src/ZF/Constructible/Datatype_absolute.thy
 author paulson Wed Jul 17 15:48:54 2002 +0200 (2002-07-17) changeset 13385 31df66ca0780 parent 13382 b37764a46b16 child 13386 f3e9e8b21aba permissions -rw-r--r--
Expressing Lset and L without using length and arity; simplifies Separation
proofs
```     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 {*lists without univ*}
```
```   110
```
```   111 lemmas datatype_univs = Inl_in_univ Inr_in_univ
```
```   112                         Pair_in_univ nat_into_univ A_into_univ
```
```   113
```
```   114 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
```
```   115 apply (rule bnd_monoI)
```
```   116  apply (intro subset_refl zero_subset_univ A_subset_univ
```
```   117 	      sum_subset_univ Sigma_subset_univ)
```
```   118 apply (rule subset_refl sum_mono Sigma_mono | assumption)+
```
```   119 done
```
```   120
```
```   121 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
```
```   122 by (intro sum_contin prod_contin id_contin const_contin)
```
```   123
```
```   124 text{*Re-expresses lists using sum and product*}
```
```   125 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
```
```   126 apply (simp add: list_def)
```
```   127 apply (rule equalityI)
```
```   128  apply (rule lfp_lowerbound)
```
```   129   prefer 2 apply (rule lfp_subset)
```
```   130  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
```
```   131  apply (simp add: Nil_def Cons_def)
```
```   132  apply blast
```
```   133 txt{*Opposite inclusion*}
```
```   134 apply (rule lfp_lowerbound)
```
```   135  prefer 2 apply (rule lfp_subset)
```
```   136 apply (clarify, subst lfp_unfold [OF list.bnd_mono])
```
```   137 apply (simp add: Nil_def Cons_def)
```
```   138 apply (blast intro: datatype_univs
```
```   139              dest: lfp_subset [THEN subsetD])
```
```   140 done
```
```   141
```
```   142 text{*Re-expresses lists using "iterates", no univ.*}
```
```   143 lemma list_eq_Union:
```
```   144      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
```
```   145 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
```
```   146
```
```   147
```
```   148 subsection {*Absoluteness for "Iterates"*}
```
```   149
```
```   150 constdefs
```
```   151
```
```   152   iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
```
```   153    "iterates_MH(M,isF,v,n,g,z) ==
```
```   154         is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
```
```   155                     n, z)"
```
```   156
```
```   157   iterates_replacement :: "[i=>o, [i,i]=>o, i] => o"
```
```   158    "iterates_replacement(M,isF,v) ==
```
```   159       \<forall>n[M]. n\<in>nat -->
```
```   160          wfrec_replacement(M, iterates_MH(M,isF,v), Memrel(succ(n)))"
```
```   161
```
```   162 lemma (in M_axioms) iterates_MH_abs:
```
```   163   "[| relativize1(M,isF,F); M(n); M(g); M(z) |]
```
```   164    ==> iterates_MH(M,isF,v,n,g,z) <-> z = nat_case(v, \<lambda>m. F(g`m), n)"
```
```   165 by (simp add: nat_case_abs [of _ "\<lambda>m. F(g ` m)"]
```
```   166               relativize1_def iterates_MH_def)
```
```   167
```
```   168 lemma (in M_axioms) iterates_imp_wfrec_replacement:
```
```   169   "[|relativize1(M,isF,F); n \<in> nat; iterates_replacement(M,isF,v)|]
```
```   170    ==> wfrec_replacement(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
```
```   171                        Memrel(succ(n)))"
```
```   172 by (simp add: iterates_replacement_def iterates_MH_abs)
```
```   173
```
```   174 theorem (in M_trancl) iterates_abs:
```
```   175   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
```
```   176       n \<in> nat; M(v); M(z); \<forall>x[M]. M(F(x)) |]
```
```   177    ==> is_wfrec(M, iterates_MH(M,isF,v), Memrel(succ(n)), n, z) <->
```
```   178        z = iterates(F,n,v)"
```
```   179 apply (frule iterates_imp_wfrec_replacement, assumption+)
```
```   180 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
```
```   181                  relativize2_def iterates_MH_abs
```
```   182                  iterates_nat_def recursor_def transrec_def
```
```   183                  eclose_sing_Ord_eq nat_into_M
```
```   184          trans_wfrec_abs [of _ _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
```
```   185 done
```
```   186
```
```   187
```
```   188 lemma (in M_wfrank) iterates_closed [intro,simp]:
```
```   189   "[| iterates_replacement(M,isF,v); relativize1(M,isF,F);
```
```   190       n \<in> nat; M(v); \<forall>x[M]. M(F(x)) |]
```
```   191    ==> M(iterates(F,n,v))"
```
```   192 apply (frule iterates_imp_wfrec_replacement, assumption+)
```
```   193 apply (simp add: wf_Memrel trans_Memrel relation_Memrel nat_into_M
```
```   194                  relativize2_def iterates_MH_abs
```
```   195                  iterates_nat_def recursor_def transrec_def
```
```   196                  eclose_sing_Ord_eq nat_into_M
```
```   197          trans_wfrec_closed [of _ _ _ "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
```
```   198 done
```
```   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 locale (open) M_datatypes = M_wfrank +
```
```   213  assumes list_replacement1:
```
```   214    "M(A) ==> iterates_replacement(M, is_list_functor(M,A), 0)"
```
```   215   and list_replacement2:
```
```   216    "M(A) ==> strong_replacement(M,
```
```   217          \<lambda>n y. n\<in>nat &
```
```   218                (\<exists>sn[M]. \<exists>msn[M]. successor(M,n,sn) & membership(M,sn,msn) &
```
```   219                is_wfrec(M, iterates_MH(M,is_list_functor(M,A), 0),
```
```   220                         msn, n, y)))"
```
```   221
```
```   222 lemma (in M_datatypes) list_replacement2':
```
```   223   "M(A) ==> strong_replacement(M, \<lambda>n y. n\<in>nat & y = (\<lambda>X. {0} + A * X)^n (0))"
```
```   224 apply (insert list_replacement2 [of A])
```
```   225 apply (rule strong_replacement_cong [THEN iffD1])
```
```   226 apply (rule conj_cong [OF iff_refl iterates_abs [of "is_list_functor(M,A)"]])
```
```   227 apply (simp_all add: list_replacement1 relativize1_def)
```
```   228 done
```
```   229
```
```   230 lemma (in M_datatypes) list_closed [intro,simp]:
```
```   231      "M(A) ==> M(list(A))"
```
```   232 apply (insert list_replacement1)
```
```   233 by  (simp add: RepFun_closed2 list_eq_Union
```
```   234                list_replacement2' relativize1_def
```
```   235                iterates_closed [of "is_list_functor(M,A)"])
```
```   236
```
```   237
```
```   238 end
```