src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author immler Tue Nov 27 13:48:40 2012 +0100 (2012-11-27) changeset 50245 dea9363887a6 parent 50105 65d5b18e1626 child 50324 0a1242d5e7d4 permissions -rw-r--r--
based countable topological basis on Countable_Set
     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   Norm_Arith

    17 begin

    18

    19 subsection {* Topological Basis *}

    20

    21 context topological_space

    22 begin

    23

    24 definition "topological_basis B =

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

    26

    27 lemma topological_basis_iff:

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

    29   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'))"

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

    31 proof safe

    32   fix O' and x::'a

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

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

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

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

    37 next

    38   assume H: ?rhs

    39   show "topological_basis B" using assms unfolding topological_basis_def

    40   proof safe

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

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

    43       by (force intro: bchoice simp: Bex_def)

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

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

    46   qed

    47 qed

    48

    49 lemma topological_basisI:

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

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

    52   shows "topological_basis B"

    53   using assms by (subst topological_basis_iff) auto

    54

    55 lemma topological_basisE:

    56   fixes O'

    57   assumes "topological_basis B"

    58   assumes "open O'"

    59   assumes "x \<in> O'"

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

    61 proof atomize_elim

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

    63   with topological_basis_iff assms

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

    65 qed

    66

    67 lemma topological_basis_open:

    68   assumes "topological_basis B"

    69   assumes "X \<in> B"

    70   shows "open X"

    71   using assms

    72   by (simp add: topological_basis_def)

    73

    74 lemma basis_dense:

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

    76   assumes "topological_basis B"

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

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

    79 proof (intro allI impI)

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

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

    82   guess B' . note B' = this

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

    84 qed

    85

    86 end

    87

    88 subsection {* Countable Basis *}

    89

    90 locale countable_basis =

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

    92   assumes is_basis: "topological_basis B"

    93   assumes countable_basis: "countable B"

    94 begin

    95

    96 lemma open_countable_basis_ex:

    97   assumes "open X"

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

    99   using assms countable_basis is_basis unfolding topological_basis_def by blast

   100

   101 lemma open_countable_basisE:

   102   assumes "open X"

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

   104   using assms open_countable_basis_ex by (atomize_elim) simp

   105

   106 lemma countable_dense_exists:

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

   108 proof -

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

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

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

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

   113 qed

   114

   115 lemma countable_dense_setE:

   116   obtains D :: "'a set"

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

   118   using countable_dense_exists by blast

   119

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

   121   therefore basis which is closed under union. *}

   122

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

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

   125

   126 lemma basis_union_closed_basis: "topological_basis union_closed_basis"

   127 proof (rule topological_basisI)

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

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

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

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

   132 next

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

   134   thus "open B'"

   135     using topological_basis_open[OF is_basis]

   136     by (auto simp: union_closed_basis_def)

   137 qed

   138

   139 lemma countable_union_closed_basis: "countable union_closed_basis"

   140   unfolding union_closed_basis_def using countable_basis by simp

   141

   142 lemmas open_union_closed_basis = topological_basis_open[OF basis_union_closed_basis]

   143

   144 lemma union_closed_basis_ex:

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

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

   147 proof -

   148   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)

   149   thus ?thesis by auto

   150 qed

   151

   152 lemma union_closed_basisE:

   153   assumes "X \<in> union_closed_basis"

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

   155

   156 lemma union_closed_basisI:

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

   158   shows "X \<in> union_closed_basis"

   159 proof -

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

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

   162 qed

   163

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

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

   166

   167 lemma union_basisI[intro]:

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

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

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

   171

   172 lemma open_imp_Union_of_incseq:

   173   assumes "open X"

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

   175 proof -

   176   from open_countable_basis_ex[OF open X]

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

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

   179   show ?thesis

   180   proof cases

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

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

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

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

   185     moreover

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

   187     proof safe

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

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

   190         by (metis countable B' from_nat_into_surj)

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

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

   193     next

   194       fix x n

   195       assume "x \<in> S n"

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

   197         by (simp add: S_def)

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

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

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

   201     qed

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

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

   204     ultimately show ?thesis by auto

   205   qed (auto simp: B')

   206 qed

   207

   208 lemma open_incseqE:

   209   assumes "open X"

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

   211   using open_imp_Union_of_incseq assms by atomize_elim

   212

   213 end

   214

   215 class countable_basis_space = topological_space +

   216   assumes ex_countable_basis:

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

   218

   219 sublocale countable_basis_space < countable_basis "SOME B. countable B \<and> topological_basis B"

   220   using someI_ex[OF ex_countable_basis] by unfold_locales safe

   221

   222 subsection {* Polish spaces *}

   223

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

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

   226

   227 class polish_space = complete_space + countable_basis_space

   228

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

   230

   231 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))"

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

   233   morphisms "openin" "topology"

   234   unfolding istopology_def by blast

   235

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

   237   using openin[of U] by blast

   238

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

   240   using topology_inverse[unfolded mem_Collect_eq] .

   241

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

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

   244

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

   246 proof-

   247   { assume "T1=T2"

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

   249   moreover

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

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

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

   253     hence "T1 = T2" unfolding openin_inverse .

   254   }

   255   ultimately show ?thesis by blast

   256 qed

   257

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

   259

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

   261

   262 subsubsection {* Main properties of open sets *}

   263

   264 lemma openin_clauses:

   265   fixes U :: "'a topology"

   266   shows "openin U {}"

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

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

   269   using openin[of U] unfolding istopology_def mem_Collect_eq

   270   by fast+

   271

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

   273   unfolding topspace_def by blast

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

   275

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

   277   using openin_clauses by simp

   278

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

   280   using openin_clauses by simp

   281

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

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

   284

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

   286

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

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

   289 proof

   290   assume ?lhs

   291   then show ?rhs by auto

   292 next

   293   assume H: ?rhs

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

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

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

   297   finally show "openin U S" .

   298 qed

   299

   300

   301 subsubsection {* Closed sets *}

   302

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

   304

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

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

   307 lemma closedin_topspace[intro,simp]:

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

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

   310   by (auto simp add: Diff_Un closedin_def)

   311

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

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

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

   315

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

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

   318

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

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

   321   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   322   apply (metis openin_subset subset_eq)

   323   done

   324

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

   326   by (simp add: openin_closedin_eq)

   327

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

   329 proof-

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

   331     by (auto simp add: topspace_def openin_subset)

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

   333 qed

   334

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

   336 proof-

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

   338     by (auto simp add: topspace_def )

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

   340 qed

   341

   342 subsubsection {* Subspace topology *}

   343

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

   345

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

   347   (is "istopology ?L")

   348 proof-

   349   have "?L {}" by blast

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

   351     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

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

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

   354   moreover

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

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

   357       apply (rule set_eqI)

   358       apply (simp add: Ball_def image_iff)

   359       by metis

   360     from K[unfolded th0 subset_image_iff]

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

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

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

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

   365   ultimately show ?thesis

   366     unfolding subset_eq mem_Collect_eq istopology_def by blast

   367 qed

   368

   369 lemma openin_subtopology:

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

   371   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   372   by auto

   373

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

   375   by (auto simp add: topspace_def openin_subtopology)

   376

   377 lemma closedin_subtopology:

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

   379   unfolding closedin_def topspace_subtopology

   380   apply (simp add: openin_subtopology)

   381   apply (rule iffI)

   382   apply clarify

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

   384   by auto

   385

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

   387   unfolding openin_subtopology

   388   apply (rule iffI, clarify)

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

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

   391   apply auto

   392   done

   393

   394 lemma subtopology_superset:

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

   396   shows "subtopology U V = U"

   397 proof-

   398   {fix S

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

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

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

   402     moreover

   403     {assume S: "openin U S"

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

   405         using openin_subset[OF S] UV by auto}

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

   407   then show ?thesis unfolding topology_eq openin_subtopology by blast

   408 qed

   409

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

   411   by (simp add: subtopology_superset)

   412

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

   414   by (simp add: subtopology_superset)

   415

   416 subsubsection {* The standard Euclidean topology *}

   417

   418 definition

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

   420   "euclidean = topology open"

   421

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

   423   unfolding euclidean_def

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

   425   apply (rule topology_inverse[symmetric])

   426   apply (auto simp add: istopology_def)

   427   done

   428

   429 lemma topspace_euclidean: "topspace euclidean = UNIV"

   430   apply (simp add: topspace_def)

   431   apply (rule set_eqI)

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

   433

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

   435   by (simp add: topspace_euclidean topspace_subtopology)

   436

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

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

   439

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

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

   442

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

   444

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

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

   447

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

   449   by (auto simp add: openin_open)

   450

   451 lemma open_openin_trans[trans]:

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

   453   by (metis Int_absorb1  openin_open_Int)

   454

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

   456   by (auto simp add: openin_open)

   457

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

   459   by (simp add: closedin_subtopology closed_closedin Int_ac)

   460

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

   462   by (metis closedin_closed)

   463

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

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

   466   apply auto

   467   apply (frule closedin_closed_Int[of T S])

   468   by simp

   469

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

   471   by (auto simp add: closedin_closed)

   472

   473 lemma openin_euclidean_subtopology_iff:

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

   475   shows "openin (subtopology euclidean U) S

   476   \<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")

   477 proof

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

   479 next

   480   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}"

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

   482     unfolding T_def

   483     apply clarsimp

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

   485     apply (clarsimp simp add: less_diff_eq)

   486     apply (erule rev_bexI)

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

   488     apply (erule le_less_trans [OF dist_triangle])

   489     done

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

   491     unfolding T_def

   492     apply auto

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

   494     apply auto

   495     done

   496   from 1 2 show ?lhs

   497     unfolding openin_open open_dist by fast

   498 qed

   499

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

   501

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

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

   504   unfolding open_openin openin_open by blast

   505

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

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

   508

   509 lemma closedin_trans[trans]:

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

   511            closedin (subtopology euclidean U) T

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

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

   514

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

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

   517

   518

   519 subsection {* Open and closed balls *}

   520

   521 definition

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

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

   524

   525 definition

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

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

   528

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

   530   by (simp add: ball_def)

   531

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

   533   by (simp add: cball_def)

   534

   535 lemma mem_ball_0:

   536   fixes x :: "'a::real_normed_vector"

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

   538   by (simp add: dist_norm)

   539

   540 lemma mem_cball_0:

   541   fixes x :: "'a::real_normed_vector"

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

   543   by (simp add: dist_norm)

   544

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

   546   by simp

   547

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

   549   by simp

   550

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

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

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

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

   555   by (simp add: set_eq_iff) arith

   556

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

   558   by (simp add: set_eq_iff)

   559

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

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

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

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

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

   565

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

   567   unfolding open_dist ball_def mem_Collect_eq Ball_def

   568   unfolding dist_commute

   569   apply clarify

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

   571   using dist_triangle_alt[where z=x]

   572   apply (clarsimp simp add: diff_less_iff)

   573   apply atomize

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

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

   576   by arith

   577

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

   579   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   580

   581 lemma openE[elim?]:

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

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

   584   using assms unfolding open_contains_ball by auto

   585

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

   587   by (metis open_contains_ball subset_eq centre_in_ball)

   588

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

   590   unfolding mem_ball set_eq_iff

   591   apply (simp add: not_less)

   592   by (metis zero_le_dist order_trans dist_self)

   593

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

   595

   596 lemma rational_boxes:

   597   fixes x :: "'a\<Colon>ordered_euclidean_space"

   598   assumes "0 < e"

   599   shows "\<exists>a b. (\<forall>i. a $$i \<in> \<rat>) \<and> (\<forall>i. b$$ i \<in> \<rat>) \<and> x \<in> {a <..< b} \<and> {a <..< b} \<subseteq> ball x e"

   600 proof -

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

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

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

   604   proof

   605     fix i from Rats_dense_in_real[of "x $$i - e'" "x$$ i"] e

   606     show "?th i" by auto

   607   qed

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

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

   610   proof

   611     fix i from Rats_dense_in_real[of "x $$i" "x$$ i + e'"] e

   612     show "?th i" by auto

   613   qed

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

   615   { fix y :: 'a assume *: "Chi a < y" "y < Chi b"

   616     have "dist x y = sqrt (\<Sum>i<DIM('a). (dist (x $$i) (y$$ i))\<twosuperior>)"

   617       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   619     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   620       fix i assume i: "i \<in> {..<DIM('a)}"

   621       have "a i < y$$i \<and> y$$i < b i" using * i eucl_less[where 'a='a] by auto

   622       moreover have "a i < x$$i" "x$$i - a i < e'" using a by auto

   623       moreover have "x$$i < b i" "b i - x$$i < e'" using b by auto

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

   625       then have "dist (x $$i) (y$$ i) < e/sqrt (real (DIM('a)))"

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

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

   628         by (rule power_strict_mono) auto

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

   630         by (simp add: power_divide)

   631     qed auto

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

   633     finally have "dist x y < e" . }

   634   with a b show ?thesis

   635     apply (rule_tac exI[of _ "Chi a"])

   636     apply (rule_tac exI[of _ "Chi b"])

   637     using eucl_less[where 'a='a] by auto

   638 qed

   639

   640 lemma ex_rat_list:

   641   fixes x :: "'a\<Colon>ordered_euclidean_space"

   642   assumes "\<And> i. x $$i \<in> \<rat>"   643 shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x$$ i)"

   644 proof -

   645   have "\<forall>i. \<exists>r. x $$i = of_rat r" using assms unfolding Rats_def by blast   646 from choice[OF this] guess r ..   647 then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])   648 qed   649   650 lemma open_UNION:   651 fixes M :: "'a\<Colon>ordered_euclidean_space set"   652 assumes "open M"   653 shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}   654 (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"   655 (is "M = UNION ?idx ?box")   656 proof safe   657 fix x assume "x \<in> M"   658 obtain e where e: "e > 0" "ball x e \<subseteq> M"   659 using openE[OF assms x \<in> M] by auto   660 then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a$$ i \<in> \<rat>" "\<And>i. b $$i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"   661 using rational_boxes[OF e(1)] by blast   662 then obtain p q where pq: "length p = DIM ('a)"   663 "length q = DIM ('a)"   664 "\<forall> i < DIM ('a). of_rat (p ! i) = a$$ i \<and> of_rat (q ! i) = b $$i"   665 using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast   666 hence p: "Chi (of_rat \<circ> op ! p) = a"   667 using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]   668 unfolding o_def by auto   669 from pq have q: "Chi (of_rat \<circ> op ! q) = b"   670 using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]   671 unfolding o_def by auto   672 have "x \<in> ?box (p, q)"   673 using p q ab by auto   674 thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto   675 qed auto   676   677 subsection{* Connectedness *}   678   679 definition "connected S \<longleftrightarrow>   680 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})   681 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"   682   683 lemma connected_local:   684 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.   685 openin (subtopology euclidean S) e1 \<and>   686 openin (subtopology euclidean S) e2 \<and>   687 S \<subseteq> e1 \<union> e2 \<and>   688 e1 \<inter> e2 = {} \<and>   689 ~(e1 = {}) \<and>   690 ~(e2 = {}))"   691 unfolding connected_def openin_open by (safe, blast+)   692   693 lemma exists_diff:   694 fixes P :: "'a set \<Rightarrow> bool"   695 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")   696 proof-   697 {assume "?lhs" hence ?rhs by blast }   698 moreover   699 {fix S assume H: "P S"   700 have "S = - (- S)" by auto   701 with H have "P (- (- S))" by metis }   702 ultimately show ?thesis by metis   703 qed   704   705 lemma connected_clopen: "connected S \<longleftrightarrow>   706 (\<forall>T. openin (subtopology euclidean S) T \<and>   707 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")   708 proof-   709 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> {})"   710 unfolding connected_def openin_open closedin_closed   711 apply (subst exists_diff) by blast   712 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> {})"   713 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis   714   715 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'))"   716 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")   717 unfolding connected_def openin_open closedin_closed by auto   718 {fix e2   719 {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)"   720 by auto}   721 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}   722 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast   723 then show ?thesis unfolding th0 th1 by simp   724 qed   725   726 lemma connected_empty[simp, intro]: "connected {}"   727 by (simp add: connected_def)   728   729   730 subsection{* Limit points *}   731   732 definition (in topological_space)   733 islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where   734 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"   735   736 lemma islimptI:   737 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"   738 shows "x islimpt S"   739 using assms unfolding islimpt_def by auto   740   741 lemma islimptE:   742 assumes "x islimpt S" and "x \<in> T" and "open T"   743 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"   744 using assms unfolding islimpt_def by auto   745   746 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"   747 unfolding islimpt_def eventually_at_topological by auto   748   749 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"   750 unfolding islimpt_def by fast   751   752 lemma islimpt_approachable:   753 fixes x :: "'a::metric_space"   754 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"   755 unfolding islimpt_iff_eventually eventually_at by fast   756   757 lemma islimpt_approachable_le:   758 fixes x :: "'a::metric_space"   759 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"   760 unfolding islimpt_approachable   761 using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",   762 THEN arg_cong [where f=Not]]   763 by (simp add: Bex_def conj_commute conj_left_commute)   764   765 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"   766 unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)   767   768 text {* A perfect space has no isolated points. *}   769   770 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"   771 unfolding islimpt_UNIV_iff by (rule not_open_singleton)   772   773 lemma perfect_choose_dist:   774 fixes x :: "'a::{perfect_space, metric_space}"   775 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"   776 using islimpt_UNIV [of x]   777 by (simp add: islimpt_approachable)   778   779 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"   780 unfolding closed_def   781 apply (subst open_subopen)   782 apply (simp add: islimpt_def subset_eq)   783 by (metis ComplE ComplI)   784   785 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"   786 unfolding islimpt_def by auto   787   788 lemma finite_set_avoid:   789 fixes a :: "'a::metric_space"   790 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"   791 proof(induct rule: finite_induct[OF fS])   792 case 1 thus ?case by (auto intro: zero_less_one)   793 next   794 case (2 x F)   795 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast   796 {assume "x = a" hence ?case using d by auto }   797 moreover   798 {assume xa: "x\<noteq>a"   799 let ?d = "min d (dist a x)"   800 have dp: "?d > 0" using xa d(1) using dist_nz by auto   801 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto   802 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }   803 ultimately show ?case by blast   804 qed   805   806 lemma islimpt_finite:   807 fixes S :: "'a::metric_space set"   808 assumes fS: "finite S" shows "\<not> a islimpt S"   809 unfolding islimpt_approachable   810 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)   811   812 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"   813 apply (rule iffI)   814 defer   815 apply (metis Un_upper1 Un_upper2 islimpt_subset)   816 unfolding islimpt_def   817 apply (rule ccontr, clarsimp, rename_tac A B)   818 apply (drule_tac x="A \<inter> B" in spec)   819 apply (auto simp add: open_Int)   820 done   821   822 lemma discrete_imp_closed:   823 fixes S :: "'a::metric_space set"   824 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"   825 shows "closed S"   826 proof-   827 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"   828 from e have e2: "e/2 > 0" by arith   829 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast   830 let ?m = "min (e/2) (dist x y) "   831 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])   832 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast   833 have th: "dist z y < e" using z y   834 by (intro dist_triangle_lt [where z=x], simp)   835 from d[rule_format, OF y(1) z(1) th] y z   836 have False by (auto simp add: dist_commute)}   837 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])   838 qed   839   840   841 subsection {* Interior of a Set *}   842   843 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"   844   845 lemma interiorI [intro?]:   846 assumes "open T" and "x \<in> T" and "T \<subseteq> S"   847 shows "x \<in> interior S"   848 using assms unfolding interior_def by fast   849   850 lemma interiorE [elim?]:   851 assumes "x \<in> interior S"   852 obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"   853 using assms unfolding interior_def by fast   854   855 lemma open_interior [simp, intro]: "open (interior S)"   856 by (simp add: interior_def open_Union)   857   858 lemma interior_subset: "interior S \<subseteq> S"   859 by (auto simp add: interior_def)   860   861 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"   862 by (auto simp add: interior_def)   863   864 lemma interior_open: "open S \<Longrightarrow> interior S = S"   865 by (intro equalityI interior_subset interior_maximal subset_refl)   866   867 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"   868 by (metis open_interior interior_open)   869   870 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"   871 by (metis interior_maximal interior_subset subset_trans)   872   873 lemma interior_empty [simp]: "interior {} = {}"   874 using open_empty by (rule interior_open)   875   876 lemma interior_UNIV [simp]: "interior UNIV = UNIV"   877 using open_UNIV by (rule interior_open)   878   879 lemma interior_interior [simp]: "interior (interior S) = interior S"   880 using open_interior by (rule interior_open)   881   882 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"   883 by (auto simp add: interior_def)   884   885 lemma interior_unique:   886 assumes "T \<subseteq> S" and "open T"   887 assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"   888 shows "interior S = T"   889 by (intro equalityI assms interior_subset open_interior interior_maximal)   890   891 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"   892 by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1   893 Int_lower2 interior_maximal interior_subset open_Int open_interior)   894   895 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"   896 using open_contains_ball_eq [where S="interior S"]   897 by (simp add: open_subset_interior)   898   899 lemma interior_limit_point [intro]:   900 fixes x :: "'a::perfect_space"   901 assumes x: "x \<in> interior S" shows "x islimpt S"   902 using x islimpt_UNIV [of x]   903 unfolding interior_def islimpt_def   904 apply (clarsimp, rename_tac T T')   905 apply (drule_tac x="T \<inter> T'" in spec)   906 apply (auto simp add: open_Int)   907 done   908   909 lemma interior_closed_Un_empty_interior:   910 assumes cS: "closed S" and iT: "interior T = {}"   911 shows "interior (S \<union> T) = interior S"   912 proof   913 show "interior S \<subseteq> interior (S \<union> T)"   914 by (rule interior_mono, rule Un_upper1)   915 next   916 show "interior (S \<union> T) \<subseteq> interior S"   917 proof   918 fix x assume "x \<in> interior (S \<union> T)"   919 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..   920 show "x \<in> interior S"   921 proof (rule ccontr)   922 assume "x \<notin> interior S"   923 with x \<in> R open R obtain y where "y \<in> R - S"   924 unfolding interior_def by fast   925 from open R closed S have "open (R - S)" by (rule open_Diff)   926 from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast   927 from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}   928 show "False" unfolding interior_def by fast   929 qed   930 qed   931 qed   932   933 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"   934 proof (rule interior_unique)   935 show "interior A \<times> interior B \<subseteq> A \<times> B"   936 by (intro Sigma_mono interior_subset)   937 show "open (interior A \<times> interior B)"   938 by (intro open_Times open_interior)   939 fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"   940 proof (safe)   941 fix x y assume "(x, y) \<in> T"   942 then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"   943 using open T unfolding open_prod_def by fast   944 hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"   945 using T \<subseteq> A \<times> B by auto   946 thus "x \<in> interior A" and "y \<in> interior B"   947 by (auto intro: interiorI)   948 qed   949 qed   950   951   952 subsection {* Closure of a Set *}   953   954 definition "closure S = S \<union> {x | x. x islimpt S}"   955   956 lemma interior_closure: "interior S = - (closure (- S))"   957 unfolding interior_def closure_def islimpt_def by auto   958   959 lemma closure_interior: "closure S = - interior (- S)"   960 unfolding interior_closure by simp   961   962 lemma closed_closure[simp, intro]: "closed (closure S)"   963 unfolding closure_interior by (simp add: closed_Compl)   964   965 lemma closure_subset: "S \<subseteq> closure S"   966 unfolding closure_def by simp   967   968 lemma closure_hull: "closure S = closed hull S"   969 unfolding hull_def closure_interior interior_def by auto   970   971 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"   972 unfolding closure_hull using closed_Inter by (rule hull_eq)   973   974 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"   975 unfolding closure_eq .   976   977 lemma closure_closure [simp]: "closure (closure S) = closure S"   978 unfolding closure_hull by (rule hull_hull)   979   980 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"   981 unfolding closure_hull by (rule hull_mono)   982   983 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"   984 unfolding closure_hull by (rule hull_minimal)   985   986 lemma closure_unique:   987 assumes "S \<subseteq> T" and "closed T"   988 assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"   989 shows "closure S = T"   990 using assms unfolding closure_hull by (rule hull_unique)   991   992 lemma closure_empty [simp]: "closure {} = {}"   993 using closed_empty by (rule closure_closed)   994   995 lemma closure_UNIV [simp]: "closure UNIV = UNIV"   996 using closed_UNIV by (rule closure_closed)   997   998 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"   999 unfolding closure_interior by simp   1000   1001 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"   1002 using closure_empty closure_subset[of S]   1003 by blast   1004   1005 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"   1006 using closure_eq[of S] closure_subset[of S]   1007 by simp   1008   1009 lemma open_inter_closure_eq_empty:   1010 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"   1011 using open_subset_interior[of S "- T"]   1012 using interior_subset[of "- T"]   1013 unfolding closure_interior   1014 by auto   1015   1016 lemma open_inter_closure_subset:   1017 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"   1018 proof   1019 fix x   1020 assume as: "open S" "x \<in> S \<inter> closure T"   1021 { assume *:"x islimpt T"   1022 have "x islimpt (S \<inter> T)"   1023 proof (rule islimptI)   1024 fix A   1025 assume "x \<in> A" "open A"   1026 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"   1027 by (simp_all add: open_Int)   1028 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"   1029 by (rule islimptE)   1030 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"   1031 by simp_all   1032 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..   1033 qed   1034 }   1035 then show "x \<in> closure (S \<inter> T)" using as   1036 unfolding closure_def   1037 by blast   1038 qed   1039   1040 lemma closure_complement: "closure (- S) = - interior S"   1041 unfolding closure_interior by simp   1042   1043 lemma interior_complement: "interior (- S) = - closure S"   1044 unfolding closure_interior by simp   1045   1046 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"   1047 proof (rule closure_unique)   1048 show "A \<times> B \<subseteq> closure A \<times> closure B"   1049 by (intro Sigma_mono closure_subset)   1050 show "closed (closure A \<times> closure B)"   1051 by (intro closed_Times closed_closure)   1052 fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"   1053 apply (simp add: closed_def open_prod_def, clarify)   1054 apply (rule ccontr)   1055 apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)   1056 apply (simp add: closure_interior interior_def)   1057 apply (drule_tac x=C in spec)   1058 apply (drule_tac x=D in spec)   1059 apply auto   1060 done   1061 qed   1062   1063   1064 subsection {* Frontier (aka boundary) *}   1065   1066 definition "frontier S = closure S - interior S"   1067   1068 lemma frontier_closed: "closed(frontier S)"   1069 by (simp add: frontier_def closed_Diff)   1070   1071 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"   1072 by (auto simp add: frontier_def interior_closure)   1073   1074 lemma frontier_straddle:   1075 fixes a :: "'a::metric_space"   1076 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))"   1077 unfolding frontier_def closure_interior   1078 by (auto simp add: mem_interior subset_eq ball_def)   1079   1080 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"   1081 by (metis frontier_def closure_closed Diff_subset)   1082   1083 lemma frontier_empty[simp]: "frontier {} = {}"   1084 by (simp add: frontier_def)   1085   1086 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"   1087 proof-   1088 { assume "frontier S \<subseteq> S"   1089 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto   1090 hence "closed S" using closure_subset_eq by auto   1091 }   1092 thus ?thesis using frontier_subset_closed[of S] ..   1093 qed   1094   1095 lemma frontier_complement: "frontier(- S) = frontier S"   1096 by (auto simp add: frontier_def closure_complement interior_complement)   1097   1098 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"   1099 using frontier_complement frontier_subset_eq[of "- S"]   1100 unfolding open_closed by auto   1101   1102 subsection {* Filters and the eventually true'' quantifier *}   1103   1104 definition   1105 at_infinity :: "'a::real_normed_vector filter" where   1106 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"   1107   1108 definition   1109 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"   1110 (infixr "indirection" 70) where   1111 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"   1112   1113 text{* Prove That They are all filters. *}   1114   1115 lemma eventually_at_infinity:   1116 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"   1117 unfolding at_infinity_def   1118 proof (rule eventually_Abs_filter, rule is_filter.intro)   1119 fix P Q :: "'a \<Rightarrow> bool"   1120 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"   1121 then obtain r s where   1122 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto   1123 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp   1124 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..   1125 qed auto   1126   1127 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}   1128   1129 lemma trivial_limit_within:   1130 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"   1131 proof   1132 assume "trivial_limit (at a within S)"   1133 thus "\<not> a islimpt S"   1134 unfolding trivial_limit_def   1135 unfolding eventually_within eventually_at_topological   1136 unfolding islimpt_def   1137 apply (clarsimp simp add: set_eq_iff)   1138 apply (rename_tac T, rule_tac x=T in exI)   1139 apply (clarsimp, drule_tac x=y in bspec, simp_all)   1140 done   1141 next   1142 assume "\<not> a islimpt S"   1143 thus "trivial_limit (at a within S)"   1144 unfolding trivial_limit_def   1145 unfolding eventually_within eventually_at_topological   1146 unfolding islimpt_def   1147 apply clarsimp   1148 apply (rule_tac x=T in exI)   1149 apply auto   1150 done   1151 qed   1152   1153 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"   1154 using trivial_limit_within [of a UNIV] by simp   1155   1156 lemma trivial_limit_at:   1157 fixes a :: "'a::perfect_space"   1158 shows "\<not> trivial_limit (at a)"   1159 by (rule at_neq_bot)   1160   1161 lemma trivial_limit_at_infinity:   1162 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"   1163 unfolding trivial_limit_def eventually_at_infinity   1164 apply clarsimp   1165 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)   1166 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)   1167 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])   1168 apply (drule_tac x=UNIV in spec, simp)   1169 done   1170   1171 text {* Some property holds "sufficiently close" to the limit point. *}   1172   1173 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)   1174 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"   1175 unfolding eventually_at dist_nz by auto   1176   1177 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>   1178 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"   1179 unfolding eventually_within eventually_at dist_nz by auto   1180   1181 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>   1182 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")   1183 unfolding eventually_within   1184 by auto (metis dense order_le_less_trans)   1185   1186 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"   1187 unfolding trivial_limit_def   1188 by (auto elim: eventually_rev_mp)   1189   1190 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"   1191 by simp   1192   1193 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"   1194 by (simp add: filter_eq_iff)   1195   1196 text{* Combining theorems for "eventually" *}   1197   1198 lemma eventually_rev_mono:   1199 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"   1200 using eventually_mono [of P Q] by fast   1201   1202 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"   1203 by (simp add: eventually_False)   1204   1205   1206 subsection {* Limits *}   1207   1208 text{* Notation Lim to avoid collition with lim defined in analysis *}   1209   1210 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"   1211 where "Lim A f = (THE l. (f ---> l) A)"   1212   1213 lemma Lim:   1214 "(f ---> l) net \<longleftrightarrow>   1215 trivial_limit net \<or>   1216 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"   1217 unfolding tendsto_iff trivial_limit_eq by auto   1218   1219 text{* Show that they yield usual definitions in the various cases. *}   1220   1221 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>   1222 (\<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)"   1223 by (auto simp add: tendsto_iff eventually_within_le)   1224   1225 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>   1226 (\<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)"   1227 by (auto simp add: tendsto_iff eventually_within)   1228   1229 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>   1230 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"   1231 by (auto simp add: tendsto_iff eventually_at)   1232   1233 lemma Lim_at_infinity:   1234 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"   1235 by (auto simp add: tendsto_iff eventually_at_infinity)   1236   1237 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"   1238 by (rule topological_tendstoI, auto elim: eventually_rev_mono)   1239   1240 text{* The expected monotonicity property. *}   1241   1242 lemma Lim_within_empty: "(f ---> l) (net within {})"   1243 unfolding tendsto_def Limits.eventually_within by simp   1244   1245 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"   1246 unfolding tendsto_def Limits.eventually_within   1247 by (auto elim!: eventually_elim1)   1248   1249 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"   1250 shows "(f ---> l) (net within (S \<union> T))"   1251 using assms unfolding tendsto_def Limits.eventually_within   1252 apply clarify   1253 apply (drule spec, drule (1) mp, drule (1) mp)   1254 apply (drule spec, drule (1) mp, drule (1) mp)   1255 apply (auto elim: eventually_elim2)   1256 done   1257   1258 lemma Lim_Un_univ:   1259 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV   1260 ==> (f ---> l) net"   1261 by (metis Lim_Un within_UNIV)   1262   1263 text{* Interrelations between restricted and unrestricted limits. *}   1264   1265 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"   1266 (* FIXME: rename *)   1267 unfolding tendsto_def Limits.eventually_within   1268 apply (clarify, drule spec, drule (1) mp, drule (1) mp)   1269 by (auto elim!: eventually_elim1)   1270   1271 lemma eventually_within_interior:   1272 assumes "x \<in> interior S"   1273 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")   1274 proof-   1275 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..   1276 { assume "?lhs"   1277 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"   1278 unfolding Limits.eventually_within Limits.eventually_at_topological   1279 by auto   1280 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"   1281 by auto   1282 then have "?rhs"   1283 unfolding Limits.eventually_at_topological by auto   1284 } moreover   1285 { assume "?rhs" hence "?lhs"   1286 unfolding Limits.eventually_within   1287 by (auto elim: eventually_elim1)   1288 } ultimately   1289 show "?thesis" ..   1290 qed   1291   1292 lemma at_within_interior:   1293 "x \<in> interior S \<Longrightarrow> at x within S = at x"   1294 by (simp add: filter_eq_iff eventually_within_interior)   1295   1296 lemma at_within_open:   1297 "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"   1298 by (simp only: at_within_interior interior_open)   1299   1300 lemma Lim_within_open:   1301 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"   1302 assumes"a \<in> S" "open S"   1303 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"   1304 using assms by (simp only: at_within_open)   1305   1306 lemma Lim_within_LIMSEQ:   1307 fixes a :: "'a::metric_space"   1308 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"   1309 shows "(X ---> L) (at a within T)"   1310 using assms unfolding tendsto_def [where l=L]   1311 by (simp add: sequentially_imp_eventually_within)   1312   1313 lemma Lim_right_bound:   1314 fixes f :: "real \<Rightarrow> real"   1315 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"   1316 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"   1317 shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"   1318 proof cases   1319 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)   1320 next   1321 assume [simp]: "{x<..} \<inter> I \<noteq> {}"   1322 show ?thesis   1323 proof (rule Lim_within_LIMSEQ, safe)   1324 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"   1325   1326 show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"   1327 proof (rule LIMSEQ_I, rule ccontr)   1328 fix r :: real assume "0 < r"   1329 with Inf_close[of "f  ({x<..} \<inter> I)" r]   1330 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto   1331 from x < y have "0 < y - x" by auto   1332 from S(2)[THEN LIMSEQ_D, OF this]   1333 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto   1334   1335 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"   1336 moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"   1337 using S bnd by (intro Inf_lower[where z=K]) auto   1338 ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"   1339 by (auto simp: not_less field_simps)   1340 with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y   1341 show False by auto   1342 qed   1343 qed   1344 qed   1345   1346 text{* Another limit point characterization. *}   1347   1348 lemma islimpt_sequential:   1349 fixes x :: "'a::metric_space"   1350 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"   1351 (is "?lhs = ?rhs")   1352 proof   1353 assume ?lhs   1354 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"   1355 unfolding islimpt_approachable   1356 using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto   1357 let ?I = "\<lambda>n. inverse (real (Suc n))"   1358 have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp   1359 moreover have "(\<lambda>n. f (?I n)) ----> x"   1360 proof (rule metric_tendsto_imp_tendsto)   1361 show "?I ----> 0"   1362 by (rule LIMSEQ_inverse_real_of_nat)   1363 show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"   1364 by (simp add: norm_conv_dist [symmetric] less_imp_le f)   1365 qed   1366 ultimately show ?rhs by fast   1367 next   1368 assume ?rhs   1369 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   1370 { fix e::real assume "e>0"   1371 then obtain N where "dist (f N) x < e" using f(2) by auto   1372 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto   1373 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto   1374 }   1375 thus ?lhs unfolding islimpt_approachable by auto   1376 qed   1377   1378 lemma Lim_inv: (* TODO: delete *)   1379 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"   1380 assumes "(f ---> l) A" and "l \<noteq> 0"   1381 shows "((inverse o f) ---> inverse l) A"   1382 unfolding o_def using assms by (rule tendsto_inverse)   1383   1384 lemma Lim_null:   1385 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1386 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"   1387 by (simp add: Lim dist_norm)   1388   1389 lemma Lim_null_comparison:   1390 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1391 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"   1392 shows "(f ---> 0) net"   1393 proof (rule metric_tendsto_imp_tendsto)   1394 show "(g ---> 0) net" by fact   1395 show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"   1396 using assms(1) by (rule eventually_elim1, simp add: dist_norm)   1397 qed   1398   1399 lemma Lim_transform_bound:   1400 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1401 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"   1402 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"   1403 shows "(f ---> 0) net"   1404 using assms(1) tendsto_norm_zero [OF assms(2)]   1405 by (rule Lim_null_comparison)   1406   1407 text{* Deducing things about the limit from the elements. *}   1408   1409 lemma Lim_in_closed_set:   1410 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"   1411 shows "l \<in> S"   1412 proof (rule ccontr)   1413 assume "l \<notin> S"   1414 with closed S have "open (- S)" "l \<in> - S"   1415 by (simp_all add: open_Compl)   1416 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"   1417 by (rule topological_tendstoD)   1418 with assms(2) have "eventually (\<lambda>x. False) net"   1419 by (rule eventually_elim2) simp   1420 with assms(3) show "False"   1421 by (simp add: eventually_False)   1422 qed   1423   1424 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}   1425   1426 lemma Lim_dist_ubound:   1427 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"   1428 shows "dist a l <= e"   1429 proof-   1430 have "dist a l \<in> {..e}"   1431 proof (rule Lim_in_closed_set)   1432 show "closed {..e}" by simp   1433 show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)   1434 show "\<not> trivial_limit net" by fact   1435 show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)   1436 qed   1437 thus ?thesis by simp   1438 qed   1439   1440 lemma Lim_norm_ubound:   1441 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1442 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"   1443 shows "norm(l) <= e"   1444 proof-   1445 have "norm l \<in> {..e}"   1446 proof (rule Lim_in_closed_set)   1447 show "closed {..e}" by simp   1448 show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)   1449 show "\<not> trivial_limit net" by fact   1450 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)   1451 qed   1452 thus ?thesis by simp   1453 qed   1454   1455 lemma Lim_norm_lbound:   1456 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1457 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"   1458 shows "e \<le> norm l"   1459 proof-   1460 have "norm l \<in> {e..}"   1461 proof (rule Lim_in_closed_set)   1462 show "closed {e..}" by simp   1463 show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)   1464 show "\<not> trivial_limit net" by fact   1465 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)   1466 qed   1467 thus ?thesis by simp   1468 qed   1469   1470 text{* Uniqueness of the limit, when nontrivial. *}   1471   1472 lemma tendsto_Lim:   1473 fixes f :: "'a \<Rightarrow> 'b::t2_space"   1474 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"   1475 unfolding Lim_def using tendsto_unique[of net f] by auto   1476   1477 text{* Limit under bilinear function *}   1478   1479 lemma Lim_bilinear:   1480 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"   1481 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"   1482 using bounded_bilinear h (f ---> l) net (g ---> m) net   1483 by (rule bounded_bilinear.tendsto)   1484   1485 text{* These are special for limits out of the same vector space. *}   1486   1487 lemma Lim_within_id: "(id ---> a) (at a within s)"   1488 unfolding id_def by (rule tendsto_ident_at_within)   1489   1490 lemma Lim_at_id: "(id ---> a) (at a)"   1491 unfolding id_def by (rule tendsto_ident_at)   1492   1493 lemma Lim_at_zero:   1494 fixes a :: "'a::real_normed_vector"   1495 fixes l :: "'b::topological_space"   1496 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")   1497 using LIM_offset_zero LIM_offset_zero_cancel ..   1498   1499 text{* It's also sometimes useful to extract the limit point from the filter. *}   1500   1501 definition   1502 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where   1503 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"   1504   1505 lemma netlimit_within:   1506 assumes "\<not> trivial_limit (at a within S)"   1507 shows "netlimit (at a within S) = a"   1508 unfolding netlimit_def   1509 apply (rule some_equality)   1510 apply (rule Lim_at_within)   1511 apply (rule tendsto_ident_at)   1512 apply (erule tendsto_unique [OF assms])   1513 apply (rule Lim_at_within)   1514 apply (rule tendsto_ident_at)   1515 done   1516   1517 lemma netlimit_at:   1518 fixes a :: "'a::{perfect_space,t2_space}"   1519 shows "netlimit (at a) = a"   1520 using netlimit_within [of a UNIV] by simp   1521   1522 lemma lim_within_interior:   1523 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"   1524 by (simp add: at_within_interior)   1525   1526 lemma netlimit_within_interior:   1527 fixes x :: "'a::{t2_space,perfect_space}"   1528 assumes "x \<in> interior S"   1529 shows "netlimit (at x within S) = x"   1530 using assms by (simp add: at_within_interior netlimit_at)   1531   1532 text{* Transformation of limit. *}   1533   1534 lemma Lim_transform:   1535 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"   1536 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"   1537 shows "(g ---> l) net"   1538 using tendsto_diff [OF assms(2) assms(1)] by simp   1539   1540 lemma Lim_transform_eventually:   1541 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"   1542 apply (rule topological_tendstoI)   1543 apply (drule (2) topological_tendstoD)   1544 apply (erule (1) eventually_elim2, simp)   1545 done   1546   1547 lemma Lim_transform_within:   1548 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"   1549 and "(f ---> l) (at x within S)"   1550 shows "(g ---> l) (at x within S)"   1551 proof (rule Lim_transform_eventually)   1552 show "eventually (\<lambda>x. f x = g x) (at x within S)"   1553 unfolding eventually_within   1554 using assms(1,2) by auto   1555 show "(f ---> l) (at x within S)" by fact   1556 qed   1557   1558 lemma Lim_transform_at:   1559 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"   1560 and "(f ---> l) (at x)"   1561 shows "(g ---> l) (at x)"   1562 proof (rule Lim_transform_eventually)   1563 show "eventually (\<lambda>x. f x = g x) (at x)"   1564 unfolding eventually_at   1565 using assms(1,2) by auto   1566 show "(f ---> l) (at x)" by fact   1567 qed   1568   1569 text{* Common case assuming being away from some crucial point like 0. *}   1570   1571 lemma Lim_transform_away_within:   1572 fixes a b :: "'a::t1_space"   1573 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"   1574 and "(f ---> l) (at a within S)"   1575 shows "(g ---> l) (at a within S)"   1576 proof (rule Lim_transform_eventually)   1577 show "(f ---> l) (at a within S)" by fact   1578 show "eventually (\<lambda>x. f x = g x) (at a within S)"   1579 unfolding Limits.eventually_within eventually_at_topological   1580 by (rule exI [where x="- {b}"], simp add: open_Compl assms)   1581 qed   1582   1583 lemma Lim_transform_away_at:   1584 fixes a b :: "'a::t1_space"   1585 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"   1586 and fl: "(f ---> l) (at a)"   1587 shows "(g ---> l) (at a)"   1588 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl   1589 by simp   1590   1591 text{* Alternatively, within an open set. *}   1592   1593 lemma Lim_transform_within_open:   1594 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"   1595 and "(f ---> l) (at a)"   1596 shows "(g ---> l) (at a)"   1597 proof (rule Lim_transform_eventually)   1598 show "eventually (\<lambda>x. f x = g x) (at a)"   1599 unfolding eventually_at_topological   1600 using assms(1,2,3) by auto   1601 show "(f ---> l) (at a)" by fact   1602 qed   1603   1604 text{* A congruence rule allowing us to transform limits assuming not at point. *}   1605   1606 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)   1607   1608 lemma Lim_cong_within(*[cong add]*):   1609 assumes "a = b" "x = y" "S = T"   1610 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"   1611 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"   1612 unfolding tendsto_def Limits.eventually_within eventually_at_topological   1613 using assms by simp   1614   1615 lemma Lim_cong_at(*[cong add]*):   1616 assumes "a = b" "x = y"   1617 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"   1618 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"   1619 unfolding tendsto_def eventually_at_topological   1620 using assms by simp   1621   1622 text{* Useful lemmas on closure and set of possible sequential limits.*}   1623   1624 lemma closure_sequential:   1625 fixes l :: "'a::metric_space"   1626 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")   1627 proof   1628 assume "?lhs" moreover   1629 { assume "l \<in> S"   1630 hence "?rhs" using tendsto_const[of l sequentially] by auto   1631 } moreover   1632 { assume "l islimpt S"   1633 hence "?rhs" unfolding islimpt_sequential by auto   1634 } ultimately   1635 show "?rhs" unfolding closure_def by auto   1636 next   1637 assume "?rhs"   1638 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto   1639 qed   1640   1641 lemma closed_sequential_limits:   1642 fixes S :: "'a::metric_space set"   1643 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"   1644 unfolding closed_limpt   1645 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]   1646 by metis   1647   1648 lemma closure_approachable:   1649 fixes S :: "'a::metric_space set"   1650 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"   1651 apply (auto simp add: closure_def islimpt_approachable)   1652 by (metis dist_self)   1653   1654 lemma closed_approachable:   1655 fixes S :: "'a::metric_space set"   1656 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"   1657 by (metis closure_closed closure_approachable)   1658   1659 subsection {* Infimum Distance *}   1660   1661 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"   1662   1663 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"   1664 by (simp add: infdist_def)   1665   1666 lemma infdist_nonneg:   1667 shows "0 \<le> infdist x A"   1668 using assms by (auto simp add: infdist_def)   1669   1670 lemma infdist_le:   1671 assumes "a \<in> A"   1672 assumes "d = dist x a"   1673 shows "infdist x A \<le> d"   1674 using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)   1675   1676 lemma infdist_zero[simp]:   1677 assumes "a \<in> A" shows "infdist a A = 0"   1678 proof -   1679 from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto   1680 with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto   1681 qed   1682   1683 lemma infdist_triangle:   1684 shows "infdist x A \<le> infdist y A + dist x y"   1685 proof cases   1686 assume "A = {}" thus ?thesis by (simp add: infdist_def)   1687 next   1688 assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto   1689 have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"   1690 proof   1691 from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp   1692 fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"   1693 then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto   1694 show "infdist x A \<le> d"   1695 unfolding infdist_notempty[OF A \<noteq> {}]   1696 proof (rule Inf_lower2)   1697 show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto   1698 show "dist x a \<le> d" unfolding d by (rule dist_triangle)   1699 fix d assume "d \<in> {dist x a |a. a \<in> A}"   1700 then obtain a where "a \<in> A" "d = dist x a" by auto   1701 thus "infdist x A \<le> d" by (rule infdist_le)   1702 qed   1703 qed   1704 also have "\<dots> = dist x y + infdist y A"   1705 proof (rule Inf_eq, safe)   1706 fix a assume "a \<in> A"   1707 thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)   1708 next   1709 fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"   1710 hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A   1711 by (intro Inf_greatest) (auto simp: field_simps)   1712 thus "i \<le> dist x y + infdist y A" by simp   1713 qed   1714 finally show ?thesis by simp   1715 qed   1716   1717 lemma   1718 in_closure_iff_infdist_zero:   1719 assumes "A \<noteq> {}"   1720 shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"   1721 proof   1722 assume "x \<in> closure A"   1723 show "infdist x A = 0"   1724 proof (rule ccontr)   1725 assume "infdist x A \<noteq> 0"   1726 with infdist_nonneg[of x A] have "infdist x A > 0" by auto   1727 hence "ball x (infdist x A) \<inter> closure A = {}" apply auto   1728 by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute   1729 eucl_less_not_refl euclidean_trans(2) infdist_le)   1730 hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)   1731 thus False using x \<in> closure A by simp   1732 qed   1733 next   1734 assume x: "infdist x A = 0"   1735 then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)   1736 show "x \<in> closure A" unfolding closure_approachable   1737 proof (safe, rule ccontr)   1738 fix e::real assume "0 < e"   1739 assume "\<not> (\<exists>y\<in>A. dist y x < e)"   1740 hence "infdist x A \<ge> e" using a \<in> A   1741 unfolding infdist_def   1742 by (force intro: Inf_greatest simp: dist_commute)   1743 with x 0 < e show False by auto   1744 qed   1745 qed   1746   1747 lemma   1748 in_closed_iff_infdist_zero:   1749 assumes "closed A" "A \<noteq> {}"   1750 shows "x \<in> A \<longleftrightarrow> infdist x A = 0"   1751 proof -   1752 have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"   1753 by (rule in_closure_iff_infdist_zero) fact   1754 with assms show ?thesis by simp   1755 qed   1756   1757 lemma tendsto_infdist [tendsto_intros]:   1758 assumes f: "(f ---> l) F"   1759 shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"   1760 proof (rule tendstoI)   1761 fix e ::real assume "0 < e"   1762 from tendstoD[OF f this]   1763 show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"   1764 proof (eventually_elim)   1765 fix x   1766 from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]   1767 have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"   1768 by (simp add: dist_commute dist_real_def)   1769 also assume "dist (f x) l < e"   1770 finally show "dist (infdist (f x) A) (infdist l A) < e" .   1771 qed   1772 qed   1773   1774 text{* Some other lemmas about sequences. *}   1775   1776 lemma sequentially_offset:   1777 assumes "eventually (\<lambda>i. P i) sequentially"   1778 shows "eventually (\<lambda>i. P (i + k)) sequentially"   1779 using assms unfolding eventually_sequentially by (metis trans_le_add1)   1780   1781 lemma seq_offset:   1782 assumes "(f ---> l) sequentially"   1783 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"   1784 using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)   1785   1786 lemma seq_offset_neg:   1787 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"   1788 apply (rule topological_tendstoI)   1789 apply (drule (2) topological_tendstoD)   1790 apply (simp only: eventually_sequentially)   1791 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")   1792 apply metis   1793 by arith   1794   1795 lemma seq_offset_rev:   1796 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"   1797 by (rule LIMSEQ_offset) (* FIXME: redundant *)   1798   1799 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"   1800 using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)   1801   1802 subsection {* More properties of closed balls *}   1803   1804 lemma closed_cball: "closed (cball x e)"   1805 unfolding cball_def closed_def   1806 unfolding Collect_neg_eq [symmetric] not_le   1807 apply (clarsimp simp add: open_dist, rename_tac y)   1808 apply (rule_tac x="dist x y - e" in exI, clarsimp)   1809 apply (rename_tac x')   1810 apply (cut_tac x=x and y=x' and z=y in dist_triangle)   1811 apply simp   1812 done   1813   1814 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"   1815 proof-   1816 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"   1817 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)   1818 } moreover   1819 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"   1820 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto   1821 } ultimately   1822 show ?thesis unfolding open_contains_ball by auto   1823 qed   1824   1825 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"   1826 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)   1827   1828 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"   1829 apply (simp add: interior_def, safe)   1830 apply (force simp add: open_contains_cball)   1831 apply (rule_tac x="ball x e" in exI)   1832 apply (simp add: subset_trans [OF ball_subset_cball])   1833 done   1834   1835 lemma islimpt_ball:   1836 fixes x y :: "'a::{real_normed_vector,perfect_space}"   1837 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")   1838 proof   1839 assume "?lhs"   1840 { assume "e \<le> 0"   1841 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto   1842 have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto   1843 }   1844 hence "e > 0" by (metis not_less)   1845 moreover   1846 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   1847 ultimately show "?rhs" by auto   1848 next   1849 assume "?rhs" hence "e>0" by auto   1850 { fix d::real assume "d>0"   1851 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1852 proof(cases "d \<le> dist x y")   1853 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1854 proof(cases "x=y")   1855 case True hence False using d \<le> dist x y d>0 by auto   1856 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto   1857 next   1858 case False   1859   1860 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))   1861 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"   1862 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto   1863 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"   1864 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]   1865 unfolding scaleR_minus_left scaleR_one   1866 by (auto simp add: norm_minus_commute)   1867 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"   1868 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]   1869 unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto   1870 also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)   1871 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto   1872   1873 moreover   1874   1875 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"   1876 using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)   1877 moreover   1878 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel   1879 using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]   1880 unfolding dist_norm by auto   1881 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   1882 qed   1883 next   1884 case False hence "d > dist x y" by auto   1885 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1886 proof(cases "x=y")   1887 case True   1888 obtain z where **: "z \<noteq> y" "dist z y < min e d"   1889 using perfect_choose_dist[of "min e d" y]   1890 using d > 0 e>0 by auto   1891 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1892 unfolding x = y   1893 using z \<noteq> y **   1894 by (rule_tac x=z in bexI, auto simp add: dist_commute)   1895 next   1896 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1897 using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)   1898 qed   1899 qed }   1900 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto   1901 qed   1902   1903 lemma closure_ball_lemma:   1904 fixes x y :: "'a::real_normed_vector"   1905 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"   1906 proof (rule islimptI)   1907 fix T assume "y \<in> T" "open T"   1908 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"   1909 unfolding open_dist by fast   1910 (* choose point between x and y, within distance r of y. *)   1911 def k \<equiv> "min 1 (r / (2 * dist x y))"   1912 def z \<equiv> "y + scaleR k (x - y)"   1913 have z_def2: "z = x + scaleR (1 - k) (y - x)"   1914 unfolding z_def by (simp add: algebra_simps)   1915 have "dist z y < r"   1916 unfolding z_def k_def using 0 < r   1917 by (simp add: dist_norm min_def)   1918 hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp   1919 have "dist x z < dist x y"   1920 unfolding z_def2 dist_norm   1921 apply (simp add: norm_minus_commute)   1922 apply (simp only: dist_norm [symmetric])   1923 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)   1924 apply (rule mult_strict_right_mono)   1925 apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)   1926 apply (simp add: zero_less_dist_iff x \<noteq> y)   1927 done   1928 hence "z \<in> ball x (dist x y)" by simp   1929 have "z \<noteq> y"   1930 unfolding z_def k_def using x \<noteq> y 0 < r   1931 by (simp add: min_def)   1932 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"   1933 using z \<in> ball x (dist x y) z \<in> T z \<noteq> y   1934 by fast   1935 qed   1936   1937 lemma closure_ball:   1938 fixes x :: "'a::real_normed_vector"   1939 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"   1940 apply (rule equalityI)   1941 apply (rule closure_minimal)   1942 apply (rule ball_subset_cball)   1943 apply (rule closed_cball)   1944 apply (rule subsetI, rename_tac y)   1945 apply (simp add: le_less [where 'a=real])   1946 apply (erule disjE)   1947 apply (rule subsetD [OF closure_subset], simp)   1948 apply (simp add: closure_def)   1949 apply clarify   1950 apply (rule closure_ball_lemma)   1951 apply (simp add: zero_less_dist_iff)   1952 done   1953   1954 (* In a trivial vector space, this fails for e = 0. *)   1955 lemma interior_cball:   1956 fixes x :: "'a::{real_normed_vector, perfect_space}"   1957 shows "interior (cball x e) = ball x e"   1958 proof(cases "e\<ge>0")   1959 case False note cs = this   1960 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover   1961 { fix y assume "y \<in> cball x e"   1962 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }   1963 hence "cball x e = {}" by auto   1964 hence "interior (cball x e) = {}" using interior_empty by auto   1965 ultimately show ?thesis by blast   1966 next   1967 case True note cs = this   1968 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover   1969 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"   1970 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast   1971   1972 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"   1973 using perfect_choose_dist [of d] by auto   1974 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)   1975 hence xa_cball:"xa \<in> cball x e" using as(1) by auto   1976   1977 hence "y \<in> ball x e" proof(cases "x = y")   1978 case True   1979 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)   1980 thus "y \<in> ball x e" using x = y  by simp   1981 next   1982 case False   1983 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm   1984 using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto   1985 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast   1986 have "y - x \<noteq> 0" using x \<noteq> y by auto   1987 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]   1988 using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto   1989   1990 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)"   1991 by (auto simp add: dist_norm algebra_simps)   1992 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"   1993 by (auto simp add: algebra_simps)   1994 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"   1995 using ** by auto   1996 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)   1997 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)   1998 thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto   1999 qed }   2000 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto   2001 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto   2002 qed   2003   2004 lemma frontier_ball:   2005 fixes a :: "'a::real_normed_vector"   2006 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"   2007 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)   2008 apply (simp add: set_eq_iff)   2009 by arith   2010   2011 lemma frontier_cball:   2012 fixes a :: "'a::{real_normed_vector, perfect_space}"   2013 shows "frontier(cball a e) = {x. dist a x = e}"   2014 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)   2015 apply (simp add: set_eq_iff)   2016 by arith   2017   2018 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"   2019 apply (simp add: set_eq_iff not_le)   2020 by (metis zero_le_dist dist_self order_less_le_trans)   2021 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)   2022   2023 lemma cball_eq_sing:   2024 fixes x :: "'a::{metric_space,perfect_space}"   2025 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"   2026 proof (rule linorder_cases)   2027 assume e: "0 < e"   2028 obtain a where "a \<noteq> x" "dist a x < e"   2029 using perfect_choose_dist [OF e] by auto   2030 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)   2031 with e show ?thesis by (auto simp add: set_eq_iff)   2032 qed auto   2033   2034 lemma cball_sing:   2035 fixes x :: "'a::metric_space"   2036 shows "e = 0 ==> cball x e = {x}"   2037 by (auto simp add: set_eq_iff)   2038   2039   2040 subsection {* Boundedness *}   2041   2042 (* FIXME: This has to be unified with BSEQ!! *)   2043 definition (in metric_space)   2044 bounded :: "'a set \<Rightarrow> bool" where   2045 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"   2046   2047 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"   2048 unfolding bounded_def   2049 apply safe   2050 apply (rule_tac x="dist a x + e" in exI, clarify)   2051 apply (drule (1) bspec)   2052 apply (erule order_trans [OF dist_triangle add_left_mono])   2053 apply auto   2054 done   2055   2056 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"   2057 unfolding bounded_any_center [where a=0]   2058 by (simp add: dist_norm)   2059   2060 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"   2061 unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)   2062 using assms by auto   2063   2064 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)   2065 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"   2066 by (metis bounded_def subset_eq)   2067   2068 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"   2069 by (metis bounded_subset interior_subset)   2070   2071 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"   2072 proof-   2073 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto   2074 { fix y assume "y \<in> closure S"   2075 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"   2076 unfolding closure_sequential by auto   2077 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp   2078 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"   2079 by (rule eventually_mono, simp add: f(1))   2080 have "dist x y \<le> a"   2081 apply (rule Lim_dist_ubound [of sequentially f])   2082 apply (rule trivial_limit_sequentially)   2083 apply (rule f(2))   2084 apply fact   2085 done   2086 }   2087 thus ?thesis unfolding bounded_def by auto   2088 qed   2089   2090 lemma bounded_cball[simp,intro]: "bounded (cball x e)"   2091 apply (simp add: bounded_def)   2092 apply (rule_tac x=x in exI)   2093 apply (rule_tac x=e in exI)   2094 apply auto   2095 done   2096   2097 lemma bounded_ball[simp,intro]: "bounded(ball x e)"   2098 by (metis ball_subset_cball bounded_cball bounded_subset)   2099   2100 lemma finite_imp_bounded[intro]:   2101 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"   2102 proof-   2103 { fix a and F :: "'a set" assume as:"bounded F"   2104 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto   2105 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto   2106 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)   2107 }   2108 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto   2109 qed   2110   2111 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"   2112 apply (auto simp add: bounded_def)   2113 apply (rename_tac x y r s)   2114 apply (rule_tac x=x in exI)   2115 apply (rule_tac x="max r (dist x y + s)" in exI)   2116 apply (rule ballI, rename_tac z, safe)   2117 apply (drule (1) bspec, simp)   2118 apply (drule (1) bspec)   2119 apply (rule min_max.le_supI2)   2120 apply (erule order_trans [OF dist_triangle add_left_mono])   2121 done   2122   2123 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"   2124 by (induct rule: finite_induct[of F], auto)   2125   2126 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"   2127 apply (simp add: bounded_iff)   2128 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")   2129 by metis arith   2130   2131 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"   2132 by (metis Int_lower1 Int_lower2 bounded_subset)   2133   2134 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"   2135 apply (metis Diff_subset bounded_subset)   2136 done   2137   2138 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"   2139 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)   2140   2141 lemma not_bounded_UNIV[simp, intro]:   2142 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"   2143 proof(auto simp add: bounded_pos not_le)   2144 obtain x :: 'a where "x \<noteq> 0"   2145 using perfect_choose_dist [OF zero_less_one] by fast   2146 fix b::real assume b: "b >0"   2147 have b1: "b +1 \<ge> 0" using b by simp   2148 with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"   2149 by (simp add: norm_sgn)   2150 then show "\<exists>x::'a. b < norm x" ..   2151 qed   2152   2153 lemma bounded_linear_image:   2154 assumes "bounded S" "bounded_linear f"   2155 shows "bounded(f  S)"   2156 proof-   2157 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto   2158 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)   2159 { fix x assume "x\<in>S"   2160 hence "norm x \<le> b" using b by auto   2161 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)   2162 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)   2163 }   2164 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)   2165 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)   2166 qed   2167   2168 lemma bounded_scaling:   2169 fixes S :: "'a::real_normed_vector set"   2170 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"   2171 apply (rule bounded_linear_image, assumption)   2172 apply (rule bounded_linear_scaleR_right)   2173 done   2174   2175 lemma bounded_translation:   2176 fixes S :: "'a::real_normed_vector set"   2177 assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"   2178 proof-   2179 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto   2180 { fix x assume "x\<in>S"   2181 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto   2182 }   2183 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]   2184 by (auto intro!: exI[of _ "b + norm a"])   2185 qed   2186   2187   2188 text{* Some theorems on sups and infs using the notion "bounded". *}   2189   2190 lemma bounded_real:   2191 fixes S :: "real set"   2192 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"   2193 by (simp add: bounded_iff)   2194   2195 lemma bounded_has_Sup:   2196 fixes S :: "real set"   2197 assumes "bounded S" "S \<noteq> {}"   2198 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"   2199 proof   2200 fix x assume "x\<in>S"   2201 thus "x \<le> Sup S"   2202 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)   2203 next   2204 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms   2205 by (metis SupInf.Sup_least)   2206 qed   2207   2208 lemma Sup_insert:   2209 fixes S :: "real set"   2210 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"   2211 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)   2212   2213 lemma Sup_insert_finite:   2214 fixes S :: "real set"   2215 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"   2216 apply (rule Sup_insert)   2217 apply (rule finite_imp_bounded)   2218 by simp   2219   2220 lemma bounded_has_Inf:   2221 fixes S :: "real set"   2222 assumes "bounded S" "S \<noteq> {}"   2223 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"   2224 proof   2225 fix x assume "x\<in>S"   2226 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto   2227 thus "x \<ge> Inf S" using x\<in>S   2228 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)   2229 next   2230 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms   2231 by (metis SupInf.Inf_greatest)   2232 qed   2233   2234 lemma Inf_insert:   2235 fixes S :: "real set"   2236 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"   2237 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)   2238 lemma Inf_insert_finite:   2239 fixes S :: "real set"   2240 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"   2241 by (rule Inf_insert, rule finite_imp_bounded, simp)   2242   2243 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)   2244 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"   2245 apply (frule isGlb_isLb)   2246 apply (frule_tac x = y in isGlb_isLb)   2247 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)   2248 done   2249   2250   2251 subsection {* Equivalent versions of compactness *}   2252   2253 subsubsection{* Sequential compactness *}   2254   2255 definition   2256 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)   2257 "compact S \<longleftrightarrow>   2258 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>   2259 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"   2260   2261 lemma compactI:   2262 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"   2263 shows "compact S"   2264 unfolding compact_def using assms by fast   2265   2266 lemma compactE:   2267 assumes "compact S" "\<forall>n. f n \<in> S"   2268 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"   2269 using assms unfolding compact_def by fast   2270   2271 text {*   2272 A metric space (or topological vector space) is said to have the   2273 Heine-Borel property if every closed and bounded subset is compact.   2274 *}   2275   2276 class heine_borel = metric_space +   2277 assumes bounded_imp_convergent_subsequence:   2278 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s   2279 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2280   2281 lemma bounded_closed_imp_compact:   2282 fixes s::"'a::heine_borel set"   2283 assumes "bounded s" and "closed s" shows "compact s"   2284 proof (unfold compact_def, clarify)   2285 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"   2286 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"   2287 using bounded_imp_convergent_subsequence [OF bounded s \<forall>n. f n \<in> s] by auto   2288 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp   2289 have "l \<in> s" using closed s fr l   2290 unfolding closed_sequential_limits by blast   2291 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2292 using l \<in> s r l by blast   2293 qed   2294   2295 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"   2296 proof(induct n)   2297 show "0 \<le> r 0" by auto   2298 next   2299 fix n assume "n \<le> r n"   2300 moreover have "r n < r (Suc n)"   2301 using assms [unfolded subseq_def] by auto   2302 ultimately show "Suc n \<le> r (Suc n)" by auto   2303 qed   2304   2305 lemma eventually_subseq:   2306 assumes r: "subseq r"   2307 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"   2308 unfolding eventually_sequentially   2309 by (metis subseq_bigger [OF r] le_trans)   2310   2311 lemma lim_subseq:   2312 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"   2313 unfolding tendsto_def eventually_sequentially o_def   2314 by (metis subseq_bigger le_trans)   2315   2316 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"   2317 unfolding Ex1_def   2318 apply (rule_tac x="nat_rec e f" in exI)   2319 apply (rule conjI)+   2320 apply (rule def_nat_rec_0, simp)   2321 apply (rule allI, rule def_nat_rec_Suc, simp)   2322 apply (rule allI, rule impI, rule ext)   2323 apply (erule conjE)   2324 apply (induct_tac x)   2325 apply simp   2326 apply (erule_tac x="n" in allE)   2327 apply (simp)   2328 done   2329   2330 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"   2331 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"   2332 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"   2333 proof-   2334 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto   2335 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto   2336 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"   2337 { fix n::nat   2338 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto   2339 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto   2340 with n have "s N \<le> t - e" using e>0 by auto   2341 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 }   2342 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto   2343 hence False using isLub_le_isUb[OF t, of "t - e"] and e>0 by auto }   2344 thus ?thesis by blast   2345 qed   2346   2347 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"   2348 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"   2349 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"   2350 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]   2351 unfolding monoseq_def incseq_def   2352 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]   2353 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto   2354   2355 (* TODO: merge this lemma with the ones above *)   2356 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"   2357 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"   2358 shows "\<exists>l. (s ---> l) sequentially"   2359 proof-   2360 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto   2361 { fix m::nat   2362 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"   2363 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)   2364 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }   2365 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto   2366 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]   2367 unfolding monoseq_def by auto   2368 thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="l" in exI)   2369 unfolding dist_norm by auto   2370 qed   2371   2372 lemma compact_real_lemma:   2373 assumes "\<forall>n::nat. abs(s n) \<le> b"   2374 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"   2375 proof-   2376 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"   2377 using seq_monosub[of s] by auto   2378 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms   2379 unfolding tendsto_iff dist_norm eventually_sequentially by auto   2380 qed   2381   2382 instance real :: heine_borel   2383 proof   2384 fix s :: "real set" and f :: "nat \<Rightarrow> real"   2385 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"   2386 then obtain b where b: "\<forall>n. abs (f n) \<le> b"   2387 unfolding bounded_iff by auto   2388 obtain l :: real and r :: "nat \<Rightarrow> nat" where   2389 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"   2390 using compact_real_lemma [OF b] by auto   2391 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2392 by auto   2393 qed   2394   2395 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x$$ i)  s)"

  2396   apply (erule bounded_linear_image)

  2397   apply (rule bounded_linear_euclidean_component)

  2398   done

  2399

  2400 lemma compact_lemma:

  2401   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  2402   assumes "bounded s" and "\<forall>n. f n \<in> s"

  2403   shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>

  2404         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2405 proof

  2406   fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"

  2407   have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto

  2408   hence "\<exists>l::'a. \<exists>r. subseq r \<and>

  2409       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2410   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  2411   next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto

  2412     have s': "bounded ((\<lambda>x. x $$k)  s)" using bounded s by (rule bounded_component)   2413 obtain l1::"'a" and r1 where r1:"subseq r1" and   2414 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n)$$ i) (l1 $$i) < e) sequentially"   2415 using insert(3) using insert(4) by auto   2416 have f': "\<forall>n. f (r1 n)$$ k \<in> (\<lambda>x. x $$k)  s" using \<forall>n. f n \<in> s by simp   2417 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i))$$ k) ---> l2) sequentially"

  2418       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

  2419     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  2420       using r1 and r2 unfolding r_def o_def subseq_def by auto

  2421     moreover

  2422     def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"   2423 { fix e::real assume "e>0"   2424 from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n)$$ i) (l1 $$i) < e) sequentially" by blast   2425 from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n))$$ k) l2 < e) sequentially" by (rule tendstoD)

  2426       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$i) (l1$$ i) < e) sequentially"

  2427         by (rule eventually_subseq)

  2428       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$i) (l$$ i) < e) sequentially"

  2429         using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def

  2430         using insert.prems by auto

  2431     }

  2432     ultimately show ?case by auto

  2433   qed

  2434   thus "\<exists>l::'a. \<exists>r. subseq r \<and>

  2435       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2436     apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe

  2437     apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe

  2438     apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)

  2439     apply(erule_tac x=i in ballE)

  2440   proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a

  2441     assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"

  2442     hence *:"i\<ge>DIM('a)" by auto

  2443     thus "dist (f (r n) $$i) (l$$ i) < e" using e by auto

  2444   qed

  2445 qed

  2446

  2447 instance euclidean_space \<subseteq> heine_borel

  2448 proof

  2449   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"

  2450   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2451   then obtain l::'a and r where r: "subseq r"

  2452     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$i) (l$$ i) < e) sequentially"

  2453     using compact_lemma [OF s f] by blast

  2454   let ?d = "{..<DIM('a)}"

  2455   { fix e::real assume "e>0"

  2456     hence "0 < e / (real_of_nat (card ?d))"

  2457       using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto

  2458     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$i) (l$$ i) < e / (real_of_nat (card ?d))) sequentially"

  2459       by simp

  2460     moreover

  2461     { fix n assume n: "\<forall>i. dist (f (r n) $$i) (l$$ i) < e / (real_of_nat (card ?d))"

  2462       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$i) (l$$ i))"

  2463         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  2464       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"

  2465         apply(rule setsum_strict_mono) using n by auto

  2466       finally have "dist (f (r n)) l < e" unfolding setsum_constant

  2467         using DIM_positive[where 'a='a] by auto

  2468     }

  2469     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  2470       by (rule eventually_elim1)

  2471   }

  2472   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  2473   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  2474 qed

  2475

  2476 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  2477 unfolding bounded_def

  2478 apply clarify

  2479 apply (rule_tac x="a" in exI)

  2480 apply (rule_tac x="e" in exI)

  2481 apply clarsimp

  2482 apply (drule (1) bspec)

  2483 apply (simp add: dist_Pair_Pair)

  2484 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  2485 done

  2486

  2487 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  2488 unfolding bounded_def

  2489 apply clarify

  2490 apply (rule_tac x="b" in exI)

  2491 apply (rule_tac x="e" in exI)

  2492 apply clarsimp

  2493 apply (drule (1) bspec)

  2494 apply (simp add: dist_Pair_Pair)

  2495 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  2496 done

  2497

  2498 instance prod :: (heine_borel, heine_borel) heine_borel

  2499 proof

  2500   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"

  2501   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2502   from s have s1: "bounded (fst  s)" by (rule bounded_fst)

  2503   from f have f1: "\<forall>n. fst (f n) \<in> fst  s" by simp

  2504   obtain l1 r1 where r1: "subseq r1"

  2505     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"

  2506     using bounded_imp_convergent_subsequence [OF s1 f1]

  2507     unfolding o_def by fast

  2508   from s have s2: "bounded (snd  s)" by (rule bounded_snd)

  2509   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd  s" by simp

  2510   obtain l2 r2 where r2: "subseq r2"

  2511     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  2512     using bounded_imp_convergent_subsequence [OF s2 f2]

  2513     unfolding o_def by fast

  2514   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  2515     using lim_subseq [OF r2 l1] unfolding o_def .

  2516   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  2517     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  2518   have r: "subseq (r1 \<circ> r2)"

  2519     using r1 r2 unfolding subseq_def by simp

  2520   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2521     using l r by fast

  2522 qed

  2523

  2524 subsubsection{* Completeness *}

  2525

  2526 lemma cauchy_def:

  2527   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  2528 unfolding Cauchy_def by blast

  2529

  2530 definition

  2531   complete :: "'a::metric_space set \<Rightarrow> bool" where

  2532   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f

  2533                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"

  2534

  2535 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  2536 proof-

  2537   { assume ?rhs

  2538     { fix e::real

  2539       assume "e>0"

  2540       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  2541         by (erule_tac x="e/2" in allE) auto

  2542       { fix n m

  2543         assume nm:"N \<le> m \<and> N \<le> n"

  2544         hence "dist (s m) (s n) < e" using N

  2545           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  2546           by blast

  2547       }

  2548       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  2549         by blast

  2550     }

  2551     hence ?lhs

  2552       unfolding cauchy_def

  2553       by blast

  2554   }

  2555   thus ?thesis

  2556     unfolding cauchy_def

  2557     using dist_triangle_half_l

  2558     by blast

  2559 qed

  2560

  2561 lemma convergent_imp_cauchy:

  2562  "(s ---> l) sequentially ==> Cauchy s"

  2563 proof(simp only: cauchy_def, rule, rule)

  2564   fix e::real assume "e>0" "(s ---> l) sequentially"

  2565   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

  2566   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

  2567 qed

  2568

  2569 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  2570 proof-

  2571   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

  2572   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  2573   moreover

  2574   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  2575   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  2576     unfolding bounded_any_center [where a="s N"] by auto

  2577   ultimately show "?thesis"

  2578     unfolding bounded_any_center [where a="s N"]

  2579     apply(rule_tac x="max a 1" in exI) apply auto

  2580     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  2581 qed

  2582

  2583 lemma compact_imp_complete: assumes "compact s" shows "complete s"

  2584 proof-

  2585   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  2586     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

  2587

  2588     note lr' = subseq_bigger [OF lr(2)]

  2589

  2590     { fix e::real assume "e>0"

  2591       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

  2592       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

  2593       { fix n::nat assume n:"n \<ge> max N M"

  2594         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  2595         moreover have "r n \<ge> N" using lr'[of n] n by auto

  2596         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  2597         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)  }

  2598       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  2599     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  2600   thus ?thesis unfolding complete_def by auto

  2601 qed

  2602

  2603 instance heine_borel < complete_space

  2604 proof

  2605   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2606   hence "bounded (range f)"

  2607     by (rule cauchy_imp_bounded)

  2608   hence "compact (closure (range f))"

  2609     using bounded_closed_imp_compact [of "closure (range f)"] by auto

  2610   hence "complete (closure (range f))"

  2611     by (rule compact_imp_complete)

  2612   moreover have "\<forall>n. f n \<in> closure (range f)"

  2613     using closure_subset [of "range f"] by auto

  2614   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  2615     using Cauchy f unfolding complete_def by auto

  2616   then show "convergent f"

  2617     unfolding convergent_def by auto

  2618 qed

  2619

  2620 instance euclidean_space \<subseteq> banach ..

  2621

  2622 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  2623 proof(simp add: complete_def, rule, rule)

  2624   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2625   hence "convergent f" by (rule Cauchy_convergent)

  2626   thus "\<exists>l. f ----> l" unfolding convergent_def .

  2627 qed

  2628

  2629 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  2630 proof -

  2631   { fix x assume "x islimpt s"

  2632     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  2633       unfolding islimpt_sequential by auto

  2634     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  2635       using complete s[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto

  2636     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  2637   }

  2638   thus "closed s" unfolding closed_limpt by auto

  2639 qed

  2640

  2641 lemma complete_eq_closed:

  2642   fixes s :: "'a::complete_space set"

  2643   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  2644 proof

  2645   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  2646 next

  2647   assume ?rhs

  2648   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  2649     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  2650     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  }

  2651   thus ?lhs unfolding complete_def by auto

  2652 qed

  2653

  2654 lemma convergent_eq_cauchy:

  2655   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  2656   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  2657   unfolding Cauchy_convergent_iff convergent_def ..

  2658

  2659 lemma convergent_imp_bounded:

  2660   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  2661   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  2662   by (intro cauchy_imp_bounded convergent_imp_cauchy)

  2663

  2664 subsubsection{* Total boundedness *}

  2665

  2666 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  2667   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  2668 declare helper_1.simps[simp del]

  2669

  2670 lemma compact_imp_totally_bounded:

  2671   assumes "compact s"

  2672   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  2673 proof(rule, rule, rule ccontr)

  2674   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)"

  2675   def x \<equiv> "helper_1 s e"

  2676   { fix n

  2677     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  2678     proof(induct_tac rule:nat_less_induct)

  2679       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  2680       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  2681       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

  2682       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  2683       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  2684         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  2685       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  2686     qed }

  2687   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  2688   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

  2689   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto

  2690   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

  2691   show False

  2692     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  2693     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  2694     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  2695 qed

  2696

  2697 subsubsection{* Heine-Borel theorem *}

  2698

  2699 text {* Following Burkill \& Burkill vol. 2. *}

  2700

  2701 lemma heine_borel_lemma: fixes s::"'a::metric_space set"

  2702   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"

  2703   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"

  2704 proof(rule ccontr)

  2705   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"

  2706   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto

  2707   { fix n::nat

  2708     have "1 / real (n + 1) > 0" by auto

  2709     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 }

  2710   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

  2711   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)"

  2712     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

  2713

  2714   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"

  2715     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto

  2716

  2717   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto

  2718   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"

  2719     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto

  2720

  2721   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"

  2722     using lr[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto

  2723

  2724   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and e>0 by auto

  2725   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"

  2726     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2

  2727     using subseq_bigger[OF r, of "N1 + N2"] by auto

  2728

  2729   def x \<equiv> "(f (r (N1 + N2)))"

  2730   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def

  2731     using f[THEN spec[where x="r (N1 + N2)"]] using b\<in>t by auto

  2732   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto

  2733   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto

  2734

  2735   have "dist x l < e/2" using N1 unfolding x_def o_def by auto

  2736   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)

  2737

  2738   thus False using e and y\<notin>b by auto

  2739 qed

  2740

  2741 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  2742                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"

  2743 proof clarify

  2744   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"

  2745   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

  2746   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto

  2747   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

  2748   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast

  2749

  2750   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

  2751   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" by auto

  2752

  2753   have "finite (bb  k)" using k(1) by auto

  2754   moreover

  2755   { fix x assume "x\<in>s"

  2756     hence "x\<in>\<Union>(\<lambda>x. ball x e)  k" using k(3)  unfolding subset_eq by auto

  2757     hence "\<exists>X\<in>bb  k. x \<in> X" using bb k(2) by blast

  2758     hence "x \<in> \<Union>(bb  k)" using  Union_iff[of x "bb  k"] by auto

  2759   }

  2760   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

  2761 qed

  2762

  2763 subsubsection {* Bolzano-Weierstrass property *}

  2764

  2765 lemma heine_borel_imp_bolzano_weierstrass:

  2766   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'))"

  2767           "infinite t"  "t \<subseteq> s"

  2768   shows "\<exists>x \<in> s. x islimpt t"

  2769 proof(rule ccontr)

  2770   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2771   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

  2772     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

  2773   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2774     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2775   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2776   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2777     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

  2778     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  }

  2779   hence "inj_on f t" unfolding inj_on_def by simp

  2780   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2781   moreover

  2782   { fix x assume "x\<in>t" "f x \<notin> g"

  2783     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2784     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2785     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2786     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2787   hence "f  t \<subseteq> g" by auto

  2788   ultimately show False using g(2) using finite_subset by auto

  2789 qed

  2790

  2791 subsubsection {* Complete the chain of compactness variants *}

  2792

  2793 lemma islimpt_range_imp_convergent_subsequence:

  2794   fixes f :: "nat \<Rightarrow> 'a::metric_space"

  2795   assumes "l islimpt (range f)"

  2796   shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2797 proof (intro exI conjI)

  2798   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2799     using assms unfolding islimpt_def

  2800     by (drule_tac x="ball l e" in spec)

  2801        (auto simp add: zero_less_dist_iff dist_commute)

  2802

  2803   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2804   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"

  2805     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])

  2806   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"

  2807     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])

  2808   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"

  2809     unfolding t_def by (simp add: Least_le)

  2810   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"

  2811     unfolding t_def by (drule not_less_Least) simp

  2812   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"

  2813     apply (rule t_le)

  2814     apply (erule f_t_neq)

  2815     apply (erule (1) less_le_trans [OF f_t_closer])

  2816     done

  2817   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"

  2818     by (drule f_t_closer) auto

  2819   have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"

  2820     apply (subst less_le)

  2821     apply (rule conjI)

  2822     apply (rule t_antimono)

  2823     apply (erule f_t_neq)

  2824     apply (erule f_t_closer [THEN less_imp_le])

  2825     apply (rule t_dist_f_neq [symmetric])

  2826     apply (erule f_t_neq)

  2827     done

  2828   have dist_f_t_less':

  2829     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"

  2830     apply (simp add: le_less)

  2831     apply (erule disjE)

  2832     apply (rule less_trans)

  2833     apply (erule f_t_closer)

  2834     apply (rule le_less_trans)

  2835     apply (erule less_tD)

  2836     apply (erule f_t_neq)

  2837     apply (erule f_t_closer)

  2838     apply (erule subst)

  2839     apply (erule f_t_closer)

  2840     done

  2841

  2842   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"

  2843   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"

  2844     unfolding r_def by simp_all

  2845   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"

  2846     by (induct_tac n) (simp_all add: r_simps f_t_neq)

  2847

  2848   show "subseq r"

  2849     unfolding subseq_Suc_iff

  2850     apply (rule allI)

  2851     apply (case_tac n)

  2852     apply (simp_all add: r_simps)

  2853     apply (rule t_less, rule zero_less_one)

  2854     apply (rule t_less, rule f_r_neq)

  2855     done

  2856   show "((f \<circ> r) ---> l) sequentially"

  2857     unfolding LIMSEQ_def o_def

  2858     apply (clarify, rename_tac e, rule_tac x="t e" in exI, clarify)

  2859     apply (drule le_trans, rule seq_suble [OF subseq r])

  2860     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)

  2861     done

  2862 qed

  2863

  2864 lemma finite_range_imp_infinite_repeats:

  2865   fixes f :: "nat \<Rightarrow> 'a"

  2866   assumes "finite (range f)"

  2867   shows "\<exists>k. infinite {n. f n = k}"

  2868 proof -

  2869   { fix A :: "'a set" assume "finite A"

  2870     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"

  2871     proof (induct)

  2872       case empty thus ?case by simp

  2873     next

  2874       case (insert x A)

  2875      show ?case

  2876       proof (cases "finite {n. f n = x}")

  2877         case True

  2878         with infinite {n. f n \<in> insert x A}

  2879         have "infinite {n. f n \<in> A}" by simp

  2880         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)

  2881       next

  2882         case False thus "\<exists>k. infinite {n. f n = k}" ..

  2883       qed

  2884     qed

  2885   } note H = this

  2886   from assms show "\<exists>k. infinite {n. f n = k}"

  2887     by (rule H) simp

  2888 qed

  2889

  2890 lemma bolzano_weierstrass_imp_compact:

  2891   fixes s :: "'a::metric_space set"

  2892   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2893   shows "compact s"

  2894 proof -

  2895   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  2896     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2897     proof (cases "finite (range f)")

  2898       case True

  2899       hence "\<exists>l. infinite {n. f n = l}"

  2900         by (rule finite_range_imp_infinite_repeats)

  2901       then obtain l where "infinite {n. f n = l}" ..

  2902       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"

  2903         by (rule infinite_enumerate)

  2904       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto

  2905       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2906         unfolding o_def by (simp add: fr tendsto_const)

  2907       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2908         by - (rule exI)

  2909       from f have "\<forall>n. f (r n) \<in> s" by simp

  2910       hence "l \<in> s" by (simp add: fr)

  2911       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2912         by (rule rev_bexI) fact

  2913     next

  2914       case False

  2915       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto

  2916       then obtain l where "l \<in> s" "l islimpt (range f)" ..

  2917       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2918         using l islimpt (range f)

  2919         by (rule islimpt_range_imp_convergent_subsequence)

  2920       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  2921     qed

  2922   }

  2923   thus ?thesis unfolding compact_def by auto

  2924 qed

  2925

  2926 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where

  2927   "helper_2 beyond 0 = beyond 0" |

  2928   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"

  2929

  2930 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"

  2931   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2932   shows "bounded s"

  2933 proof(rule ccontr)

  2934   assume "\<not> bounded s"

  2935   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"

  2936     unfolding bounded_any_center [where a=undefined]

  2937     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto

  2938   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"

  2939     unfolding linorder_not_le by auto

  2940   def x \<equiv> "helper_2 beyond"

  2941

  2942   { fix m n ::nat assume "m<n"

  2943     hence "dist undefined (x m) + 1 < dist undefined (x n)"

  2944     proof(induct n)

  2945       case 0 thus ?case by auto

  2946     next

  2947       case (Suc n)

  2948       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"

  2949         unfolding x_def and helper_2.simps

  2950         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto

  2951       thus ?case proof(cases "m < n")

  2952         case True thus ?thesis using Suc and * by auto

  2953       next

  2954         case False hence "m = n" using Suc(2) by auto

  2955         thus ?thesis using * by auto

  2956       qed

  2957     qed  } note * = this

  2958   { fix m n ::nat assume "m\<noteq>n"

  2959     have "1 < dist (x m) (x n)"

  2960     proof(cases "m<n")

  2961       case True

  2962       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto

  2963       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith

  2964     next

  2965       case False hence "n<m" using m\<noteq>n by auto

  2966       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto

  2967       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith

  2968     qed  } note ** = this

  2969   { fix a b assume "x a = x b" "a \<noteq> b"

  2970     hence False using **[of a b] by auto  }

  2971   hence "inj x" unfolding inj_on_def by auto

  2972   moreover

  2973   { fix n::nat

  2974     have "x n \<in> s"

  2975     proof(cases "n = 0")

  2976       case True thus ?thesis unfolding x_def using beyond by auto

  2977     next

  2978       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto

  2979       thus ?thesis unfolding x_def using beyond by auto

  2980     qed  }

  2981   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

  2982

  2983   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto

  2984   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

  2985   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"]]

  2986     unfolding dist_nz by auto

  2987   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto

  2988 qed

  2989

  2990 lemma sequence_infinite_lemma:

  2991   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2992   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2993   shows "infinite (range f)"

  2994 proof

  2995   assume "finite (range f)"

  2996   hence "closed (range f)" by (rule finite_imp_closed)

  2997   hence "open (- range f)" by (rule open_Compl)

  2998   from assms(1) have "l \<in> - range f" by auto

  2999   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  3000     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  3001   thus False unfolding eventually_sequentially by auto

  3002 qed

  3003

  3004 lemma closure_insert:

  3005   fixes x :: "'a::t1_space"

  3006   shows "closure (insert x s) = insert x (closure s)"

  3007 apply (rule closure_unique)

  3008 apply (rule insert_mono [OF closure_subset])

  3009 apply (rule closed_insert [OF closed_closure])

  3010 apply (simp add: closure_minimal)

  3011 done

  3012

  3013 lemma islimpt_insert:

  3014   fixes x :: "'a::t1_space"

  3015   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  3016 proof

  3017   assume *: "x islimpt (insert a s)"

  3018   show "x islimpt s"

  3019   proof (rule islimptI)

  3020     fix t assume t: "x \<in> t" "open t"

  3021     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  3022     proof (cases "x = a")

  3023       case True

  3024       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  3025         using * t by (rule islimptE)

  3026       with x = a show ?thesis by auto

  3027     next

  3028       case False

  3029       with t have t': "x \<in> t - {a}" "open (t - {a})"

  3030         by (simp_all add: open_Diff)

  3031       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  3032         using * t' by (rule islimptE)

  3033       thus ?thesis by auto

  3034     qed

  3035   qed

  3036 next

  3037   assume "x islimpt s" thus "x islimpt (insert a s)"

  3038     by (rule islimpt_subset) auto

  3039 qed

  3040

  3041 lemma islimpt_union_finite:

  3042   fixes x :: "'a::t1_space"

  3043   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3044 by (induct set: finite, simp_all add: islimpt_insert)

  3045

  3046 lemma sequence_unique_limpt:

  3047   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3048   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  3049   shows "l' = l"

  3050 proof (rule ccontr)

  3051   assume "l' \<noteq> l"

  3052   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3053     using hausdorff [OF l' \<noteq> l] by auto

  3054   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3055     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3056   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3057     unfolding eventually_sequentially by auto

  3058

  3059   have "UNIV = {..<N} \<union> {N..}" by auto

  3060   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  3061   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  3062   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  3063   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3064     using l' \<in> s open s by (rule islimptE)

  3065   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  3066   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  3067   with s \<inter> t = {} show False by simp

  3068 qed

  3069

  3070 lemma bolzano_weierstrass_imp_closed:

  3071   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)

  3072   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3073   shows "closed s"

  3074 proof-

  3075   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3076     hence "l \<in> s"

  3077     proof(cases "\<forall>n. x n \<noteq> l")

  3078       case False thus "l\<in>s" using as(1) by auto

  3079     next

  3080       case True note cas = this

  3081       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  3082       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  3083       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  3084     qed  }

  3085   thus ?thesis unfolding closed_sequential_limits by fast

  3086 qed

  3087

  3088 text {* Hence express everything as an equivalence. *}

  3089

  3090 lemma compact_eq_heine_borel:

  3091   fixes s :: "'a::metric_space set"

  3092   shows "compact s \<longleftrightarrow>

  3093            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  3094                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")

  3095 proof

  3096   assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)

  3097 next

  3098   assume ?rhs

  3099   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"

  3100     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])

  3101   thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3102 qed

  3103

  3104 lemma compact_eq_bolzano_weierstrass:

  3105   fixes s :: "'a::metric_space set"

  3106   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3107 proof

  3108   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3109 next

  3110   assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3111 qed

  3112

  3113 lemma nat_approx_posE:

  3114   fixes e::real

  3115   assumes "0 < e"

  3116   obtains n::nat where "1 / (Suc n) < e"

  3117 proof atomize_elim

  3118   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3119     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3120   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3121     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3122   also have "\<dots> = e" by simp

  3123   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3124 qed

  3125

  3126 lemma compact_eq_totally_bounded:

  3127   shows "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3128 proof (safe intro!: compact_imp_complete)

  3129   fix e::real

  3130   def f \<equiv> "(\<lambda>x::'a. ball x e)  UNIV"

  3131   assume "0 < e" "compact s"

  3132   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3133     by (simp add: compact_eq_heine_borel)

  3134   moreover

  3135   have d0: "\<And>x::'a. dist x x < e" using 0 < e by simp

  3136   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f" by (auto simp: f_def intro!: d0)

  3137   ultimately have "(\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')" ..

  3138   then guess K .. note K = this

  3139   have "\<forall>K'\<in>K. \<exists>k. K' = ball k e" using K by (auto simp: f_def)

  3140   then obtain k where "\<And>K'. K' \<in> K \<Longrightarrow> K' = ball (k K') e" unfolding bchoice_iff by blast

  3141   thus "\<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" using K

  3142     by (intro exI[where x="k  K"]) (auto simp: f_def)

  3143 next

  3144   assume assms: "complete s" "\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k"

  3145   show "compact s"

  3146   proof cases

  3147     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3148   next

  3149     assume "s \<noteq> {}"

  3150     show ?thesis

  3151       unfolding compact_def

  3152     proof safe

  3153       fix f::"nat \<Rightarrow> _" assume "\<forall>n. f n \<in> s" hence f: "\<And>n. f n \<in> s" by simp

  3154       from assms have "\<forall>e. \<exists>k. e>0 \<longrightarrow> finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))" by simp

  3155       then obtain K where

  3156         K: "\<And>e. e > 0 \<Longrightarrow> finite (K e) \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3157         unfolding choice_iff by blast

  3158       {

  3159         fix e::real and f' have f': "\<And>n::nat. (f o f') n \<in> s" using f by auto

  3160         assume "e > 0"

  3161         from K[OF this] have K: "finite (K e)" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3162           by simp_all

  3163         have "\<exists>k\<in>(K e). \<exists>r. subseq r \<and> (\<forall>i. (f o f' o r) i \<in> ball k e)"

  3164         proof (rule ccontr)

  3165           from K have "finite (K e)" "K e \<noteq> {}" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3166             using s \<noteq> {}

  3167             by auto

  3168           moreover

  3169           assume "\<not> (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f \<circ> f' o r) i \<in> ball k e))"

  3170           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

  3171           ultimately

  3172           show False using f'

  3173           proof (induct arbitrary: s f f' rule: finite_ne_induct)

  3174             case (singleton x)

  3175             have "\<exists>i. (f \<circ> f' o id) i \<notin> ball x e" by (rule singleton) (auto simp: subseq_def)

  3176             thus ?case using singleton by (auto simp: ball_def)

  3177           next

  3178             case (insert x A)

  3179             show ?case

  3180             proof cases

  3181               have inf_ms: "infinite ((f o f') - s)" using insert by (simp add: vimage_def)

  3182               have "infinite ((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A)))"

  3183                 using insert by (intro infinite_super[OF _ inf_ms]) auto

  3184               also have "((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A))) =

  3185                 {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

  3186               finally have "infinite \<dots>" .

  3187               moreover assume "finite {m. (f o f') m \<in> ball x e}"

  3188               ultimately have inf: "infinite {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by blast

  3189               hence "A \<noteq> {}" by auto then obtain k where "k \<in> A" by auto

  3190               def r \<equiv> "enumerate {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3191               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3192                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3193               hence "subseq r" by (simp add: subseq_def)

  3194               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3195                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3196               show False

  3197               proof (rule insert)

  3198                 show "\<Union>(\<lambda>x. ball x e)  A \<subseteq> \<Union>(\<lambda>x. ball x e)  A" by simp

  3199                 fix k s assume "k \<in> A" "subseq s"

  3200                 thus "\<exists>i. (f o f' o r o s) i \<notin> ball k e" using subseq r

  3201                   by (subst (2) o_assoc[symmetric]) (intro insert(6) subseq_o, simp_all)

  3202               next

  3203                 fix n show "(f \<circ> f' o r) n \<in> \<Union>(\<lambda>x. ball x e)  A" using r_in_set by auto

  3204               qed

  3205             next

  3206               assume inf: "infinite {m. (f o f') m \<in> ball x e}"

  3207               def r \<equiv> "enumerate {m. (f o f') m \<in> ball x e}"

  3208               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3209                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3210               hence "subseq r" by (simp add: subseq_def)

  3211               from insert(6)[OF insertI1 this] obtain i where "(f o f') (r i) \<notin> ball x e" by auto

  3212               moreover

  3213               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> ball x e}"

  3214                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3215               hence "(f o f') (r i) \<in> ball x e" by simp

  3216               ultimately show False by simp

  3217             qed

  3218           qed

  3219         qed

  3220       }

  3221       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

  3222       let ?e = "\<lambda>n. 1 / real (Suc n)"

  3223       let ?P = "\<lambda>n s. \<exists>k\<in>K (?e n). (\<forall>i. (f o s) i \<in> ball k (?e n))"

  3224       interpret subseqs ?P using ex by unfold_locales force

  3225       from complete s have limI: "\<And>f. (\<And>n. f n \<in> s) \<Longrightarrow> Cauchy f \<Longrightarrow> (\<exists>l\<in>s. f ----> l)"

  3226         by (simp add: complete_def)

  3227       have "\<exists>l\<in>s. (f o diagseq) ----> l"

  3228       proof (intro limI metric_CauchyI)

  3229         fix e::real assume "0 < e" hence "0 < e / 2" by auto

  3230         from nat_approx_posE[OF this] guess n . note n = this

  3231         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) n) < e"

  3232         proof (rule exI[where x="Suc n"], safe)

  3233           fix m mm assume "Suc n \<le> m" "Suc n \<le> mm"

  3234           let ?e = "1 / real (Suc n)"

  3235           from reducer_reduces[of n] obtain k where

  3236             "k\<in>K ?e"  "\<And>i. (f o seqseq (Suc n)) i \<in> ball k ?e"

  3237             unfolding seqseq_reducer by auto

  3238           moreover

  3239           note diagseq_sub[OF Suc n \<le> m] diagseq_sub[OF Suc n \<le> mm]

  3240           ultimately have "{(f o diagseq) m, (f o diagseq) mm} \<subseteq> ball k ?e" by auto

  3241           also have "\<dots> \<subseteq> ball k (e / 2)" using n by (intro subset_ball) simp

  3242           finally

  3243           have "dist k ((f \<circ> diagseq) m) + dist k ((f \<circ> diagseq) mm) < e / 2 + e /2"

  3244             by (intro add_strict_mono) auto

  3245           hence "dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k < e"

  3246             by (simp add: dist_commute)

  3247           moreover have "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) \<le>

  3248             dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k"

  3249             by (rule dist_triangle2)

  3250           ultimately show "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) < e"

  3251             by simp

  3252         qed

  3253       next

  3254         fix n show "(f o diagseq) n \<in> s" using f by simp

  3255       qed

  3256       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l" using subseq_diagseq by auto

  3257     qed

  3258   qed

  3259 qed

  3260

  3261 lemma compact_eq_bounded_closed:

  3262   fixes s :: "'a::heine_borel set"

  3263   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3264 proof

  3265   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto

  3266 next

  3267   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto

  3268 qed

  3269

  3270 lemma compact_imp_bounded:

  3271   fixes s :: "'a::metric_space set"

  3272   shows "compact s ==> bounded s"

  3273 proof -

  3274   assume "compact s"

  3275   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')"

  3276     by (rule compact_imp_heine_borel)

  3277   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"

  3278     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3279   thus "bounded s"

  3280     by (rule bolzano_weierstrass_imp_bounded)

  3281 qed

  3282

  3283 lemma compact_imp_closed:

  3284   fixes s :: "'a::metric_space set"

  3285   shows "compact s ==> closed s"

  3286 proof -

  3287   assume "compact s"

  3288   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')"

  3289     by (rule compact_imp_heine_borel)

  3290   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"

  3291     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3292   thus "closed s"

  3293     by (rule bolzano_weierstrass_imp_closed)

  3294 qed

  3295

  3296 text{* In particular, some common special cases. *}

  3297

  3298 lemma compact_empty[simp]:

  3299  "compact {}"

  3300   unfolding compact_def

  3301   by simp

  3302

  3303 lemma compact_union [intro]:

  3304   assumes "compact s" and "compact t"

  3305   shows "compact (s \<union> t)"

  3306 proof (rule compactI)

  3307   fix f :: "nat \<Rightarrow> 'a"

  3308   assume "\<forall>n. f n \<in> s \<union> t"

  3309   hence "infinite {n. f n \<in> s \<union> t}" by simp

  3310   hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp

  3311   thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3312   proof

  3313     assume "infinite {n. f n \<in> s}"

  3314     from infinite_enumerate [OF this]

  3315     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto

  3316     obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3317       using compact s \<forall>n. (f \<circ> q) n \<in> s by (rule compactE)

  3318     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3319       using subseq q by (simp_all add: subseq_o o_assoc)

  3320     thus ?thesis by auto

  3321   next

  3322     assume "infinite {n. f n \<in> t}"

  3323     from infinite_enumerate [OF this]

  3324     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto

  3325     obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3326       using compact t \<forall>n. (f \<circ> q) n \<in> t by (rule compactE)

  3327     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3328       using subseq q by (simp_all add: subseq_o o_assoc)

  3329     thus ?thesis by auto

  3330   qed

  3331 qed

  3332

  3333 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3334   by (induct set: finite) auto

  3335

  3336 lemma compact_UN [intro]:

  3337   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3338   unfolding SUP_def by (rule compact_Union) auto

  3339

  3340 lemma compact_inter_closed [intro]:

  3341   assumes "compact s" and "closed t"

  3342   shows "compact (s \<inter> t)"

  3343 proof (rule compactI)

  3344   fix f :: "nat \<Rightarrow> 'a"

  3345   assume "\<forall>n. f n \<in> s \<inter> t"

  3346   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all

  3347   obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3348     using compact s \<forall>n. f n \<in> s by (rule compactE)

  3349   moreover

  3350   from closed t \<forall>n. f n \<in> t ((f \<circ> r) ---> l) sequentially have "l \<in> t"

  3351     unfolding closed_sequential_limits o_def by fast

  3352   ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3353     by auto

  3354 qed

  3355

  3356 lemma closed_inter_compact [intro]:

  3357   assumes "closed s" and "compact t"

  3358   shows "compact (s \<inter> t)"

  3359   using compact_inter_closed [of t s] assms

  3360   by (simp add: Int_commute)

  3361

  3362 lemma compact_inter [intro]:

  3363   assumes "compact s" and "compact t"

  3364   shows "compact (s \<inter> t)"

  3365   using assms by (intro compact_inter_closed compact_imp_closed)

  3366

  3367 lemma compact_sing [simp]: "compact {a}"

  3368   unfolding compact_def o_def subseq_def

  3369   by (auto simp add: tendsto_const)

  3370

  3371 lemma compact_insert [simp]:

  3372   assumes "compact s" shows "compact (insert x s)"

  3373 proof -

  3374   have "compact ({x} \<union> s)"

  3375     using compact_sing assms by (rule compact_union)

  3376   thus ?thesis by simp

  3377 qed

  3378

  3379 lemma finite_imp_compact:

  3380   shows "finite s \<Longrightarrow> compact s"

  3381   by (induct set: finite) simp_all

  3382

  3383 lemma compact_cball[simp]:

  3384   fixes x :: "'a::heine_borel"

  3385   shows "compact(cball x e)"

  3386   using compact_eq_bounded_closed bounded_cball closed_cball

  3387   by blast

  3388

  3389 lemma compact_frontier_bounded[intro]:

  3390   fixes s :: "'a::heine_borel set"

  3391   shows "bounded s ==> compact(frontier s)"

  3392   unfolding frontier_def

  3393   using compact_eq_bounded_closed

  3394   by blast

  3395

  3396 lemma compact_frontier[intro]:

  3397   fixes s :: "'a::heine_borel set"

  3398   shows "compact s ==> compact (frontier s)"

  3399   using compact_eq_bounded_closed compact_frontier_bounded

  3400   by blast

  3401

  3402 lemma frontier_subset_compact:

  3403   fixes s :: "'a::heine_borel set"

  3404   shows "compact s ==> frontier s \<subseteq> s"

  3405   using frontier_subset_closed compact_eq_bounded_closed

  3406   by blast

  3407

  3408 lemma open_delete:

  3409   fixes s :: "'a::t1_space set"

  3410   shows "open s \<Longrightarrow> open (s - {x})"

  3411   by (simp add: open_Diff)

  3412

  3413 text{* Finite intersection property. I could make it an equivalence in fact. *}

  3414

  3415 lemma compact_imp_fip:

  3416   assumes "compact s"  "\<forall>t \<in> f. closed t"

  3417         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"

  3418   shows "s \<inter> (\<Inter> f) \<noteq> {}"

  3419 proof

  3420   assume as:"s \<inter> (\<Inter> f) = {}"

  3421   hence "s \<subseteq> \<Union> uminus  f" by auto

  3422   moreover have "Ball (uminus  f) open" using open_Diff closed_Diff using assms(2) by auto

  3423   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

  3424   hence "finite (uminus  f') \<and> uminus  f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)

  3425   hence "s \<inter> \<Inter>uminus  f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus  f'"]] by auto

  3426   thus False using f'(3) unfolding subset_eq and Union_iff by blast

  3427 qed

  3428

  3429

  3430 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3431

  3432 lemma bounded_closed_nest:

  3433   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3434   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3435   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3436 proof-

  3437   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

  3438   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto

  3439

  3440   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3441     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3442

  3443   { fix n::nat

  3444     { fix e::real assume "e>0"

  3445       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3446       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3447       moreover

  3448       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto

  3449       hence "(x \<circ> r) (max N n) \<in> s n"

  3450         using x apply(erule_tac x=n in allE)

  3451         using x apply(erule_tac x="r (max N n)" in allE)

  3452         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto

  3453       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3454     }

  3455     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3456   }

  3457   thus ?thesis by auto

  3458 qed

  3459

  3460 text {* Decreasing case does not even need compactness, just completeness. *}

  3461

  3462 lemma decreasing_closed_nest:

  3463   assumes "\<forall>n. closed(s n)"

  3464           "\<forall>n. (s n \<noteq> {})"

  3465           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3466           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3467   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3468 proof-

  3469   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3470   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3471   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3472   { fix e::real assume "e>0"

  3473     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3474     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3475       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3476       hence "dist (t m) (t n) < e" using N by auto

  3477     }

  3478     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3479   }

  3480   hence  "Cauchy t" unfolding cauchy_def by auto

  3481   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3482   { fix n::nat

  3483     { fix e::real assume "e>0"

  3484       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3485       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

  3486       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3487     }

  3488     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3489   }

  3490   then show ?thesis by auto

  3491 qed

  3492

  3493 text {* Strengthen it to the intersection actually being a singleton. *}

  3494

  3495 lemma decreasing_closed_nest_sing:

  3496   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3497   assumes "\<forall>n. closed(s n)"

  3498           "\<forall>n. s n \<noteq> {}"

  3499           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3500           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3501   shows "\<exists>a. \<Inter>(range s) = {a}"

  3502 proof-

  3503   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3504   { fix b assume b:"b \<in> \<Inter>(range s)"

  3505     { fix e::real assume "e>0"

  3506       hence "dist a b < e" using assms(4 )using b using a by blast

  3507     }

  3508     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3509   }

  3510   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3511   thus ?thesis ..

  3512 qed

  3513

  3514 text{* Cauchy-type criteria for uniform convergence. *}

  3515

  3516 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows

  3517  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3518   (\<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")

  3519 proof(rule)

  3520   assume ?lhs

  3521   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

  3522   { fix e::real assume "e>0"

  3523     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

  3524     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3525       hence "dist (s m x) (s n x) < e"

  3526         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3527         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3528         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3529     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  }

  3530   thus ?rhs by auto

  3531 next

  3532   assume ?rhs

  3533   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

  3534   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3535     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3536   { fix e::real assume "e>0"

  3537     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"

  3538       using ?rhs[THEN spec[where x="e/2"]] by auto

  3539     { fix x assume "P x"

  3540       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3541         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3542       fix n::nat assume "n\<ge>N"

  3543       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]]

  3544         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)  }

  3545     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3546   thus ?lhs by auto

  3547 qed

  3548

  3549 lemma uniformly_cauchy_imp_uniformly_convergent:

  3550   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"

  3551   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"

  3552           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3553   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3554 proof-

  3555   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"

  3556     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3557   moreover

  3558   { fix x assume "P x"

  3559     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3560       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3561   ultimately show ?thesis by auto

  3562 qed

  3563

  3564

  3565 subsection {* Continuity *}

  3566

  3567 text {* Define continuity over a net to take in restrictions of the set. *}

  3568

  3569 definition

  3570   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3571   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"

  3572

  3573 lemma continuous_trivial_limit:

  3574  "trivial_limit net ==> continuous net f"

  3575   unfolding continuous_def tendsto_def trivial_limit_eq by auto

  3576

  3577 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"

  3578   unfolding continuous_def

  3579   unfolding tendsto_def

  3580   using netlimit_within[of x s]

  3581   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)

  3582

  3583 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"

  3584   using continuous_within [of x UNIV f] by simp

  3585

  3586 lemma continuous_at_within:

  3587   assumes "continuous (at x) f"  shows "continuous (at x within s) f"

  3588   using assms unfolding continuous_at continuous_within

  3589   by (rule Lim_at_within)

  3590

  3591 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3592

  3593 lemma continuous_within_eps_delta:

  3594   "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)"

  3595   unfolding continuous_within and Lim_within

  3596   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3597

  3598 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3599                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3600   using continuous_within_eps_delta [of x UNIV f] by simp

  3601

  3602 text{* Versions in terms of open balls. *}

  3603

  3604 lemma continuous_within_ball:

  3605  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3606                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3607 proof

  3608   assume ?lhs

  3609   { fix e::real assume "e>0"

  3610     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"

  3611       using ?lhs[unfolded continuous_within Lim_within] by auto

  3612     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3613       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3614         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3615     }

  3616     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)  }

  3617   thus ?rhs by auto

  3618 next

  3619   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3620     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3621 qed

  3622

  3623 lemma continuous_at_ball:

  3624   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3625 proof

  3626   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3627     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)

  3628     unfolding dist_nz[THEN sym] by auto

  3629 next

  3630   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3631     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)

  3632 qed

  3633

  3634 text{* Define setwise continuity in terms of limits within the set. *}

  3635

  3636 definition

  3637   continuous_on ::

  3638     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3639 where

  3640   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"

  3641

  3642 lemma continuous_on_topological:

  3643   "continuous_on s f \<longleftrightarrow>

  3644     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  3645       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  3646 unfolding continuous_on_def tendsto_def

  3647 unfolding Limits.eventually_within eventually_at_topological

  3648 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  3649

  3650 lemma continuous_on_iff:

  3651   "continuous_on s f \<longleftrightarrow>

  3652     (\<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)"

  3653 unfolding continuous_on_def Lim_within

  3654 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3655 apply (rename_tac y, case_tac "y = x", simp)

  3656 apply (simp add: dist_nz)

  3657 done

  3658

  3659 definition

  3660   uniformly_continuous_on ::

  3661     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3662 where

  3663   "uniformly_continuous_on s f \<longleftrightarrow>

  3664     (\<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)"

  3665

  3666 text{* Some simple consequential lemmas. *}

  3667

  3668 lemma uniformly_continuous_imp_continuous:

  3669  " uniformly_continuous_on s f ==> continuous_on s f"

  3670   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3671

  3672 lemma continuous_at_imp_continuous_within:

  3673  "continuous (at x) f ==> continuous (at x within s) f"

  3674   unfolding continuous_within continuous_at using Lim_at_within by auto

  3675

  3676 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3677 unfolding tendsto_def by (simp add: trivial_limit_eq)

  3678

  3679 lemma continuous_at_imp_continuous_on:

  3680   assumes "\<forall>x\<in>s. continuous (at x) f"

  3681   shows "continuous_on s f"

  3682 unfolding continuous_on_def

  3683 proof

  3684   fix x assume "x \<in> s"

  3685   with assms have *: "(f ---> f (netlimit (at x))) (at x)"

  3686     unfolding continuous_def by simp

  3687   have "(f ---> f x) (at x)"

  3688   proof (cases "trivial_limit (at x)")

  3689     case True thus ?thesis

  3690       by (rule Lim_trivial_limit)

  3691   next

  3692     case False

  3693     hence 1: "netlimit (at x) = x"

  3694       using netlimit_within [of x UNIV] by simp

  3695     with * show ?thesis by simp

  3696   qed

  3697   thus "(f ---> f x) (at x within s)"

  3698     by (rule Lim_at_within)

  3699 qed

  3700

  3701 lemma continuous_on_eq_continuous_within:

  3702   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"

  3703 unfolding continuous_on_def continuous_def

  3704 apply (rule ball_cong [OF refl])

  3705 apply (case_tac "trivial_limit (at x within s)")

  3706 apply (simp add: Lim_trivial_limit)

  3707 apply (simp add: netlimit_within)

  3708 done

  3709

  3710 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3711

  3712 lemma continuous_on_eq_continuous_at:

  3713   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"

  3714   by (auto simp add: continuous_on continuous_at Lim_within_open)

  3715

  3716 lemma continuous_within_subset:

  3717  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3718              ==> continuous (at x within t) f"

  3719   unfolding continuous_within by(metis Lim_within_subset)

  3720

  3721 lemma continuous_on_subset:

  3722   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"

  3723   unfolding continuous_on by (metis subset_eq Lim_within_subset)

  3724

  3725 lemma continuous_on_interior:

  3726   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3727   by (erule interiorE, drule (1) continuous_on_subset,

  3728     simp add: continuous_on_eq_continuous_at)

  3729

  3730 lemma continuous_on_eq:

  3731   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3732   unfolding continuous_on_def tendsto_def Limits.eventually_within

  3733   by simp

  3734

  3735 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3736

  3737 lemma continuous_within_sequentially:

  3738   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3739   shows "continuous (at a within s) f \<longleftrightarrow>

  3740                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3741                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3742 proof

  3743   assume ?lhs

  3744   { 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"

  3745     fix T::"'b set" assume "open T" and "f a \<in> T"

  3746     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"

  3747       unfolding continuous_within tendsto_def eventually_within by auto

  3748     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3749       using x(2) d>0 by simp

  3750     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3751     proof eventually_elim

  3752       case (elim n) thus ?case

  3753         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3754     qed

  3755   }

  3756   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3757 next

  3758   assume ?rhs thus ?lhs

  3759     unfolding continuous_within tendsto_def [where l="f a"]

  3760     by (simp add: sequentially_imp_eventually_within)

  3761 qed

  3762

  3763 lemma continuous_at_sequentially:

  3764   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3765   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  3766                   --> ((f o x) ---> f a) sequentially)"

  3767   using continuous_within_sequentially[of a UNIV f] by simp

  3768

  3769 lemma continuous_on_sequentially:

  3770   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3771   shows "continuous_on s f \<longleftrightarrow>

  3772     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  3773                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  3774 proof

  3775   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  3776 next

  3777   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  3778 qed

  3779

  3780 lemma uniformly_continuous_on_sequentially:

  3781   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  3782                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  3783                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  3784 proof

  3785   assume ?lhs

  3786   { 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"

  3787     { fix e::real assume "e>0"

  3788       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"

  3789         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  3790       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

  3791       { fix n assume "n\<ge>N"

  3792         hence "dist (f (x n)) (f (y n)) < e"

  3793           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

  3794           unfolding dist_commute by simp  }

  3795       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  3796     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  3797   thus ?rhs by auto

  3798 next

  3799   assume ?rhs

  3800   { assume "\<not> ?lhs"

  3801     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

  3802     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"

  3803       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

  3804       by (auto simp add: dist_commute)

  3805     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  3806     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  3807     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"

  3808       unfolding x_def and y_def using fa by auto

  3809     { fix e::real assume "e>0"

  3810       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

  3811       { fix n::nat assume "n\<ge>N"

  3812         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  3813         also have "\<dots> < e" using N by auto

  3814         finally have "inverse (real n + 1) < e" by auto

  3815         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  3816       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  3817     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

  3818     hence False using fxy and e>0 by auto  }

  3819   thus ?lhs unfolding uniformly_continuous_on_def by blast

  3820 qed

  3821

  3822 text{* The usual transformation theorems. *}

  3823

  3824 lemma continuous_transform_within:

  3825   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3826   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  3827           "continuous (at x within s) f"

  3828   shows "continuous (at x within s) g"

  3829 unfolding continuous_within

  3830 proof (rule Lim_transform_within)

  3831   show "0 < d" by fact

  3832   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  3833     using assms(3) by auto

  3834   have "f x = g x"

  3835     using assms(1,2,3) by auto

  3836   thus "(f ---> g x) (at x within s)"

  3837     using assms(4) unfolding continuous_within by simp

  3838 qed

  3839

  3840 lemma continuous_transform_at:

  3841   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3842   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  3843           "continuous (at x) f"

  3844   shows "continuous (at x) g"

  3845   using continuous_transform_within [of d x UNIV f g] assms by simp

  3846

  3847 subsubsection {* Structural rules for pointwise continuity *}

  3848

  3849 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"

  3850   unfolding continuous_within by (rule tendsto_ident_at_within)

  3851

  3852 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"

  3853   unfolding continuous_at by (rule tendsto_ident_at)

  3854

  3855 lemma continuous_const: "continuous F (\<lambda>x. c)"

  3856   unfolding continuous_def by (rule tendsto_const)

  3857

  3858 lemma continuous_dist:

  3859   assumes "continuous F f" and "continuous F g"

  3860   shows "continuous F (\<lambda>x. dist (f x) (g x))"

  3861   using assms unfolding continuous_def by (rule tendsto_dist)

  3862

  3863 lemma continuous_infdist:

  3864   assumes "continuous F f"

  3865   shows "continuous F (\<lambda>x. infdist (f x) A)"

  3866   using assms unfolding continuous_def by (rule tendsto_infdist)

  3867

  3868 lemma continuous_norm:

  3869   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"

  3870   unfolding continuous_def by (rule tendsto_norm)

  3871

  3872 lemma continuous_infnorm:

  3873   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  3874   unfolding continuous_def by (rule tendsto_infnorm)

  3875

  3876 lemma continuous_add:

  3877   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3878   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"

  3879   unfolding continuous_def by (rule tendsto_add)

  3880

  3881 lemma continuous_minus:

  3882   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3883   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"

  3884   unfolding continuous_def by (rule tendsto_minus)

  3885

  3886 lemma continuous_diff:

  3887   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3888   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"

  3889   unfolding continuous_def by (rule tendsto_diff)

  3890

  3891 lemma continuous_scaleR:

  3892   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3893   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"

  3894   unfolding continuous_def by (rule tendsto_scaleR)

  3895

  3896 lemma continuous_mult:

  3897   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"

  3898   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"

  3899   unfolding continuous_def by (rule tendsto_mult)

  3900

  3901 lemma continuous_inner:

  3902   assumes "continuous F f" and "continuous F g"

  3903   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  3904   using assms unfolding continuous_def by (rule tendsto_inner)

  3905

  3906 lemma continuous_euclidean_component:

  3907   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $$i)"   3908 unfolding continuous_def by (rule tendsto_euclidean_component)   3909   3910 lemma continuous_inverse:   3911 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3912 assumes "continuous F f" and "f (netlimit F) \<noteq> 0"   3913 shows "continuous F (\<lambda>x. inverse (f x))"   3914 using assms unfolding continuous_def by (rule tendsto_inverse)   3915   3916 lemma continuous_at_within_inverse:   3917 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3918 assumes "continuous (at a within s) f" and "f a \<noteq> 0"   3919 shows "continuous (at a within s) (\<lambda>x. inverse (f x))"   3920 using assms unfolding continuous_within by (rule tendsto_inverse)   3921   3922 lemma continuous_at_inverse:   3923 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3924 assumes "continuous (at a) f" and "f a \<noteq> 0"   3925 shows "continuous (at a) (\<lambda>x. inverse (f x))"   3926 using assms unfolding continuous_at by (rule tendsto_inverse)   3927   3928 lemmas continuous_intros = continuous_at_id continuous_within_id   3929 continuous_const continuous_dist continuous_norm continuous_infnorm   3930 continuous_add continuous_minus continuous_diff   3931 continuous_scaleR continuous_mult   3932 continuous_inner continuous_euclidean_component   3933 continuous_at_inverse continuous_at_within_inverse   3934   3935 subsubsection {* Structural rules for setwise continuity *}   3936   3937 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"   3938 unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)   3939   3940 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"   3941 unfolding continuous_on_def by (auto intro: tendsto_intros)   3942   3943 lemma continuous_on_norm:   3944 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"   3945 unfolding continuous_on_def by (fast intro: tendsto_norm)   3946   3947 lemma continuous_on_infnorm:   3948 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"   3949 unfolding continuous_on by (fast intro: tendsto_infnorm)   3950   3951 lemma continuous_on_minus:   3952 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3953 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"   3954 unfolding continuous_on_def by (auto intro: tendsto_intros)   3955   3956 lemma continuous_on_add:   3957 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3958 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3959 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"   3960 unfolding continuous_on_def by (auto intro: tendsto_intros)   3961   3962 lemma continuous_on_diff:   3963 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3964 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3965 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"   3966 unfolding continuous_on_def by (auto intro: tendsto_intros)   3967   3968 lemma (in bounded_linear) continuous_on:   3969 "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"   3970 unfolding continuous_on_def by (fast intro: tendsto)   3971   3972 lemma (in bounded_bilinear) continuous_on:   3973 "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"   3974 unfolding continuous_on_def by (fast intro: tendsto)   3975   3976 lemma continuous_on_scaleR:   3977 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3978 assumes "continuous_on s f" and "continuous_on s g"   3979 shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"   3980 using bounded_bilinear_scaleR assms   3981 by (rule bounded_bilinear.continuous_on)   3982   3983 lemma continuous_on_mult:   3984 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"   3985 assumes "continuous_on s f" and "continuous_on s g"   3986 shows "continuous_on s (\<lambda>x. f x * g x)"   3987 using bounded_bilinear_mult assms   3988 by (rule bounded_bilinear.continuous_on)   3989   3990 lemma continuous_on_inner:   3991 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"   3992 assumes "continuous_on s f" and "continuous_on s g"   3993 shows "continuous_on s (\<lambda>x. inner (f x) (g x))"   3994 using bounded_bilinear_inner assms   3995 by (rule bounded_bilinear.continuous_on)   3996   3997 lemma continuous_on_euclidean_component:   3998 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x$$ i)"

  3999   using bounded_linear_euclidean_component

  4000   by (rule bounded_linear.continuous_on)

  4001

  4002 lemma continuous_on_inverse:

  4003   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"

  4004   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

  4005   shows "continuous_on s (\<lambda>x. inverse (f x))"

  4006   using assms unfolding continuous_on by (fast intro: tendsto_inverse)

  4007

  4008 subsubsection {* Structural rules for uniform continuity *}

  4009

  4010 lemma uniformly_continuous_on_id:

  4011   shows "uniformly_continuous_on s (\<lambda>x. x)"

  4012   unfolding uniformly_continuous_on_def by auto

  4013

  4014 lemma uniformly_continuous_on_const:

  4015   shows "uniformly_continuous_on s (\<lambda>x. c)"

  4016   unfolding uniformly_continuous_on_def by simp

  4017

  4018 lemma uniformly_continuous_on_dist:

  4019   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4020   assumes "uniformly_continuous_on s f"

  4021   assumes "uniformly_continuous_on s g"

  4022   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4023 proof -

  4024   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4025       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4026       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4027       by arith

  4028   } note le = this

  4029   { fix x y

  4030     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4031     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4032     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4033       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4034         simp add: le)

  4035   }

  4036   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  4037     unfolding dist_real_def by simp

  4038 qed

  4039

  4040 lemma uniformly_continuous_on_norm:

  4041   assumes "uniformly_continuous_on s f"

  4042   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4043   unfolding norm_conv_dist using assms

  4044   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4045

  4046 lemma (in bounded_linear) uniformly_continuous_on:

  4047   assumes "uniformly_continuous_on s g"

  4048   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4049   using assms unfolding uniformly_continuous_on_sequentially

  4050   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4051   by (auto intro: tendsto_zero)

  4052

  4053 lemma uniformly_continuous_on_cmul:

  4054   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4055   assumes "uniformly_continuous_on s f"

  4056   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4057   using bounded_linear_scaleR_right assms

  4058   by (rule bounded_linear.uniformly_continuous_on)

  4059

  4060 lemma dist_minus:

  4061   fixes x y :: "'a::real_normed_vector"

  4062   shows "dist (- x) (- y) = dist x y"

  4063   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4064

  4065 lemma uniformly_continuous_on_minus:

  4066   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4067   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4068   unfolding uniformly_continuous_on_def dist_minus .

  4069

  4070 lemma uniformly_continuous_on_add:

  4071   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4072   assumes "uniformly_continuous_on s f"

  4073   assumes "uniformly_continuous_on s g"

  4074   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4075   using assms unfolding uniformly_continuous_on_sequentially

  4076   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4077   by (auto intro: tendsto_add_zero)

  4078

  4079 lemma uniformly_continuous_on_diff:

  4080   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4081   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4082   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4083   unfolding ab_diff_minus using assms

  4084   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4085

  4086 text{* Continuity of all kinds is preserved under composition. *}

  4087

  4088 lemma continuous_within_topological:

  4089   "continuous (at x within s) f \<longleftrightarrow>

  4090     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  4091       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  4092 unfolding continuous_within

  4093 unfolding tendsto_def Limits.eventually_within eventually_at_topological

  4094 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  4095

  4096 lemma continuous_within_compose:

  4097   assumes "continuous (at x within s) f"

  4098   assumes "continuous (at (f x) within f  s) g"

  4099   shows "continuous (at x within s) (g o f)"

  4100 using assms unfolding continuous_within_topological by simp metis

  4101

  4102 lemma continuous_at_compose:

  4103   assumes "continuous (at x) f" and "continuous (at (f x)) g"

  4104   shows "continuous (at x) (g o f)"

  4105 proof-

  4106   have "continuous (at (f x) within range f) g" using assms(2)

  4107     using continuous_within_subset[of "f x" UNIV g "range f"] by simp

  4108   thus ?thesis using assms(1)

  4109     using continuous_within_compose[of x UNIV f g] by simp

  4110 qed

  4111

  4112 lemma continuous_on_compose:

  4113   "continuous_on s f \<Longrightarrow> continuous_on (f  s) g \<Longrightarrow> continuous_on s (g o f)"

  4114   unfolding continuous_on_topological by simp metis

  4115

  4116 lemma uniformly_continuous_on_compose:

  4117   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4118   shows "uniformly_continuous_on s (g o f)"

  4119 proof-

  4120   { fix e::real assume "e>0"

  4121     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

  4122     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

  4123     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  }

  4124   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4125 qed

  4126

  4127 lemmas continuous_on_intros = continuous_on_id continuous_on_const

  4128   continuous_on_compose continuous_on_norm continuous_on_infnorm

  4129   continuous_on_add continuous_on_minus continuous_on_diff

  4130   continuous_on_scaleR continuous_on_mult continuous_on_inverse

  4131   continuous_on_inner continuous_on_euclidean_component

  4132   uniformly_continuous_on_id uniformly_continuous_on_const

  4133   uniformly_continuous_on_dist uniformly_continuous_on_norm

  4134   uniformly_continuous_on_compose uniformly_continuous_on_add

  4135   uniformly_continuous_on_minus uniformly_continuous_on_diff

  4136   uniformly_continuous_on_cmul

  4137

  4138 text{* Continuity in terms of open preimages. *}

  4139

  4140 lemma continuous_at_open:

  4141   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)))"

  4142 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  4143 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4144

  4145 lemma continuous_on_open:

  4146   shows "continuous_on s f \<longleftrightarrow>

  4147         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4148             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4149 proof (safe)

  4150   fix t :: "'b set"

  4151   assume 1: "continuous_on s f"

  4152   assume 2: "openin (subtopology euclidean (f  s)) t"

  4153   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  4154     unfolding openin_open by auto

  4155   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  4156   have "open U" unfolding U_def by (simp add: open_Union)

  4157   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  4158   proof (intro ballI iffI)

  4159     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  4160       unfolding U_def t by auto

  4161   next

  4162     fix x assume "x \<in> s" and "f x \<in> t"

  4163     hence "x \<in> s" and "f x \<in> B"

  4164       unfolding t by auto

  4165     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  4166       unfolding t continuous_on_topological by metis

  4167     then show "x \<in> U"

  4168       unfolding U_def by auto

  4169   qed

  4170   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  4171   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4172     unfolding openin_open by fast

  4173 next

  4174   assume "?rhs" show "continuous_on s f"

  4175   unfolding continuous_on_topological

  4176   proof (clarify)

  4177     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  4178     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  4179       unfolding openin_open using open B by auto

  4180     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  4181       using ?rhs by fast

  4182     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  4183       unfolding openin_open using x \<in> s and f x \<in> B by auto

  4184   qed

  4185 qed

  4186

  4187 text {* Similarly in terms of closed sets. *}

  4188

  4189 lemma continuous_on_closed:

  4190   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")

  4191 proof

  4192   assume ?lhs

  4193   { fix t

  4194     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4195     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  4196     assume as:"closedin (subtopology euclidean (f  s)) t"

  4197     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  4198     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"]]

  4199       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  4200   thus ?rhs by auto

  4201 next

  4202   assume ?rhs

  4203   { fix t

  4204     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4205     assume as:"openin (subtopology euclidean (f  s)) t"

  4206     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  4207       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  4208   thus ?lhs unfolding continuous_on_open by auto

  4209 qed

  4210

  4211 text {* Half-global and completely global cases. *}

  4212

  4213 lemma continuous_open_in_preimage:

  4214   assumes "continuous_on s f"  "open t"

  4215   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4216 proof-

  4217   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

  4218   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4219     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4220   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4221 qed

  4222

  4223 lemma continuous_closed_in_preimage:

  4224   assumes "continuous_on s f"  "closed t"

  4225   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4226 proof-

  4227   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

  4228   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4229     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4230   thus ?thesis

  4231     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4232 qed

  4233

  4234 lemma continuous_open_preimage:

  4235   assumes "continuous_on s f" "open s" "open t"

  4236   shows "open {x \<in> s. f x \<in> t}"

  4237 proof-

  4238   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4239     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4240   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4241 qed

  4242

  4243 lemma continuous_closed_preimage:

  4244   assumes "continuous_on s f" "closed s" "closed t"

  4245   shows "closed {x \<in> s. f x \<in> t}"

  4246 proof-

  4247   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4248     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4249   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4250 qed

  4251

  4252 lemma continuous_open_preimage_univ:

  4253   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4254   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4255

  4256 lemma continuous_closed_preimage_univ:

  4257   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4258   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4259

  4260 lemma continuous_open_vimage:

  4261   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4262   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4263

  4264 lemma continuous_closed_vimage:

  4265   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4266   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4267

  4268 lemma interior_image_subset:

  4269   assumes "\<forall>x. continuous (at x) f" "inj f"

  4270   shows "interior (f  s) \<subseteq> f  (interior s)"

  4271 proof

  4272   fix x assume "x \<in> interior (f  s)"

  4273   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4274   hence "x \<in> f  s" by auto

  4275   then obtain y where y: "y \<in> s" "x = f y" by auto

  4276   have "open (vimage f T)"

  4277     using assms(1) open T by (rule continuous_open_vimage)

  4278   moreover have "y \<in> vimage f T"

  4279     using x = f y x \<in> T by simp

  4280   moreover have "vimage f T \<subseteq> s"

  4281     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4282   ultimately have "y \<in> interior s" ..

  4283   with x = f y show "x \<in> f  interior s" ..

  4284 qed

  4285

  4286 text {* Equality of continuous functions on closure and related results. *}

  4287

  4288 lemma continuous_closed_in_preimage_constant:

  4289   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4290   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4291   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4292

  4293 lemma continuous_closed_preimage_constant:

  4294   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4295   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4296   using continuous_closed_preimage[of s f "{a}"] by auto

  4297

  4298 lemma continuous_constant_on_closure:

  4299   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4300   assumes "continuous_on (closure s) f"

  4301           "\<forall>x \<in> s. f x = a"

  4302   shows "\<forall>x \<in> (closure s). f x = a"

  4303     using continuous_closed_preimage_constant[of "closure s" f a]

  4304     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4305

  4306 lemma image_closure_subset:

  4307   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4308   shows "f  (closure s) \<subseteq> t"

  4309 proof-

  4310   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4311   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4312     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4313   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4314     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4315   thus ?thesis by auto

  4316 qed

  4317

  4318 lemma continuous_on_closure_norm_le:

  4319   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4320   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4321   shows "norm(f x) \<le> b"

  4322 proof-

  4323   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4324   show ?thesis

  4325     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4326     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4327 qed

  4328

  4329 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4330

  4331 lemma continuous_within_avoid:

  4332   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4333   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"

  4334   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4335 proof-

  4336   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"

  4337     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto

  4338   { fix y assume " y\<in>s"  "dist x y < d"

  4339     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]

  4340       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }

  4341   thus ?thesis using d>0 by auto

  4342 qed

  4343

  4344 lemma continuous_at_avoid:

  4345   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4346   assumes "continuous (at x) f" and "f x \<noteq> a"

  4347   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4348   using assms continuous_within_avoid[of x UNIV f a] by simp

  4349

  4350 lemma continuous_on_avoid:

  4351   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4352   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4353   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4354 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

  4355

  4356 lemma continuous_on_open_avoid:

  4357   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4358   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4359   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4360 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

  4361

  4362 text {* Proving a function is constant by proving open-ness of level set. *}

  4363

  4364 lemma continuous_levelset_open_in_cases:

  4365   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4366   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4367         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4368         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4369 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4370

  4371 lemma continuous_levelset_open_in:

  4372   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4373   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4374         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4375         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4376 using continuous_levelset_open_in_cases[of s f ]

  4377 by meson

  4378

  4379 lemma continuous_levelset_open:

  4380   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4381   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4382   shows "\<forall>x \<in> s. f x = a"

  4383 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4384

  4385 text {* Some arithmetical combinations (more to prove). *}

  4386

  4387 lemma open_scaling[intro]:

  4388   fixes s :: "'a::real_normed_vector set"

  4389   assumes "c \<noteq> 0"  "open s"

  4390   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4391 proof-

  4392   { fix x assume "x \<in> s"

  4393     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

  4394     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4395     moreover

  4396     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4397       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4398         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4399           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4400       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  }

  4401     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  }

  4402   thus ?thesis unfolding open_dist by auto

  4403 qed

  4404

  4405 lemma minus_image_eq_vimage:

  4406   fixes A :: "'a::ab_group_add set"

  4407   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4408   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4409

  4410 lemma open_negations:

  4411   fixes s :: "'a::real_normed_vector set"

  4412   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4413   unfolding scaleR_minus1_left [symmetric]

  4414   by (rule open_scaling, auto)

  4415

  4416 lemma open_translation:

  4417   fixes s :: "'a::real_normed_vector set"

  4418   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4419 proof-

  4420   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4421       by (intro continuous_diff continuous_at_id continuous_const) }

  4422   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4423   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4424 qed

  4425

  4426 lemma open_affinity:

  4427   fixes s :: "'a::real_normed_vector set"

  4428   assumes "open s"  "c \<noteq> 0"

  4429   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4430 proof-

  4431   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4432   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4433   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4434 qed

  4435

  4436 lemma interior_translation:

  4437   fixes s :: "'a::real_normed_vector set"

  4438   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4439 proof (rule set_eqI, rule)

  4440   fix x assume "x \<in> interior (op + a  s)"

  4441   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4442   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

  4443   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

  4444 next

  4445   fix x assume "x \<in> op + a  interior s"

  4446   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

  4447   { fix z have *:"a + y - z = y + a - z" by auto

  4448     assume "z\<in>ball x e"

  4449     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

  4450     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4451   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4452   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4453 qed

  4454

  4455 text {* Topological properties of linear functions. *}

  4456

  4457 lemma linear_lim_0:

  4458   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4459 proof-

  4460   interpret f: bounded_linear f by fact

  4461   have "(f ---> f 0) (at 0)"

  4462     using tendsto_ident_at by (rule f.tendsto)

  4463   thus ?thesis unfolding f.zero .

  4464 qed

  4465

  4466 lemma linear_continuous_at:

  4467   assumes "bounded_linear f"  shows "continuous (at a) f"

  4468   unfolding continuous_at using assms

  4469   apply (rule bounded_linear.tendsto)

  4470   apply (rule tendsto_ident_at)

  4471   done

  4472

  4473 lemma linear_continuous_within:

  4474   shows "bounded_linear f ==> continuous (at x within s) f"

  4475   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4476

  4477 lemma linear_continuous_on:

  4478   shows "bounded_linear f ==> continuous_on s f"

  4479   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4480

  4481 text {* Also bilinear functions, in composition form. *}

  4482

  4483 lemma bilinear_continuous_at_compose:

  4484   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4485         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4486   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4487

  4488 lemma bilinear_continuous_within_compose:

  4489   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4490         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4491   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4492

  4493 lemma bilinear_continuous_on_compose:

  4494   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4495              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4496   unfolding continuous_on_def

  4497   by (fast elim: bounded_bilinear.tendsto)

  4498

  4499 text {* Preservation of compactness and connectedness under continuous function. *}

  4500

  4501 lemma compact_continuous_image:

  4502   assumes "continuous_on s f"  "compact s"

  4503   shows "compact(f  s)"

  4504 proof-

  4505   { fix x assume x:"\<forall>n::nat. x n \<in> f  s"

  4506     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

  4507     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

  4508     { fix e::real assume "e>0"

  4509       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

  4510       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

  4511       { 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  }

  4512       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }

  4513     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  }

  4514   thus ?thesis unfolding compact_def by auto

  4515 qed

  4516

  4517 lemma connected_continuous_image:

  4518   assumes "continuous_on s f"  "connected s"

  4519   shows "connected(f  s)"

  4520 proof-

  4521   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4522     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4523       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4524       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4525       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4526     hence False using as(1,2)

  4527       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4528   thus ?thesis unfolding connected_clopen by auto

  4529 qed

  4530

  4531 text {* Continuity implies uniform continuity on a compact domain. *}

  4532

  4533 lemma compact_uniformly_continuous:

  4534   assumes "continuous_on s f"  "compact s"

  4535   shows "uniformly_continuous_on s f"

  4536 proof-

  4537     { fix x assume x:"x\<in>s"

  4538       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

  4539       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  }

  4540     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

  4541     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)"

  4542       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

  4543

  4544   { fix e::real assume "e>0"

  4545

  4546     { 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  }

  4547     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto

  4548     moreover

  4549     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }

  4550     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

  4551

  4552     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"

  4553       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

  4554       hence "x\<in>ball z (d z (e / 2))" using ea>0 unfolding subset_eq by auto

  4555       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

  4556         by (auto  simp add: dist_commute)

  4557       moreover have "y\<in>ball z (d z (e / 2))" using as and ea>0 and z[unfolded subset_eq]

  4558         by (auto simp add: dist_commute)

  4559       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

  4560         by (auto  simp add: dist_commute)

  4561       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]

  4562         by (auto simp add: dist_commute)  }

  4563     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  }

  4564   thus ?thesis unfolding uniformly_continuous_on_def by auto

  4565 qed

  4566

  4567 text{* Continuity of inverse function on compact domain. *}

  4568

  4569 lemma continuous_on_inv:

  4570   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"

  4571     (* TODO: can this be generalized more? *)

  4572   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4573   shows "continuous_on (f  s) g"

  4574 proof-

  4575   have *:"g  f  s = s" using assms(3) by (auto simp add: image_iff)

  4576   { fix t assume t:"closedin (subtopology euclidean (g  f  s)) t"

  4577     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto

  4578     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]

  4579       unfolding T(2) and Int_left_absorb by auto

  4580     moreover have "compact (s \<inter> T)"

  4581       using assms(2) unfolding compact_eq_bounded_closed

  4582       using bounded_subset[of s "s \<inter> T"] and T(1) by auto

  4583     ultimately have "closed (f  t)" using T(1) unfolding T(2)

  4584       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto

  4585     moreover have "{x \<in> f  s. g x \<in> t} = f  s \<inter> f  t" using assms(3) unfolding T(2) by auto

  4586     ultimately have "closedin (subtopology euclidean (f  s)) {x \<in> f  s. g x \<in> t}"

  4587       unfolding closedin_closed by auto  }

  4588   thus ?thesis unfolding continuous_on_closed by auto

  4589 qed

  4590

  4591 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4592

  4593 lemma continuous_uniform_limit:

  4594   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4595   assumes "\<not> trivial_limit F"

  4596   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4597   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4598   shows "continuous_on s g"

  4599 proof-

  4600   { fix x and e::real assume "x\<in>s" "e>0"

  4601     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4602       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4603     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4604     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4605       using assms(1) by blast

  4606     have "e / 3 > 0" using e>0 by auto

  4607     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"

  4608       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4609     { fix y assume "y \<in> s" and "dist y x < d"

  4610       hence "dist (f n y) (f n x) < e / 3"

  4611         by (rule d [rule_format])

  4612       hence "dist (f n y) (g x) < 2 * e / 3"

  4613         using dist_triangle [of "f n y" "g x" "f n x"]

  4614         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4615         by auto

  4616       hence "dist (g y) (g x) < e"

  4617         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4618         using dist_triangle3 [of "g y" "g x" "f n y"]

  4619         by auto }

  4620     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4621       using d>0 by auto }

  4622   thus ?thesis unfolding continuous_on_iff by auto

  4623 qed

  4624

  4625

  4626 subsection {* Topological stuff lifted from and dropped to R *}

  4627

  4628 lemma open_real:

  4629   fixes s :: "real set" shows

  4630  "open s \<longleftrightarrow>

  4631         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4632   unfolding open_dist dist_norm by simp

  4633

  4634 lemma islimpt_approachable_real:

  4635   fixes s :: "real set"

  4636   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4637   unfolding islimpt_approachable dist_norm by simp

  4638

  4639 lemma closed_real:

  4640   fixes s :: "real set"

  4641   shows "closed s \<longleftrightarrow>

  4642         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4643             --> x \<in> s)"

  4644   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4645

  4646 lemma continuous_at_real_range:

  4647   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4648   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4649         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4650   unfolding continuous_at unfolding Lim_at

  4651   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4652   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

  4653   apply(erule_tac x=e in allE) by auto

  4654

  4655 lemma continuous_on_real_range:

  4656   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4657   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))"

  4658   unfolding continuous_on_iff dist_norm by simp

  4659

  4660 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4661

  4662 lemma compact_attains_sup:

  4663   fixes s :: "real set"

  4664   assumes "compact s"  "s \<noteq> {}"

  4665   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"

  4666 proof-

  4667   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4668   { 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"

  4669     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto

  4670     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto

  4671     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using e>0 by auto  }

  4672   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]

  4673     apply(rule_tac x="Sup s" in bexI) by auto

  4674 qed

  4675

  4676 lemma Inf:

  4677   fixes S :: "real set"

  4678   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"

  4679 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)

  4680

  4681 lemma compact_attains_inf:

  4682   fixes s :: "real set"

  4683   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"

  4684 proof-

  4685   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4686   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"

  4687       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"

  4688     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto

  4689     moreover

  4690     { fix x assume "x \<in> s"

  4691       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto

  4692       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) x\<in>s unfolding * by auto }

  4693     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto

  4694     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using e>0 by auto  }

  4695   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]

  4696     apply(rule_tac x="Inf s" in bexI) by auto

  4697 qed

  4698

  4699 lemma continuous_attains_sup:

  4700   fixes f :: "'a::metric_space \<Rightarrow> real"

  4701   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4702         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"

  4703   using compact_attains_sup[of "f  s"]

  4704   using compact_continuous_image[of s f] by auto

  4705

  4706 lemma continuous_attains_inf:

  4707   fixes f :: "'a::metric_space \<Rightarrow> real"

  4708   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4709         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"

  4710   using compact_attains_inf[of "f  s"]

  4711   using compact_continuous_image[of s f] by auto

  4712

  4713 lemma distance_attains_sup:

  4714   assumes "compact s" "s \<noteq> {}"

  4715   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"

  4716 proof (rule continuous_attains_sup [OF assms])

  4717   { fix x assume "x\<in>s"

  4718     have "(dist a ---> dist a x) (at x within s)"

  4719       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  4720   }

  4721   thus "continuous_on s (dist a)"

  4722     unfolding continuous_on ..

  4723 qed

  4724

  4725 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4726

  4727 lemma distance_attains_inf:

  4728   fixes a :: "'a::heine_borel"

  4729   assumes "closed s"  "s \<noteq> {}"

  4730   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"

  4731 proof-

  4732   from assms(2) obtain b where "b\<in>s" by auto

  4733   let ?B = "cball a (dist b a) \<inter> s"

  4734   have "b \<in> ?B" using b\<in>s by (simp add: dist_commute)

  4735   hence "?B \<noteq> {}" by auto

  4736   moreover

  4737   { fix x assume "x\<in>?B"

  4738     fix e::real assume "e>0"

  4739     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"

  4740       from as have "\<bar>dist a x' - dist a x\<bar> < e"

  4741         unfolding abs_less_iff minus_diff_eq

  4742         using dist_triangle2 [of a x' x]

  4743         using dist_triangle [of a x x']

  4744         by arith

  4745     }

  4746     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"

  4747       using e>0 by auto

  4748   }

  4749   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"

  4750     unfolding continuous_on Lim_within dist_norm real_norm_def

  4751     by fast

  4752   moreover have "compact ?B"

  4753     using compact_cball[of a "dist b a"]

  4754     unfolding compact_eq_bounded_closed

  4755     using bounded_Int and closed_Int and assms(1) by auto

  4756   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"

  4757     using continuous_attains_inf[of ?B "dist a"] by fastforce

  4758   thus ?thesis by fastforce

  4759 qed

  4760

  4761

  4762 subsection {* Pasted sets *}

  4763

  4764 lemma bounded_Times:

  4765   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4766 proof-

  4767   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4768     using assms [unfolded bounded_def] by auto

  4769   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4770     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4771   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4772 qed

  4773

  4774 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4775 by (induct x) simp

  4776

  4777 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"

  4778 unfolding compact_def

  4779 apply clarify

  4780 apply (drule_tac x="fst \<circ> f" in spec)

  4781 apply (drule mp, simp add: mem_Times_iff)

  4782 apply (clarify, rename_tac l1 r1)

  4783 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4784 apply (drule mp, simp add: mem_Times_iff)

  4785 apply (clarify, rename_tac l2 r2)

  4786 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4787 apply (rule_tac x="r1 \<circ> r2" in exI)

  4788 apply (rule conjI, simp add: subseq_def)

  4789 apply (drule_tac r=r2 in lim_subseq [rotated], assumption)

  4790 apply (drule (1) tendsto_Pair) back

  4791 apply (simp add: o_def)

  4792 done

  4793

  4794 text{* Hence some useful properties follow quite easily. *}

  4795

  4796 lemma compact_scaling:

  4797   fixes s :: "'a::real_normed_vector set"

  4798   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4799 proof-

  4800   let ?f = "\<lambda>x. scaleR c x"

  4801   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4802   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4803     using linear_continuous_at[OF *] assms by auto

  4804 qed

  4805

  4806 lemma compact_negations:

  4807   fixes s :: "'a::real_normed_vector set"

  4808   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4809   using compact_scaling [OF assms, of "- 1"] by auto

  4810

  4811 lemma compact_sums:

  4812   fixes s t :: "'a::real_normed_vector set"

  4813   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4814 proof-

  4815   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4816     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4817   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4818     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4819   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4820 qed

  4821

  4822 lemma compact_differences:

  4823   fixes s t :: "'a::real_normed_vector set"

  4824   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4825 proof-

  4826   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4827     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4828   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4829 qed

  4830

  4831 lemma compact_translation:

  4832   fixes s :: "'a::real_normed_vector set"

  4833   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4834 proof-

  4835   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4836   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4837 qed

  4838

  4839 lemma compact_affinity:

  4840   fixes s :: "'a::real_normed_vector set"

  4841   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4842 proof-

  4843   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4844   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4845 qed

  4846

  4847 text {* Hence we get the following. *}

  4848

  4849 lemma compact_sup_maxdistance:

  4850   fixes s :: "'a::real_normed_vector set"

  4851   assumes "compact s"  "s \<noteq> {}"

  4852   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"

  4853 proof-

  4854   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using s \<noteq> {} by auto

  4855   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"

  4856     using compact_differences[OF assms(1) assms(1)]

  4857     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

  4858   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto

  4859   thus ?thesis using x(2)[unfolded x = a - b] by blast

  4860 qed

  4861

  4862 text {* We can state this in terms of diameter of a set. *}

  4863

  4864 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"

  4865   (* TODO: generalize to class metric_space *)

  4866

  4867 lemma diameter_bounded:

  4868   assumes "bounded s"

  4869   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4870         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"

  4871 proof-

  4872   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"

  4873   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto

  4874   { fix x y assume "x \<in> s" "y \<in> s"

  4875     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)  }

  4876   note * = this

  4877   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto

  4878     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

  4879       by simp (blast del: Sup_upper intro!: * Sup_upper) }

  4880   moreover

  4881   { fix d::real assume "d>0" "d < diameter s"

  4882     hence "s\<noteq>{}" unfolding diameter_def by auto

  4883     have "\<exists>d' \<in> ?D. d' > d"

  4884     proof(rule ccontr)

  4885       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"

  4886       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)

  4887       thus False using d < diameter s s\<noteq>{}

  4888         apply (auto simp add: diameter_def)

  4889         apply (drule Sup_real_iff [THEN [2] rev_iffD2])

  4890         apply (auto, force)

  4891         done

  4892     qed

  4893     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }

  4894   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4895         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto

  4896 qed

  4897

  4898 lemma diameter_bounded_bound:

  4899  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"

  4900   using diameter_bounded by blast

  4901

  4902 lemma diameter_compact_attained:

  4903   fixes s :: "'a::real_normed_vector set"

  4904   assumes "compact s"  "s \<noteq> {}"

  4905   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"

  4906 proof-

  4907   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4908   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

  4909   hence "diameter s \<le> norm (x - y)"

  4910     unfolding diameter_def by clarsimp (rule Sup_least, fast+)

  4911   thus ?thesis

  4912     by (metis b diameter_bounded_bound order_antisym xys)

  4913 qed

  4914

  4915 text {* Related results with closure as the conclusion. *}

  4916

  4917 lemma closed_scaling:

  4918   fixes s :: "'a::real_normed_vector set"

  4919   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4920 proof(cases "s={}")

  4921   case True thus ?thesis by auto

  4922 next

  4923   case False

  4924   show ?thesis

  4925   proof(cases "c=0")

  4926     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4927     case True thus ?thesis apply auto unfolding * by auto

  4928   next

  4929     case False

  4930     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4931       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4932           using as(1)[THEN spec[where x=n]]

  4933           using c\<noteq>0 by auto

  4934       }

  4935       moreover

  4936       { fix e::real assume "e>0"

  4937         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4938         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4939           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4940         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4941           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4942           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4943       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4944       ultimately have "l \<in> scaleR c  s"

  4945         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"]]

  4946         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4947     thus ?thesis unfolding closed_sequential_limits by fast

  4948   qed

  4949 qed

  4950

  4951 lemma closed_negations:

  4952   fixes s :: "'a::real_normed_vector set"

  4953   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4954   using closed_scaling[OF assms, of "- 1"] by simp

  4955

  4956 lemma compact_closed_sums:

  4957   fixes s :: "'a::real_normed_vector set"

  4958   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4959 proof-

  4960   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4961   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4962     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"

  4963       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4964     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4965       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4966     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4967       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto

  4968     hence "l - l' \<in> t"

  4969       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4970       using f(3) by auto

  4971     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

  4972   }

  4973   thus ?thesis unfolding closed_sequential_limits by fast

  4974 qed

  4975

  4976 lemma closed_compact_sums:

  4977   fixes s t :: "'a::real_normed_vector set"

  4978   assumes "closed s"  "compact t"

  4979   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4980 proof-

  4981   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

  4982     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4983   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4984 qed

  4985

  4986 lemma compact_closed_differences:

  4987   fixes s t :: "'a::real_normed_vector set"

  4988   assumes "compact s"  "closed t"

  4989   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4990 proof-

  4991   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4992     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4993   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4994 qed

  4995

  4996 lemma closed_compact_differences:

  4997   fixes s t :: "'a::real_normed_vector set"

  4998   assumes "closed s" "compact t"

  4999   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5000 proof-

  5001   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  5002     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5003  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  5004 qed

  5005

  5006 lemma closed_translation:

  5007   fixes a :: "'a::real_normed_vector"

  5008   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  5009 proof-

  5010   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  5011   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  5012 qed

  5013

  5014 lemma translation_Compl:

  5015   fixes a :: "'a::ab_group_add"

  5016   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5017   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  5018

  5019 lemma translation_UNIV:

  5020   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  5021   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  5022

  5023 lemma translation_diff:

  5024   fixes a :: "'a::ab_group_add"

  5025   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5026   by auto

  5027

  5028 lemma closure_translation:

  5029   fixes a :: "'a::real_normed_vector"

  5030   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5031 proof-

  5032   have *:"op + a  (- s) = - op + a  s"

  5033     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  5034   show ?thesis unfolding closure_interior translation_Compl

  5035     using interior_translation[of a "- s"] unfolding * by auto

  5036 qed

  5037

  5038 lemma frontier_translation:

  5039   fixes a :: "'a::real_normed_vector"

  5040   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5041   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  5042

  5043

  5044 subsection {* Separation between points and sets *}

  5045

  5046 lemma separate_point_closed:

  5047   fixes s :: "'a::heine_borel set"

  5048   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  5049 proof(cases "s = {}")

  5050   case True

  5051   thus ?thesis by(auto intro!: exI[where x=1])

  5052 next

  5053   case False

  5054   assume "closed s" "a \<notin> s"

  5055   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

  5056   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5057 qed

  5058

  5059 lemma separate_compact_closed:

  5060   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5061     (* TODO: does this generalize to heine_borel? *)

  5062   assumes "compact s" and "closed t" and "s \<inter> t = {}"

  5063   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5064 proof-

  5065   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto

  5066   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"

  5067     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto

  5068   { fix x y assume "x\<in>s" "y\<in>t"

  5069     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto

  5070     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute

  5071       by (auto  simp add: dist_commute)

  5072     hence "d \<le> dist x y" unfolding dist_norm by auto  }

  5073   thus ?thesis using d>0 by auto

  5074 qed

  5075

  5076 lemma separate_closed_compact:

  5077   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5078   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5079   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5080 proof-

  5081   have *:"t \<inter> s = {}" using assms(3) by auto

  5082   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5083     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5084     by (auto simp add: dist_commute)

  5085 qed

  5086

  5087

  5088 subsection {* Intervals *}

  5089

  5090 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5091   "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and

  5092   "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"

  5093   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5094

  5095 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5096   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"

  5097   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"

  5098   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5099

  5100 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5101  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and

  5102  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)

  5103 proof-

  5104   { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"

  5105     hence "a $$i < x$$ i \<and> x $$i < b$$ i" unfolding mem_interval by auto

  5106     hence "a$$i < b$$i" by auto

  5107     hence False using as by auto  }

  5108   moreover

  5109   { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"

  5110     let ?x = "(1/2) *\<^sub>R (a + b)"

  5111     { fix i assume i:"i<DIM('a)"

  5112       have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto

  5113       hence "a$$i < ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i < b$$i"

  5114         unfolding euclidean_simps by auto }

  5115     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5116   ultimately show ?th1 by blast

  5117

  5118   { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"

  5119     hence "a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i" unfolding mem_interval by auto

  5120     hence "a$$i \<le> b$$i" by auto

  5121     hence False using as by auto  }

  5122   moreover

  5123   { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"

  5124     let ?x = "(1/2) *\<^sub>R (a + b)"

  5125     { fix i assume i:"i<DIM('a)"

  5126       have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto

  5127       hence "a$$i \<le> ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i \<le> b$$i"

  5128         unfolding euclidean_simps by auto }

  5129     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5130   ultimately show ?th2 by blast

  5131 qed

  5132

  5133 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5134   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and

  5135   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"

  5136   unfolding interval_eq_empty[of a b] by fastforce+

  5137

  5138 lemma interval_sing:

  5139   fixes a :: "'a::ordered_euclidean_space"

  5140   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5141   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5142   by (auto simp add: euclidean_eq[where 'a='a] eq_commute

  5143     eucl_less[where 'a='a] eucl_le[where 'a='a])

  5144

  5145 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5146  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  5147  "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5148  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  5149  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  5150   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5151   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5152

  5153 lemma interval_open_subset_closed:

  5154   fixes a :: "'a::ordered_euclidean_space"

  5155   shows "{a<..<b} \<subseteq> {a .. b}"

  5156   unfolding subset_eq [unfolded Ball_def] mem_interval

  5157   by (fast intro: less_imp_le)

  5158

  5159 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5160  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th1) and

  5161  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i \<le> d$$i) --> (\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i)" (is ?th2) and

  5162  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th3) and

  5163  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). c$$i < d$$i) --> (\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i)" (is ?th4)

  5164 proof-

  5165   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5166   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)

  5167   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"

  5168     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5169     fix i assume i:"i<DIM('a)"

  5170     (** TODO combine the following two parts as done in the HOL_light version. **)

  5171     { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"   5172 assume as2: "a$$i > c$$i"   5173 { fix j assume j:"j<DIM('a)"   5174 hence "c$$ j < ?x $$j \<and> ?x$$ j < d $$j"   5175 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i   5176 by (auto simp add: as2) }   5177 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto   5178 moreover   5179 have "?x\<notin>{a .. b}"   5180 unfolding mem_interval apply auto apply(rule_tac x=i in exI)   5181 using as(2)[THEN spec[where x=i]] and as2 i   5182 by auto   5183 ultimately have False using as by auto }   5184 hence "a$$i \<le> c$$i" by(rule ccontr)auto   5185 moreover   5186 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"

  5187       assume as2: "b$$i < d$$i"

  5188       { fix j assume "j<DIM('a)"

  5189         hence "d $$j > ?x$$ j \<and> ?x $$j > c$$ j"

  5190           apply(cases "j=i") using as(2)[THEN spec[where x=j]]

  5191           by (auto simp add: as2)  }

  5192       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5193       moreover

  5194       have "?x\<notin>{a .. b}"

  5195         unfolding mem_interval apply auto apply(rule_tac x=i in exI)

  5196         using as(2)[THEN spec[where x=i]] and as2 using i

  5197         by auto

  5198       ultimately have False using as by auto  }

  5199     hence "b$$i \<ge> d$$i" by(rule ccontr)auto

  5200     ultimately

  5201     have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto

  5202   } note part1 = this

  5203   show ?th3 unfolding subset_eq and Ball_def and mem_interval

  5204     apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval

  5205     prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastforce)+

  5206   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"

  5207     fix i assume i:"i<DIM('a)"

  5208     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5209     hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto  } note * = this

  5210   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5211     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5212     apply auto by(erule_tac x=i in allE, simp)+

  5213 qed

  5214

  5215 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5216   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i < c$$i \<or> b$$i < c$$i \<or> d$$i < a$$i))" (is ?th1) and

  5217   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i < a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th2) and

  5218   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i < c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th3) and

  5219   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i<DIM('a). (b$$i \<le> a$$i \<or> d$$i \<le> c$$i \<or> b$$i \<le> c$$i \<or> d$$i \<le> a$$i))" (is ?th4)

  5220 proof-

  5221   let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"

  5222   note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False

  5223   show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5224     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5225   show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5226     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5227   show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5228     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5229   show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5230     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5231 qed

  5232

  5233 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5234  "{a .. b} \<inter> {c .. d} =  {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"

  5235   unfolding set_eq_iff and Int_iff and mem_interval

  5236   by auto

  5237

  5238 (* Moved interval_open_subset_closed a bit upwards *)

  5239

  5240 lemma open_interval[intro]:

  5241   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5242 proof-

  5243   have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i})"   5244 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI   5245 linear_continuous_at bounded_linear_euclidean_component   5246 open_real_greaterThanLessThan)   5247 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i}) = {a<..<b}"

  5248     by (auto simp add: eucl_less [where 'a='a])

  5249   finally show "open {a<..<b}" .

  5250 qed

  5251

  5252 lemma closed_interval[intro]:

  5253   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5254 proof-

  5255   have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i})"   5256 by (intro closed_INT ballI continuous_closed_vimage allI   5257 linear_continuous_at bounded_linear_euclidean_component   5258 closed_real_atLeastAtMost)   5259 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i}) = {a .. b}"

  5260     by (auto simp add: eucl_le [where 'a='a])

  5261   finally show "closed {a .. b}" .

  5262 qed

  5263

  5264 lemma interior_closed_interval [intro]:

  5265   fixes a b :: "'a::ordered_euclidean_space"

  5266   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5267 proof(rule subset_antisym)

  5268   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5269     by (rule interior_maximal)

  5270 next

  5271   { fix x assume "x \<in> interior {a..b}"

  5272     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5273     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

  5274     { fix i assume i:"i<DIM('a)"

  5275       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"

  5276            "dist (x + (e / 2) *\<^sub>R basis i) x < e"

  5277         unfolding dist_norm apply auto

  5278         unfolding norm_minus_cancel using norm_basis and e>0 by auto

  5279       hence "a $$i \<le> (x - (e / 2) *\<^sub>R basis i)$$ i"

  5280                      "(x + (e / 2) *\<^sub>R basis i) $$i \<le> b$$ i"

  5281         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]

  5282         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]

  5283         unfolding mem_interval using i by blast+

  5284       hence "a $$i < x$$ i" and "x $$i < b$$ i" unfolding euclidean_simps

  5285         unfolding basis_component using e>0 i by auto  }

  5286     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5287   thus "?L \<subseteq> ?R" ..

  5288 qed

  5289

  5290 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5291 proof-

  5292   let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"

  5293   { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i"

  5294     { fix i assume "i<DIM('a)"

  5295       hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }   5296 hence "(\<Sum>i<DIM('a). \<bar>x$$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5297     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5298   thus ?thesis unfolding interval and bounded_iff by auto

  5299 qed

  5300

  5301 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5302  "bounded {a .. b} \<and> bounded {a<..<b}"

  5303   using bounded_closed_interval[of a b]

  5304   using interval_open_subset_closed[of a b]

  5305   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5306   by simp

  5307

  5308 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5309  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5310   using bounded_interval[of a b] by auto

  5311

  5312 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5313   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]

  5314   by auto

  5315

  5316 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5317   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5318 proof-

  5319   { fix i assume "i<DIM('a)"

  5320     hence "a $$i < ((1 / 2) *\<^sub>R (a + b))$$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$i < b$$ i"

  5321       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]

  5322       unfolding euclidean_simps by auto  }

  5323   thus ?thesis unfolding mem_interval by auto

  5324 qed

  5325

  5326 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5327   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5328   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5329 proof-

  5330   { fix i assume i:"i<DIM('a)"

  5331     have "a $$i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp   5332 also have "\<dots> < e * x$$ i + (1 - e) * y $$i" apply(rule add_less_le_mono)   5333 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all   5334 using x unfolding mem_interval using i apply simp   5335 using y unfolding mem_interval using i apply simp   5336 done   5337 finally have "a$$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i" unfolding euclidean_simps by auto   5338 moreover {   5339 have "b$$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp

  5340     also have "\<dots> > e * x $$i + (1 - e) * y$$ i" apply(rule add_less_le_mono)

  5341       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5342       using x unfolding mem_interval using i apply simp

  5343       using y unfolding mem_interval using i apply simp

  5344       done

  5345     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i < b$$ i" unfolding euclidean_simps by auto

  5346     } ultimately have "a $$i < (e *\<^sub>R x + (1 - e) *\<^sub>R y)$$ i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i < b$$ i" by auto }

  5347   thus ?thesis unfolding mem_interval by auto

  5348 qed

  5349

  5350 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5351   assumes "{a<..<b} \<noteq> {}"

  5352   shows "closure {a<..<b} = {a .. b}"

  5353 proof-

  5354   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5355   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5356   { fix x assume as:"x \<in> {a .. b}"

  5357     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5358     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5359       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5360       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5361         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5362         by (auto simp add: algebra_simps)

  5363       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

  5364       hence False using fn unfolding f_def using xc by auto  }

  5365     moreover

  5366     { assume "\<not> (f ---> x) sequentially"

  5367       { fix e::real assume "e>0"

  5368         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

  5369         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5370         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)

  5371         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5372       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5373         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5374       hence "(f ---> x) sequentially" unfolding f_def

  5375         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]

  5376         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  }

  5377     ultimately have "x \<in> closure {a<..<b}"

  5378       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5379   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5380 qed

  5381

  5382 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5383   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5384 proof-

  5385   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5386   def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"

  5387   { fix x assume "x\<in>s"

  5388     fix i assume i:"i<DIM('a)"

  5389     hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF x\<in>s]

  5390       and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto  }

  5391   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5392 qed

  5393

  5394 lemma bounded_subset_open_interval:

  5395   fixes s :: "('a::ordered_euclidean_space) set"

  5396   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5397   by (auto dest!: bounded_subset_open_interval_symmetric)

  5398

  5399 lemma bounded_subset_closed_interval_symmetric:

  5400   fixes s :: "('a::ordered_euclidean_space) set"

  5401   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5402 proof-

  5403   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5404   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5405 qed

  5406

  5407 lemma bounded_subset_closed_interval:

  5408   fixes s :: "('a::ordered_euclidean_space) set"

  5409   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5410   using bounded_subset_closed_interval_symmetric[of s] by auto

  5411

  5412 lemma frontier_closed_interval:

  5413   fixes a b :: "'a::ordered_euclidean_space"

  5414   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5415   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5416

  5417 lemma frontier_open_interval:

  5418   fixes a b :: "'a::ordered_euclidean_space"

  5419   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5420 proof(cases "{a<..<b} = {}")

  5421   case True thus ?thesis using frontier_empty by auto

  5422 next

  5423   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5424 qed

  5425

  5426 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5427   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5428   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5429

  5430

  5431 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5432

  5433 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5434   shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"

  5435 proof-

  5436   { fix i assume i:"i<DIM('a)"

  5437     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). x $$i \<le> b$$ i}. x' \<noteq> x \<and> dist x' x < e"

  5438     { assume "x$$i > b$$i"

  5439       then obtain y where "y $$i \<le> b$$ i"  "y \<noteq> x"  "dist y x < x$$i - b$$i"

  5440         using x[THEN spec[where x="x$$i - b$$i"]] using i by auto

  5441       hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i

  5442         by auto   }

  5443     hence "x$$i \<le> b$$i" by(rule ccontr)auto  }

  5444   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5445 qed

  5446

  5447 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5448   shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"

  5449 proof-

  5450   { fix i assume i:"i<DIM('a)"

  5451     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i<DIM('a). a $$i \<le> x$$ i}. x' \<noteq> x \<and> dist x' x < e"

  5452     { assume "a$$i > x$$i"

  5453       then obtain y where "a $$i \<le> y$$ i"  "y \<noteq> x"  "dist y x < a$$i - x$$i"

  5454         using x[THEN spec[where x="a$$i - x$$i"]] i by auto

  5455       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto   }

  5456     hence "a$$i \<le> x$$i" by(rule ccontr)auto  }

  5457   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5458 qed

  5459

  5460 instance ordered_euclidean_space \<subseteq> countable_basis_space

  5461 proof

  5462   def to_cube \<equiv> "\<lambda>(a, b). {Chi (real_of_rat \<circ> op ! a)<..<Chi (real_of_rat \<circ> op ! b)}::'a set"

  5463   def B \<equiv> "(\<lambda>n. (to_cube (from_nat n)::'a set))  UNIV"

  5464   have "countable B" unfolding B_def by simp

  5465   moreover

  5466   have "Ball B open" unfolding B_def

  5467   proof safe

  5468     fix n show "open (to_cube (from_nat n))"

  5469       by (cases "from_nat n::rat list \<times> rat list")

  5470          (simp add: open_interval to_cube_def)

  5471   qed

  5472   moreover have "(\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = x))"

  5473   proof safe

  5474     fix x::"'a set" assume "open x"

  5475     def lists \<equiv> "{(a, b) |a b. to_cube (a, b) \<subseteq> x}"

  5476     from open_UNION[OF open x]

  5477     have "\<Union>(to_cube  lists) = x" unfolding lists_def to_cube_def

  5478      by simp

  5479     moreover have "to_cube  lists \<subseteq> B"

  5480     proof

  5481       fix x assume "x \<in> to_cube  lists"

  5482       then obtain l where "l \<in> lists" "x = to_cube l" by auto

  5483       thus "x \<in> B" by (auto simp add: B_def intro!: image_eqI[where x="to_nat l"])

  5484     qed

  5485     ultimately

  5486     show "\<exists>B'\<subseteq>B. \<Union>B' = x" by blast

  5487   qed

  5488   ultimately

  5489   show "\<exists>B::'a set set. countable B \<and> topological_basis B" unfolding topological_basis_def by blast

  5490 qed

  5491

  5492 instance ordered_euclidean_space \<subseteq> polish_space ..

  5493

  5494 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5495

  5496 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5497   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i<DIM('a). ((a$$i \<le> x$$i \<and> x$$i \<le> b$$i) \<or> (b$$i \<le> x$$i \<and> x$$i \<le> a$$i))) \<longrightarrow> x \<in> s)"

  5498

  5499 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5500   "is_interval {a<..<b}" (is ?th2) proof -

  5501   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5502     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5503

  5504 lemma is_interval_empty:

  5505  "is_interval {}"

  5506   unfolding is_interval_def

  5507   by simp

  5508

  5509 lemma is_interval_univ:

  5510  "is_interval UNIV"

  5511   unfolding is_interval_def

  5512   by simp

  5513

  5514

  5515 subsection {* Closure of halfspaces and hyperplanes *}

  5516

  5517 lemma isCont_open_vimage:

  5518   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5519 proof -

  5520   from assms(1) have "continuous_on UNIV f"

  5521     unfolding isCont_def continuous_on_def within_UNIV by simp

  5522   hence "open {x \<in> UNIV. f x \<in> s}"

  5523     using open_UNIV open s by (rule continuous_open_preimage)

  5524   thus "open (f - s)"

  5525     by (simp add: vimage_def)

  5526 qed

  5527

  5528 lemma isCont_closed_vimage:

  5529   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5530   using assms unfolding closed_def vimage_Compl [symmetric]

  5531   by (rule isCont_open_vimage)

  5532

  5533 lemma open_Collect_less:

  5534   fixes f g :: "'a::topological_space \<Rightarrow> real"

  5535   assumes f: "\<And>x. isCont f x"

  5536   assumes g: "\<And>x. isCont g x"

  5537   shows "open {x. f x < g x}"

  5538 proof -

  5539   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5540     using isCont_diff [OF g f] open_real_greaterThan

  5541     by (rule isCont_open_vimage)

  5542   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5543     by auto

  5544   finally show ?thesis .

  5545 qed

  5546

  5547 lemma closed_Collect_le:

  5548   fixes f g :: "'a::topological_space \<Rightarrow> real"

  5549   assumes f: "\<And>x. isCont f x"

  5550   assumes g: "\<And>x. isCont g x"

  5551   shows "closed {x. f x \<le> g x}"

  5552 proof -

  5553   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5554     using isCont_diff [OF g f] closed_real_atLeast

  5555     by (rule isCont_closed_vimage)

  5556   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5557     by auto

  5558   finally show ?thesis .

  5559 qed

  5560

  5561 lemma closed_Collect_eq:

  5562   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  5563   assumes f: "\<And>x. isCont f x"

  5564   assumes g: "\<And>x. isCont g x"

  5565   shows "closed {x. f x = g x}"

  5566 proof -

  5567   have "open {(x::'b, y::'b). x \<noteq> y}"

  5568     unfolding open_prod_def by (auto dest!: hausdorff)

  5569   hence "closed {(x::'b, y::'b). x = y}"

  5570     unfolding closed_def split_def Collect_neg_eq .

  5571   with isCont_Pair [OF f g]

  5572   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5573     by (rule isCont_closed_vimage)

  5574   also have "\<dots> = {x. f x = g x}" by auto

  5575   finally show ?thesis .

  5576 qed

  5577

  5578 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5579   unfolding continuous_at by (intro tendsto_intros)

  5580

  5581 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$i)"   5582 unfolding euclidean_component_def by (rule continuous_at_inner)   5583   5584 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"   5585 by (simp add: closed_Collect_le)   5586   5587 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"   5588 by (simp add: closed_Collect_le)   5589   5590 lemma closed_hyperplane: "closed {x. inner a x = b}"   5591 by (simp add: closed_Collect_eq)   5592   5593 lemma closed_halfspace_component_le:   5594 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"

  5595   by (simp add: closed_Collect_le)

  5596

  5597 lemma closed_halfspace_component_ge:

  5598   shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"   5599 by (simp add: closed_Collect_le)   5600   5601 text {* Openness of halfspaces. *}   5602   5603 lemma open_halfspace_lt: "open {x. inner a x < b}"   5604 by (simp add: open_Collect_less)   5605   5606 lemma open_halfspace_gt: "open {x. inner a x > b}"   5607 by (simp add: open_Collect_less)   5608   5609 lemma open_halfspace_component_lt:   5610 shows "open {x::'a::euclidean_space. x$$i < a}"

  5611   by (simp add: open_Collect_less)

  5612

  5613 lemma open_halfspace_component_gt:

  5614   shows "open {x::'a::euclidean_space. x$$i > a}"   5615 by (simp add: open_Collect_less)   5616   5617 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}   5618   5619 lemma eucl_lessThan_eq_halfspaces:   5620 fixes a :: "'a\<Colon>ordered_euclidean_space"   5621 shows "{..<a} = (\<Inter>i<DIM('a). {x. x$$ i < a $$i})"   5622 by (auto simp: eucl_less[where 'a='a])   5623   5624 lemma eucl_greaterThan_eq_halfspaces:   5625 fixes a :: "'a\<Colon>ordered_euclidean_space"   5626 shows "{a<..} = (\<Inter>i<DIM('a). {x. a$$ i < x $$i})"   5627 by (auto simp: eucl_less[where 'a='a])   5628   5629 lemma eucl_atMost_eq_halfspaces:   5630 fixes a :: "'a\<Colon>ordered_euclidean_space"   5631 shows "{.. a} = (\<Inter>i<DIM('a). {x. x$$ i \<le> a $$i})"   5632 by (auto simp: eucl_le[where 'a='a])   5633   5634 lemma eucl_atLeast_eq_halfspaces:   5635 fixes a :: "'a\<Colon>ordered_euclidean_space"   5636 shows "{a ..} = (\<Inter>i<DIM('a). {x. a$$ i \<le> x $$i})"   5637 by (auto simp: eucl_le[where 'a='a])   5638   5639 lemma open_eucl_lessThan[simp, intro]:   5640 fixes a :: "'a\<Colon>ordered_euclidean_space"   5641 shows "open {..< a}"   5642 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)   5643   5644 lemma open_eucl_greaterThan[simp, intro]:   5645 fixes a :: "'a\<Colon>ordered_euclidean_space"   5646 shows "open {a <..}"   5647 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)   5648   5649 lemma closed_eucl_atMost[simp, intro]:   5650 fixes a :: "'a\<Colon>ordered_euclidean_space"   5651 shows "closed {.. a}"   5652 unfolding eucl_atMost_eq_halfspaces   5653 by (simp add: closed_INT closed_Collect_le)   5654   5655 lemma closed_eucl_atLeast[simp, intro]:   5656 fixes a :: "'a\<Colon>ordered_euclidean_space"   5657 shows "closed {a ..}"   5658 unfolding eucl_atLeast_eq_halfspaces   5659 by (simp add: closed_INT closed_Collect_le)   5660   5661 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x$$ i) - S)"

  5662   by (auto intro!: continuous_open_vimage)

  5663

  5664 text {* This gives a simple derivation of limit component bounds. *}

  5665

  5666 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5667   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$$i \<le> b) net"   5668 shows "l$$i \<le> b"

  5669 proof-

  5670   { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"   5671 unfolding euclidean_component_def by auto } note * = this   5672 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *   5673 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto   5674 qed   5675   5676 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"   5677 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"

  5678   shows "b \<le> l$$i"   5679 proof-   5680 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"

  5681       unfolding euclidean_component_def by auto  } note * = this

  5682   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *

  5683     using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto

  5684 qed

  5685

  5686 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5687   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"   5688 shows "l$$i = b"

  5689   using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto

  5690 text{* Limits relative to a union.                                               *}

  5691

  5692 lemma eventually_within_Un:

  5693   "eventually P (net within (s \<union> t)) \<longleftrightarrow>

  5694     eventually P (net within s) \<and> eventually P (net within t)"

  5695   unfolding Limits.eventually_within

  5696   by (auto elim!: eventually_rev_mp)

  5697

  5698 lemma Lim_within_union:

  5699  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>

  5700   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"

  5701   unfolding tendsto_def

  5702   by (auto simp add: eventually_within_Un)

  5703

  5704 lemma Lim_topological:

  5705  "(f ---> l) net \<longleftrightarrow>

  5706         trivial_limit net \<or>

  5707         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5708   unfolding tendsto_def trivial_limit_eq by auto

  5709

  5710 lemma continuous_on_union:

  5711   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"

  5712   shows "continuous_on (s \<union> t) f"

  5713   using assms unfolding continuous_on Lim_within_union

  5714   unfolding Lim_topological trivial_limit_within closed_limpt by auto

  5715

  5716 lemma continuous_on_cases:

  5717   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"

  5718           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"

  5719   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  5720 proof-

  5721   let ?h = "(\<lambda>x. if P x then f x else g x)"

  5722   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto

  5723   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto

  5724   moreover

  5725   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto

  5726   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto

  5727   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto

  5728 qed

  5729

  5730

  5731 text{* Some more convenient intermediate-value theorem formulations.             *}

  5732

  5733 lemma connected_ivt_hyperplane:

  5734   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5735   shows "\<exists>z \<in> s. inner a z = b"

  5736 proof(rule ccontr)

  5737   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5738   let ?A = "{x. inner a x < b}"

  5739   let ?B = "{x. inner a x > b}"

  5740   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5741   moreover have "?A \<inter> ?B = {}" by auto

  5742   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5743   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

  5744 qed

  5745

  5746 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5747  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$$k \<le> a \<Longrightarrow> a \<le> y$$k \<Longrightarrow> (\<exists>z\<in>s.  zk = a)"

  5748   using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]

  5749   unfolding euclidean_component_def by auto

  5750

  5751

  5752 subsection {* Homeomorphisms *}

  5753

  5754 definition "homeomorphism s t f g \<equiv>

  5755      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5756      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5757

  5758 definition

  5759   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"

  5760     (i`