src/HOL/Library/Continuity.thy
 author wenzelm Thu Oct 16 22:44:24 2008 +0200 (2008-10-16) changeset 28615 4c8fa015ec7f parent 27487 c8a6ce181805 child 30663 0b6aff7451b2 permissions -rw-r--r--
explicit SORT_CONSTRAINT for proofs depending implicitly on certain sorts;
```     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 Plain "~~/src/HOL/Relation_Power"
```
```    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 lemma SUP_nat_conv:
```
```    23   "(SUP n. M n) = sup (M 0) (SUP n. M(Suc n))"
```
```    24 apply(rule order_antisym)
```
```    25  apply(rule SUP_leI)
```
```    26  apply(case_tac n)
```
```    27   apply simp
```
```    28  apply (fast intro:le_SUPI le_supI2)
```
```    29 apply(simp)
```
```    30 apply (blast intro:SUP_leI le_SUPI)
```
```    31 done
```
```    32
```
```    33 lemma continuous_mono: fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
```
```    34   assumes "continuous F" shows "mono F"
```
```    35 proof
```
```    36   fix A B :: "'a" assume "A <= B"
```
```    37   let ?C = "%i::nat. if i=0 then A else B"
```
```    38   have "chain ?C" using `A <= B` by(simp add:chain_def)
```
```    39   have "F B = sup (F A) (F B)"
```
```    40   proof -
```
```    41     have "sup A B = B" using `A <= B` by (simp add:sup_absorb2)
```
```    42     hence "F B = F(SUP i. ?C i)" by (subst SUP_nat_conv) simp
```
```    43     also have "\<dots> = (SUP i. F(?C i))"
```
```    44       using `chain ?C` `continuous F` by(simp add:continuous_def)
```
```    45     also have "\<dots> = sup (F A) (F B)" by (subst SUP_nat_conv) simp
```
```    46     finally show ?thesis .
```
```    47   qed
```
```    48   thus "F A \<le> F B" by(subst le_iff_sup, simp)
```
```    49 qed
```
```    50
```
```    51 lemma continuous_lfp:
```
```    52  assumes "continuous F" shows "lfp F = (SUP i. (F^i) bot)"
```
```    53 proof -
```
```    54   note mono = continuous_mono[OF `continuous F`]
```
```    55   { fix i have "(F^i) bot \<le> lfp F"
```
```    56     proof (induct i)
```
```    57       show "(F^0) bot \<le> lfp F" by simp
```
```    58     next
```
```    59       case (Suc i)
```
```    60       have "(F^(Suc i)) bot = F((F^i) bot)" by simp
```
```    61       also have "\<dots> \<le> F(lfp F)" by(rule monoD[OF mono Suc])
```
```    62       also have "\<dots> = lfp F" by(simp add:lfp_unfold[OF mono, symmetric])
```
```    63       finally show ?case .
```
```    64     qed }
```
```    65   hence "(SUP i. (F^i) bot) \<le> lfp F" by (blast intro!:SUP_leI)
```
```    66   moreover have "lfp F \<le> (SUP i. (F^i) bot)" (is "_ \<le> ?U")
```
```    67   proof (rule lfp_lowerbound)
```
```    68     have "chain(%i. (F^i) bot)"
```
```    69     proof -
```
```    70       { fix i have "(F^i) bot \<le> (F^(Suc i)) bot"
```
```    71 	proof (induct i)
```
```    72 	  case 0 show ?case by simp
```
```    73 	next
```
```    74 	  case Suc thus ?case using monoD[OF mono Suc] by auto
```
```    75 	qed }
```
```    76       thus ?thesis by(auto simp add:chain_def)
```
```    77     qed
```
```    78     hence "F ?U = (SUP i. (F^(i+1)) bot)" using `continuous F` by (simp add:continuous_def)
```
```    79     also have "\<dots> \<le> ?U" by(fast intro:SUP_leI le_SUPI)
```
```    80     finally show "F ?U \<le> ?U" .
```
```    81   qed
```
```    82   ultimately show ?thesis by (blast intro:order_antisym)
```
```    83 qed
```
```    84
```
```    85 text{* The following development is just for sets but presents an up
```
```    86 and a down version of chains and continuity and covers @{const gfp}. *}
```
```    87
```
```    88
```
```    89 subsection "Chains"
```
```    90
```
```    91 definition
```
```    92   up_chain :: "(nat => 'a set) => bool" where
```
```    93   "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
```
```    94
```
```    95 lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
```
```    96   by (simp add: up_chain_def)
```
```    97
```
```    98 lemma up_chainD: "up_chain F ==> F i \<subseteq> F (Suc i)"
```
```    99   by (simp add: up_chain_def)
```
```   100
```
```   101 lemma up_chain_less_mono:
```
```   102     "up_chain F ==> x < y ==> F x \<subseteq> F y"
```
```   103   apply (induct y)
```
```   104    apply (blast dest: up_chainD elim: less_SucE)+
```
```   105   done
```
```   106
```
```   107 lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> F y"
```
```   108   apply (drule le_imp_less_or_eq)
```
```   109   apply (blast dest: up_chain_less_mono)
```
```   110   done
```
```   111
```
```   112
```
```   113 definition
```
```   114   down_chain :: "(nat => 'a set) => bool" where
```
```   115   "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
```
```   116
```
```   117 lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
```
```   118   by (simp add: down_chain_def)
```
```   119
```
```   120 lemma down_chainD: "down_chain F ==> F (Suc i) \<subseteq> F i"
```
```   121   by (simp add: down_chain_def)
```
```   122
```
```   123 lemma down_chain_less_mono:
```
```   124     "down_chain F ==> x < y ==> F y \<subseteq> F x"
```
```   125   apply (induct y)
```
```   126    apply (blast dest: down_chainD elim: less_SucE)+
```
```   127   done
```
```   128
```
```   129 lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> F x"
```
```   130   apply (drule le_imp_less_or_eq)
```
```   131   apply (blast dest: down_chain_less_mono)
```
```   132   done
```
```   133
```
```   134
```
```   135 subsection "Continuity"
```
```   136
```
```   137 definition
```
```   138   up_cont :: "('a set => 'a set) => bool" where
```
```   139   "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
```
```   140
```
```   141 lemma up_contI:
```
```   142   "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
```
```   143 apply (unfold up_cont_def)
```
```   144 apply blast
```
```   145 done
```
```   146
```
```   147 lemma up_contD:
```
```   148   "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
```
```   149 apply (unfold up_cont_def)
```
```   150 apply auto
```
```   151 done
```
```   152
```
```   153
```
```   154 lemma up_cont_mono: "up_cont f ==> mono f"
```
```   155 apply (rule monoI)
```
```   156 apply (drule_tac F = "\<lambda>i. if i = 0 then x else y" in up_contD)
```
```   157  apply (rule up_chainI)
```
```   158  apply simp
```
```   159 apply (drule Un_absorb1)
```
```   160 apply (auto simp add: nat_not_singleton)
```
```   161 done
```
```   162
```
```   163
```
```   164 definition
```
```   165   down_cont :: "('a set => 'a set) => bool" where
```
```   166   "down_cont f =
```
```   167     (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
```
```   168
```
```   169 lemma down_contI:
```
```   170   "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
```
```   171     down_cont f"
```
```   172   apply (unfold down_cont_def)
```
```   173   apply blast
```
```   174   done
```
```   175
```
```   176 lemma down_contD: "down_cont f ==> down_chain F ==>
```
```   177     f (Inter (range F)) = Inter (f ` range F)"
```
```   178   apply (unfold down_cont_def)
```
```   179   apply auto
```
```   180   done
```
```   181
```
```   182 lemma down_cont_mono: "down_cont f ==> mono f"
```
```   183 apply (rule monoI)
```
```   184 apply (drule_tac F = "\<lambda>i. if i = 0 then y else x" in down_contD)
```
```   185  apply (rule down_chainI)
```
```   186  apply simp
```
```   187 apply (drule Int_absorb1)
```
```   188 apply auto
```
```   189 apply (auto simp add: nat_not_singleton)
```
```   190 done
```
```   191
```
```   192
```
```   193 subsection "Iteration"
```
```   194
```
```   195 definition
```
```   196   up_iterate :: "('a set => 'a set) => nat => 'a set" where
```
```   197   "up_iterate f n = (f^n) {}"
```
```   198
```
```   199 lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
```
```   200   by (simp add: up_iterate_def)
```
```   201
```
```   202 lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
```
```   203   by (simp add: up_iterate_def)
```
```   204
```
```   205 lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
```
```   206   apply (rule up_chainI)
```
```   207   apply (induct_tac i)
```
```   208    apply simp+
```
```   209   apply (erule (1) monoD)
```
```   210   done
```
```   211
```
```   212 lemma UNION_up_iterate_is_fp:
```
```   213   "up_cont F ==>
```
```   214     F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
```
```   215   apply (frule up_cont_mono [THEN up_iterate_chain])
```
```   216   apply (drule (1) up_contD)
```
```   217   apply simp
```
```   218   apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
```
```   219   apply (case_tac xa)
```
```   220    apply auto
```
```   221   done
```
```   222
```
```   223 lemma UNION_up_iterate_lowerbound:
```
```   224     "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
```
```   225   apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> P)")
```
```   226    apply fast
```
```   227   apply (induct_tac i)
```
```   228   prefer 2 apply (drule (1) monoD)
```
```   229    apply auto
```
```   230   done
```
```   231
```
```   232 lemma UNION_up_iterate_is_lfp:
```
```   233     "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
```
```   234   apply (rule set_eq_subset [THEN iffD2])
```
```   235   apply (rule conjI)
```
```   236    prefer 2
```
```   237    apply (drule up_cont_mono)
```
```   238    apply (rule UNION_up_iterate_lowerbound)
```
```   239     apply assumption
```
```   240    apply (erule lfp_unfold [symmetric])
```
```   241   apply (rule lfp_lowerbound)
```
```   242   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
```
```   243   apply (erule UNION_up_iterate_is_fp [symmetric])
```
```   244   done
```
```   245
```
```   246
```
```   247 definition
```
```   248   down_iterate :: "('a set => 'a set) => nat => 'a set" where
```
```   249   "down_iterate f n = (f^n) UNIV"
```
```   250
```
```   251 lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
```
```   252   by (simp add: down_iterate_def)
```
```   253
```
```   254 lemma down_iterate_Suc [simp]:
```
```   255     "down_iterate f (Suc i) = f (down_iterate f i)"
```
```   256   by (simp add: down_iterate_def)
```
```   257
```
```   258 lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
```
```   259   apply (rule down_chainI)
```
```   260   apply (induct_tac i)
```
```   261    apply simp+
```
```   262   apply (erule (1) monoD)
```
```   263   done
```
```   264
```
```   265 lemma INTER_down_iterate_is_fp:
```
```   266   "down_cont F ==>
```
```   267     F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
```
```   268   apply (frule down_cont_mono [THEN down_iterate_chain])
```
```   269   apply (drule (1) down_contD)
```
```   270   apply simp
```
```   271   apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
```
```   272   apply (case_tac xa)
```
```   273    apply auto
```
```   274   done
```
```   275
```
```   276 lemma INTER_down_iterate_upperbound:
```
```   277     "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
```
```   278   apply (subgoal_tac "(!!i. P \<subseteq> down_iterate F i)")
```
```   279    apply fast
```
```   280   apply (induct_tac i)
```
```   281   prefer 2 apply (drule (1) monoD)
```
```   282    apply auto
```
```   283   done
```
```   284
```
```   285 lemma INTER_down_iterate_is_gfp:
```
```   286     "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
```
```   287   apply (rule set_eq_subset [THEN iffD2])
```
```   288   apply (rule conjI)
```
```   289    apply (drule down_cont_mono)
```
```   290    apply (rule INTER_down_iterate_upperbound)
```
```   291     apply assumption
```
```   292    apply (erule gfp_unfold [symmetric])
```
```   293   apply (rule gfp_upperbound)
```
```   294   apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
```
```   295   apply (erule INTER_down_iterate_is_fp)
```
```   296   done
```
```   297
```
```   298 end
```