src/ZF/Constructible/Datatype_absolute.thy
 author paulson Wed Jul 17 16:41:32 2002 +0200 (2002-07-17) changeset 13386 f3e9e8b21aba parent 13385 31df66ca0780 child 13395 4eb948d1eb4e permissions -rw-r--r--
Formulas (and lists) in M (and L!)
```     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 locale (open) M_datatypes = M_wfrank +
```
```   272  assumes list_replacement1:
```
```   273    "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
```
```   274   and list_replacement2:
```
```   275    "M(A) ==> strong_replacement(M,
```
```   276          \<lambda>n y. n\<in>nat &
```
```   277                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   278                is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0),
```
```   279                         msn, n, y)))"
```
```   280   and formula_replacement1:
```
```   281    "iterates_replacement(M, is_formula_functor(M), 0)"
```
```   282   and formula_replacement2:
```
```   283    "strong_replacement(M,
```
```   284          \<lambda>n y. n\<in>nat &
```
```   285                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   286                is_wfrec(M, iterates_MH(M,is_formula_functor(M), 0),
```
```   287                         msn, n, y)))"
```
```   288
```
```   289 lemma (in M_datatypes) list_replacement2':
```
```   290   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
```
```   291 apply (insert list_replacement2 [of A])
```
```   292 apply (rule strong_replacement_cong [THEN iffD1])
```
```   293 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
```
```   294 apply (simp_all add: list_replacement1 relativize1_def)
```
```   295 done
```
```   296
```
```   297 lemma (in M_datatypes) list_closed [intro,simp]:
```
```   298      "M(A) ==> M(list(A))"
```
```   299 apply (insert list_replacement1)
```
```   300 by  (simp add: RepFun_closed2 list_eq_Union
```
```   301                list_replacement2' relativize1_def
```
```   302                iterates_closed [of "is_list_functor(M,A)"])
```
```   303
```
```   304
```
```   305 lemma (in M_datatypes) formula_replacement2':
```
```   306   "strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. ((nat*nat) + (nat*nat)) + (X + (X*X + X)))^n (0))"
```
```   307 apply (insert formula_replacement2)
```
```   308 apply (rule strong_replacement_cong [THEN iffD1])
```
```   309 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_formula_functor(M)"]])
```
```   310 apply (simp_all add: formula_replacement1 relativize1_def)
```
```   311 done
```
```   312
```
```   313 lemma (in M_datatypes) formula_closed [intro,simp]:
```
```   314      "M(formula)"
```
```   315 apply (insert formula_replacement1)
```
```   316 apply  (simp add: RepFun_closed2 formula_eq_Union
```
```   317                   formula_replacement2' relativize1_def
```
```   318                   iterates_closed [of "is_formula_functor(M)"])
```
```   319 done
```
```   320
```
```   321 end
```