src/ZF/Constructible/Datatype_absolute.thy
 author paulson Fri Jul 19 13:29:22 2002 +0200 (2002-07-19) changeset 13397 6e5f4d911435 parent 13395 4eb948d1eb4e child 13398 1cadd412da48 permissions -rw-r--r--
Absoluteness of the function "nth"
```     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 + 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 + 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 + 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 Neg_def And_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 Neg_def And_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 + 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]. \<exists>X4[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) & is_sum(M,X,X3,X4) &
```
```   261           is_sum(M,natnatsum,X4,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 + 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_N(A,x),
```
```   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
```
```   383
```
```   384 subsubsection{*Absoluteness of the List Construction*}
```
```   385
```
```   386 lemma (in M_datatypes) list_replacement2':
```
```   387   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
```
```   388 apply (insert list_replacement2 [of A])
```
```   389 apply (rule strong_replacement_cong [THEN iffD1])
```
```   390 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
```
```   391 apply (simp_all add: list_replacement1 relativize1_def)
```
```   392 done
```
```   393
```
```   394 lemma (in M_datatypes) list_closed [intro,simp]:
```
```   395      "M(A) ==> M(list(A))"
```
```   396 apply (insert list_replacement1)
```
```   397 by  (simp add: RepFun_closed2 list_eq_Union
```
```   398                list_replacement2' relativize1_def
```
```   399                iterates_closed [of "is_list_functor(M,A)"])
```
```   400
```
```   401 lemma (in M_datatypes) list_N_abs [simp]:
```
```   402      "[|M(A); n\<in>nat; M(Z)|]
```
```   403       ==> is_list_N(M,A,n,Z) <-> Z = list_N(A,n)"
```
```   404 apply (insert list_replacement1)
```
```   405 apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
```
```   406                  iterates_abs [of "is_list_functor(M,A)" _ "\<lambda>X. {0} + A*X"])
```
```   407 done
```
```   408
```
```   409 lemma (in M_datatypes) list_N_closed [intro,simp]:
```
```   410      "[|M(A); n\<in>nat|] ==> M(list_N(A,n))"
```
```   411 apply (insert list_replacement1)
```
```   412 apply (simp add: is_list_N_def list_N_def relativize1_def nat_into_M
```
```   413                  iterates_closed [of "is_list_functor(M,A)"])
```
```   414 done
```
```   415
```
```   416 lemma (in M_datatypes) mem_list_abs [simp]:
```
```   417      "M(A) ==> mem_list(M,A,l) <-> l \<in> list(A)"
```
```   418 apply (insert list_replacement1)
```
```   419 apply (simp add: mem_list_def list_N_def relativize1_def list_eq_Union
```
```   420                  iterates_closed [of "is_list_functor(M,A)"])
```
```   421 done
```
```   422
```
```   423 lemma (in M_datatypes) list_abs [simp]:
```
```   424      "[|M(A); M(Z)|] ==> is_list(M,A,Z) <-> Z = list(A)"
```
```   425 apply (simp add: is_list_def, safe)
```
```   426 apply (rule M_equalityI, simp_all)
```
```   427 done
```
```   428
```
```   429 subsubsection{*Absoluteness of Formulas*}
```
```   430
```
```   431 lemma (in M_datatypes) formula_replacement2':
```
```   432   "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))"
```
```   433 apply (insert formula_replacement2)
```
```   434 apply (rule strong_replacement_cong [THEN iffD1])
```
```   435 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
```
```   436 apply (simp_all add: formula_replacement1 relativize1_def)
```
```   437 done
```
```   438
```
```   439 lemma (in M_datatypes) formula_closed [intro,simp]:
```
```   440      "M(formula)"
```
```   441 apply (insert formula_replacement1)
```
```   442 apply  (simp add: RepFun_closed2 formula_eq_Union
```
```   443                   formula_replacement2' relativize1_def
```
```   444                   iterates_closed [of "is_formula_functor(M)"])
```
```   445 done
```
```   446
```
```   447 lemma (in M_datatypes) is_formula_n_abs [simp]:
```
```   448      "[|n\<in>nat; M(Z)|]
```
```   449       ==> is_formula_n(M,n,Z) <->
```
```   450           Z = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0)"
```
```   451 apply (insert formula_replacement1)
```
```   452 apply (simp add: is_formula_n_def relativize1_def nat_into_M
```
```   453                  iterates_abs [of "is_formula_functor(M)" _
```
```   454                         "\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X))"])
```
```   455 done
```
```   456
```
```   457 lemma (in M_datatypes) mem_formula_abs [simp]:
```
```   458      "mem_formula(M,l) <-> l \<in> formula"
```
```   459 apply (insert formula_replacement1)
```
```   460 apply (simp add: mem_formula_def relativize1_def formula_eq_Union
```
```   461                  iterates_closed [of "is_formula_functor(M)"])
```
```   462 done
```
```   463
```
```   464 lemma (in M_datatypes) formula_abs [simp]:
```
```   465      "[|M(Z)|] ==> is_formula(M,Z) <-> Z = formula"
```
```   466 apply (simp add: is_formula_def, safe)
```
```   467 apply (rule M_equalityI, simp_all)
```
```   468 done
```
```   469
```
```   470
```
```   471 subsection{*Absoluteness for Some List Operators*}
```
```   472
```
```   473 subsection{*Absoluteness for @{text \<epsilon>}-Closure: the @{term eclose} Operator*}
```
```   474
```
```   475 text{*Re-expresses eclose using "iterates"*}
```
```   476 lemma eclose_eq_Union:
```
```   477      "eclose(A) = (\<Union>n\<in>nat. Union^n (A))"
```
```   478 apply (simp add: eclose_def)
```
```   479 apply (rule UN_cong)
```
```   480 apply (rule refl)
```
```   481 apply (induct_tac n)
```
```   482 apply (simp add: nat_rec_0)
```
```   483 apply (simp add: nat_rec_succ)
```
```   484 done
```
```   485
```
```   486 constdefs
```
```   487   is_eclose_n :: "[i=>o,i,i,i] => o"
```
```   488     "is_eclose_n(M,A,n,Z) ==
```
```   489       \<exists>sn[M]. \<exists>msn[M].
```
```   490        successor(M,n,sn) & membership(M,sn,msn) &
```
```   491        is_wfrec(M, iterates_MH(M, big_union(M), A), msn, n, Z)"
```
```   492
```
```   493   mem_eclose :: "[i=>o,i,i] => o"
```
```   494     "mem_eclose(M,A,l) ==
```
```   495       \<exists>n[M]. \<exists>eclosen[M].
```
```   496        finite_ordinal(M,n) & is_eclose_n(M,A,n,eclosen) & l \<in> eclosen"
```
```   497
```
```   498   is_eclose :: "[i=>o,i,i] => o"
```
```   499     "is_eclose(M,A,Z) == \<forall>u[M]. u \<in> Z <-> mem_eclose(M,A,u)"
```
```   500
```
```   501
```
```   502 locale (open) M_eclose = M_wfrank +
```
```   503  assumes eclose_replacement1:
```
```   504    "M(A) ==> iterates_replacement(M, big_union(M), A)"
```
```   505   and eclose_replacement2:
```
```   506    "M(A) ==> strong_replacement(M,
```
```   507          \<lambda>n y. n\<in>nat &
```
```   508                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   509                is_wfrec(M, iterates_MH(M,big_union(M), A),
```
```   510                         msn, n, y)))"
```
```   511
```
```   512 lemma (in M_eclose) eclose_replacement2':
```
```   513   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = Union^n (A))"
```
```   514 apply (insert eclose_replacement2 [of A])
```
```   515 apply (rule strong_replacement_cong [THEN iffD1])
```
```   516 apply (rule conj_cong [OF iff_refl iterates_abs [of "big_union(M)"]])
```
```   517 apply (simp_all add: eclose_replacement1 relativize1_def)
```
```   518 done
```
```   519
```
```   520 lemma (in M_eclose) eclose_closed [intro,simp]:
```
```   521      "M(A) ==> M(eclose(A))"
```
```   522 apply (insert eclose_replacement1)
```
```   523 by  (simp add: RepFun_closed2 eclose_eq_Union
```
```   524                eclose_replacement2' relativize1_def
```
```   525                iterates_closed [of "big_union(M)"])
```
```   526
```
```   527 lemma (in M_eclose) is_eclose_n_abs [simp]:
```
```   528      "[|M(A); n\<in>nat; M(Z)|] ==> is_eclose_n(M,A,n,Z) <-> Z = Union^n (A)"
```
```   529 apply (insert eclose_replacement1)
```
```   530 apply (simp add: is_eclose_n_def relativize1_def nat_into_M
```
```   531                  iterates_abs [of "big_union(M)" _ "Union"])
```
```   532 done
```
```   533
```
```   534 lemma (in M_eclose) mem_eclose_abs [simp]:
```
```   535      "M(A) ==> mem_eclose(M,A,l) <-> l \<in> eclose(A)"
```
```   536 apply (insert eclose_replacement1)
```
```   537 apply (simp add: mem_eclose_def relativize1_def eclose_eq_Union
```
```   538                  iterates_closed [of "big_union(M)"])
```
```   539 done
```
```   540
```
```   541 lemma (in M_eclose) eclose_abs [simp]:
```
```   542      "[|M(A); M(Z)|] ==> is_eclose(M,A,Z) <-> Z = eclose(A)"
```
```   543 apply (simp add: is_eclose_def, safe)
```
```   544 apply (rule M_equalityI, simp_all)
```
```   545 done
```
```   546
```
```   547
```
```   548
```
```   549
```
```   550 subsection {*Absoluteness for @{term transrec}*}
```
```   551
```
```   552
```
```   553 text{* @{term "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"} *}
```
```   554 constdefs
```
```   555
```
```   556   is_transrec :: "[i=>o, [i,i,i]=>o, i, i] => o"
```
```   557    "is_transrec(M,MH,a,z) ==
```
```   558       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
```
```   559        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
```
```   560        is_wfrec(M,MH,mesa,a,z)"
```
```   561
```
```   562   transrec_replacement :: "[i=>o, [i,i,i]=>o, i] => o"
```
```   563    "transrec_replacement(M,MH,a) ==
```
```   564       \<exists>sa[M]. \<exists>esa[M]. \<exists>mesa[M].
```
```   565        upair(M,a,a,sa) & is_eclose(M,sa,esa) & membership(M,esa,mesa) &
```
```   566        wfrec_replacement(M,MH,mesa)"
```
```   567
```
```   568 text{*The condition @{term "Ord(i)"} lets us use the simpler
```
```   569   @{text "trans_wfrec_abs"} rather than @{text "trans_wfrec_abs"},
```
```   570   which I haven't even proved yet. *}
```
```   571 theorem (in M_eclose) transrec_abs:
```
```   572   "[|Ord(i);  M(i);  M(z);
```
```   573      transrec_replacement(M,MH,i);  relativize2(M,MH,H);
```
```   574      \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   575    ==> is_transrec(M,MH,i,z) <-> z = transrec(i,H)"
```
```   576 by (simp add: trans_wfrec_abs transrec_replacement_def is_transrec_def
```
```   577        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
```
```   578
```
```   579
```
```   580 theorem (in M_eclose) transrec_closed:
```
```   581      "[|Ord(i);  M(i);  M(z);
```
```   582 	transrec_replacement(M,MH,i);  relativize2(M,MH,H);
```
```   583 	\<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g))|]
```
```   584       ==> M(transrec(i,H))"
```
```   585 by (simp add: trans_wfrec_closed transrec_replacement_def is_transrec_def
```
```   586        transrec_def eclose_sing_Ord_eq wf_Memrel trans_Memrel relation_Memrel)
```
```   587
```
```   588
```
```   589
```
```   590 subsection{*Absoluteness for the List Operator @{term length}*}
```
```   591 constdefs
```
```   592
```
```   593   is_length :: "[i=>o,i,i,i] => o"
```
```   594     "is_length(M,A,l,n) ==
```
```   595        \<exists>sn[M]. \<exists>list_n[M]. \<exists>list_sn[M].
```
```   596         is_list_N(M,A,n,list_n) & l \<notin> list_n &
```
```   597         successor(M,n,sn) & is_list_N(M,A,sn,list_sn) & l \<in> list_sn"
```
```   598
```
```   599
```
```   600 lemma (in M_datatypes) length_abs [simp]:
```
```   601      "[|M(A); l \<in> list(A); n \<in> nat|] ==> is_length(M,A,l,n) <-> n = length(l)"
```
```   602 apply (subgoal_tac "M(l) & M(n)")
```
```   603  prefer 2 apply (blast dest: transM)
```
```   604 apply (simp add: is_length_def)
```
```   605 apply (blast intro: list_imp_list_N nat_into_Ord list_N_imp_eq_length
```
```   606              dest: list_N_imp_length_lt)
```
```   607 done
```
```   608
```
```   609 text{*Proof is trivial since @{term length} returns natural numbers.*}
```
```   610 lemma (in M_triv_axioms) length_closed [intro,simp]:
```
```   611      "l \<in> list(A) ==> M(length(l))"
```
```   612 by (simp add: nat_into_M )
```
```   613
```
```   614
```
```   615 subsection {*Absoluteness for @{term nth}*}
```
```   616
```
```   617 lemma nth_eq_hd_iterates_tl [rule_format]:
```
```   618      "xs \<in> list(A) ==> \<forall>n \<in> nat. nth(n,xs) = hd' (tl'^n (xs))"
```
```   619 apply (induct_tac xs)
```
```   620 apply (simp add: iterates_tl_Nil hd'_Nil, clarify)
```
```   621 apply (erule natE)
```
```   622 apply (simp add: hd'_Cons)
```
```   623 apply (simp add: tl'_Cons iterates_commute)
```
```   624 done
```
```   625
```
```   626 lemma (in M_axioms) iterates_tl'_closed:
```
```   627      "[|n \<in> nat; M(x)|] ==> M(tl'^n (x))"
```
```   628 apply (induct_tac n, simp)
```
```   629 apply (simp add: tl'_Cons tl'_closed)
```
```   630 done
```
```   631
```
```   632 locale (open) M_nth = M_datatypes +
```
```   633  assumes nth_replacement1:
```
```   634    "M(xs) ==> iterates_replacement(M, %l t. is_tl(M,l,t), xs)"
```
```   635
```
```   636 text{*Immediate by type-checking*}
```
```   637 lemma (in M_datatypes) nth_closed [intro,simp]:
```
```   638      "[|xs \<in> list(A); n \<in> nat; M(A)|] ==> M(nth(n,xs))"
```
```   639 apply (case_tac "n < length(xs)")
```
```   640  apply (blast intro: nth_type transM)
```
```   641 apply (simp add: not_lt_iff_le nth_eq_0)
```
```   642 done
```
```   643
```
```   644 constdefs
```
```   645   is_nth :: "[i=>o,i,i,i] => o"
```
```   646     "is_nth(M,n,l,Z) ==
```
```   647       \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M].
```
```   648        successor(M,n,sn) & membership(M,sn,msn) &
```
```   649        is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
```
```   650        is_hd(M,X,Z)"
```
```   651
```
```   652 lemma (in M_nth) nth_abs [simp]:
```
```   653      "[|M(A); n \<in> nat; l \<in> list(A); M(Z)|]
```
```   654       ==> is_nth(M,n,l,Z) <-> Z = nth(n,l)"
```
```   655 apply (subgoal_tac "M(l)")
```
```   656  prefer 2 apply (blast intro: transM)
```
```   657 apply (insert nth_replacement1)
```
```   658 apply (simp add: is_nth_def nth_eq_hd_iterates_tl nat_into_M
```
```   659                  tl'_closed iterates_tl'_closed
```
```   660                  iterates_abs [OF _ relativize1_tl])
```
```   661 done
```
```   662
```
```   663
```
```   664 end
```