src/ZF/Constructible/Datatype_absolute.thy
author paulson
Thu Jul 04 10:54:04 2002 +0200 (2002-07-04)
changeset 13293 09276ee04361
parent 13269 3ba9be497c33
child 13306 6eebcddee32b
permissions -rw-r--r--
tweaks
     1 theory Datatype_absolute = Formula + WF_absolute:
     2 
     3 
     4 subsection{*The lfp of a continuous function can be expressed as a union*}
     5 
     6 constdefs
     7   contin :: "[i=>i]=>o"
     8    "contin(h) == (\<forall>A. A\<noteq>0 --> h(\<Union>A) = (\<Union>X\<in>A. h(X)))"
     9 
    10 lemma bnd_mono_iterates_subset: "[|bnd_mono(D, h); n \<in> nat|] ==> h^n (0) <= D"
    11 apply (induct_tac n) 
    12  apply (simp_all add: bnd_mono_def, blast) 
    13 done
    14 
    15 
    16 lemma contin_iterates_eq: 
    17     "contin(h) \<Longrightarrow> h(\<Union>n\<in>nat. h^n (0)) = (\<Union>n\<in>nat. h^n (0))"
    18 apply (simp add: contin_def) 
    19 apply (rule trans) 
    20 apply (rule equalityI) 
    21  apply (simp_all add: UN_subset_iff) 
    22  apply safe
    23  apply (erule_tac [2] natE) 
    24   apply (rule_tac a="succ(x)" in UN_I) 
    25    apply simp_all 
    26 apply blast 
    27 done
    28 
    29 lemma lfp_subset_Union:
    30      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) <= (\<Union>n\<in>nat. h^n(0))"
    31 apply (rule lfp_lowerbound) 
    32  apply (simp add: contin_iterates_eq) 
    33 apply (simp add: contin_def bnd_mono_iterates_subset UN_subset_iff) 
    34 done
    35 
    36 lemma Union_subset_lfp:
    37      "bnd_mono(D,h) ==> (\<Union>n\<in>nat. h^n(0)) <= lfp(D,h)"
    38 apply (simp add: UN_subset_iff)
    39 apply (rule ballI)  
    40 apply (induct_tac x, simp_all) 
    41 apply (rule subset_trans [of _ "h(lfp(D,h))"])
    42  apply (blast dest: bnd_monoD2 [OF _ _ lfp_subset] )  
    43 apply (erule lfp_lemma2) 
    44 done
    45 
    46 lemma lfp_eq_Union:
    47      "[|bnd_mono(D, h); contin(h)|] ==> lfp(D,h) = (\<Union>n\<in>nat. h^n(0))"
    48 by (blast del: subsetI 
    49           intro: lfp_subset_Union Union_subset_lfp)
    50 
    51 
    52 subsection {*lists without univ*}
    53 
    54 lemmas datatype_univs = A_into_univ Inl_in_univ Inr_in_univ 
    55                         Pair_in_univ zero_in_univ
    56 
    57 lemma list_fun_bnd_mono: "bnd_mono(univ(A), \<lambda>X. {0} + A*X)"
    58 apply (rule bnd_monoI)
    59  apply (intro subset_refl zero_subset_univ A_subset_univ 
    60 	      sum_subset_univ Sigma_subset_univ) 
    61  apply (blast intro!: subset_refl sum_mono Sigma_mono del: subsetI)
    62 done
    63 
    64 lemma list_fun_contin: "contin(\<lambda>X. {0} + A*X)"
    65 by (simp add: contin_def, blast)
    66 
    67 text{*Re-expresses lists using sum and product*}
    68 lemma list_eq_lfp2: "list(A) = lfp(univ(A), \<lambda>X. {0} + A*X)"
    69 apply (simp add: list_def) 
    70 apply (rule equalityI) 
    71  apply (rule lfp_lowerbound) 
    72   prefer 2 apply (rule lfp_subset)
    73  apply (clarify, subst lfp_unfold [OF list_fun_bnd_mono])
    74  apply (simp add: Nil_def Cons_def)
    75  apply blast 
    76 txt{*Opposite inclusion*}
    77 apply (rule lfp_lowerbound) 
    78  prefer 2 apply (rule lfp_subset) 
    79 apply (clarify, subst lfp_unfold [OF list.bnd_mono]) 
    80 apply (simp add: Nil_def Cons_def)
    81 apply (blast intro: datatype_univs
    82              dest: lfp_subset [THEN subsetD])
    83 done
    84 
    85 text{*Re-expresses lists using "iterates", no univ.*}
    86 lemma list_eq_Union:
    87      "list(A) = (\<Union>n\<in>nat. (\<lambda>X. {0} + A*X) ^ n (0))"
    88 by (simp add: list_eq_lfp2 lfp_eq_Union list_fun_bnd_mono list_fun_contin)
    89 
    90 
    91 subsection {*Absoluteness for "Iterates"*}
    92 
    93 lemma (in M_trancl) iterates_relativize:
    94   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
    95      strong_replacement(M, 
    96        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
    97               is_recfun (Memrel(succ(n)), x,
    98                          \<lambda>n f. nat_case(v, \<lambda>m. F(f`m), n), g) &
    99               y = nat_case(v, \<lambda>m. F(g`m), x))|] 
   100    ==> iterates(F,n,v) = z <-> 
   101        (\<exists>g[M]. is_recfun(Memrel(succ(n)), n, 
   102                              \<lambda>n g. nat_case(v, \<lambda>m. F(g`m), n), g) &
   103             z = nat_case(v, \<lambda>m. F(g`m), n))"
   104 by (simp add: iterates_nat_def recursor_def transrec_def 
   105               eclose_sing_Ord_eq trans_wfrec_relativize nat_into_M
   106               wf_Memrel trans_Memrel relation_Memrel nat_case_closed)
   107 
   108 
   109 lemma (in M_wfrank) iterates_closed [intro,simp]:
   110   "[|n \<in> nat; M(v); \<forall>x[M]. M(F(x));
   111      strong_replacement(M, 
   112        \<lambda>x z. \<exists>y[M]. \<exists>g[M]. pair(M, x, y, z) &
   113               is_recfun (Memrel(succ(n)), x,
   114                          \<lambda>n f. nat_case(v, \<lambda>m. F(f`m), n), g) &
   115               y = nat_case(v, \<lambda>m. F(g`m), x))|] 
   116    ==> M(iterates(F,n,v))"
   117 by (simp add: iterates_nat_def recursor_def transrec_def 
   118               eclose_sing_Ord_eq trans_wfrec_closed nat_into_M
   119               wf_Memrel trans_Memrel relation_Memrel nat_case_closed)
   120 
   121 
   122 
   123 locale M_datatypes = M_wfrank +
   124 (*THEY NEED RELATIVIZATION*)
   125   assumes list_replacement1: 
   126 	   "[|M(A); n \<in> nat|] ==> 
   127 	    strong_replacement(M, 
   128 	      \<lambda>x z. \<exists>y[M]. \<exists>g[M]. \<exists>sucn[M]. \<exists>memr[M]. 
   129                      pair(M,x,y,z) & successor(M,n,sucn) & 
   130                      membership(M,sucn,memr) &
   131 		     is_recfun (memr, x,
   132 				\<lambda>n f. nat_case(0, \<lambda>m. {0} + A \<times> f`m, n), g) &
   133 		     y = nat_case(0, \<lambda>m. {0} + A \<times> g`m, x))"
   134       and list_replacement2': 
   135            "M(A) ==> strong_replacement(M, \<lambda>x y. y = (\<lambda>X. {0} + A \<times> X)^x (0))"
   136 
   137 
   138 lemma (in M_datatypes) list_replacement1':
   139   "[|M(A); n \<in> nat|]
   140    ==> strong_replacement
   141 	  (M, \<lambda>x y. \<exists>z[M]. y = \<langle>x,z\<rangle> &
   142                (\<exists>g[M]. is_recfun (Memrel(succ(n)), x,
   143 		          \<lambda>n f. nat_case(0, \<lambda>m. {0} + A \<times> f`m, n), g) &
   144  	       z = nat_case(0, \<lambda>m. {0} + A \<times> g ` m, x)))"
   145 by (insert list_replacement1, simp add: nat_into_M) 
   146 
   147 
   148 lemma (in M_datatypes) list_closed [intro,simp]:
   149      "M(A) ==> M(list(A))"
   150 by (simp add: list_eq_Union list_replacement1' list_replacement2')
   151 
   152 
   153 end