src/HOL/Library/Continuity.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 46508 ec9630fe9ca7 child 54257 5c7a3b6b05a9 permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/Continuity.thy
```
```     2     Author:     David von Oheimb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Continuity and iterations (of set transformers) *}
```
```     6
```
```     7 theory Continuity
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 subsection {* Continuity for complete lattices *}
```
```    12
```
```    13 definition
```
```    14   chain :: "(nat \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
```
```    15   "chain M \<longleftrightarrow> (\<forall>i. M i \<le> M (Suc i))"
```
```    16
```
```    17 definition
```
```    18   continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
```
```    19   "continuous F \<longleftrightarrow> (\<forall>M. chain M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
```
```    20
```
```    21 lemma SUP_nat_conv:
```
```    22   "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
```
```    23 apply(rule order_antisym)
```
```    24  apply(rule SUP_least)
```
```    25  apply(case_tac n)
```
```    26   apply simp
```
```    27  apply (fast intro:SUP_upper le_supI2)
```
```    28 apply(simp)
```
```    29 apply (blast intro:SUP_least SUP_upper)
```
```    30 done
```
```    31
```
```    32 lemma continuous_mono: fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
```
```    33   assumes "continuous F" shows "mono F"
```
```    34 proof
```
```    35   fix A B :: "'a" assume "A <= B"
```
```    36   let ?C = "%i::nat. if i=0 then A else B"
```
```    37   have "chain ?C" using `A <= B` by(simp add:chain_def)
```
```    38   have "F B = sup (F A) (F B)"
```
```    39   proof -
```
```    40     have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
```
```    41     hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
```
```    42     also have "\<dots> = (SUP i. F(?C i))"
```
```    43       using `chain ?C` `continuous F` by(simp add:continuous_def)
```
```    44     also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
```
```    45     finally show ?thesis .
```
```    46   qed
```
```    47   thus "F A \<le> F B" by(subst le_iff_sup, simp)
```
```    48 qed
```
```    49
```
```    50 lemma continuous_lfp:
```
```    51  assumes "continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
```
```    52 proof -
```
```    53   note mono = continuous_mono[OF `continuous F`]
```
```    54   { fix i have "(F ^^ i) bot \<le> lfp F"
```
```    55     proof (induct i)
```
```    56       show "(F ^^ 0) bot \<le> lfp F" by simp
```
```    57     next
```
```    58       case (Suc i)
```
```    59       have "(F ^^ Suc i) bot = F((F ^^ i) bot)" by simp
```
```    60       also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
```
```    61       also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
```
```    62       finally show ?case .
```
```    63     qed }
```
```    64   hence "(SUP i. (F ^^ i) bot) \<le> lfp F" by (blast intro!:SUP_least)
```
```    65   moreover have "lfp F \<le> (SUP i. (F ^^ i) bot)" (is "_ \<le> ?U")
```
```    66   proof (rule lfp_lowerbound)
```
```    67     have "chain(%i. (F ^^ i) bot)"
```
```    68     proof -
```
```    69       { fix i have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
```
```    70         proof (induct i)
```
```    71           case 0 show ?case by simp
```
```    72         next
```
```    73           case Suc thus ?case using monoD[OF mono Suc] by auto
```
```    74         qed }
```
```    75       thus ?thesis by(auto simp add:chain_def)
```
```    76     qed
```
```    77     hence "F ?U = (SUP i. (F ^^ (i+1)) bot)" using `continuous F` by (simp add:continuous_def)
```
```    78     also have "\<dots> \<le> ?U" by(fast intro:SUP_least SUP_upper)
```
```    79     finally show "F ?U \<le> ?U" .
```
```    80   qed
```
```    81   ultimately show ?thesis by (blast intro:order_antisym)
```
```    82 qed
```
```    83
```
```    84 text{* The following development is just for sets but presents an up
```
```    85 and a down version of chains and continuity and covers @{const gfp}. *}
```
```    86
```
```    87
```
```    88 subsection "Chains"
```
```    89
```
```    90 definition
```
```    91   up_chain :: "(nat => 'a set) => bool" where
```
```    92   "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
```
```    93
```
```    94 lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
```
```    95   by (simp add: up_chain_def)
```
```    96
```
```    97 lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
```
```    98   by (simp add: up_chain_def)
```
```    99
```
```   100 lemma up_chain_less_mono:
```
```   101     "up_chain F ==> x < y ==> F x \<subseteq> F y"
```
```   102   apply (induct y)
```
```   103    apply (blast dest: up_chainD elim: less_SucE)+
```
```   104   done
```
```   105
```
```   106 lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
```
```   107   apply (drule le_imp_less_or_eq)
```
```   108   apply (blast dest: up_chain_less_mono)
```
```   109   done
```
```   110
```
```   111
```
```   112 definition
```
```   113   down_chain :: "(nat => 'a set) => bool" where
```
```   114   "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
```
```   115
```
```   116 lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
```
```   117   by (simp add: down_chain_def)
```
```   118
```
```   119 lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
```
```   120   by (simp add: down_chain_def)
```
```   121
```
```   122 lemma down_chain_less_mono:
```
```   123     "down_chain F ==> x < y ==> F y \<subseteq> F x"
```
```   124   apply (induct y)
```
```   125    apply (blast dest: down_chainD elim: less_SucE)+
```
```   126   done
```
```   127
```
```   128 lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
```
```   129   apply (drule le_imp_less_or_eq)
```
```   130   apply (blast dest: down_chain_less_mono)
```
```   131   done
```
```   132
```
```   133
```
```   134 subsection "Continuity"
```
```   135
```
```   136 definition
```
```   137   up_cont :: "('a set => 'a set) => bool" where
```
```   138   "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
```
```   139
```
```   140 lemma up_contI:
```
```   141   "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
```
```   142 apply (unfold up_cont_def)
```
```   143 apply blast
```
```   144 done
```
```   145
```
```   146 lemma up_contD:
```
```   147   "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
```
```   148 apply (unfold up_cont_def)
```
```   149 apply auto
```
```   150 done
```
```   151
```
```   152
```
```   153 lemma up_cont_mono: "up_cont f ==> mono f"
```
```   154 apply (rule monoI)
```
```   155 apply (drule_tac F = "\<lambda>i. if i = 0 then x else y" in up_contD)
```
```   156  apply (rule up_chainI)
```
```   157  apply simp
```
```   158 apply (drule Un_absorb1)
```
```   159 apply (auto split:split_if_asm)
```
```   160 done
```
```   161
```
```   162
```
```   163 definition
```
```   164   down_cont :: "('a set => 'a set) => bool" where
```
```   165   "down_cont f =
```
```   166     (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
```
```   167
```
```   168 lemma down_contI:
```
```   169   "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
```
```   170     down_cont f"
```
```   171   apply (unfold down_cont_def)
```
```   172   apply blast
```
```   173   done
```
```   174
```
```   175 lemma down_contD: "down_cont f ==> down_chain F ==>
```
```   176     f (Inter (range F)) = Inter (f ` range F)"
```
```   177   apply (unfold down_cont_def)
```
```   178   apply auto
```
```   179   done
```
```   180
```
```   181 lemma down_cont_mono: "down_cont f ==> mono f"
```
```   182 apply (rule monoI)
```
```   183 apply (drule_tac F = "\<lambda>i. if i = 0 then y else x" in down_contD)
```
```   184  apply (rule down_chainI)
```
```   185  apply simp
```
```   186 apply (drule Int_absorb1)
```
```   187 apply (auto split:split_if_asm)
```
```   188 done
```
```   189
```
```   190
```
```   191 subsection "Iteration"
```
```   192
```
```   193 definition
```
```   194   up_iterate :: "('a set => 'a set) => nat => 'a set" where
```
```   195   "up_iterate f n = (f ^^ n) {}"
```
```   196
```
```   197 lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
```
```   198   by (simp add: up_iterate_def)
```
```   199
```
```   200 lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
```
```   201   by (simp add: up_iterate_def)
```
```   202
```
```   203 lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
```
```   204   apply (rule up_chainI)
```
```   205   apply (induct_tac i)
```
```   206    apply simp+
```
```   207   apply (erule (1) monoD)
```
```   208   done
```
```   209
```
```   210 lemma UNION_up_iterate_is_fp:
```
```   211   "up_cont F ==>
```
```   212     F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
```
```   213   apply (frule up_cont_mono [THEN up_iterate_chain])
```
```   214   apply (drule (1) up_contD)
```
```   215   apply simp
```
```   216   apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
```
```   217   apply (case_tac xa)
```
```   218    apply auto
```
```   219   done
```
```   220
```
```   221 lemma UNION_up_iterate_lowerbound:
```
```   222     "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
```
```   223   apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
```
```   224    apply fast
```
```   225   apply (induct_tac i)
```
```   226   prefer 2 apply (drule (1) monoD)
```
```   227    apply auto
```
```   228   done
```
```   229
```
```   230 lemma UNION_up_iterate_is_lfp:
```
```   231     "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
```
```   232   apply (rule set_eq_subset [THEN iffD2])
```
```   233   apply (rule conjI)
```
```   234    prefer 2
```
```   235    apply (drule up_cont_mono)
```
```   236    apply (rule UNION_up_iterate_lowerbound)
```
```   237     apply assumption
```
```   238    apply (erule lfp_unfold [symmetric])
```
```   239   apply (rule lfp_lowerbound)
```
```   240   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
```
```   241   apply (erule UNION_up_iterate_is_fp [symmetric])
```
```   242   done
```
```   243
```
```   244
```
```   245 definition
```
```   246   down_iterate :: "('a set => 'a set) => nat => 'a set" where
```
```   247   "down_iterate f n = (f ^^ n) UNIV"
```
```   248
```
```   249 lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
```
```   250   by (simp add: down_iterate_def)
```
```   251
```
```   252 lemma down_iterate_Suc [simp]:
```
```   253     "down_iterate f (Suc i) = f (down_iterate f i)"
```
```   254   by (simp add: down_iterate_def)
```
```   255
```
```   256 lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
```
```   257   apply (rule down_chainI)
```
```   258   apply (induct_tac i)
```
```   259    apply simp+
```
```   260   apply (erule (1) monoD)
```
```   261   done
```
```   262
```
```   263 lemma INTER_down_iterate_is_fp:
```
```   264   "down_cont F ==>
```
```   265     F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
```
```   266   apply (frule down_cont_mono [THEN down_iterate_chain])
```
```   267   apply (drule (1) down_contD)
```
```   268   apply simp
```
```   269   apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
```
```   270   apply (case_tac xa)
```
```   271    apply auto
```
```   272   done
```
```   273
```
```   274 lemma INTER_down_iterate_upperbound:
```
```   275     "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
```
```   276   apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
```
```   277    apply fast
```
```   278   apply (induct_tac i)
```
```   279   prefer 2 apply (drule (1) monoD)
```
```   280    apply auto
```
```   281   done
```
```   282
```
```   283 lemma INTER_down_iterate_is_gfp:
```
```   284     "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
```
```   285   apply (rule set_eq_subset [THEN iffD2])
```
```   286   apply (rule conjI)
```
```   287    apply (drule down_cont_mono)
```
```   288    apply (rule INTER_down_iterate_upperbound)
```
```   289     apply assumption
```
```   290    apply (erule gfp_unfold [symmetric])
```
```   291   apply (rule gfp_upperbound)
```
```   292   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
```
```   293   apply (erule INTER_down_iterate_is_fp)
```
```   294   done
```
```   295
```
```   296 end
```