src/HOL/Library/Continuity.thy
author chaieb
Mon Jun 11 11:06:04 2007 +0200 (2007-06-11)
changeset 23315 df3a7e9ebadb
parent 22452 8a86fd2a1bf0
child 23752 15839159f8b6
permissions -rw-r--r--
tuned Proof
     1 (*  Title:      HOL/Library/Continuity.thy
     2     ID:         $Id$
     3     Author:     David von Oheimb, TU Muenchen
     4 *)
     5 
     6 header {* Continuity and iterations (of set transformers) *}
     7 
     8 theory Continuity
     9 imports Main
    10 begin
    11 
    12 subsection {* Continuity for complete lattices *}
    13 
    14 definition
    15   chain :: "(nat \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
    16   "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
    17 
    18 definition
    19   continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
    20   "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
    21 
    22 abbreviation
    23   bot :: "'a::complete_lattice" where
    24   "bot \<equiv> Sup {}"
    25 
    26 lemma SUP_nat_conv:
    27   "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
    28 apply(rule order_antisym)
    29  apply(rule SUP_leI)
    30  apply(case_tac n)
    31   apply simp
    32  apply (fast intro:le_SUPI le_supI2)
    33 apply(simp)
    34 apply (blast intro:SUP_leI le_SUPI)
    35 done
    36 
    37 lemma continuous_mono: fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
    38   assumes "continuous F" shows "mono F"
    39 proof
    40   fix A B :: "'a" assume "A <= B"
    41   let ?C = "%i::nat. if i=0 then A else B"
    42   have "chain ?C" using `A <= B` by(simp add:chain_def)
    43   have "F B = sup (F A) (F B)"
    44   proof -
    45     have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
    46     hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
    47     also have "\<dots> = (SUP i. F(?C i))"
    48       using `chain ?C` `continuous F` by(simp add:continuous_def)
    49     also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
    50     finally show ?thesis .
    51   qed
    52   thus "F A \<le> F B" by(subst le_iff_sup, simp)
    53 qed
    54 
    55 lemma continuous_lfp:
    56  assumes "continuous F" shows "lfp F = (SUP i. (F^i) bot)"
    57 proof -
    58   note mono = continuous_mono[OF `continuous F`]
    59   { fix i have "(F^i) bot \<le> lfp F"
    60     proof (induct i)
    61       show "(F^0) bot \<le> lfp F" by simp
    62     next
    63       case (Suc i)
    64       have "(F^(Suc i)) bot = F((F^i) bot)" by simp
    65       also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
    66       also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
    67       finally show ?case .
    68     qed }
    69   hence "(SUP i. (F^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
    70   moreover have "lfp F \<le> (SUP i. (F^i) bot)" (is "_ \<le> ?U")
    71   proof (rule lfp_lowerbound)
    72     have "chain(%i. (F^i) bot)"
    73     proof -
    74       { fix i have "(F^i) bot \<le> (F^(Suc i)) bot"
    75 	proof (induct i)
    76 	  case 0 show ?case by simp
    77 	next
    78 	  case Suc thus ?case using monoD[OF mono Suc] by auto
    79 	qed }
    80       thus ?thesis by(auto simp add:chain_def)
    81     qed
    82     hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
    83     also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
    84     finally show "F ?U \<le> ?U" .
    85   qed
    86   ultimately show ?thesis by (blast intro:order_antisym)
    87 qed
    88 
    89 text{* The following development is just for sets but presents an up
    90 and a down version of chains and continuity and covers @{const gfp}. *}
    91 
    92 
    93 subsection "Chains"
    94 
    95 definition
    96   up_chain :: "(nat => 'a set) => bool" where
    97   "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
    98 
    99 lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
   100   by (simp add: up_chain_def)
   101 
   102 lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
   103   by (simp add: up_chain_def)
   104 
   105 lemma up_chain_less_mono:
   106     "up_chain F ==> x < y ==> F x \<subseteq> F y"
   107   apply (induct y)
   108    apply (blast dest: up_chainD elim: less_SucE)+
   109   done
   110 
   111 lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
   112   apply (drule le_imp_less_or_eq)
   113   apply (blast dest: up_chain_less_mono)
   114   done
   115 
   116 
   117 definition
   118   down_chain :: "(nat => 'a set) => bool" where
   119   "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
   120 
   121 lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
   122   by (simp add: down_chain_def)
   123 
   124 lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
   125   by (simp add: down_chain_def)
   126 
   127 lemma down_chain_less_mono:
   128     "down_chain F ==> x < y ==> F y \<subseteq> F x"
   129   apply (induct y)
   130    apply (blast dest: down_chainD elim: less_SucE)+
   131   done
   132 
   133 lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
   134   apply (drule le_imp_less_or_eq)
   135   apply (blast dest: down_chain_less_mono)
   136   done
   137 
   138 
   139 subsection "Continuity"
   140 
   141 definition
   142   up_cont :: "('a set => 'a set) => bool" where
   143   "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
   144 
   145 lemma up_contI:
   146     "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
   147   apply (unfold up_cont_def)
   148   apply blast
   149   done
   150 
   151 lemma up_contD:
   152     "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
   153   apply (unfold up_cont_def)
   154   apply auto
   155   done
   156 
   157 
   158 lemma up_cont_mono: "up_cont f ==> mono f"
   159   apply (rule monoI)
   160   apply (drule_tac F = "\<lambda>i. if i = 0 then A else B" in up_contD)
   161    apply (rule up_chainI)
   162    apply  simp+
   163   apply (drule Un_absorb1)
   164   apply (auto simp add: nat_not_singleton)
   165   done
   166 
   167 
   168 definition
   169   down_cont :: "('a set => 'a set) => bool" where
   170   "down_cont f =
   171     (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
   172 
   173 lemma down_contI:
   174   "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
   175     down_cont f"
   176   apply (unfold down_cont_def)
   177   apply blast
   178   done
   179 
   180 lemma down_contD: "down_cont f ==> down_chain F ==>
   181     f (Inter (range F)) = Inter (f ` range F)"
   182   apply (unfold down_cont_def)
   183   apply auto
   184   done
   185 
   186 lemma down_cont_mono: "down_cont f ==> mono f"
   187   apply (rule monoI)
   188   apply (drule_tac F = "\<lambda>i. if i = 0 then B else A" in down_contD)
   189    apply (rule down_chainI)
   190    apply simp+
   191   apply (drule Int_absorb1)
   192   apply (auto simp add: nat_not_singleton)
   193   done
   194 
   195 
   196 subsection "Iteration"
   197 
   198 definition
   199   up_iterate :: "('a set => 'a set) => nat => 'a set" where
   200   "up_iterate f n = (f^n) {}"
   201 
   202 lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
   203   by (simp add: up_iterate_def)
   204 
   205 lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
   206   by (simp add: up_iterate_def)
   207 
   208 lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
   209   apply (rule up_chainI)
   210   apply (induct_tac i)
   211    apply simp+
   212   apply (erule (1) monoD)
   213   done
   214 
   215 lemma UNION_up_iterate_is_fp:
   216   "up_cont F ==>
   217     F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
   218   apply (frule up_cont_mono [THEN up_iterate_chain])
   219   apply (drule (1) up_contD)
   220   apply simp
   221   apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
   222   apply (case_tac xa)
   223    apply auto
   224   done
   225 
   226 lemma UNION_up_iterate_lowerbound:
   227     "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
   228   apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
   229    apply fast
   230   apply (induct_tac i)
   231   prefer 2 apply (drule (1) monoD)
   232    apply auto
   233   done
   234 
   235 lemma UNION_up_iterate_is_lfp:
   236     "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
   237   apply (rule set_eq_subset [THEN iffD2])
   238   apply (rule conjI)
   239    prefer 2
   240    apply (drule up_cont_mono)
   241    apply (rule UNION_up_iterate_lowerbound)
   242     apply assumption
   243    apply (erule lfp_unfold [symmetric])
   244   apply (rule lfp_lowerbound)
   245   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
   246   apply (erule UNION_up_iterate_is_fp [symmetric])
   247   done
   248 
   249 
   250 definition
   251   down_iterate :: "('a set => 'a set) => nat => 'a set" where
   252   "down_iterate f n = (f^n) UNIV"
   253 
   254 lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
   255   by (simp add: down_iterate_def)
   256 
   257 lemma down_iterate_Suc [simp]:
   258     "down_iterate f (Suc i) = f (down_iterate f i)"
   259   by (simp add: down_iterate_def)
   260 
   261 lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
   262   apply (rule down_chainI)
   263   apply (induct_tac i)
   264    apply simp+
   265   apply (erule (1) monoD)
   266   done
   267 
   268 lemma INTER_down_iterate_is_fp:
   269   "down_cont F ==>
   270     F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
   271   apply (frule down_cont_mono [THEN down_iterate_chain])
   272   apply (drule (1) down_contD)
   273   apply simp
   274   apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
   275   apply (case_tac xa)
   276    apply auto
   277   done
   278 
   279 lemma INTER_down_iterate_upperbound:
   280     "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
   281   apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
   282    apply fast
   283   apply (induct_tac i)
   284   prefer 2 apply (drule (1) monoD)
   285    apply auto
   286   done
   287 
   288 lemma INTER_down_iterate_is_gfp:
   289     "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
   290   apply (rule set_eq_subset [THEN iffD2])
   291   apply (rule conjI)
   292    apply (drule down_cont_mono)
   293    apply (rule INTER_down_iterate_upperbound)
   294     apply assumption
   295    apply (erule gfp_unfold [symmetric])
   296   apply (rule gfp_upperbound)
   297   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
   298   apply (erule INTER_down_iterate_is_fp)
   299   done
   300 
   301 end