src/ZF/Constructible/Formula.thy
author paulson
Mon Jun 24 11:59:21 2002 +0200 (2002-06-24)
changeset 13245 714f7a423a15
parent 13223 45be08fbdcff
child 13269 3ba9be497c33
permissions -rw-r--r--
development and tweaks
     1 header {* First-Order Formulas and the Definition of the Class L *}
     2 
     3 theory Formula = Main:
     4 
     5 
     6 (*??for Bool.thy**)
     7 constdefs bool_of_o :: "o=>i"
     8    "bool_of_o(P) == (if P then 1 else 0)"
     9 
    10 lemma [simp]: "bool_of_o(True) = 1"
    11 by (simp add: bool_of_o_def) 
    12 
    13 lemma [simp]: "bool_of_o(False) = 0"
    14 by (simp add: bool_of_o_def) 
    15 
    16 lemma [simp,TC]: "bool_of_o(P) \<in> bool"
    17 by (simp add: bool_of_o_def) 
    18 
    19 lemma [simp]: "(bool_of_o(P) = 1) <-> P"
    20 by (simp add: bool_of_o_def) 
    21 
    22 lemma [simp]: "(bool_of_o(P) = 0) <-> ~P"
    23 by (simp add: bool_of_o_def) 
    24 
    25 (*????????????????CardinalArith *)
    26 
    27 lemma Finite_Vset: "i \<in> nat ==> Finite(Vset(i))";
    28 apply (erule nat_induct)
    29  apply (simp add: Vfrom_0) 
    30 apply (simp add: Vset_succ) 
    31 done
    32 
    33 (*???Ordinal maybe, but some lemmas seem to be in CardinalArith??*)
    34 text{*Every ordinal is exceeded by some limit ordinal.*}
    35 lemma Ord_imp_greater_Limit: "Ord(i) ==> \<exists>k. i<k & Limit(k)"
    36 apply (rule_tac x="i ++ nat" in exI) 
    37 apply (blast intro: oadd_LimitI  oadd_lt_self  Limit_nat [THEN Limit_has_0])
    38 done
    39 
    40 lemma Ord2_imp_greater_Limit: "[|Ord(i); Ord(j)|] ==> \<exists>k. i<k & j<k & Limit(k)"
    41 apply (insert Ord_Un [of i j, THEN Ord_imp_greater_Limit]) 
    42 apply (simp add: Un_least_lt_iff) 
    43 done
    44 
    45 
    46 
    47 (*Internalized formulas of FOL. De Bruijn representation. 
    48   Unbound variables get their denotations from an environment.*)
    49 
    50 consts   formula :: i
    51 datatype
    52   "formula" = Member ("x: nat", "y: nat")
    53             | Equal  ("x: nat", "y: nat")
    54             | Neg ("p: formula")
    55             | And ("p: formula", "q: formula")
    56             | Forall ("p: formula")
    57 
    58 declare formula.intros [TC]
    59 
    60 constdefs Or :: "[i,i]=>i"
    61     "Or(p,q) == Neg(And(Neg(p),Neg(q)))"
    62 
    63 constdefs Implies :: "[i,i]=>i"
    64     "Implies(p,q) == Neg(And(p,Neg(q)))"
    65 
    66 constdefs Exists :: "i=>i"
    67     "Exists(p) == Neg(Forall(Neg(p)))";
    68 
    69 lemma Or_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> Or(p,q) \<in> formula"
    70 by (simp add: Or_def) 
    71 
    72 lemma Implies_type [TC]:
    73      "[| p \<in> formula; q \<in> formula |] ==> Implies(p,q) \<in> formula"
    74 by (simp add: Implies_def) 
    75 
    76 lemma Exists_type [TC]: "p \<in> formula ==> Exists(p) \<in> formula"
    77 by (simp add: Exists_def) 
    78 
    79 
    80 consts   satisfies :: "[i,i]=>i"
    81 primrec (*explicit lambda is required because the environment varies*)
    82   "satisfies(A,Member(x,y)) = 
    83       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) \<in> nth(y,env)))"
    84 
    85   "satisfies(A,Equal(x,y)) = 
    86       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) = nth(y,env)))"
    87 
    88   "satisfies(A,Neg(p)) = 
    89       (\<lambda>env \<in> list(A). not(satisfies(A,p)`env))"
    90 
    91   "satisfies(A,And(p,q)) =
    92       (\<lambda>env \<in> list(A). (satisfies(A,p)`env) and (satisfies(A,q)`env))"
    93 
    94   "satisfies(A,Forall(p)) = 
    95       (\<lambda>env \<in> list(A). bool_of_o (\<forall>x\<in>A. satisfies(A,p) ` (Cons(x,env)) = 1))"
    96 
    97 
    98 lemma "p \<in> formula ==> satisfies(A,p) \<in> list(A) -> bool"
    99 by (induct_tac p, simp_all) 
   100 
   101 syntax sats :: "[i,i,i] => o"
   102 translations "sats(A,p,env)" == "satisfies(A,p)`env = 1"
   103 
   104 lemma [simp]:
   105   "env \<in> list(A) 
   106    ==> sats(A, Member(x,y), env) <-> nth(x,env) \<in> nth(y,env)"
   107 by simp
   108 
   109 lemma [simp]:
   110   "env \<in> list(A) 
   111    ==> sats(A, Equal(x,y), env) <-> nth(x,env) = nth(y,env)"
   112 by simp
   113 
   114 lemma sats_Neg_iff [simp]:
   115   "env \<in> list(A) 
   116    ==> sats(A, Neg(p), env) <-> ~ sats(A,p,env)"
   117 by (simp add: Bool.not_def cond_def) 
   118 
   119 lemma sats_And_iff [simp]:
   120   "env \<in> list(A) 
   121    ==> (sats(A, And(p,q), env)) <-> sats(A,p,env) & sats(A,q,env)"
   122 by (simp add: Bool.and_def cond_def) 
   123 
   124 lemma sats_Forall_iff [simp]:
   125   "env \<in> list(A) 
   126    ==> sats(A, Forall(p), env) <-> (\<forall>x\<in>A. sats(A, p, Cons(x,env)))"
   127 by simp
   128 
   129 declare satisfies.simps [simp del]; 
   130 
   131 (**** DIVIDING LINE BETWEEN PRIMITIVE AND DERIVED CONNECTIVES ****)
   132 
   133 lemma sats_Or_iff [simp]:
   134   "env \<in> list(A) 
   135    ==> (sats(A, Or(p,q), env)) <-> sats(A,p,env) | sats(A,q,env)"
   136 by (simp add: Or_def)
   137 
   138 lemma sats_Implies_iff [simp]:
   139   "env \<in> list(A) 
   140    ==> (sats(A, Implies(p,q), env)) <-> (sats(A,p,env) --> sats(A,q,env))"
   141 apply (simp add: Implies_def, blast) 
   142 done
   143 
   144 lemma sats_Exists_iff [simp]:
   145   "env \<in> list(A) 
   146    ==> sats(A, Exists(p), env) <-> (\<exists>x\<in>A. sats(A, p, Cons(x,env)))"
   147 by (simp add: Exists_def)
   148 
   149 
   150 
   151 
   152 (*pretty but unnecessary
   153 constdefs sat     :: "[i,i] => o"
   154   "sat(A,p) == satisfies(A,p)`[] = 1"
   155 
   156 syntax "_sat"  :: "[i,i] => o"    (infixl "|=" 50)
   157 translations "A |= p" == "sat(A,p)"
   158 
   159 lemma [simp]: "(A |= Neg(p)) <-> ~ (A |= p)"
   160 by (simp add: sat_def)
   161 
   162 lemma [simp]: "(A |= And(p,q)) <-> A|=p & A|=q"
   163 by (simp add: sat_def)
   164 *) 
   165 
   166 
   167 constdefs incr_var :: "[i,i]=>i"
   168     "incr_var(x,lev) == if x<lev then x else succ(x)"
   169 
   170 lemma incr_var_lt: "x<lev ==> incr_var(x,lev) = x"
   171 by (simp add: incr_var_def)
   172 
   173 lemma incr_var_le: "lev\<le>x ==> incr_var(x,lev) = succ(x)"
   174 apply (simp add: incr_var_def) 
   175 apply (blast dest: lt_trans1) 
   176 done
   177 
   178 consts   incr_bv :: "i=>i"
   179 primrec
   180   "incr_bv(Member(x,y)) = 
   181       (\<lambda>lev \<in> nat. Member (incr_var(x,lev), incr_var(y,lev)))"
   182 
   183   "incr_bv(Equal(x,y)) = 
   184       (\<lambda>lev \<in> nat. Equal (incr_var(x,lev), incr_var(y,lev)))"
   185 
   186   "incr_bv(Neg(p)) = 
   187       (\<lambda>lev \<in> nat. Neg(incr_bv(p)`lev))"
   188 
   189   "incr_bv(And(p,q)) =
   190       (\<lambda>lev \<in> nat. And (incr_bv(p)`lev, incr_bv(q)`lev))"
   191 
   192   "incr_bv(Forall(p)) = 
   193       (\<lambda>lev \<in> nat. Forall (incr_bv(p) ` succ(lev)))"
   194 
   195 
   196 constdefs incr_boundvars :: "i => i"
   197     "incr_boundvars(p) == incr_bv(p)`0"
   198 
   199 
   200 lemma [TC]: "x \<in> nat ==> incr_var(x,lev) \<in> nat"
   201 by (simp add: incr_var_def) 
   202 
   203 lemma incr_bv_type [TC]: "p \<in> formula ==> incr_bv(p) \<in> nat -> formula"
   204 by (induct_tac p, simp_all) 
   205 
   206 lemma incr_boundvars_type [TC]: "p \<in> formula ==> incr_boundvars(p) \<in> formula"
   207 by (simp add: incr_boundvars_def) 
   208 
   209 (*Obviously DPow is closed under complements and finite intersections and
   210 unions.  Needs an inductive lemma to allow two lists of parameters to 
   211 be combined.*)
   212 
   213 lemma sats_incr_bv_iff [rule_format]:
   214   "[| p \<in> formula; env \<in> list(A); x \<in> A |]
   215    ==> \<forall>bvs \<in> list(A). 
   216            sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x,env)) <-> 
   217            sats(A, p, bvs@env)"
   218 apply (induct_tac p)
   219 apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type)
   220 apply (auto simp add: diff_succ not_lt_iff_le)
   221 done
   222 
   223 (*UNUSED*)
   224 lemma sats_incr_boundvars_iff:
   225   "[| p \<in> formula; env \<in> list(A); x \<in> A |]
   226    ==> sats(A, incr_boundvars(p), Cons(x,env)) <-> sats(A, p, env)"
   227 apply (insert sats_incr_bv_iff [of p env A x Nil])
   228 apply (simp add: incr_boundvars_def) 
   229 done
   230 
   231 (*UNUSED
   232 lemma formula_add_params [rule_format]:
   233   "[| p \<in> formula; n \<in> nat |]
   234    ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
   235          length(bvs) = n --> 
   236          sats(A, iterates(incr_boundvars,n,p), bvs@env) <-> sats(A, p, env)"
   237 apply (induct_tac n, simp, clarify) 
   238 apply (erule list.cases)
   239 apply (auto simp add: sats_incr_boundvars_iff)  
   240 done
   241 *)
   242 
   243 consts   arity :: "i=>i"
   244 primrec
   245   "arity(Member(x,y)) = succ(x) \<union> succ(y)"
   246 
   247   "arity(Equal(x,y)) = succ(x) \<union> succ(y)"
   248 
   249   "arity(Neg(p)) = arity(p)"
   250 
   251   "arity(And(p,q)) = arity(p) \<union> arity(q)"
   252 
   253   "arity(Forall(p)) = nat_case3(0, %x. x, arity(p))"
   254 
   255 
   256 lemma arity_type [TC]: "p \<in> formula ==> arity(p) \<in> nat"
   257 by (induct_tac p, simp_all) 
   258 
   259 lemma arity_Or [simp]: "arity(Or(p,q)) = arity(p) \<union> arity(q)"
   260 by (simp add: Or_def) 
   261 
   262 lemma arity_Implies [simp]: "arity(Implies(p,q)) = arity(p) \<union> arity(q)"
   263 by (simp add: Implies_def) 
   264 
   265 lemma arity_Exists [simp]: "arity(Exists(p)) = nat_case3(0, %x. x, arity(p))"
   266 by (simp add: Exists_def) 
   267 
   268 
   269 lemma arity_sats_iff [rule_format]:
   270   "[| p \<in> formula; extra \<in> list(A) |]
   271    ==> \<forall>env \<in> list(A). 
   272            arity(p) \<le> length(env) --> 
   273            sats(A, p, env @ extra) <-> sats(A, p, env)"
   274 apply (induct_tac p)
   275 apply (simp_all add: nth_append Un_least_lt_iff arity_type 
   276                 split: split_nat_case3, auto) 
   277 done
   278 
   279 lemma arity_sats1_iff:
   280   "[| arity(p) \<le> succ(length(env)); p \<in> formula; x \<in> A; env \<in> list(A); 
   281     extra \<in> list(A) |]
   282    ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
   283 apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
   284 apply simp 
   285 done
   286 
   287 (*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
   288 lemma incr_var_lemma:
   289      "[| x \<in> nat; y \<in> nat; lev \<le> x |]
   290       ==> succ(x) \<union> incr_var(y,lev) = succ(x \<union> y)"
   291 apply (simp add: incr_var_def Ord_Un_if, auto)
   292   apply (blast intro: leI)
   293  apply (simp add: not_lt_iff_le)  
   294  apply (blast intro: le_anti_sym) 
   295 apply (blast dest: lt_trans2) 
   296 done
   297 
   298 lemma incr_And_lemma:
   299      "y < x ==> y \<union> succ(x) = succ(x \<union> y)"
   300 apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff) 
   301 apply (blast dest: lt_asym) 
   302 done
   303 
   304 lemma arity_incr_bv_lemma [rule_format]:
   305   "p \<in> formula 
   306    ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) = 
   307                  (if n < arity(p) then succ(arity(p)) else arity(p))"
   308 apply (induct_tac p) 
   309 apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
   310                      succ_Un_distrib [symmetric] incr_var_lt incr_var_le
   311                      Un_commute incr_var_lemma arity_type 
   312             split: split_nat_case3) 
   313 (*left with the And case*)
   314 apply safe
   315  apply (blast intro: incr_And_lemma lt_trans1) 
   316 apply (subst incr_And_lemma)
   317  apply (blast intro:  lt_trans1) 
   318 apply (simp add:  Un_commute)
   319 done
   320 
   321 lemma arity_incr_boundvars_eq:
   322   "p \<in> formula
   323    ==> arity(incr_boundvars(p)) =
   324         (if 0 < arity(p) then succ(arity(p)) else arity(p))"
   325 apply (insert arity_incr_bv_lemma [of p 0])
   326 apply (simp add: incr_boundvars_def) 
   327 done
   328 
   329 lemma arity_iterates_incr_boundvars_eq:
   330   "[| p \<in> formula; n \<in> nat |]
   331    ==> arity(incr_boundvars^n(p)) =
   332          (if 0 < arity(p) then n #+ arity(p) else arity(p))"
   333 apply (induct_tac n) 
   334 apply (simp_all add: arity_incr_boundvars_eq not_lt_iff_le) 
   335 done
   336 
   337 
   338 (**** TRYING INCR_BV1 AGAIN ****)
   339 
   340 constdefs incr_bv1 :: "i => i"
   341     "incr_bv1(p) == incr_bv(p)`1"
   342 
   343 
   344 lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
   345 by (simp add: incr_bv1_def) 
   346 
   347 (*For renaming all but the bound variable at level 0*)
   348 lemma sats_incr_bv1_iff [rule_format]:
   349   "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
   350    ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <-> 
   351        sats(A, p, Cons(x,env))"
   352 apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
   353 apply (simp add: incr_bv1_def) 
   354 done
   355 
   356 lemma formula_add_params1 [rule_format]:
   357   "[| p \<in> formula; n \<in> nat; x \<in> A |]
   358    ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A). 
   359           length(bvs) = n --> 
   360           sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <-> 
   361           sats(A, p, Cons(x,env))"
   362 apply (induct_tac n, simp, clarify) 
   363 apply (erule list.cases)
   364 apply (simp_all add: sats_incr_bv1_iff) 
   365 done
   366 
   367 
   368 lemma arity_incr_bv1_eq:
   369   "p \<in> formula
   370    ==> arity(incr_bv1(p)) =
   371         (if 1 < arity(p) then succ(arity(p)) else arity(p))"
   372 apply (insert arity_incr_bv_lemma [of p 1])
   373 apply (simp add: incr_bv1_def) 
   374 done
   375 
   376 lemma arity_iterates_incr_bv1_eq:
   377   "[| p \<in> formula; n \<in> nat |]
   378    ==> arity(incr_bv1^n(p)) =
   379          (if 1 < arity(p) then n #+ arity(p) else arity(p))"
   380 apply (induct_tac n) 
   381 apply (simp_all add: arity_incr_bv1_eq )
   382 apply (simp add: not_lt_iff_le)
   383 apply (blast intro: le_trans add_le_self2 arity_type) 
   384 done
   385 
   386 
   387 (*Definable powerset operation: Kunen's definition 1.1, page 165.*)
   388 constdefs DPow :: "i => i"
   389   "DPow(A) == {X \<in> Pow(A). 
   390                \<exists>env \<in> list(A). \<exists>p \<in> formula. 
   391                  arity(p) \<le> succ(length(env)) & 
   392                  X = {x\<in>A. sats(A, p, Cons(x,env))}}"
   393 
   394 lemma DPowI:
   395   "[|X <= A;  env \<in> list(A);  p \<in> formula; 
   396      arity(p) \<le> succ(length(env))|]
   397    ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
   398 by (simp add: DPow_def, blast) 
   399 
   400 lemma DPowD:
   401   "X \<in> DPow(A) 
   402    ==> X <= A &
   403        (\<exists>env \<in> list(A). 
   404         \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) & 
   405                       X = {x\<in>A. sats(A, p, Cons(x,env))})"
   406 by (simp add: DPow_def) 
   407 
   408 lemmas DPow_imp_subset = DPowD [THEN conjunct1]
   409 
   410 (*Lemma 1.2*)
   411 lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |] 
   412        ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
   413 by (blast intro: DPowI)
   414 
   415 lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
   416 by (simp add: DPow_def, blast)
   417 
   418 lemma empty_in_DPow: "0 \<in> DPow(A)"
   419 apply (simp add: DPow_def)
   420 apply (rule_tac x="Nil" in bexI) 
   421  apply (rule_tac x="Neg(Equal(0,0))" in bexI) 
   422   apply (auto simp add: Un_least_lt_iff) 
   423 done
   424 
   425 lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
   426 apply (simp add: DPow_def, clarify, auto) 
   427 apply (rule bexI) 
   428  apply (rule_tac x="Neg(p)" in bexI) 
   429   apply auto 
   430 done
   431 
   432 lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
   433 apply (simp add: DPow_def, auto) 
   434 apply (rename_tac envp p envq q) 
   435 apply (rule_tac x="envp@envq" in bexI) 
   436  apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
   437   apply typecheck
   438 apply (rule conjI) 
   439 (*finally check the arity!*)
   440  apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
   441  apply (force intro: add_le_self le_trans) 
   442 apply (simp add: arity_sats1_iff formula_add_params1, blast) 
   443 done
   444 
   445 lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
   446 apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))") 
   447 apply (simp add: Int_in_DPow Compl_in_DPow) 
   448 apply (simp add: DPow_def, blast) 
   449 done
   450 
   451 lemma singleton_in_DPow: "x \<in> A ==> {x} \<in> DPow(A)"
   452 apply (simp add: DPow_def)
   453 apply (rule_tac x="Cons(x,Nil)" in bexI) 
   454  apply (rule_tac x="Equal(0,1)" in bexI) 
   455   apply typecheck
   456 apply (force simp add: succ_Un_distrib [symmetric])  
   457 done
   458 
   459 lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
   460 apply (rule cons_eq [THEN subst]) 
   461 apply (blast intro: singleton_in_DPow Un_in_DPow) 
   462 done
   463 
   464 (*Part of Lemma 1.3*)
   465 lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
   466 apply (erule Fin.induct) 
   467  apply (rule empty_in_DPow) 
   468 apply (blast intro: cons_in_DPow) 
   469 done
   470 
   471 (*DPow is not monotonic.  For example, let A be some non-constructible set
   472   of natural numbers, and let B be nat.  Then A<=B and obviously A : DPow(A)
   473   but A ~: DPow(B).*)
   474 lemma DPow_mono: "A : DPow(B) ==> DPow(A) <= DPow(B)"
   475 apply (simp add: DPow_def, auto) 
   476 (*must use the formula defining A in B to relativize the new formula...*)
   477 oops
   478 
   479 lemma DPow_0: "DPow(0) = {0}" 
   480 by (blast intro: empty_in_DPow dest: DPow_imp_subset)
   481 
   482 lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)" 
   483 by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
   484 
   485 lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
   486 apply (rule equalityI) 
   487 apply (rule DPow_subset_Pow) 
   488 apply (erule Finite_Pow_subset_Pow) 
   489 done
   490 
   491 (*This may be true but the proof looks difficult, requiring relativization 
   492 lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
   493 apply (rule equalityI, safe)
   494 oops
   495 *)
   496 
   497 subsection{* Constant Lset: Levels of the Constructible Universe *}
   498 
   499 constdefs Lset :: "i=>i"
   500     "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
   501 
   502 text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
   503 lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
   504 by (subst Lset_def [THEN def_transrec], simp)
   505 
   506 lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
   507 by (subst Lset, blast)
   508 
   509 lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
   510 apply (insert Lset [of x]) 
   511 apply (blast intro: elim: equalityE) 
   512 done
   513 
   514 subsubsection{* Transitivity *}
   515 
   516 lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
   517 apply (simp add: Transset_def DPow_def)
   518 apply (rule_tac x="[X]" in bexI) 
   519  apply (rule_tac x="Member(0,1)" in bexI) 
   520   apply (auto simp add: Un_least_lt_iff) 
   521 done
   522 
   523 lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
   524 apply clarify  
   525 apply (simp add: Transset_def)
   526 apply (blast intro: elem_subset_in_DPow) 
   527 done
   528 
   529 lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
   530 apply (simp add: Transset_def) 
   531 apply (blast intro: elem_subset_in_DPow dest: DPowD) 
   532 done
   533 
   534 text{*Kunen's VI, 1.6 (a)*}
   535 lemma Transset_Lset: "Transset(Lset(i))"
   536 apply (rule_tac a=i in eps_induct)
   537 apply (subst Lset)
   538 apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
   539 done
   540 
   541 subsubsection{* Monotonicity *}
   542 
   543 text{*Kunen's VI, 1.6 (b)*}
   544 lemma Lset_mono [rule_format]:
   545      "ALL j. i<=j --> Lset(i) <= Lset(j)"
   546 apply (rule_tac a=i in eps_induct)
   547 apply (rule impI [THEN allI])
   548 apply (subst Lset)
   549 apply (subst Lset, blast) 
   550 done
   551 
   552 text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
   553 lemma Lset_mono_mem [rule_format]:
   554      "ALL j. i:j --> Lset(i) <= Lset(j)"
   555 apply (rule_tac a=i in eps_induct)
   556 apply (rule impI [THEN allI])
   557 apply (subst Lset, auto) 
   558 apply (rule rev_bexI, assumption)
   559 apply (blast intro: elem_subset_in_DPow dest: LsetD DPowD) 
   560 done
   561 
   562 subsubsection{* 0, successor and limit equations fof Lset *}
   563 
   564 lemma Lset_0 [simp]: "Lset(0) = 0"
   565 by (subst Lset, blast)
   566 
   567 lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
   568 by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
   569 
   570 lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
   571 apply (subst Lset, rule UN_least)
   572 apply (erule succE) 
   573  apply blast 
   574 apply clarify
   575 apply (rule elem_subset_in_DPow)
   576  apply (subst Lset)
   577  apply blast 
   578 apply (blast intro: dest: DPowD Lset_mono_mem) 
   579 done
   580 
   581 lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
   582 by (intro equalityI Lset_succ_subset1 Lset_succ_subset2) 
   583 
   584 lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
   585 apply (subst Lset)
   586 apply (rule equalityI)
   587  txt{*first inclusion*}
   588  apply (rule UN_least)
   589  apply (erule UnionE)
   590  apply (rule subset_trans)
   591   apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
   592 txt{*opposite inclusion*}
   593 apply (rule UN_least)
   594 apply (subst Lset, blast)
   595 done
   596 
   597 subsubsection{* Lset applied to Limit ordinals *}
   598 
   599 lemma Limit_Lset_eq:
   600     "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
   601 by (simp add: Lset_Union [symmetric] Limit_Union_eq)
   602 
   603 lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a : Lset(i)"
   604 by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
   605 
   606 lemma Limit_LsetE:
   607     "[| a: Lset(i);  ~R ==> Limit(i);
   608         !!x. [| x<i;  a: Lset(x) |] ==> R
   609      |] ==> R"
   610 apply (rule classical)
   611 apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
   612   prefer 2 apply assumption
   613  apply blast 
   614 apply (blast intro: ltI  Limit_is_Ord)
   615 done
   616 
   617 subsubsection{* Basic closure properties *}
   618 
   619 lemma zero_in_Lset: "y:x ==> 0 : Lset(x)"
   620 by (subst Lset, blast intro: empty_in_DPow)
   621 
   622 lemma notin_Lset: "x \<notin> Lset(x)"
   623 apply (rule_tac a=x in eps_induct)
   624 apply (subst Lset)
   625 apply (blast dest: DPowD)  
   626 done
   627 
   628 
   629 
   630 text{*Kunen's VI, 1.9 (b)*}
   631 
   632 constdefs subset_fm :: "[i,i]=>i"
   633     "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
   634 
   635 lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
   636 by (simp add: subset_fm_def) 
   637 
   638 lemma arity_subset_fm [simp]:
   639      "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
   640 by (simp add: subset_fm_def succ_Un_distrib [symmetric]) 
   641 
   642 lemma sats_subset_fm [simp]:
   643    "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
   644     ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
   645 apply (frule lt_nat_in_nat, erule length_type) 
   646 apply (simp add: subset_fm_def Transset_def) 
   647 apply (blast intro: nth_type ) 
   648 done
   649 
   650 constdefs transset_fm :: "i=>i"
   651    "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
   652 
   653 lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
   654 by (simp add: transset_fm_def) 
   655 
   656 lemma arity_transset_fm [simp]:
   657      "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
   658 by (simp add: transset_fm_def succ_Un_distrib [symmetric]) 
   659 
   660 lemma sats_transset_fm [simp]:
   661    "[|x < length(env); env \<in> list(A); Transset(A)|]
   662     ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
   663 apply (frule lt_nat_in_nat, erule length_type) 
   664 apply (simp add: transset_fm_def Transset_def) 
   665 apply (blast intro: nth_type ) 
   666 done
   667 
   668 constdefs ordinal_fm :: "i=>i"
   669    "ordinal_fm(x) == 
   670       And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
   671 
   672 lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
   673 by (simp add: ordinal_fm_def) 
   674 
   675 lemma arity_ordinal_fm [simp]:
   676      "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
   677 by (simp add: ordinal_fm_def succ_Un_distrib [symmetric]) 
   678 
   679 lemma sats_ordinal_fm [simp]:
   680    "[|x < length(env); env \<in> list(A); Transset(A)|]
   681     ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
   682 apply (frule lt_nat_in_nat, erule length_type) 
   683 apply (simp add: ordinal_fm_def Ord_def Transset_def)
   684 apply (blast intro: nth_type ) 
   685 done
   686 
   687 text{*The subset consisting of the ordinals is definable.*}
   688 lemma Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
   689 apply (simp add: DPow_def Collect_subset) 
   690 apply (rule_tac x="Nil" in bexI) 
   691  apply (rule_tac x="ordinal_fm(0)" in bexI) 
   692 apply (simp_all add: sats_ordinal_fm)
   693 done 
   694 
   695 lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
   696 apply (erule trans_induct3)
   697   apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
   698 txt{*The successor case remains.*} 
   699 apply (rule equalityI)
   700 txt{*First inclusion*}
   701  apply clarify  
   702  apply (erule Ord_linear_lt, assumption) 
   703    apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset]) 
   704   apply blast 
   705  apply (blast dest: ltD)
   706 txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
   707 apply auto
   708 txt{*Key case: *}
   709   apply (erule subst, rule Ords_in_DPow [OF Transset_Lset]) 
   710  apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE) 
   711 apply (blast intro: Ord_in_Ord) 
   712 done
   713 
   714 
   715 lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
   716 by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
   717 
   718 lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
   719 apply (simp add: Lset_succ)
   720 apply (subst Ords_of_Lset_eq [symmetric], assumption, 
   721        rule Ords_in_DPow [OF Transset_Lset]) 
   722 done
   723 
   724 subsubsection{* Unions *}
   725 
   726 lemma Union_in_Lset:
   727      "X \<in> Lset(j) ==> Union(X) \<in> Lset(succ(j))"
   728 apply (insert Transset_Lset)
   729 apply (rule LsetI [OF succI1])
   730 apply (simp add: Transset_def DPow_def) 
   731 apply (intro conjI, blast)
   732 txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
   733 apply (rule_tac x="Cons(X,Nil)" in bexI) 
   734  apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI) 
   735   apply typecheck
   736 apply (simp add: succ_Un_distrib [symmetric], blast) 
   737 done
   738 
   739 lemma Union_in_LLimit:
   740      "[| X: Lset(i);  Limit(i) |] ==> Union(X) : Lset(i)"
   741 apply (rule Limit_LsetE, assumption+)
   742 apply (blast intro: Limit_has_succ lt_LsetI Union_in_Lset)
   743 done
   744 
   745 subsubsection{* Finite sets and ordered pairs *}
   746 
   747 lemma singleton_in_Lset: "a: Lset(i) ==> {a} : Lset(succ(i))"
   748 by (simp add: Lset_succ singleton_in_DPow) 
   749 
   750 lemma doubleton_in_Lset:
   751      "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} : Lset(succ(i))"
   752 by (simp add: Lset_succ empty_in_DPow cons_in_DPow) 
   753 
   754 lemma Pair_in_Lset:
   755     "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> : Lset(succ(succ(i)))"
   756 apply (unfold Pair_def)
   757 apply (blast intro: doubleton_in_Lset) 
   758 done
   759 
   760 lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
   761 
   762 lemma singleton_in_LLimit:
   763     "[| a: Lset(i);  Limit(i) |] ==> {a} : Lset(i)"
   764 apply (erule Limit_LsetE, assumption)
   765 apply (erule singleton_in_Lset [THEN lt_LsetI])
   766 apply (blast intro: Limit_has_succ) 
   767 done
   768 
   769 lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
   770 lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
   771 
   772 text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
   773 lemma doubleton_in_LLimit:
   774     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} : Lset(i)"
   775 apply (erule Limit_LsetE, assumption)
   776 apply (erule Limit_LsetE, assumption)
   777 apply (blast intro:  lt_LsetI [OF doubleton_in_Lset]
   778                      Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
   779 done
   780 
   781 lemma Pair_in_LLimit:
   782     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> : Lset(i)"
   783 txt{*Infer that a, b occur at ordinals x,xa < i.*}
   784 apply (erule Limit_LsetE, assumption)
   785 apply (erule Limit_LsetE, assumption)
   786 txt{*Infer that succ(succ(x Un xa)) < i *}
   787 apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
   788                     Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
   789 done
   790 
   791 lemma product_LLimit: "Limit(i) ==> Lset(i) * Lset(i) <= Lset(i)"
   792 by (blast intro: Pair_in_LLimit)
   793 
   794 lemmas Sigma_subset_LLimit = subset_trans [OF Sigma_mono product_LLimit]
   795 
   796 lemma nat_subset_LLimit: "Limit(i) ==> nat \<subseteq> Lset(i)"
   797 by (blast dest: Ord_subset_Lset nat_le_Limit le_imp_subset Limit_is_Ord)
   798 
   799 lemma nat_into_LLimit: "[| n: nat;  Limit(i) |] ==> n : Lset(i)"
   800 by (blast intro: nat_subset_LLimit [THEN subsetD])
   801 
   802 
   803 subsubsection{* Closure under disjoint union *}
   804 
   805 lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
   806 
   807 lemma one_in_LLimit: "Limit(i) ==> 1 : Lset(i)"
   808 by (blast intro: nat_into_LLimit)
   809 
   810 lemma Inl_in_LLimit:
   811     "[| a: Lset(i); Limit(i) |] ==> Inl(a) : Lset(i)"
   812 apply (unfold Inl_def)
   813 apply (blast intro: zero_in_LLimit Pair_in_LLimit)
   814 done
   815 
   816 lemma Inr_in_LLimit:
   817     "[| b: Lset(i); Limit(i) |] ==> Inr(b) : Lset(i)"
   818 apply (unfold Inr_def)
   819 apply (blast intro: one_in_LLimit Pair_in_LLimit)
   820 done
   821 
   822 lemma sum_LLimit: "Limit(i) ==> Lset(i) + Lset(i) <= Lset(i)"
   823 by (blast intro!: Inl_in_LLimit Inr_in_LLimit)
   824 
   825 lemmas sum_subset_LLimit = subset_trans [OF sum_mono sum_LLimit]
   826 
   827 
   828 text{*The constructible universe and its rank function*}
   829 constdefs
   830   L :: "i=>o" --{*Kunen's definition VI, 1.5, page 167*}
   831     "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
   832   
   833   lrank :: "i=>i" --{*Kunen's definition VI, 1.7*}
   834     "lrank(x) == \<mu>i. x \<in> Lset(succ(i))"
   835 
   836 lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
   837 by (simp add: L_def, blast)
   838 
   839 lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
   840 by (simp add: L_def)
   841 
   842 lemma Ord_lrank [simp]: "Ord(lrank(a))"
   843 by (simp add: lrank_def)
   844 
   845 lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
   846 apply (erule trans_induct3)
   847   apply simp   
   848  apply (simp only: lrank_def) 
   849  apply (blast intro: Least_le) 
   850 apply (simp_all add: Limit_Lset_eq) 
   851 apply (blast intro: ltI Limit_is_Ord lt_trans) 
   852 done
   853 
   854 text{*Kunen's VI, 1.8, and the proof is much less trivial than the text
   855 would suggest.  For a start it need the previous lemma, proved by induction.*}
   856 lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
   857 apply (simp add: L_def, auto) 
   858  apply (blast intro: Lset_lrank_lt) 
   859  apply (unfold lrank_def) 
   860 apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD]) 
   861 apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption) 
   862 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
   863 done
   864 
   865 lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
   866 by (simp add: Lset_iff_lrank_lt)
   867 
   868 text{*Kunen's VI, 1.9 (a)*}
   869 lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
   870 apply (unfold lrank_def) 
   871 apply (rule Least_equality) 
   872   apply (erule Ord_in_Lset) 
   873  apply assumption
   874 apply (insert notin_Lset [of i]) 
   875 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD]) 
   876 done
   877 
   878 
   879 lemma Ord_in_L: "Ord(i) ==> L(i)"
   880 by (blast intro: Ord_in_Lset L_I)
   881 
   882 text{*This is lrank(lrank(a)) = lrank(a) *}
   883 declare Ord_lrank [THEN lrank_of_Ord, simp]
   884 
   885 text{*Kunen's VI, 1.10 *}
   886 lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
   887 apply (simp add: Lset_succ DPow_def) 
   888 apply (rule_tac x="Nil" in bexI) 
   889  apply (rule_tac x="Equal(0,0)" in bexI) 
   890 apply auto 
   891 done
   892 
   893 lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
   894 apply (unfold lrank_def) 
   895 apply (rule Least_equality) 
   896   apply (rule Lset_in_Lset_succ) 
   897  apply assumption
   898 apply clarify 
   899 apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
   900  apply (blast dest: mem_irrefl) 
   901 apply (blast intro!: le_imp_subset Lset_mono) 
   902 done
   903 
   904 text{*Kunen's VI, 1.11 *}
   905 lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
   906 apply (erule trans_induct)
   907 apply (subst Lset) 
   908 apply (subst Vset) 
   909 apply (rule UN_mono [OF subset_refl]) 
   910 apply (rule subset_trans [OF DPow_subset_Pow]) 
   911 apply (rule Pow_mono, blast) 
   912 done
   913 
   914 text{*Kunen's VI, 1.12 *}
   915 lemma Lset_subset_Vset: "i \<in> nat ==> Lset(i) = Vset(i)";
   916 apply (erule nat_induct)
   917  apply (simp add: Vfrom_0) 
   918 apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow) 
   919 done
   920 
   921 subsection{*For L to satisfy the ZF axioms*}
   922 
   923 theorem Union_in_L: "L(X) ==> L(Union(X))"
   924 apply (simp add: L_def, clarify) 
   925 apply (drule Ord_imp_greater_Limit) 
   926 apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord) 
   927 done
   928 
   929 theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
   930 apply (simp add: L_def, clarify) 
   931 apply (drule Ord2_imp_greater_Limit, assumption) 
   932 apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord) 
   933 done
   934 
   935 subsubsection{*For L to satisfy Powerset *}
   936 
   937 lemma LPow_env_typing:
   938      "[| y : Lset(i); Ord(i); y \<subseteq> X |] ==> y \<in> (\<Union>y\<in>Pow(X). Lset(succ(lrank(y))))"
   939 by (auto intro: L_I iff: Lset_succ_lrank_iff) 
   940 
   941 lemma LPow_in_Lset:
   942      "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
   943 apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
   944 apply simp 
   945 apply (rule LsetI [OF succI1])
   946 apply (simp add: DPow_def) 
   947 apply (intro conjI, clarify) 
   948 apply (rule_tac a="x" in UN_I, simp+)  
   949 txt{*Now to create the formula @{term "y \<subseteq> X"} *}
   950 apply (rule_tac x="Cons(X,Nil)" in bexI) 
   951  apply (rule_tac x="subset_fm(0,1)" in bexI) 
   952   apply typecheck
   953 apply (rule conjI) 
   954 apply (simp add: succ_Un_distrib [symmetric]) 
   955 apply (rule equality_iffI) 
   956 apply (simp add: Transset_UN [OF Transset_Lset] list.Cons [OF LPow_env_typing])
   957 apply (auto intro: L_I iff: Lset_succ_lrank_iff) 
   958 done
   959 
   960 theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
   961 by (blast intro: L_I dest: L_D LPow_in_Lset)
   962 
   963 end