src/ZF/Constructible/Datatype_absolute.thy
 author paulson Thu Jul 25 18:29:04 2002 +0200 (2002-07-25) changeset 13423 7ec771711c09 parent 13422 af9bc8d87a75 child 13428 99e52e78eb65 permissions -rw-r--r--
More lemmas, working towards relativization of "satisfies"
```     1 header {*Absoluteness Properties for Recursive Datatypes*}
```
```     2
```
```     3 theory Datatype_absolute = Formula + WF_absolute:
```
```     4
```
```     5
```
```     6 subsection{*The lfp of a continuous function can be expressed as a union*}
```
```     7
```
```     8 constdefs
```
```     9   directed :: "i=>o"
```
```    10    "directed(A) == A\<noteq>0 & (\<forall>x\<in>A. \<forall>y\<in>A. x \<union> y \<in> A)"
```
```    11
```
```    12   contin :: "(i=>i) => o"
```
```    13    "contin(h) == (\<forall>A. directed(A) --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
```
```    14
```
```    15 lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
```
```    16 apply (induct_tac n)
```
```    17  apply (simp_all add: bnd_mono_def, blast)
```
```    18 done
```
```    19
```
```    20 lemma bnd_mono_increasing [rule_format]:
```
```    21      "[|i \<in> nat; j \<in> nat; bnd_mono(D,h)|] ==> i \<le> j --> h^i(0) \<subseteq> h^j(0)"
```
```    22 apply (rule_tac m=i and n=j in diff_induct, simp_all)
```
```    23 apply (blast del: subsetI
```
```    24 	     intro: bnd_mono_iterates_subset bnd_monoD2 [of concl: h])
```
```    25 done
```
```    26
```
```    27 lemma directed_iterates: "bnd_mono(D,h) ==> directed({h^n (0). n\<in>nat})"
```
```    28 apply (simp add: directed_def, clarify)
```
```    29 apply (rename_tac i j)
```
```    30 apply (rule_tac x="i \<union> j" in bexI)
```
```    31 apply (rule_tac i = i and j = j in Ord_linear_le)
```
```    32 apply (simp_all add: subset_Un_iff [THEN iffD1] le_imp_subset
```
```    33                      subset_Un_iff2 [THEN iffD1])
```
```    34 apply (simp_all add: subset_Un_iff [THEN iff_sym] bnd_mono_increasing
```
```    35                      subset_Un_iff2 [THEN iff_sym])
```
```    36 done
```
```    37
```
```    38
```
```    39 lemma contin_iterates_eq:
```
```    40     "[|bnd_mono(D, h); contin(h)|]
```
```    41      ==> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
```
```    42 apply (simp add: contin_def directed_iterates)
```
```    43 apply (rule trans)
```
```    44 apply (rule equalityI)
```
```    45  apply (simp_all add: UN_subset_iff)
```
```    46  apply safe
```
```    47  apply (erule_tac [2] natE)
```
```    48   apply (rule_tac a="succ(x)" in UN_I)
```
```    49    apply simp_all
```
```    50 apply blast
```
```    51 done
```
```    52
```
```    53 lemma lfp_subset_Union:
```
```    54      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
```
```    55 apply (rule lfp_lowerbound)
```
```    56  apply (simp add: contin_iterates_eq)
```
```    57 apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff)
```
```    58 done
```
```    59
```
```    60 lemma Union_subset_lfp:
```
```    61      "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
```
```    62 apply (simp add: UN_subset_iff)
```
```    63 apply (rule ballI)
```
```    64 apply (induct_tac n, simp_all)
```
```    65 apply (rule subset_trans [of _ "h(lfp(D,h))"])
```
```    66  apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset])
```
```    67 apply (erule lfp_lemma2)
```
```    68 done
```
```    69
```
```    70 lemma lfp_eq_Union:
```
```    71      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
```
```    72 by (blast del: subsetI
```
```    73           intro: lfp_subset_Union Union_subset_lfp)
```
```    74
```
```    75
```
```    76 subsubsection{*Some Standard Datatype Constructions Preserve Continuity*}
```
```    77
```
```    78 lemma contin_imp_mono: "[|X\<subseteq>Y; contin(F)|] ==> F(X) \<subseteq> F(Y)"
```
```    79 apply (simp add: contin_def)
```
```    80 apply (drule_tac x="{X,Y}" in spec)
```
```    81 apply (simp add: directed_def subset_Un_iff2 Un_commute)
```
```    82 done
```
```    83
```
```    84 lemma sum_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) + G(X))"
```
```    85 by (simp add: contin_def, blast)
```
```    86
```
```    87 lemma prod_contin: "[|contin(F); contin(G)|] ==> contin(\<lambda>X. F(X) * G(X))"
```
```    88 apply (subgoal_tac "\<forall>B C. F(B) \<subseteq> F(B \<union> C)")
```
```    89  prefer 2 apply (simp add: Un_upper1 contin_imp_mono)
```
```    90 apply (subgoal_tac "\<forall>B C. G(C) \<subseteq> G(B \<union> C)")
```
```    91  prefer 2 apply (simp add: Un_upper2 contin_imp_mono)
```
```    92 apply (simp add: contin_def, clarify)
```
```    93 apply (rule equalityI)
```
```    94  prefer 2 apply blast
```
```    95 apply clarify
```
```    96 apply (rename_tac B C)
```
```    97 apply (rule_tac a="B \<union> C" in UN_I)
```
```    98  apply (simp add: directed_def, blast)
```
```    99 done
```
```   100
```
```   101 lemma const_contin: "contin(\<lambda>X. A)"
```
```   102 by (simp add: contin_def directed_def)
```
```   103
```
```   104 lemma id_contin: "contin(\<lambda>X. X)"
```
```   105 by (simp add: contin_def)
```
```   106
```
```   107
```
```   108
```
```   109 subsection {*Absoluteness for "Iterates"*}
```
```   110
```
```   111 constdefs
```
```   112
```
```   113   iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   114    "iterates_MH(M,isF,v,n,g,z) ==
```
```   115         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   116                     n, z)"
```
```   117
```
```   118   iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"
```
```   119    "iterates_replacement(M,isF,v) ==
```
```   120       \<forall>n[M]. n\<in>nat -->
```
```   121          wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
```
```   122
```
```   123 lemma (in M_axioms) iterates_MH_abs:
```
```   124   "[| relativize1(M,isF,F); M(n); M(g); M(z) |]
```
```   125    ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
```
```   126 by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
```
```   127               relativize1_def iterates_MH_def)
```
```   128
```
```   129 lemma (in M_axioms) iterates_imp_wfrec_replacement:
```
```   130   "[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|]
```
```   131    ==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
```
```   132                        Memrel(succ(n)))"
```
```   133 by (simp add: iterates_replacement_def iterates_MH_abs)
```
```   134
```
```   135 theorem (in M_trancl) iterates_abs:
```
```   136   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
```
```   137       n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |]
```
```   138    ==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->
```
```   139        z = iterates(F,n,v)"
```
```   140 apply (frule iterates_imp_wfrec_replacement, assumption+)
```
```   141 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
```
```   142                  relativize2_def iterates_MH_abs
```
```   143                  iterates_nat_def recursor_def transrec_def
```
```   144                  eclose_sing_Ord_eq nat_into_M
```
```   145          trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
```
```   146 done
```
```   147
```
```   148
```
```   149 lemma (in M_wfrank) iterates_closed [intro,simp]:
```
```   150   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
```
```   151       n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |]
```
```   152    ==> M(iterates(F,n,v))"
```
```   153 apply (frule iterates_imp_wfrec_replacement, assumption+)
```
```   154 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
```
```   155                  relativize2_def iterates_MH_abs
```
```   156                  iterates_nat_def recursor_def transrec_def
```
```   157                  eclose_sing_Ord_eq nat_into_M
```
```   158          trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
```
```   159 done
```
```   160
```
```   161
```
```   162 subsection {*lists without univ*}
```
```   163
```
```   164 lemmas datatype_univs = Inl_in_univ Inr_in_univ
```
```   165                         Pair_in_univ nat_into_univ A_into_univ
```
```   166
```
```   167 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
```
```   168 apply (rule bnd_monoI)
```
```   169  apply (intro subset_refl zero_subset_univ A_subset_univ
```
```   170 	      sum_subset_univ Sigma_subset_univ)
```
```   171 apply (rule subset_refl sum_mono Sigma_mono | assumption)+
```
```   172 done
```
```   173
```
```   174 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
```
```   175 by (intro sum_contin prod_contin id_contin const_contin)
```
```   176
```
```   177 text{*Re-expresses lists using sum and product*}
```
```   178 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
```
```   179 apply (simp add: list_def)
```
```   180 apply (rule equalityI)
```
```   181  apply (rule lfp_lowerbound)
```
```   182   prefer 2 apply (rule lfp_subset)
```
```   183  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
```
```   184  apply (simp add: Nil_def Cons_def)
```
```   185  apply blast
```
```   186 txt{*Opposite inclusion*}
```
```   187 apply (rule lfp_lowerbound)
```
```   188  prefer 2 apply (rule lfp_subset)
```
```   189 apply (clarify, subst lfp_unfold [OF list.bnd_mono])
```
```   190 apply (simp add: Nil_def Cons_def)
```
```   191 apply (blast intro: datatype_univs
```
```   192              dest: lfp_subset [THEN subsetD])
```
```   193 done
```
```   194
```
```   195 text{*Re-expresses lists using "iterates", no univ.*}
```
```   196 lemma list_eq_Union:
```
```   197      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
```
```   198 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
```
```   199
```
```   200
```
```   201 constdefs
```
```   202   is_list_functor :: "[i=>o,i,i,i] => o"
```
```   203     "is_list_functor(M,A,X,Z) ==
```
```   204         \<exists>n1[M]. \<exists>AX[M].
```
```   205          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
```
```   206
```
```   207 lemma (in M_axioms) list_functor_abs [simp]:
```
```   208      "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
```
```   209 by (simp add: is_list_functor_def singleton_0 nat_into_M)
```
```   210
```
```   211
```
```   212 subsection {*formulas without univ*}
```
```   213
```
```   214 lemma formula_fun_bnd_mono:
```
```   215      "bnd_mono(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"
```
```   216 apply (rule bnd_monoI)
```
```   217  apply (intro subset_refl zero_subset_univ A_subset_univ
```
```   218 	      sum_subset_univ Sigma_subset_univ nat_subset_univ)
```
```   219 apply (rule subset_refl sum_mono Sigma_mono | assumption)+
```
```   220 done
```
```   221
```
```   222 lemma formula_fun_contin:
```
```   223      "contin(\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"
```
```   224 by (intro sum_contin prod_contin id_contin const_contin)
```
```   225
```
```   226
```
```   227 text{*Re-expresses formulas using sum and product*}
```
```   228 lemma formula_eq_lfp2:
```
```   229     "formula = lfp(univ(0), \<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))"
```
```   230 apply (simp add: formula_def)
```
```   231 apply (rule equalityI)
```
```   232  apply (rule lfp_lowerbound)
```
```   233   prefer 2 apply (rule lfp_subset)
```
```   234  apply (clarify, subst lfp_unfold [OF formula_fun_bnd_mono])
```
```   235  apply (simp add: Member_def Equal_def Nand_def Forall_def)
```
```   236  apply blast
```
```   237 txt{*Opposite inclusion*}
```
```   238 apply (rule lfp_lowerbound)
```
```   239  prefer 2 apply (rule lfp_subset, clarify)
```
```   240 apply (subst lfp_unfold [OF formula.bnd_mono, simplified])
```
```   241 apply (simp add: Member_def Equal_def Nand_def Forall_def)
```
```   242 apply (elim sumE SigmaE, simp_all)
```
```   243 apply (blast intro: datatype_univs dest: lfp_subset [THEN subsetD])+
```
```   244 done
```
```   245
```
```   246 text{*Re-expresses formulas using "iterates", no univ.*}
```
```   247 lemma formula_eq_Union:
```
```   248      "formula =
```
```   249       (\<Union>n\<in>nat. (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0))"
```
```   250 by (simp add: formula_eq_lfp2 lfp_eq_Union formula_fun_bnd_mono
```
```   251               formula_fun_contin)
```
```   252
```
```   253
```
```   254 constdefs
```
```   255   is_formula_functor :: "[i=>o,i,i] => o"
```
```   256     "is_formula_functor(M,X,Z) ==
```
```   257         \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
```
```   258           omega(M,nat') & cartprod(M,nat',nat',natnat) &
```
```   259           is_sum(M,natnat,natnat,natnatsum) &
```
```   260           cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
```
```   261           is_sum(M,natnatsum,X3,Z)"
```
```   262
```
```   263 lemma (in M_axioms) formula_functor_abs [simp]:
```
```   264      "[| M(X); M(Z) |]
```
```   265       ==> is_formula_functor(M,X,Z) <->
```
```   266           Z = ((nat*nat) + (nat*nat)) + (X*X + X)"
```
```   267 by (simp add: is_formula_functor_def)
```
```   268
```
```   269
```
```   270 subsection{*@{term M} Contains the List and Formula Datatypes*}
```
```   271
```
```   272 constdefs
```
```   273   list_N :: "[i,i] => i"
```
```   274     "list_N(A,n) == (\<lambda>X. {0} + A * X)^n (0)"
```
```   275
```
```   276 lemma Nil_in_list_N [simp]: "[] \<in> list_N(A,succ(n))"
```
```   277 by (simp add: list_N_def Nil_def)
```
```   278
```
```   279 lemma Cons_in_list_N [simp]:
```
```   280      "Cons(a,l) \<in> list_N(A,succ(n)) <-> a\<in>A & l \<in> list_N(A,n)"
```
```   281 by (simp add: list_N_def Cons_def)
```
```   282
```
```   283 text{*These two aren't simprules because they reveal the underlying
```
```   284 list representation.*}
```
```   285 lemma list_N_0: "list_N(A,0) = 0"
```
```   286 by (simp add: list_N_def)
```
```   287
```
```   288 lemma list_N_succ: "list_N(A,succ(n)) = {0} + A * (list_N(A,n))"
```
```   289 by (simp add: list_N_def)
```
```   290
```
```   291 lemma list_N_imp_list:
```
```   292   "[| l \<in> list_N(A,n); n \<in> nat |] ==> l \<in> list(A)"
```
```   293 by (force simp add: list_eq_Union list_N_def)
```
```   294
```
```   295 lemma list_N_imp_length_lt [rule_format]:
```
```   296      "n \<in> nat ==> \<forall>l \<in> list_N(A,n). length(l) < n"
```
```   297 apply (induct_tac n)
```
```   298 apply (auto simp add: list_N_0 list_N_succ
```
```   299                       Nil_def [symmetric] Cons_def [symmetric])
```
```   300 done
```
```   301
```
```   302 lemma list_imp_list_N [rule_format]:
```
```   303      "l \<in> list(A) ==> \<forall>n\<in>nat. length(l) < n --> l \<in> list_N(A, n)"
```
```   304 apply (induct_tac l)
```
```   305 apply (force elim: natE)+
```
```   306 done
```
```   307
```
```   308 lemma list_N_imp_eq_length:
```
```   309       "[|n \<in> nat; l \<notin> list_N(A, n); l \<in> list_N(A, succ(n))|]
```
```   310        ==> n = length(l)"
```
```   311 apply (rule le_anti_sym)
```
```   312  prefer 2 apply (simp add: list_N_imp_length_lt)
```
```   313 apply (frule list_N_imp_list, simp)
```
```   314 apply (simp add: not_lt_iff_le [symmetric])
```
```   315 apply (blast intro: list_imp_list_N)
```
```   316 done
```
```   317
```
```   318 text{*Express @{term list_rec} without using @{term rank} or @{term Vset},
```
```   319 neither of which is absolute.*}
```
```   320 lemma (in M_triv_axioms) list_rec_eq:
```
```   321   "l \<in> list(A) ==>
```
```   322    list_rec(a,g,l) =
```
```   323    transrec (succ(length(l)),
```
```   324       \<lambda>x h. Lambda (list(A),
```
```   325                     list_case' (a,
```
```   326                            \<lambda>a l. g(a, l, h ` succ(length(l)) ` l)))) ` l"
```
```   327 apply (induct_tac l)
```
```   328 apply (subst transrec, simp)
```
```   329 apply (subst transrec)
```
```   330 apply (simp add: list_imp_list_N)
```
```   331 done
```
```   332
```
```   333 constdefs
```
```   334   is_list_N :: "[i=>o,i,i,i] => o"
```
```   335     "is_list_N(M,A,n,Z) ==
```
```   336       \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
```
```   337        empty(M,zero) &
```
```   338        successor(M,n,sn) & membership(M,sn,msn) &
```
```   339        is_wfrec(M, iterates_MH(M, is_list_functor(M,A),zero), msn, n, Z)"
```
```   340
```
```   341   mem_list :: "[i=>o,i,i] => o"
```
```   342     "mem_list(M,A,l) ==
```
```   343       \<exists>n[M]. \<exists>listn[M].
```
```   344        finite_ordinal(M,n) & is_list_N(M,A,n,listn) & l \<in> listn"
```
```   345
```
```   346   is_list :: "[i=>o,i,i] => o"
```
```   347     "is_list(M,A,Z) == \<forall>l[M]. l \<in> Z <-> mem_list(M,A,l)"
```
```   348
```
```   349 constdefs
```
```   350   is_formula_n :: "[i=>o,i,i] => o"
```
```   351     "is_formula_n(M,n,Z) ==
```
```   352       \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
```
```   353        empty(M,zero) &
```
```   354        successor(M,n,sn) & membership(M,sn,msn) &
```
```   355        is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"
```
```   356
```
```   357   mem_formula :: "[i=>o,i] => o"
```
```   358     "mem_formula(M,p) ==
```
```   359       \<exists>n[M]. \<exists>formn[M].
```
```   360        finite_ordinal(M,n) & is_formula_n(M,n,formn) & p \<in> formn"
```
```   361
```
```   362   is_formula :: "[i=>o,i] => o"
```
```   363     "is_formula(M,Z) == \<forall>p[M]. p \<in> Z <-> mem_formula(M,p)"
```
```   364
```
```   365 locale (open) M_datatypes = M_wfrank +
```
```   366  assumes list_replacement1:
```
```   367    "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
```
```   368   and list_replacement2:
```
```   369    "M(A) ==> strong_replacement(M,
```
```   370          \<lambda>n y. n\<in>nat &
```
```   371                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   372                is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0),
```
```   373                         msn, n, y)))"
```
```   374   and formula_replacement1:
```
```   375    "iterates_replacement(M, is_formula_functor(M), 0)"
```
```   376   and formula_replacement2:
```
```   377    "strong_replacement(M,
```
```   378          \<lambda>n y. n\<in>nat &
```
```   379                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   380                is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0),
```
```   381                         msn, n, y)))"
```
```   382   and nth_replacement:
```
```   383    "M(l) ==> iterates_replacement(M, %l t. is_tl(M,l,t), l)"
```
```   384
```
```   385
```
```   386 subsubsection{*Absoluteness of the List Construction*}
```
```   387
```
```   388 lemma (in M_datatypes) list_replacement2':
```
```   389   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
```
```   390 apply (insert list_replacement2 [of A])
```
```   391 apply (rule strong_replacement_cong [THEN iffD1])
```
```   392 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
```
```   393 apply (simp_all add: list_replacement1 relativize1_def)
```
```   394 done
```
```   395
```
```   396 lemma (in M_datatypes) list_closed [intro,simp]:
```
```   397      "M(A) ==> M(list(A))"
```
```   398 apply (insert list_replacement1)
```
```   399 by  (simp add: RepFun_closed2 list_eq_Union
```
```   400                list_replacement2' relativize1_def
```
```   401                iterates_closed [of "is_list_functor(M,A)"])
```
```   402
```
```   403 text{*WARNING: use only with @{text "dest:"} or with variables fixed!*}
```
```   404 lemmas (in M_datatypes) list_into_M = transM [OF _ list_closed]
```
```   405
```
```   406 lemma (in M_datatypes) list_N_abs [simp]:
```
```   407      "[|M(A); n\<in>nat; M(Z)|]
```
```   408       ==> is_list_N(M,A,n,Z) <-> Z = list_N(A,n)"
```
```   409 apply (insert list_replacement1)
```
```   410 apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
```
```   411                  iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
```
```   412 done
```
```   413
```
```   414 lemma (in M_datatypes) list_N_closed [intro,simp]:
```
```   415      "[|M(A); n\<in>nat|] ==> M(list_N(A,n))"
```
```   416 apply (insert list_replacement1)
```
```   417 apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
```
```   418                  iterates_closed [of "is_list_functor(M,A)"])
```
```   419 done
```
```   420
```
```   421 lemma (in M_datatypes) mem_list_abs [simp]:
```
```   422      "M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"
```
```   423 apply (insert list_replacement1)
```
```   424 apply (simp add: mem_list_def list_N_def relativize1_def list_eq_Union
```
```   425                  iterates_closed [of "is_list_functor(M,A)"])
```
```   426 done
```
```   427
```
```   428 lemma (in M_datatypes) list_abs [simp]:
```
```   429      "[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"
```
```   430 apply (simp add: is_list_def, safe)
```
```   431 apply (rule M_equalityI, simp_all)
```
```   432 done
```
```   433
```
```   434 subsubsection{*Absoluteness of Formulas*}
```
```   435
```
```   436 lemma (in M_datatypes) formula_replacement2':
```
```   437   "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))^n (0))"
```
```   438 apply (insert formula_replacement2)
```
```   439 apply (rule strong_replacement_cong [THEN iffD1])
```
```   440 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
```
```   441 apply (simp_all add: formula_replacement1 relativize1_def)
```
```   442 done
```
```   443
```
```   444 lemma (in M_datatypes) formula_closed [intro,simp]:
```
```   445      "M(formula)"
```
```   446 apply (insert formula_replacement1)
```
```   447 apply  (simp add: RepFun_closed2 formula_eq_Union
```
```   448                   formula_replacement2' relativize1_def
```
```   449                   iterates_closed [of "is_formula_functor(M)"])
```
```   450 done
```
```   451
```
```   452 lemmas (in M_datatypes) formula_into_M = transM [OF _ formula_closed]
```
```   453
```
```   454 lemma (in M_datatypes) is_formula_n_abs [simp]:
```
```   455      "[|n\<in>nat; M(Z)|]
```
```   456       ==> is_formula_n(M,n,Z) <->
```
```   457           Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X))^n (0)"
```
```   458 apply (insert formula_replacement1)
```
```   459 apply (simp add: is_formula_n_def relativize1_def nat_into_M
```
```   460                  iterates_abs [of "is_formula_functor(M)" _
```
```   461                         "\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)"])
```
```   462 done
```
```   463
```
```   464 lemma (in M_datatypes) mem_formula_abs [simp]:
```
```   465      "mem_formula(M,l) <-> l \<in> formula"
```
```   466 apply (insert formula_replacement1)
```
```   467 apply (simp add: mem_formula_def relativize1_def formula_eq_Union
```
```   468                  iterates_closed [of "is_formula_functor(M)"])
```
```   469 done
```
```   470
```
```   471 lemma (in M_datatypes) formula_abs [simp]:
```
```   472      "[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"
```
```   473 apply (simp add: is_formula_def, safe)
```
```   474 apply (rule M_equalityI, simp_all)
```
```   475 done
```
```   476
```
```   477
```
```   478 subsection{*Absoluteness for Some List Operators*}
```
```   479
```
```   480 subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}
```
```   481
```
```   482 text{*Re-expresses eclose using "iterates"*}
```
```   483 lemma eclose_eq_Union:
```
```   484      "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
```
```   485 apply (simp add: eclose_def)
```
```   486 apply (rule UN_cong)
```
```   487 apply (rule refl)
```
```   488 apply (induct_tac n)
```
```   489 apply (simp add: nat_rec_0)
```
```   490 apply (simp add: nat_rec_succ)
```
```   491 done
```
```   492
```
```   493 constdefs
```
```   494   is_eclose_n :: "[i=>o,i,i,i] => o"
```
```   495     "is_eclose_n(M,A,n,Z) ==
```
```   496       \<exists>sn[M]. \<exists>msn[M].
```
```   497        successor(M,n,sn) & membership(M,sn,msn) &
```
```   498        is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"
```
```   499
```
```   500   mem_eclose :: "[i=>o,i,i] => o"
```
```   501     "mem_eclose(M,A,l) ==
```
```   502       \<exists>n[M]. \<exists>eclosen[M].
```
```   503        finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
```
```   504
```
```   505   is_eclose :: "[i=>o,i,i] => o"
```
```   506     "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"
```
```   507
```
```   508
```
```   509 locale (open) M_eclose = M_datatypes +
```
```   510  assumes eclose_replacement1:
```
```   511    "M(A) ==> iterates_replacement(M, big_union(M), A)"
```
```   512   and eclose_replacement2:
```
```   513    "M(A) ==> strong_replacement(M,
```
```   514          \<lambda>n y. n\<in>nat &
```
```   515                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   516                is_wfrec(M, iterates_MH(M,big_union(M), A),
```
```   517                         msn, n, y)))"
```
```   518
```
```   519 lemma (in M_eclose) eclose_replacement2':
```
```   520   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
```
```   521 apply (insert eclose_replacement2 [of A])
```
```   522 apply (rule strong_replacement_cong [THEN iffD1])
```
```   523 apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]])
```
```   524 apply (simp_all add: eclose_replacement1 relativize1_def)
```
```   525 done
```
```   526
```
```   527 lemma (in M_eclose) eclose_closed [intro,simp]:
```
```   528      "M(A) ==> M(eclose(A))"
```
```   529 apply (insert eclose_replacement1)
```
```   530 by  (simp add: RepFun_closed2 eclose_eq_Union
```
```   531                eclose_replacement2' relativize1_def
```
```   532                iterates_closed [of "big_union(M)"])
```
```   533
```
```   534 lemma (in M_eclose) is_eclose_n_abs [simp]:
```
```   535      "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"
```
```   536 apply (insert eclose_replacement1)
```
```   537 apply (simp add: is_eclose_n_def relativize1_def nat_into_M
```
```   538                  iterates_abs [of "big_union(M)" _ "Union"])
```
```   539 done
```
```   540
```
```   541 lemma (in M_eclose) mem_eclose_abs [simp]:
```
```   542      "M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"
```
```   543 apply (insert eclose_replacement1)
```
```   544 apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union
```
```   545                  iterates_closed [of "big_union(M)"])
```
```   546 done
```
```   547
```
```   548 lemma (in M_eclose) eclose_abs [simp]:
```
```   549      "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"
```
```   550 apply (simp add: is_eclose_def, safe)
```
```   551 apply (rule M_equalityI, simp_all)
```
```   552 done
```
```   553
```
```   554
```
```   555
```
```   556
```
```   557 subsection {*Absoluteness for @{term transrec}*}
```
```   558
```
```   559
```
```   560 text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}
```
```   561 constdefs
```
```   562
```
```   563   is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
```
```   564    "is_transrec(M,MH,a,z) ==
```
```   565       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
```
```   566        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
```
```   567        is_wfrec(M,MH,mesa,a,z)"
```
```   568
```
```   569   transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"
```
```   570    "transrec_replacement(M,MH,a) ==
```
```   571       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
```
```   572        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
```
```   573        wfrec_replacement(M,MH,mesa)"
```
```   574
```
```   575 text{*The condition @{term "Ord(i)"} lets us use the simpler
```
```   576   @{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},
```
```   577   which I haven't even proved yet. *}
```
```   578 theorem (in M_eclose) transrec_abs:
```
```   579   "[|transrec_replacement(M,MH,i);  relativize2(M,MH,H);
```
```   580      Ord(i);  M(i);  M(z);
```
```   581      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   582    ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)"
```
```   583 apply (rotate_tac 2)
```
```   584 apply (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
```
```   585        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
```
```   586 done
```
```   587
```
```   588
```
```   589 theorem (in M_eclose) transrec_closed:
```
```   590      "[|transrec_replacement(M,MH,i);  relativize2(M,MH,H);
```
```   591 	Ord(i);  M(i);
```
```   592 	\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   593       ==> M(transrec(i,H))"
```
```   594 apply (rotate_tac 2)
```
```   595 apply (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
```
```   596        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
```
```   597 done
```
```   598
```
```   599
```
```   600 subsection{*Absoluteness for the List Operator @{term length}*}
```
```   601 constdefs
```
```   602
```
```   603   is_length :: "[i=>o,i,i,i] => o"
```
```   604     "is_length(M,A,l,n) ==
```
```   605        \<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M].
```
```   606         is_list_N(M,A,n,list_n) & l \<notin> list_n &
```
```   607         successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn"
```
```   608
```
```   609
```
```   610 lemma (in M_datatypes) length_abs [simp]:
```
```   611      "[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) <-> n = length(l)"
```
```   612 apply (subgoal_tac "M(l) & M(n)")
```
```   613  prefer 2 apply (blast dest: transM)
```
```   614 apply (simp add: is_length_def)
```
```   615 apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length
```
```   616              dest: list_N_imp_length_lt)
```
```   617 done
```
```   618
```
```   619 text{*Proof is trivial since @{term length} returns natural numbers.*}
```
```   620 lemma (in M_triv_axioms) length_closed [intro,simp]:
```
```   621      "l \<in> list(A) ==> M(length(l))"
```
```   622 by (simp add: nat_into_M)
```
```   623
```
```   624
```
```   625 subsection {*Absoluteness for @{term nth}*}
```
```   626
```
```   627 lemma nth_eq_hd_iterates_tl [rule_format]:
```
```   628      "xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))"
```
```   629 apply (induct_tac xs)
```
```   630 apply (simp add: iterates_tl_Nil hd'_Nil, clarify)
```
```   631 apply (erule natE)
```
```   632 apply (simp add: hd'_Cons)
```
```   633 apply (simp add: tl'_Cons iterates_commute)
```
```   634 done
```
```   635
```
```   636 lemma (in M_axioms) iterates_tl'_closed:
```
```   637      "[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"
```
```   638 apply (induct_tac n, simp)
```
```   639 apply (simp add: tl'_Cons tl'_closed)
```
```   640 done
```
```   641
```
```   642 text{*Immediate by type-checking*}
```
```   643 lemma (in M_datatypes) nth_closed [intro,simp]:
```
```   644      "[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))"
```
```   645 apply (case_tac "n < length(xs)")
```
```   646  apply (blast intro: nth_type transM)
```
```   647 apply (simp add: not_lt_iff_le nth_eq_0)
```
```   648 done
```
```   649
```
```   650 constdefs
```
```   651   is_nth :: "[i=>o,i,i,i] => o"
```
```   652     "is_nth(M,n,l,Z) ==
```
```   653       \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M].
```
```   654        successor(M,n,sn) & membership(M,sn,msn) &
```
```   655        is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
```
```   656        is_hd(M,X,Z)"
```
```   657
```
```   658 lemma (in M_datatypes) nth_abs [simp]:
```
```   659      "[|M(A); n \<in> nat; l \<in> list(A); M(Z)|]
```
```   660       ==> is_nth(M,n,l,Z) <-> Z = nth(n,l)"
```
```   661 apply (subgoal_tac "M(l)")
```
```   662  prefer 2 apply (blast intro: transM)
```
```   663 apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M
```
```   664                  tl'_closed iterates_tl'_closed
```
```   665                  iterates_abs [OF _ relativize1_tl] nth_replacement)
```
```   666 done
```
```   667
```
```   668
```
```   669 subsection{*Relativization and Absoluteness for the @{term formula} Constructors*}
```
```   670
```
```   671 constdefs
```
```   672   is_Member :: "[i=>o,i,i,i] => o"
```
```   673      --{* because @{term "Member(x,y) \<equiv> Inl(Inl(\<langle>x,y\<rangle>))"}*}
```
```   674     "is_Member(M,x,y,Z) ==
```
```   675 	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inl(M,u,Z)"
```
```   676
```
```   677 lemma (in M_triv_axioms) Member_abs [simp]:
```
```   678      "[|M(x); M(y); M(Z)|] ==> is_Member(M,x,y,Z) <-> (Z = Member(x,y))"
```
```   679 by (simp add: is_Member_def Member_def)
```
```   680
```
```   681 lemma (in M_triv_axioms) Member_in_M_iff [iff]:
```
```   682      "M(Member(x,y)) <-> M(x) & M(y)"
```
```   683 by (simp add: Member_def)
```
```   684
```
```   685 constdefs
```
```   686   is_Equal :: "[i=>o,i,i,i] => o"
```
```   687      --{* because @{term "Equal(x,y) \<equiv> Inl(Inr(\<langle>x,y\<rangle>))"}*}
```
```   688     "is_Equal(M,x,y,Z) ==
```
```   689 	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inr(M,p,u) & is_Inl(M,u,Z)"
```
```   690
```
```   691 lemma (in M_triv_axioms) Equal_abs [simp]:
```
```   692      "[|M(x); M(y); M(Z)|] ==> is_Equal(M,x,y,Z) <-> (Z = Equal(x,y))"
```
```   693 by (simp add: is_Equal_def Equal_def)
```
```   694
```
```   695 lemma (in M_triv_axioms) Equal_in_M_iff [iff]: "M(Equal(x,y)) <-> M(x) & M(y)"
```
```   696 by (simp add: Equal_def)
```
```   697
```
```   698 constdefs
```
```   699   is_Nand :: "[i=>o,i,i,i] => o"
```
```   700      --{* because @{term "Nand(x,y) \<equiv> Inr(Inl(\<langle>x,y\<rangle>))"}*}
```
```   701     "is_Nand(M,x,y,Z) ==
```
```   702 	\<exists>p[M]. \<exists>u[M]. pair(M,x,y,p) & is_Inl(M,p,u) & is_Inr(M,u,Z)"
```
```   703
```
```   704 lemma (in M_triv_axioms) Nand_abs [simp]:
```
```   705      "[|M(x); M(y); M(Z)|] ==> is_Nand(M,x,y,Z) <-> (Z = Nand(x,y))"
```
```   706 by (simp add: is_Nand_def Nand_def)
```
```   707
```
```   708 lemma (in M_triv_axioms) Nand_in_M_iff [iff]: "M(Nand(x,y)) <-> M(x) & M(y)"
```
```   709 by (simp add: Nand_def)
```
```   710
```
```   711 constdefs
```
```   712   is_Forall :: "[i=>o,i,i] => o"
```
```   713      --{* because @{term "Forall(x) \<equiv> Inr(Inr(p))"}*}
```
```   714     "is_Forall(M,p,Z) == \<exists>u[M]. is_Inr(M,p,u) & is_Inr(M,u,Z)"
```
```   715
```
```   716 lemma (in M_triv_axioms) Forall_abs [simp]:
```
```   717      "[|M(x); M(Z)|] ==> is_Forall(M,x,Z) <-> (Z = Forall(x))"
```
```   718 by (simp add: is_Forall_def Forall_def)
```
```   719
```
```   720 lemma (in M_triv_axioms) Forall_in_M_iff [iff]: "M(Forall(x)) <-> M(x)"
```
```   721 by (simp add: Forall_def)
```
```   722
```
```   723
```
```   724 subsection {*Absoluteness for @{term formula_rec}*}
```
```   725
```
```   726 subsubsection{*@{term is_formula_case}: relativization of @{term formula_case}*}
```
```   727
```
```   728 constdefs
```
```   729
```
```   730  is_formula_case ::
```
```   731     "[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o"
```
```   732   --{*no constraint on non-formulas*}
```
```   733   "is_formula_case(M, is_a, is_b, is_c, is_d, p, z) ==
```
```   734       (\<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> is_Member(M,x,y,p) --> is_a(x,y,z)) &
```
```   735       (\<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> is_Equal(M,x,y,p) --> is_b(x,y,z)) &
```
```   736       (\<forall>x[M]. \<forall>y[M]. x\<in>formula --> y\<in>formula -->
```
```   737                      is_Nand(M,x,y,p) --> is_c(x,y,z)) &
```
```   738       (\<forall>x[M]. x\<in>formula --> is_Forall(M,x,p) --> is_d(x,z))"
```
```   739
```
```   740 lemma (in M_datatypes) formula_case_abs [simp]:
```
```   741      "[| Relativize2(M,nat,nat,is_a,a); Relativize2(M,nat,nat,is_b,b);
```
```   742          Relativize2(M,formula,formula,is_c,c); Relativize1(M,formula,is_d,d);
```
```   743          p \<in> formula; M(z) |]
```
```   744       ==> is_formula_case(M,is_a,is_b,is_c,is_d,p,z) <->
```
```   745           z = formula_case(a,b,c,d,p)"
```
```   746 apply (simp add: formula_into_M is_formula_case_def)
```
```   747 apply (erule formula.cases)
```
```   748    apply (simp_all add: Relativize1_def Relativize2_def)
```
```   749 done
```
```   750
```
```   751
```
```   752 subsubsection{*@{term quasiformula}: For Case-Splitting with @{term formula_case'}*}
```
```   753
```
```   754 constdefs
```
```   755
```
```   756   quasiformula :: "i => o"
```
```   757     "quasiformula(p) ==
```
```   758 	(\<exists>x y. p = Member(x,y)) |
```
```   759 	(\<exists>x y. p = Equal(x,y)) |
```
```   760 	(\<exists>x y. p = Nand(x,y)) |
```
```   761 	(\<exists>x. p = Forall(x))"
```
```   762
```
```   763   is_quasiformula :: "[i=>o,i] => o"
```
```   764     "is_quasiformula(M,p) ==
```
```   765 	(\<exists>x[M]. \<exists>y[M]. is_Member(M,x,y,p)) |
```
```   766 	(\<exists>x[M]. \<exists>y[M]. is_Equal(M,x,y,p)) |
```
```   767 	(\<exists>x[M]. \<exists>y[M]. is_Nand(M,x,y,p)) |
```
```   768 	(\<exists>x[M]. is_Forall(M,x,p))"
```
```   769
```
```   770 lemma [iff]: "quasiformula(Member(x,y))"
```
```   771 by (simp add: quasiformula_def)
```
```   772
```
```   773 lemma [iff]: "quasiformula(Equal(x,y))"
```
```   774 by (simp add: quasiformula_def)
```
```   775
```
```   776 lemma [iff]: "quasiformula(Nand(x,y))"
```
```   777 by (simp add: quasiformula_def)
```
```   778
```
```   779 lemma [iff]: "quasiformula(Forall(x))"
```
```   780 by (simp add: quasiformula_def)
```
```   781
```
```   782 lemma formula_imp_quasiformula: "p \<in> formula ==> quasiformula(p)"
```
```   783 by (erule formula.cases, simp_all)
```
```   784
```
```   785 lemma (in M_triv_axioms) quasiformula_abs [simp]:
```
```   786      "M(z) ==> is_quasiformula(M,z) <-> quasiformula(z)"
```
```   787 by (auto simp add: is_quasiformula_def quasiformula_def)
```
```   788
```
```   789 constdefs
```
```   790
```
```   791   formula_case' :: "[[i,i]=>i, [i,i]=>i, [i,i]=>i, i=>i, i] => i"
```
```   792     --{*A version of @{term formula_case} that's always defined.*}
```
```   793     "formula_case'(a,b,c,d,p) ==
```
```   794        if quasiformula(p) then formula_case(a,b,c,d,p) else 0"
```
```   795
```
```   796   is_formula_case' ::
```
```   797       "[i=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i,i]=>o, [i,i]=>o, i, i] => o"
```
```   798     --{*Returns 0 for non-formulas*}
```
```   799     "is_formula_case'(M, is_a, is_b, is_c, is_d, p, z) ==
```
```   800 	(\<forall>x[M]. \<forall>y[M]. is_Member(M,x,y,p) --> is_a(x,y,z)) &
```
```   801 	(\<forall>x[M]. \<forall>y[M]. is_Equal(M,x,y,p) --> is_b(x,y,z)) &
```
```   802 	(\<forall>x[M]. \<forall>y[M]. is_Nand(M,x,y,p) --> is_c(x,y,z)) &
```
```   803 	(\<forall>x[M]. is_Forall(M,x,p) --> is_d(x,z)) &
```
```   804         (is_quasiformula(M,p) | empty(M,z))"
```
```   805
```
```   806 subsubsection{*@{term formula_case'}, the Modified Version of @{term formula_case}*}
```
```   807
```
```   808 lemma formula_case'_Member [simp]:
```
```   809      "formula_case'(a,b,c,d,Member(x,y)) = a(x,y)"
```
```   810 by (simp add: formula_case'_def)
```
```   811
```
```   812 lemma formula_case'_Equal [simp]:
```
```   813      "formula_case'(a,b,c,d,Equal(x,y)) = b(x,y)"
```
```   814 by (simp add: formula_case'_def)
```
```   815
```
```   816 lemma formula_case'_Nand [simp]:
```
```   817      "formula_case'(a,b,c,d,Nand(x,y)) = c(x,y)"
```
```   818 by (simp add: formula_case'_def)
```
```   819
```
```   820 lemma formula_case'_Forall [simp]:
```
```   821      "formula_case'(a,b,c,d,Forall(x)) = d(x)"
```
```   822 by (simp add: formula_case'_def)
```
```   823
```
```   824 lemma non_formula_case: "~ quasiformula(x) ==> formula_case'(a,b,c,d,x) = 0"
```
```   825 by (simp add: quasiformula_def formula_case'_def)
```
```   826
```
```   827 lemma formula_case'_eq_formula_case [simp]:
```
```   828      "p \<in> formula ==>formula_case'(a,b,c,d,p) = formula_case(a,b,c,d,p)"
```
```   829 by (erule formula.cases, simp_all)
```
```   830
```
```   831 lemma (in M_axioms) formula_case'_closed [intro,simp]:
```
```   832   "[|M(p); \<forall>x[M]. \<forall>y[M]. M(a(x,y));
```
```   833            \<forall>x[M]. \<forall>y[M]. M(b(x,y));
```
```   834            \<forall>x[M]. \<forall>y[M]. M(c(x,y));
```
```   835            \<forall>x[M]. M(d(x))|] ==> M(formula_case'(a,b,c,d,p))"
```
```   836 apply (case_tac "quasiformula(p)")
```
```   837  apply (simp add: quasiformula_def, force)
```
```   838 apply (simp add: non_formula_case)
```
```   839 done
```
```   840
```
```   841 text{*Compared with @{text formula_case_closed'}, the premise on @{term p} is
```
```   842       stronger while the other premises are weaker, incorporating typing
```
```   843       information.*}
```
```   844 lemma (in M_datatypes) formula_case_closed [intro,simp]:
```
```   845   "[|p \<in> formula;
```
```   846      \<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> M(a(x,y));
```
```   847      \<forall>x[M]. \<forall>y[M]. x\<in>nat --> y\<in>nat --> M(b(x,y));
```
```   848      \<forall>x[M]. \<forall>y[M]. x\<in>formula --> y\<in>formula --> M(c(x,y));
```
```   849      \<forall>x[M]. x\<in>formula --> M(d(x))|] ==> M(formula_case(a,b,c,d,p))"
```
```   850 by (erule formula.cases, simp_all)
```
```   851
```
```   852 lemma (in M_triv_axioms) formula_case'_abs [simp]:
```
```   853      "[| relativize2(M,is_a,a); relativize2(M,is_b,b);
```
```   854          relativize2(M,is_c,c); relativize1(M,is_d,d); M(p); M(z) |]
```
```   855       ==> is_formula_case'(M,is_a,is_b,is_c,is_d,p,z) <->
```
```   856           z = formula_case'(a,b,c,d,p)"
```
```   857 apply (case_tac "quasiformula(p)")
```
```   858  prefer 2
```
```   859  apply (simp add: is_formula_case'_def non_formula_case)
```
```   860  apply (force simp add: quasiformula_def)
```
```   861 apply (simp add: quasiformula_def is_formula_case'_def)
```
```   862 apply (elim disjE exE)
```
```   863  apply (simp_all add: relativize1_def relativize2_def)
```
```   864 done
```
```   865
```
```   866
```
```   867 subsubsection{*Towards Absoluteness of @{term formula_rec}*}
```
```   868
```
```   869 consts   depth :: "i=>i"
```
```   870 primrec
```
```   871   "depth(Member(x,y)) = 0"
```
```   872   "depth(Equal(x,y))  = 0"
```
```   873   "depth(Nand(p,q)) = succ(depth(p) \<union> depth(q))"
```
```   874   "depth(Forall(p)) = succ(depth(p))"
```
```   875
```
```   876 lemma depth_type [TC]: "p \<in> formula ==> depth(p) \<in> nat"
```
```   877 by (induct_tac p, simp_all)
```
```   878
```
```   879
```
```   880 constdefs
```
```   881   formula_N :: "i => i"
```
```   882     "formula_N(n) == (\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)) ^ n (0)"
```
```   883
```
```   884 lemma Member_in_formula_N [simp]:
```
```   885      "Member(x,y) \<in> formula_N(succ(n)) <-> x \<in> nat & y \<in> nat"
```
```   886 by (simp add: formula_N_def Member_def)
```
```   887
```
```   888 lemma Equal_in_formula_N [simp]:
```
```   889      "Equal(x,y) \<in> formula_N(succ(n)) <-> x \<in> nat & y \<in> nat"
```
```   890 by (simp add: formula_N_def Equal_def)
```
```   891
```
```   892 lemma Nand_in_formula_N [simp]:
```
```   893      "Nand(x,y) \<in> formula_N(succ(n)) <-> x \<in> formula_N(n) & y \<in> formula_N(n)"
```
```   894 by (simp add: formula_N_def Nand_def)
```
```   895
```
```   896 lemma Forall_in_formula_N [simp]:
```
```   897      "Forall(x) \<in> formula_N(succ(n)) <-> x \<in> formula_N(n)"
```
```   898 by (simp add: formula_N_def Forall_def)
```
```   899
```
```   900 text{*These two aren't simprules because they reveal the underlying
```
```   901 formula representation.*}
```
```   902 lemma formula_N_0: "formula_N(0) = 0"
```
```   903 by (simp add: formula_N_def)
```
```   904
```
```   905 lemma formula_N_succ:
```
```   906      "formula_N(succ(n)) =
```
```   907       ((nat*nat) + (nat*nat)) + (formula_N(n) * formula_N(n) + formula_N(n))"
```
```   908 by (simp add: formula_N_def)
```
```   909
```
```   910 lemma formula_N_imp_formula:
```
```   911   "[| p \<in> formula_N(n); n \<in> nat |] ==> p \<in> formula"
```
```   912 by (force simp add: formula_eq_Union formula_N_def)
```
```   913
```
```   914 lemma formula_N_imp_depth_lt [rule_format]:
```
```   915      "n \<in> nat ==> \<forall>p \<in> formula_N(n). depth(p) < n"
```
```   916 apply (induct_tac n)
```
```   917 apply (auto simp add: formula_N_0 formula_N_succ
```
```   918                       depth_type formula_N_imp_formula Un_least_lt_iff
```
```   919                       Member_def [symmetric] Equal_def [symmetric]
```
```   920                       Nand_def [symmetric] Forall_def [symmetric])
```
```   921 done
```
```   922
```
```   923 lemma formula_imp_formula_N [rule_format]:
```
```   924      "p \<in> formula ==> \<forall>n\<in>nat. depth(p) < n --> p \<in> formula_N(n)"
```
```   925 apply (induct_tac p)
```
```   926 apply (simp_all add: succ_Un_distrib Un_least_lt_iff)
```
```   927 apply (force elim: natE)+
```
```   928 done
```
```   929
```
```   930 lemma formula_N_imp_eq_depth:
```
```   931       "[|n \<in> nat; p \<notin> formula_N(n); p \<in> formula_N(succ(n))|]
```
```   932        ==> n = depth(p)"
```
```   933 apply (rule le_anti_sym)
```
```   934  prefer 2 apply (simp add: formula_N_imp_depth_lt)
```
```   935 apply (frule formula_N_imp_formula, simp)
```
```   936 apply (simp add: not_lt_iff_le [symmetric])
```
```   937 apply (blast intro: formula_imp_formula_N)
```
```   938 done
```
```   939
```
```   940
```
```   941
```
```   942 lemma formula_N_mono [rule_format]:
```
```   943   "[| m \<in> nat; n \<in> nat |] ==> m\<le>n --> formula_N(m) \<subseteq> formula_N(n)"
```
```   944 apply (rule_tac m = m and n = n in diff_induct)
```
```   945 apply (simp_all add: formula_N_0 formula_N_succ, blast)
```
```   946 done
```
```   947
```
```   948 lemma formula_N_distrib:
```
```   949   "[| m \<in> nat; n \<in> nat |] ==> formula_N(m \<union> n) = formula_N(m) \<union> formula_N(n)"
```
```   950 apply (rule_tac i = m and j = n in Ord_linear_le, auto)
```
```   951 apply (simp_all add: subset_Un_iff [THEN iffD1] subset_Un_iff2 [THEN iffD1]
```
```   952                      le_imp_subset formula_N_mono)
```
```   953 done
```
```   954
```
```   955 text{*Express @{term formula_rec} without using @{term rank} or @{term Vset},
```
```   956 neither of which is absolute.*}
```
```   957 lemma (in M_triv_axioms) formula_rec_eq:
```
```   958   "p \<in> formula ==>
```
```   959    formula_rec(a,b,c,d,p) =
```
```   960    transrec (succ(depth(p)),
```
```   961       \<lambda>x h. Lambda (formula,
```
```   962              formula_case' (a, b,
```
```   963                 \<lambda>u v. c(u, v, h ` succ(depth(u)) ` u,
```
```   964                               h ` succ(depth(v)) ` v),
```
```   965                 \<lambda>u. d(u, h ` succ(depth(u)) ` u))))
```
```   966    ` p"
```
```   967 apply (induct_tac p)
```
```   968    txt{*Base case for @{term Member}*}
```
```   969    apply (subst transrec, simp add: formula.intros)
```
```   970   txt{*Base case for @{term Equal}*}
```
```   971   apply (subst transrec, simp add: formula.intros)
```
```   972  txt{*Inductive step for @{term Nand}*}
```
```   973  apply (subst transrec)
```
```   974  apply (simp add: succ_Un_distrib formula.intros)
```
```   975 txt{*Inductive step for @{term Forall}*}
```
```   976 apply (subst transrec)
```
```   977 apply (simp add: formula_imp_formula_N formula.intros)
```
```   978 done
```
```   979
```
```   980
```
```   981 constdefs
```
```   982   is_formula_N :: "[i=>o,i,i] => o"
```
```   983     "is_formula_N(M,n,Z) ==
```
```   984       \<exists>zero[M]. \<exists>sn[M]. \<exists>msn[M].
```
```   985        empty(M,zero) &
```
```   986        successor(M,n,sn) & membership(M,sn,msn) &
```
```   987        is_wfrec(M, iterates_MH(M, is_formula_functor(M),zero), msn, n, Z)"
```
```   988
```
```   989
```
```   990 lemma (in M_datatypes) formula_N_abs [simp]:
```
```   991      "[|n\<in>nat; M(Z)|]
```
```   992       ==> is_formula_N(M,n,Z) <-> Z = formula_N(n)"
```
```   993 apply (insert formula_replacement1)
```
```   994 apply (simp add: is_formula_N_def formula_N_def relativize1_def nat_into_M
```
```   995                  iterates_abs [of "is_formula_functor(M)" _
```
```   996                                   "\<lambda>X. ((nat*nat) + (nat*nat)) + (X*X + X)"])
```
```   997 done
```
```   998
```
```   999 lemma (in M_datatypes) formula_N_closed [intro,simp]:
```
```  1000      "n\<in>nat ==> M(formula_N(n))"
```
```  1001 apply (insert formula_replacement1)
```
```  1002 apply (simp add: is_formula_N_def formula_N_def relativize1_def nat_into_M
```
```  1003                  iterates_closed [of "is_formula_functor(M)"])
```
```  1004 done
```
```  1005
```
```  1006 subsection{*Absoluteness for the Formula Operator @{term depth}*}
```
```  1007 constdefs
```
```  1008
```
```  1009   is_depth :: "[i=>o,i,i] => o"
```
```  1010     "is_depth(M,p,n) ==
```
```  1011        \<exists>sn[M]. \<exists>formula_n[M]. \<exists>formula_sn[M].
```
```  1012         is_formula_N(M,n,formula_n) & p \<notin> formula_n &
```
```  1013         successor(M,n,sn) & is_formula_N(M,sn,formula_sn) & p \<in> formula_sn"
```
```  1014
```
```  1015
```
```  1016 lemma (in M_datatypes) depth_abs [simp]:
```
```  1017      "[|p \<in> formula; n \<in> nat|] ==> is_depth(M,p,n) <-> n = depth(p)"
```
```  1018 apply (subgoal_tac "M(p) & M(n)")
```
```  1019  prefer 2 apply (blast dest: transM)
```
```  1020 apply (simp add: is_depth_def)
```
```  1021 apply (blast intro: formula_imp_formula_N nat_into_Ord formula_N_imp_eq_depth
```
```  1022              dest: formula_N_imp_depth_lt)
```
```  1023 done
```
```  1024
```
```  1025 text{*Proof is trivial since @{term depth} returns natural numbers.*}
```
```  1026 lemma (in M_triv_axioms) depth_closed [intro,simp]:
```
```  1027      "p \<in> formula ==> M(depth(p))"
```
```  1028 by (simp add: nat_into_M)
```
```  1029
```
```  1030 end
```