src/ZF/Constructible/Datatype_absolute.thy
author paulson
Mon Jul 01 18:16:18 2002 +0200 (2002-07-01)
changeset 13268 240509babf00
child 13269 3ba9be497c33
permissions -rw-r--r--
more use of relativized quantifiers
list_closed
     1 theory Datatype_absolute = WF_absolute:
     2 
     3 (*Epsilon.thy*)
     4 lemma succ_subset_eclose_sing: "succ(i) <= eclose({i})"
     5 apply (insert arg_subset_eclose [of "{i}"], simp) 
     6 apply (frule eclose_subset, blast) 
     7 done
     8 
     9 lemma eclose_sing_Ord_eq: "Ord(i) ==> eclose({i}) = succ(i)"
    10 apply (rule equalityI)
    11 apply (erule eclose_sing_Ord)  
    12 apply (rule succ_subset_eclose_sing) 
    13 done
    14 
    15 (*Ordinal.thy*)
    16 lemma relation_Memrel: "relation(Memrel(A))"
    17 by (simp add: relation_def Memrel_def, blast)
    18 
    19 lemma (in M_axioms) nat_case_closed:
    20   "[|M(k); M(a); \<forall>m[M]. M(b(m))|] ==> M(nat_case(a,b,k))"
    21 apply (case_tac "k=0", simp) 
    22 apply (case_tac "\<exists>m. k = succ(m)")
    23 apply force 
    24 apply (simp add: nat_case_def) 
    25 done
    26 
    27 
    28 subsection{*The lfp of a continuous function can be expressed as a union*}
    29 
    30 constdefs
    31   contin :: "[i=>i]=>o"
    32    "contin(h) == (\<forall>A. A\<noteq>0 --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
    33 
    34 lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
    35 apply (induct_tac n) 
    36  apply (simp_all add: bnd_mono_def, blast) 
    37 done
    38 
    39 
    40 lemma contin_iterates_eq: 
    41     "contin(h) \<Longrightarrow> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
    42 apply (simp add: contin_def) 
    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 x, 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 subsection {*lists without univ*}
    77 
    78 lemmas datatype_univs = A_into_univ Inl_in_univ Inr_in_univ 
    79                         Pair_in_univ zero_in_univ
    80 
    81 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
    82 apply (rule bnd_monoI)
    83  apply (intro subset_refl zero_subset_univ A_subset_univ 
    84 	      sum_subset_univ Sigma_subset_univ) 
    85  apply (blast intro!: subset_refl sum_mono Sigma_mono del: subsetI)
    86 done
    87 
    88 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
    89 by (simp add: contin_def, blast)
    90 
    91 text{*Re-expresses lists using sum and product*}
    92 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
    93 apply (simp add: list_def) 
    94 apply (rule equalityI) 
    95  apply (rule lfp_lowerbound) 
    96   prefer 2 apply (rule lfp_subset)
    97  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
    98  apply (simp add: Nil_def Cons_def)
    99  apply blast 
   100 txt{*Opposite inclusion*}
   101 apply (rule lfp_lowerbound) 
   102  prefer 2 apply (rule lfp_subset) 
   103 apply (clarify, subst lfp_unfold [OF list.bnd_mono]) 
   104 apply (simp add: Nil_def Cons_def)
   105 apply (blast intro: datatype_univs
   106              dest: lfp_subset [THEN subsetD])
   107 done
   108 
   109 text{*Re-expresses lists using "iterates", no univ.*}
   110 lemma list_eq_Union:
   111      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
   112 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
   113 
   114 
   115 subsection {*Absoluteness for "Iterates"*}
   116 
   117 lemma (in M_trancl) iterates_relativize:
   118   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
   119      strong_replacement(M, 
   120        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
   121               is_recfun (Memrel(succ(n)), x,
   122                          \<lambda>n f. nat_case(v, \<lambda>m. F(f`m), n), g) &
   123               y = nat_case(v, \<lambda>m. F(g`m), x))|] 
   124    ==> iterates(F,n,v) = z <-> 
   125        (\<exists>g[M]. is_recfun(Memrel(succ(n)), n, 
   126                              \<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n), g) &
   127             z = nat_case(v, \<lambda>m. F(g`m), n))"
   128 by (simp add: iterates_nat_def recursor_def transrec_def 
   129               eclose_sing_Ord_eq trans_wfrec_relativize nat_into_M
   130               wf_Memrel trans_Memrel relation_Memrel nat_case_closed)
   131 
   132 
   133 lemma (in M_wfrank) iterates_closed [intro,simp]:
   134   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
   135      strong_replacement(M, 
   136        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
   137               is_recfun (Memrel(succ(n)), x,
   138                          \<lambda>n f. nat_case(v, \<lambda>m. F(f`m), n), g) &
   139               y = nat_case(v, \<lambda>m. F(g`m), x))|] 
   140    ==> M(iterates(F,n,v))"
   141 by (simp add: iterates_nat_def recursor_def transrec_def 
   142               eclose_sing_Ord_eq trans_wfrec_closed nat_into_M
   143               wf_Memrel trans_Memrel relation_Memrel nat_case_closed)
   144 
   145 
   146 locale M_datatypes = M_wfrank +
   147 (*THEY NEED RELATIVIZATION*)
   148   assumes list_replacement1: 
   149 	   "[|M(A); n \<in> nat|] ==> 
   150 	    strong_replacement(M, 
   151 	      \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
   152 		     is_recfun (Memrel(succ(n)), x,
   153 				\<lambda>n f. nat_case(0, \<lambda>m. {0} + A \<times> f`m, n), g) &
   154 		     y = nat_case(0, \<lambda>m. {0} + A \<times> g`m, x))"
   155       and list_replacement2': 
   156            "M(A) ==> strong_replacement(M, \<lambda>x y. y = (\<lambda>X. {0} + A \<times> X)^x (0))"
   157 
   158 
   159 lemma (in M_datatypes) list_replacement1':
   160   "[|M(A); n \<in> nat|]
   161    ==> strong_replacement
   162 	  (M, \<lambda>x y. \<exists>z[M]. \<exists>g[M]. y = \<langle>x, z\<rangle> &
   163                is_recfun (Memrel(succ(n)), x,
   164 		          \<lambda>n f. nat_case(0, \<lambda>m. {0} + A \<times> f ` m, n), g) &
   165  	       z = nat_case(0, \<lambda>m. {0} + A \<times> g ` m, x))"
   166 by (insert list_replacement1, simp) 
   167 
   168 
   169 lemma (in M_datatypes) list_closed [intro,simp]:
   170      "M(A) ==> M(list(A))"
   171 by (simp add: list_eq_Union list_replacement1' list_replacement2')
   172 
   173 end