src/ZF/Constructible/Formula.thy
 author wenzelm Fri Nov 17 02:20:03 2006 +0100 (2006-11-17) changeset 21404 eb85850d3eb7 parent 21233 5a5c8ea5f66a child 32960 69916a850301 permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
```     1 (*  Title:      ZF/Constructible/Formula.thy
```
```     2     ID: \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4 *)
```
```     5
```
```     6 header {* First-Order Formulas and the Definition of the Class L *}
```
```     7
```
```     8 theory Formula imports Main begin
```
```     9
```
```    10 subsection{*Internalized formulas of FOL*}
```
```    11
```
```    12 text{*De Bruijn representation.
```
```    13   Unbound variables get their denotations from an environment.*}
```
```    14
```
```    15 consts   formula :: i
```
```    16 datatype
```
```    17   "formula" = Member ("x: nat", "y: nat")
```
```    18             | Equal  ("x: nat", "y: nat")
```
```    19             | Nand ("p: formula", "q: formula")
```
```    20             | Forall ("p: formula")
```
```    21
```
```    22 declare formula.intros [TC]
```
```    23
```
```    24 definition
```
```    25   Neg :: "i=>i" where
```
```    26   "Neg(p) == Nand(p,p)"
```
```    27
```
```    28 definition
```
```    29   And :: "[i,i]=>i" where
```
```    30   "And(p,q) == Neg(Nand(p,q))"
```
```    31
```
```    32 definition
```
```    33   Or :: "[i,i]=>i" where
```
```    34   "Or(p,q) == Nand(Neg(p),Neg(q))"
```
```    35
```
```    36 definition
```
```    37   Implies :: "[i,i]=>i" where
```
```    38   "Implies(p,q) == Nand(p,Neg(q))"
```
```    39
```
```    40 definition
```
```    41   Iff :: "[i,i]=>i" where
```
```    42   "Iff(p,q) == And(Implies(p,q), Implies(q,p))"
```
```    43
```
```    44 definition
```
```    45   Exists :: "i=>i" where
```
```    46   "Exists(p) == Neg(Forall(Neg(p)))";
```
```    47
```
```    48 lemma Neg_type [TC]: "p \<in> formula ==> Neg(p) \<in> formula"
```
```    49 by (simp add: Neg_def)
```
```    50
```
```    51 lemma And_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> And(p,q) \<in> formula"
```
```    52 by (simp add: And_def)
```
```    53
```
```    54 lemma Or_type [TC]: "[| p \<in> formula; q \<in> formula |] ==> Or(p,q) \<in> formula"
```
```    55 by (simp add: Or_def)
```
```    56
```
```    57 lemma Implies_type [TC]:
```
```    58      "[| p \<in> formula; q \<in> formula |] ==> Implies(p,q) \<in> formula"
```
```    59 by (simp add: Implies_def)
```
```    60
```
```    61 lemma Iff_type [TC]:
```
```    62      "[| p \<in> formula; q \<in> formula |] ==> Iff(p,q) \<in> formula"
```
```    63 by (simp add: Iff_def)
```
```    64
```
```    65 lemma Exists_type [TC]: "p \<in> formula ==> Exists(p) \<in> formula"
```
```    66 by (simp add: Exists_def)
```
```    67
```
```    68
```
```    69 consts   satisfies :: "[i,i]=>i"
```
```    70 primrec (*explicit lambda is required because the environment varies*)
```
```    71   "satisfies(A,Member(x,y)) =
```
```    72       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) \<in> nth(y,env)))"
```
```    73
```
```    74   "satisfies(A,Equal(x,y)) =
```
```    75       (\<lambda>env \<in> list(A). bool_of_o (nth(x,env) = nth(y,env)))"
```
```    76
```
```    77   "satisfies(A,Nand(p,q)) =
```
```    78       (\<lambda>env \<in> list(A). not ((satisfies(A,p)`env) and (satisfies(A,q)`env)))"
```
```    79
```
```    80   "satisfies(A,Forall(p)) =
```
```    81       (\<lambda>env \<in> list(A). bool_of_o (\<forall>x\<in>A. satisfies(A,p) ` (Cons(x,env)) = 1))"
```
```    82
```
```    83
```
```    84 lemma "p \<in> formula ==> satisfies(A,p) \<in> list(A) -> bool"
```
```    85 by (induct set: formula) simp_all
```
```    86
```
```    87 abbreviation
```
```    88   sats :: "[i,i,i] => o" where
```
```    89   "sats(A,p,env) == satisfies(A,p)`env = 1"
```
```    90
```
```    91 lemma [simp]:
```
```    92   "env \<in> list(A)
```
```    93    ==> sats(A, Member(x,y), env) <-> nth(x,env) \<in> nth(y,env)"
```
```    94 by simp
```
```    95
```
```    96 lemma [simp]:
```
```    97   "env \<in> list(A)
```
```    98    ==> sats(A, Equal(x,y), env) <-> nth(x,env) = nth(y,env)"
```
```    99 by simp
```
```   100
```
```   101 lemma sats_Nand_iff [simp]:
```
```   102   "env \<in> list(A)
```
```   103    ==> (sats(A, Nand(p,q), env)) <-> ~ (sats(A,p,env) & sats(A,q,env))"
```
```   104 by (simp add: Bool.and_def Bool.not_def cond_def)
```
```   105
```
```   106 lemma sats_Forall_iff [simp]:
```
```   107   "env \<in> list(A)
```
```   108    ==> sats(A, Forall(p), env) <-> (\<forall>x\<in>A. sats(A, p, Cons(x,env)))"
```
```   109 by simp
```
```   110
```
```   111 declare satisfies.simps [simp del];
```
```   112
```
```   113 subsection{*Dividing line between primitive and derived connectives*}
```
```   114
```
```   115 lemma sats_Neg_iff [simp]:
```
```   116   "env \<in> list(A)
```
```   117    ==> sats(A, Neg(p), env) <-> ~ sats(A,p,env)"
```
```   118 by (simp add: Neg_def)
```
```   119
```
```   120 lemma sats_And_iff [simp]:
```
```   121   "env \<in> list(A)
```
```   122    ==> (sats(A, And(p,q), env)) <-> sats(A,p,env) & sats(A,q,env)"
```
```   123 by (simp add: And_def)
```
```   124
```
```   125 lemma sats_Or_iff [simp]:
```
```   126   "env \<in> list(A)
```
```   127    ==> (sats(A, Or(p,q), env)) <-> sats(A,p,env) | sats(A,q,env)"
```
```   128 by (simp add: Or_def)
```
```   129
```
```   130 lemma sats_Implies_iff [simp]:
```
```   131   "env \<in> list(A)
```
```   132    ==> (sats(A, Implies(p,q), env)) <-> (sats(A,p,env) --> sats(A,q,env))"
```
```   133 by (simp add: Implies_def, blast)
```
```   134
```
```   135 lemma sats_Iff_iff [simp]:
```
```   136   "env \<in> list(A)
```
```   137    ==> (sats(A, Iff(p,q), env)) <-> (sats(A,p,env) <-> sats(A,q,env))"
```
```   138 by (simp add: Iff_def, blast)
```
```   139
```
```   140 lemma sats_Exists_iff [simp]:
```
```   141   "env \<in> list(A)
```
```   142    ==> sats(A, Exists(p), env) <-> (\<exists>x\<in>A. sats(A, p, Cons(x,env)))"
```
```   143 by (simp add: Exists_def)
```
```   144
```
```   145
```
```   146 subsubsection{*Derived rules to help build up formulas*}
```
```   147
```
```   148 lemma mem_iff_sats:
```
```   149       "[| nth(i,env) = x; nth(j,env) = y; env \<in> list(A)|]
```
```   150        ==> (x\<in>y) <-> sats(A, Member(i,j), env)"
```
```   151 by (simp add: satisfies.simps)
```
```   152
```
```   153 lemma equal_iff_sats:
```
```   154       "[| nth(i,env) = x; nth(j,env) = y; env \<in> list(A)|]
```
```   155        ==> (x=y) <-> sats(A, Equal(i,j), env)"
```
```   156 by (simp add: satisfies.simps)
```
```   157
```
```   158 lemma not_iff_sats:
```
```   159       "[| P <-> sats(A,p,env); env \<in> list(A)|]
```
```   160        ==> (~P) <-> sats(A, Neg(p), env)"
```
```   161 by simp
```
```   162
```
```   163 lemma conj_iff_sats:
```
```   164       "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
```
```   165        ==> (P & Q) <-> sats(A, And(p,q), env)"
```
```   166 by (simp add: sats_And_iff)
```
```   167
```
```   168 lemma disj_iff_sats:
```
```   169       "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
```
```   170        ==> (P | Q) <-> sats(A, Or(p,q), env)"
```
```   171 by (simp add: sats_Or_iff)
```
```   172
```
```   173 lemma iff_iff_sats:
```
```   174       "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
```
```   175        ==> (P <-> Q) <-> sats(A, Iff(p,q), env)"
```
```   176 by (simp add: sats_Forall_iff)
```
```   177
```
```   178 lemma imp_iff_sats:
```
```   179       "[| P <-> sats(A,p,env); Q <-> sats(A,q,env); env \<in> list(A)|]
```
```   180        ==> (P --> Q) <-> sats(A, Implies(p,q), env)"
```
```   181 by (simp add: sats_Forall_iff)
```
```   182
```
```   183 lemma ball_iff_sats:
```
```   184       "[| !!x. x\<in>A ==> P(x) <-> sats(A, p, Cons(x, env)); env \<in> list(A)|]
```
```   185        ==> (\<forall>x\<in>A. P(x)) <-> sats(A, Forall(p), env)"
```
```   186 by (simp add: sats_Forall_iff)
```
```   187
```
```   188 lemma bex_iff_sats:
```
```   189       "[| !!x. x\<in>A ==> P(x) <-> sats(A, p, Cons(x, env)); env \<in> list(A)|]
```
```   190        ==> (\<exists>x\<in>A. P(x)) <-> sats(A, Exists(p), env)"
```
```   191 by (simp add: sats_Exists_iff)
```
```   192
```
```   193 lemmas FOL_iff_sats =
```
```   194         mem_iff_sats equal_iff_sats not_iff_sats conj_iff_sats
```
```   195         disj_iff_sats imp_iff_sats iff_iff_sats imp_iff_sats ball_iff_sats
```
```   196         bex_iff_sats
```
```   197
```
```   198
```
```   199 subsection{*Arity of a Formula: Maximum Free de Bruijn Index*}
```
```   200
```
```   201 consts   arity :: "i=>i"
```
```   202 primrec
```
```   203   "arity(Member(x,y)) = succ(x) \<union> succ(y)"
```
```   204
```
```   205   "arity(Equal(x,y)) = succ(x) \<union> succ(y)"
```
```   206
```
```   207   "arity(Nand(p,q)) = arity(p) \<union> arity(q)"
```
```   208
```
```   209   "arity(Forall(p)) = Arith.pred(arity(p))"
```
```   210
```
```   211
```
```   212 lemma arity_type [TC]: "p \<in> formula ==> arity(p) \<in> nat"
```
```   213 by (induct_tac p, simp_all)
```
```   214
```
```   215 lemma arity_Neg [simp]: "arity(Neg(p)) = arity(p)"
```
```   216 by (simp add: Neg_def)
```
```   217
```
```   218 lemma arity_And [simp]: "arity(And(p,q)) = arity(p) \<union> arity(q)"
```
```   219 by (simp add: And_def)
```
```   220
```
```   221 lemma arity_Or [simp]: "arity(Or(p,q)) = arity(p) \<union> arity(q)"
```
```   222 by (simp add: Or_def)
```
```   223
```
```   224 lemma arity_Implies [simp]: "arity(Implies(p,q)) = arity(p) \<union> arity(q)"
```
```   225 by (simp add: Implies_def)
```
```   226
```
```   227 lemma arity_Iff [simp]: "arity(Iff(p,q)) = arity(p) \<union> arity(q)"
```
```   228 by (simp add: Iff_def, blast)
```
```   229
```
```   230 lemma arity_Exists [simp]: "arity(Exists(p)) = Arith.pred(arity(p))"
```
```   231 by (simp add: Exists_def)
```
```   232
```
```   233
```
```   234 lemma arity_sats_iff [rule_format]:
```
```   235   "[| p \<in> formula; extra \<in> list(A) |]
```
```   236    ==> \<forall>env \<in> list(A).
```
```   237            arity(p) \<le> length(env) -->
```
```   238            sats(A, p, env @ extra) <-> sats(A, p, env)"
```
```   239 apply (induct_tac p)
```
```   240 apply (simp_all add: Arith.pred_def nth_append Un_least_lt_iff nat_imp_quasinat
```
```   241                 split: split_nat_case, auto)
```
```   242 done
```
```   243
```
```   244 lemma arity_sats1_iff:
```
```   245   "[| arity(p) \<le> succ(length(env)); p \<in> formula; x \<in> A; env \<in> list(A);
```
```   246       extra \<in> list(A) |]
```
```   247    ==> sats(A, p, Cons(x, env @ extra)) <-> sats(A, p, Cons(x, env))"
```
```   248 apply (insert arity_sats_iff [of p extra A "Cons(x,env)"])
```
```   249 apply simp
```
```   250 done
```
```   251
```
```   252
```
```   253 subsection{*Renaming Some de Bruijn Variables*}
```
```   254
```
```   255 definition
```
```   256   incr_var :: "[i,i]=>i" where
```
```   257   "incr_var(x,nq) == if x<nq then x else succ(x)"
```
```   258
```
```   259 lemma incr_var_lt: "x<nq ==> incr_var(x,nq) = x"
```
```   260 by (simp add: incr_var_def)
```
```   261
```
```   262 lemma incr_var_le: "nq\<le>x ==> incr_var(x,nq) = succ(x)"
```
```   263 apply (simp add: incr_var_def)
```
```   264 apply (blast dest: lt_trans1)
```
```   265 done
```
```   266
```
```   267 consts   incr_bv :: "i=>i"
```
```   268 primrec
```
```   269   "incr_bv(Member(x,y)) =
```
```   270       (\<lambda>nq \<in> nat. Member (incr_var(x,nq), incr_var(y,nq)))"
```
```   271
```
```   272   "incr_bv(Equal(x,y)) =
```
```   273       (\<lambda>nq \<in> nat. Equal (incr_var(x,nq), incr_var(y,nq)))"
```
```   274
```
```   275   "incr_bv(Nand(p,q)) =
```
```   276       (\<lambda>nq \<in> nat. Nand (incr_bv(p)`nq, incr_bv(q)`nq))"
```
```   277
```
```   278   "incr_bv(Forall(p)) =
```
```   279       (\<lambda>nq \<in> nat. Forall (incr_bv(p) ` succ(nq)))"
```
```   280
```
```   281
```
```   282 lemma [TC]: "x \<in> nat ==> incr_var(x,nq) \<in> nat"
```
```   283 by (simp add: incr_var_def)
```
```   284
```
```   285 lemma incr_bv_type [TC]: "p \<in> formula ==> incr_bv(p) \<in> nat -> formula"
```
```   286 by (induct_tac p, simp_all)
```
```   287
```
```   288 text{*Obviously, @{term DPow} is closed under complements and finite
```
```   289 intersections and unions.  Needs an inductive lemma to allow two lists of
```
```   290 parameters to be combined.*}
```
```   291
```
```   292 lemma sats_incr_bv_iff [rule_format]:
```
```   293   "[| p \<in> formula; env \<in> list(A); x \<in> A |]
```
```   294    ==> \<forall>bvs \<in> list(A).
```
```   295            sats(A, incr_bv(p) ` length(bvs), bvs @ Cons(x,env)) <->
```
```   296            sats(A, p, bvs@env)"
```
```   297 apply (induct_tac p)
```
```   298 apply (simp_all add: incr_var_def nth_append succ_lt_iff length_type)
```
```   299 apply (auto simp add: diff_succ not_lt_iff_le)
```
```   300 done
```
```   301
```
```   302
```
```   303 (*the following two lemmas prevent huge case splits in arity_incr_bv_lemma*)
```
```   304 lemma incr_var_lemma:
```
```   305      "[| x \<in> nat; y \<in> nat; nq \<le> x |]
```
```   306       ==> succ(x) \<union> incr_var(y,nq) = succ(x \<union> y)"
```
```   307 apply (simp add: incr_var_def Ord_Un_if, auto)
```
```   308   apply (blast intro: leI)
```
```   309  apply (simp add: not_lt_iff_le)
```
```   310  apply (blast intro: le_anti_sym)
```
```   311 apply (blast dest: lt_trans2)
```
```   312 done
```
```   313
```
```   314 lemma incr_And_lemma:
```
```   315      "y < x ==> y \<union> succ(x) = succ(x \<union> y)"
```
```   316 apply (simp add: Ord_Un_if lt_Ord lt_Ord2 succ_lt_iff)
```
```   317 apply (blast dest: lt_asym)
```
```   318 done
```
```   319
```
```   320 lemma arity_incr_bv_lemma [rule_format]:
```
```   321   "p \<in> formula
```
```   322    ==> \<forall>n \<in> nat. arity (incr_bv(p) ` n) =
```
```   323                  (if n < arity(p) then succ(arity(p)) else arity(p))"
```
```   324 apply (induct_tac p)
```
```   325 apply (simp_all add: imp_disj not_lt_iff_le Un_least_lt_iff lt_Un_iff le_Un_iff
```
```   326                      succ_Un_distrib [symmetric] incr_var_lt incr_var_le
```
```   327                      Un_commute incr_var_lemma Arith.pred_def nat_imp_quasinat
```
```   328             split: split_nat_case)
```
```   329  txt{*the Forall case reduces to linear arithmetic*}
```
```   330  prefer 2
```
```   331  apply clarify
```
```   332  apply (blast dest: lt_trans1)
```
```   333 txt{*left with the And case*}
```
```   334 apply safe
```
```   335  apply (blast intro: incr_And_lemma lt_trans1)
```
```   336 apply (subst incr_And_lemma)
```
```   337  apply (blast intro: lt_trans1)
```
```   338 apply (simp add: Un_commute)
```
```   339 done
```
```   340
```
```   341
```
```   342 subsection{*Renaming all but the First de Bruijn Variable*}
```
```   343
```
```   344 definition
```
```   345   incr_bv1 :: "i => i" where
```
```   346   "incr_bv1(p) == incr_bv(p)`1"
```
```   347
```
```   348
```
```   349 lemma incr_bv1_type [TC]: "p \<in> formula ==> incr_bv1(p) \<in> formula"
```
```   350 by (simp add: incr_bv1_def)
```
```   351
```
```   352 (*For renaming all but the bound variable at level 0*)
```
```   353 lemma sats_incr_bv1_iff:
```
```   354   "[| p \<in> formula; env \<in> list(A); x \<in> A; y \<in> A |]
```
```   355    ==> sats(A, incr_bv1(p), Cons(x, Cons(y, env))) <->
```
```   356        sats(A, p, Cons(x,env))"
```
```   357 apply (insert sats_incr_bv_iff [of p env A y "Cons(x,Nil)"])
```
```   358 apply (simp add: incr_bv1_def)
```
```   359 done
```
```   360
```
```   361 lemma formula_add_params1 [rule_format]:
```
```   362   "[| p \<in> formula; n \<in> nat; x \<in> A |]
```
```   363    ==> \<forall>bvs \<in> list(A). \<forall>env \<in> list(A).
```
```   364           length(bvs) = n -->
```
```   365           sats(A, iterates(incr_bv1, n, p), Cons(x, bvs@env)) <->
```
```   366           sats(A, p, Cons(x,env))"
```
```   367 apply (induct_tac n, simp, clarify)
```
```   368 apply (erule list.cases)
```
```   369 apply (simp_all add: sats_incr_bv1_iff)
```
```   370 done
```
```   371
```
```   372
```
```   373 lemma arity_incr_bv1_eq:
```
```   374   "p \<in> formula
```
```   375    ==> arity(incr_bv1(p)) =
```
```   376         (if 1 < arity(p) then succ(arity(p)) else arity(p))"
```
```   377 apply (insert arity_incr_bv_lemma [of p 1])
```
```   378 apply (simp add: incr_bv1_def)
```
```   379 done
```
```   380
```
```   381 lemma arity_iterates_incr_bv1_eq:
```
```   382   "[| p \<in> formula; n \<in> nat |]
```
```   383    ==> arity(incr_bv1^n(p)) =
```
```   384          (if 1 < arity(p) then n #+ arity(p) else arity(p))"
```
```   385 apply (induct_tac n)
```
```   386 apply (simp_all add: arity_incr_bv1_eq)
```
```   387 apply (simp add: not_lt_iff_le)
```
```   388 apply (blast intro: le_trans add_le_self2 arity_type)
```
```   389 done
```
```   390
```
```   391
```
```   392
```
```   393 subsection{*Definable Powerset*}
```
```   394
```
```   395 text{*The definable powerset operation: Kunen's definition VI 1.1, page 165.*}
```
```   396 definition
```
```   397   DPow :: "i => i" where
```
```   398   "DPow(A) == {X \<in> Pow(A).
```
```   399                \<exists>env \<in> list(A). \<exists>p \<in> formula.
```
```   400                  arity(p) \<le> succ(length(env)) &
```
```   401                  X = {x\<in>A. sats(A, p, Cons(x,env))}}"
```
```   402
```
```   403 lemma DPowI:
```
```   404   "[|env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
```
```   405    ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
```
```   406 by (simp add: DPow_def, blast)
```
```   407
```
```   408 text{*With this rule we can specify @{term p} later.*}
```
```   409 lemma DPowI2 [rule_format]:
```
```   410   "[|\<forall>x\<in>A. P(x) <-> sats(A, p, Cons(x,env));
```
```   411      env \<in> list(A);  p \<in> formula;  arity(p) \<le> succ(length(env))|]
```
```   412    ==> {x\<in>A. P(x)} \<in> DPow(A)"
```
```   413 by (simp add: DPow_def, blast)
```
```   414
```
```   415 lemma DPowD:
```
```   416   "X \<in> DPow(A)
```
```   417    ==> X <= A &
```
```   418        (\<exists>env \<in> list(A).
```
```   419         \<exists>p \<in> formula. arity(p) \<le> succ(length(env)) &
```
```   420                       X = {x\<in>A. sats(A, p, Cons(x,env))})"
```
```   421 by (simp add: DPow_def)
```
```   422
```
```   423 lemmas DPow_imp_subset = DPowD [THEN conjunct1]
```
```   424
```
```   425 (*Kunen's Lemma VI 1.2*)
```
```   426 lemma "[| p \<in> formula; env \<in> list(A); arity(p) \<le> succ(length(env)) |]
```
```   427        ==> {x\<in>A. sats(A, p, Cons(x,env))} \<in> DPow(A)"
```
```   428 by (blast intro: DPowI)
```
```   429
```
```   430 lemma DPow_subset_Pow: "DPow(A) <= Pow(A)"
```
```   431 by (simp add: DPow_def, blast)
```
```   432
```
```   433 lemma empty_in_DPow: "0 \<in> DPow(A)"
```
```   434 apply (simp add: DPow_def)
```
```   435 apply (rule_tac x=Nil in bexI)
```
```   436  apply (rule_tac x="Neg(Equal(0,0))" in bexI)
```
```   437   apply (auto simp add: Un_least_lt_iff)
```
```   438 done
```
```   439
```
```   440 lemma Compl_in_DPow: "X \<in> DPow(A) ==> (A-X) \<in> DPow(A)"
```
```   441 apply (simp add: DPow_def, clarify, auto)
```
```   442 apply (rule bexI)
```
```   443  apply (rule_tac x="Neg(p)" in bexI)
```
```   444   apply auto
```
```   445 done
```
```   446
```
```   447 lemma Int_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Int Y \<in> DPow(A)"
```
```   448 apply (simp add: DPow_def, auto)
```
```   449 apply (rename_tac envp p envq q)
```
```   450 apply (rule_tac x="envp@envq" in bexI)
```
```   451  apply (rule_tac x="And(p, iterates(incr_bv1,length(envp),q))" in bexI)
```
```   452   apply typecheck
```
```   453 apply (rule conjI)
```
```   454 (*finally check the arity!*)
```
```   455  apply (simp add: arity_iterates_incr_bv1_eq length_app Un_least_lt_iff)
```
```   456  apply (force intro: add_le_self le_trans)
```
```   457 apply (simp add: arity_sats1_iff formula_add_params1, blast)
```
```   458 done
```
```   459
```
```   460 lemma Un_in_DPow: "[| X \<in> DPow(A); Y \<in> DPow(A) |] ==> X Un Y \<in> DPow(A)"
```
```   461 apply (subgoal_tac "X Un Y = A - ((A-X) Int (A-Y))")
```
```   462 apply (simp add: Int_in_DPow Compl_in_DPow)
```
```   463 apply (simp add: DPow_def, blast)
```
```   464 done
```
```   465
```
```   466 lemma singleton_in_DPow: "a \<in> A ==> {a} \<in> DPow(A)"
```
```   467 apply (simp add: DPow_def)
```
```   468 apply (rule_tac x="Cons(a,Nil)" in bexI)
```
```   469  apply (rule_tac x="Equal(0,1)" in bexI)
```
```   470   apply typecheck
```
```   471 apply (force simp add: succ_Un_distrib [symmetric])
```
```   472 done
```
```   473
```
```   474 lemma cons_in_DPow: "[| a \<in> A; X \<in> DPow(A) |] ==> cons(a,X) \<in> DPow(A)"
```
```   475 apply (rule cons_eq [THEN subst])
```
```   476 apply (blast intro: singleton_in_DPow Un_in_DPow)
```
```   477 done
```
```   478
```
```   479 (*Part of Lemma 1.3*)
```
```   480 lemma Fin_into_DPow: "X \<in> Fin(A) ==> X \<in> DPow(A)"
```
```   481 apply (erule Fin.induct)
```
```   482  apply (rule empty_in_DPow)
```
```   483 apply (blast intro: cons_in_DPow)
```
```   484 done
```
```   485
```
```   486 text{*@{term DPow} is not monotonic.  For example, let @{term A} be some
```
```   487 non-constructible set of natural numbers, and let @{term B} be @{term nat}.
```
```   488 Then @{term "A<=B"} and obviously @{term "A \<in> DPow(A)"} but @{term "A ~:
```
```   489 DPow(B)"}.*}
```
```   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 lemma Finite_Pow_subset_Pow: "Finite(A) ==> Pow(A) <= DPow(A)"
```
```   498 by (blast intro: Fin_into_DPow Finite_into_Fin Fin_subset)
```
```   499
```
```   500 lemma Finite_DPow_eq_Pow: "Finite(A) ==> DPow(A) = Pow(A)"
```
```   501 apply (rule equalityI)
```
```   502 apply (rule DPow_subset_Pow)
```
```   503 apply (erule Finite_Pow_subset_Pow)
```
```   504 done
```
```   505
```
```   506
```
```   507 subsection{*Internalized Formulas for the Ordinals*}
```
```   508
```
```   509 text{*The @{text sats} theorems below differ from the usual form in that they
```
```   510 include an element of absoluteness.  That is, they relate internalized
```
```   511 formulas to real concepts such as the subset relation, rather than to the
```
```   512 relativized concepts defined in theory @{text Relative}.  This lets us prove
```
```   513 the theorem as @{text Ords_in_DPow} without first having to instantiate the
```
```   514 locale @{text M_trivial}.  Note that the present theory does not even take
```
```   515 @{text Relative} as a parent.*}
```
```   516
```
```   517 subsubsection{*The subset relation*}
```
```   518
```
```   519 definition
```
```   520   subset_fm :: "[i,i]=>i" where
```
```   521   "subset_fm(x,y) == Forall(Implies(Member(0,succ(x)), Member(0,succ(y))))"
```
```   522
```
```   523 lemma subset_type [TC]: "[| x \<in> nat; y \<in> nat |] ==> subset_fm(x,y) \<in> formula"
```
```   524 by (simp add: subset_fm_def)
```
```   525
```
```   526 lemma arity_subset_fm [simp]:
```
```   527      "[| x \<in> nat; y \<in> nat |] ==> arity(subset_fm(x,y)) = succ(x) \<union> succ(y)"
```
```   528 by (simp add: subset_fm_def succ_Un_distrib [symmetric])
```
```   529
```
```   530 lemma sats_subset_fm [simp]:
```
```   531    "[|x < length(env); y \<in> nat; env \<in> list(A); Transset(A)|]
```
```   532     ==> sats(A, subset_fm(x,y), env) <-> nth(x,env) \<subseteq> nth(y,env)"
```
```   533 apply (frule lt_length_in_nat, assumption)
```
```   534 apply (simp add: subset_fm_def Transset_def)
```
```   535 apply (blast intro: nth_type)
```
```   536 done
```
```   537
```
```   538 subsubsection{*Transitive sets*}
```
```   539
```
```   540 definition
```
```   541   transset_fm :: "i=>i" where
```
```   542   "transset_fm(x) == Forall(Implies(Member(0,succ(x)), subset_fm(0,succ(x))))"
```
```   543
```
```   544 lemma transset_type [TC]: "x \<in> nat ==> transset_fm(x) \<in> formula"
```
```   545 by (simp add: transset_fm_def)
```
```   546
```
```   547 lemma arity_transset_fm [simp]:
```
```   548      "x \<in> nat ==> arity(transset_fm(x)) = succ(x)"
```
```   549 by (simp add: transset_fm_def succ_Un_distrib [symmetric])
```
```   550
```
```   551 lemma sats_transset_fm [simp]:
```
```   552    "[|x < length(env); env \<in> list(A); Transset(A)|]
```
```   553     ==> sats(A, transset_fm(x), env) <-> Transset(nth(x,env))"
```
```   554 apply (frule lt_nat_in_nat, erule length_type)
```
```   555 apply (simp add: transset_fm_def Transset_def)
```
```   556 apply (blast intro: nth_type)
```
```   557 done
```
```   558
```
```   559 subsubsection{*Ordinals*}
```
```   560
```
```   561 definition
```
```   562   ordinal_fm :: "i=>i" where
```
```   563   "ordinal_fm(x) ==
```
```   564     And(transset_fm(x), Forall(Implies(Member(0,succ(x)), transset_fm(0))))"
```
```   565
```
```   566 lemma ordinal_type [TC]: "x \<in> nat ==> ordinal_fm(x) \<in> formula"
```
```   567 by (simp add: ordinal_fm_def)
```
```   568
```
```   569 lemma arity_ordinal_fm [simp]:
```
```   570      "x \<in> nat ==> arity(ordinal_fm(x)) = succ(x)"
```
```   571 by (simp add: ordinal_fm_def succ_Un_distrib [symmetric])
```
```   572
```
```   573 lemma sats_ordinal_fm:
```
```   574    "[|x < length(env); env \<in> list(A); Transset(A)|]
```
```   575     ==> sats(A, ordinal_fm(x), env) <-> Ord(nth(x,env))"
```
```   576 apply (frule lt_nat_in_nat, erule length_type)
```
```   577 apply (simp add: ordinal_fm_def Ord_def Transset_def)
```
```   578 apply (blast intro: nth_type)
```
```   579 done
```
```   580
```
```   581 text{*The subset consisting of the ordinals is definable.  Essential lemma for
```
```   582 @{text Ord_in_Lset}.  This result is the objective of the present subsection.*}
```
```   583 theorem Ords_in_DPow: "Transset(A) ==> {x \<in> A. Ord(x)} \<in> DPow(A)"
```
```   584 apply (simp add: DPow_def Collect_subset)
```
```   585 apply (rule_tac x=Nil in bexI)
```
```   586  apply (rule_tac x="ordinal_fm(0)" in bexI)
```
```   587 apply (simp_all add: sats_ordinal_fm)
```
```   588 done
```
```   589
```
```   590
```
```   591 subsection{* Constant Lset: Levels of the Constructible Universe *}
```
```   592
```
```   593 definition
```
```   594   Lset :: "i=>i" where
```
```   595   "Lset(i) == transrec(i, %x f. \<Union>y\<in>x. DPow(f`y))"
```
```   596
```
```   597 definition
```
```   598   L :: "i=>o" where --{*Kunen's definition VI 1.5, page 167*}
```
```   599   "L(x) == \<exists>i. Ord(i) & x \<in> Lset(i)"
```
```   600
```
```   601 text{*NOT SUITABLE FOR REWRITING -- RECURSIVE!*}
```
```   602 lemma Lset: "Lset(i) = (UN j:i. DPow(Lset(j)))"
```
```   603 by (subst Lset_def [THEN def_transrec], simp)
```
```   604
```
```   605 lemma LsetI: "[|y\<in>x; A \<in> DPow(Lset(y))|] ==> A \<in> Lset(x)";
```
```   606 by (subst Lset, blast)
```
```   607
```
```   608 lemma LsetD: "A \<in> Lset(x) ==> \<exists>y\<in>x. A \<in> DPow(Lset(y))";
```
```   609 apply (insert Lset [of x])
```
```   610 apply (blast intro: elim: equalityE)
```
```   611 done
```
```   612
```
```   613 subsubsection{* Transitivity *}
```
```   614
```
```   615 lemma elem_subset_in_DPow: "[|X \<in> A; X \<subseteq> A|] ==> X \<in> DPow(A)"
```
```   616 apply (simp add: Transset_def DPow_def)
```
```   617 apply (rule_tac x="[X]" in bexI)
```
```   618  apply (rule_tac x="Member(0,1)" in bexI)
```
```   619   apply (auto simp add: Un_least_lt_iff)
```
```   620 done
```
```   621
```
```   622 lemma Transset_subset_DPow: "Transset(A) ==> A <= DPow(A)"
```
```   623 apply clarify
```
```   624 apply (simp add: Transset_def)
```
```   625 apply (blast intro: elem_subset_in_DPow)
```
```   626 done
```
```   627
```
```   628 lemma Transset_DPow: "Transset(A) ==> Transset(DPow(A))"
```
```   629 apply (simp add: Transset_def)
```
```   630 apply (blast intro: elem_subset_in_DPow dest: DPowD)
```
```   631 done
```
```   632
```
```   633 text{*Kunen's VI 1.6 (a)*}
```
```   634 lemma Transset_Lset: "Transset(Lset(i))"
```
```   635 apply (rule_tac a=i in eps_induct)
```
```   636 apply (subst Lset)
```
```   637 apply (blast intro!: Transset_Union_family Transset_Un Transset_DPow)
```
```   638 done
```
```   639
```
```   640 lemma mem_Lset_imp_subset_Lset: "a \<in> Lset(i) ==> a \<subseteq> Lset(i)"
```
```   641 apply (insert Transset_Lset)
```
```   642 apply (simp add: Transset_def)
```
```   643 done
```
```   644
```
```   645 subsubsection{* Monotonicity *}
```
```   646
```
```   647 text{*Kunen's VI 1.6 (b)*}
```
```   648 lemma Lset_mono [rule_format]:
```
```   649      "ALL j. i<=j --> Lset(i) <= Lset(j)"
```
```   650 proof (induct i rule: eps_induct, intro allI impI)
```
```   651   fix x j
```
```   652   assume "\<forall>y\<in>x. \<forall>j. y \<subseteq> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)"
```
```   653      and "x \<subseteq> j"
```
```   654   thus "Lset(x) \<subseteq> Lset(j)"
```
```   655     by (force simp add: Lset [of x] Lset [of j])
```
```   656 qed
```
```   657
```
```   658 text{*This version lets us remove the premise @{term "Ord(i)"} sometimes.*}
```
```   659 lemma Lset_mono_mem [rule_format]:
```
```   660      "ALL j. i:j --> Lset(i) <= Lset(j)"
```
```   661 proof (induct i rule: eps_induct, intro allI impI)
```
```   662   fix x j
```
```   663   assume "\<forall>y\<in>x. \<forall>j. y \<in> j \<longrightarrow> Lset(y) \<subseteq> Lset(j)"
```
```   664      and "x \<in> j"
```
```   665   thus "Lset(x) \<subseteq> Lset(j)"
```
```   666     by (force simp add: Lset [of j]
```
```   667               intro!: bexI intro: elem_subset_in_DPow dest: LsetD DPowD)
```
```   668 qed
```
```   669
```
```   670
```
```   671 text{*Useful with Reflection to bump up the ordinal*}
```
```   672 lemma subset_Lset_ltD: "[|A \<subseteq> Lset(i); i < j|] ==> A \<subseteq> Lset(j)"
```
```   673 by (blast dest: ltD [THEN Lset_mono_mem])
```
```   674
```
```   675 subsubsection{* 0, successor and limit equations for Lset *}
```
```   676
```
```   677 lemma Lset_0 [simp]: "Lset(0) = 0"
```
```   678 by (subst Lset, blast)
```
```   679
```
```   680 lemma Lset_succ_subset1: "DPow(Lset(i)) <= Lset(succ(i))"
```
```   681 by (subst Lset, rule succI1 [THEN RepFunI, THEN Union_upper])
```
```   682
```
```   683 lemma Lset_succ_subset2: "Lset(succ(i)) <= DPow(Lset(i))"
```
```   684 apply (subst Lset, rule UN_least)
```
```   685 apply (erule succE)
```
```   686  apply blast
```
```   687 apply clarify
```
```   688 apply (rule elem_subset_in_DPow)
```
```   689  apply (subst Lset)
```
```   690  apply blast
```
```   691 apply (blast intro: dest: DPowD Lset_mono_mem)
```
```   692 done
```
```   693
```
```   694 lemma Lset_succ: "Lset(succ(i)) = DPow(Lset(i))"
```
```   695 by (intro equalityI Lset_succ_subset1 Lset_succ_subset2)
```
```   696
```
```   697 lemma Lset_Union [simp]: "Lset(\<Union>(X)) = (\<Union>y\<in>X. Lset(y))"
```
```   698 apply (subst Lset)
```
```   699 apply (rule equalityI)
```
```   700  txt{*first inclusion*}
```
```   701  apply (rule UN_least)
```
```   702  apply (erule UnionE)
```
```   703  apply (rule subset_trans)
```
```   704   apply (erule_tac [2] UN_upper, subst Lset, erule UN_upper)
```
```   705 txt{*opposite inclusion*}
```
```   706 apply (rule UN_least)
```
```   707 apply (subst Lset, blast)
```
```   708 done
```
```   709
```
```   710 subsubsection{* Lset applied to Limit ordinals *}
```
```   711
```
```   712 lemma Limit_Lset_eq:
```
```   713     "Limit(i) ==> Lset(i) = (\<Union>y\<in>i. Lset(y))"
```
```   714 by (simp add: Lset_Union [symmetric] Limit_Union_eq)
```
```   715
```
```   716 lemma lt_LsetI: "[| a: Lset(j);  j<i |] ==> a \<in> Lset(i)"
```
```   717 by (blast dest: Lset_mono [OF le_imp_subset [OF leI]])
```
```   718
```
```   719 lemma Limit_LsetE:
```
```   720     "[| a: Lset(i);  ~R ==> Limit(i);
```
```   721         !!x. [| x<i;  a: Lset(x) |] ==> R
```
```   722      |] ==> R"
```
```   723 apply (rule classical)
```
```   724 apply (rule Limit_Lset_eq [THEN equalityD1, THEN subsetD, THEN UN_E])
```
```   725   prefer 2 apply assumption
```
```   726  apply blast
```
```   727 apply (blast intro: ltI  Limit_is_Ord)
```
```   728 done
```
```   729
```
```   730 subsubsection{* Basic closure properties *}
```
```   731
```
```   732 lemma zero_in_Lset: "y:x ==> 0 \<in> Lset(x)"
```
```   733 by (subst Lset, blast intro: empty_in_DPow)
```
```   734
```
```   735 lemma notin_Lset: "x \<notin> Lset(x)"
```
```   736 apply (rule_tac a=x in eps_induct)
```
```   737 apply (subst Lset)
```
```   738 apply (blast dest: DPowD)
```
```   739 done
```
```   740
```
```   741
```
```   742 subsection{*Constructible Ordinals: Kunen's VI 1.9 (b)*}
```
```   743
```
```   744 lemma Ords_of_Lset_eq: "Ord(i) ==> {x\<in>Lset(i). Ord(x)} = i"
```
```   745 apply (erule trans_induct3)
```
```   746   apply (simp_all add: Lset_succ Limit_Lset_eq Limit_Union_eq)
```
```   747 txt{*The successor case remains.*}
```
```   748 apply (rule equalityI)
```
```   749 txt{*First inclusion*}
```
```   750  apply clarify
```
```   751  apply (erule Ord_linear_lt, assumption)
```
```   752    apply (blast dest: DPow_imp_subset ltD notE [OF notin_Lset])
```
```   753   apply blast
```
```   754  apply (blast dest: ltD)
```
```   755 txt{*Opposite inclusion, @{term "succ(x) \<subseteq> DPow(Lset(x)) \<inter> ON"}*}
```
```   756 apply auto
```
```   757 txt{*Key case: *}
```
```   758   apply (erule subst, rule Ords_in_DPow [OF Transset_Lset])
```
```   759  apply (blast intro: elem_subset_in_DPow dest: OrdmemD elim: equalityE)
```
```   760 apply (blast intro: Ord_in_Ord)
```
```   761 done
```
```   762
```
```   763
```
```   764 lemma Ord_subset_Lset: "Ord(i) ==> i \<subseteq> Lset(i)"
```
```   765 by (subst Ords_of_Lset_eq [symmetric], assumption, fast)
```
```   766
```
```   767 lemma Ord_in_Lset: "Ord(i) ==> i \<in> Lset(succ(i))"
```
```   768 apply (simp add: Lset_succ)
```
```   769 apply (subst Ords_of_Lset_eq [symmetric], assumption,
```
```   770        rule Ords_in_DPow [OF Transset_Lset])
```
```   771 done
```
```   772
```
```   773 lemma Ord_in_L: "Ord(i) ==> L(i)"
```
```   774 by (simp add: L_def, blast intro: Ord_in_Lset)
```
```   775
```
```   776 subsubsection{* Unions *}
```
```   777
```
```   778 lemma Union_in_Lset:
```
```   779      "X \<in> Lset(i) ==> Union(X) \<in> Lset(succ(i))"
```
```   780 apply (insert Transset_Lset)
```
```   781 apply (rule LsetI [OF succI1])
```
```   782 apply (simp add: Transset_def DPow_def)
```
```   783 apply (intro conjI, blast)
```
```   784 txt{*Now to create the formula @{term "\<exists>y. y \<in> X \<and> x \<in> y"} *}
```
```   785 apply (rule_tac x="Cons(X,Nil)" in bexI)
```
```   786  apply (rule_tac x="Exists(And(Member(0,2), Member(1,0)))" in bexI)
```
```   787   apply typecheck
```
```   788 apply (simp add: succ_Un_distrib [symmetric], blast)
```
```   789 done
```
```   790
```
```   791 theorem Union_in_L: "L(X) ==> L(Union(X))"
```
```   792 by (simp add: L_def, blast dest: Union_in_Lset)
```
```   793
```
```   794 subsubsection{* Finite sets and ordered pairs *}
```
```   795
```
```   796 lemma singleton_in_Lset: "a: Lset(i) ==> {a} \<in> Lset(succ(i))"
```
```   797 by (simp add: Lset_succ singleton_in_DPow)
```
```   798
```
```   799 lemma doubleton_in_Lset:
```
```   800      "[| a: Lset(i);  b: Lset(i) |] ==> {a,b} \<in> Lset(succ(i))"
```
```   801 by (simp add: Lset_succ empty_in_DPow cons_in_DPow)
```
```   802
```
```   803 lemma Pair_in_Lset:
```
```   804     "[| a: Lset(i);  b: Lset(i); Ord(i) |] ==> <a,b> \<in> Lset(succ(succ(i)))"
```
```   805 apply (unfold Pair_def)
```
```   806 apply (blast intro: doubleton_in_Lset)
```
```   807 done
```
```   808
```
```   809 lemmas Lset_UnI1 = Un_upper1 [THEN Lset_mono [THEN subsetD], standard]
```
```   810 lemmas Lset_UnI2 = Un_upper2 [THEN Lset_mono [THEN subsetD], standard]
```
```   811
```
```   812 text{*Hard work is finding a single j:i such that {a,b}<=Lset(j)*}
```
```   813 lemma doubleton_in_LLimit:
```
```   814     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> {a,b} \<in> Lset(i)"
```
```   815 apply (erule Limit_LsetE, assumption)
```
```   816 apply (erule Limit_LsetE, assumption)
```
```   817 apply (blast intro: lt_LsetI [OF doubleton_in_Lset]
```
```   818                     Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
```
```   819 done
```
```   820
```
```   821 theorem doubleton_in_L: "[| L(a); L(b) |] ==> L({a, b})"
```
```   822 apply (simp add: L_def, clarify)
```
```   823 apply (drule Ord2_imp_greater_Limit, assumption)
```
```   824 apply (blast intro: lt_LsetI doubleton_in_LLimit Limit_is_Ord)
```
```   825 done
```
```   826
```
```   827 lemma Pair_in_LLimit:
```
```   828     "[| a: Lset(i);  b: Lset(i);  Limit(i) |] ==> <a,b> \<in> Lset(i)"
```
```   829 txt{*Infer that a, b occur at ordinals x,xa < i.*}
```
```   830 apply (erule Limit_LsetE, assumption)
```
```   831 apply (erule Limit_LsetE, assumption)
```
```   832 txt{*Infer that succ(succ(x Un xa)) < i *}
```
```   833 apply (blast intro: lt_Ord lt_LsetI [OF Pair_in_Lset]
```
```   834                     Lset_UnI1 Lset_UnI2 Limit_has_succ Un_least_lt)
```
```   835 done
```
```   836
```
```   837
```
```   838
```
```   839 text{*The rank function for the constructible universe*}
```
```   840 definition
```
```   841   lrank :: "i=>i" where --{*Kunen's definition VI 1.7*}
```
```   842   "lrank(x) == \<mu> i. x \<in> Lset(succ(i))"
```
```   843
```
```   844 lemma L_I: "[|x \<in> Lset(i); Ord(i)|] ==> L(x)"
```
```   845 by (simp add: L_def, blast)
```
```   846
```
```   847 lemma L_D: "L(x) ==> \<exists>i. Ord(i) & x \<in> Lset(i)"
```
```   848 by (simp add: L_def)
```
```   849
```
```   850 lemma Ord_lrank [simp]: "Ord(lrank(a))"
```
```   851 by (simp add: lrank_def)
```
```   852
```
```   853 lemma Lset_lrank_lt [rule_format]: "Ord(i) ==> x \<in> Lset(i) --> lrank(x) < i"
```
```   854 apply (erule trans_induct3)
```
```   855   apply simp
```
```   856  apply (simp only: lrank_def)
```
```   857  apply (blast intro: Least_le)
```
```   858 apply (simp_all add: Limit_Lset_eq)
```
```   859 apply (blast intro: ltI Limit_is_Ord lt_trans)
```
```   860 done
```
```   861
```
```   862 text{*Kunen's VI 1.8.  The proof is much harder than the text would
```
```   863 suggest.  For a start, it needs the previous lemma, which is proved by
```
```   864 induction.*}
```
```   865 lemma Lset_iff_lrank_lt: "Ord(i) ==> x \<in> Lset(i) <-> L(x) & lrank(x) < i"
```
```   866 apply (simp add: L_def, auto)
```
```   867  apply (blast intro: Lset_lrank_lt)
```
```   868  apply (unfold lrank_def)
```
```   869 apply (drule succI1 [THEN Lset_mono_mem, THEN subsetD])
```
```   870 apply (drule_tac P="\<lambda>i. x \<in> Lset(succ(i))" in LeastI, assumption)
```
```   871 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
```
```   872 done
```
```   873
```
```   874 lemma Lset_succ_lrank_iff [simp]: "x \<in> Lset(succ(lrank(x))) <-> L(x)"
```
```   875 by (simp add: Lset_iff_lrank_lt)
```
```   876
```
```   877 text{*Kunen's VI 1.9 (a)*}
```
```   878 lemma lrank_of_Ord: "Ord(i) ==> lrank(i) = i"
```
```   879 apply (unfold lrank_def)
```
```   880 apply (rule Least_equality)
```
```   881   apply (erule Ord_in_Lset)
```
```   882  apply assumption
```
```   883 apply (insert notin_Lset [of i])
```
```   884 apply (blast intro!: le_imp_subset Lset_mono [THEN subsetD])
```
```   885 done
```
```   886
```
```   887
```
```   888 text{*This is lrank(lrank(a)) = lrank(a) *}
```
```   889 declare Ord_lrank [THEN lrank_of_Ord, simp]
```
```   890
```
```   891 text{*Kunen's VI 1.10 *}
```
```   892 lemma Lset_in_Lset_succ: "Lset(i) \<in> Lset(succ(i))";
```
```   893 apply (simp add: Lset_succ DPow_def)
```
```   894 apply (rule_tac x=Nil in bexI)
```
```   895  apply (rule_tac x="Equal(0,0)" in bexI)
```
```   896 apply auto
```
```   897 done
```
```   898
```
```   899 lemma lrank_Lset: "Ord(i) ==> lrank(Lset(i)) = i"
```
```   900 apply (unfold lrank_def)
```
```   901 apply (rule Least_equality)
```
```   902   apply (rule Lset_in_Lset_succ)
```
```   903  apply assumption
```
```   904 apply clarify
```
```   905 apply (subgoal_tac "Lset(succ(ia)) <= Lset(i)")
```
```   906  apply (blast dest: mem_irrefl)
```
```   907 apply (blast intro!: le_imp_subset Lset_mono)
```
```   908 done
```
```   909
```
```   910 text{*Kunen's VI 1.11 *}
```
```   911 lemma Lset_subset_Vset: "Ord(i) ==> Lset(i) <= Vset(i)";
```
```   912 apply (erule trans_induct)
```
```   913 apply (subst Lset)
```
```   914 apply (subst Vset)
```
```   915 apply (rule UN_mono [OF subset_refl])
```
```   916 apply (rule subset_trans [OF DPow_subset_Pow])
```
```   917 apply (rule Pow_mono, blast)
```
```   918 done
```
```   919
```
```   920 text{*Kunen's VI 1.12 *}
```
```   921 lemma Lset_subset_Vset': "i \<in> nat ==> Lset(i) = Vset(i)";
```
```   922 apply (erule nat_induct)
```
```   923  apply (simp add: Vfrom_0)
```
```   924 apply (simp add: Lset_succ Vset_succ Finite_Vset Finite_DPow_eq_Pow)
```
```   925 done
```
```   926
```
```   927 text{*Every set of constructible sets is included in some @{term Lset}*}
```
```   928 lemma subset_Lset:
```
```   929      "(\<forall>x\<in>A. L(x)) ==> \<exists>i. Ord(i) & A \<subseteq> Lset(i)"
```
```   930 by (rule_tac x = "\<Union>x\<in>A. succ(lrank(x))" in exI, force)
```
```   931
```
```   932 lemma subset_LsetE:
```
```   933      "[|\<forall>x\<in>A. L(x);
```
```   934         !!i. [|Ord(i); A \<subseteq> Lset(i)|] ==> P|]
```
```   935       ==> P"
```
```   936 by (blast dest: subset_Lset)
```
```   937
```
```   938 subsubsection{*For L to satisfy the Powerset axiom *}
```
```   939
```
```   940 lemma LPow_env_typing:
```
```   941     "[| y \<in> Lset(i); Ord(i); y \<subseteq> X |]
```
```   942      ==> \<exists>z \<in> Pow(X). y \<in> Lset(succ(lrank(z)))"
```
```   943 by (auto intro: L_I iff: Lset_succ_lrank_iff)
```
```   944
```
```   945 lemma LPow_in_Lset:
```
```   946      "[|X \<in> Lset(i); Ord(i)|] ==> \<exists>j. Ord(j) & {y \<in> Pow(X). L(y)} \<in> Lset(j)"
```
```   947 apply (rule_tac x="succ(\<Union>y \<in> Pow(X). succ(lrank(y)))" in exI)
```
```   948 apply simp
```
```   949 apply (rule LsetI [OF succI1])
```
```   950 apply (simp add: DPow_def)
```
```   951 apply (intro conjI, clarify)
```
```   952  apply (rule_tac a=x in UN_I, simp+)
```
```   953 txt{*Now to create the formula @{term "y \<subseteq> X"} *}
```
```   954 apply (rule_tac x="Cons(X,Nil)" in bexI)
```
```   955  apply (rule_tac x="subset_fm(0,1)" in bexI)
```
```   956   apply typecheck
```
```   957  apply (rule conjI)
```
```   958 apply (simp add: succ_Un_distrib [symmetric])
```
```   959 apply (rule equality_iffI)
```
```   960 apply (simp add: Transset_UN [OF Transset_Lset] LPow_env_typing)
```
```   961 apply (auto intro: L_I iff: Lset_succ_lrank_iff)
```
```   962 done
```
```   963
```
```   964 theorem LPow_in_L: "L(X) ==> L({y \<in> Pow(X). L(y)})"
```
```   965 by (blast intro: L_I dest: L_D LPow_in_Lset)
```
```   966
```
```   967
```
```   968 subsection{*Eliminating @{term arity} from the Definition of @{term Lset}*}
```
```   969
```
```   970 lemma nth_zero_eq_0: "n \<in> nat ==> nth(n,[0]) = 0"
```
```   971 by (induct_tac n, auto)
```
```   972
```
```   973 lemma sats_app_0_iff [rule_format]:
```
```   974   "[| p \<in> formula; 0 \<in> A |]
```
```   975    ==> \<forall>env \<in> list(A). sats(A,p, env@[0]) <-> sats(A,p,env)"
```
```   976 apply (induct_tac p)
```
```   977 apply (simp_all del: app_Cons add: app_Cons [symmetric]
```
```   978 		add: nth_zero_eq_0 nth_append not_lt_iff_le nth_eq_0)
```
```   979 done
```
```   980
```
```   981 lemma sats_app_zeroes_iff:
```
```   982   "[| p \<in> formula; 0 \<in> A; env \<in> list(A); n \<in> nat |]
```
```   983    ==> sats(A,p,env @ repeat(0,n)) <-> sats(A,p,env)"
```
```   984 apply (induct_tac n, simp)
```
```   985 apply (simp del: repeat.simps
```
```   986             add: repeat_succ_app sats_app_0_iff app_assoc [symmetric])
```
```   987 done
```
```   988
```
```   989 lemma exists_bigger_env:
```
```   990   "[| p \<in> formula; 0 \<in> A; env \<in> list(A) |]
```
```   991    ==> \<exists>env' \<in> list(A). arity(p) \<le> succ(length(env')) &
```
```   992               (\<forall>a\<in>A. sats(A,p,Cons(a,env')) <-> sats(A,p,Cons(a,env)))"
```
```   993 apply (rule_tac x="env @ repeat(0,arity(p))" in bexI)
```
```   994 apply (simp del: app_Cons add: app_Cons [symmetric]
```
```   995 	    add: length_repeat sats_app_zeroes_iff, typecheck)
```
```   996 done
```
```   997
```
```   998
```
```   999 text{*A simpler version of @{term DPow}: no arity check!*}
```
```  1000 definition
```
```  1001   DPow' :: "i => i" where
```
```  1002   "DPow'(A) == {X \<in> Pow(A).
```
```  1003                 \<exists>env \<in> list(A). \<exists>p \<in> formula.
```
```  1004                     X = {x\<in>A. sats(A, p, Cons(x,env))}}"
```
```  1005
```
```  1006 lemma DPow_subset_DPow': "DPow(A) <= DPow'(A)";
```
```  1007 by (simp add: DPow_def DPow'_def, blast)
```
```  1008
```
```  1009 lemma DPow'_0: "DPow'(0) = {0}"
```
```  1010 by (auto simp add: DPow'_def)
```
```  1011
```
```  1012 lemma DPow'_subset_DPow: "0 \<in> A ==> DPow'(A) \<subseteq> DPow(A)"
```
```  1013 apply (auto simp add: DPow'_def DPow_def)
```
```  1014 apply (frule exists_bigger_env, assumption+, force)
```
```  1015 done
```
```  1016
```
```  1017 lemma DPow_eq_DPow': "Transset(A) ==> DPow(A) = DPow'(A)"
```
```  1018 apply (drule Transset_0_disj)
```
```  1019 apply (erule disjE)
```
```  1020  apply (simp add: DPow'_0 Finite_DPow_eq_Pow)
```
```  1021 apply (rule equalityI)
```
```  1022  apply (rule DPow_subset_DPow')
```
```  1023 apply (erule DPow'_subset_DPow)
```
```  1024 done
```
```  1025
```
```  1026 text{*And thus we can relativize @{term Lset} without bothering with
```
```  1027       @{term arity} and @{term length}*}
```
```  1028 lemma Lset_eq_transrec_DPow': "Lset(i) = transrec(i, %x f. \<Union>y\<in>x. DPow'(f`y))"
```
```  1029 apply (rule_tac a=i in eps_induct)
```
```  1030 apply (subst Lset)
```
```  1031 apply (subst transrec)
```
```  1032 apply (simp only: DPow_eq_DPow' [OF Transset_Lset], simp)
```
```  1033 done
```
```  1034
```
```  1035 text{*With this rule we can specify @{term p} later and don't worry about
```
```  1036       arities at all!*}
```
```  1037 lemma DPow_LsetI [rule_format]:
```
```  1038   "[|\<forall>x\<in>Lset(i). P(x) <-> sats(Lset(i), p, Cons(x,env));
```
```  1039      env \<in> list(Lset(i));  p \<in> formula|]
```
```  1040    ==> {x\<in>Lset(i). P(x)} \<in> DPow(Lset(i))"
```
```  1041 by (simp add: DPow_eq_DPow' [OF Transset_Lset] DPow'_def, blast)
```
```  1042
```
```  1043 end
```