A. A \<noteq> Empty \<longrightarrow> (? x. Elem x A & (! y. (y, x) \<in> R \<longrightarrow> Not (Elem y A)))"

   706   set_like :: "(ZF * ZF) set \<Rightarrow> bool"

   707   "set_like R == ! y. Ext R y \<in> range explode"

   708   wfzf :: "(ZF * ZF) set \<Rightarrow> bool"

   709   "wfzf R == regular R & set_like R"

   711 lemma regular_Elem: "regular is_Elem_of"

   712   by (simp add: regular_def is_Elem_of_def Regularity)

   714 lemma set_like_Elem: "set_like is_Elem_of"

   715   by (auto simp add: set_like_def image_def Ext_Elem)

   717 lemma wfzf_is_Elem_of: "wfzf is_Elem_of"

   718   by (auto simp add: wfzf_def regular_Elem set_like_Elem)

   720 constdefs

   721   SeqSum :: "(nat \<Rightarrow> ZF) \<Rightarrow> ZF"

   722   "SeqSum f == Sum (Repl Nat (f o Nat2nat))"

   724 lemma SeqSum: "Elem x (SeqSum f) = (? n. Elem x (f n))"

   725   apply (auto simp add: SeqSum_def Sum Repl)

   726   apply (rule_tac x = "f n" in exI)

   727   apply auto

   728   done

   730 constdefs

   731   Ext_ZF :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"

   732   "Ext_ZF R s == implode (Ext R s)"

   734 lemma Elem_implode: "A \<in> range explode \<Longrightarrow> Elem x (implode A) = (x \<in> A)"

   735   apply (auto)

   736   apply (simp_all add: explode_def)

   737   done

   739 lemma Elem_Ext_ZF: "set_like R \<Longrightarrow> Elem x (Ext_ZF R s) = ((x,s) \<in> R)"

   740   apply (simp add: Ext_ZF_def)

   741   apply (subst Elem_implode)

   742   apply (simp add: set_like_def)

   743   apply (simp add: Ext_def)

   744   done

   746 consts

   747   Ext_ZF_n :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> nat \<Rightarrow> ZF"

   749 primrec

   750   "Ext_ZF_n R s 0 = Ext_ZF R s"

   751   "Ext_ZF_n R s (Suc n) = Sum (Repl (Ext_ZF_n R s n) (Ext_ZF R))"

   753 constdefs

   754   Ext_ZF_hull :: "(ZF * ZF) set \<Rightarrow> ZF \<Rightarrow> ZF"

   755   "Ext_ZF_hull R s == SeqSum (Ext_ZF_n R s)"

   757 lemma Elem_Ext_ZF_hull:

   758   assumes set_like_R: "set_like R"

   759   shows "Elem x (Ext_ZF_hull R S) = (? n. Elem x (Ext_ZF_n R S n))"

   760   by (simp add: Ext_ZF_hull_def SeqSum)

   762 lemma Elem_Elem_Ext_ZF_hull:

   763   assumes set_like_R: "set_like R"

   764           and x_hull: "Elem x (Ext_ZF_hull R S)"

   765           and y_R_x: "(y, x) \<in> R"

   766   shows "Elem y (Ext_ZF_hull R S)"

   767 proof -

   768   from Elem_Ext_ZF_hull[OF set_like_R] x_hull

   769   have "? n. Elem x (Ext_ZF_n R S n)" by auto

   770   then obtain n where n:"Elem x (Ext_ZF_n R S n)" ..

   771   with y_R_x have "Elem y (Ext_ZF_n R S (Suc n))"

   772     apply (auto simp add: Repl Sum)

   773     apply (rule_tac x="Ext_ZF R x" in exI)

   774     apply (auto simp add: Elem_Ext_ZF[OF set_like_R])

   775     done

   776   with Elem_Ext_ZF_hull[OF set_like_R, where x=y] show ?thesis

   777     by (auto simp del: Ext_ZF_n.simps)

   778 qed

   780 lemma wfzf_minimal:

   781   assumes hyps: "wfzf R" "C \<noteq> {}"

   782   shows "\<exists>x. x \<in> C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> C)"

   783 proof -

   784   from hyps have "\<exists>S. S \<in> C" by auto

   785   then obtain S where S:"S \<in> C" by auto

   786   let ?T = "Sep (Ext_ZF_hull R S) (\<lambda> s. s \<in> C)"

   787   from hyps have set_like_R: "set_like R" by (simp add: wfzf_def)

   788   show ?thesis

   789   proof (cases "?T = Empty")

   790     case True

   791     then have "\<forall> z. \<not> (Elem z (Sep (Ext_ZF R S) (\<lambda> s. s \<in> C)))"

   792       apply (auto simp add: Ext Empty Sep Ext_ZF_hull_def SeqSum)

   793       apply (erule_tac x="z" in allE, auto)

   794       apply (erule_tac x=0 in allE, auto)

   795       done

   796     then show ?thesis

   797       apply (rule_tac exI[where x=S])

   798       apply (auto simp add: Sep Empty S)

   799       apply (erule_tac x=y in allE)

   800       apply (simp add: set_like_R Elem_Ext_ZF)

   801       done

   802   next

   803     case False

   804     from hyps have regular_R: "regular R" by (simp add: wfzf_def)

   805     from

   806       regular_R[simplified regular_def, rule_format, OF False, simplified Sep]

   807       Elem_Elem_Ext_ZF_hull[OF set_like_R]

   808     show ?thesis by blast

   809   qed

   810 qed

   812 lemma wfzf_implies_wf: "wfzf R \<Longrightarrow> wf R"

   813 proof (subst wf_def, rule allI)

   814   assume wfzf: "wfzf R"

   815   fix P :: "ZF \<Rightarrow> bool"

   816   let ?C = "{x. P x}"

   817   {

   818     assume induct: "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x)"

   819     let ?C = "{x. \<not> (P x)}"

   820     have "?C = {}"

   821     proof (rule ccontr)

   822       assume C: "?C \<noteq> {}"

   823       from

   824         wfzf_minimal[OF wfzf C]

   825       obtain x where x: "x \<in> ?C \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> ?C)" ..

   826       then have "P x"

   827         apply (rule_tac induct[rule_format])

   828         apply auto

   829         done

   830       with x show "False" by auto

   831     qed

   832     then have "! x. P x" by auto

   833   }

   834   then show "(\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (! x. P x)" by blast

   835 qed

   837 lemma wf_is_Elem_of: "wf is_Elem_of"

   838   by (auto simp add: wfzf_is_Elem_of wfzf_implies_wf)

   840 lemma in_Ext_RTrans_implies_Elem_Ext_ZF_hull:

   841   "set_like R \<Longrightarrow> x \<in> (Ext (R^+) s) \<Longrightarrow> Elem x (Ext_ZF_hull R s)"

   842   apply (simp add: Ext_def Elem_Ext_ZF_hull)

   843   apply (erule converse_trancl_induct[where r="R"])

   844   apply (rule exI[where x=0])

   845   apply (simp add: Elem_Ext_ZF)

   846   apply auto

   847   apply (rule_tac x="Suc n" in exI)

   848   apply (simp add: Sum Repl)

   849   apply (rule_tac x="Ext_ZF R z" in exI)

   850   apply (auto simp add: Elem_Ext_ZF)

   851   done

   853 lemma implodeable_Ext_trancl: "set_like R \<Longrightarrow> set_like (R^+)"

   854   apply (subst set_like_def)

   855   apply (auto simp add: image_def)

   856   apply (rule_tac x="Sep (Ext_ZF_hull R y) (\<lambda> z. z \<in> (Ext (R^+) y))" in exI)

   857   apply (auto simp add: explode_def Sep set_ext

   858     in_Ext_RTrans_implies_Elem_Ext_ZF_hull)

   859   done

   861 lemma Elem_Ext_ZF_hull_implies_in_Ext_RTrans[rule_format]:

   862   "set_like R \<Longrightarrow> ! x. Elem x (Ext_ZF_n R s n) \<longrightarrow> x \<in> (Ext (R^+) s)"

   863   apply (induct_tac n)

   864   apply (auto simp add: Elem_Ext_ZF Ext_def Sum Repl)

   865   done

   867 lemma "set_like R \<Longrightarrow> Ext_ZF (R^+) s = Ext_ZF_hull R s"

   868   apply (frule implodeable_Ext_trancl)

   869   apply (auto simp add: Ext)

   870   apply (erule in_Ext_RTrans_implies_Elem_Ext_ZF_hull)

   871   apply (simp add: Elem_Ext_ZF Ext_def)

   872   apply (auto simp add: Elem_Ext_ZF Elem_Ext_ZF_hull)

   873   apply (erule Elem_Ext_ZF_hull_implies_in_Ext_RTrans[simplified Ext_def, simplified], assumption)

   874   done

   876 lemma wf_implies_regular: "wf R \<Longrightarrow> regular R"

   877 proof (simp add: regular_def, rule allI)

   878   assume wf: "wf R"

   879   fix A

   880   show "A \<noteq> Empty \<longrightarrow> (\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A))"

   881   proof

   882     assume A: "A \<noteq> Empty"

   883     then have "? x. x \<in> explode A"

   884       by (auto simp add: explode_def Ext Empty)

   885     then obtain x where x:"x \<in> explode A" ..

   886     from iffD1[OF wf_eq_minimal wf, rule_format, where Q="explode A", OF x]

   887     obtain z where "z \<in> explode A \<and> (\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> explode A)" by auto

   888     then show "\<exists>x. Elem x A \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> \<not> Elem y A)"

   889       apply (rule_tac exI[where x = z])

   890       apply (simp add: explode_def)

   891       done

   892   qed

   893 qed

   895 lemma wf_eq_wfzf: "(wf R \<and> set_like R) = wfzf R"

   896   apply (auto simp add: wfzf_implies_wf)

   897   apply (auto simp add: wfzf_def wf_implies_regular)

   898   done

   900 lemma wfzf_trancl: "wfzf R \<Longrightarrow> wfzf (R^+)"

   901   by (auto simp add: wf_eq_wfzf[symmetric] implodeable_Ext_trancl wf_trancl)

   903 lemma Ext_subset_mono: "R \<subseteq> S \<Longrightarrow> Ext R y \<subseteq> Ext S y"

   904   by (auto simp add: Ext_def)

   906 lemma set_like_subset: "set_like R \<Longrightarrow> S \<subseteq> R \<Longrightarrow> set_like S"

   907   apply (auto simp add: set_like_def)

   908   apply (erule_tac x=y in allE)

   909   apply (drule_tac y=y in Ext_subset_mono)

   910   apply (auto simp add: image_def)

   911   apply (rule_tac x="Sep x (% z. z \<in> (Ext S y))" in exI)

   912   apply (auto simp add: explode_def Sep)

   913   done

   915 lemma wfzf_subset: "wfzf S \<Longrightarrow> R \<subseteq> S \<Longrightarrow> wfzf R"

   916   by (auto intro: set_like_subset wf_subset simp add: wf_eq_wfzf[symmetric])

   918 end

author	nipkow
	Sun, 18 Oct 2009 12:07:56 +0200
changeset 32989	c28279b29ff1
parent 32962	69916a850301
parent 32988	d1d4d7a08a66
child 33057	764547b68538
permissions	-rw-r--r--