src/ZF/Constructible/Datatype_absolute.thy
 author paulson Fri Jul 12 11:24:40 2002 +0200 (2002-07-12) changeset 13352 3cd767f8d78b parent 13350 626b79677dfa child 13353 1800e7134d2e permissions -rw-r--r--
new definitions of fun_apply and M_is_recfun
```     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   contin :: "[i=>i]=>o"
```
```    10    "contin(h) == (\<forall>A. A\<noteq>0 --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
```
```    11
```
```    12 lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
```
```    13 apply (induct_tac n)
```
```    14  apply (simp_all add: bnd_mono_def, blast)
```
```    15 done
```
```    16
```
```    17
```
```    18 lemma contin_iterates_eq:
```
```    19     "contin(h) \<Longrightarrow> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
```
```    20 apply (simp add: contin_def)
```
```    21 apply (rule trans)
```
```    22 apply (rule equalityI)
```
```    23  apply (simp_all add: UN_subset_iff)
```
```    24  apply safe
```
```    25  apply (erule_tac [2] natE)
```
```    26   apply (rule_tac a="succ(x)" in UN_I)
```
```    27    apply simp_all
```
```    28 apply blast
```
```    29 done
```
```    30
```
```    31 lemma lfp_subset_Union:
```
```    32      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
```
```    33 apply (rule lfp_lowerbound)
```
```    34  apply (simp add: contin_iterates_eq)
```
```    35 apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff)
```
```    36 done
```
```    37
```
```    38 lemma Union_subset_lfp:
```
```    39      "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
```
```    40 apply (simp add: UN_subset_iff)
```
```    41 apply (rule ballI)
```
```    42 apply (induct_tac n, simp_all)
```
```    43 apply (rule subset_trans [of _ "h(lfp(D,h))"])
```
```    44  apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] )
```
```    45 apply (erule lfp_lemma2)
```
```    46 done
```
```    47
```
```    48 lemma lfp_eq_Union:
```
```    49      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
```
```    50 by (blast del: subsetI
```
```    51           intro: lfp_subset_Union Union_subset_lfp)
```
```    52
```
```    53
```
```    54 subsection {*lists without univ*}
```
```    55
```
```    56 lemmas datatype_univs = A_into_univ Inl_in_univ Inr_in_univ
```
```    57                         Pair_in_univ zero_in_univ
```
```    58
```
```    59 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
```
```    60 apply (rule bnd_monoI)
```
```    61  apply (intro subset_refl zero_subset_univ A_subset_univ
```
```    62 	      sum_subset_univ Sigma_subset_univ)
```
```    63  apply (blast intro!: subset_refl sum_mono Sigma_mono del: subsetI)
```
```    64 done
```
```    65
```
```    66 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
```
```    67 by (simp add: contin_def, blast)
```
```    68
```
```    69 text{*Re-expresses lists using sum and product*}
```
```    70 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
```
```    71 apply (simp add: list_def)
```
```    72 apply (rule equalityI)
```
```    73  apply (rule lfp_lowerbound)
```
```    74   prefer 2 apply (rule lfp_subset)
```
```    75  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
```
```    76  apply (simp add: Nil_def Cons_def)
```
```    77  apply blast
```
```    78 txt{*Opposite inclusion*}
```
```    79 apply (rule lfp_lowerbound)
```
```    80  prefer 2 apply (rule lfp_subset)
```
```    81 apply (clarify, subst lfp_unfold [OF list.bnd_mono])
```
```    82 apply (simp add: Nil_def Cons_def)
```
```    83 apply (blast intro: datatype_univs
```
```    84              dest: lfp_subset [THEN subsetD])
```
```    85 done
```
```    86
```
```    87 text{*Re-expresses lists using "iterates", no univ.*}
```
```    88 lemma list_eq_Union:
```
```    89      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
```
```    90 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
```
```    91
```
```    92
```
```    93 subsection {*Absoluteness for "Iterates"*}
```
```    94
```
```    95 lemma (in M_trancl) iterates_relativize:
```
```    96   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
```
```    97      strong_replacement(M,
```
```    98        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M,x,y,z) &
```
```    99               M_is_recfun(M, \<lambda>n f z. z = nat_case(v, \<lambda>m. F(f`m), n),
```
```   100                           Memrel(succ(n)), x, g) &
```
```   101               y = nat_case(v, \<lambda>m. F(g`m), x))|]
```
```   102    ==> iterates(F,n,v) = z <->
```
```   103        (\<exists>g[M]. is_recfun(Memrel(succ(n)), n,
```
```   104                              \<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n), g) &
```
```   105             z = nat_case(v, \<lambda>m. F(g`m), n))"
```
```   106 by (simp add: iterates_nat_def recursor_def transrec_def
```
```   107               eclose_sing_Ord_eq trans_wfrec_relativize nat_into_M
```
```   108               wf_Memrel trans_Memrel relation_Memrel
```
```   109               is_recfun_abs [of "\<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n)"])
```
```   110
```
```   111 lemma (in M_wfrank) iterates_closed [intro,simp]:
```
```   112   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
```
```   113      strong_replacement(M,
```
```   114        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
```
```   115               is_recfun (Memrel(succ(n)), x,
```
```   116                          \<lambda>n f. nat_case(v, \<lambda>m. F(f`m), n), g) &
```
```   117               y = nat_case(v, \<lambda>m. F(g`m), x))|]
```
```   118    ==> M(iterates(F,n,v))"
```
```   119 by (simp add: iterates_nat_def recursor_def transrec_def
```
```   120               eclose_sing_Ord_eq trans_wfrec_closed nat_into_M
```
```   121               wf_Memrel trans_Memrel relation_Memrel nat_case_closed)
```
```   122
```
```   123
```
```   124 constdefs
```
```   125   is_list_functor :: "[i=>o,i,i,i] => o"
```
```   126     "is_list_functor(M,A,X,Z) ==
```
```   127         \<exists>n1[M]. \<exists>AX[M].
```
```   128          number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)"
```
```   129
```
```   130   list_functor_case :: "[i=>o,i,i,i,i] => o"
```
```   131     --{*Abbreviation for the definition of lists below*}
```
```   132     "list_functor_case(M,A,g,x,y) ==
```
```   133         is_nat_case(M, 0,
```
```   134              \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & is_list_functor(M,A,gm,u),
```
```   135              x, y)"
```
```   136
```
```   137 lemma (in M_axioms) list_functor_abs [simp]:
```
```   138      "[| M(A); M(X); M(Z) |] ==> is_list_functor(M,A,X,Z) <-> (Z = {0} + A*X)"
```
```   139 by (simp add: is_list_functor_def singleton_0 nat_into_M)
```
```   140
```
```   141 lemma (in M_axioms) list_functor_case_abs:
```
```   142      "[| M(A); M(n); M(y); M(g) |]
```
```   143       ==> list_functor_case(M,A,g,n,y) <->
```
```   144           y = nat_case(0, \<lambda>m. {0} + A * g`m, n)"
```
```   145 by (simp add: list_functor_case_def nat_into_M)
```
```   146
```
```   147
```
```   148 locale M_datatypes = M_wfrank +
```
```   149   assumes list_replacement1:
```
```   150        "[|M(A); n \<in> nat|] ==>
```
```   151 	strong_replacement(M,
```
```   152 	  \<lambda>x z. \<exists>y[M]. \<exists>g[M]. \<exists>sucn[M]. \<exists>memr[M].
```
```   153 		 pair(M,x,y,z) & successor(M,n,sucn) &
```
```   154 		 membership(M,sucn,memr) &
```
```   155 		 M_is_recfun(M, \<lambda>n f z. list_functor_case(M,A,f,n,z),
```
```   156                              memr, x, g) &
```
```   157                  list_functor_case(M,A,g,x,y))"
```
```   158 (*THEY NEED RELATIVIZATION*)
```
```   159       and list_replacement2:
```
```   160            "M(A) ==> strong_replacement(M, \<lambda>x y. y = (\<lambda>X. {0} + A * X)^x (0))"
```
```   161
```
```   162
```
```   163
```
```   164 lemma (in M_datatypes) list_replacement1':
```
```   165   "[|M(A); n \<in> nat|]
```
```   166    ==> strong_replacement
```
```   167 	  (M, \<lambda>x z. \<exists>y[M]. z = \<langle>x,y\<rangle> &
```
```   168                (\<exists>g[M]. is_recfun (Memrel(succ(n)), x,
```
```   169 		          \<lambda>n f. nat_case(0, \<lambda>m. {0} + A * f`m, n), g) &
```
```   170  	       y = nat_case(0, \<lambda>m. {0} + A * g ` m, x)))"
```
```   171 apply (insert list_replacement1 [of A n], simp add: nat_into_M)
```
```   172 apply (simp add: nat_into_M list_functor_case_abs
```
```   173                  is_recfun_abs [of "\<lambda>n f. nat_case(0, \<lambda>m. {0} + A * f`m, n)"])
```
```   174 done
```
```   175
```
```   176 lemma (in M_datatypes) list_replacement2':
```
```   177   "M(A) ==> strong_replacement(M, \<lambda>x y. y = (\<lambda>X. {0} + A * X)^x (0))"
```
```   178 by (insert list_replacement2, simp add: nat_into_M)
```
```   179
```
```   180
```
```   181 lemma (in M_datatypes) list_closed [intro,simp]:
```
```   182      "M(A) ==> M(list(A))"
```
```   183 by (simp add: list_eq_Union list_replacement1' list_replacement2')
```
```   184
```
```   185
```
```   186 end
```