src/ZF/Constructible/Formula.thy
 author paulson Wed Jun 19 11:48:01 2002 +0200 (2002-06-19) changeset 13223 45be08fbdcff child 13245 714f7a423a15 permissions -rw-r--r--
new theory of inner models
```     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 (*????????????????Cardinal.ML*)
```
```    26 lemma Finite_cons_iff [iff]:  "Finite(cons(y,x)) <-> Finite(x)"
```
```    27 by (blast intro: Finite_cons subset_Finite)
```
```    28
```
```    29 lemma Finite_succ_iff [iff]:  "Finite(succ(x)) <-> Finite(x)"
```
```    30 by (simp add: succ_def)
```
```    31
```
```    32 declare Finite_0 [simp]
```
```    33
```
```    34 lemma Finite_RepFun: "Finite(A) ==> Finite(RepFun(A,f))"
```
```    35 by (erule Finite_induct, simp_all)
```
```    36
```
```    37 lemma Finite_RepFun_lemma [rule_format]:
```
```    38      "[|Finite(x); !!x y. f(x)=f(y) ==> x=y|]
```
```    39       ==> \<forall>A. x = RepFun(A,f) --> Finite(A)"
```
```    40 apply (erule Finite_induct)
```
```    41  apply clarify
```
```    42  apply (case_tac "A=0", simp)
```
```    43  apply (blast del: allE, clarify)
```
```    44 apply (subgoal_tac "\<exists>z\<in>A. x = f(z)")
```
```    45  prefer 2 apply (blast del: allE elim: equalityE, clarify)
```
```    46 apply (subgoal_tac "B = {f(u) . u \<in> A - {z}}")
```
```    47  apply (blast intro: Diff_sing_Finite)
```
```    48 apply (thin_tac "\<forall>A. ?P(A) --> Finite(A)")
```
```    49 apply (rule equalityI)
```
```    50  apply (blast intro: elim: equalityE)
```
```    51 apply (blast intro: elim: equalityCE)
```
```    52 done
```
```    53
```
```    54 text{*I don't know why, but if the premise is expressed using meta-connectives
```
```    55 then  the simplifier cannot prove it automatically in conditional rewriting.*}
```
```    56 lemma Finite_RepFun_iff:
```
```    57      "(\<forall>x y. f(x)=f(y) --> x=y) ==> Finite(RepFun(A,f)) <-> Finite(A)"
```
```    58 by (blast intro: Finite_RepFun Finite_RepFun_lemma [of _ f])
```
```    59
```
```    60 lemma Finite_Pow: "Finite(A) ==> Finite(Pow(A))"
```
```    61 apply (erule Finite_induct)
```
```    62 apply (simp_all add: Pow_insert Finite_Un Finite_RepFun)
```
```    63 done
```
```    64
```
```    65 lemma Finite_Pow_imp_Finite: "Finite(Pow(A)) ==> Finite(A)"
```
```    66 apply (subgoal_tac "Finite({{x} . x \<in> A})")
```
```    67  apply (simp add: Finite_RepFun_iff )
```
```    68 apply (blast intro: subset_Finite)
```
```    69 done
```
```    70
```
```    71 lemma Finite_Pow_iff [iff]: "Finite(Pow(A)) <-> Finite(A)"
```
```    72 by (blast intro: Finite_Pow Finite_Pow_imp_Finite)
```
```    73
```
```    74 lemma Finite_Vset: "i \<in> nat ==> Finite(Vset(i))";
```
```    75 apply (erule nat_induct)
```
```    76  apply (simp add: Vfrom_0)
```
```    77 apply (simp add: Vset_succ)
```
```    78 done
```
```    79
```
```    80 (*???Ordinal maybe, but some lemmas seem to be in CardinalArith??*)
```
```    81 text{*Every ordinal is exceeded by some limit ordinal.*}
```
```    82 lemma Ord_imp_greater_Limit: "Ord(i) ==> \<exists>k. i<k & Limit(k)"
```
```    83 apply (rule_tac x="i ++ nat" in exI)
```
```    84 apply (blast intro: oadd_LimitI  oadd_lt_self  Limit_nat [THEN Limit_has_0])
```
```    85 done
```
```    86
```
```    87 lemma Ord2_imp_greater_Limit: "[|Ord(i); Ord(j)|] ==> \<exists>k. i<k & j<k & Limit(k)"
```
```    88 apply (insert Ord_Un [of i j, THEN Ord_imp_greater_Limit])
```
```    89 apply (simp add: Un_least_lt_iff)
```
```    90 done
```
```    91
```
```    92
```
```    93
```
```    94 (*Internalized formulas of FOL. De Bruijn representation.
```
```    95   Unbound variables get their denotations from an environment.*)
```
```    96
```
```    97 consts   formula :: i
```
```    98 datatype
```
```    99   "formula" = Member ("x: nat", "y: nat")
```
```   100             | Equal  ("x: nat", "y: nat")
```
```   101             | Neg ("p: formula")
```
```   102             | And ("p: formula", "q: formula")
```
```   103             | Forall ("p: formula")
```
```   104
```
```   105 declare formula.intros [TC]
```
```   106
```
```   107 constdefs Or :: "[i,i]=>i"
```
```   108     "Or(p,q) == Neg(And(Neg(p),Neg(q)))"
```
```   109
```
```   110 constdefs Implies :: "[i,i]=>i"
```
```   111     "Implies(p,q) == Neg(And(p,Neg(q)))"
```
```   112
```
```   113 constdefs Exists :: "i=>i"
```
```   114     "Exists(p) == Neg(Forall(Neg(p)))";
```
```   115
```
```   116 lemma Or_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> Or(p,q) \<in> formula"
```
```   117 by (simp add: Or_def)
```
```   118
```
```   119 lemma Implies_type [TC]:
```
```   120      "[| p \<in> formula; q \<in> formula |] ==> Implies(p,q) \<in> formula"
```
```   121 by (simp add: Implies_def)
```
```   122
```
```   123 lemma Exists_type [TC]: "p \<in> formula ==> Exists(p) \<in> formula"
```
```   124 by (simp add: Exists_def)
```
```   125
```
```   126
```
```   127 consts   satisfies :: "[i,i]=>i"
```
```   128 primrec (*explicit lambda is required because the environment varies*)
```
```   129   "satisfies(A,Member(x,y)) =
```
```   130       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) \<in> nth(y,env)))"
```
```   131
```
```   132   "satisfies(A,Equal(x,y)) =
```
```   133       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) = nth(y,env)))"
```
```   134
```
```   135   "satisfies(A,Neg(p)) =
```
```   136       (\<lambda>env \<in> list(A). not(satisfies(A,p)`env))"
```
```   137
```
```   138   "satisfies(A,And(p,q)) =
```
```   139       (\<lambda>env \<in> list(A). (satisfies(A,p)`env) and (satisfies(A,q)`env))"
```
```   140
```
```   141   "satisfies(A,Forall(p)) =
```
```   142       (\<lambda>env \<in> list(A). bool_of_o (\<forall>x\<in>A. satisfies(A,p) ` (Cons(x,env)) = 1))"
```
```   143
```
```   144
```
```   145 lemma "p \<in> formula ==> satisfies(A,p) \<in> list(A) -> bool"
```
```   146 by (induct_tac p, simp_all)
```
```   147
```
```   148 syntax sats :: "[i,i,i] => o"
```
```   149 translations "sats(A,p,env)" == "satisfies(A,p)`env = 1"
```
```   150
```
```   151 lemma [simp]:
```
```   152   "env \<in> list(A)
```
```   153    ==> sats(A, Member(x,y), env) <-> nth(x,env) \<in> nth(y,env)"
```
```   154 by simp
```
```   155
```
```   156 lemma [simp]:
```
```   157   "env \<in> list(A)
```
```   158    ==> sats(A, Equal(x,y), env) <-> nth(x,env) = nth(y,env)"
```
```   159 by simp
```
```   160
```
```   161 lemma sats_Neg_iff [simp]:
```
```   162   "env \<in> list(A)
```
```   163    ==> sats(A, Neg(p), env) <-> ~ sats(A,p,env)"
```
```   164 by (simp add: Bool.not_def cond_def)
```
```   165
```
```   166 lemma sats_And_iff [simp]:
```
```   167   "env \<in> list(A)
```
```   168    ==> (sats(A, And(p,q), env)) <-> sats(A,p,env) & sats(A,q,env)"
```
```   169 by (simp add: Bool.and_def cond_def)
```
```   170
```
```   171 lemma sats_Forall_iff [simp]:
```
```   172   "env \<in> list(A)
```
```   173    ==> sats(A, Forall(p), env) <-> (\<forall>x\<in>A. sats(A, p, Cons(x,env)))"
```
```   174 by simp
```
```   175
```
```   176 declare satisfies.simps [simp del];
```
```   177
```
```   178 (**** DIVIDING LINE BETWEEN PRIMITIVE AND DERIVED CONNECTIVES ****)
```
```   179
```
```   180 lemma sats_Or_iff [simp]:
```
```   181   "env \<in> list(A)
```
```   182    ==> (sats(A, Or(p,q), env)) <-> sats(A,p,env) | sats(A,q,env)"
```
```   183 by (simp add: Or_def)
```
```   184
```
```   185 lemma sats_Implies_iff [simp]:
```
```   186   "env \<in> list(A)
```
```   187    ==> (sats(A, Implies(p,q), env)) <-> (sats(A,p,env) --> sats(A,q,env))"
```
```   188 apply (simp add: Implies_def, blast)
```
```   189 done
```
```   190
```
```   191 lemma sats_Exists_iff [simp]:
```
```   192   "env \<in> list(A)
```
```   193    ==> sats(A, Exists(p), env) <-> (\<exists>x\<in>A. sats(A, p, Cons(x,env)))"
```
```   194 by (simp add: Exists_def)
```
```   195
```
```   196
```
```   197
```
```   198
```
```   199 (*pretty but unnecessary
```
```   200 constdefs sat     :: "[i,i] => o"
```
```   201   "sat(A,p) == satisfies(A,p)`[] = 1"
```
```   202
```
```   203 syntax "_sat"  :: "[i,i] => o"    (infixl "|=" 50)
```
```   204 translations "A |= p" == "sat(A,p)"
```
```   205
```
```   206 lemma [simp]: "(A |= Neg(p)) <-> ~ (A |= p)"
```
```   207 by (simp add: sat_def)
```
```   208
```
```   209 lemma [simp]: "(A |= And(p,q)) <-> A|=p & A|=q"
```
```   210 by (simp add: sat_def)
```
```   211 *)
```
```   212
```
```   213
```
```   214 constdefs incr_var :: "[i,i]=>i"
```
```   215     "incr_var(x,lev) == if x<lev then x else succ(x)"
```
```   216
```
```   217 lemma incr_var_lt: "x<lev ==> incr_var(x,lev) = x"
```
```   218 by (simp add: incr_var_def)
```
```   219
```
```   220 lemma incr_var_le: "lev\<le>x ==> incr_var(x,lev) = succ(x)"
```
```   221 apply (simp add: incr_var_def)
```
```   222 apply (blast dest: lt_trans1)
```
```   223 done
```
```   224
```
```   225 consts   incr_bv :: "i=>i"
```
```   226 primrec
```
```   227   "incr_bv(Member(x,y)) =
```
```   228       (\<lambda>lev \<in> nat. Member (incr_var(x,lev), incr_var(y,lev)))"
```
```   229
```
```   230   "incr_bv(Equal(x,y)) =
```
```   231       (\<lambda>lev \<in> nat. Equal (incr_var(x,lev), incr_var(y,lev)))"
```
```   232
```
```   233   "incr_bv(Neg(p)) =
```
```   234       (\<lambda>lev \<in> nat. Neg(incr_bv(p)`lev))"
```
```   235
```
```   236   "incr_bv(And(p,q)) =
```
```   237       (\<lambda>lev \<in> nat. And (incr_bv(p)`lev, incr_bv(q)`lev))"
```
```   238
```
```   239   "incr_bv(Forall(p)) =
```
```   240       (\<lambda>lev \<in> nat. Forall (incr_bv(p) ` succ(lev)))"
```
```   241
```
```   242
```
```   243 constdefs incr_boundvars :: "i => i"
```
```   244     "incr_boundvars(p) == incr_bv(p)`0"
```
```   245
```
```   246
```
```   247 lemma [TC]: "x \<in> nat ==> incr_var(x,lev) \<in> nat"
```
```   248 by (simp add: incr_var_def)
```
```   249
```
```   250 lemma incr_bv_type [TC]: "p \<in> formula ==> incr_bv(p) \<in> nat -> formula"
```
```   251 by (induct_tac p, simp_all)
```
```   252
```
```   253 lemma incr_boundvars_type [TC]: "p \<in> formula ==> incr_boundvars(p) \<in> formula"
```
```   254 by (simp add: incr_boundvars_def)
```
```   255
```
```   256 (*Obviously DPow is closed under complements and finite intersections and
```
```   257 unions.  Needs an inductive lemma to allow two lists of parameters to
```
```   258 be combined.*)
```
```   259
```
```   260 lemma sats_incr_bv_iff [rule_format]:
```
```   261   "[| p \<in> formula; env \<in> list(A); x \<in> A |]
```
```   262    ==> \<forall>bvs \<in> list(A).
```
```   263            sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x,env)) <->
```
```   264            sats(A, p, bvs@env)"
```
```   265 apply (induct_tac p)
```
```   266 apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type)
```
```   267 apply (auto simp add: diff_succ not_lt_iff_le)
```
```   268 done
```
```   269
```
```   270 (*UNUSED*)
```
```   271 lemma sats_incr_boundvars_iff:
```
```   272   "[| p \<in> formula; env \<in> list(A); x \<in> A |]
```
```   273    ==> sats(A, incr_boundvars(p), Cons(x,env)) <-> sats(A, p, env)"
```
```   274 apply (insert sats_incr_bv_iff [of p env A x Nil])
```
```   275 apply (simp add: incr_boundvars_def)
```
```   276 done
```
```   277
```
```   278 (*UNUSED
```
```   279 lemma formula_add_params [rule_format]:
```
```   280   "[| p \<in> formula; n \<in> nat |]
```
```   281    ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A).
```
```   282          length(bvs) = n -->
```
```   283          sats(A, iterates(incr_boundvars,n,p), bvs@env) <-> sats(A, p, env)"
```
```   284 apply (induct_tac n, simp, clarify)
```
```   285 apply (erule list.cases)
```
```   286 apply (auto simp add: sats_incr_boundvars_iff)
```
```   287 done
```
```   288 *)
```
```   289
```
```   290 consts   arity :: "i=>i"
```
```   291 primrec
```
```   292   "arity(Member(x,y)) = succ(x) \<union> succ(y)"
```
```   293
```
```   294   "arity(Equal(x,y)) = succ(x) \<union> succ(y)"
```
```   295
```
```   296   "arity(Neg(p)) = arity(p)"
```
```   297
```
```   298   "arity(And(p,q)) = arity(p) \<union> arity(q)"
```
```   299
```
```   300   "arity(Forall(p)) = nat_case3(0, %x. x, arity(p))"
```
```   301
```
```   302
```
```   303 lemma arity_type [TC]: "p \<in> formula ==> arity(p) \<in> nat"
```
```   304 by (induct_tac p, simp_all)
```
```   305
```
```   306 lemma arity_Or [simp]: "arity(Or(p,q)) = arity(p) \<union> arity(q)"
```
```   307 by (simp add: Or_def)
```
```   308
```
```   309 lemma arity_Implies [simp]: "arity(Implies(p,q)) = arity(p) \<union> arity(q)"
```
```   310 by (simp add: Implies_def)
```
```   311
```
```   312 lemma arity_Exists [simp]: "arity(Exists(p)) = nat_case3(0, %x. x, arity(p))"
```
```   313 by (simp add: Exists_def)
```
```   314
```
```   315
```
```   316 lemma arity_sats_iff [rule_format]:
```
```   317   "[| p \<in> formula; extra \<in> list(A) |]
```
```   318    ==> \<forall>env \<in> list(A).
```
```   319            arity(p) \<le> length(env) -->
```
```   320            sats(A, p, env @ extra) <-> sats(A, p, env)"
```
```   321 apply (induct_tac p)
```
```   322 apply (simp_all add: nth_append Un_least_lt_iff arity_type
```
```   323                 split: split_nat_case3, auto)
```
```   324 done
```
```   325
```
```   326 lemma arity_sats1_iff:
```
```   327   "[| arity(p) \<le> succ(length(env)); p \<in> formula; x \<in> A; env \<in> list(A);
```
```   328     extra \<in> list(A) |]
```
```   329    ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
```
```   330 apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
```
```   331 apply simp
```
```   332 done
```
```   333
```
```   334 (*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
```
```   335 lemma incr_var_lemma:
```
```   336      "[| x \<in> nat; y \<in> nat; lev \<le> x |]
```
```   337       ==> succ(x) \<union> incr_var(y,lev) = succ(x \<union> y)"
```
```   338 apply (simp add: incr_var_def Ord_Un_if, auto)
```
```   339   apply (blast intro: leI)
```
```   340  apply (simp add: not_lt_iff_le)
```
```   341  apply (blast intro: le_anti_sym)
```
```   342 apply (blast dest: lt_trans2)
```
```   343 done
```
```   344
```
```   345 lemma incr_And_lemma:
```
```   346      "y < x ==> y \<union> succ(x) = succ(x \<union> y)"
```
```   347 apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff)
```
```   348 apply (blast dest: lt_asym)
```
```   349 done
```
```   350
```
```   351 lemma arity_incr_bv_lemma [rule_format]:
```
```   352   "p \<in> formula
```
```   353    ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) =
```
```   354                  (if n < arity(p) then succ(arity(p)) else arity(p))"
```
```   355 apply (induct_tac p)
```
```   356 apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
```
```   357                      succ_Un_distrib [symmetric] incr_var_lt incr_var_le
```
```   358                      Un_commute incr_var_lemma arity_type
```
```   359             split: split_nat_case3)
```
```   360 (*left with the And case*)
```
```   361 apply safe
```
```   362  apply (blast intro: incr_And_lemma lt_trans1)
```
```   363 apply (subst incr_And_lemma)
```
```   364  apply (blast intro:  lt_trans1)
```
```   365 apply (simp add:  Un_commute)
```
```   366 done
```
```   367
```
```   368 lemma arity_incr_boundvars_eq:
```
```   369   "p \<in> formula
```
```   370    ==> arity(incr_boundvars(p)) =
```
```   371         (if 0 < arity(p) then succ(arity(p)) else arity(p))"
```
```   372 apply (insert arity_incr_bv_lemma [of p 0])
```
```   373 apply (simp add: incr_boundvars_def)
```
```   374 done
```
```   375
```
```   376 lemma arity_iterates_incr_boundvars_eq:
```
```   377   "[| p \<in> formula; n \<in> nat |]
```
```   378    ==> arity(incr_boundvars^n(p)) =
```
```   379          (if 0 < arity(p) then n #+ arity(p) else arity(p))"
```
```   380 apply (induct_tac n)
```
```   381 apply (simp_all add: arity_incr_boundvars_eq not_lt_iff_le)
```
```   382 done
```
```   383
```
```   384
```
```   385 (**** TRYING INCR_BV1 AGAIN ****)
```
```   386
```
```   387 constdefs incr_bv1 :: "i => i"
```
```   388     "incr_bv1(p) == incr_bv(p)`1"
```
```   389
```
```   390
```
```   391 lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
```
```   392 by (simp add: incr_bv1_def)
```
```   393
```
```   394 (*For renaming all but the bound variable at level 0*)
```
```   395 lemma sats_incr_bv1_iff [rule_format]:
```
```   396   "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
```
```   397    ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <->
```
```   398        sats(A, p, Cons(x,env))"
```
```   399 apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
```
```   400 apply (simp add: incr_bv1_def)
```
```   401 done
```
```   402
```
```   403 lemma formula_add_params1 [rule_format]:
```
```   404   "[| p \<in> formula; n \<in> nat; x \<in> A |]
```
```   405    ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A).
```
```   406           length(bvs) = n -->
```
```   407           sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <->
```
```   408           sats(A, p, Cons(x,env))"
```
```   409 apply (induct_tac n, simp, clarify)
```
```   410 apply (erule list.cases)
```
```   411 apply (simp_all add: sats_incr_bv1_iff)
```
```   412 done
```
```   413
```
```   414
```
```   415 lemma arity_incr_bv1_eq:
```
```   416   "p \<in> formula
```
```   417    ==> arity(incr_bv1(p)) =
```
```   418         (if 1 < arity(p) then succ(arity(p)) else arity(p))"
```
```   419 apply (insert arity_incr_bv_lemma [of p 1])
```
```   420 apply (simp add: incr_bv1_def)
```
```   421 done
```
```   422
```
```   423 lemma arity_iterates_incr_bv1_eq:
```
```   424   "[| p \<in> formula; n \<in> nat |]
```
```   425    ==> arity(incr_bv1^n(p)) =
```
```   426          (if 1 < arity(p) then n #+ arity(p) else arity(p))"
```
```   427 apply (induct_tac n)
```
```   428 apply (simp_all add: arity_incr_bv1_eq )
```
```   429 apply (simp add: not_lt_iff_le)
```
```   430 apply (blast intro: le_trans add_le_self2 arity_type)
```
```   431 done
```
```   432
```
```   433
```
```   434 (*Definable powerset operation: Kunen's definition 1.1, page 165.*)
```
```   435 constdefs DPow :: "i => i"
```
```   436   "DPow(A) == {X \<in> Pow(A).
```
```   437                \<exists>env \<in> list(A). \<exists>p \<in> formula.
```
```   438                  arity(p) \<le> succ(length(env)) &
```
```   439                  X = {x\<in>A. sats(A, p, Cons(x,env))}}"
```
```   440
```
```   441 lemma DPowI:
```
```   442   "[|X <= A;  env \<in> list(A);  p \<in> formula;
```
```   443      arity(p) \<le> succ(length(env))|]
```
```   444    ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
```
```   445 by (simp add: DPow_def, blast)
```
```   446
```
```   447 lemma DPowD:
```
```   448   "X \<in> DPow(A)
```
```   449    ==> X <= A &
```
```   450        (\<exists>env \<in> list(A).
```
```   451         \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) &
```
```   452                       X = {x\<in>A. sats(A, p, Cons(x,env))})"
```
```   453 by (simp add: DPow_def)
```
```   454
```
```   455 lemmas DPow_imp_subset = DPowD [THEN conjunct1]
```
```   456
```
```   457 (*Lemma 1.2*)
```
```   458 lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |]
```
```   459        ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
```
```   460 by (blast intro: DPowI)
```
```   461
```
```   462 lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
```
```   463 by (simp add: DPow_def, blast)
```
```   464
```
```   465 lemma empty_in_DPow: "0 \<in> DPow(A)"
```
```   466 apply (simp add: DPow_def)
```
```   467 apply (rule_tac x="Nil" in bexI)
```
```   468  apply (rule_tac x="Neg(Equal(0,0))" in bexI)
```
```   469   apply (auto simp add: Un_least_lt_iff)
```
```   470 done
```
```   471
```
```   472 lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
```
```   473 apply (simp add: DPow_def, clarify, auto)
```
```   474 apply (rule bexI)
```
```   475  apply (rule_tac x="Neg(p)" in bexI)
```
```   476   apply auto
```
```   477 done
```
```   478
```
```   479 lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
```
```   480 apply (simp add: DPow_def, auto)
```
```   481 apply (rename_tac envp p envq q)
```
```   482 apply (rule_tac x="envp@envq" in bexI)
```
```   483  apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
```
```   484   apply typecheck
```
```   485 apply (rule conjI)
```
```   486 (*finally check the arity!*)
```
```   487  apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
```
```   488  apply (force intro: add_le_self le_trans)
```
```   489 apply (simp add: arity_sats1_iff formula_add_params1, blast)
```
```   490 done
```
```   491
```
```   492 lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
```
```   493 apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))")
```
```   494 apply (simp add: Int_in_DPow Compl_in_DPow)
```
```   495 apply (simp add: DPow_def, blast)
```
```   496 done
```
```   497
```
```   498 lemma singleton_in_DPow: "x \<in> A ==> {x} \<in> DPow(A)"
```
```   499 apply (simp add: DPow_def)
```
```   500 apply (rule_tac x="Cons(x,Nil)" in bexI)
```
```   501  apply (rule_tac x="Equal(0,1)" in bexI)
```
```   502   apply typecheck
```
```   503 apply (force simp add: succ_Un_distrib [symmetric])
```
```   504 done
```
```   505
```
```   506 lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
```
```   507 apply (rule cons_eq [THEN subst])
```
```   508 apply (blast intro: singleton_in_DPow Un_in_DPow)
```
```   509 done
```
```   510
```
```   511 (*Part of Lemma 1.3*)
```
```   512 lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
```
```   513 apply (erule Fin.induct)
```
```   514  apply (rule empty_in_DPow)
```
```   515 apply (blast intro: cons_in_DPow)
```
```   516 done
```
```   517
```
```   518 (*DPow is not monotonic.  For example, let A be some non-constructible set
```
```   519   of natural numbers, and let B be nat.  Then A<=B and obviously A : DPow(A)
```
```   520   but A ~: DPow(B).*)
```
```   521 lemma DPow_mono: "A : DPow(B) ==> DPow(A) <= DPow(B)"
```
```   522 apply (simp add: DPow_def, auto)
```
```   523 (*must use the formula defining A in B to relativize the new formula...*)
```
```   524 oops
```
```   525
```
```   526 lemma DPow_0: "DPow(0) = {0}"
```
```   527 by (blast intro: empty_in_DPow dest: DPow_imp_subset)
```
```   528
```
```   529 lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)"
```
```   530 by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
```
```   531
```
```   532 lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
```
```   533 apply (rule equalityI)
```
```   534 apply (rule DPow_subset_Pow)
```
```   535 apply (erule Finite_Pow_subset_Pow)
```
```   536 done
```
```   537
```
```   538 (*This may be true but the proof looks difficult, requiring relativization
```
```   539 lemma DPow_insert: "DPow (cons(a,A)) = DPow(A) Un {cons(a,X) . X: DPow(A)}"
```
```   540 apply (rule equalityI, safe)
```
```   541 oops
```
```   542 *)
```
```   543
```
```   544 subsection{* Constant Lset: Levels of the Constructible Universe *}
```
```   545
```
```   546 constdefs Lset :: "i=>i"
```
```   547     "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
```
```   548
```
```   549 text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
```
```   550 lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
```
```   551 by (subst Lset_def [THEN def_transrec], simp)
```
```   552
```
```   553 lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
```
```   554 by (subst Lset, blast)
```
```   555
```
```   556 lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
```
```   557 apply (insert Lset [of x])
```
```   558 apply (blast intro: elim: equalityE)
```
```   559 done
```
```   560
```
```   561 subsubsection{* Transitivity *}
```
```   562
```
```   563 lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
```
```   564 apply (simp add: Transset_def DPow_def)
```
```   565 apply (rule_tac x="[X]" in bexI)
```
```   566  apply (rule_tac x="Member(0,1)" in bexI)
```
```   567   apply (auto simp add: Un_least_lt_iff)
```
```   568 done
```
```   569
```
```   570 lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
```
```   571 apply clarify
```
```   572 apply (simp add: Transset_def)
```
```   573 apply (blast intro: elem_subset_in_DPow)
```
```   574 done
```
```   575
```
```   576 lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
```
```   577 apply (simp add: Transset_def)
```
```   578 apply (blast intro: elem_subset_in_DPow dest: DPowD)
```
```   579 done
```
```   580
```
```   581 text{*Kunen's VI, 1.6 (a)*}
```
```   582 lemma Transset_Lset: "Transset(Lset(i))"
```
```   583 apply (rule_tac a=i in eps_induct)
```
```   584 apply (subst Lset)
```
```   585 apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
```
```   586 done
```
```   587
```
```   588 subsubsection{* Monotonicity *}
```
```   589
```
```   590 text{*Kunen's VI, 1.6 (b)*}
```
```   591 lemma Lset_mono [rule_format]:
```
```   592      "ALL j. i<=j --> Lset(i) <= Lset(j)"
```
```   593 apply (rule_tac a=i in eps_induct)
```
```   594 apply (rule impI [THEN allI])
```
```   595 apply (subst Lset)
```
```   596 apply (subst Lset, blast)
```
```   597 done
```
```   598
```
```   599 text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
```
```   600 lemma Lset_mono_mem [rule_format]:
```
```   601      "ALL j. i:j --> Lset(i) <= Lset(j)"
```
```   602 apply (rule_tac a=i in eps_induct)
```
```   603 apply (rule impI [THEN allI])
```
```   604 apply (subst Lset, auto)
```
```   605 apply (rule rev_bexI, assumption)
```
```   606 apply (blast intro: elem_subset_in_DPow dest: LsetD DPowD)
```
```   607 done
```
```   608
```
```   609 subsubsection{* 0, successor and limit equations fof Lset *}
```
```   610
```
```   611 lemma Lset_0 [simp]: "Lset(0) = 0"
```
```   612 by (subst Lset, blast)
```
```   613
```
```   614 lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
```
```   615 by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
```
```   616
```
```   617 lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
```
```   618 apply (subst Lset, rule UN_least)
```
```   619 apply (erule succE)
```
```   620  apply blast
```
```   621 apply clarify
```
```   622 apply (rule elem_subset_in_DPow)
```
```   623  apply (subst Lset)
```
```   624  apply blast
```
```   625 apply (blast intro: dest: DPowD Lset_mono_mem)
```
```   626 done
```
```   627
```
```   628 lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
```
```   629 by (intro equalityI Lset_succ_subset1 Lset_succ_subset2)
```
```   630
```
```   631 lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
```
```   632 apply (subst Lset)
```
```   633 apply (rule equalityI)
```
```   634  txt{*first inclusion*}
```
```   635  apply (rule UN_least)
```
```   636  apply (erule UnionE)
```
```   637  apply (rule subset_trans)
```
```   638   apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
```
```   639 txt{*opposite inclusion*}
```
```   640 apply (rule UN_least)
```
```   641 apply (subst Lset, blast)
```
```   642 done
```
```   643
```
```   644 subsubsection{* Lset applied to Limit ordinals *}
```
```   645
```
```   646 lemma Limit_Lset_eq:
```
```   647     "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
```
```   648 by (simp add: Lset_Union [symmetric] Limit_Union_eq)
```
```   649
```
```   650 lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a : Lset(i)"
```
```   651 by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
```
```   652
```
```   653 lemma Limit_LsetE:
```
```   654     "[| a: Lset(i);  ~R ==> Limit(i);
```
```   655         !!x. [| x<i;  a: Lset(x) |] ==> R
```
```   656      |] ==> R"
```
```   657 apply (rule classical)
```
```   658 apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
```
```   659   prefer 2 apply assumption
```
```   660  apply blast
```
```   661 apply (blast intro: ltI  Limit_is_Ord)
```
```   662 done
```
```   663
```
```   664 subsubsection{* Basic closure properties *}
```
```   665
```
```   666 lemma zero_in_Lset: "y:x ==> 0 : Lset(x)"
```
```   667 by (subst Lset, blast intro: empty_in_DPow)
```
```   668
```
```   669 lemma notin_Lset: "x \<notin> Lset(x)"
```
```   670 apply (rule_tac a=x in eps_induct)
```
```   671 apply (subst Lset)
```
```   672 apply (blast dest: DPowD)
```
```   673 done
```
```   674
```
```   675
```
```   676
```
```   677 text{*Kunen's VI, 1.9 (b)*}
```
```   678
```
```   679 constdefs subset_fm :: "[i,i]=>i"
```
```   680     "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
```
```   681
```
```   682 lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
```
```   683 by (simp add: subset_fm_def)
```
```   684
```
```   685 lemma arity_subset_fm [simp]:
```
```   686      "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   687 by (simp add: subset_fm_def succ_Un_distrib [symmetric])
```
```   688
```
```   689 lemma sats_subset_fm [simp]:
```
```   690    "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
```
```   691     ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
```
```   692 apply (frule lt_nat_in_nat, erule length_type)
```
```   693 apply (simp add: subset_fm_def Transset_def)
```
```   694 apply (blast intro: nth_type )
```
```   695 done
```
```   696
```
```   697 constdefs transset_fm :: "i=>i"
```
```   698    "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
```
```   699
```
```   700 lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
```
```   701 by (simp add: transset_fm_def)
```
```   702
```
```   703 lemma arity_transset_fm [simp]:
```
```   704      "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
```
```   705 by (simp add: transset_fm_def succ_Un_distrib [symmetric])
```
```   706
```
```   707 lemma sats_transset_fm [simp]:
```
```   708    "[|x < length(env); env \<in> list(A); Transset(A)|]
```
```   709     ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
```
```   710 apply (frule lt_nat_in_nat, erule length_type)
```
```   711 apply (simp add: transset_fm_def Transset_def)
```
```   712 apply (blast intro: nth_type )
```
```   713 done
```
```   714
```
```   715 constdefs ordinal_fm :: "i=>i"
```
```   716    "ordinal_fm(x) ==
```
```   717       And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
```
```   718
```
```   719 lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
```
```   720 by (simp add: ordinal_fm_def)
```
```   721
```
```   722 lemma arity_ordinal_fm [simp]:
```
```   723      "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
```
```   724 by (simp add: ordinal_fm_def succ_Un_distrib [symmetric])
```
```   725
```
```   726 lemma sats_ordinal_fm [simp]:
```
```   727    "[|x < length(env); env \<in> list(A); Transset(A)|]
```
```   728     ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
```
```   729 apply (frule lt_nat_in_nat, erule length_type)
```
```   730 apply (simp add: ordinal_fm_def Ord_def Transset_def)
```
```   731 apply (blast intro: nth_type )
```
```   732 done
```
```   733
```
```   734 text{*The subset consisting of the ordinals is definable.*}
```
```   735 lemma Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
```
```   736 apply (simp add: DPow_def Collect_subset)
```
```   737 apply (rule_tac x="Nil" in bexI)
```
```   738  apply (rule_tac x="ordinal_fm(0)" in bexI)
```
```   739 apply (simp_all add: sats_ordinal_fm)
```
```   740 done
```
```   741
```
```   742 lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
```
```   743 apply (erule trans_induct3)
```
```   744   apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
```
```   745 txt{*The successor case remains.*}
```
```   746 apply (rule equalityI)
```
```   747 txt{*First inclusion*}
```
```   748  apply clarify
```
```   749  apply (erule Ord_linear_lt, assumption)
```
```   750    apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset])
```
```   751   apply blast
```
```   752  apply (blast dest: ltD)
```
```   753 txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
```
```   754 apply auto
```
```   755 txt{*Key case: *}
```
```   756   apply (erule subst, rule Ords_in_DPow [OF Transset_Lset])
```
```   757  apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE)
```
```   758 apply (blast intro: Ord_in_Ord)
```
```   759 done
```
```   760
```
```   761
```
```   762 lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
```
```   763 by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
```
```   764
```
```   765 lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
```
```   766 apply (simp add: Lset_succ)
```
```   767 apply (subst Ords_of_Lset_eq [symmetric], assumption,
```
```   768        rule Ords_in_DPow [OF Transset_Lset])
```
```   769 done
```
```   770
```
```   771 subsubsection{* Unions *}
```
```   772
```
```   773 lemma Union_in_Lset:
```
```   774      "X \<in> Lset(j) ==> Union(X) \<in> Lset(succ(j))"
```
```   775 apply (insert Transset_Lset)
```
```   776 apply (rule LsetI [OF succI1])
```
```   777 apply (simp add: Transset_def DPow_def)
```
```   778 apply (intro conjI, blast)
```
```   779 txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
```
```   780 apply (rule_tac x="Cons(X,Nil)" in bexI)
```
```   781  apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI)
```
```   782   apply typecheck
```
```   783 apply (simp add: succ_Un_distrib [symmetric], blast)
```
```   784 done
```
```   785
```
```   786 lemma Union_in_LLimit:
```
```   787      "[| X: Lset(i);  Limit(i) |] ==> Union(X) : Lset(i)"
```
```   788 apply (rule Limit_LsetE, assumption+)
```
```   789 apply (blast intro: Limit_has_succ lt_LsetI Union_in_Lset)
```
```   790 done
```
```   791
```
```   792 subsubsection{* Finite sets and ordered pairs *}
```
```   793
```
```   794 lemma singleton_in_Lset: "a: Lset(i) ==> {a} : Lset(succ(i))"
```
```   795 by (simp add: Lset_succ singleton_in_DPow)
```
```   796
```
```   797 lemma doubleton_in_Lset:
```
```   798      "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} : Lset(succ(i))"
```
```   799 by (simp add: Lset_succ empty_in_DPow cons_in_DPow)
```
```   800
```
```   801 lemma Pair_in_Lset:
```
```   802     "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> : Lset(succ(succ(i)))"
```
```   803 apply (unfold Pair_def)
```
```   804 apply (blast intro: doubleton_in_Lset)
```
```   805 done
```
```   806
```
```   807 lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
```
```   808
```
```   809 lemma singleton_in_LLimit:
```
```   810     "[| a: Lset(i);  Limit(i) |] ==> {a} : Lset(i)"
```
```   811 apply (erule Limit_LsetE, assumption)
```
```   812 apply (erule singleton_in_Lset [THEN lt_LsetI])
```
```   813 apply (blast intro: Limit_has_succ)
```
```   814 done
```
```   815
```
```   816 lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
```
```   817 lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
```
```   818
```
```   819 text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
```
```   820 lemma doubleton_in_LLimit:
```
```   821     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} : Lset(i)"
```
```   822 apply (erule Limit_LsetE, assumption)
```
```   823 apply (erule Limit_LsetE, assumption)
```
```   824 apply (blast intro:  lt_LsetI [OF doubleton_in_Lset]
```
```   825                      Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
```
```   826 done
```
```   827
```
```   828 lemma Pair_in_LLimit:
```
```   829     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> : Lset(i)"
```
```   830 txt{*Infer that a, b occur at ordinals x,xa < i.*}
```
```   831 apply (erule Limit_LsetE, assumption)
```
```   832 apply (erule Limit_LsetE, assumption)
```
```   833 txt{*Infer that succ(succ(x Un xa)) < i *}
```
```   834 apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
```
```   835                     Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
```
```   836 done
```
```   837
```
```   838 lemma product_LLimit: "Limit(i) ==> Lset(i) * Lset(i) <= Lset(i)"
```
```   839 by (blast intro: Pair_in_LLimit)
```
```   840
```
```   841 lemmas Sigma_subset_LLimit = subset_trans [OF Sigma_mono product_LLimit]
```
```   842
```
```   843 lemma nat_subset_LLimit: "Limit(i) ==> nat \<subseteq> Lset(i)"
```
```   844 by (blast dest: Ord_subset_Lset nat_le_Limit le_imp_subset Limit_is_Ord)
```
```   845
```
```   846 lemma nat_into_LLimit: "[| n: nat;  Limit(i) |] ==> n : Lset(i)"
```
```   847 by (blast intro: nat_subset_LLimit [THEN subsetD])
```
```   848
```
```   849
```
```   850 subsubsection{* Closure under disjoint union *}
```
```   851
```
```   852 lemmas zero_in_LLimit = Limit_has_0 [THEN ltD, THEN zero_in_Lset, standard]
```
```   853
```
```   854 lemma one_in_LLimit: "Limit(i) ==> 1 : Lset(i)"
```
```   855 by (blast intro: nat_into_LLimit)
```
```   856
```
```   857 lemma Inl_in_LLimit:
```
```   858     "[| a: Lset(i); Limit(i) |] ==> Inl(a) : Lset(i)"
```
```   859 apply (unfold Inl_def)
```
```   860 apply (blast intro: zero_in_LLimit Pair_in_LLimit)
```
```   861 done
```
```   862
```
```   863 lemma Inr_in_LLimit:
```
```   864     "[| b: Lset(i); Limit(i) |] ==> Inr(b) : Lset(i)"
```
```   865 apply (unfold Inr_def)
```
```   866 apply (blast intro: one_in_LLimit Pair_in_LLimit)
```
```   867 done
```
```   868
```
```   869 lemma sum_LLimit: "Limit(i) ==> Lset(i) + Lset(i) <= Lset(i)"
```
```   870 by (blast intro!: Inl_in_LLimit Inr_in_LLimit)
```
```   871
```
```   872 lemmas sum_subset_LLimit = subset_trans [OF sum_mono sum_LLimit]
```
```   873
```
```   874
```
```   875 text{*The constructible universe and its rank function*}
```
```   876 constdefs
```
```   877   L :: "i=>o" --{*Kunen's definition VI, 1.5, page 167*}
```
```   878     "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
```
```   879
```
```   880   lrank :: "i=>i" --{*Kunen's definition VI, 1.7*}
```
```   881     "lrank(x) == \<mu>i. x \<in> Lset(succ(i))"
```
```   882
```
```   883 lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
```
```   884 by (simp add: L_def, blast)
```
```   885
```
```   886 lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
```
```   887 by (simp add: L_def)
```
```   888
```
```   889 lemma Ord_lrank [simp]: "Ord(lrank(a))"
```
```   890 by (simp add: lrank_def)
```
```   891
```
```   892 lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
```
```   893 apply (erule trans_induct3)
```
```   894   apply simp
```
```   895  apply (simp only: lrank_def)
```
```   896  apply (blast intro: Least_le)
```
```   897 apply (simp_all add: Limit_Lset_eq)
```
```   898 apply (blast intro: ltI Limit_is_Ord lt_trans)
```
```   899 done
```
```   900
```
```   901 text{*Kunen's VI, 1.8, and the proof is much less trivial than the text
```
```   902 would suggest.  For a start it need the previous lemma, proved by induction.*}
```
```   903 lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
```
```   904 apply (simp add: L_def, auto)
```
```   905  apply (blast intro: Lset_lrank_lt)
```
```   906  apply (unfold lrank_def)
```
```   907 apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD])
```
```   908 apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption)
```
```   909 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
```
```   910 done
```
```   911
```
```   912 lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
```
```   913 by (simp add: Lset_iff_lrank_lt)
```
```   914
```
```   915 text{*Kunen's VI, 1.9 (a)*}
```
```   916 lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
```
```   917 apply (unfold lrank_def)
```
```   918 apply (rule Least_equality)
```
```   919   apply (erule Ord_in_Lset)
```
```   920  apply assumption
```
```   921 apply (insert notin_Lset [of i])
```
```   922 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
```
```   923 done
```
```   924
```
```   925 text{*This is lrank(lrank(a)) = lrank(a) *}
```
```   926 declare Ord_lrank [THEN lrank_of_Ord, simp]
```
```   927
```
```   928 text{*Kunen's VI, 1.10 *}
```
```   929 lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
```
```   930 apply (simp add: Lset_succ DPow_def)
```
```   931 apply (rule_tac x="Nil" in bexI)
```
```   932  apply (rule_tac x="Equal(0,0)" in bexI)
```
```   933 apply auto
```
```   934 done
```
```   935
```
```   936 lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
```
```   937 apply (unfold lrank_def)
```
```   938 apply (rule Least_equality)
```
```   939   apply (rule Lset_in_Lset_succ)
```
```   940  apply assumption
```
```   941 apply clarify
```
```   942 apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
```
```   943  apply (blast dest: mem_irrefl)
```
```   944 apply (blast intro!: le_imp_subset Lset_mono)
```
```   945 done
```
```   946
```
```   947 text{*Kunen's VI, 1.11 *}
```
```   948 lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
```
```   949 apply (erule trans_induct)
```
```   950 apply (subst Lset)
```
```   951 apply (subst Vset)
```
```   952 apply (rule UN_mono [OF subset_refl])
```
```   953 apply (rule subset_trans [OF DPow_subset_Pow])
```
```   954 apply (rule Pow_mono, blast)
```
```   955 done
```
```   956
```
```   957 text{*Kunen's VI, 1.12 *}
```
```   958 lemma Lset_subset_Vset: "i \<in> nat ==> Lset(i) = Vset(i)";
```
```   959 apply (erule nat_induct)
```
```   960  apply (simp add: Vfrom_0)
```
```   961 apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow)
```
```   962 done
```
```   963
```
```   964 subsection{*For L to satisfy the ZF axioms*}
```
```   965
```
```   966 lemma Union_in_L: "L(X) ==> L(Union(X))"
```
```   967 apply (simp add: L_def, clarify)
```
```   968 apply (drule Ord_imp_greater_Limit)
```
```   969 apply (blast intro: lt_LsetI Union_in_LLimit Limit_is_Ord)
```
```   970 done
```
```   971
```
```   972 lemma doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
```
```   973 apply (simp add: L_def, clarify)
```
```   974 apply (drule Ord2_imp_greater_Limit, assumption)
```
```   975 apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord)
```
```   976 done
```
```   977
```
```   978 subsubsection{*For L to satisfy Powerset *}
```
```   979
```
```   980 lemma LPow_env_typing:
```
```   981      "[| y : Lset(i); Ord(i); y \<subseteq> X |] ==> y \<in> (\<Union>y\<in>Pow(X). Lset(succ(lrank(y))))"
```
```   982 by (auto intro: L_I iff: Lset_succ_lrank_iff)
```
```   983
```
```   984 lemma LPow_in_Lset:
```
```   985      "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
```
```   986 apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
```
```   987 apply simp
```
```   988 apply (rule LsetI [OF succI1])
```
```   989 apply (simp add: DPow_def)
```
```   990 apply (intro conjI, clarify)
```
```   991 apply (rule_tac a="x" in UN_I, simp+)
```
```   992 txt{*Now to create the formula @{term "y \<subseteq> X"} *}
```
```   993 apply (rule_tac x="Cons(X,Nil)" in bexI)
```
```   994  apply (rule_tac x="subset_fm(0,1)" in bexI)
```
```   995   apply typecheck
```
```   996 apply (rule conjI)
```
```   997 apply (simp add: succ_Un_distrib [symmetric])
```
```   998 apply (rule equality_iffI)
```
```   999 apply (simp add: Transset_UN [OF Transset_Lset] list.Cons [OF LPow_env_typing])
```
```  1000 apply (auto intro: L_I iff: Lset_succ_lrank_iff)
```
```  1001 done
```
```  1002
```
```  1003 lemma LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
```
```  1004 by (blast intro: L_I dest: L_D LPow_in_Lset)
```
```  1005
```
```  1006 end
```