src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author hoelzl Mon Jan 14 17:29:04 2013 +0100 (2013-01-14) changeset 50881 ae630bab13da parent 50526 899c9c4e4a4c child 50882 a382bf90867e permissions -rw-r--r--
renamed countable_basis_space to second_countable_topology
     1 (*  title:      HOL/Library/Topology_Euclidian_Space.thy

     2     Author:     Amine Chaieb, University of Cambridge

     3     Author:     Robert Himmelmann, TU Muenchen

     4     Author:     Brian Huffman, Portland State University

     5 *)

     6

     7 header {* Elementary topology in Euclidean space. *}

     8

     9 theory Topology_Euclidean_Space

    10 imports

    11   SEQ

    12   "~~/src/HOL/Library/Diagonal_Subsequence"

    13   "~~/src/HOL/Library/Countable_Set"

    14   Linear_Algebra

    15   "~~/src/HOL/Library/Glbs"

    16   "~~/src/HOL/Library/FuncSet"

    17   Norm_Arith

    18 begin

    19

    20 lemma countable_PiE:

    21   "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"

    22   by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)

    23

    24 lemma countable_rat: "countable \<rat>"

    25   unfolding Rats_def by auto

    26

    27 subsection {* Topological Basis *}

    28

    29 context topological_space

    30 begin

    31

    32 definition "topological_basis B =

    33   ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> Union B' = x)))"

    34

    35 lemma topological_basis_iff:

    36   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"

    37   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"

    38     (is "_ \<longleftrightarrow> ?rhs")

    39 proof safe

    40   fix O' and x::'a

    41   assume H: "topological_basis B" "open O'" "x \<in> O'"

    42   hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)

    43   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto

    44   thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto

    45 next

    46   assume H: ?rhs

    47   show "topological_basis B" using assms unfolding topological_basis_def

    48   proof safe

    49     fix O'::"'a set" assume "open O'"

    50     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"

    51       by (force intro: bchoice simp: Bex_def)

    52     thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"

    53       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])

    54   qed

    55 qed

    56

    57 lemma topological_basisI:

    58   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"

    59   assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"

    60   shows "topological_basis B"

    61   using assms by (subst topological_basis_iff) auto

    62

    63 lemma topological_basisE:

    64   fixes O'

    65   assumes "topological_basis B"

    66   assumes "open O'"

    67   assumes "x \<in> O'"

    68   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"

    69 proof atomize_elim

    70   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)

    71   with topological_basis_iff assms

    72   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)

    73 qed

    74

    75 lemma topological_basis_open:

    76   assumes "topological_basis B"

    77   assumes "X \<in> B"

    78   shows "open X"

    79   using assms

    80   by (simp add: topological_basis_def)

    81

    82 lemma basis_dense:

    83   fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"

    84   assumes "topological_basis B"

    85   assumes choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"

    86   shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"

    87 proof (intro allI impI)

    88   fix X::"'a set" assume "open X" "X \<noteq> {}"

    89   from topological_basisE[OF topological_basis B open X choosefrom_basis[OF X \<noteq> {}]]

    90   guess B' . note B' = this

    91   thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)

    92 qed

    93

    94 end

    95

    96 subsection {* Countable Basis *}

    97

    98 locale countable_basis =

    99   fixes B::"'a::topological_space set set"

   100   assumes is_basis: "topological_basis B"

   101   assumes countable_basis: "countable B"

   102 begin

   103

   104 lemma open_countable_basis_ex:

   105   assumes "open X"

   106   shows "\<exists>B' \<subseteq> B. X = Union B'"

   107   using assms countable_basis is_basis unfolding topological_basis_def by blast

   108

   109 lemma open_countable_basisE:

   110   assumes "open X"

   111   obtains B' where "B' \<subseteq> B" "X = Union B'"

   112   using assms open_countable_basis_ex by (atomize_elim) simp

   113

   114 lemma countable_dense_exists:

   115   shows "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"

   116 proof -

   117   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"

   118   have "countable (?f  B)" using countable_basis by simp

   119   with basis_dense[OF is_basis, of ?f] show ?thesis

   120     by (intro exI[where x="?f  B"]) (metis (mono_tags) all_not_in_conv imageI someI)

   121 qed

   122

   123 lemma countable_dense_setE:

   124   obtains D :: "'a set"

   125   where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"

   126   using countable_dense_exists by blast

   127

   128 text {* Construction of an increasing sequence approximating open sets,

   129   therefore basis which is closed under union. *}

   130

   131 definition union_closed_basis::"'a set set" where

   132   "union_closed_basis = (\<lambda>l. \<Union>set l)  lists B"

   133

   134 lemma basis_union_closed_basis: "topological_basis union_closed_basis"

   135 proof (rule topological_basisI)

   136   fix O' and x::'a assume "open O'" "x \<in> O'"

   137   from topological_basisE[OF is_basis this] guess B' . note B' = this

   138   thus "\<exists>B'\<in>union_closed_basis. x \<in> B' \<and> B' \<subseteq> O'" unfolding union_closed_basis_def

   139     by (auto intro!: bexI[where x="[B']"])

   140 next

   141   fix B' assume "B' \<in> union_closed_basis"

   142   thus "open B'"

   143     using topological_basis_open[OF is_basis]

   144     by (auto simp: union_closed_basis_def)

   145 qed

   146

   147 lemma countable_union_closed_basis: "countable union_closed_basis"

   148   unfolding union_closed_basis_def using countable_basis by simp

   149

   150 lemmas open_union_closed_basis = topological_basis_open[OF basis_union_closed_basis]

   151

   152 lemma union_closed_basis_ex:

   153  assumes X: "X \<in> union_closed_basis"

   154  shows "\<exists>B'. finite B' \<and> X = \<Union>B' \<and> B' \<subseteq> B"

   155 proof -

   156   from X obtain l where "\<And>x. x\<in>set l \<Longrightarrow> x\<in>B" "X = \<Union>set l" by (auto simp: union_closed_basis_def)

   157   thus ?thesis by auto

   158 qed

   159

   160 lemma union_closed_basisE:

   161   assumes "X \<in> union_closed_basis"

   162   obtains B' where "finite B'" "X = \<Union>B'" "B' \<subseteq> B" using union_closed_basis_ex[OF assms] by blast

   163

   164 lemma union_closed_basisI:

   165   assumes "finite B'" "X = \<Union>B'" "B' \<subseteq> B"

   166   shows "X \<in> union_closed_basis"

   167 proof -

   168   from finite_list[OF finite B'] guess l ..

   169   thus ?thesis using assms unfolding union_closed_basis_def by (auto intro!: image_eqI[where x=l])

   170 qed

   171

   172 lemma empty_basisI[intro]: "{} \<in> union_closed_basis"

   173   by (rule union_closed_basisI[of "{}"]) auto

   174

   175 lemma union_basisI[intro]:

   176   assumes "X \<in> union_closed_basis" "Y \<in> union_closed_basis"

   177   shows "X \<union> Y \<in> union_closed_basis"

   178   using assms by (auto intro: union_closed_basisI elim!:union_closed_basisE)

   179

   180 lemma open_imp_Union_of_incseq:

   181   assumes "open X"

   182   shows "\<exists>S. incseq S \<and> (\<Union>j. S j) = X \<and> range S \<subseteq> union_closed_basis"

   183 proof -

   184   from open_countable_basis_ex[OF open X]

   185   obtain B' where B': "B'\<subseteq>B" "X = \<Union>B'" by auto

   186   from this(1) countable_basis have "countable B'" by (rule countable_subset)

   187   show ?thesis

   188   proof cases

   189     assume "B' \<noteq> {}"

   190     def S \<equiv> "\<lambda>n. \<Union>i\<in>{0..n}. from_nat_into B' i"

   191     have S:"\<And>n. S n = \<Union>{from_nat_into B' i|i. i\<in>{0..n}}" unfolding S_def by force

   192     have "incseq S" by (force simp: S_def incseq_Suc_iff)

   193     moreover

   194     have "(\<Union>j. S j) = X" unfolding B'

   195     proof safe

   196       fix x X assume "X \<in> B'" "x \<in> X"

   197       then obtain n where "X = from_nat_into B' n"

   198         by (metis countable B' from_nat_into_surj)

   199       also have "\<dots> \<subseteq> S n" by (auto simp: S_def)

   200       finally show "x \<in> (\<Union>j. S j)" using x \<in> X by auto

   201     next

   202       fix x n

   203       assume "x \<in> S n"

   204       also have "\<dots> = (\<Union>i\<in>{0..n}. from_nat_into B' i)"

   205         by (simp add: S_def)

   206       also have "\<dots> \<subseteq> (\<Union>i. from_nat_into B' i)" by auto

   207       also have "\<dots> \<subseteq> \<Union>B'" using B' \<noteq> {} by (auto intro: from_nat_into)

   208       finally show "x \<in> \<Union>B'" .

   209     qed

   210     moreover have "range S \<subseteq> union_closed_basis" using B'

   211       by (auto intro!: union_closed_basisI[OF _ S] simp: from_nat_into B' \<noteq> {})

   212     ultimately show ?thesis by auto

   213   qed (auto simp: B')

   214 qed

   215

   216 lemma open_incseqE:

   217   assumes "open X"

   218   obtains S where "incseq S" "(\<Union>j. S j) = X" "range S \<subseteq> union_closed_basis"

   219   using open_imp_Union_of_incseq assms by atomize_elim

   220

   221 end

   222

   223 class second_countable_topology = topological_space +

   224   assumes ex_countable_basis:

   225     "\<exists>B::'a::topological_space set set. countable B \<and> topological_basis B"

   226

   227 sublocale second_countable_topology < countable_basis "SOME B. countable B \<and> topological_basis B"

   228   using someI_ex[OF ex_countable_basis] by unfold_locales safe

   229

   230 subsection {* Polish spaces *}

   231

   232 text {* Textbooks define Polish spaces as completely metrizable.

   233   We assume the topology to be complete for a given metric. *}

   234

   235 class polish_space = complete_space + second_countable_topology

   236

   237 subsection {* General notion of a topology as a value *}

   238

   239 definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"

   240 typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"

   241   morphisms "openin" "topology"

   242   unfolding istopology_def by blast

   243

   244 lemma istopology_open_in[intro]: "istopology(openin U)"

   245   using openin[of U] by blast

   246

   247 lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"

   248   using topology_inverse[unfolded mem_Collect_eq] .

   249

   250 lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"

   251   using topology_inverse[of U] istopology_open_in[of "topology U"] by auto

   252

   253 lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"

   254 proof-

   255   { assume "T1=T2"

   256     hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }

   257   moreover

   258   { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"

   259     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)

   260     hence "topology (openin T1) = topology (openin T2)" by simp

   261     hence "T1 = T2" unfolding openin_inverse .

   262   }

   263   ultimately show ?thesis by blast

   264 qed

   265

   266 text{* Infer the "universe" from union of all sets in the topology. *}

   267

   268 definition "topspace T =  \<Union>{S. openin T S}"

   269

   270 subsubsection {* Main properties of open sets *}

   271

   272 lemma openin_clauses:

   273   fixes U :: "'a topology"

   274   shows "openin U {}"

   275   "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"

   276   "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"

   277   using openin[of U] unfolding istopology_def mem_Collect_eq

   278   by fast+

   279

   280 lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"

   281   unfolding topspace_def by blast

   282 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)

   283

   284 lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"

   285   using openin_clauses by simp

   286

   287 lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"

   288   using openin_clauses by simp

   289

   290 lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"

   291   using openin_Union[of "{S,T}" U] by auto

   292

   293 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)

   294

   295 lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"

   296   (is "?lhs \<longleftrightarrow> ?rhs")

   297 proof

   298   assume ?lhs

   299   then show ?rhs by auto

   300 next

   301   assume H: ?rhs

   302   let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"

   303   have "openin U ?t" by (simp add: openin_Union)

   304   also have "?t = S" using H by auto

   305   finally show "openin U S" .

   306 qed

   307

   308

   309 subsubsection {* Closed sets *}

   310

   311 definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"

   312

   313 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)

   314 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)

   315 lemma closedin_topspace[intro,simp]:

   316   "closedin U (topspace U)" by (simp add: closedin_def)

   317 lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"

   318   by (auto simp add: Diff_Un closedin_def)

   319

   320 lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto

   321 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"

   322   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto

   323

   324 lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"

   325   using closedin_Inter[of "{S,T}" U] by auto

   326

   327 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast

   328 lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"

   329   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   330   apply (metis openin_subset subset_eq)

   331   done

   332

   333 lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"

   334   by (simp add: openin_closedin_eq)

   335

   336 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"

   337 proof-

   338   have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT

   339     by (auto simp add: topspace_def openin_subset)

   340   then show ?thesis using oS cT by (auto simp add: closedin_def)

   341 qed

   342

   343 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"

   344 proof-

   345   have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT

   346     by (auto simp add: topspace_def )

   347   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)

   348 qed

   349

   350 subsubsection {* Subspace topology *}

   351

   352 definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

   353

   354 lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

   355   (is "istopology ?L")

   356 proof-

   357   have "?L {}" by blast

   358   {fix A B assume A: "?L A" and B: "?L B"

   359     from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast

   360     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+

   361     then have "?L (A \<inter> B)" by blast}

   362   moreover

   363   {fix K assume K: "K \<subseteq> Collect ?L"

   364     have th0: "Collect ?L = (\<lambda>S. S \<inter> V)  Collect (openin U)"

   365       apply (rule set_eqI)

   366       apply (simp add: Ball_def image_iff)

   367       by metis

   368     from K[unfolded th0 subset_image_iff]

   369     obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V)  Sk" by blast

   370     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto

   371     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)

   372     ultimately have "?L (\<Union>K)" by blast}

   373   ultimately show ?thesis

   374     unfolding subset_eq mem_Collect_eq istopology_def by blast

   375 qed

   376

   377 lemma openin_subtopology:

   378   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"

   379   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   380   by auto

   381

   382 lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"

   383   by (auto simp add: topspace_def openin_subtopology)

   384

   385 lemma closedin_subtopology:

   386   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"

   387   unfolding closedin_def topspace_subtopology

   388   apply (simp add: openin_subtopology)

   389   apply (rule iffI)

   390   apply clarify

   391   apply (rule_tac x="topspace U - T" in exI)

   392   by auto

   393

   394 lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"

   395   unfolding openin_subtopology

   396   apply (rule iffI, clarify)

   397   apply (frule openin_subset[of U])  apply blast

   398   apply (rule exI[where x="topspace U"])

   399   apply auto

   400   done

   401

   402 lemma subtopology_superset:

   403   assumes UV: "topspace U \<subseteq> V"

   404   shows "subtopology U V = U"

   405 proof-

   406   {fix S

   407     {fix T assume T: "openin U T" "S = T \<inter> V"

   408       from T openin_subset[OF T(1)] UV have eq: "S = T" by blast

   409       have "openin U S" unfolding eq using T by blast}

   410     moreover

   411     {assume S: "openin U S"

   412       hence "\<exists>T. openin U T \<and> S = T \<inter> V"

   413         using openin_subset[OF S] UV by auto}

   414     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}

   415   then show ?thesis unfolding topology_eq openin_subtopology by blast

   416 qed

   417

   418 lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

   419   by (simp add: subtopology_superset)

   420

   421 lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

   422   by (simp add: subtopology_superset)

   423

   424 subsubsection {* The standard Euclidean topology *}

   425

   426 definition

   427   euclidean :: "'a::topological_space topology" where

   428   "euclidean = topology open"

   429

   430 lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"

   431   unfolding euclidean_def

   432   apply (rule cong[where x=S and y=S])

   433   apply (rule topology_inverse[symmetric])

   434   apply (auto simp add: istopology_def)

   435   done

   436

   437 lemma topspace_euclidean: "topspace euclidean = UNIV"

   438   apply (simp add: topspace_def)

   439   apply (rule set_eqI)

   440   by (auto simp add: open_openin[symmetric])

   441

   442 lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

   443   by (simp add: topspace_euclidean topspace_subtopology)

   444

   445 lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"

   446   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

   447

   448 lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"

   449   by (simp add: open_openin openin_subopen[symmetric])

   450

   451 text {* Basic "localization" results are handy for connectedness. *}

   452

   453 lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"

   454   by (auto simp add: openin_subtopology open_openin[symmetric])

   455

   456 lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"

   457   by (auto simp add: openin_open)

   458

   459 lemma open_openin_trans[trans]:

   460  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"

   461   by (metis Int_absorb1  openin_open_Int)

   462

   463 lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"

   464   by (auto simp add: openin_open)

   465

   466 lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"

   467   by (simp add: closedin_subtopology closed_closedin Int_ac)

   468

   469 lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"

   470   by (metis closedin_closed)

   471

   472 lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"

   473   apply (subgoal_tac "S \<inter> T = T" )

   474   apply auto

   475   apply (frule closedin_closed_Int[of T S])

   476   by simp

   477

   478 lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"

   479   by (auto simp add: closedin_closed)

   480

   481 lemma openin_euclidean_subtopology_iff:

   482   fixes S U :: "'a::metric_space set"

   483   shows "openin (subtopology euclidean U) S

   484   \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")

   485 proof

   486   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast

   487 next

   488   def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"

   489   have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"

   490     unfolding T_def

   491     apply clarsimp

   492     apply (rule_tac x="d - dist x a" in exI)

   493     apply (clarsimp simp add: less_diff_eq)

   494     apply (erule rev_bexI)

   495     apply (rule_tac x=d in exI, clarify)

   496     apply (erule le_less_trans [OF dist_triangle])

   497     done

   498   assume ?rhs hence 2: "S = U \<inter> T"

   499     unfolding T_def

   500     apply auto

   501     apply (drule (1) bspec, erule rev_bexI)

   502     apply auto

   503     done

   504   from 1 2 show ?lhs

   505     unfolding openin_open open_dist by fast

   506 qed

   507

   508 text {* These "transitivity" results are handy too *}

   509

   510 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T

   511   \<Longrightarrow> openin (subtopology euclidean U) S"

   512   unfolding open_openin openin_open by blast

   513

   514 lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"

   515   by (auto simp add: openin_open intro: openin_trans)

   516

   517 lemma closedin_trans[trans]:

   518  "closedin (subtopology euclidean T) S \<Longrightarrow>

   519            closedin (subtopology euclidean U) T

   520            ==> closedin (subtopology euclidean U) S"

   521   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)

   522

   523 lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"

   524   by (auto simp add: closedin_closed intro: closedin_trans)

   525

   526

   527 subsection {* Open and closed balls *}

   528

   529 definition

   530   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

   531   "ball x e = {y. dist x y < e}"

   532

   533 definition

   534   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

   535   "cball x e = {y. dist x y \<le> e}"

   536

   537 lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"

   538   by (simp add: ball_def)

   539

   540 lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"

   541   by (simp add: cball_def)

   542

   543 lemma mem_ball_0:

   544   fixes x :: "'a::real_normed_vector"

   545   shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"

   546   by (simp add: dist_norm)

   547

   548 lemma mem_cball_0:

   549   fixes x :: "'a::real_normed_vector"

   550   shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"

   551   by (simp add: dist_norm)

   552

   553 lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"

   554   by simp

   555

   556 lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"

   557   by simp

   558

   559 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)

   560 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)

   561 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)

   562 lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"

   563   by (simp add: set_eq_iff) arith

   564

   565 lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"

   566   by (simp add: set_eq_iff)

   567

   568 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"

   569   "(a::real) - b < 0 \<longleftrightarrow> a < b"

   570   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+

   571 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"

   572   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+

   573

   574 lemma open_ball[intro, simp]: "open (ball x e)"

   575   unfolding open_dist ball_def mem_Collect_eq Ball_def

   576   unfolding dist_commute

   577   apply clarify

   578   apply (rule_tac x="e - dist xa x" in exI)

   579   using dist_triangle_alt[where z=x]

   580   apply (clarsimp simp add: diff_less_iff)

   581   apply atomize

   582   apply (erule_tac x="y" in allE)

   583   apply (erule_tac x="xa" in allE)

   584   by arith

   585

   586 lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"

   587   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   588

   589 lemma openE[elim?]:

   590   assumes "open S" "x\<in>S"

   591   obtains e where "e>0" "ball x e \<subseteq> S"

   592   using assms unfolding open_contains_ball by auto

   593

   594 lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

   595   by (metis open_contains_ball subset_eq centre_in_ball)

   596

   597 lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"

   598   unfolding mem_ball set_eq_iff

   599   apply (simp add: not_less)

   600   by (metis zero_le_dist order_trans dist_self)

   601

   602 lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp

   603

   604 lemma euclidean_dist_l2:

   605   fixes x y :: "'a :: euclidean_space"

   606   shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"

   607   unfolding dist_norm norm_eq_sqrt_inner setL2_def

   608   by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

   609

   610 definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"

   611

   612 lemma rational_boxes:

   613   fixes x :: "'a\<Colon>euclidean_space"

   614   assumes "0 < e"

   615   shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"

   616 proof -

   617   def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"

   618   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)

   619   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")

   620   proof

   621     fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto

   622   qed

   623   from choice[OF this] guess a .. note a = this

   624   have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")

   625   proof

   626     fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto

   627   qed

   628   from choice[OF this] guess b .. note b = this

   629   let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"

   630   show ?thesis

   631   proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)

   632     fix y :: 'a assume *: "y \<in> box ?a ?b"

   633     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"

   634       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

   635     also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"

   636     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   637       fix i :: "'a" assume i: "i \<in> Basis"

   638       have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)

   639       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto

   640       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto

   641       ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto

   642       then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"

   643         unfolding e'_def by (auto simp: dist_real_def)

   644       then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"

   645         by (rule power_strict_mono) auto

   646       then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"

   647         by (simp add: power_divide)

   648     qed auto

   649     also have "\<dots> = e" using 0 < e by (simp add: real_eq_of_nat)

   650     finally show "y \<in> ball x e" by (auto simp: ball_def)

   651   qed (insert a b, auto simp: box_def)

   652 qed

   653

   654 lemma open_UNION_box:

   655   fixes M :: "'a\<Colon>euclidean_space set"

   656   assumes "open M"

   657   defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"

   658   defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"

   659   defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"

   660   shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"

   661 proof safe

   662   fix x assume "x \<in> M"

   663   obtain e where e: "e > 0" "ball x e \<subseteq> M"

   664     using openE[OF open M x \<in> M] by auto

   665   moreover then obtain a b where ab: "x \<in> box a b"

   666     "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"

   667     using rational_boxes[OF e(1)] by metis

   668   ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"

   669      by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])

   670         (auto simp: euclidean_representation I_def a'_def b'_def)

   671 qed (auto simp: I_def)

   672

   673 subsection{* Connectedness *}

   674

   675 definition "connected S \<longleftrightarrow>

   676   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})

   677   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"

   678

   679 lemma connected_local:

   680  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.

   681                  openin (subtopology euclidean S) e1 \<and>

   682                  openin (subtopology euclidean S) e2 \<and>

   683                  S \<subseteq> e1 \<union> e2 \<and>

   684                  e1 \<inter> e2 = {} \<and>

   685                  ~(e1 = {}) \<and>

   686                  ~(e2 = {}))"

   687 unfolding connected_def openin_open by (safe, blast+)

   688

   689 lemma exists_diff:

   690   fixes P :: "'a set \<Rightarrow> bool"

   691   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

   692 proof-

   693   {assume "?lhs" hence ?rhs by blast }

   694   moreover

   695   {fix S assume H: "P S"

   696     have "S = - (- S)" by auto

   697     with H have "P (- (- S))" by metis }

   698   ultimately show ?thesis by metis

   699 qed

   700

   701 lemma connected_clopen: "connected S \<longleftrightarrow>

   702         (\<forall>T. openin (subtopology euclidean S) T \<and>

   703             closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

   704 proof-

   705   have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"

   706     unfolding connected_def openin_open closedin_closed

   707     apply (subst exists_diff) by blast

   708   hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"

   709     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis

   710

   711   have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"

   712     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

   713     unfolding connected_def openin_open closedin_closed by auto

   714   {fix e2

   715     {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"

   716         by auto}

   717     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}

   718   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast

   719   then show ?thesis unfolding th0 th1 by simp

   720 qed

   721

   722 lemma connected_empty[simp, intro]: "connected {}"

   723   by (simp add: connected_def)

   724

   725

   726 subsection{* Limit points *}

   727

   728 definition (in topological_space)

   729   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where

   730   "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"

   731

   732 lemma islimptI:

   733   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

   734   shows "x islimpt S"

   735   using assms unfolding islimpt_def by auto

   736

   737 lemma islimptE:

   738   assumes "x islimpt S" and "x \<in> T" and "open T"

   739   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

   740   using assms unfolding islimpt_def by auto

   741

   742 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

   743   unfolding islimpt_def eventually_at_topological by auto

   744

   745 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"

   746   unfolding islimpt_def by fast

   747

   748 lemma islimpt_approachable:

   749   fixes x :: "'a::metric_space"

   750   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

   751   unfolding islimpt_iff_eventually eventually_at by fast

   752

   753 lemma islimpt_approachable_le:

   754   fixes x :: "'a::metric_space"

   755   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"

   756   unfolding islimpt_approachable

   757   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

   758     THEN arg_cong [where f=Not]]

   759   by (simp add: Bex_def conj_commute conj_left_commute)

   760

   761 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

   762   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

   763

   764 text {* A perfect space has no isolated points. *}

   765

   766 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

   767   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   768

   769 lemma perfect_choose_dist:

   770   fixes x :: "'a::{perfect_space, metric_space}"

   771   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

   772 using islimpt_UNIV [of x]

   773 by (simp add: islimpt_approachable)

   774

   775 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

   776   unfolding closed_def

   777   apply (subst open_subopen)

   778   apply (simp add: islimpt_def subset_eq)

   779   by (metis ComplE ComplI)

   780

   781 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

   782   unfolding islimpt_def by auto

   783

   784 lemma finite_set_avoid:

   785   fixes a :: "'a::metric_space"

   786   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"

   787 proof(induct rule: finite_induct[OF fS])

   788   case 1 thus ?case by (auto intro: zero_less_one)

   789 next

   790   case (2 x F)

   791   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast

   792   {assume "x = a" hence ?case using d by auto  }

   793   moreover

   794   {assume xa: "x\<noteq>a"

   795     let ?d = "min d (dist a x)"

   796     have dp: "?d > 0" using xa d(1) using dist_nz by auto

   797     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto

   798     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }

   799   ultimately show ?case by blast

   800 qed

   801

   802 lemma islimpt_finite:

   803   fixes S :: "'a::metric_space set"

   804   assumes fS: "finite S" shows "\<not> a islimpt S"

   805   unfolding islimpt_approachable

   806   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)

   807

   808 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

   809   apply (rule iffI)

   810   defer

   811   apply (metis Un_upper1 Un_upper2 islimpt_subset)

   812   unfolding islimpt_def

   813   apply (rule ccontr, clarsimp, rename_tac A B)

   814   apply (drule_tac x="A \<inter> B" in spec)

   815   apply (auto simp add: open_Int)

   816   done

   817

   818 lemma discrete_imp_closed:

   819   fixes S :: "'a::metric_space set"

   820   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

   821   shows "closed S"

   822 proof-

   823   {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

   824     from e have e2: "e/2 > 0" by arith

   825     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast

   826     let ?m = "min (e/2) (dist x y) "

   827     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])

   828     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast

   829     have th: "dist z y < e" using z y

   830       by (intro dist_triangle_lt [where z=x], simp)

   831     from d[rule_format, OF y(1) z(1) th] y z

   832     have False by (auto simp add: dist_commute)}

   833   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])

   834 qed

   835

   836

   837 subsection {* Interior of a Set *}

   838

   839 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

   840

   841 lemma interiorI [intro?]:

   842   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

   843   shows "x \<in> interior S"

   844   using assms unfolding interior_def by fast

   845

   846 lemma interiorE [elim?]:

   847   assumes "x \<in> interior S"

   848   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

   849   using assms unfolding interior_def by fast

   850

   851 lemma open_interior [simp, intro]: "open (interior S)"

   852   by (simp add: interior_def open_Union)

   853

   854 lemma interior_subset: "interior S \<subseteq> S"

   855   by (auto simp add: interior_def)

   856

   857 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

   858   by (auto simp add: interior_def)

   859

   860 lemma interior_open: "open S \<Longrightarrow> interior S = S"

   861   by (intro equalityI interior_subset interior_maximal subset_refl)

   862

   863 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

   864   by (metis open_interior interior_open)

   865

   866 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

   867   by (metis interior_maximal interior_subset subset_trans)

   868

   869 lemma interior_empty [simp]: "interior {} = {}"

   870   using open_empty by (rule interior_open)

   871

   872 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

   873   using open_UNIV by (rule interior_open)

   874

   875 lemma interior_interior [simp]: "interior (interior S) = interior S"

   876   using open_interior by (rule interior_open)

   877

   878 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

   879   by (auto simp add: interior_def)

   880

   881 lemma interior_unique:

   882   assumes "T \<subseteq> S" and "open T"

   883   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

   884   shows "interior S = T"

   885   by (intro equalityI assms interior_subset open_interior interior_maximal)

   886

   887 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

   888   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

   889     Int_lower2 interior_maximal interior_subset open_Int open_interior)

   890

   891 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

   892   using open_contains_ball_eq [where S="interior S"]

   893   by (simp add: open_subset_interior)

   894

   895 lemma interior_limit_point [intro]:

   896   fixes x :: "'a::perfect_space"

   897   assumes x: "x \<in> interior S" shows "x islimpt S"

   898   using x islimpt_UNIV [of x]

   899   unfolding interior_def islimpt_def

   900   apply (clarsimp, rename_tac T T')

   901   apply (drule_tac x="T \<inter> T'" in spec)

   902   apply (auto simp add: open_Int)

   903   done

   904

   905 lemma interior_closed_Un_empty_interior:

   906   assumes cS: "closed S" and iT: "interior T = {}"

   907   shows "interior (S \<union> T) = interior S"

   908 proof

   909   show "interior S \<subseteq> interior (S \<union> T)"

   910     by (rule interior_mono, rule Un_upper1)

   911 next

   912   show "interior (S \<union> T) \<subseteq> interior S"

   913   proof

   914     fix x assume "x \<in> interior (S \<union> T)"

   915     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

   916     show "x \<in> interior S"

   917     proof (rule ccontr)

   918       assume "x \<notin> interior S"

   919       with x \<in> R open R obtain y where "y \<in> R - S"

   920         unfolding interior_def by fast

   921       from open R closed S have "open (R - S)" by (rule open_Diff)

   922       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

   923       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

   924       show "False" unfolding interior_def by fast

   925     qed

   926   qed

   927 qed

   928

   929 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

   930 proof (rule interior_unique)

   931   show "interior A \<times> interior B \<subseteq> A \<times> B"

   932     by (intro Sigma_mono interior_subset)

   933   show "open (interior A \<times> interior B)"

   934     by (intro open_Times open_interior)

   935   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

   936   proof (safe)

   937     fix x y assume "(x, y) \<in> T"

   938     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

   939       using open T unfolding open_prod_def by fast

   940     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

   941       using T \<subseteq> A \<times> B by auto

   942     thus "x \<in> interior A" and "y \<in> interior B"

   943       by (auto intro: interiorI)

   944   qed

   945 qed

   946

   947

   948 subsection {* Closure of a Set *}

   949

   950 definition "closure S = S \<union> {x | x. x islimpt S}"

   951

   952 lemma interior_closure: "interior S = - (closure (- S))"

   953   unfolding interior_def closure_def islimpt_def by auto

   954

   955 lemma closure_interior: "closure S = - interior (- S)"

   956   unfolding interior_closure by simp

   957

   958 lemma closed_closure[simp, intro]: "closed (closure S)"

   959   unfolding closure_interior by (simp add: closed_Compl)

   960

   961 lemma closure_subset: "S \<subseteq> closure S"

   962   unfolding closure_def by simp

   963

   964 lemma closure_hull: "closure S = closed hull S"

   965   unfolding hull_def closure_interior interior_def by auto

   966

   967 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

   968   unfolding closure_hull using closed_Inter by (rule hull_eq)

   969

   970 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

   971   unfolding closure_eq .

   972

   973 lemma closure_closure [simp]: "closure (closure S) = closure S"

   974   unfolding closure_hull by (rule hull_hull)

   975

   976 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

   977   unfolding closure_hull by (rule hull_mono)

   978

   979 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

   980   unfolding closure_hull by (rule hull_minimal)

   981

   982 lemma closure_unique:

   983   assumes "S \<subseteq> T" and "closed T"

   984   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

   985   shows "closure S = T"

   986   using assms unfolding closure_hull by (rule hull_unique)

   987

   988 lemma closure_empty [simp]: "closure {} = {}"

   989   using closed_empty by (rule closure_closed)

   990

   991 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

   992   using closed_UNIV by (rule closure_closed)

   993

   994 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

   995   unfolding closure_interior by simp

   996

   997 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

   998   using closure_empty closure_subset[of S]

   999   by blast

  1000

  1001 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1002   using closure_eq[of S] closure_subset[of S]

  1003   by simp

  1004

  1005 lemma open_inter_closure_eq_empty:

  1006   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1007   using open_subset_interior[of S "- T"]

  1008   using interior_subset[of "- T"]

  1009   unfolding closure_interior

  1010   by auto

  1011

  1012 lemma open_inter_closure_subset:

  1013   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1014 proof

  1015   fix x

  1016   assume as: "open S" "x \<in> S \<inter> closure T"

  1017   { assume *:"x islimpt T"

  1018     have "x islimpt (S \<inter> T)"

  1019     proof (rule islimptI)

  1020       fix A

  1021       assume "x \<in> A" "open A"

  1022       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1023         by (simp_all add: open_Int)

  1024       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1025         by (rule islimptE)

  1026       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1027         by simp_all

  1028       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1029     qed

  1030   }

  1031   then show "x \<in> closure (S \<inter> T)" using as

  1032     unfolding closure_def

  1033     by blast

  1034 qed

  1035

  1036 lemma closure_complement: "closure (- S) = - interior S"

  1037   unfolding closure_interior by simp

  1038

  1039 lemma interior_complement: "interior (- S) = - closure S"

  1040   unfolding closure_interior by simp

  1041

  1042 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1043 proof (rule closure_unique)

  1044   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1045     by (intro Sigma_mono closure_subset)

  1046   show "closed (closure A \<times> closure B)"

  1047     by (intro closed_Times closed_closure)

  1048   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

  1049     apply (simp add: closed_def open_prod_def, clarify)

  1050     apply (rule ccontr)

  1051     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1052     apply (simp add: closure_interior interior_def)

  1053     apply (drule_tac x=C in spec)

  1054     apply (drule_tac x=D in spec)

  1055     apply auto

  1056     done

  1057 qed

  1058

  1059

  1060 subsection {* Frontier (aka boundary) *}

  1061

  1062 definition "frontier S = closure S - interior S"

  1063

  1064 lemma frontier_closed: "closed(frontier S)"

  1065   by (simp add: frontier_def closed_Diff)

  1066

  1067 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1068   by (auto simp add: frontier_def interior_closure)

  1069

  1070 lemma frontier_straddle:

  1071   fixes a :: "'a::metric_space"

  1072   shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"

  1073   unfolding frontier_def closure_interior

  1074   by (auto simp add: mem_interior subset_eq ball_def)

  1075

  1076 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1077   by (metis frontier_def closure_closed Diff_subset)

  1078

  1079 lemma frontier_empty[simp]: "frontier {} = {}"

  1080   by (simp add: frontier_def)

  1081

  1082 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1083 proof-

  1084   { assume "frontier S \<subseteq> S"

  1085     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

  1086     hence "closed S" using closure_subset_eq by auto

  1087   }

  1088   thus ?thesis using frontier_subset_closed[of S] ..

  1089 qed

  1090

  1091 lemma frontier_complement: "frontier(- S) = frontier S"

  1092   by (auto simp add: frontier_def closure_complement interior_complement)

  1093

  1094 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1095   using frontier_complement frontier_subset_eq[of "- S"]

  1096   unfolding open_closed by auto

  1097

  1098 subsection {* Filters and the eventually true'' quantifier *}

  1099

  1100 definition

  1101   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1102     (infixr "indirection" 70) where

  1103   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1104

  1105 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1106

  1107 lemma trivial_limit_within:

  1108   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1109 proof

  1110   assume "trivial_limit (at a within S)"

  1111   thus "\<not> a islimpt S"

  1112     unfolding trivial_limit_def

  1113     unfolding eventually_within eventually_at_topological

  1114     unfolding islimpt_def

  1115     apply (clarsimp simp add: set_eq_iff)

  1116     apply (rename_tac T, rule_tac x=T in exI)

  1117     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1118     done

  1119 next

  1120   assume "\<not> a islimpt S"

  1121   thus "trivial_limit (at a within S)"

  1122     unfolding trivial_limit_def

  1123     unfolding eventually_within eventually_at_topological

  1124     unfolding islimpt_def

  1125     apply clarsimp

  1126     apply (rule_tac x=T in exI)

  1127     apply auto

  1128     done

  1129 qed

  1130

  1131 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1132   using trivial_limit_within [of a UNIV] by simp

  1133

  1134 lemma trivial_limit_at:

  1135   fixes a :: "'a::perfect_space"

  1136   shows "\<not> trivial_limit (at a)"

  1137   by (rule at_neq_bot)

  1138

  1139 lemma trivial_limit_at_infinity:

  1140   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1141   unfolding trivial_limit_def eventually_at_infinity

  1142   apply clarsimp

  1143   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1144    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1145   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1146   apply (drule_tac x=UNIV in spec, simp)

  1147   done

  1148

  1149 text {* Some property holds "sufficiently close" to the limit point. *}

  1150

  1151 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)

  1152   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1153 unfolding eventually_at dist_nz by auto

  1154

  1155 lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *)

  1156   "eventually P (at a within S) \<longleftrightarrow>

  1157         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1158   by (rule eventually_within_less)

  1159

  1160 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

  1161   unfolding trivial_limit_def

  1162   by (auto elim: eventually_rev_mp)

  1163

  1164 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1165   by simp

  1166

  1167 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1168   by (simp add: filter_eq_iff)

  1169

  1170 text{* Combining theorems for "eventually" *}

  1171

  1172 lemma eventually_rev_mono:

  1173   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1174 using eventually_mono [of P Q] by fast

  1175

  1176 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

  1177   by (simp add: eventually_False)

  1178

  1179

  1180 subsection {* Limits *}

  1181

  1182 text{* Notation Lim to avoid collition with lim defined in analysis *}

  1183

  1184 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"

  1185   where "Lim A f = (THE l. (f ---> l) A)"

  1186

  1187 lemma Lim:

  1188  "(f ---> l) net \<longleftrightarrow>

  1189         trivial_limit net \<or>

  1190         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1191   unfolding tendsto_iff trivial_limit_eq by auto

  1192

  1193 text{* Show that they yield usual definitions in the various cases. *}

  1194

  1195 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1196            (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"

  1197   by (auto simp add: tendsto_iff eventually_within_le)

  1198

  1199 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1200         (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1201   by (auto simp add: tendsto_iff eventually_within)

  1202

  1203 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1204         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1205   by (auto simp add: tendsto_iff eventually_at)

  1206

  1207 lemma Lim_at_infinity:

  1208   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1209   by (auto simp add: tendsto_iff eventually_at_infinity)

  1210

  1211 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1212   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1213

  1214 text{* The expected monotonicity property. *}

  1215

  1216 lemma Lim_within_empty: "(f ---> l) (net within {})"

  1217   unfolding tendsto_def Limits.eventually_within by simp

  1218

  1219 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"

  1220   unfolding tendsto_def Limits.eventually_within

  1221   by (auto elim!: eventually_elim1)

  1222

  1223 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"

  1224   shows "(f ---> l) (net within (S \<union> T))"

  1225   using assms unfolding tendsto_def Limits.eventually_within

  1226   apply clarify

  1227   apply (drule spec, drule (1) mp, drule (1) mp)

  1228   apply (drule spec, drule (1) mp, drule (1) mp)

  1229   apply (auto elim: eventually_elim2)

  1230   done

  1231

  1232 lemma Lim_Un_univ:

  1233  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV

  1234         ==> (f ---> l) net"

  1235   by (metis Lim_Un within_UNIV)

  1236

  1237 text{* Interrelations between restricted and unrestricted limits. *}

  1238

  1239 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"

  1240   (* FIXME: rename *)

  1241   unfolding tendsto_def Limits.eventually_within

  1242   apply (clarify, drule spec, drule (1) mp, drule (1) mp)

  1243   by (auto elim!: eventually_elim1)

  1244

  1245 lemma eventually_within_interior:

  1246   assumes "x \<in> interior S"

  1247   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1248 proof-

  1249   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1250   { assume "?lhs"

  1251     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1252       unfolding Limits.eventually_within Limits.eventually_at_topological

  1253       by auto

  1254     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1255       by auto

  1256     then have "?rhs"

  1257       unfolding Limits.eventually_at_topological by auto

  1258   } moreover

  1259   { assume "?rhs" hence "?lhs"

  1260       unfolding Limits.eventually_within

  1261       by (auto elim: eventually_elim1)

  1262   } ultimately

  1263   show "?thesis" ..

  1264 qed

  1265

  1266 lemma at_within_interior:

  1267   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1268   by (simp add: filter_eq_iff eventually_within_interior)

  1269

  1270 lemma at_within_open:

  1271   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"

  1272   by (simp only: at_within_interior interior_open)

  1273

  1274 lemma Lim_within_open:

  1275   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

  1276   assumes"a \<in> S" "open S"

  1277   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

  1278   using assms by (simp only: at_within_open)

  1279

  1280 lemma Lim_within_LIMSEQ:

  1281   fixes a :: "'a::metric_space"

  1282   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1283   shows "(X ---> L) (at a within T)"

  1284   using assms unfolding tendsto_def [where l=L]

  1285   by (simp add: sequentially_imp_eventually_within)

  1286

  1287 lemma Lim_right_bound:

  1288   fixes f :: "real \<Rightarrow> real"

  1289   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"

  1290   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1291   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1292 proof cases

  1293   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1294 next

  1295   assume [simp]: "{x<..} \<inter> I \<noteq> {}"

  1296   show ?thesis

  1297   proof (rule Lim_within_LIMSEQ, safe)

  1298     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"

  1299

  1300     show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"

  1301     proof (rule LIMSEQ_I, rule ccontr)

  1302       fix r :: real assume "0 < r"

  1303       with Inf_close[of "f  ({x<..} \<inter> I)" r]

  1304       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto

  1305       from x < y have "0 < y - x" by auto

  1306       from S(2)[THEN LIMSEQ_D, OF this]

  1307       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto

  1308

  1309       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"

  1310       moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1311         using S bnd by (intro Inf_lower[where z=K]) auto

  1312       ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1313         by (auto simp: not_less field_simps)

  1314       with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y

  1315       show False by auto

  1316     qed

  1317   qed

  1318 qed

  1319

  1320 text{* Another limit point characterization. *}

  1321

  1322 lemma islimpt_sequential:

  1323   fixes x :: "'a::metric_space"

  1324   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"

  1325     (is "?lhs = ?rhs")

  1326 proof

  1327   assume ?lhs

  1328   then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"

  1329     unfolding islimpt_approachable

  1330     using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto

  1331   let ?I = "\<lambda>n. inverse (real (Suc n))"

  1332   have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp

  1333   moreover have "(\<lambda>n. f (?I n)) ----> x"

  1334   proof (rule metric_tendsto_imp_tendsto)

  1335     show "?I ----> 0"

  1336       by (rule LIMSEQ_inverse_real_of_nat)

  1337     show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"

  1338       by (simp add: norm_conv_dist [symmetric] less_imp_le f)

  1339   qed

  1340   ultimately show ?rhs by fast

  1341 next

  1342   assume ?rhs

  1343   then obtain f::"nat\<Rightarrow>'a"  where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding LIMSEQ_def by auto

  1344   { fix e::real assume "e>0"

  1345     then obtain N where "dist (f N) x < e" using f(2) by auto

  1346     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto

  1347     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto

  1348   }

  1349   thus ?lhs unfolding islimpt_approachable by auto

  1350 qed

  1351

  1352 lemma Lim_inv: (* TODO: delete *)

  1353   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1354   assumes "(f ---> l) A" and "l \<noteq> 0"

  1355   shows "((inverse o f) ---> inverse l) A"

  1356   unfolding o_def using assms by (rule tendsto_inverse)

  1357

  1358 lemma Lim_null:

  1359   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1360   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1361   by (simp add: Lim dist_norm)

  1362

  1363 lemma Lim_null_comparison:

  1364   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1365   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1366   shows "(f ---> 0) net"

  1367 proof (rule metric_tendsto_imp_tendsto)

  1368   show "(g ---> 0) net" by fact

  1369   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1370     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1371 qed

  1372

  1373 lemma Lim_transform_bound:

  1374   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1375   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1376   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1377   shows "(f ---> 0) net"

  1378   using assms(1) tendsto_norm_zero [OF assms(2)]

  1379   by (rule Lim_null_comparison)

  1380

  1381 text{* Deducing things about the limit from the elements. *}

  1382

  1383 lemma Lim_in_closed_set:

  1384   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1385   shows "l \<in> S"

  1386 proof (rule ccontr)

  1387   assume "l \<notin> S"

  1388   with closed S have "open (- S)" "l \<in> - S"

  1389     by (simp_all add: open_Compl)

  1390   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1391     by (rule topological_tendstoD)

  1392   with assms(2) have "eventually (\<lambda>x. False) net"

  1393     by (rule eventually_elim2) simp

  1394   with assms(3) show "False"

  1395     by (simp add: eventually_False)

  1396 qed

  1397

  1398 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1399

  1400 lemma Lim_dist_ubound:

  1401   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1402   shows "dist a l <= e"

  1403 proof-

  1404   have "dist a l \<in> {..e}"

  1405   proof (rule Lim_in_closed_set)

  1406     show "closed {..e}" by simp

  1407     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1408     show "\<not> trivial_limit net" by fact

  1409     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1410   qed

  1411   thus ?thesis by simp

  1412 qed

  1413

  1414 lemma Lim_norm_ubound:

  1415   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1416   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1417   shows "norm(l) <= e"

  1418 proof-

  1419   have "norm l \<in> {..e}"

  1420   proof (rule Lim_in_closed_set)

  1421     show "closed {..e}" by simp

  1422     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1423     show "\<not> trivial_limit net" by fact

  1424     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1425   qed

  1426   thus ?thesis by simp

  1427 qed

  1428

  1429 lemma Lim_norm_lbound:

  1430   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1431   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1432   shows "e \<le> norm l"

  1433 proof-

  1434   have "norm l \<in> {e..}"

  1435   proof (rule Lim_in_closed_set)

  1436     show "closed {e..}" by simp

  1437     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1438     show "\<not> trivial_limit net" by fact

  1439     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1440   qed

  1441   thus ?thesis by simp

  1442 qed

  1443

  1444 text{* Uniqueness of the limit, when nontrivial. *}

  1445

  1446 lemma tendsto_Lim:

  1447   fixes f :: "'a \<Rightarrow> 'b::t2_space"

  1448   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"

  1449   unfolding Lim_def using tendsto_unique[of net f] by auto

  1450

  1451 text{* Limit under bilinear function *}

  1452

  1453 lemma Lim_bilinear:

  1454   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1455   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1456 using bounded_bilinear h (f ---> l) net (g ---> m) net

  1457 by (rule bounded_bilinear.tendsto)

  1458

  1459 text{* These are special for limits out of the same vector space. *}

  1460

  1461 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1462   unfolding id_def by (rule tendsto_ident_at_within)

  1463

  1464 lemma Lim_at_id: "(id ---> a) (at a)"

  1465   unfolding id_def by (rule tendsto_ident_at)

  1466

  1467 lemma Lim_at_zero:

  1468   fixes a :: "'a::real_normed_vector"

  1469   fixes l :: "'b::topological_space"

  1470   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1471   using LIM_offset_zero LIM_offset_zero_cancel ..

  1472

  1473 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1474

  1475 definition

  1476   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where

  1477   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"

  1478

  1479 lemma netlimit_within:

  1480   assumes "\<not> trivial_limit (at a within S)"

  1481   shows "netlimit (at a within S) = a"

  1482 unfolding netlimit_def

  1483 apply (rule some_equality)

  1484 apply (rule Lim_at_within)

  1485 apply (rule tendsto_ident_at)

  1486 apply (erule tendsto_unique [OF assms])

  1487 apply (rule Lim_at_within)

  1488 apply (rule tendsto_ident_at)

  1489 done

  1490

  1491 lemma netlimit_at:

  1492   fixes a :: "'a::{perfect_space,t2_space}"

  1493   shows "netlimit (at a) = a"

  1494   using netlimit_within [of a UNIV] by simp

  1495

  1496 lemma lim_within_interior:

  1497   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1498   by (simp add: at_within_interior)

  1499

  1500 lemma netlimit_within_interior:

  1501   fixes x :: "'a::{t2_space,perfect_space}"

  1502   assumes "x \<in> interior S"

  1503   shows "netlimit (at x within S) = x"

  1504 using assms by (simp add: at_within_interior netlimit_at)

  1505

  1506 text{* Transformation of limit. *}

  1507

  1508 lemma Lim_transform:

  1509   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1510   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1511   shows "(g ---> l) net"

  1512   using tendsto_diff [OF assms(2) assms(1)] by simp

  1513

  1514 lemma Lim_transform_eventually:

  1515   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1516   apply (rule topological_tendstoI)

  1517   apply (drule (2) topological_tendstoD)

  1518   apply (erule (1) eventually_elim2, simp)

  1519   done

  1520

  1521 lemma Lim_transform_within:

  1522   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1523   and "(f ---> l) (at x within S)"

  1524   shows "(g ---> l) (at x within S)"

  1525 proof (rule Lim_transform_eventually)

  1526   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1527     unfolding eventually_within

  1528     using assms(1,2) by auto

  1529   show "(f ---> l) (at x within S)" by fact

  1530 qed

  1531

  1532 lemma Lim_transform_at:

  1533   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1534   and "(f ---> l) (at x)"

  1535   shows "(g ---> l) (at x)"

  1536 proof (rule Lim_transform_eventually)

  1537   show "eventually (\<lambda>x. f x = g x) (at x)"

  1538     unfolding eventually_at

  1539     using assms(1,2) by auto

  1540   show "(f ---> l) (at x)" by fact

  1541 qed

  1542

  1543 text{* Common case assuming being away from some crucial point like 0. *}

  1544

  1545 lemma Lim_transform_away_within:

  1546   fixes a b :: "'a::t1_space"

  1547   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1548   and "(f ---> l) (at a within S)"

  1549   shows "(g ---> l) (at a within S)"

  1550 proof (rule Lim_transform_eventually)

  1551   show "(f ---> l) (at a within S)" by fact

  1552   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1553     unfolding Limits.eventually_within eventually_at_topological

  1554     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1555 qed

  1556

  1557 lemma Lim_transform_away_at:

  1558   fixes a b :: "'a::t1_space"

  1559   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1560   and fl: "(f ---> l) (at a)"

  1561   shows "(g ---> l) (at a)"

  1562   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl

  1563   by simp

  1564

  1565 text{* Alternatively, within an open set. *}

  1566

  1567 lemma Lim_transform_within_open:

  1568   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1569   and "(f ---> l) (at a)"

  1570   shows "(g ---> l) (at a)"

  1571 proof (rule Lim_transform_eventually)

  1572   show "eventually (\<lambda>x. f x = g x) (at a)"

  1573     unfolding eventually_at_topological

  1574     using assms(1,2,3) by auto

  1575   show "(f ---> l) (at a)" by fact

  1576 qed

  1577

  1578 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1579

  1580 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1581

  1582 lemma Lim_cong_within(*[cong add]*):

  1583   assumes "a = b" "x = y" "S = T"

  1584   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1585   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1586   unfolding tendsto_def Limits.eventually_within eventually_at_topological

  1587   using assms by simp

  1588

  1589 lemma Lim_cong_at(*[cong add]*):

  1590   assumes "a = b" "x = y"

  1591   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1592   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1593   unfolding tendsto_def eventually_at_topological

  1594   using assms by simp

  1595

  1596 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1597

  1598 lemma closure_sequential:

  1599   fixes l :: "'a::metric_space"

  1600   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1601 proof

  1602   assume "?lhs" moreover

  1603   { assume "l \<in> S"

  1604     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1605   } moreover

  1606   { assume "l islimpt S"

  1607     hence "?rhs" unfolding islimpt_sequential by auto

  1608   } ultimately

  1609   show "?rhs" unfolding closure_def by auto

  1610 next

  1611   assume "?rhs"

  1612   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

  1613 qed

  1614

  1615 lemma closed_sequential_limits:

  1616   fixes S :: "'a::metric_space set"

  1617   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1618   unfolding closed_limpt

  1619   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1620   by metis

  1621

  1622 lemma closure_approachable:

  1623   fixes S :: "'a::metric_space set"

  1624   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1625   apply (auto simp add: closure_def islimpt_approachable)

  1626   by (metis dist_self)

  1627

  1628 lemma closed_approachable:

  1629   fixes S :: "'a::metric_space set"

  1630   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1631   by (metis closure_closed closure_approachable)

  1632

  1633 subsection {* Infimum Distance *}

  1634

  1635 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1636

  1637 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1638   by (simp add: infdist_def)

  1639

  1640 lemma infdist_nonneg:

  1641   shows "0 \<le> infdist x A"

  1642   using assms by (auto simp add: infdist_def)

  1643

  1644 lemma infdist_le:

  1645   assumes "a \<in> A"

  1646   assumes "d = dist x a"

  1647   shows "infdist x A \<le> d"

  1648   using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)

  1649

  1650 lemma infdist_zero[simp]:

  1651   assumes "a \<in> A" shows "infdist a A = 0"

  1652 proof -

  1653   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  1654   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  1655 qed

  1656

  1657 lemma infdist_triangle:

  1658   shows "infdist x A \<le> infdist y A + dist x y"

  1659 proof cases

  1660   assume "A = {}" thus ?thesis by (simp add: infdist_def)

  1661 next

  1662   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto

  1663   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1664   proof

  1665     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp

  1666     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1667     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto

  1668     show "infdist x A \<le> d"

  1669       unfolding infdist_notempty[OF A \<noteq> {}]

  1670     proof (rule Inf_lower2)

  1671       show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto

  1672       show "dist x a \<le> d" unfolding d by (rule dist_triangle)

  1673       fix d assume "d \<in> {dist x a |a. a \<in> A}"

  1674       then obtain a where "a \<in> A" "d = dist x a" by auto

  1675       thus "infdist x A \<le> d" by (rule infdist_le)

  1676     qed

  1677   qed

  1678   also have "\<dots> = dist x y + infdist y A"

  1679   proof (rule Inf_eq, safe)

  1680     fix a assume "a \<in> A"

  1681     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)

  1682   next

  1683     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1684     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  1685       by (intro Inf_greatest) (auto simp: field_simps)

  1686     thus "i \<le> dist x y + infdist y A" by simp

  1687   qed

  1688   finally show ?thesis by simp

  1689 qed

  1690

  1691 lemma

  1692   in_closure_iff_infdist_zero:

  1693   assumes "A \<noteq> {}"

  1694   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1695 proof

  1696   assume "x \<in> closure A"

  1697   show "infdist x A = 0"

  1698   proof (rule ccontr)

  1699     assume "infdist x A \<noteq> 0"

  1700     with infdist_nonneg[of x A] have "infdist x A > 0" by auto

  1701     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto

  1702       by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  1703         eucl_less_not_refl euclidean_trans(2) infdist_le)

  1704     hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  1705     thus False using x \<in> closure A by simp

  1706   qed

  1707 next

  1708   assume x: "infdist x A = 0"

  1709   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)

  1710   show "x \<in> closure A" unfolding closure_approachable

  1711   proof (safe, rule ccontr)

  1712     fix e::real assume "0 < e"

  1713     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1714     hence "infdist x A \<ge> e" using a \<in> A

  1715       unfolding infdist_def

  1716       by (force simp: dist_commute)

  1717     with x 0 < e show False by auto

  1718   qed

  1719 qed

  1720

  1721 lemma

  1722   in_closed_iff_infdist_zero:

  1723   assumes "closed A" "A \<noteq> {}"

  1724   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1725 proof -

  1726   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1727     by (rule in_closure_iff_infdist_zero) fact

  1728   with assms show ?thesis by simp

  1729 qed

  1730

  1731 lemma tendsto_infdist [tendsto_intros]:

  1732   assumes f: "(f ---> l) F"

  1733   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  1734 proof (rule tendstoI)

  1735   fix e ::real assume "0 < e"

  1736   from tendstoD[OF f this]

  1737   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1738   proof (eventually_elim)

  1739     fix x

  1740     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1741     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1742       by (simp add: dist_commute dist_real_def)

  1743     also assume "dist (f x) l < e"

  1744     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1745   qed

  1746 qed

  1747

  1748 text{* Some other lemmas about sequences. *}

  1749

  1750 lemma sequentially_offset:

  1751   assumes "eventually (\<lambda>i. P i) sequentially"

  1752   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1753   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1754

  1755 lemma seq_offset:

  1756   assumes "(f ---> l) sequentially"

  1757   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1758   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1759

  1760 lemma seq_offset_neg:

  1761   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1762   apply (rule topological_tendstoI)

  1763   apply (drule (2) topological_tendstoD)

  1764   apply (simp only: eventually_sequentially)

  1765   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1766   apply metis

  1767   by arith

  1768

  1769 lemma seq_offset_rev:

  1770   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1771   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1772

  1773 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1774   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1775

  1776 subsection {* More properties of closed balls *}

  1777

  1778 lemma closed_cball: "closed (cball x e)"

  1779 unfolding cball_def closed_def

  1780 unfolding Collect_neg_eq [symmetric] not_le

  1781 apply (clarsimp simp add: open_dist, rename_tac y)

  1782 apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1783 apply (rename_tac x')

  1784 apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1785 apply simp

  1786 done

  1787

  1788 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1789 proof-

  1790   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1791     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1792   } moreover

  1793   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1794     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto

  1795   } ultimately

  1796   show ?thesis unfolding open_contains_ball by auto

  1797 qed

  1798

  1799 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1800   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1801

  1802 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1803   apply (simp add: interior_def, safe)

  1804   apply (force simp add: open_contains_cball)

  1805   apply (rule_tac x="ball x e" in exI)

  1806   apply (simp add: subset_trans [OF ball_subset_cball])

  1807   done

  1808

  1809 lemma islimpt_ball:

  1810   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1811   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1812 proof

  1813   assume "?lhs"

  1814   { assume "e \<le> 0"

  1815     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1816     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1817   }

  1818   hence "e > 0" by (metis not_less)

  1819   moreover

  1820   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] ?lhs unfolding closed_limpt by auto

  1821   ultimately show "?rhs" by auto

  1822 next

  1823   assume "?rhs" hence "e>0"  by auto

  1824   { fix d::real assume "d>0"

  1825     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1826     proof(cases "d \<le> dist x y")

  1827       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1828       proof(cases "x=y")

  1829         case True hence False using d \<le> dist x y d>0 by auto

  1830         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1831       next

  1832         case False

  1833

  1834         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1835               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1836           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1837         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1838           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1839           unfolding scaleR_minus_left scaleR_one

  1840           by (auto simp add: norm_minus_commute)

  1841         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1842           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1843           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1844         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1845         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1846

  1847         moreover

  1848

  1849         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  1850           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  1851         moreover

  1852         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  1853           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  1854           unfolding dist_norm by auto

  1855         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto

  1856       qed

  1857     next

  1858       case False hence "d > dist x y" by auto

  1859       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1860       proof(cases "x=y")

  1861         case True

  1862         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  1863           using perfect_choose_dist[of "min e d" y]

  1864           using d > 0 e>0 by auto

  1865         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1866           unfolding x = y

  1867           using z \<noteq> y **

  1868           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  1869       next

  1870         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1871           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  1872       qed

  1873     qed  }

  1874   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  1875 qed

  1876

  1877 lemma closure_ball_lemma:

  1878   fixes x y :: "'a::real_normed_vector"

  1879   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  1880 proof (rule islimptI)

  1881   fix T assume "y \<in> T" "open T"

  1882   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  1883     unfolding open_dist by fast

  1884   (* choose point between x and y, within distance r of y. *)

  1885   def k \<equiv> "min 1 (r / (2 * dist x y))"

  1886   def z \<equiv> "y + scaleR k (x - y)"

  1887   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  1888     unfolding z_def by (simp add: algebra_simps)

  1889   have "dist z y < r"

  1890     unfolding z_def k_def using 0 < r

  1891     by (simp add: dist_norm min_def)

  1892   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  1893   have "dist x z < dist x y"

  1894     unfolding z_def2 dist_norm

  1895     apply (simp add: norm_minus_commute)

  1896     apply (simp only: dist_norm [symmetric])

  1897     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  1898     apply (rule mult_strict_right_mono)

  1899     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  1900     apply (simp add: zero_less_dist_iff x \<noteq> y)

  1901     done

  1902   hence "z \<in> ball x (dist x y)" by simp

  1903   have "z \<noteq> y"

  1904     unfolding z_def k_def using x \<noteq> y 0 < r

  1905     by (simp add: min_def)

  1906   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  1907     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  1908     by fast

  1909 qed

  1910

  1911 lemma closure_ball:

  1912   fixes x :: "'a::real_normed_vector"

  1913   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  1914 apply (rule equalityI)

  1915 apply (rule closure_minimal)

  1916 apply (rule ball_subset_cball)

  1917 apply (rule closed_cball)

  1918 apply (rule subsetI, rename_tac y)

  1919 apply (simp add: le_less [where 'a=real])

  1920 apply (erule disjE)

  1921 apply (rule subsetD [OF closure_subset], simp)

  1922 apply (simp add: closure_def)

  1923 apply clarify

  1924 apply (rule closure_ball_lemma)

  1925 apply (simp add: zero_less_dist_iff)

  1926 done

  1927

  1928 (* In a trivial vector space, this fails for e = 0. *)

  1929 lemma interior_cball:

  1930   fixes x :: "'a::{real_normed_vector, perfect_space}"

  1931   shows "interior (cball x e) = ball x e"

  1932 proof(cases "e\<ge>0")

  1933   case False note cs = this

  1934   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  1935   { fix y assume "y \<in> cball x e"

  1936     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  1937   hence "cball x e = {}" by auto

  1938   hence "interior (cball x e) = {}" using interior_empty by auto

  1939   ultimately show ?thesis by blast

  1940 next

  1941   case True note cs = this

  1942   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  1943   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  1944     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  1945

  1946     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  1947       using perfect_choose_dist [of d] by auto

  1948     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  1949     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  1950

  1951     hence "y \<in> ball x e" proof(cases "x = y")

  1952       case True

  1953       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

  1954       thus "y \<in> ball x e" using x = y  by simp

  1955     next

  1956       case False

  1957       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  1958         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  1959       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

  1960       have "y - x \<noteq> 0" using x \<noteq> y by auto

  1961       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

  1962         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  1963

  1964       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  1965         by (auto simp add: dist_norm algebra_simps)

  1966       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1967         by (auto simp add: algebra_simps)

  1968       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  1969         using ** by auto

  1970       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

  1971       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  1972       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  1973     qed  }

  1974   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  1975   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  1976 qed

  1977

  1978 lemma frontier_ball:

  1979   fixes a :: "'a::real_normed_vector"

  1980   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  1981   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  1982   apply (simp add: set_eq_iff)

  1983   by arith

  1984

  1985 lemma frontier_cball:

  1986   fixes a :: "'a::{real_normed_vector, perfect_space}"

  1987   shows "frontier(cball a e) = {x. dist a x = e}"

  1988   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  1989   apply (simp add: set_eq_iff)

  1990   by arith

  1991

  1992 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  1993   apply (simp add: set_eq_iff not_le)

  1994   by (metis zero_le_dist dist_self order_less_le_trans)

  1995 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

  1996

  1997 lemma cball_eq_sing:

  1998   fixes x :: "'a::{metric_space,perfect_space}"

  1999   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2000 proof (rule linorder_cases)

  2001   assume e: "0 < e"

  2002   obtain a where "a \<noteq> x" "dist a x < e"

  2003     using perfect_choose_dist [OF e] by auto

  2004   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  2005   with e show ?thesis by (auto simp add: set_eq_iff)

  2006 qed auto

  2007

  2008 lemma cball_sing:

  2009   fixes x :: "'a::metric_space"

  2010   shows "e = 0 ==> cball x e = {x}"

  2011   by (auto simp add: set_eq_iff)

  2012

  2013

  2014 subsection {* Boundedness *}

  2015

  2016   (* FIXME: This has to be unified with BSEQ!! *)

  2017 definition (in metric_space)

  2018   bounded :: "'a set \<Rightarrow> bool" where

  2019   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2020

  2021 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2022 unfolding bounded_def

  2023 apply safe

  2024 apply (rule_tac x="dist a x + e" in exI, clarify)

  2025 apply (drule (1) bspec)

  2026 apply (erule order_trans [OF dist_triangle add_left_mono])

  2027 apply auto

  2028 done

  2029

  2030 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2031 unfolding bounded_any_center [where a=0]

  2032 by (simp add: dist_norm)

  2033

  2034 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

  2035   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

  2036   using assms by auto

  2037

  2038 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)

  2039 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2040   by (metis bounded_def subset_eq)

  2041

  2042 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2043   by (metis bounded_subset interior_subset)

  2044

  2045 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

  2046 proof-

  2047   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

  2048   { fix y assume "y \<in> closure S"

  2049     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2050       unfolding closure_sequential by auto

  2051     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2052     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2053       by (rule eventually_mono, simp add: f(1))

  2054     have "dist x y \<le> a"

  2055       apply (rule Lim_dist_ubound [of sequentially f])

  2056       apply (rule trivial_limit_sequentially)

  2057       apply (rule f(2))

  2058       apply fact

  2059       done

  2060   }

  2061   thus ?thesis unfolding bounded_def by auto

  2062 qed

  2063

  2064 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2065   apply (simp add: bounded_def)

  2066   apply (rule_tac x=x in exI)

  2067   apply (rule_tac x=e in exI)

  2068   apply auto

  2069   done

  2070

  2071 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2072   by (metis ball_subset_cball bounded_cball bounded_subset)

  2073

  2074 lemma finite_imp_bounded[intro]:

  2075   fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"

  2076 proof-

  2077   { fix a and F :: "'a set" assume as:"bounded F"

  2078     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto

  2079     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto

  2080     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)

  2081   }

  2082   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto

  2083 qed

  2084

  2085 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2086   apply (auto simp add: bounded_def)

  2087   apply (rename_tac x y r s)

  2088   apply (rule_tac x=x in exI)

  2089   apply (rule_tac x="max r (dist x y + s)" in exI)

  2090   apply (rule ballI, rename_tac z, safe)

  2091   apply (drule (1) bspec, simp)

  2092   apply (drule (1) bspec)

  2093   apply (rule min_max.le_supI2)

  2094   apply (erule order_trans [OF dist_triangle add_left_mono])

  2095   done

  2096

  2097 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2098   by (induct rule: finite_induct[of F], auto)

  2099

  2100 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2101   apply (simp add: bounded_iff)

  2102   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2103   by metis arith

  2104

  2105 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2106   by (metis Int_lower1 Int_lower2 bounded_subset)

  2107

  2108 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2109 apply (metis Diff_subset bounded_subset)

  2110 done

  2111

  2112 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"

  2113   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)

  2114

  2115 lemma not_bounded_UNIV[simp, intro]:

  2116   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2117 proof(auto simp add: bounded_pos not_le)

  2118   obtain x :: 'a where "x \<noteq> 0"

  2119     using perfect_choose_dist [OF zero_less_one] by fast

  2120   fix b::real  assume b: "b >0"

  2121   have b1: "b +1 \<ge> 0" using b by simp

  2122   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2123     by (simp add: norm_sgn)

  2124   then show "\<exists>x::'a. b < norm x" ..

  2125 qed

  2126

  2127 lemma bounded_linear_image:

  2128   assumes "bounded S" "bounded_linear f"

  2129   shows "bounded(f  S)"

  2130 proof-

  2131   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2132   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2133   { fix x assume "x\<in>S"

  2134     hence "norm x \<le> b" using b by auto

  2135     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

  2136       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2137   }

  2138   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

  2139     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

  2140 qed

  2141

  2142 lemma bounded_scaling:

  2143   fixes S :: "'a::real_normed_vector set"

  2144   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2145   apply (rule bounded_linear_image, assumption)

  2146   apply (rule bounded_linear_scaleR_right)

  2147   done

  2148

  2149 lemma bounded_translation:

  2150   fixes S :: "'a::real_normed_vector set"

  2151   assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"

  2152 proof-

  2153   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2154   { fix x assume "x\<in>S"

  2155     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

  2156   }

  2157   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

  2158     by (auto intro!: exI[of _ "b + norm a"])

  2159 qed

  2160

  2161

  2162 text{* Some theorems on sups and infs using the notion "bounded". *}

  2163

  2164 lemma bounded_real:

  2165   fixes S :: "real set"

  2166   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2167   by (simp add: bounded_iff)

  2168

  2169 lemma bounded_has_Sup:

  2170   fixes S :: "real set"

  2171   assumes "bounded S" "S \<noteq> {}"

  2172   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  2173 proof

  2174   fix x assume "x\<in>S"

  2175   thus "x \<le> Sup S"

  2176     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)

  2177 next

  2178   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  2179     by (metis SupInf.Sup_least)

  2180 qed

  2181

  2182 lemma Sup_insert:

  2183   fixes S :: "real set"

  2184   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2185 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)

  2186

  2187 lemma Sup_insert_finite:

  2188   fixes S :: "real set"

  2189   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2190   apply (rule Sup_insert)

  2191   apply (rule finite_imp_bounded)

  2192   by simp

  2193

  2194 lemma bounded_has_Inf:

  2195   fixes S :: "real set"

  2196   assumes "bounded S"  "S \<noteq> {}"

  2197   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  2198 proof

  2199   fix x assume "x\<in>S"

  2200   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

  2201   thus "x \<ge> Inf S" using x\<in>S

  2202     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)

  2203 next

  2204   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  2205     by (metis SupInf.Inf_greatest)

  2206 qed

  2207

  2208 lemma Inf_insert:

  2209   fixes S :: "real set"

  2210   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2211 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)

  2212 lemma Inf_insert_finite:

  2213   fixes S :: "real set"

  2214   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2215   by (rule Inf_insert, rule finite_imp_bounded, simp)

  2216

  2217 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)

  2218 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"

  2219   apply (frule isGlb_isLb)

  2220   apply (frule_tac x = y in isGlb_isLb)

  2221   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)

  2222   done

  2223

  2224

  2225 subsection {* Equivalent versions of compactness *}

  2226

  2227 subsubsection{* Sequential compactness *}

  2228

  2229 definition

  2230   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)

  2231   "compact S \<longleftrightarrow>

  2232    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  2233        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  2234

  2235 lemma compactI:

  2236   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  2237   shows "compact S"

  2238   unfolding compact_def using assms by fast

  2239

  2240 lemma compactE:

  2241   assumes "compact S" "\<forall>n. f n \<in> S"

  2242   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  2243   using assms unfolding compact_def by fast

  2244

  2245 text {*

  2246   A metric space (or topological vector space) is said to have the

  2247   Heine-Borel property if every closed and bounded subset is compact.

  2248 *}

  2249

  2250 class heine_borel = metric_space +

  2251   assumes bounded_imp_convergent_subsequence:

  2252     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s

  2253       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2254

  2255 lemma bounded_closed_imp_compact:

  2256   fixes s::"'a::heine_borel set"

  2257   assumes "bounded s" and "closed s" shows "compact s"

  2258 proof (unfold compact_def, clarify)

  2259   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  2260   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  2261     using bounded_imp_convergent_subsequence [OF bounded s \<forall>n. f n \<in> s] by auto

  2262   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  2263   have "l \<in> s" using closed s fr l

  2264     unfolding closed_sequential_limits by blast

  2265   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2266     using l \<in> s r l by blast

  2267 qed

  2268

  2269 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"

  2270 proof(induct n)

  2271   show "0 \<le> r 0" by auto

  2272 next

  2273   fix n assume "n \<le> r n"

  2274   moreover have "r n < r (Suc n)"

  2275     using assms [unfolded subseq_def] by auto

  2276   ultimately show "Suc n \<le> r (Suc n)" by auto

  2277 qed

  2278

  2279 lemma eventually_subseq:

  2280   assumes r: "subseq r"

  2281   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"

  2282 unfolding eventually_sequentially

  2283 by (metis subseq_bigger [OF r] le_trans)

  2284

  2285 lemma lim_subseq:

  2286   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"

  2287 unfolding tendsto_def eventually_sequentially o_def

  2288 by (metis subseq_bigger le_trans)

  2289

  2290 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"

  2291   unfolding Ex1_def

  2292   apply (rule_tac x="nat_rec e f" in exI)

  2293   apply (rule conjI)+

  2294 apply (rule def_nat_rec_0, simp)

  2295 apply (rule allI, rule def_nat_rec_Suc, simp)

  2296 apply (rule allI, rule impI, rule ext)

  2297 apply (erule conjE)

  2298 apply (induct_tac x)

  2299 apply simp

  2300 apply (erule_tac x="n" in allE)

  2301 apply (simp)

  2302 done

  2303

  2304 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"

  2305   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"

  2306   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"

  2307 proof-

  2308   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto

  2309   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto

  2310   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"

  2311     { fix n::nat

  2312       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto

  2313       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto

  2314       with n have "s N \<le> t - e" using e>0 by auto

  2315       hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using n\<le>N by auto  }

  2316     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto

  2317     hence False using isLub_le_isUb[OF t, of "t - e"] and e>0 by auto  }

  2318   thus ?thesis by blast

  2319 qed

  2320

  2321 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"

  2322   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"

  2323   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"

  2324   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]

  2325   unfolding monoseq_def incseq_def

  2326   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]

  2327   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto

  2328

  2329 (* TODO: merge this lemma with the ones above *)

  2330 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"

  2331   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"

  2332   shows "\<exists>l. (s ---> l) sequentially"

  2333 proof-

  2334   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto

  2335   { fix m::nat

  2336     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"

  2337       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)

  2338       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }

  2339   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto

  2340   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]

  2341     unfolding monoseq_def by auto

  2342   thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="l" in exI)

  2343     unfolding dist_norm  by auto

  2344 qed

  2345

  2346 lemma compact_real_lemma:

  2347   assumes "\<forall>n::nat. abs(s n) \<le> b"

  2348   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"

  2349 proof-

  2350   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"

  2351     using seq_monosub[of s] by auto

  2352   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms

  2353     unfolding tendsto_iff dist_norm eventually_sequentially by auto

  2354 qed

  2355

  2356 instance real :: heine_borel

  2357 proof

  2358   fix s :: "real set" and f :: "nat \<Rightarrow> real"

  2359   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2360   then obtain b where b: "\<forall>n. abs (f n) \<le> b"

  2361     unfolding bounded_iff by auto

  2362   obtain l :: real and r :: "nat \<Rightarrow> nat" where

  2363     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  2364     using compact_real_lemma [OF b] by auto

  2365   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2366     by auto

  2367 qed

  2368

  2369 lemma compact_lemma:

  2370   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  2371   assumes "bounded s" and "\<forall>n. f n \<in> s"

  2372   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  2373         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  2374 proof safe

  2375   fix d :: "'a set" assume d: "d \<subseteq> Basis"

  2376   with finite_Basis have "finite d" by (blast intro: finite_subset)

  2377   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  2378       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  2379   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  2380   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto

  2381     have s': "bounded ((\<lambda>x. x \<bullet> k)  s)" using bounded s

  2382       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  2383     obtain l1::"'a" and r1 where r1:"subseq r1" and

  2384       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  2385       using insert(3) using insert(4) by auto

  2386     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  s" using \<forall>n. f n \<in> s by simp

  2387     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  2388       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

  2389     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  2390       using r1 and r2 unfolding r_def o_def subseq_def by auto

  2391     moreover

  2392     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  2393     { fix e::real assume "e>0"

  2394       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast

  2395       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)

  2396       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  2397         by (rule eventually_subseq)

  2398       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  2399         using N1' N2

  2400         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  2401     }

  2402     ultimately show ?case by auto

  2403   qed

  2404 qed

  2405

  2406 instance euclidean_space \<subseteq> heine_borel

  2407 proof

  2408   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"

  2409   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2410   then obtain l::'a and r where r: "subseq r"

  2411     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  2412     using compact_lemma [OF s f] by blast

  2413   { fix e::real assume "e>0"

  2414     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)

  2415     with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"

  2416       by simp

  2417     moreover

  2418     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  2419       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  2420         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  2421       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  2422         apply(rule setsum_strict_mono) using n by auto

  2423       finally have "dist (f (r n)) l < e"

  2424         by auto

  2425     }

  2426     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  2427       by (rule eventually_elim1)

  2428   }

  2429   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  2430   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  2431 qed

  2432

  2433 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  2434 unfolding bounded_def

  2435 apply clarify

  2436 apply (rule_tac x="a" in exI)

  2437 apply (rule_tac x="e" in exI)

  2438 apply clarsimp

  2439 apply (drule (1) bspec)

  2440 apply (simp add: dist_Pair_Pair)

  2441 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  2442 done

  2443

  2444 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  2445 unfolding bounded_def

  2446 apply clarify

  2447 apply (rule_tac x="b" in exI)

  2448 apply (rule_tac x="e" in exI)

  2449 apply clarsimp

  2450 apply (drule (1) bspec)

  2451 apply (simp add: dist_Pair_Pair)

  2452 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  2453 done

  2454

  2455 instance prod :: (heine_borel, heine_borel) heine_borel

  2456 proof

  2457   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"

  2458   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2459   from s have s1: "bounded (fst  s)" by (rule bounded_fst)

  2460   from f have f1: "\<forall>n. fst (f n) \<in> fst  s" by simp

  2461   obtain l1 r1 where r1: "subseq r1"

  2462     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"

  2463     using bounded_imp_convergent_subsequence [OF s1 f1]

  2464     unfolding o_def by fast

  2465   from s have s2: "bounded (snd  s)" by (rule bounded_snd)

  2466   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd  s" by simp

  2467   obtain l2 r2 where r2: "subseq r2"

  2468     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  2469     using bounded_imp_convergent_subsequence [OF s2 f2]

  2470     unfolding o_def by fast

  2471   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  2472     using lim_subseq [OF r2 l1] unfolding o_def .

  2473   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  2474     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  2475   have r: "subseq (r1 \<circ> r2)"

  2476     using r1 r2 unfolding subseq_def by simp

  2477   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2478     using l r by fast

  2479 qed

  2480

  2481 subsubsection{* Completeness *}

  2482

  2483 lemma cauchy_def:

  2484   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  2485 unfolding Cauchy_def by blast

  2486

  2487 definition

  2488   complete :: "'a::metric_space set \<Rightarrow> bool" where

  2489   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f

  2490                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"

  2491

  2492 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  2493 proof-

  2494   { assume ?rhs

  2495     { fix e::real

  2496       assume "e>0"

  2497       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  2498         by (erule_tac x="e/2" in allE) auto

  2499       { fix n m

  2500         assume nm:"N \<le> m \<and> N \<le> n"

  2501         hence "dist (s m) (s n) < e" using N

  2502           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  2503           by blast

  2504       }

  2505       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  2506         by blast

  2507     }

  2508     hence ?lhs

  2509       unfolding cauchy_def

  2510       by blast

  2511   }

  2512   thus ?thesis

  2513     unfolding cauchy_def

  2514     using dist_triangle_half_l

  2515     by blast

  2516 qed

  2517

  2518 lemma convergent_imp_cauchy:

  2519  "(s ---> l) sequentially ==> Cauchy s"

  2520 proof(simp only: cauchy_def, rule, rule)

  2521   fix e::real assume "e>0" "(s ---> l) sequentially"

  2522   then obtain N::nat where N:"\<forall>n\<ge>N. dist (s n) l < e/2" unfolding LIMSEQ_def by(erule_tac x="e/2" in allE) auto

  2523   thus "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"  using dist_triangle_half_l[of _ l e _] by (rule_tac x=N in exI) auto

  2524 qed

  2525

  2526 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  2527 proof-

  2528   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto

  2529   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  2530   moreover

  2531   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  2532   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  2533     unfolding bounded_any_center [where a="s N"] by auto

  2534   ultimately show "?thesis"

  2535     unfolding bounded_any_center [where a="s N"]

  2536     apply(rule_tac x="max a 1" in exI) apply auto

  2537     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  2538 qed

  2539

  2540 lemma compact_imp_complete: assumes "compact s" shows "complete s"

  2541 proof-

  2542   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  2543     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast

  2544

  2545     note lr' = subseq_bigger [OF lr(2)]

  2546

  2547     { fix e::real assume "e>0"

  2548       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using e>0 apply (erule_tac x="e/2" in allE) by auto

  2549       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using e>0 by auto

  2550       { fix n::nat assume n:"n \<ge> max N M"

  2551         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  2552         moreover have "r n \<ge> N" using lr'[of n] n by auto

  2553         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  2554         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }

  2555       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  2556     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  2557   thus ?thesis unfolding complete_def by auto

  2558 qed

  2559

  2560 instance heine_borel < complete_space

  2561 proof

  2562   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2563   hence "bounded (range f)"

  2564     by (rule cauchy_imp_bounded)

  2565   hence "compact (closure (range f))"

  2566     using bounded_closed_imp_compact [of "closure (range f)"] by auto

  2567   hence "complete (closure (range f))"

  2568     by (rule compact_imp_complete)

  2569   moreover have "\<forall>n. f n \<in> closure (range f)"

  2570     using closure_subset [of "range f"] by auto

  2571   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  2572     using Cauchy f unfolding complete_def by auto

  2573   then show "convergent f"

  2574     unfolding convergent_def by auto

  2575 qed

  2576

  2577 instance euclidean_space \<subseteq> banach ..

  2578

  2579 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  2580 proof(simp add: complete_def, rule, rule)

  2581   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2582   hence "convergent f" by (rule Cauchy_convergent)

  2583   thus "\<exists>l. f ----> l" unfolding convergent_def .

  2584 qed

  2585

  2586 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  2587 proof -

  2588   { fix x assume "x islimpt s"

  2589     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  2590       unfolding islimpt_sequential by auto

  2591     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  2592       using complete s[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto

  2593     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  2594   }

  2595   thus "closed s" unfolding closed_limpt by auto

  2596 qed

  2597

  2598 lemma complete_eq_closed:

  2599   fixes s :: "'a::complete_space set"

  2600   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  2601 proof

  2602   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  2603 next

  2604   assume ?rhs

  2605   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  2606     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  2607     hence "\<exists>l\<in>s. (f ---> l) sequentially" using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }

  2608   thus ?lhs unfolding complete_def by auto

  2609 qed

  2610

  2611 lemma convergent_eq_cauchy:

  2612   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  2613   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  2614   unfolding Cauchy_convergent_iff convergent_def ..

  2615

  2616 lemma convergent_imp_bounded:

  2617   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  2618   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  2619   by (intro cauchy_imp_bounded convergent_imp_cauchy)

  2620

  2621 subsubsection{* Total boundedness *}

  2622

  2623 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  2624   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  2625 declare helper_1.simps[simp del]

  2626

  2627 lemma compact_imp_totally_bounded:

  2628   assumes "compact s"

  2629   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  2630 proof(rule, rule, rule ccontr)

  2631   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k)"

  2632   def x \<equiv> "helper_1 s e"

  2633   { fix n

  2634     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  2635     proof(induct_tac rule:nat_less_induct)

  2636       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  2637       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  2638       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x  {0 ..< n}" in allE) using as by auto

  2639       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  2640       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  2641         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  2642       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  2643     qed }

  2644   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  2645   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto

  2646   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto

  2647   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using e>0 by auto

  2648   show False

  2649     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  2650     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  2651     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  2652 qed

  2653

  2654 subsubsection{* Heine-Borel theorem *}

  2655

  2656 text {* Following Burkill \& Burkill vol. 2. *}

  2657

  2658 lemma heine_borel_lemma: fixes s::"'a::metric_space set"

  2659   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"

  2660   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"

  2661 proof(rule ccontr)

  2662   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"

  2663   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto

  2664   { fix n::nat

  2665     have "1 / real (n + 1) > 0" by auto

  2666     hence "\<exists>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> (ball x (inverse (real (n+1))) \<subseteq> xa))" using cont unfolding Bex_def by auto }

  2667   hence "\<forall>n::nat. \<exists>x. x \<in> s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)" by auto

  2668   then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"

  2669     using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto

  2670

  2671   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"

  2672     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto

  2673

  2674   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto

  2675   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"

  2676     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto

  2677

  2678   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"

  2679     using lr[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto

  2680

  2681   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and e>0 by auto

  2682   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"

  2683     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2

  2684     using subseq_bigger[OF r, of "N1 + N2"] by auto

  2685

  2686   def x \<equiv> "(f (r (N1 + N2)))"

  2687   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def

  2688     using f[THEN spec[where x="r (N1 + N2)"]] using b\<in>t by auto

  2689   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto

  2690   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto

  2691

  2692   have "dist x l < e/2" using N1 unfolding x_def o_def by auto

  2693   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)

  2694

  2695   thus False using e and y\<notin>b by auto

  2696 qed

  2697

  2698 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  2699                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"

  2700 proof clarify

  2701   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"

  2702   then obtain e::real where "e>0" and "\<forall>x\<in>s. \<exists>b\<in>f. ball x e \<subseteq> b" using heine_borel_lemma[of s f] by auto

  2703   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto

  2704   hence "\<exists>bb. \<forall>x\<in>s. bb x \<in>f \<and> ball x e \<subseteq> bb x" using bchoice[of s "\<lambda>x b. b\<in>f \<and> ball x e \<subseteq> b"] by auto

  2705   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast

  2706

  2707   from compact s have  "\<exists> k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" using compact_imp_totally_bounded[of s] e>0 by auto

  2708   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" by auto

  2709

  2710   have "finite (bb  k)" using k(1) by auto

  2711   moreover

  2712   { fix x assume "x\<in>s"

  2713     hence "x\<in>\<Union>(\<lambda>x. ball x e)  k" using k(3)  unfolding subset_eq by auto

  2714     hence "\<exists>X\<in>bb  k. x \<in> X" using bb k(2) by blast

  2715     hence "x \<in> \<Union>(bb  k)" using  Union_iff[of x "bb  k"] by auto

  2716   }

  2717   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f'" using bb k(2) by (rule_tac x="bb  k" in exI) auto

  2718 qed

  2719

  2720 subsubsection {* Bolzano-Weierstrass property *}

  2721

  2722 lemma heine_borel_imp_bolzano_weierstrass:

  2723   assumes "\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f) --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f'))"

  2724           "infinite t"  "t \<subseteq> s"

  2725   shows "\<exists>x \<in> s. x islimpt t"

  2726 proof(rule ccontr)

  2727   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2728   then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def

  2729     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto

  2730   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2731     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2732   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2733   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2734     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and t\<subseteq>s by auto

  2735     hence "x = y" using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto  }

  2736   hence "inj_on f t" unfolding inj_on_def by simp

  2737   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2738   moreover

  2739   { fix x assume "x\<in>t" "f x \<notin> g"

  2740     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2741     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2742     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2743     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2744   hence "f  t \<subseteq> g" by auto

  2745   ultimately show False using g(2) using finite_subset by auto

  2746 qed

  2747

  2748 subsubsection {* Complete the chain of compactness variants *}

  2749

  2750 lemma islimpt_range_imp_convergent_subsequence:

  2751   fixes f :: "nat \<Rightarrow> 'a::metric_space"

  2752   assumes "l islimpt (range f)"

  2753   shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2754 proof (intro exI conjI)

  2755   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2756     using assms unfolding islimpt_def

  2757     by (drule_tac x="ball l e" in spec)

  2758        (auto simp add: zero_less_dist_iff dist_commute)

  2759

  2760   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2761   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"

  2762     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])

  2763   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"

  2764     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])

  2765   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"

  2766     unfolding t_def by (simp add: Least_le)

  2767   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"

  2768     unfolding t_def by (drule not_less_Least) simp

  2769   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"

  2770     apply (rule t_le)

  2771     apply (erule f_t_neq)

  2772     apply (erule (1) less_le_trans [OF f_t_closer])

  2773     done

  2774   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"

  2775     by (drule f_t_closer) auto

  2776   have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"

  2777     apply (subst less_le)

  2778     apply (rule conjI)

  2779     apply (rule t_antimono)

  2780     apply (erule f_t_neq)

  2781     apply (erule f_t_closer [THEN less_imp_le])

  2782     apply (rule t_dist_f_neq [symmetric])

  2783     apply (erule f_t_neq)

  2784     done

  2785   have dist_f_t_less':

  2786     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"

  2787     apply (simp add: le_less)

  2788     apply (erule disjE)

  2789     apply (rule less_trans)

  2790     apply (erule f_t_closer)

  2791     apply (rule le_less_trans)

  2792     apply (erule less_tD)

  2793     apply (erule f_t_neq)

  2794     apply (erule f_t_closer)

  2795     apply (erule subst)

  2796     apply (erule f_t_closer)

  2797     done

  2798

  2799   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"

  2800   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"

  2801     unfolding r_def by simp_all

  2802   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"

  2803     by (induct_tac n) (simp_all add: r_simps f_t_neq)

  2804

  2805   show "subseq r"

  2806     unfolding subseq_Suc_iff

  2807     apply (rule allI)

  2808     apply (case_tac n)

  2809     apply (simp_all add: r_simps)

  2810     apply (rule t_less, rule zero_less_one)

  2811     apply (rule t_less, rule f_r_neq)

  2812     done

  2813   show "((f \<circ> r) ---> l) sequentially"

  2814     unfolding LIMSEQ_def o_def

  2815     apply (clarify, rename_tac e, rule_tac x="t e" in exI, clarify)

  2816     apply (drule le_trans, rule seq_suble [OF subseq r])

  2817     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)

  2818     done

  2819 qed

  2820

  2821 lemma finite_range_imp_infinite_repeats:

  2822   fixes f :: "nat \<Rightarrow> 'a"

  2823   assumes "finite (range f)"

  2824   shows "\<exists>k. infinite {n. f n = k}"

  2825 proof -

  2826   { fix A :: "'a set" assume "finite A"

  2827     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"

  2828     proof (induct)

  2829       case empty thus ?case by simp

  2830     next

  2831       case (insert x A)

  2832      show ?case

  2833       proof (cases "finite {n. f n = x}")

  2834         case True

  2835         with infinite {n. f n \<in> insert x A}

  2836         have "infinite {n. f n \<in> A}" by simp

  2837         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)

  2838       next

  2839         case False thus "\<exists>k. infinite {n. f n = k}" ..

  2840       qed

  2841     qed

  2842   } note H = this

  2843   from assms show "\<exists>k. infinite {n. f n = k}"

  2844     by (rule H) simp

  2845 qed

  2846

  2847 lemma bolzano_weierstrass_imp_compact:

  2848   fixes s :: "'a::metric_space set"

  2849   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2850   shows "compact s"

  2851 proof -

  2852   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  2853     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2854     proof (cases "finite (range f)")

  2855       case True

  2856       hence "\<exists>l. infinite {n. f n = l}"

  2857         by (rule finite_range_imp_infinite_repeats)

  2858       then obtain l where "infinite {n. f n = l}" ..

  2859       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"

  2860         by (rule infinite_enumerate)

  2861       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto

  2862       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2863         unfolding o_def by (simp add: fr tendsto_const)

  2864       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2865         by - (rule exI)

  2866       from f have "\<forall>n. f (r n) \<in> s" by simp

  2867       hence "l \<in> s" by (simp add: fr)

  2868       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2869         by (rule rev_bexI) fact

  2870     next

  2871       case False

  2872       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto

  2873       then obtain l where "l \<in> s" "l islimpt (range f)" ..

  2874       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2875         using l islimpt (range f)

  2876         by (rule islimpt_range_imp_convergent_subsequence)

  2877       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  2878     qed

  2879   }

  2880   thus ?thesis unfolding compact_def by auto

  2881 qed

  2882

  2883 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where

  2884   "helper_2 beyond 0 = beyond 0" |

  2885   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"

  2886

  2887 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"

  2888   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2889   shows "bounded s"

  2890 proof(rule ccontr)

  2891   assume "\<not> bounded s"

  2892   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"

  2893     unfolding bounded_any_center [where a=undefined]

  2894     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto

  2895   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"

  2896     unfolding linorder_not_le by auto

  2897   def x \<equiv> "helper_2 beyond"

  2898

  2899   { fix m n ::nat assume "m<n"

  2900     hence "dist undefined (x m) + 1 < dist undefined (x n)"

  2901     proof(induct n)

  2902       case 0 thus ?case by auto

  2903     next

  2904       case (Suc n)

  2905       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"

  2906         unfolding x_def and helper_2.simps

  2907         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto

  2908       thus ?case proof(cases "m < n")

  2909         case True thus ?thesis using Suc and * by auto

  2910       next

  2911         case False hence "m = n" using Suc(2) by auto

  2912         thus ?thesis using * by auto

  2913       qed

  2914     qed  } note * = this

  2915   { fix m n ::nat assume "m\<noteq>n"

  2916     have "1 < dist (x m) (x n)"

  2917     proof(cases "m<n")

  2918       case True

  2919       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto

  2920       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith

  2921     next

  2922       case False hence "n<m" using m\<noteq>n by auto

  2923       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto

  2924       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith

  2925     qed  } note ** = this

  2926   { fix a b assume "x a = x b" "a \<noteq> b"

  2927     hence False using **[of a b] by auto  }

  2928   hence "inj x" unfolding inj_on_def by auto

  2929   moreover

  2930   { fix n::nat

  2931     have "x n \<in> s"

  2932     proof(cases "n = 0")

  2933       case True thus ?thesis unfolding x_def using beyond by auto

  2934     next

  2935       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto

  2936       thus ?thesis unfolding x_def using beyond by auto

  2937     qed  }

  2938   ultimately have "infinite (range x) \<and> range x \<subseteq> s" unfolding x_def using range_inj_infinite[of "helper_2 beyond"] using beyond(1) by auto

  2939

  2940   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto

  2941   then obtain y where "x y \<noteq> l" and y:"dist (x y) l < 1/2" unfolding islimpt_approachable apply(erule_tac x="1/2" in allE) by auto

  2942   then obtain z where "x z \<noteq> l" and z:"dist (x z) l < dist (x y) l" using l[unfolded islimpt_approachable, THEN spec[where x="dist (x y) l"]]

  2943     unfolding dist_nz by auto

  2944   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto

  2945 qed

  2946

  2947 lemma sequence_infinite_lemma:

  2948   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2949   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2950   shows "infinite (range f)"

  2951 proof

  2952   assume "finite (range f)"

  2953   hence "closed (range f)" by (rule finite_imp_closed)

  2954   hence "open (- range f)" by (rule open_Compl)

  2955   from assms(1) have "l \<in> - range f" by auto

  2956   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2957     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2958   thus False unfolding eventually_sequentially by auto

  2959 qed

  2960

  2961 lemma closure_insert:

  2962   fixes x :: "'a::t1_space"

  2963   shows "closure (insert x s) = insert x (closure s)"

  2964 apply (rule closure_unique)

  2965 apply (rule insert_mono [OF closure_subset])

  2966 apply (rule closed_insert [OF closed_closure])

  2967 apply (simp add: closure_minimal)

  2968 done

  2969

  2970 lemma islimpt_insert:

  2971   fixes x :: "'a::t1_space"

  2972   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2973 proof

  2974   assume *: "x islimpt (insert a s)"

  2975   show "x islimpt s"

  2976   proof (rule islimptI)

  2977     fix t assume t: "x \<in> t" "open t"

  2978     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2979     proof (cases "x = a")

  2980       case True

  2981       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2982         using * t by (rule islimptE)

  2983       with x = a show ?thesis by auto

  2984     next

  2985       case False

  2986       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2987         by (simp_all add: open_Diff)

  2988       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2989         using * t' by (rule islimptE)

  2990       thus ?thesis by auto

  2991     qed

  2992   qed

  2993 next

  2994   assume "x islimpt s" thus "x islimpt (insert a s)"

  2995     by (rule islimpt_subset) auto

  2996 qed

  2997

  2998 lemma islimpt_union_finite:

  2999   fixes x :: "'a::t1_space"

  3000   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3001 by (induct set: finite, simp_all add: islimpt_insert)

  3002

  3003 lemma sequence_unique_limpt:

  3004   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3005   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  3006   shows "l' = l"

  3007 proof (rule ccontr)

  3008   assume "l' \<noteq> l"

  3009   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3010     using hausdorff [OF l' \<noteq> l] by auto

  3011   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3012     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3013   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3014     unfolding eventually_sequentially by auto

  3015

  3016   have "UNIV = {..<N} \<union> {N..}" by auto

  3017   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  3018   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  3019   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  3020   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3021     using l' \<in> s open s by (rule islimptE)

  3022   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  3023   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  3024   with s \<inter> t = {} show False by simp

  3025 qed

  3026

  3027 lemma bolzano_weierstrass_imp_closed:

  3028   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)

  3029   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3030   shows "closed s"

  3031 proof-

  3032   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3033     hence "l \<in> s"

  3034     proof(cases "\<forall>n. x n \<noteq> l")

  3035       case False thus "l\<in>s" using as(1) by auto

  3036     next

  3037       case True note cas = this

  3038       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  3039       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  3040       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  3041     qed  }

  3042   thus ?thesis unfolding closed_sequential_limits by fast

  3043 qed

  3044

  3045 text {* Hence express everything as an equivalence. *}

  3046

  3047 lemma compact_eq_heine_borel:

  3048   fixes s :: "'a::metric_space set"

  3049   shows "compact s \<longleftrightarrow>

  3050            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  3051                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")

  3052 proof

  3053   assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)

  3054 next

  3055   assume ?rhs

  3056   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"

  3057     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])

  3058   thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3059 qed

  3060

  3061 lemma compact_eq_bolzano_weierstrass:

  3062   fixes s :: "'a::metric_space set"

  3063   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3064 proof

  3065   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3066 next

  3067   assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3068 qed

  3069

  3070 lemma nat_approx_posE:

  3071   fixes e::real

  3072   assumes "0 < e"

  3073   obtains n::nat where "1 / (Suc n) < e"

  3074 proof atomize_elim

  3075   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3076     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3077   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3078     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3079   also have "\<dots> = e" by simp

  3080   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3081 qed

  3082

  3083 lemma compact_eq_totally_bounded:

  3084   shows "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3085 proof (safe intro!: compact_imp_complete)

  3086   fix e::real

  3087   def f \<equiv> "(\<lambda>x::'a. ball x e)  UNIV"

  3088   assume "0 < e" "compact s"

  3089   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3090     by (simp add: compact_eq_heine_borel)

  3091   moreover

  3092   have d0: "\<And>x::'a. dist x x < e" using 0 < e by simp

  3093   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f" by (auto simp: f_def intro!: d0)

  3094   ultimately have "(\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')" ..

  3095   then guess K .. note K = this

  3096   have "\<forall>K'\<in>K. \<exists>k. K' = ball k e" using K by (auto simp: f_def)

  3097   then obtain k where "\<And>K'. K' \<in> K \<Longrightarrow> K' = ball (k K') e" unfolding bchoice_iff by blast

  3098   thus "\<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" using K

  3099     by (intro exI[where x="k  K"]) (auto simp: f_def)

  3100 next

  3101   assume assms: "complete s" "\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k"

  3102   show "compact s"

  3103   proof cases

  3104     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3105   next

  3106     assume "s \<noteq> {}"

  3107     show ?thesis

  3108       unfolding compact_def

  3109     proof safe

  3110       fix f::"nat \<Rightarrow> _" assume "\<forall>n. f n \<in> s" hence f: "\<And>n. f n \<in> s" by simp

  3111       from assms have "\<forall>e. \<exists>k. e>0 \<longrightarrow> finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))" by simp

  3112       then obtain K where

  3113         K: "\<And>e. e > 0 \<Longrightarrow> finite (K e) \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3114         unfolding choice_iff by blast

  3115       {

  3116         fix e::real and f' have f': "\<And>n::nat. (f o f') n \<in> s" using f by auto

  3117         assume "e > 0"

  3118         from K[OF this] have K: "finite (K e)" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3119           by simp_all

  3120         have "\<exists>k\<in>(K e). \<exists>r. subseq r \<and> (\<forall>i. (f o f' o r) i \<in> ball k e)"

  3121         proof (rule ccontr)

  3122           from K have "finite (K e)" "K e \<noteq> {}" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3123             using s \<noteq> {}

  3124             by auto

  3125           moreover

  3126           assume "\<not> (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f \<circ> f' o r) i \<in> ball k e))"

  3127           hence "\<And>r k. k \<in> K e \<Longrightarrow> subseq r \<Longrightarrow> (\<exists>i. (f o f' o r) i \<notin> ball k e)" by simp

  3128           ultimately

  3129           show False using f'

  3130           proof (induct arbitrary: s f f' rule: finite_ne_induct)

  3131             case (singleton x)

  3132             have "\<exists>i. (f \<circ> f' o id) i \<notin> ball x e" by (rule singleton) (auto simp: subseq_def)

  3133             thus ?case using singleton by (auto simp: ball_def)

  3134           next

  3135             case (insert x A)

  3136             show ?case

  3137             proof cases

  3138               have inf_ms: "infinite ((f o f') - s)" using insert by (simp add: vimage_def)

  3139               have "infinite ((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A)))"

  3140                 using insert by (intro infinite_super[OF _ inf_ms]) auto

  3141               also have "((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A))) =

  3142                 {m. (f o f') m \<in> ball x e} \<union> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by auto

  3143               finally have "infinite \<dots>" .

  3144               moreover assume "finite {m. (f o f') m \<in> ball x e}"

  3145               ultimately have inf: "infinite {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by blast

  3146               hence "A \<noteq> {}" by auto then obtain k where "k \<in> A" by auto

  3147               def r \<equiv> "enumerate {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3148               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3149                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3150               hence "subseq r" by (simp add: subseq_def)

  3151               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3152                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3153               show False

  3154               proof (rule insert)

  3155                 show "\<Union>(\<lambda>x. ball x e)  A \<subseteq> \<Union>(\<lambda>x. ball x e)  A" by simp

  3156                 fix k s assume "k \<in> A" "subseq s"

  3157                 thus "\<exists>i. (f o f' o r o s) i \<notin> ball k e" using subseq r

  3158                   by (subst (2) o_assoc[symmetric]) (intro insert(6) subseq_o, simp_all)

  3159               next

  3160                 fix n show "(f \<circ> f' o r) n \<in> \<Union>(\<lambda>x. ball x e)  A" using r_in_set by auto

  3161               qed

  3162             next

  3163               assume inf: "infinite {m. (f o f') m \<in> ball x e}"

  3164               def r \<equiv> "enumerate {m. (f o f') m \<in> ball x e}"

  3165               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3166                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3167               hence "subseq r" by (simp add: subseq_def)

  3168               from insert(6)[OF insertI1 this] obtain i where "(f o f') (r i) \<notin> ball x e" by auto

  3169               moreover

  3170               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> ball x e}"

  3171                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3172               hence "(f o f') (r i) \<in> ball x e" by simp

  3173               ultimately show False by simp

  3174             qed

  3175           qed

  3176         qed

  3177       }

  3178       hence ex: "\<forall>f'. \<forall>e > 0. (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f o f' \<circ> r) i \<in> ball k e))" by simp

  3179       let ?e = "\<lambda>n. 1 / real (Suc n)"

  3180       let ?P = "\<lambda>n s. \<exists>k\<in>K (?e n). (\<forall>i. (f o s) i \<in> ball k (?e n))"

  3181       interpret subseqs ?P using ex by unfold_locales force

  3182       from complete s have limI: "\<And>f. (\<And>n. f n \<in> s) \<Longrightarrow> Cauchy f \<Longrightarrow> (\<exists>l\<in>s. f ----> l)"

  3183         by (simp add: complete_def)

  3184       have "\<exists>l\<in>s. (f o diagseq) ----> l"

  3185       proof (intro limI metric_CauchyI)

  3186         fix e::real assume "0 < e" hence "0 < e / 2" by auto

  3187         from nat_approx_posE[OF this] guess n . note n = this

  3188         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) n) < e"

  3189         proof (rule exI[where x="Suc n"], safe)

  3190           fix m mm assume "Suc n \<le> m" "Suc n \<le> mm"

  3191           let ?e = "1 / real (Suc n)"

  3192           from reducer_reduces[of n] obtain k where

  3193             "k\<in>K ?e"  "\<And>i. (f o seqseq (Suc n)) i \<in> ball k ?e"

  3194             unfolding seqseq_reducer by auto

  3195           moreover

  3196           note diagseq_sub[OF Suc n \<le> m] diagseq_sub[OF Suc n \<le> mm]

  3197           ultimately have "{(f o diagseq) m, (f o diagseq) mm} \<subseteq> ball k ?e" by auto

  3198           also have "\<dots> \<subseteq> ball k (e / 2)" using n by (intro subset_ball) simp

  3199           finally

  3200           have "dist k ((f \<circ> diagseq) m) + dist k ((f \<circ> diagseq) mm) < e / 2 + e /2"

  3201             by (intro add_strict_mono) auto

  3202           hence "dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k < e"

  3203             by (simp add: dist_commute)

  3204           moreover have "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) \<le>

  3205             dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k"

  3206             by (rule dist_triangle2)

  3207           ultimately show "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) < e"

  3208             by simp

  3209         qed

  3210       next

  3211         fix n show "(f o diagseq) n \<in> s" using f by simp

  3212       qed

  3213       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l" using subseq_diagseq by auto

  3214     qed

  3215   qed

  3216 qed

  3217

  3218 lemma compact_eq_bounded_closed:

  3219   fixes s :: "'a::heine_borel set"

  3220   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3221 proof

  3222   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto

  3223 next

  3224   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto

  3225 qed

  3226

  3227 lemma compact_imp_bounded:

  3228   fixes s :: "'a::metric_space set"

  3229   shows "compact s ==> bounded s"

  3230 proof -

  3231   assume "compact s"

  3232   hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3233     by (rule compact_imp_heine_borel)

  3234   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"

  3235     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3236   thus "bounded s"

  3237     by (rule bolzano_weierstrass_imp_bounded)

  3238 qed

  3239

  3240 lemma compact_imp_closed:

  3241   fixes s :: "'a::metric_space set"

  3242   shows "compact s ==> closed s"

  3243 proof -

  3244   assume "compact s"

  3245   hence "\<forall>f. (\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3246     by (rule compact_imp_heine_borel)

  3247   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"

  3248     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3249   thus "closed s"

  3250     by (rule bolzano_weierstrass_imp_closed)

  3251 qed

  3252

  3253 text{* In particular, some common special cases. *}

  3254

  3255 lemma compact_empty[simp]:

  3256  "compact {}"

  3257   unfolding compact_def

  3258   by simp

  3259

  3260 lemma compact_union [intro]:

  3261   assumes "compact s" and "compact t"

  3262   shows "compact (s \<union> t)"

  3263 proof (rule compactI)

  3264   fix f :: "nat \<Rightarrow> 'a"

  3265   assume "\<forall>n. f n \<in> s \<union> t"

  3266   hence "infinite {n. f n \<in> s \<union> t}" by simp

  3267   hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp

  3268   thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3269   proof

  3270     assume "infinite {n. f n \<in> s}"

  3271     from infinite_enumerate [OF this]

  3272     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto

  3273     obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3274       using compact s \<forall>n. (f \<circ> q) n \<in> s by (rule compactE)

  3275     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3276       using subseq q by (simp_all add: subseq_o o_assoc)

  3277     thus ?thesis by auto

  3278   next

  3279     assume "infinite {n. f n \<in> t}"

  3280     from infinite_enumerate [OF this]

  3281     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto

  3282     obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3283       using compact t \<forall>n. (f \<circ> q) n \<in> t by (rule compactE)

  3284     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3285       using subseq q by (simp_all add: subseq_o o_assoc)

  3286     thus ?thesis by auto

  3287   qed

  3288 qed

  3289

  3290 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3291   by (induct set: finite) auto

  3292

  3293 lemma compact_UN [intro]:

  3294   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3295   unfolding SUP_def by (rule compact_Union) auto

  3296

  3297 lemma compact_inter_closed [intro]:

  3298   assumes "compact s" and "closed t"

  3299   shows "compact (s \<inter> t)"

  3300 proof (rule compactI)

  3301   fix f :: "nat \<Rightarrow> 'a"

  3302   assume "\<forall>n. f n \<in> s \<inter> t"

  3303   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all

  3304   obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3305     using compact s \<forall>n. f n \<in> s by (rule compactE)

  3306   moreover

  3307   from closed t \<forall>n. f n \<in> t ((f \<circ> r) ---> l) sequentially have "l \<in> t"

  3308     unfolding closed_sequential_limits o_def by fast

  3309   ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3310     by auto

  3311 qed

  3312

  3313 lemma closed_inter_compact [intro]:

  3314   assumes "closed s" and "compact t"

  3315   shows "compact (s \<inter> t)"

  3316   using compact_inter_closed [of t s] assms

  3317   by (simp add: Int_commute)

  3318

  3319 lemma compact_inter [intro]:

  3320   assumes "compact s" and "compact t"

  3321   shows "compact (s \<inter> t)"

  3322   using assms by (intro compact_inter_closed compact_imp_closed)

  3323

  3324 lemma compact_sing [simp]: "compact {a}"

  3325   unfolding compact_def o_def subseq_def

  3326   by (auto simp add: tendsto_const)

  3327

  3328 lemma compact_insert [simp]:

  3329   assumes "compact s" shows "compact (insert x s)"

  3330 proof -

  3331   have "compact ({x} \<union> s)"

  3332     using compact_sing assms by (rule compact_union)

  3333   thus ?thesis by simp

  3334 qed

  3335

  3336 lemma finite_imp_compact:

  3337   shows "finite s \<Longrightarrow> compact s"

  3338   by (induct set: finite) simp_all

  3339

  3340 lemma compact_cball[simp]:

  3341   fixes x :: "'a::heine_borel"

  3342   shows "compact(cball x e)"

  3343   using compact_eq_bounded_closed bounded_cball closed_cball

  3344   by blast

  3345

  3346 lemma compact_frontier_bounded[intro]:

  3347   fixes s :: "'a::heine_borel set"

  3348   shows "bounded s ==> compact(frontier s)"

  3349   unfolding frontier_def

  3350   using compact_eq_bounded_closed

  3351   by blast

  3352

  3353 lemma compact_frontier[intro]:

  3354   fixes s :: "'a::heine_borel set"

  3355   shows "compact s ==> compact (frontier s)"

  3356   using compact_eq_bounded_closed compact_frontier_bounded

  3357   by blast

  3358

  3359 lemma frontier_subset_compact:

  3360   fixes s :: "'a::heine_borel set"

  3361   shows "compact s ==> frontier s \<subseteq> s"

  3362   using frontier_subset_closed compact_eq_bounded_closed

  3363   by blast

  3364

  3365 lemma open_delete:

  3366   fixes s :: "'a::t1_space set"

  3367   shows "open s \<Longrightarrow> open (s - {x})"

  3368   by (simp add: open_Diff)

  3369

  3370 text{* Finite intersection property. I could make it an equivalence in fact. *}

  3371

  3372 lemma compact_imp_fip:

  3373   assumes "compact s"  "\<forall>t \<in> f. closed t"

  3374         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"

  3375   shows "s \<inter> (\<Inter> f) \<noteq> {}"

  3376 proof

  3377   assume as:"s \<inter> (\<Inter> f) = {}"

  3378   hence "s \<subseteq> \<Union> uminus  f" by auto

  3379   moreover have "Ball (uminus  f) open" using open_Diff closed_Diff using assms(2) by auto

  3380   ultimately obtain f' where f':"f' \<subseteq> uminus  f"  "finite f'"  "s \<subseteq> \<Union>f'" using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="(\<lambda>t. - t)  f"]] by auto

  3381   hence "finite (uminus  f') \<and> uminus  f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)

  3382   hence "s \<inter> \<Inter>uminus  f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus  f'"]] by auto

  3383   thus False using f'(3) unfolding subset_eq and Union_iff by blast

  3384 qed

  3385

  3386

  3387 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3388

  3389 lemma bounded_closed_nest:

  3390   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3391   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3392   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3393 proof-

  3394   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto

  3395   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto

  3396

  3397   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3398     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3399

  3400   { fix n::nat

  3401     { fix e::real assume "e>0"

  3402       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3403       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3404       moreover

  3405       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto

  3406       hence "(x \<circ> r) (max N n) \<in> s n"

  3407         using x apply(erule_tac x=n in allE)

  3408         using x apply(erule_tac x="r (max N n)" in allE)

  3409         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto

  3410       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3411     }

  3412     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3413   }

  3414   thus ?thesis by auto

  3415 qed

  3416

  3417 text {* Decreasing case does not even need compactness, just completeness. *}

  3418

  3419 lemma decreasing_closed_nest:

  3420   assumes "\<forall>n. closed(s n)"

  3421           "\<forall>n. (s n \<noteq> {})"

  3422           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3423           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3424   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3425 proof-

  3426   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3427   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3428   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3429   { fix e::real assume "e>0"

  3430     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3431     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3432       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3433       hence "dist (t m) (t n) < e" using N by auto

  3434     }

  3435     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3436   }

  3437   hence  "Cauchy t" unfolding cauchy_def by auto

  3438   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3439   { fix n::nat

  3440     { fix e::real assume "e>0"

  3441       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3442       have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto

  3443       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3444     }

  3445     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3446   }

  3447   then show ?thesis by auto

  3448 qed

  3449

  3450 text {* Strengthen it to the intersection actually being a singleton. *}

  3451

  3452 lemma decreasing_closed_nest_sing:

  3453   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3454   assumes "\<forall>n. closed(s n)"

  3455           "\<forall>n. s n \<noteq> {}"

  3456           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3457           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3458   shows "\<exists>a. \<Inter>(range s) = {a}"

  3459 proof-

  3460   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3461   { fix b assume b:"b \<in> \<Inter>(range s)"

  3462     { fix e::real assume "e>0"

  3463       hence "dist a b < e" using assms(4 )using b using a by blast

  3464     }

  3465     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3466   }

  3467   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3468   thus ?thesis ..

  3469 qed

  3470

  3471 text{* Cauchy-type criteria for uniform convergence. *}

  3472

  3473 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows

  3474  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3475   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")

  3476 proof(rule)

  3477   assume ?lhs

  3478   then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto

  3479   { fix e::real assume "e>0"

  3480     then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto

  3481     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3482       hence "dist (s m x) (s n x) < e"

  3483         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3484         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3485         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3486     hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e"  by auto  }

  3487   thus ?rhs by auto

  3488 next

  3489   assume ?rhs

  3490   hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto

  3491   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3492     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3493   { fix e::real assume "e>0"

  3494     then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"

  3495       using ?rhs[THEN spec[where x="e/2"]] by auto

  3496     { fix x assume "P x"

  3497       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3498         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3499       fix n::nat assume "n\<ge>N"

  3500       hence "dist(s n x)(l x) < e"  using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  3501         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }

  3502     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3503   thus ?lhs by auto

  3504 qed

  3505

  3506 lemma uniformly_cauchy_imp_uniformly_convergent:

  3507   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"

  3508   assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"

  3509           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3510   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3511 proof-

  3512   obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"

  3513     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3514   moreover

  3515   { fix x assume "P x"

  3516     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3517       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3518   ultimately show ?thesis by auto

  3519 qed

  3520

  3521

  3522 subsection {* Continuity *}

  3523

  3524 text {* Define continuity over a net to take in restrictions of the set. *}

  3525

  3526 definition

  3527   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3528   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"

  3529

  3530 lemma continuous_trivial_limit:

  3531  "trivial_limit net ==> continuous net f"

  3532   unfolding continuous_def tendsto_def trivial_limit_eq by auto

  3533

  3534 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"

  3535   unfolding continuous_def

  3536   unfolding tendsto_def

  3537   using netlimit_within[of x s]

  3538   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)

  3539

  3540 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"

  3541   using continuous_within [of x UNIV f] by simp

  3542

  3543 lemma continuous_at_within:

  3544   assumes "continuous (at x) f"  shows "continuous (at x within s) f"

  3545   using assms unfolding continuous_at continuous_within

  3546   by (rule Lim_at_within)

  3547

  3548 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3549

  3550 lemma continuous_within_eps_delta:

  3551   "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"

  3552   unfolding continuous_within and Lim_within

  3553   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3554

  3555 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3556                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3557   using continuous_within_eps_delta [of x UNIV f] by simp

  3558

  3559 text{* Versions in terms of open balls. *}

  3560

  3561 lemma continuous_within_ball:

  3562  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3563                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3564 proof

  3565   assume ?lhs

  3566   { fix e::real assume "e>0"

  3567     then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"

  3568       using ?lhs[unfolded continuous_within Lim_within] by auto

  3569     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3570       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3571         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3572     }

  3573     hence "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e" using d>0 unfolding subset_eq ball_def by (auto simp add: dist_commute)  }

  3574   thus ?rhs by auto

  3575 next

  3576   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3577     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3578 qed

  3579

  3580 lemma continuous_at_ball:

  3581   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3582 proof

  3583   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3584     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)

  3585     unfolding dist_nz[THEN sym] by auto

  3586 next

  3587   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3588     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)

  3589 qed

  3590

  3591 text{* Define setwise continuity in terms of limits within the set. *}

  3592

  3593 definition

  3594   continuous_on ::

  3595     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3596 where

  3597   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"

  3598

  3599 lemma continuous_on_topological:

  3600   "continuous_on s f \<longleftrightarrow>

  3601     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  3602       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  3603 unfolding continuous_on_def tendsto_def

  3604 unfolding Limits.eventually_within eventually_at_topological

  3605 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  3606

  3607 lemma continuous_on_iff:

  3608   "continuous_on s f \<longleftrightarrow>

  3609     (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  3610 unfolding continuous_on_def Lim_within

  3611 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3612 apply (rename_tac y, case_tac "y = x", simp)

  3613 apply (simp add: dist_nz)

  3614 done

  3615

  3616 definition

  3617   uniformly_continuous_on ::

  3618     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3619 where

  3620   "uniformly_continuous_on s f \<longleftrightarrow>

  3621     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  3622

  3623 text{* Some simple consequential lemmas. *}

  3624

  3625 lemma uniformly_continuous_imp_continuous:

  3626  " uniformly_continuous_on s f ==> continuous_on s f"

  3627   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3628

  3629 lemma continuous_at_imp_continuous_within:

  3630  "continuous (at x) f ==> continuous (at x within s) f"

  3631   unfolding continuous_within continuous_at using Lim_at_within by auto

  3632

  3633 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3634 unfolding tendsto_def by (simp add: trivial_limit_eq)

  3635

  3636 lemma continuous_at_imp_continuous_on:

  3637   assumes "\<forall>x\<in>s. continuous (at x) f"

  3638   shows "continuous_on s f"

  3639 unfolding continuous_on_def

  3640 proof

  3641   fix x assume "x \<in> s"

  3642   with assms have *: "(f ---> f (netlimit (at x))) (at x)"

  3643     unfolding continuous_def by simp

  3644   have "(f ---> f x) (at x)"

  3645   proof (cases "trivial_limit (at x)")

  3646     case True thus ?thesis

  3647       by (rule Lim_trivial_limit)

  3648   next

  3649     case False

  3650     hence 1: "netlimit (at x) = x"

  3651       using netlimit_within [of x UNIV] by simp

  3652     with * show ?thesis by simp

  3653   qed

  3654   thus "(f ---> f x) (at x within s)"

  3655     by (rule Lim_at_within)

  3656 qed

  3657

  3658 lemma continuous_on_eq_continuous_within:

  3659   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"

  3660 unfolding continuous_on_def continuous_def

  3661 apply (rule ball_cong [OF refl])

  3662 apply (case_tac "trivial_limit (at x within s)")

  3663 apply (simp add: Lim_trivial_limit)

  3664 apply (simp add: netlimit_within)

  3665 done

  3666

  3667 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3668

  3669 lemma continuous_on_eq_continuous_at:

  3670   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"

  3671   by (auto simp add: continuous_on continuous_at Lim_within_open)

  3672

  3673 lemma continuous_within_subset:

  3674  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3675              ==> continuous (at x within t) f"

  3676   unfolding continuous_within by(metis Lim_within_subset)

  3677

  3678 lemma continuous_on_subset:

  3679   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"

  3680   unfolding continuous_on by (metis subset_eq Lim_within_subset)

  3681

  3682 lemma continuous_on_interior:

  3683   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3684   by (erule interiorE, drule (1) continuous_on_subset,

  3685     simp add: continuous_on_eq_continuous_at)

  3686

  3687 lemma continuous_on_eq:

  3688   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3689   unfolding continuous_on_def tendsto_def Limits.eventually_within

  3690   by simp

  3691

  3692 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3693

  3694 lemma continuous_within_sequentially:

  3695   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3696   shows "continuous (at a within s) f \<longleftrightarrow>

  3697                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3698                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3699 proof

  3700   assume ?lhs

  3701   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  3702     fix T::"'b set" assume "open T" and "f a \<in> T"

  3703     with ?lhs obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"

  3704       unfolding continuous_within tendsto_def eventually_within by auto

  3705     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3706       using x(2) d>0 by simp

  3707     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3708     proof eventually_elim

  3709       case (elim n) thus ?case

  3710         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3711     qed

  3712   }

  3713   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3714 next

  3715   assume ?rhs thus ?lhs

  3716     unfolding continuous_within tendsto_def [where l="f a"]

  3717     by (simp add: sequentially_imp_eventually_within)

  3718 qed

  3719

  3720 lemma continuous_at_sequentially:

  3721   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3722   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  3723                   --> ((f o x) ---> f a) sequentially)"

  3724   using continuous_within_sequentially[of a UNIV f] by simp

  3725

  3726 lemma continuous_on_sequentially:

  3727   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3728   shows "continuous_on s f \<longleftrightarrow>

  3729     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  3730                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  3731 proof

  3732   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  3733 next

  3734   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  3735 qed

  3736

  3737 lemma uniformly_continuous_on_sequentially:

  3738   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  3739                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  3740                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  3741 proof

  3742   assume ?lhs

  3743   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  3744     { fix e::real assume "e>0"

  3745       then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  3746         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  3747       obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  3748       { fix n assume "n\<ge>N"

  3749         hence "dist (f (x n)) (f (y n)) < e"

  3750           using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y

  3751           unfolding dist_commute by simp  }

  3752       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  3753     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  3754   thus ?rhs by auto

  3755 next

  3756   assume ?rhs

  3757   { assume "\<not> ?lhs"

  3758     then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto

  3759     then obtain fa where fa:"\<forall>x.  0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"

  3760       using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def

  3761       by (auto simp add: dist_commute)

  3762     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  3763     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  3764     have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  3765       unfolding x_def and y_def using fa by auto

  3766     { fix e::real assume "e>0"

  3767       then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto

  3768       { fix n::nat assume "n\<ge>N"

  3769         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  3770         also have "\<dots> < e" using N by auto

  3771         finally have "inverse (real n + 1) < e" by auto

  3772         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  3773       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  3774     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto

  3775     hence False using fxy and e>0 by auto  }

  3776   thus ?lhs unfolding uniformly_continuous_on_def by blast

  3777 qed

  3778

  3779 text{* The usual transformation theorems. *}

  3780

  3781 lemma continuous_transform_within:

  3782   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3783   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  3784           "continuous (at x within s) f"

  3785   shows "continuous (at x within s) g"

  3786 unfolding continuous_within

  3787 proof (rule Lim_transform_within)

  3788   show "0 < d" by fact

  3789   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  3790     using assms(3) by auto

  3791   have "f x = g x"

  3792     using assms(1,2,3) by auto

  3793   thus "(f ---> g x) (at x within s)"

  3794     using assms(4) unfolding continuous_within by simp

  3795 qed

  3796

  3797 lemma continuous_transform_at:

  3798   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3799   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  3800           "continuous (at x) f"

  3801   shows "continuous (at x) g"

  3802   using continuous_transform_within [of d x UNIV f g] assms by simp

  3803

  3804 subsubsection {* Structural rules for pointwise continuity *}

  3805

  3806 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"

  3807   unfolding continuous_within by (rule tendsto_ident_at_within)

  3808

  3809 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"

  3810   unfolding continuous_at by (rule tendsto_ident_at)

  3811

  3812 lemma continuous_const: "continuous F (\<lambda>x. c)"

  3813   unfolding continuous_def by (rule tendsto_const)

  3814

  3815 lemma continuous_dist:

  3816   assumes "continuous F f" and "continuous F g"

  3817   shows "continuous F (\<lambda>x. dist (f x) (g x))"

  3818   using assms unfolding continuous_def by (rule tendsto_dist)

  3819

  3820 lemma continuous_infdist:

  3821   assumes "continuous F f"

  3822   shows "continuous F (\<lambda>x. infdist (f x) A)"

  3823   using assms unfolding continuous_def by (rule tendsto_infdist)

  3824

  3825 lemma continuous_norm:

  3826   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"

  3827   unfolding continuous_def by (rule tendsto_norm)

  3828

  3829 lemma continuous_infnorm:

  3830   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  3831   unfolding continuous_def by (rule tendsto_infnorm)

  3832

  3833 lemma continuous_add:

  3834   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3835   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"

  3836   unfolding continuous_def by (rule tendsto_add)

  3837

  3838 lemma continuous_minus:

  3839   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3840   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"

  3841   unfolding continuous_def by (rule tendsto_minus)

  3842

  3843 lemma continuous_diff:

  3844   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3845   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"

  3846   unfolding continuous_def by (rule tendsto_diff)

  3847

  3848 lemma continuous_scaleR:

  3849   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3850   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"

  3851   unfolding continuous_def by (rule tendsto_scaleR)

  3852

  3853 lemma continuous_mult:

  3854   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"

  3855   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"

  3856   unfolding continuous_def by (rule tendsto_mult)

  3857

  3858 lemma continuous_inner:

  3859   assumes "continuous F f" and "continuous F g"

  3860   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  3861   using assms unfolding continuous_def by (rule tendsto_inner)

  3862

  3863 lemma continuous_inverse:

  3864   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  3865   assumes "continuous F f" and "f (netlimit F) \<noteq> 0"

  3866   shows "continuous F (\<lambda>x. inverse (f x))"

  3867   using assms unfolding continuous_def by (rule tendsto_inverse)

  3868

  3869 lemma continuous_at_within_inverse:

  3870   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  3871   assumes "continuous (at a within s) f" and "f a \<noteq> 0"

  3872   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"

  3873   using assms unfolding continuous_within by (rule tendsto_inverse)

  3874

  3875 lemma continuous_at_inverse:

  3876   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  3877   assumes "continuous (at a) f" and "f a \<noteq> 0"

  3878   shows "continuous (at a) (\<lambda>x. inverse (f x))"

  3879   using assms unfolding continuous_at by (rule tendsto_inverse)

  3880

  3881 lemmas continuous_intros = continuous_at_id continuous_within_id

  3882   continuous_const continuous_dist continuous_norm continuous_infnorm

  3883   continuous_add continuous_minus continuous_diff continuous_scaleR continuous_mult

  3884   continuous_inner continuous_at_inverse continuous_at_within_inverse

  3885

  3886 subsubsection {* Structural rules for setwise continuity *}

  3887

  3888 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"

  3889   unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)

  3890

  3891 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"

  3892   unfolding continuous_on_def by (auto intro: tendsto_intros)

  3893

  3894 lemma continuous_on_norm:

  3895   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"

  3896   unfolding continuous_on_def by (fast intro: tendsto_norm)

  3897

  3898 lemma continuous_on_infnorm:

  3899   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  3900   unfolding continuous_on by (fast intro: tendsto_infnorm)

  3901

  3902 lemma continuous_on_minus:

  3903   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  3904   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"

  3905   unfolding continuous_on_def by (auto intro: tendsto_intros)

  3906

  3907 lemma continuous_on_add:

  3908   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  3909   shows "continuous_on s f \<Longrightarrow> continuous_on s g

  3910            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"

  3911   unfolding continuous_on_def by (auto intro: tendsto_intros)

  3912

  3913 lemma continuous_on_diff:

  3914   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  3915   shows "continuous_on s f \<Longrightarrow> continuous_on s g

  3916            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"

  3917   unfolding continuous_on_def by (auto intro: tendsto_intros)

  3918

  3919 lemma (in bounded_linear) continuous_on:

  3920   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"

  3921   unfolding continuous_on_def by (fast intro: tendsto)

  3922

  3923 lemma (in bounded_bilinear) continuous_on:

  3924   "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"

  3925   unfolding continuous_on_def by (fast intro: tendsto)

  3926

  3927 lemma continuous_on_scaleR:

  3928   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  3929   assumes "continuous_on s f" and "continuous_on s g"

  3930   shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"

  3931   using bounded_bilinear_scaleR assms

  3932   by (rule bounded_bilinear.continuous_on)

  3933

  3934 lemma continuous_on_mult:

  3935   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"

  3936   assumes "continuous_on s f" and "continuous_on s g"

  3937   shows "continuous_on s (\<lambda>x. f x * g x)"

  3938   using bounded_bilinear_mult assms

  3939   by (rule bounded_bilinear.continuous_on)

  3940

  3941 lemma continuous_on_inner:

  3942   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  3943   assumes "continuous_on s f" and "continuous_on s g"

  3944   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  3945   using bounded_bilinear_inner assms

  3946   by (rule bounded_bilinear.continuous_on)

  3947

  3948 lemma continuous_on_inverse:

  3949   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"

  3950   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

  3951   shows "continuous_on s (\<lambda>x. inverse (f x))"

  3952   using assms unfolding continuous_on by (fast intro: tendsto_inverse)

  3953

  3954 subsubsection {* Structural rules for uniform continuity *}

  3955

  3956 lemma uniformly_continuous_on_id:

  3957   shows "uniformly_continuous_on s (\<lambda>x. x)"

  3958   unfolding uniformly_continuous_on_def by auto

  3959

  3960 lemma uniformly_continuous_on_const:

  3961   shows "uniformly_continuous_on s (\<lambda>x. c)"

  3962   unfolding uniformly_continuous_on_def by simp

  3963

  3964 lemma uniformly_continuous_on_dist:

  3965   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  3966   assumes "uniformly_continuous_on s f"

  3967   assumes "uniformly_continuous_on s g"

  3968   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  3969 proof -

  3970   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  3971       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  3972       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  3973       by arith

  3974   } note le = this

  3975   { fix x y

  3976     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  3977     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  3978     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  3979       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  3980         simp add: le)

  3981   }

  3982   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  3983     unfolding dist_real_def by simp

  3984 qed

  3985

  3986 lemma uniformly_continuous_on_norm:

  3987   assumes "uniformly_continuous_on s f"

  3988   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  3989   unfolding norm_conv_dist using assms

  3990   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  3991

  3992 lemma (in bounded_linear) uniformly_continuous_on:

  3993   assumes "uniformly_continuous_on s g"

  3994   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  3995   using assms unfolding uniformly_continuous_on_sequentially

  3996   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  3997   by (auto intro: tendsto_zero)

  3998

  3999 lemma uniformly_continuous_on_cmul:

  4000   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4001   assumes "uniformly_continuous_on s f"

  4002   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4003   using bounded_linear_scaleR_right assms

  4004   by (rule bounded_linear.uniformly_continuous_on)

  4005

  4006 lemma dist_minus:

  4007   fixes x y :: "'a::real_normed_vector"

  4008   shows "dist (- x) (- y) = dist x y"

  4009   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4010

  4011 lemma uniformly_continuous_on_minus:

  4012   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4013   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4014   unfolding uniformly_continuous_on_def dist_minus .

  4015

  4016 lemma uniformly_continuous_on_add:

  4017   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4018   assumes "uniformly_continuous_on s f"

  4019   assumes "uniformly_continuous_on s g"

  4020   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4021   using assms unfolding uniformly_continuous_on_sequentially

  4022   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4023   by (auto intro: tendsto_add_zero)

  4024

  4025 lemma uniformly_continuous_on_diff:

  4026   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4027   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4028   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4029   unfolding ab_diff_minus using assms

  4030   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4031

  4032 text{* Continuity of all kinds is preserved under composition. *}

  4033

  4034 lemma continuous_within_topological:

  4035   "continuous (at x within s) f \<longleftrightarrow>

  4036     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  4037       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  4038 unfolding continuous_within

  4039 unfolding tendsto_def Limits.eventually_within eventually_at_topological

  4040 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  4041

  4042 lemma continuous_within_compose:

  4043   assumes "continuous (at x within s) f"

  4044   assumes "continuous (at (f x) within f  s) g"

  4045   shows "continuous (at x within s) (g o f)"

  4046 using assms unfolding continuous_within_topological by simp metis

  4047

  4048 lemma continuous_at_compose:

  4049   assumes "continuous (at x) f" and "continuous (at (f x)) g"

  4050   shows "continuous (at x) (g o f)"

  4051 proof-

  4052   have "continuous (at (f x) within range f) g" using assms(2)

  4053     using continuous_within_subset[of "f x" UNIV g "range f"] by simp

  4054   thus ?thesis using assms(1)

  4055     using continuous_within_compose[of x UNIV f g] by simp

  4056 qed

  4057

  4058 lemma continuous_on_compose:

  4059   "continuous_on s f \<Longrightarrow> continuous_on (f  s) g \<Longrightarrow> continuous_on s (g o f)"

  4060   unfolding continuous_on_topological by simp metis

  4061

  4062 lemma uniformly_continuous_on_compose:

  4063   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4064   shows "uniformly_continuous_on s (g o f)"

  4065 proof-

  4066   { fix e::real assume "e>0"

  4067     then obtain d where "d>0" and d:"\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto

  4068     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using d>0 using assms(1) unfolding uniformly_continuous_on_def by auto

  4069     hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using d>0 using d by auto  }

  4070   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4071 qed

  4072

  4073 lemmas continuous_on_intros = continuous_on_id continuous_on_const

  4074   continuous_on_compose continuous_on_norm continuous_on_infnorm

  4075   continuous_on_add continuous_on_minus continuous_on_diff

  4076   continuous_on_scaleR continuous_on_mult continuous_on_inverse

  4077   continuous_on_inner

  4078   uniformly_continuous_on_id uniformly_continuous_on_const

  4079   uniformly_continuous_on_dist uniformly_continuous_on_norm

  4080   uniformly_continuous_on_compose uniformly_continuous_on_add

  4081   uniformly_continuous_on_minus uniformly_continuous_on_diff

  4082   uniformly_continuous_on_cmul

  4083

  4084 text{* Continuity in terms of open preimages. *}

  4085

  4086 lemma continuous_at_open:

  4087   shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"

  4088 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  4089 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4090

  4091 lemma continuous_on_open:

  4092   shows "continuous_on s f \<longleftrightarrow>

  4093         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4094             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4095 proof (safe)

  4096   fix t :: "'b set"

  4097   assume 1: "continuous_on s f"

  4098   assume 2: "openin (subtopology euclidean (f  s)) t"

  4099   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  4100     unfolding openin_open by auto

  4101   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  4102   have "open U" unfolding U_def by (simp add: open_Union)

  4103   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  4104   proof (intro ballI iffI)

  4105     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  4106       unfolding U_def t by auto

  4107   next

  4108     fix x assume "x \<in> s" and "f x \<in> t"

  4109     hence "x \<in> s" and "f x \<in> B"

  4110       unfolding t by auto

  4111     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  4112       unfolding t continuous_on_topological by metis

  4113     then show "x \<in> U"

  4114       unfolding U_def by auto

  4115   qed

  4116   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  4117   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4118     unfolding openin_open by fast

  4119 next

  4120   assume "?rhs" show "continuous_on s f"

  4121   unfolding continuous_on_topological

  4122   proof (clarify)

  4123     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  4124     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  4125       unfolding openin_open using open B by auto

  4126     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  4127       using ?rhs by fast

  4128     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  4129       unfolding openin_open using x \<in> s and f x \<in> B by auto

  4130   qed

  4131 qed

  4132

  4133 text {* Similarly in terms of closed sets. *}

  4134

  4135 lemma continuous_on_closed:

  4136   shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f  s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4137 proof

  4138   assume ?lhs

  4139   { fix t

  4140     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4141     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  4142     assume as:"closedin (subtopology euclidean (f  s)) t"

  4143     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  4144     hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?lhs[unfolded continuous_on_open, THEN spec[where x="(f  s) - t"]]

  4145       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  4146   thus ?rhs by auto

  4147 next

  4148   assume ?rhs

  4149   { fix t

  4150     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4151     assume as:"openin (subtopology euclidean (f  s)) t"

  4152     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  4153       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  4154   thus ?lhs unfolding continuous_on_open by auto

  4155 qed

  4156

  4157 text {* Half-global and completely global cases. *}

  4158

  4159 lemma continuous_open_in_preimage:

  4160   assumes "continuous_on s f"  "open t"

  4161   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4162 proof-

  4163   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto

  4164   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4165     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4166   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4167 qed

  4168

  4169 lemma continuous_closed_in_preimage:

  4170   assumes "continuous_on s f"  "closed t"

  4171   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4172 proof-

  4173   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto

  4174   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4175     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4176   thus ?thesis

  4177     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4178 qed

  4179

  4180 lemma continuous_open_preimage:

  4181   assumes "continuous_on s f" "open s" "open t"

  4182   shows "open {x \<in> s. f x \<in> t}"

  4183 proof-

  4184   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4185     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4186   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4187 qed

  4188

  4189 lemma continuous_closed_preimage:

  4190   assumes "continuous_on s f" "closed s" "closed t"

  4191   shows "closed {x \<in> s. f x \<in> t}"

  4192 proof-

  4193   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4194     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4195   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4196 qed

  4197

  4198 lemma continuous_open_preimage_univ:

  4199   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4200   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4201

  4202 lemma continuous_closed_preimage_univ:

  4203   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4204   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4205

  4206 lemma continuous_open_vimage:

  4207   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4208   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4209

  4210 lemma continuous_closed_vimage:

  4211   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4212   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4213

  4214 lemma interior_image_subset:

  4215   assumes "\<forall>x. continuous (at x) f" "inj f"

  4216   shows "interior (f  s) \<subseteq> f  (interior s)"

  4217 proof

  4218   fix x assume "x \<in> interior (f  s)"

  4219   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4220   hence "x \<in> f  s" by auto

  4221   then obtain y where y: "y \<in> s" "x = f y" by auto

  4222   have "open (vimage f T)"

  4223     using assms(1) open T by (rule continuous_open_vimage)

  4224   moreover have "y \<in> vimage f T"

  4225     using x = f y x \<in> T by simp

  4226   moreover have "vimage f T \<subseteq> s"

  4227     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4228   ultimately have "y \<in> interior s" ..

  4229   with x = f y show "x \<in> f  interior s" ..

  4230 qed

  4231

  4232 text {* Equality of continuous functions on closure and related results. *}

  4233

  4234 lemma continuous_closed_in_preimage_constant:

  4235   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4236   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4237   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4238

  4239 lemma continuous_closed_preimage_constant:

  4240   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4241   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4242   using continuous_closed_preimage[of s f "{a}"] by auto

  4243

  4244 lemma continuous_constant_on_closure:

  4245   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4246   assumes "continuous_on (closure s) f"

  4247           "\<forall>x \<in> s. f x = a"

  4248   shows "\<forall>x \<in> (closure s). f x = a"

  4249     using continuous_closed_preimage_constant[of "closure s" f a]

  4250     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4251

  4252 lemma image_closure_subset:

  4253   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4254   shows "f  (closure s) \<subseteq> t"

  4255 proof-

  4256   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4257   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4258     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4259   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4260     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4261   thus ?thesis by auto

  4262 qed

  4263

  4264 lemma continuous_on_closure_norm_le:

  4265   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4266   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4267   shows "norm(f x) \<le> b"

  4268 proof-

  4269   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4270   show ?thesis

  4271     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4272     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4273 qed

  4274

  4275 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4276

  4277 lemma continuous_within_avoid:

  4278   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4279   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"

  4280   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4281 proof-

  4282   obtain d where "d>0" and d:"\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < dist (f x) a"

  4283     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto

  4284   { fix y assume " y\<in>s"  "dist x y < d"

  4285     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]

  4286       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }

  4287   thus ?thesis using d>0 by auto

  4288 qed

  4289

  4290 lemma continuous_at_avoid:

  4291   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4292   assumes "continuous (at x) f" and "f x \<noteq> a"

  4293   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4294   using assms continuous_within_avoid[of x UNIV f a] by simp

  4295

  4296 lemma continuous_on_avoid:

  4297   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4298   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4299   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4300 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(2,3) by auto

  4301

  4302 lemma continuous_on_open_avoid:

  4303   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4304   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4305   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4306 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(3,4) by auto

  4307

  4308 text {* Proving a function is constant by proving open-ness of level set. *}

  4309

  4310 lemma continuous_levelset_open_in_cases:

  4311   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4312   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4313         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4314         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4315 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4316

  4317 lemma continuous_levelset_open_in:

  4318   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4319   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4320         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4321         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4322 using continuous_levelset_open_in_cases[of s f ]

  4323 by meson

  4324

  4325 lemma continuous_levelset_open:

  4326   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4327   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4328   shows "\<forall>x \<in> s. f x = a"

  4329 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4330

  4331 text {* Some arithmetical combinations (more to prove). *}

  4332

  4333 lemma open_scaling[intro]:

  4334   fixes s :: "'a::real_normed_vector set"

  4335   assumes "c \<noteq> 0"  "open s"

  4336   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4337 proof-

  4338   { fix x assume "x \<in> s"

  4339     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto

  4340     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4341     moreover

  4342     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4343       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4344         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4345           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4346       hence "y \<in> op *\<^sub>R c  s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }

  4347     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s" apply(rule_tac x="e * abs c" in exI) by auto  }

  4348   thus ?thesis unfolding open_dist by auto

  4349 qed

  4350

  4351 lemma minus_image_eq_vimage:

  4352   fixes A :: "'a::ab_group_add set"

  4353   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4354   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4355

  4356 lemma open_negations:

  4357   fixes s :: "'a::real_normed_vector set"

  4358   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4359   unfolding scaleR_minus1_left [symmetric]

  4360   by (rule open_scaling, auto)

  4361

  4362 lemma open_translation:

  4363   fixes s :: "'a::real_normed_vector set"

  4364   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4365 proof-

  4366   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4367       by (intro continuous_diff continuous_at_id continuous_const) }

  4368   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4369   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4370 qed

  4371

  4372 lemma open_affinity:

  4373   fixes s :: "'a::real_normed_vector set"

  4374   assumes "open s"  "c \<noteq> 0"

  4375   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4376 proof-

  4377   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4378   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4379   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4380 qed

  4381

  4382 lemma interior_translation:

  4383   fixes s :: "'a::real_normed_vector set"

  4384   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4385 proof (rule set_eqI, rule)

  4386   fix x assume "x \<in> interior (op + a  s)"

  4387   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4388   hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto

  4389   thus "x \<in> op + a  interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using e > 0 by auto

  4390 next

  4391   fix x assume "x \<in> op + a  interior s"

  4392   then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto

  4393   { fix z have *:"a + y - z = y + a - z" by auto

  4394     assume "z\<in>ball x e"

  4395     hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto

  4396     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4397   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4398   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4399 qed

  4400

  4401 text {* Topological properties of linear functions. *}

  4402

  4403 lemma linear_lim_0:

  4404   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4405 proof-

  4406   interpret f: bounded_linear f by fact

  4407   have "(f ---> f 0) (at 0)"

  4408     using tendsto_ident_at by (rule f.tendsto)

  4409   thus ?thesis unfolding f.zero .

  4410 qed

  4411

  4412 lemma linear_continuous_at:

  4413   assumes "bounded_linear f"  shows "continuous (at a) f"

  4414   unfolding continuous_at using assms

  4415   apply (rule bounded_linear.tendsto)

  4416   apply (rule tendsto_ident_at)

  4417   done

  4418

  4419 lemma linear_continuous_within:

  4420   shows "bounded_linear f ==> continuous (at x within s) f"

  4421   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4422

  4423 lemma linear_continuous_on:

  4424   shows "bounded_linear f ==> continuous_on s f"

  4425   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4426

  4427 text {* Also bilinear functions, in composition form. *}

  4428

  4429 lemma bilinear_continuous_at_compose:

  4430   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4431         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4432   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4433

  4434 lemma bilinear_continuous_within_compose:

  4435   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4436         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4437   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4438

  4439 lemma bilinear_continuous_on_compose:

  4440   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4441              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4442   unfolding continuous_on_def

  4443   by (fast elim: bounded_bilinear.tendsto)

  4444

  4445 text {* Preservation of compactness and connectedness under continuous function. *}

  4446

  4447 lemma compact_continuous_image:

  4448   assumes "continuous_on s f"  "compact s"

  4449   shows "compact(f  s)"

  4450 proof-

  4451   { fix x assume x:"\<forall>n::nat. x n \<in> f  s"

  4452     then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto

  4453     then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto

  4454     { fix e::real assume "e>0"

  4455       then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=l], OF l\<in>s] by auto

  4456       then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded LIMSEQ_def, THEN spec[where x=d]] by auto

  4457       { fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto  }

  4458       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }

  4459     hence "\<exists>l\<in>f  s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding LIMSEQ_def using r lr l\<in>s by auto  }

  4460   thus ?thesis unfolding compact_def by auto

  4461 qed

  4462

  4463 lemma connected_continuous_image:

  4464   assumes "continuous_on s f"  "connected s"

  4465   shows "connected(f  s)"

  4466 proof-

  4467   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4468     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4469       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4470       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4471       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4472     hence False using as(1,2)

  4473       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4474   thus ?thesis unfolding connected_clopen by auto

  4475 qed

  4476

  4477 text {* Continuity implies uniform continuity on a compact domain. *}

  4478

  4479 lemma compact_uniformly_continuous:

  4480   assumes "continuous_on s f"  "compact s"

  4481   shows "uniformly_continuous_on s f"

  4482 proof-

  4483     { fix x assume x:"x\<in>s"

  4484       hence "\<forall>xa. \<exists>y. 0 < xa \<longrightarrow> (y > 0 \<and> (\<forall>x'\<in>s. dist x' x < y \<longrightarrow> dist (f x') (f x) < xa))" using assms(1)[unfolded continuous_on_iff, THEN bspec[where x=x]] by auto

  4485       hence "\<exists>fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)" using choice[of "\<lambda>e d. e>0 \<longrightarrow> d>0 \<and>(\<forall>x'\<in>s. (dist x' x < d \<longrightarrow> dist (f x') (f x) < e))"] by auto  }

  4486     then have "\<forall>x\<in>s. \<exists>y. \<forall>xa. 0 < xa \<longrightarrow> (\<forall>x'\<in>s. y xa > 0 \<and> (dist x' x < y xa \<longrightarrow> dist (f x') (f x) < xa))" by auto

  4487     then obtain d where d:"\<forall>e>0. \<forall>x\<in>s. \<forall>x'\<in>s. d x e > 0 \<and> (dist x' x < d x e \<longrightarrow> dist (f x') (f x) < e)"

  4488       using bchoice[of s "\<lambda>x fa. \<forall>xa>0. \<forall>x'\<in>s. fa xa > 0 \<and> (dist x' x < fa xa \<longrightarrow> dist (f x') (f x) < xa)"] by blast

  4489

  4490   { fix e::real assume "e>0"

  4491

  4492     { fix x assume "x\<in>s" hence "x \<in> ball x (d x (e / 2))" unfolding centre_in_ball using d[THEN spec[where x="e/2"]] using e>0 by auto  }

  4493     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto

  4494     moreover

  4495     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }

  4496     ultimately obtain ea where "ea>0" and ea:"\<forall>x\<in>s. \<exists>b\<in>{ball x (d x (e / 2)) |x. x \<in> s}. ball x ea \<subseteq> b" using heine_borel_lemma[OF assms(2), of "{ball x (d x (e / 2)) | x. x\<in>s }"] by auto

  4497

  4498     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"

  4499       obtain z where "z\<in>s" and z:"ball x ea \<subseteq> ball z (d z (e / 2))" using ea[THEN bspec[where x=x]] and x\<in>s by auto

  4500       hence "x\<in>ball z (d z (e / 2))" using ea>0 unfolding subset_eq by auto

  4501       hence "dist (f z) (f x) < e / 2" using d[THEN spec[where x="e/2"]] and e>0 and x\<in>s and z\<in>s

  4502         by (auto  simp add: dist_commute)

  4503       moreover have "y\<in>ball z (d z (e / 2))" using as and ea>0 and z[unfolded subset_eq]

  4504         by (auto simp add: dist_commute)

  4505       hence "dist (f z) (f y) < e / 2" using d[THEN spec[where x="e/2"]] and e>0 and y\<in>s and z\<in>s

  4506         by (auto  simp add: dist_commute)

  4507       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]

  4508         by (auto simp add: dist_commute)  }

  4509     then have "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e" using ea>0 by auto  }

  4510   thus ?thesis unfolding uniformly_continuous_on_def by auto

  4511 qed

  4512

  4513 text{* Continuity of inverse function on compact domain. *}

  4514

  4515 lemma continuous_on_inv:

  4516   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"

  4517     (* TODO: can this be generalized more? *)

  4518   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4519   shows "continuous_on (f  s) g"

  4520 proof-

  4521   have *:"g  f  s = s" using assms(3) by (auto simp add: image_iff)

  4522   { fix t assume t:"closedin (subtopology euclidean (g  f  s)) t"

  4523     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto

  4524     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]

  4525       unfolding T(2) and Int_left_absorb by auto

  4526     moreover have "compact (s \<inter> T)"

  4527       using assms(2) unfolding compact_eq_bounded_closed

  4528       using bounded_subset[of s "s \<inter> T"] and T(1) by auto

  4529     ultimately have "closed (f  t)" using T(1) unfolding T(2)

  4530       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto

  4531     moreover have "{x \<in> f  s. g x \<in> t} = f  s \<inter> f  t" using assms(3) unfolding T(2) by auto

  4532     ultimately have "closedin (subtopology euclidean (f  s)) {x \<in> f  s. g x \<in> t}"

  4533       unfolding closedin_closed by auto  }

  4534   thus ?thesis unfolding continuous_on_closed by auto

  4535 qed

  4536

  4537 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4538

  4539 lemma continuous_uniform_limit:

  4540   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4541   assumes "\<not> trivial_limit F"

  4542   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4543   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4544   shows "continuous_on s g"

  4545 proof-

  4546   { fix x and e::real assume "x\<in>s" "e>0"

  4547     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4548       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4549     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4550     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4551       using assms(1) by blast

  4552     have "e / 3 > 0" using e>0 by auto

  4553     then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"

  4554       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4555     { fix y assume "y \<in> s" and "dist y x < d"

  4556       hence "dist (f n y) (f n x) < e / 3"

  4557         by (rule d [rule_format])

  4558       hence "dist (f n y) (g x) < 2 * e / 3"

  4559         using dist_triangle [of "f n y" "g x" "f n x"]

  4560         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4561         by auto

  4562       hence "dist (g y) (g x) < e"

  4563         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4564         using dist_triangle3 [of "g y" "g x" "f n y"]

  4565         by auto }

  4566     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4567       using d>0 by auto }

  4568   thus ?thesis unfolding continuous_on_iff by auto

  4569 qed

  4570

  4571

  4572 subsection {* Topological stuff lifted from and dropped to R *}

  4573

  4574 lemma open_real:

  4575   fixes s :: "real set" shows

  4576  "open s \<longleftrightarrow>

  4577         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4578   unfolding open_dist dist_norm by simp

  4579

  4580 lemma islimpt_approachable_real:

  4581   fixes s :: "real set"

  4582   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4583   unfolding islimpt_approachable dist_norm by simp

  4584

  4585 lemma closed_real:

  4586   fixes s :: "real set"

  4587   shows "closed s \<longleftrightarrow>

  4588         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4589             --> x \<in> s)"

  4590   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4591

  4592 lemma continuous_at_real_range:

  4593   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4594   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4595         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4596   unfolding continuous_at unfolding Lim_at

  4597   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4598   apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto

  4599   apply(erule_tac x=e in allE) by auto

  4600

  4601 lemma continuous_on_real_range:

  4602   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4603   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"

  4604   unfolding continuous_on_iff dist_norm by simp

  4605

  4606 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4607

  4608 lemma compact_attains_sup:

  4609   fixes s :: "real set"

  4610   assumes "compact s"  "s \<noteq> {}"

  4611   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"

  4612 proof-

  4613   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4614   { fix e::real assume as: "\<forall>x\<in>s. x \<le> Sup s" "Sup s \<notin> s"  "0 < e" "\<forall>x'\<in>s. x' = Sup s \<or> \<not> Sup s - x' < e"

  4615     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto

  4616     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto

  4617     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using e>0 by auto  }

  4618   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]

  4619     apply(rule_tac x="Sup s" in bexI) by auto

  4620 qed

  4621

  4622 lemma Inf:

  4623   fixes S :: "real set"

  4624   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"

  4625 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)

  4626

  4627 lemma compact_attains_inf:

  4628   fixes s :: "real set"

  4629   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"

  4630 proof-

  4631   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4632   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"

  4633       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"

  4634     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto

  4635     moreover

  4636     { fix x assume "x \<in> s"

  4637       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto

  4638       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) x\<in>s unfolding * by auto }

  4639     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto

  4640     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using e>0 by auto  }

  4641   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]

  4642     apply(rule_tac x="Inf s" in bexI) by auto

  4643 qed

  4644

  4645 lemma continuous_attains_sup:

  4646   fixes f :: "'a::metric_space \<Rightarrow> real"

  4647   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4648         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"

  4649   using compact_attains_sup[of "f  s"]

  4650   using compact_continuous_image[of s f] by auto

  4651

  4652 lemma continuous_attains_inf:

  4653   fixes f :: "'a::metric_space \<Rightarrow> real"

  4654   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4655         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"

  4656   using compact_attains_inf[of "f  s"]

  4657   using compact_continuous_image[of s f] by auto

  4658

  4659 lemma distance_attains_sup:

  4660   assumes "compact s" "s \<noteq> {}"

  4661   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"

  4662 proof (rule continuous_attains_sup [OF assms])

  4663   { fix x assume "x\<in>s"

  4664     have "(dist a ---> dist a x) (at x within s)"

  4665       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  4666   }

  4667   thus "continuous_on s (dist a)"

  4668     unfolding continuous_on ..

  4669 qed

  4670

  4671 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4672

  4673 lemma distance_attains_inf:

  4674   fixes a :: "'a::heine_borel"

  4675   assumes "closed s"  "s \<noteq> {}"

  4676   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"

  4677 proof-

  4678   from assms(2) obtain b where "b\<in>s" by auto

  4679   let ?B = "cball a (dist b a) \<inter> s"

  4680   have "b \<in> ?B" using b\<in>s by (simp add: dist_commute)

  4681   hence "?B \<noteq> {}" by auto

  4682   moreover

  4683   { fix x assume "x\<in>?B"

  4684     fix e::real assume "e>0"

  4685     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"

  4686       from as have "\<bar>dist a x' - dist a x\<bar> < e"

  4687         unfolding abs_less_iff minus_diff_eq

  4688         using dist_triangle2 [of a x' x]

  4689         using dist_triangle [of a x x']

  4690         by arith

  4691     }

  4692     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"

  4693       using e>0 by auto

  4694   }

  4695   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"

  4696     unfolding continuous_on Lim_within dist_norm real_norm_def

  4697     by fast

  4698   moreover have "compact ?B"

  4699     using compact_cball[of a "dist b a"]

  4700     unfolding compact_eq_bounded_closed

  4701     using bounded_Int and closed_Int and assms(1) by auto

  4702   ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"

  4703     using continuous_attains_inf[of ?B "dist a"] by fastforce

  4704   thus ?thesis by fastforce

  4705 qed

  4706

  4707

  4708 subsection {* Pasted sets *}

  4709

  4710 lemma bounded_Times:

  4711   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4712 proof-

  4713   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4714     using assms [unfolded bounded_def] by auto

  4715   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4716     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4717   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4718 qed

  4719

  4720 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4721 by (induct x) simp

  4722

  4723 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"

  4724 unfolding compact_def

  4725 apply clarify

  4726 apply (drule_tac x="fst \<circ> f" in spec)

  4727 apply (drule mp, simp add: mem_Times_iff)

  4728 apply (clarify, rename_tac l1 r1)

  4729 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4730 apply (drule mp, simp add: mem_Times_iff)

  4731 apply (clarify, rename_tac l2 r2)

  4732 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4733 apply (rule_tac x="r1 \<circ> r2" in exI)

  4734 apply (rule conjI, simp add: subseq_def)

  4735 apply (drule_tac r=r2 in lim_subseq [rotated], assumption)

  4736 apply (drule (1) tendsto_Pair) back

  4737 apply (simp add: o_def)

  4738 done

  4739

  4740 text{* Hence some useful properties follow quite easily. *}

  4741

  4742 lemma compact_scaling:

  4743   fixes s :: "'a::real_normed_vector set"

  4744   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4745 proof-

  4746   let ?f = "\<lambda>x. scaleR c x"

  4747   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4748   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4749     using linear_continuous_at[OF *] assms by auto

  4750 qed

  4751

  4752 lemma compact_negations:

  4753   fixes s :: "'a::real_normed_vector set"

  4754   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4755   using compact_scaling [OF assms, of "- 1"] by auto

  4756

  4757 lemma compact_sums:

  4758   fixes s t :: "'a::real_normed_vector set"

  4759   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4760 proof-

  4761   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4762     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4763   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4764     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4765   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4766 qed

  4767

  4768 lemma compact_differences:

  4769   fixes s t :: "'a::real_normed_vector set"

  4770   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4771 proof-

  4772   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4773     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4774   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4775 qed

  4776

  4777 lemma compact_translation:

  4778   fixes s :: "'a::real_normed_vector set"

  4779   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4780 proof-

  4781   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4782   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4783 qed

  4784

  4785 lemma compact_affinity:

  4786   fixes s :: "'a::real_normed_vector set"

  4787   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4788 proof-

  4789   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4790   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4791 qed

  4792

  4793 text {* Hence we get the following. *}

  4794

  4795 lemma compact_sup_maxdistance:

  4796   fixes s :: "'a::real_normed_vector set"

  4797   assumes "compact s"  "s \<noteq> {}"

  4798   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"

  4799 proof-

  4800   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using s \<noteq> {} by auto

  4801   then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}"  "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"

  4802     using compact_differences[OF assms(1) assms(1)]

  4803     using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto

  4804   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto

  4805   thus ?thesis using x(2)[unfolded x = a - b] by blast

  4806 qed

  4807

  4808 text {* We can state this in terms of diameter of a set. *}

  4809

  4810 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"

  4811   (* TODO: generalize to class metric_space *)

  4812

  4813 lemma diameter_bounded:

  4814   assumes "bounded s"

  4815   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4816         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"

  4817 proof-

  4818   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"

  4819   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto

  4820   { fix x y assume "x \<in> s" "y \<in> s"

  4821     hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps)  }

  4822   note * = this

  4823   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto

  4824     have "norm(x - y) \<le> diameter s" unfolding diameter_def using s\<noteq>{} *[OF x\<in>s y\<in>s] x\<in>s y\<in>s

  4825       by simp (blast del: Sup_upper intro!: * Sup_upper) }

  4826   moreover

  4827   { fix d::real assume "d>0" "d < diameter s"

  4828     hence "s\<noteq>{}" unfolding diameter_def by auto

  4829     have "\<exists>d' \<in> ?D. d' > d"

  4830     proof(rule ccontr)

  4831       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"

  4832       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)

  4833       thus False using d < diameter s s\<noteq>{}

  4834         apply (auto simp add: diameter_def)

  4835         apply (drule Sup_real_iff [THEN [2] rev_iffD2])

  4836         apply (auto, force)

  4837         done

  4838     qed

  4839     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }

  4840   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4841         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto

  4842 qed

  4843

  4844 lemma diameter_bounded_bound:

  4845  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"

  4846   using diameter_bounded by blast

  4847

  4848 lemma diameter_compact_attained:

  4849   fixes s :: "'a::real_normed_vector set"

  4850   assumes "compact s"  "s \<noteq> {}"

  4851   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"

  4852 proof-

  4853   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4854   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto

  4855   hence "diameter s \<le> norm (x - y)"

  4856     unfolding diameter_def by clarsimp (rule Sup_least, fast+)

  4857   thus ?thesis

  4858     by (metis b diameter_bounded_bound order_antisym xys)

  4859 qed

  4860

  4861 text {* Related results with closure as the conclusion. *}

  4862

  4863 lemma closed_scaling:

  4864   fixes s :: "'a::real_normed_vector set"

  4865   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4866 proof(cases "s={}")

  4867   case True thus ?thesis by auto

  4868 next

  4869   case False

  4870   show ?thesis

  4871   proof(cases "c=0")

  4872     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4873     case True thus ?thesis apply auto unfolding * by auto

  4874   next

  4875     case False

  4876     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4877       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4878           using as(1)[THEN spec[where x=n]]

  4879           using c\<noteq>0 by auto

  4880       }

  4881       moreover

  4882       { fix e::real assume "e>0"

  4883         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4884         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4885           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4886         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4887           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4888           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4889       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4890       ultimately have "l \<in> scaleR c  s"

  4891         using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]

  4892         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4893     thus ?thesis unfolding closed_sequential_limits by fast

  4894   qed

  4895 qed

  4896

  4897 lemma closed_negations:

  4898   fixes s :: "'a::real_normed_vector set"

  4899   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4900   using closed_scaling[OF assms, of "- 1"] by simp

  4901

  4902 lemma compact_closed_sums:

  4903   fixes s :: "'a::real_normed_vector set"

  4904   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4905 proof-

  4906   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4907   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4908     from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"

  4909       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4910     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4911       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4912     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4913       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto

  4914     hence "l - l' \<in> t"

  4915       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4916       using f(3) by auto

  4917     hence "l \<in> ?S" using l' \<in> s apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto

  4918   }

  4919   thus ?thesis unfolding closed_sequential_limits by fast

  4920 qed

  4921

  4922 lemma closed_compact_sums:

  4923   fixes s t :: "'a::real_normed_vector set"

  4924   assumes "closed s"  "compact t"

  4925   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4926 proof-

  4927   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto

  4928     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4929   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4930 qed

  4931

  4932 lemma compact_closed_differences:

  4933   fixes s t :: "'a::real_normed_vector set"

  4934   assumes "compact s"  "closed t"

  4935   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4936 proof-

  4937   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4938     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4939   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4940 qed

  4941

  4942 lemma closed_compact_differences:

  4943   fixes s t :: "'a::real_normed_vector set"

  4944   assumes "closed s" "compact t"

  4945   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4946 proof-

  4947   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  4948     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4949  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  4950 qed

  4951

  4952 lemma closed_translation:

  4953   fixes a :: "'a::real_normed_vector"

  4954   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  4955 proof-

  4956   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  4957   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  4958 qed

  4959

  4960 lemma translation_Compl:

  4961   fixes a :: "'a::ab_group_add"

  4962   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  4963   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  4964

  4965 lemma translation_UNIV:

  4966   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  4967   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  4968

  4969 lemma translation_diff:

  4970   fixes a :: "'a::ab_group_add"

  4971   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  4972   by auto

  4973

  4974 lemma closure_translation:

  4975   fixes a :: "'a::real_normed_vector"

  4976   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  4977 proof-

  4978   have *:"op + a  (- s) = - op + a  s"

  4979     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  4980   show ?thesis unfolding closure_interior translation_Compl

  4981     using interior_translation[of a "- s"] unfolding * by auto

  4982 qed

  4983

  4984 lemma frontier_translation:

  4985   fixes a :: "'a::real_normed_vector"

  4986   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  4987   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  4988

  4989

  4990 subsection {* Separation between points and sets *}

  4991

  4992 lemma separate_point_closed:

  4993   fixes s :: "'a::heine_borel set"

  4994   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  4995 proof(cases "s = {}")

  4996   case True

  4997   thus ?thesis by(auto intro!: exI[where x=1])

  4998 next

  4999   case False

  5000   assume "closed s" "a \<notin> s"

  5001   then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using s \<noteq> {} distance_attains_inf [of s a] by blast

  5002   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5003 qed

  5004

  5005 lemma separate_compact_closed:

  5006   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5007     (* TODO: does this generalize to heine_borel? *)

  5008   assumes "compact s" and "closed t" and "s \<inter> t = {}"

  5009   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5010 proof-

  5011   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto

  5012   then obtain d where "d>0" and d:"\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. d \<le> dist 0 x"

  5013     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto

  5014   { fix x y assume "x\<in>s" "y\<in>t"

  5015     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto

  5016     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute

  5017       by (auto  simp add: dist_commute)

  5018     hence "d \<le> dist x y" unfolding dist_norm by auto  }

  5019   thus ?thesis using d>0 by auto

  5020 qed

  5021

  5022 lemma separate_closed_compact:

  5023   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5024   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5025   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5026 proof-

  5027   have *:"t \<inter> s = {}" using assms(3) by auto

  5028   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5029     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5030     by (auto simp add: dist_commute)

  5031 qed

  5032

  5033

  5034 subsection {* Intervals *}

  5035

  5036 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5037   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and

  5038   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5039   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5040

  5041 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5042   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5043   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  5044   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5045

  5046 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5047  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and

  5048  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  5049 proof-

  5050   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  5051     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto

  5052     hence "a\<bullet>i < b\<bullet>i" by auto

  5053     hence False using as by auto  }

  5054   moreover

  5055   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  5056     let ?x = "(1/2) *\<^sub>R (a + b)"

  5057     { fix i :: 'a assume i:"i\<in>Basis"

  5058       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5059       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  5060         by (auto simp: inner_add_left) }

  5061     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5062   ultimately show ?th1 by blast

  5063

  5064   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  5065     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto

  5066     hence "a\<bullet>i \<le> b\<bullet>i" by auto

  5067     hence False using as by auto  }

  5068   moreover

  5069   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  5070     let ?x = "(1/2) *\<^sub>R (a + b)"

  5071     { fix i :: 'a assume i:"i\<in>Basis"

  5072       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5073       hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"

  5074         by (auto simp: inner_add_left) }

  5075     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5076   ultimately show ?th2 by blast

  5077 qed

  5078

  5079 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5080   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and

  5081   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  5082   unfolding interval_eq_empty[of a b] by fastforce+

  5083

  5084 lemma interval_sing:

  5085   fixes a :: "'a::ordered_euclidean_space"

  5086   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5087   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5088   by (auto intro: euclidean_eqI simp: ex_in_conv)

  5089

  5090 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5091  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  5092  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5093  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  5094  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  5095   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5096   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5097

  5098 lemma interval_open_subset_closed:

  5099   fixes a :: "'a::ordered_euclidean_space"

  5100   shows "{a<..<b} \<subseteq> {a .. b}"

  5101   unfolding subset_eq [unfolded Ball_def] mem_interval

  5102   by (fast intro: less_imp_le)

  5103

  5104 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5105  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and

  5106  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and

  5107  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and

  5108  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)

  5109 proof-

  5110   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5111   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)

  5112   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5113     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5114     fix i :: 'a assume i:"i\<in>Basis"

  5115     (** TODO combine the following two parts as done in the HOL_light version. **)

  5116     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5117       assume as2: "a\<bullet>i > c\<bullet>i"

  5118       { fix j :: 'a assume j:"j\<in>Basis"

  5119         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  5120           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i

  5121           by (auto simp add: as2)  }

  5122       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto

  5123       moreover

  5124       have "?x\<notin>{a .. b}"

  5125         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5126         using as(2)[THEN bspec[where x=i]] and as2 i

  5127         by auto

  5128       ultimately have False using as by auto  }

  5129     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto

  5130     moreover

  5131     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5132       assume as2: "b\<bullet>i < d\<bullet>i"

  5133       { fix j :: 'a assume "j\<in>Basis"

  5134         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  5135           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]

  5136           by (auto simp add: as2) }

  5137       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5138       moreover

  5139       have "?x\<notin>{a .. b}"

  5140         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5141         using as(2)[THEN bspec[where x=i]] and as2 using i

  5142         by auto

  5143       ultimately have False using as by auto  }

  5144     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto

  5145     ultimately

  5146     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  5147   } note part1 = this

  5148   show ?th3

  5149     unfolding subset_eq and Ball_def and mem_interval

  5150     apply(rule,rule,rule,rule)

  5151     apply(rule part1)

  5152     unfolding subset_eq and Ball_def and mem_interval

  5153     prefer 4

  5154     apply auto

  5155     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+

  5156   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5157     fix i :: 'a assume i:"i\<in>Basis"

  5158     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5159     hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this

  5160   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5161     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5162     apply auto by(erule_tac x=xa in allE, simp)+

  5163 qed

  5164

  5165 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5166  "{a .. b} \<inter> {c .. d} =  {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"

  5167   unfolding set_eq_iff and Int_iff and mem_interval by auto

  5168

  5169 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5170   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and

  5171   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and

  5172   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and

  5173   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)

  5174 proof-

  5175   let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"

  5176   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  5177       (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"

  5178     by blast

  5179   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

  5180   show ?th1 unfolding * by (intro **) auto

  5181   show ?th2 unfolding * by (intro **) auto

  5182   show ?th3 unfolding * by (intro **) auto

  5183   show ?th4 unfolding * by (intro **) auto

  5184 qed

  5185

  5186 (* Moved interval_open_subset_closed a bit upwards *)

  5187

  5188 lemma open_interval[intro]:

  5189   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5190 proof-

  5191   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"

  5192     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

  5193       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

  5194   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"

  5195     by (auto simp add: eucl_less [where 'a='a])

  5196   finally show "open {a<..<b}" .

  5197 qed

  5198

  5199 lemma closed_interval[intro]:

  5200   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5201 proof-

  5202   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"

  5203     by (intro closed_INT ballI continuous_closed_vimage allI

  5204       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

  5205   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"

  5206     by (auto simp add: eucl_le [where 'a='a])

  5207   finally show "closed {a .. b}" .

  5208 qed

  5209

  5210 lemma interior_closed_interval [intro]:

  5211   fixes a b :: "'a::ordered_euclidean_space"

  5212   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5213 proof(rule subset_antisym)

  5214   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5215     by (rule interior_maximal)

  5216 next

  5217   { fix x assume "x \<in> interior {a..b}"

  5218     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5219     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto

  5220     { fix i :: 'a assume i:"i\<in>Basis"

  5221       have "dist (x - (e / 2) *\<^sub>R i) x < e"

  5222            "dist (x + (e / 2) *\<^sub>R i) x < e"

  5223         unfolding dist_norm apply auto

  5224         unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto

  5225       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"

  5226                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"

  5227         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]

  5228         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]

  5229         unfolding mem_interval using i by blast+

  5230       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"

  5231         using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }

  5232     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5233   thus "?L \<subseteq> ?R" ..

  5234 qed

  5235

  5236 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5237 proof-

  5238   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"

  5239   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  5240     { fix i :: 'a assume "i\<in>Basis"

  5241       hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }

  5242     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5243     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5244   thus ?thesis unfolding interval and bounded_iff by auto

  5245 qed

  5246

  5247 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5248  "bounded {a .. b} \<and> bounded {a<..<b}"

  5249   using bounded_closed_interval[of a b]

  5250   using interval_open_subset_closed[of a b]

  5251   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5252   by simp

  5253

  5254 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5255  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5256   using bounded_interval[of a b] by auto

  5257

  5258 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5259   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]

  5260   by auto

  5261

  5262 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5263   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5264 proof-

  5265   { fix i :: 'a assume "i\<in>Basis"

  5266     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"

  5267       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }

  5268   thus ?thesis unfolding mem_interval by auto

  5269 qed

  5270

  5271 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5272   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5273   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5274 proof-

  5275   { fix i :: 'a assume i:"i\<in>Basis"

  5276     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp

  5277     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5278       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5279       using x unfolding mem_interval using i apply simp

  5280       using y unfolding mem_interval using i apply simp

  5281       done

  5282     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto

  5283     moreover {

  5284     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp

  5285     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5286       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5287       using x unfolding mem_interval using i apply simp

  5288       using y unfolding mem_interval using i apply simp

  5289       done

  5290     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto

  5291     } ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }

  5292   thus ?thesis unfolding mem_interval by auto

  5293 qed

  5294

  5295 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5296   assumes "{a<..<b} \<noteq> {}"

  5297   shows "closure {a<..<b} = {a .. b}"

  5298 proof-

  5299   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5300   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5301   { fix x assume as:"x \<in> {a .. b}"

  5302     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5303     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5304       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5305       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5306         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5307         by (auto simp add: algebra_simps)

  5308       hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto

  5309       hence False using fn unfolding f_def using xc by auto  }

  5310     moreover

  5311     { assume "\<not> (f ---> x) sequentially"

  5312       { fix e::real assume "e>0"

  5313         hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto

  5314         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5315         hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)

  5316         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5317       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5318         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5319       hence "(f ---> x) sequentially" unfolding f_def

  5320         using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]

  5321         using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }

  5322     ultimately have "x \<in> closure {a<..<b}"

  5323       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5324   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5325 qed

  5326

  5327 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5328   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5329 proof-

  5330   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5331   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"

  5332   { fix x assume "x\<in>s"

  5333     fix i :: 'a assume i:"i\<in>Basis"

  5334     hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF x\<in>s]

  5335       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }

  5336   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5337 qed

  5338

  5339 lemma bounded_subset_open_interval:

  5340   fixes s :: "('a::ordered_euclidean_space) set"

  5341   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5342   by (auto dest!: bounded_subset_open_interval_symmetric)

  5343

  5344 lemma bounded_subset_closed_interval_symmetric:

  5345   fixes s :: "('a::ordered_euclidean_space) set"

  5346   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5347 proof-

  5348   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5349   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5350 qed

  5351

  5352 lemma bounded_subset_closed_interval:

  5353   fixes s :: "('a::ordered_euclidean_space) set"

  5354   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5355   using bounded_subset_closed_interval_symmetric[of s] by auto

  5356

  5357 lemma frontier_closed_interval:

  5358   fixes a b :: "'a::ordered_euclidean_space"

  5359   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5360   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5361

  5362 lemma frontier_open_interval:

  5363   fixes a b :: "'a::ordered_euclidean_space"

  5364   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5365 proof(cases "{a<..<b} = {}")

  5366   case True thus ?thesis using frontier_empty by auto

  5367 next

  5368   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5369 qed

  5370

  5371 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5372   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5373   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5374

  5375

  5376 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5377

  5378 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5379   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"

  5380 proof-

  5381   { fix i :: 'a assume i:"i\<in>Basis"

  5382     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"

  5383     { assume "x\<bullet>i > b\<bullet>i"

  5384       then obtain y where "y \<bullet> i \<le> b \<bullet> i"  "y \<noteq> x"  "dist y x < x\<bullet>i - b\<bullet>i"

  5385         using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto

  5386       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i

  5387         by auto }

  5388     hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto  }

  5389   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5390 qed

  5391

  5392 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5393   shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"

  5394 proof-

  5395   { fix i :: 'a assume i:"i\<in>Basis"

  5396     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"

  5397     { assume "a\<bullet>i > x\<bullet>i"

  5398       then obtain y where "a \<bullet> i \<le> y \<bullet> i"  "y \<noteq> x"  "dist y x < a\<bullet>i - x\<bullet>i"

  5399         using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto

  5400       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }

  5401     hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto  }

  5402   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5403 qed

  5404

  5405 lemma open_box: "open (box a b)"

  5406 proof -

  5407   have "open (\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i})"

  5408     by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)

  5409   also have "(\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i}) = box a b"

  5410     by (auto simp add: box_def inner_commute)

  5411   finally show ?thesis .

  5412 qed

  5413

  5414 instance euclidean_space \<subseteq> second_countable_topology

  5415 proof

  5416   def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"

  5417   then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp

  5418   def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"

  5419   then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp

  5420   def B \<equiv> "(\<lambda>f. box (a f) (b f))  (Basis \<rightarrow>\<^isub>E (\<rat> \<times> \<rat>))"

  5421

  5422   have "countable B" unfolding B_def

  5423     by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)

  5424   moreover

  5425   have "Ball B open" by (simp add: B_def open_box)

  5426   moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"

  5427   proof safe

  5428     fix A::"'a set" assume "open A"

  5429     show "\<exists>B'\<subseteq>B. \<Union>B' = A"

  5430       apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])

  5431       apply (subst (3) open_UNION_box[OF open A])

  5432       apply (auto simp add: a b B_def)

  5433       done

  5434   qed

  5435   ultimately

  5436   show "\<exists>B::'a set set. countable B \<and> topological_basis B" unfolding topological_basis_def by blast

  5437 qed

  5438

  5439 instance ordered_euclidean_space \<subseteq> polish_space ..

  5440

  5441 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5442

  5443 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5444   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"

  5445

  5446 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5447   "is_interval {a<..<b}" (is ?th2) proof -

  5448   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5449     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5450

  5451 lemma is_interval_empty:

  5452  "is_interval {}"

  5453   unfolding is_interval_def

  5454   by simp

  5455

  5456 lemma is_interval_univ:

  5457  "is_interval UNIV"

  5458   unfolding is_interval_def

  5459   by simp

  5460

  5461

  5462 subsection {* Closure of halfspaces and hyperplanes *}

  5463

  5464 lemma isCont_open_vimage:

  5465   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5466 proof -

  5467   from assms(1) have "continuous_on UNIV f"

  5468     unfolding isCont_def continuous_on_def within_UNIV by simp

  5469   hence "open {x \<in> UNIV. f x \<in> s}"

  5470     using open_UNIV open s by (rule continuous_open_preimage)

  5471   thus "open (f - s)"

  5472     by (simp add: vimage_def)

  5473 qed

  5474

  5475 lemma isCont_closed_vimage:

  5476   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5477   using assms unfolding closed_def vimage_Compl [symmetric]

  5478   by (rule isCont_open_vimage)

  5479

  5480 lemma open_Collect_less:

  5481   fixes f g :: "'a::topological_space \<Rightarrow> real"

  5482   assumes f: "\<And>x. isCont f x"

  5483   assumes g: "\<And>x. isCont g x"

  5484   shows "open {x. f x < g x}"

  5485 proof -

  5486   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5487     using isCont_diff [OF g f] open_real_greaterThan

  5488     by (rule isCont_open_vimage)

  5489   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5490     by auto

  5491   finally show ?thesis .

  5492 qed

  5493

  5494 lemma closed_Collect_le:

  5495   fixes f g :: "'a::topological_space \<Rightarrow> real"

  5496   assumes f: "\<And>x. isCont f x"

  5497   assumes g: "\<And>x. isCont g x"

  5498   shows "closed {x. f x \<le> g x}"

  5499 proof -

  5500   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5501     using isCont_diff [OF g f] closed_real_atLeast

  5502     by (rule isCont_closed_vimage)

  5503   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5504     by auto

  5505   finally show ?thesis .

  5506 qed

  5507

  5508 lemma closed_Collect_eq:

  5509   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  5510   assumes f: "\<And>x. isCont f x"

  5511   assumes g: "\<And>x. isCont g x"

  5512   shows "closed {x. f x = g x}"

  5513 proof -

  5514   have "open {(x::'b, y::'b). x \<noteq> y}"

  5515     unfolding open_prod_def by (auto dest!: hausdorff)

  5516   hence "closed {(x::'b, y::'b). x = y}"

  5517     unfolding closed_def split_def Collect_neg_eq .

  5518   with isCont_Pair [OF f g]

  5519   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5520     by (rule isCont_closed_vimage)

  5521   also have "\<dots> = {x. f x = g x}" by auto

  5522   finally show ?thesis .

  5523 qed

  5524

  5525 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5526   unfolding continuous_at by (intro tendsto_intros)

  5527

  5528 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"

  5529   by (simp add: closed_Collect_le)

  5530

  5531 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

  5532   by (simp add: closed_Collect_le)

  5533

  5534 lemma closed_hyperplane: "closed {x. inner a x = b}"

  5535   by (simp add: closed_Collect_eq)

  5536

  5537 lemma closed_halfspace_component_le:

  5538   shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

  5539   by (simp add: closed_Collect_le)

  5540

  5541 lemma closed_halfspace_component_ge:

  5542   shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

  5543   by (simp add: closed_Collect_le)

  5544

  5545 text {* Openness of halfspaces. *}

  5546

  5547 lemma open_halfspace_lt: "open {x. inner a x < b}"

  5548   by (simp add: open_Collect_less)

  5549

  5550 lemma open_halfspace_gt: "open {x. inner a x > b}"

  5551   by (simp add: open_Collect_less)

  5552

  5553 lemma open_halfspace_component_lt:

  5554   shows "open {x::'a::euclidean_space. x\<bullet>i < a}"

  5555   by (simp add: open_Collect_less)

  5556

  5557 lemma open_halfspace_component_gt:

  5558   shows "open {x::'a::euclidean_space. x\<bullet>i > a}"

  5559   by (simp add: open_Collect_less)

  5560

  5561 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

  5562

  5563 lemma eucl_lessThan_eq_halfspaces:

  5564   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5565   shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"

  5566  by (auto simp: eucl_less[where 'a='a])

  5567

  5568 lemma eucl_greaterThan_eq_halfspaces:

  5569   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5570   shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"

  5571  by (auto simp: eucl_less[where 'a='a])

  5572

  5573 lemma eucl_atMost_eq_halfspaces:

  5574   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5575   shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"

  5576  by (auto simp: eucl_le[where 'a='a])

  5577

  5578 lemma eucl_atLeast_eq_halfspaces:

  5579   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5580   shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"

  5581  by (auto simp: eucl_le[where 'a='a])

  5582

  5583 lemma open_eucl_lessThan[simp, intro]:

  5584   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5585   shows "open {..< a}"

  5586   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

  5587

  5588 lemma open_eucl_greaterThan[simp, intro]:

  5589   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5590   shows "open {a <..}"

  5591   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

  5592

  5593 lemma closed_eucl_atMost[simp, intro]:

  5594   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5595   shows "closed {.. a}"

  5596   unfolding eucl_atMost_eq_halfspaces

  5597   by (simp add: closed_INT closed_Collect_le)

  5598

  5599 lemma closed_eucl_atLeast[simp, intro]:

  5600   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5601   shows "closed {a ..}"

  5602   unfolding eucl_atLeast_eq_halfspaces

  5603   by (simp add: closed_INT closed_Collect_le)

  5604

  5605 text {* This gives a simple derivation of limit component bounds. *}

  5606

  5607 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5608   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"

  5609   shows "l\<bullet>i \<le> b"

  5610   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

  5611

  5612 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5613   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"

  5614   shows "b \<le> l\<bullet>i"

  5615   by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

  5616

  5617 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5618   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"

  5619   shows "l\<bullet>i = b"

  5620   using ev[unfolded order_eq_iff eventually_conj_iff]

  5621   using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto

  5622

  5623 text{* Limits relative to a union.                                               *}

  5624

  5625 lemma eventually_within_Un:

  5626   "eventually P (net within (s \<union> t)) \<longleftrightarrow>

  5627     eventually P (net within s) \<and> eventually P (net within t)"

  5628   unfolding Limits.eventually_within

  5629   by (auto elim!: eventually_rev_mp)

  5630

  5631 lemma Lim_within_union:

  5632  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>

  5633   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"

  5634   unfolding tendsto_def

  5635   by (auto simp add: eventually_within_Un)

  5636

  5637 lemma Lim_topological:

  5638  "(f ---> l) net \<longleftrightarrow>

  5639         trivial_limit net \<or>

  5640         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5641   unfolding tendsto_def trivial_limit_eq by auto

  5642

  5643 lemma continuous_on_union:

  5644   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"

  5645   shows "continuous_on (s \<union> t) f"

  5646   using assms unfolding continuous_on Lim_within_union

  5647   unfolding Lim_topological trivial_limit_within closed_limpt by auto

  5648

  5649 lemma continuous_on_cases:

  5650   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"

  5651           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"

  5652   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  5653 proof-

  5654   let ?h = "(\<lambda>x. if P x then f x else g x)"

  5655   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto

  5656   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto

  5657   moreover

  5658   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto

  5659   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto

  5660   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto

  5661 qed

  5662

  5663

  5664 text{* Some more convenient intermediate-value theorem formulations.             *}

  5665

  5666 lemma connected_ivt_hyperplane:

  5667   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5668   shows "\<exists>z \<in> s. inner a z = b"

  5669 proof(rule ccontr)

  5670   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5671   let ?A = "{x. inner a x < b}"

  5672   let ?B = "{x. inner a x > b}"

  5673   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5674   moreover have "?A \<inter> ?B = {}" by auto

  5675   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5676   ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto

  5677 qed

  5678

  5679 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5680  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s.  z\<bullet>k = a)"

  5681   using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute)

  5682

  5683

  5684 subsection {* Homeomorphisms *}

  5685

  5686 definition "homeomorphism s t f g \<equiv>

  5687      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5688      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5689

  5690 definition

  5691   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"

  5692     (infixr "homeomorphic" 60) where

  5693   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  5694

  5695 lemma homeomorphic_refl: "s homeomorphic s"

  5696   unfolding homeomorphic_def

  5697   unfolding homeomorphism_def

  5698   using continuous_on_id

  5699   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5700   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5701   by blast

  5702

  5703 lemma homeomorphic_sym:

  5704  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  5705 unfolding homeomorphic_def

  5706 unfolding homeomorphism_def

  5707 by blast

  5708

  5709 lemma homeomorphic_trans:

  5710   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"

  5711 proof-

  5712   obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x"  "f1  s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1  t = s" "continuous_on t g1"

  5713     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto

  5714   obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x"  "f2  t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2  u = t" "continuous_on u g2"

  5715     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto

  5716

  5717   { fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }

  5718   moreover have "(f2 \<circ> f1)  s = u" using fg1(2) fg2(2) by auto

  5719   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto

  5720   moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }

  5721   moreover have "(g1 \<circ> g2)  u = s" using fg1(5) fg2(5) by auto

  5722   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto

  5723   ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto

  5724 qed

  5725

  5726 lemma homeomorphic_minimal:

  5727  "s homeomorphic t \<longleftrightarrow>

  5728     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) =