src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author hoelzl Fri Nov 16 19:14:23 2012 +0100 (2012-11-16) changeset 50105 65d5b18e1626 parent 50104 de19856feb54 child 50245 dea9363887a6 permissions -rw-r--r--
moved (b)choice_iff(') to Hilbert_Choice
     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"

    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 end

    75

    76 subsection {* Enumerable Basis *}

    77

    78 locale enumerates_basis =

    79   fixes f::"nat \<Rightarrow> 'a::topological_space set"

    80   assumes enumerable_basis: "topological_basis (range f)"

    81 begin

    82

    83 lemma open_enumerable_basis_ex:

    84   assumes "open X"

    85   shows "\<exists>N. X = (\<Union>n\<in>N. f n)"

    86 proof -

    87   from enumerable_basis assms obtain B' where "B' \<subseteq> range f" "X = Union B'"

    88     unfolding topological_basis_def by blast

    89   hence "Union B' = (\<Union>n\<in>{n. f n \<in> B'}. f n)" by auto

    90   with X = Union B' show ?thesis by blast

    91 qed

    92

    93 lemma open_enumerable_basisE:

    94   assumes "open X"

    95   obtains N where "X = (\<Union>n\<in>N. f n)"

    96   using assms open_enumerable_basis_ex by (atomize_elim) simp

    97

    98 lemma countable_dense_set:

    99   shows "\<exists>x::nat \<Rightarrow> 'a. \<forall>y. open y \<longrightarrow> y \<noteq> {} \<longrightarrow> (\<exists>n. x n \<in> y)"

   100 proof -

   101   def x \<equiv> "\<lambda>n. (SOME x::'a. x \<in> f n)"

   102   have x: "\<And>n. f n \<noteq> ({}::'a set) \<Longrightarrow> x n \<in> f n" unfolding x_def

   103     by (rule someI_ex) auto

   104   have "\<forall>y. open y \<longrightarrow> y \<noteq> {} \<longrightarrow> (\<exists>n. x n \<in> y)"

   105   proof (intro allI impI)

   106     fix y::"'a set" assume "open y" "y \<noteq> {}"

   107     from open_enumerable_basisE[OF open y] guess N . note N = this

   108     obtain n where n: "n \<in> N" "f n \<noteq> ({}::'a set)"

   109     proof (atomize_elim, rule ccontr, clarsimp)

   110       assume "\<forall>n. n \<in> N \<longrightarrow> f n = ({}::'a set)"

   111       hence "(\<Union>n\<in>N. f n) = (\<Union>n\<in>N. {}::'a set)"

   112         by (intro UN_cong) auto

   113       hence "y = {}" unfolding N by simp

   114       with y \<noteq> {} show False by auto

   115     qed

   116     with x N n have "x n \<in> y" by auto

   117     thus "\<exists>n. x n \<in> y" ..

   118   qed

   119   thus ?thesis by blast

   120 qed

   121

   122 lemma countable_dense_setE:

   123   obtains x :: "nat \<Rightarrow> 'a"

   124   where "\<And>y. open y \<Longrightarrow> y \<noteq> {} \<Longrightarrow> \<exists>n. x n \<in> y"

   125   using countable_dense_set by blast

   126

   127 text {* Construction of an increasing sequence approximating open sets, therefore enumeration of

   128   basis which is closed under union. *}

   129

   130 definition enum_basis::"nat \<Rightarrow> 'a set"

   131   where "enum_basis n = \<Union>(set (map f (from_nat n)))"

   132

   133 lemma enum_basis_basis: "topological_basis (range enum_basis)"

   134 proof (rule topological_basisI)

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

   136   from topological_basisE[OF enumerable_basis this] guess B' . note B' = this

   137   moreover then obtain n where "B' = f n" by auto

   138   moreover hence "B' = enum_basis (to_nat [n])" by (auto simp: enum_basis_def)

   139   ultimately show "\<exists>B'\<in>range enum_basis. x \<in> B' \<and> B' \<subseteq> O'" by blast

   140 next

   141   fix B' assume "B' \<in> range enum_basis"

   142   with topological_basis_open[OF enumerable_basis]

   143   show "open B'" by (auto simp add: enum_basis_def intro!: open_UN)

   144 qed

   145

   146 lemmas open_enum_basis = topological_basis_open[OF enum_basis_basis]

   147

   148 lemma empty_basisI[intro]: "{} \<in> range enum_basis"

   149 proof

   150   show "{} = enum_basis (to_nat ([]::nat list))" by (simp add: enum_basis_def)

   151 qed rule

   152

   153 lemma union_basisI[intro]:

   154   assumes "A \<in> range enum_basis" "B \<in> range enum_basis"

   155   shows "A \<union> B \<in> range enum_basis"

   156 proof -

   157   from assms obtain a b where "A \<union> B = enum_basis a \<union> enum_basis b" by auto

   158   also have "\<dots> = enum_basis (to_nat (from_nat a @ from_nat b::nat list))"

   159     by (simp add: enum_basis_def)

   160   finally show ?thesis by simp

   161 qed

   162

   163 lemma open_imp_Union_of_incseq:

   164   assumes "open X"

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

   166 proof -

   167   interpret E: enumerates_basis enum_basis proof qed (rule enum_basis_basis)

   168   from E.open_enumerable_basis_ex[OF open X] obtain N where N: "X = (\<Union>n\<in>N. enum_basis n)" by auto

   169   hence X: "X = (\<Union>n. if n \<in> N then enum_basis n else {})" by (auto split: split_if_asm)

   170   def S \<equiv> "nat_rec (if 0 \<in> N then enum_basis 0 else {})

   171     (\<lambda>n S. if (Suc n) \<in> N then S \<union> enum_basis (Suc n) else S)"

   172   have S_simps[simp]:

   173     "S 0 = (if 0 \<in> N then enum_basis 0 else {})"

   174     "\<And>n. S (Suc n) = (if (Suc n) \<in> N then S n \<union> enum_basis (Suc n) else S n)"

   175     by (simp_all add: S_def)

   176   have "incseq S" by (rule incseq_SucI) auto

   177   moreover

   178   have "(\<Union>j. S j) = X" unfolding N

   179   proof safe

   180     fix x n assume "n \<in> N" "x \<in> enum_basis n"

   181     hence "x \<in> S n" by (cases n) auto

   182     thus "x \<in> (\<Union>j. S j)" by auto

   183   next

   184     fix x j

   185     assume "x \<in> S j"

   186     thus "x \<in> UNION N enum_basis" by (induct j) (auto split: split_if_asm)

   187   qed

   188   moreover have "range S \<subseteq> range enum_basis"

   189   proof safe

   190     fix j show "S j \<in> range enum_basis" by (induct j) auto

   191   qed

   192   ultimately show ?thesis by auto

   193 qed

   194

   195 lemma open_incseqE:

   196   assumes "open X"

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

   198   using open_imp_Union_of_incseq assms by atomize_elim

   199

   200 end

   201

   202 class enumerable_basis = topological_space +

   203   assumes ex_enum_basis: "\<exists>f::nat \<Rightarrow> 'a::topological_space set. topological_basis (range f)"

   204

   205 sublocale enumerable_basis < enumerates_basis "Eps (topological_basis o range)"

   206   unfolding o_def

   207   proof qed (rule someI_ex[OF ex_enum_basis])

   208

   209 subsection {* Polish spaces *}

   210

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

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

   213

   214 class polish_space = complete_space + enumerable_basis

   215

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

   217

   218 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))"

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

   220   morphisms "openin" "topology"

   221   unfolding istopology_def by blast

   222

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

   224   using openin[of U] by blast

   225

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

   227   using topology_inverse[unfolded mem_Collect_eq] .

   228

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

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

   231

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

   233 proof-

   234   { assume "T1=T2"

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

   236   moreover

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

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

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

   240     hence "T1 = T2" unfolding openin_inverse .

   241   }

   242   ultimately show ?thesis by blast

   243 qed

   244

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

   246

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

   248

   249 subsubsection {* Main properties of open sets *}

   250

   251 lemma openin_clauses:

   252   fixes U :: "'a topology"

   253   shows "openin U {}"

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

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

   256   using openin[of U] unfolding istopology_def mem_Collect_eq

   257   by fast+

   258

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

   260   unfolding topspace_def by blast

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

   262

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

   264   using openin_clauses by simp

   265

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

   267   using openin_clauses by simp

   268

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

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

   271

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

   273

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

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

   276 proof

   277   assume ?lhs

   278   then show ?rhs by auto

   279 next

   280   assume H: ?rhs

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

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

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

   284   finally show "openin U S" .

   285 qed

   286

   287

   288 subsubsection {* Closed sets *}

   289

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

   291

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

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

   294 lemma closedin_topspace[intro,simp]:

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

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

   297   by (auto simp add: Diff_Un closedin_def)

   298

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

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

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

   302

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

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

   305

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

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

   308   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   309   apply (metis openin_subset subset_eq)

   310   done

   311

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

   313   by (simp add: openin_closedin_eq)

   314

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

   316 proof-

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

   318     by (auto simp add: topspace_def openin_subset)

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

   320 qed

   321

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

   323 proof-

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

   325     by (auto simp add: topspace_def )

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

   327 qed

   328

   329 subsubsection {* Subspace topology *}

   330

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

   332

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

   334   (is "istopology ?L")

   335 proof-

   336   have "?L {}" by blast

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

   338     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

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

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

   341   moreover

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

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

   344       apply (rule set_eqI)

   345       apply (simp add: Ball_def image_iff)

   346       by metis

   347     from K[unfolded th0 subset_image_iff]

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

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

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

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

   352   ultimately show ?thesis

   353     unfolding subset_eq mem_Collect_eq istopology_def by blast

   354 qed

   355

   356 lemma openin_subtopology:

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

   358   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   359   by auto

   360

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

   362   by (auto simp add: topspace_def openin_subtopology)

   363

   364 lemma closedin_subtopology:

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

   366   unfolding closedin_def topspace_subtopology

   367   apply (simp add: openin_subtopology)

   368   apply (rule iffI)

   369   apply clarify

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

   371   by auto

   372

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

   374   unfolding openin_subtopology

   375   apply (rule iffI, clarify)

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

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

   378   apply auto

   379   done

   380

   381 lemma subtopology_superset:

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

   383   shows "subtopology U V = U"

   384 proof-

   385   {fix S

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

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

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

   389     moreover

   390     {assume S: "openin U S"

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

   392         using openin_subset[OF S] UV by auto}

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

   394   then show ?thesis unfolding topology_eq openin_subtopology by blast

   395 qed

   396

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

   398   by (simp add: subtopology_superset)

   399

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

   401   by (simp add: subtopology_superset)

   402

   403 subsubsection {* The standard Euclidean topology *}

   404

   405 definition

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

   407   "euclidean = topology open"

   408

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

   410   unfolding euclidean_def

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

   412   apply (rule topology_inverse[symmetric])

   413   apply (auto simp add: istopology_def)

   414   done

   415

   416 lemma topspace_euclidean: "topspace euclidean = UNIV"

   417   apply (simp add: topspace_def)

   418   apply (rule set_eqI)

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

   420

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

   422   by (simp add: topspace_euclidean topspace_subtopology)

   423

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

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

   426

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

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

   429

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

   431

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

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

   434

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

   436   by (auto simp add: openin_open)

   437

   438 lemma open_openin_trans[trans]:

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

   440   by (metis Int_absorb1  openin_open_Int)

   441

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

   443   by (auto simp add: openin_open)

   444

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

   446   by (simp add: closedin_subtopology closed_closedin Int_ac)

   447

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

   449   by (metis closedin_closed)

   450

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

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

   453   apply auto

   454   apply (frule closedin_closed_Int[of T S])

   455   by simp

   456

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

   458   by (auto simp add: closedin_closed)

   459

   460 lemma openin_euclidean_subtopology_iff:

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

   462   shows "openin (subtopology euclidean U) S

   463   \<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")

   464 proof

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

   466 next

   467   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}"

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

   469     unfolding T_def

   470     apply clarsimp

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

   472     apply (clarsimp simp add: less_diff_eq)

   473     apply (erule rev_bexI)

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

   475     apply (erule le_less_trans [OF dist_triangle])

   476     done

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

   478     unfolding T_def

   479     apply auto

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

   481     apply auto

   482     done

   483   from 1 2 show ?lhs

   484     unfolding openin_open open_dist by fast

   485 qed

   486

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

   488

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

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

   491   unfolding open_openin openin_open by blast

   492

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

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

   495

   496 lemma closedin_trans[trans]:

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

   498            closedin (subtopology euclidean U) T

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

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

   501

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

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

   504

   505

   506 subsection {* Open and closed balls *}

   507

   508 definition

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

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

   511

   512 definition

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

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

   515

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

   517   by (simp add: ball_def)

   518

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

   520   by (simp add: cball_def)

   521

   522 lemma mem_ball_0:

   523   fixes x :: "'a::real_normed_vector"

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

   525   by (simp add: dist_norm)

   526

   527 lemma mem_cball_0:

   528   fixes x :: "'a::real_normed_vector"

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

   530   by (simp add: dist_norm)

   531

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

   533   by simp

   534

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

   536   by simp

   537

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

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

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

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

   542   by (simp add: set_eq_iff) arith

   543

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

   545   by (simp add: set_eq_iff)

   546

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

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

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

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

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

   552

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

   554   unfolding open_dist ball_def mem_Collect_eq Ball_def

   555   unfolding dist_commute

   556   apply clarify

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

   558   using dist_triangle_alt[where z=x]

   559   apply (clarsimp simp add: diff_less_iff)

   560   apply atomize

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

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

   563   by arith

   564

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

   566   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   567

   568 lemma openE[elim?]:

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

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

   571   using assms unfolding open_contains_ball by auto

   572

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

   574   by (metis open_contains_ball subset_eq centre_in_ball)

   575

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

   577   unfolding mem_ball set_eq_iff

   578   apply (simp add: not_less)

   579   by (metis zero_le_dist order_trans dist_self)

   580

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

   582

   583 lemma rational_boxes:

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

   585   assumes "0 < e"

   586   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"

   587 proof -

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

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

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

   591   proof

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

   593     show "?th i" by auto

   594   qed

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

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

   597   proof

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

   599     show "?th i" by auto

   600   qed

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

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

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

   604       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   606     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

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

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

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

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

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

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

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

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

   615         by (rule power_strict_mono) auto

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

   617         by (simp add: power_divide)

   618     qed auto

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

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

   621   with a b show ?thesis

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

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

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

   625 qed

   626

   627 lemma ex_rat_list:

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

   629   assumes "\<And> i. x $$i \<in> \<rat>"   630 shows "\<exists> r. length r = DIM('a) \<and> (\<forall> i < DIM('a). of_rat (r ! i) = x$$ i)"

   631 proof -

   632   have "\<forall>i. \<exists>r. x $$i = of_rat r" using assms unfolding Rats_def by blast   633 from choice[OF this] guess r ..   634 then show ?thesis by (auto intro!: exI[of _ "map r [0 ..< DIM('a)]"])   635 qed   636   637 lemma open_UNION:   638 fixes M :: "'a\<Colon>ordered_euclidean_space set"   639 assumes "open M"   640 shows "M = UNION {(a, b) | a b. {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)} \<subseteq> M}   641 (\<lambda> (a, b). {Chi (of_rat \<circ> op ! a) <..< Chi (of_rat \<circ> op ! b)})"   642 (is "M = UNION ?idx ?box")   643 proof safe   644 fix x assume "x \<in> M"   645 obtain e where e: "e > 0" "ball x e \<subseteq> M"   646 using openE[OF assms x \<in> M] by auto   647 then obtain a b where ab: "x \<in> {a <..< b}" "\<And>i. a$$ i \<in> \<rat>" "\<And>i. b $$i \<in> \<rat>" "{a <..< b} \<subseteq> ball x e"   648 using rational_boxes[OF e(1)] by blast   649 then obtain p q where pq: "length p = DIM ('a)"   650 "length q = DIM ('a)"   651 "\<forall> i < DIM ('a). of_rat (p ! i) = a$$ i \<and> of_rat (q ! i) = b $$i"   652 using ex_rat_list[OF ab(2)] ex_rat_list[OF ab(3)] by blast   653 hence p: "Chi (of_rat \<circ> op ! p) = a"   654 using euclidean_eq[of "Chi (of_rat \<circ> op ! p)" a]   655 unfolding o_def by auto   656 from pq have q: "Chi (of_rat \<circ> op ! q) = b"   657 using euclidean_eq[of "Chi (of_rat \<circ> op ! q)" b]   658 unfolding o_def by auto   659 have "x \<in> ?box (p, q)"   660 using p q ab by auto   661 thus "x \<in> UNION ?idx ?box" using ab e p q exI[of _ p] exI[of _ q] by auto   662 qed auto   663   664 subsection{* Connectedness *}   665   666 definition "connected S \<longleftrightarrow>   667 ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})   668 \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"   669   670 lemma connected_local:   671 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.   672 openin (subtopology euclidean S) e1 \<and>   673 openin (subtopology euclidean S) e2 \<and>   674 S \<subseteq> e1 \<union> e2 \<and>   675 e1 \<inter> e2 = {} \<and>   676 ~(e1 = {}) \<and>   677 ~(e2 = {}))"   678 unfolding connected_def openin_open by (safe, blast+)   679   680 lemma exists_diff:   681 fixes P :: "'a set \<Rightarrow> bool"   682 shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")   683 proof-   684 {assume "?lhs" hence ?rhs by blast }   685 moreover   686 {fix S assume H: "P S"   687 have "S = - (- S)" by auto   688 with H have "P (- (- S))" by metis }   689 ultimately show ?thesis by metis   690 qed   691   692 lemma connected_clopen: "connected S \<longleftrightarrow>   693 (\<forall>T. openin (subtopology euclidean S) T \<and>   694 closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")   695 proof-   696 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> {})"   697 unfolding connected_def openin_open closedin_closed   698 apply (subst exists_diff) by blast   699 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> {})"   700 (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis   701   702 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'))"   703 (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")   704 unfolding connected_def openin_open closedin_closed by auto   705 {fix e2   706 {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)"   707 by auto}   708 then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}   709 then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast   710 then show ?thesis unfolding th0 th1 by simp   711 qed   712   713 lemma connected_empty[simp, intro]: "connected {}"   714 by (simp add: connected_def)   715   716   717 subsection{* Limit points *}   718   719 definition (in topological_space)   720 islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where   721 "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"   722   723 lemma islimptI:   724 assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"   725 shows "x islimpt S"   726 using assms unfolding islimpt_def by auto   727   728 lemma islimptE:   729 assumes "x islimpt S" and "x \<in> T" and "open T"   730 obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"   731 using assms unfolding islimpt_def by auto   732   733 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"   734 unfolding islimpt_def eventually_at_topological by auto   735   736 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"   737 unfolding islimpt_def by fast   738   739 lemma islimpt_approachable:   740 fixes x :: "'a::metric_space"   741 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"   742 unfolding islimpt_iff_eventually eventually_at by fast   743   744 lemma islimpt_approachable_le:   745 fixes x :: "'a::metric_space"   746 shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"   747 unfolding islimpt_approachable   748 using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",   749 THEN arg_cong [where f=Not]]   750 by (simp add: Bex_def conj_commute conj_left_commute)   751   752 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"   753 unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)   754   755 text {* A perfect space has no isolated points. *}   756   757 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"   758 unfolding islimpt_UNIV_iff by (rule not_open_singleton)   759   760 lemma perfect_choose_dist:   761 fixes x :: "'a::{perfect_space, metric_space}"   762 shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"   763 using islimpt_UNIV [of x]   764 by (simp add: islimpt_approachable)   765   766 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"   767 unfolding closed_def   768 apply (subst open_subopen)   769 apply (simp add: islimpt_def subset_eq)   770 by (metis ComplE ComplI)   771   772 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"   773 unfolding islimpt_def by auto   774   775 lemma finite_set_avoid:   776 fixes a :: "'a::metric_space"   777 assumes fS: "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"   778 proof(induct rule: finite_induct[OF fS])   779 case 1 thus ?case by (auto intro: zero_less_one)   780 next   781 case (2 x F)   782 from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast   783 {assume "x = a" hence ?case using d by auto }   784 moreover   785 {assume xa: "x\<noteq>a"   786 let ?d = "min d (dist a x)"   787 have dp: "?d > 0" using xa d(1) using dist_nz by auto   788 from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto   789 with dp xa have ?case by(auto intro!: exI[where x="?d"]) }   790 ultimately show ?case by blast   791 qed   792   793 lemma islimpt_finite:   794 fixes S :: "'a::metric_space set"   795 assumes fS: "finite S" shows "\<not> a islimpt S"   796 unfolding islimpt_approachable   797 using finite_set_avoid[OF fS, of a] by (metis dist_commute not_le)   798   799 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"   800 apply (rule iffI)   801 defer   802 apply (metis Un_upper1 Un_upper2 islimpt_subset)   803 unfolding islimpt_def   804 apply (rule ccontr, clarsimp, rename_tac A B)   805 apply (drule_tac x="A \<inter> B" in spec)   806 apply (auto simp add: open_Int)   807 done   808   809 lemma discrete_imp_closed:   810 fixes S :: "'a::metric_space set"   811 assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"   812 shows "closed S"   813 proof-   814 {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"   815 from e have e2: "e/2 > 0" by arith   816 from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast   817 let ?m = "min (e/2) (dist x y) "   818 from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])   819 from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast   820 have th: "dist z y < e" using z y   821 by (intro dist_triangle_lt [where z=x], simp)   822 from d[rule_format, OF y(1) z(1) th] y z   823 have False by (auto simp add: dist_commute)}   824 then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])   825 qed   826   827   828 subsection {* Interior of a Set *}   829   830 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"   831   832 lemma interiorI [intro?]:   833 assumes "open T" and "x \<in> T" and "T \<subseteq> S"   834 shows "x \<in> interior S"   835 using assms unfolding interior_def by fast   836   837 lemma interiorE [elim?]:   838 assumes "x \<in> interior S"   839 obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"   840 using assms unfolding interior_def by fast   841   842 lemma open_interior [simp, intro]: "open (interior S)"   843 by (simp add: interior_def open_Union)   844   845 lemma interior_subset: "interior S \<subseteq> S"   846 by (auto simp add: interior_def)   847   848 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"   849 by (auto simp add: interior_def)   850   851 lemma interior_open: "open S \<Longrightarrow> interior S = S"   852 by (intro equalityI interior_subset interior_maximal subset_refl)   853   854 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"   855 by (metis open_interior interior_open)   856   857 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"   858 by (metis interior_maximal interior_subset subset_trans)   859   860 lemma interior_empty [simp]: "interior {} = {}"   861 using open_empty by (rule interior_open)   862   863 lemma interior_UNIV [simp]: "interior UNIV = UNIV"   864 using open_UNIV by (rule interior_open)   865   866 lemma interior_interior [simp]: "interior (interior S) = interior S"   867 using open_interior by (rule interior_open)   868   869 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"   870 by (auto simp add: interior_def)   871   872 lemma interior_unique:   873 assumes "T \<subseteq> S" and "open T"   874 assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"   875 shows "interior S = T"   876 by (intro equalityI assms interior_subset open_interior interior_maximal)   877   878 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"   879 by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1   880 Int_lower2 interior_maximal interior_subset open_Int open_interior)   881   882 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"   883 using open_contains_ball_eq [where S="interior S"]   884 by (simp add: open_subset_interior)   885   886 lemma interior_limit_point [intro]:   887 fixes x :: "'a::perfect_space"   888 assumes x: "x \<in> interior S" shows "x islimpt S"   889 using x islimpt_UNIV [of x]   890 unfolding interior_def islimpt_def   891 apply (clarsimp, rename_tac T T')   892 apply (drule_tac x="T \<inter> T'" in spec)   893 apply (auto simp add: open_Int)   894 done   895   896 lemma interior_closed_Un_empty_interior:   897 assumes cS: "closed S" and iT: "interior T = {}"   898 shows "interior (S \<union> T) = interior S"   899 proof   900 show "interior S \<subseteq> interior (S \<union> T)"   901 by (rule interior_mono, rule Un_upper1)   902 next   903 show "interior (S \<union> T) \<subseteq> interior S"   904 proof   905 fix x assume "x \<in> interior (S \<union> T)"   906 then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..   907 show "x \<in> interior S"   908 proof (rule ccontr)   909 assume "x \<notin> interior S"   910 with x \<in> R open R obtain y where "y \<in> R - S"   911 unfolding interior_def by fast   912 from open R closed S have "open (R - S)" by (rule open_Diff)   913 from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast   914 from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}   915 show "False" unfolding interior_def by fast   916 qed   917 qed   918 qed   919   920 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"   921 proof (rule interior_unique)   922 show "interior A \<times> interior B \<subseteq> A \<times> B"   923 by (intro Sigma_mono interior_subset)   924 show "open (interior A \<times> interior B)"   925 by (intro open_Times open_interior)   926 fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"   927 proof (safe)   928 fix x y assume "(x, y) \<in> T"   929 then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"   930 using open T unfolding open_prod_def by fast   931 hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"   932 using T \<subseteq> A \<times> B by auto   933 thus "x \<in> interior A" and "y \<in> interior B"   934 by (auto intro: interiorI)   935 qed   936 qed   937   938   939 subsection {* Closure of a Set *}   940   941 definition "closure S = S \<union> {x | x. x islimpt S}"   942   943 lemma interior_closure: "interior S = - (closure (- S))"   944 unfolding interior_def closure_def islimpt_def by auto   945   946 lemma closure_interior: "closure S = - interior (- S)"   947 unfolding interior_closure by simp   948   949 lemma closed_closure[simp, intro]: "closed (closure S)"   950 unfolding closure_interior by (simp add: closed_Compl)   951   952 lemma closure_subset: "S \<subseteq> closure S"   953 unfolding closure_def by simp   954   955 lemma closure_hull: "closure S = closed hull S"   956 unfolding hull_def closure_interior interior_def by auto   957   958 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"   959 unfolding closure_hull using closed_Inter by (rule hull_eq)   960   961 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"   962 unfolding closure_eq .   963   964 lemma closure_closure [simp]: "closure (closure S) = closure S"   965 unfolding closure_hull by (rule hull_hull)   966   967 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"   968 unfolding closure_hull by (rule hull_mono)   969   970 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"   971 unfolding closure_hull by (rule hull_minimal)   972   973 lemma closure_unique:   974 assumes "S \<subseteq> T" and "closed T"   975 assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"   976 shows "closure S = T"   977 using assms unfolding closure_hull by (rule hull_unique)   978   979 lemma closure_empty [simp]: "closure {} = {}"   980 using closed_empty by (rule closure_closed)   981   982 lemma closure_UNIV [simp]: "closure UNIV = UNIV"   983 using closed_UNIV by (rule closure_closed)   984   985 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"   986 unfolding closure_interior by simp   987   988 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"   989 using closure_empty closure_subset[of S]   990 by blast   991   992 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"   993 using closure_eq[of S] closure_subset[of S]   994 by simp   995   996 lemma open_inter_closure_eq_empty:   997 "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"   998 using open_subset_interior[of S "- T"]   999 using interior_subset[of "- T"]   1000 unfolding closure_interior   1001 by auto   1002   1003 lemma open_inter_closure_subset:   1004 "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"   1005 proof   1006 fix x   1007 assume as: "open S" "x \<in> S \<inter> closure T"   1008 { assume *:"x islimpt T"   1009 have "x islimpt (S \<inter> T)"   1010 proof (rule islimptI)   1011 fix A   1012 assume "x \<in> A" "open A"   1013 with as have "x \<in> A \<inter> S" "open (A \<inter> S)"   1014 by (simp_all add: open_Int)   1015 with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"   1016 by (rule islimptE)   1017 hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"   1018 by simp_all   1019 thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..   1020 qed   1021 }   1022 then show "x \<in> closure (S \<inter> T)" using as   1023 unfolding closure_def   1024 by blast   1025 qed   1026   1027 lemma closure_complement: "closure (- S) = - interior S"   1028 unfolding closure_interior by simp   1029   1030 lemma interior_complement: "interior (- S) = - closure S"   1031 unfolding closure_interior by simp   1032   1033 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"   1034 proof (rule closure_unique)   1035 show "A \<times> B \<subseteq> closure A \<times> closure B"   1036 by (intro Sigma_mono closure_subset)   1037 show "closed (closure A \<times> closure B)"   1038 by (intro closed_Times closed_closure)   1039 fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"   1040 apply (simp add: closed_def open_prod_def, clarify)   1041 apply (rule ccontr)   1042 apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)   1043 apply (simp add: closure_interior interior_def)   1044 apply (drule_tac x=C in spec)   1045 apply (drule_tac x=D in spec)   1046 apply auto   1047 done   1048 qed   1049   1050   1051 subsection {* Frontier (aka boundary) *}   1052   1053 definition "frontier S = closure S - interior S"   1054   1055 lemma frontier_closed: "closed(frontier S)"   1056 by (simp add: frontier_def closed_Diff)   1057   1058 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"   1059 by (auto simp add: frontier_def interior_closure)   1060   1061 lemma frontier_straddle:   1062 fixes a :: "'a::metric_space"   1063 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))"   1064 unfolding frontier_def closure_interior   1065 by (auto simp add: mem_interior subset_eq ball_def)   1066   1067 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"   1068 by (metis frontier_def closure_closed Diff_subset)   1069   1070 lemma frontier_empty[simp]: "frontier {} = {}"   1071 by (simp add: frontier_def)   1072   1073 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"   1074 proof-   1075 { assume "frontier S \<subseteq> S"   1076 hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto   1077 hence "closed S" using closure_subset_eq by auto   1078 }   1079 thus ?thesis using frontier_subset_closed[of S] ..   1080 qed   1081   1082 lemma frontier_complement: "frontier(- S) = frontier S"   1083 by (auto simp add: frontier_def closure_complement interior_complement)   1084   1085 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"   1086 using frontier_complement frontier_subset_eq[of "- S"]   1087 unfolding open_closed by auto   1088   1089 subsection {* Filters and the eventually true'' quantifier *}   1090   1091 definition   1092 at_infinity :: "'a::real_normed_vector filter" where   1093 "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"   1094   1095 definition   1096 indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"   1097 (infixr "indirection" 70) where   1098 "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"   1099   1100 text{* Prove That They are all filters. *}   1101   1102 lemma eventually_at_infinity:   1103 "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"   1104 unfolding at_infinity_def   1105 proof (rule eventually_Abs_filter, rule is_filter.intro)   1106 fix P Q :: "'a \<Rightarrow> bool"   1107 assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"   1108 then obtain r s where   1109 "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto   1110 then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp   1111 then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..   1112 qed auto   1113   1114 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}   1115   1116 lemma trivial_limit_within:   1117 shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"   1118 proof   1119 assume "trivial_limit (at a within S)"   1120 thus "\<not> a islimpt S"   1121 unfolding trivial_limit_def   1122 unfolding eventually_within eventually_at_topological   1123 unfolding islimpt_def   1124 apply (clarsimp simp add: set_eq_iff)   1125 apply (rename_tac T, rule_tac x=T in exI)   1126 apply (clarsimp, drule_tac x=y in bspec, simp_all)   1127 done   1128 next   1129 assume "\<not> a islimpt S"   1130 thus "trivial_limit (at a within S)"   1131 unfolding trivial_limit_def   1132 unfolding eventually_within eventually_at_topological   1133 unfolding islimpt_def   1134 apply clarsimp   1135 apply (rule_tac x=T in exI)   1136 apply auto   1137 done   1138 qed   1139   1140 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"   1141 using trivial_limit_within [of a UNIV] by simp   1142   1143 lemma trivial_limit_at:   1144 fixes a :: "'a::perfect_space"   1145 shows "\<not> trivial_limit (at a)"   1146 by (rule at_neq_bot)   1147   1148 lemma trivial_limit_at_infinity:   1149 "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"   1150 unfolding trivial_limit_def eventually_at_infinity   1151 apply clarsimp   1152 apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)   1153 apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)   1154 apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])   1155 apply (drule_tac x=UNIV in spec, simp)   1156 done   1157   1158 text {* Some property holds "sufficiently close" to the limit point. *}   1159   1160 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)   1161 "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"   1162 unfolding eventually_at dist_nz by auto   1163   1164 lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>   1165 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"   1166 unfolding eventually_within eventually_at dist_nz by auto   1167   1168 lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>   1169 (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")   1170 unfolding eventually_within   1171 by auto (metis dense order_le_less_trans)   1172   1173 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"   1174 unfolding trivial_limit_def   1175 by (auto elim: eventually_rev_mp)   1176   1177 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"   1178 by simp   1179   1180 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"   1181 by (simp add: filter_eq_iff)   1182   1183 text{* Combining theorems for "eventually" *}   1184   1185 lemma eventually_rev_mono:   1186 "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"   1187 using eventually_mono [of P Q] by fast   1188   1189 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"   1190 by (simp add: eventually_False)   1191   1192   1193 subsection {* Limits *}   1194   1195 text{* Notation Lim to avoid collition with lim defined in analysis *}   1196   1197 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"   1198 where "Lim A f = (THE l. (f ---> l) A)"   1199   1200 lemma Lim:   1201 "(f ---> l) net \<longleftrightarrow>   1202 trivial_limit net \<or>   1203 (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"   1204 unfolding tendsto_iff trivial_limit_eq by auto   1205   1206 text{* Show that they yield usual definitions in the various cases. *}   1207   1208 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>   1209 (\<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)"   1210 by (auto simp add: tendsto_iff eventually_within_le)   1211   1212 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>   1213 (\<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)"   1214 by (auto simp add: tendsto_iff eventually_within)   1215   1216 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>   1217 (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"   1218 by (auto simp add: tendsto_iff eventually_at)   1219   1220 lemma Lim_at_infinity:   1221 "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"   1222 by (auto simp add: tendsto_iff eventually_at_infinity)   1223   1224 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"   1225 by (rule topological_tendstoI, auto elim: eventually_rev_mono)   1226   1227 text{* The expected monotonicity property. *}   1228   1229 lemma Lim_within_empty: "(f ---> l) (net within {})"   1230 unfolding tendsto_def Limits.eventually_within by simp   1231   1232 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"   1233 unfolding tendsto_def Limits.eventually_within   1234 by (auto elim!: eventually_elim1)   1235   1236 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"   1237 shows "(f ---> l) (net within (S \<union> T))"   1238 using assms unfolding tendsto_def Limits.eventually_within   1239 apply clarify   1240 apply (drule spec, drule (1) mp, drule (1) mp)   1241 apply (drule spec, drule (1) mp, drule (1) mp)   1242 apply (auto elim: eventually_elim2)   1243 done   1244   1245 lemma Lim_Un_univ:   1246 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow> S \<union> T = UNIV   1247 ==> (f ---> l) net"   1248 by (metis Lim_Un within_UNIV)   1249   1250 text{* Interrelations between restricted and unrestricted limits. *}   1251   1252 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"   1253 (* FIXME: rename *)   1254 unfolding tendsto_def Limits.eventually_within   1255 apply (clarify, drule spec, drule (1) mp, drule (1) mp)   1256 by (auto elim!: eventually_elim1)   1257   1258 lemma eventually_within_interior:   1259 assumes "x \<in> interior S"   1260 shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")   1261 proof-   1262 from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..   1263 { assume "?lhs"   1264 then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"   1265 unfolding Limits.eventually_within Limits.eventually_at_topological   1266 by auto   1267 with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"   1268 by auto   1269 then have "?rhs"   1270 unfolding Limits.eventually_at_topological by auto   1271 } moreover   1272 { assume "?rhs" hence "?lhs"   1273 unfolding Limits.eventually_within   1274 by (auto elim: eventually_elim1)   1275 } ultimately   1276 show "?thesis" ..   1277 qed   1278   1279 lemma at_within_interior:   1280 "x \<in> interior S \<Longrightarrow> at x within S = at x"   1281 by (simp add: filter_eq_iff eventually_within_interior)   1282   1283 lemma at_within_open:   1284 "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"   1285 by (simp only: at_within_interior interior_open)   1286   1287 lemma Lim_within_open:   1288 fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"   1289 assumes"a \<in> S" "open S"   1290 shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"   1291 using assms by (simp only: at_within_open)   1292   1293 lemma Lim_within_LIMSEQ:   1294 fixes a :: "'a::metric_space"   1295 assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"   1296 shows "(X ---> L) (at a within T)"   1297 using assms unfolding tendsto_def [where l=L]   1298 by (simp add: sequentially_imp_eventually_within)   1299   1300 lemma Lim_right_bound:   1301 fixes f :: "real \<Rightarrow> real"   1302 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"   1303 assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"   1304 shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"   1305 proof cases   1306 assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)   1307 next   1308 assume [simp]: "{x<..} \<inter> I \<noteq> {}"   1309 show ?thesis   1310 proof (rule Lim_within_LIMSEQ, safe)   1311 fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"   1312   1313 show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"   1314 proof (rule LIMSEQ_I, rule ccontr)   1315 fix r :: real assume "0 < r"   1316 with Inf_close[of "f  ({x<..} \<inter> I)" r]   1317 obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto   1318 from x < y have "0 < y - x" by auto   1319 from S(2)[THEN LIMSEQ_D, OF this]   1320 obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto   1321   1322 assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"   1323 moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"   1324 using S bnd by (intro Inf_lower[where z=K]) auto   1325 ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"   1326 by (auto simp: not_less field_simps)   1327 with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y   1328 show False by auto   1329 qed   1330 qed   1331 qed   1332   1333 text{* Another limit point characterization. *}   1334   1335 lemma islimpt_sequential:   1336 fixes x :: "'a::metric_space"   1337 shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S -{x}) \<and> (f ---> x) sequentially)"   1338 (is "?lhs = ?rhs")   1339 proof   1340 assume ?lhs   1341 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"   1342 unfolding islimpt_approachable   1343 using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto   1344 let ?I = "\<lambda>n. inverse (real (Suc n))"   1345 have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp   1346 moreover have "(\<lambda>n. f (?I n)) ----> x"   1347 proof (rule metric_tendsto_imp_tendsto)   1348 show "?I ----> 0"   1349 by (rule LIMSEQ_inverse_real_of_nat)   1350 show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"   1351 by (simp add: norm_conv_dist [symmetric] less_imp_le f)   1352 qed   1353 ultimately show ?rhs by fast   1354 next   1355 assume ?rhs   1356 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   1357 { fix e::real assume "e>0"   1358 then obtain N where "dist (f N) x < e" using f(2) by auto   1359 moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto   1360 ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto   1361 }   1362 thus ?lhs unfolding islimpt_approachable by auto   1363 qed   1364   1365 lemma Lim_inv: (* TODO: delete *)   1366 fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"   1367 assumes "(f ---> l) A" and "l \<noteq> 0"   1368 shows "((inverse o f) ---> inverse l) A"   1369 unfolding o_def using assms by (rule tendsto_inverse)   1370   1371 lemma Lim_null:   1372 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1373 shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"   1374 by (simp add: Lim dist_norm)   1375   1376 lemma Lim_null_comparison:   1377 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1378 assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"   1379 shows "(f ---> 0) net"   1380 proof (rule metric_tendsto_imp_tendsto)   1381 show "(g ---> 0) net" by fact   1382 show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"   1383 using assms(1) by (rule eventually_elim1, simp add: dist_norm)   1384 qed   1385   1386 lemma Lim_transform_bound:   1387 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1388 fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"   1389 assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net" "(g ---> 0) net"   1390 shows "(f ---> 0) net"   1391 using assms(1) tendsto_norm_zero [OF assms(2)]   1392 by (rule Lim_null_comparison)   1393   1394 text{* Deducing things about the limit from the elements. *}   1395   1396 lemma Lim_in_closed_set:   1397 assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"   1398 shows "l \<in> S"   1399 proof (rule ccontr)   1400 assume "l \<notin> S"   1401 with closed S have "open (- S)" "l \<in> - S"   1402 by (simp_all add: open_Compl)   1403 with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"   1404 by (rule topological_tendstoD)   1405 with assms(2) have "eventually (\<lambda>x. False) net"   1406 by (rule eventually_elim2) simp   1407 with assms(3) show "False"   1408 by (simp add: eventually_False)   1409 qed   1410   1411 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}   1412   1413 lemma Lim_dist_ubound:   1414 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"   1415 shows "dist a l <= e"   1416 proof-   1417 have "dist a l \<in> {..e}"   1418 proof (rule Lim_in_closed_set)   1419 show "closed {..e}" by simp   1420 show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)   1421 show "\<not> trivial_limit net" by fact   1422 show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)   1423 qed   1424 thus ?thesis by simp   1425 qed   1426   1427 lemma Lim_norm_ubound:   1428 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1429 assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"   1430 shows "norm(l) <= e"   1431 proof-   1432 have "norm l \<in> {..e}"   1433 proof (rule Lim_in_closed_set)   1434 show "closed {..e}" by simp   1435 show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)   1436 show "\<not> trivial_limit net" by fact   1437 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)   1438 qed   1439 thus ?thesis by simp   1440 qed   1441   1442 lemma Lim_norm_lbound:   1443 fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"   1444 assumes "\<not> (trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. e <= norm(f x)) net"   1445 shows "e \<le> norm l"   1446 proof-   1447 have "norm l \<in> {e..}"   1448 proof (rule Lim_in_closed_set)   1449 show "closed {e..}" by simp   1450 show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)   1451 show "\<not> trivial_limit net" by fact   1452 show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)   1453 qed   1454 thus ?thesis by simp   1455 qed   1456   1457 text{* Uniqueness of the limit, when nontrivial. *}   1458   1459 lemma tendsto_Lim:   1460 fixes f :: "'a \<Rightarrow> 'b::t2_space"   1461 shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"   1462 unfolding Lim_def using tendsto_unique[of net f] by auto   1463   1464 text{* Limit under bilinear function *}   1465   1466 lemma Lim_bilinear:   1467 assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"   1468 shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"   1469 using bounded_bilinear h (f ---> l) net (g ---> m) net   1470 by (rule bounded_bilinear.tendsto)   1471   1472 text{* These are special for limits out of the same vector space. *}   1473   1474 lemma Lim_within_id: "(id ---> a) (at a within s)"   1475 unfolding id_def by (rule tendsto_ident_at_within)   1476   1477 lemma Lim_at_id: "(id ---> a) (at a)"   1478 unfolding id_def by (rule tendsto_ident_at)   1479   1480 lemma Lim_at_zero:   1481 fixes a :: "'a::real_normed_vector"   1482 fixes l :: "'b::topological_space"   1483 shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")   1484 using LIM_offset_zero LIM_offset_zero_cancel ..   1485   1486 text{* It's also sometimes useful to extract the limit point from the filter. *}   1487   1488 definition   1489 netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where   1490 "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"   1491   1492 lemma netlimit_within:   1493 assumes "\<not> trivial_limit (at a within S)"   1494 shows "netlimit (at a within S) = a"   1495 unfolding netlimit_def   1496 apply (rule some_equality)   1497 apply (rule Lim_at_within)   1498 apply (rule tendsto_ident_at)   1499 apply (erule tendsto_unique [OF assms])   1500 apply (rule Lim_at_within)   1501 apply (rule tendsto_ident_at)   1502 done   1503   1504 lemma netlimit_at:   1505 fixes a :: "'a::{perfect_space,t2_space}"   1506 shows "netlimit (at a) = a"   1507 using netlimit_within [of a UNIV] by simp   1508   1509 lemma lim_within_interior:   1510 "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"   1511 by (simp add: at_within_interior)   1512   1513 lemma netlimit_within_interior:   1514 fixes x :: "'a::{t2_space,perfect_space}"   1515 assumes "x \<in> interior S"   1516 shows "netlimit (at x within S) = x"   1517 using assms by (simp add: at_within_interior netlimit_at)   1518   1519 text{* Transformation of limit. *}   1520   1521 lemma Lim_transform:   1522 fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"   1523 assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"   1524 shows "(g ---> l) net"   1525 using tendsto_diff [OF assms(2) assms(1)] by simp   1526   1527 lemma Lim_transform_eventually:   1528 "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"   1529 apply (rule topological_tendstoI)   1530 apply (drule (2) topological_tendstoD)   1531 apply (erule (1) eventually_elim2, simp)   1532 done   1533   1534 lemma Lim_transform_within:   1535 assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"   1536 and "(f ---> l) (at x within S)"   1537 shows "(g ---> l) (at x within S)"   1538 proof (rule Lim_transform_eventually)   1539 show "eventually (\<lambda>x. f x = g x) (at x within S)"   1540 unfolding eventually_within   1541 using assms(1,2) by auto   1542 show "(f ---> l) (at x within S)" by fact   1543 qed   1544   1545 lemma Lim_transform_at:   1546 assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"   1547 and "(f ---> l) (at x)"   1548 shows "(g ---> l) (at x)"   1549 proof (rule Lim_transform_eventually)   1550 show "eventually (\<lambda>x. f x = g x) (at x)"   1551 unfolding eventually_at   1552 using assms(1,2) by auto   1553 show "(f ---> l) (at x)" by fact   1554 qed   1555   1556 text{* Common case assuming being away from some crucial point like 0. *}   1557   1558 lemma Lim_transform_away_within:   1559 fixes a b :: "'a::t1_space"   1560 assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"   1561 and "(f ---> l) (at a within S)"   1562 shows "(g ---> l) (at a within S)"   1563 proof (rule Lim_transform_eventually)   1564 show "(f ---> l) (at a within S)" by fact   1565 show "eventually (\<lambda>x. f x = g x) (at a within S)"   1566 unfolding Limits.eventually_within eventually_at_topological   1567 by (rule exI [where x="- {b}"], simp add: open_Compl assms)   1568 qed   1569   1570 lemma Lim_transform_away_at:   1571 fixes a b :: "'a::t1_space"   1572 assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"   1573 and fl: "(f ---> l) (at a)"   1574 shows "(g ---> l) (at a)"   1575 using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl   1576 by simp   1577   1578 text{* Alternatively, within an open set. *}   1579   1580 lemma Lim_transform_within_open:   1581 assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"   1582 and "(f ---> l) (at a)"   1583 shows "(g ---> l) (at a)"   1584 proof (rule Lim_transform_eventually)   1585 show "eventually (\<lambda>x. f x = g x) (at a)"   1586 unfolding eventually_at_topological   1587 using assms(1,2,3) by auto   1588 show "(f ---> l) (at a)" by fact   1589 qed   1590   1591 text{* A congruence rule allowing us to transform limits assuming not at point. *}   1592   1593 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)   1594   1595 lemma Lim_cong_within(*[cong add]*):   1596 assumes "a = b" "x = y" "S = T"   1597 assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"   1598 shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"   1599 unfolding tendsto_def Limits.eventually_within eventually_at_topological   1600 using assms by simp   1601   1602 lemma Lim_cong_at(*[cong add]*):   1603 assumes "a = b" "x = y"   1604 assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"   1605 shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"   1606 unfolding tendsto_def eventually_at_topological   1607 using assms by simp   1608   1609 text{* Useful lemmas on closure and set of possible sequential limits.*}   1610   1611 lemma closure_sequential:   1612 fixes l :: "'a::metric_space"   1613 shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")   1614 proof   1615 assume "?lhs" moreover   1616 { assume "l \<in> S"   1617 hence "?rhs" using tendsto_const[of l sequentially] by auto   1618 } moreover   1619 { assume "l islimpt S"   1620 hence "?rhs" unfolding islimpt_sequential by auto   1621 } ultimately   1622 show "?rhs" unfolding closure_def by auto   1623 next   1624 assume "?rhs"   1625 thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto   1626 qed   1627   1628 lemma closed_sequential_limits:   1629 fixes S :: "'a::metric_space set"   1630 shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"   1631 unfolding closed_limpt   1632 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]   1633 by metis   1634   1635 lemma closure_approachable:   1636 fixes S :: "'a::metric_space set"   1637 shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"   1638 apply (auto simp add: closure_def islimpt_approachable)   1639 by (metis dist_self)   1640   1641 lemma closed_approachable:   1642 fixes S :: "'a::metric_space set"   1643 shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"   1644 by (metis closure_closed closure_approachable)   1645   1646 subsection {* Infimum Distance *}   1647   1648 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"   1649   1650 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"   1651 by (simp add: infdist_def)   1652   1653 lemma infdist_nonneg:   1654 shows "0 \<le> infdist x A"   1655 using assms by (auto simp add: infdist_def)   1656   1657 lemma infdist_le:   1658 assumes "a \<in> A"   1659 assumes "d = dist x a"   1660 shows "infdist x A \<le> d"   1661 using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)   1662   1663 lemma infdist_zero[simp]:   1664 assumes "a \<in> A" shows "infdist a A = 0"   1665 proof -   1666 from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto   1667 with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto   1668 qed   1669   1670 lemma infdist_triangle:   1671 shows "infdist x A \<le> infdist y A + dist x y"   1672 proof cases   1673 assume "A = {}" thus ?thesis by (simp add: infdist_def)   1674 next   1675 assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto   1676 have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"   1677 proof   1678 from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp   1679 fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"   1680 then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto   1681 show "infdist x A \<le> d"   1682 unfolding infdist_notempty[OF A \<noteq> {}]   1683 proof (rule Inf_lower2)   1684 show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto   1685 show "dist x a \<le> d" unfolding d by (rule dist_triangle)   1686 fix d assume "d \<in> {dist x a |a. a \<in> A}"   1687 then obtain a where "a \<in> A" "d = dist x a" by auto   1688 thus "infdist x A \<le> d" by (rule infdist_le)   1689 qed   1690 qed   1691 also have "\<dots> = dist x y + infdist y A"   1692 proof (rule Inf_eq, safe)   1693 fix a assume "a \<in> A"   1694 thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)   1695 next   1696 fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"   1697 hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A   1698 by (intro Inf_greatest) (auto simp: field_simps)   1699 thus "i \<le> dist x y + infdist y A" by simp   1700 qed   1701 finally show ?thesis by simp   1702 qed   1703   1704 lemma   1705 in_closure_iff_infdist_zero:   1706 assumes "A \<noteq> {}"   1707 shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"   1708 proof   1709 assume "x \<in> closure A"   1710 show "infdist x A = 0"   1711 proof (rule ccontr)   1712 assume "infdist x A \<noteq> 0"   1713 with infdist_nonneg[of x A] have "infdist x A > 0" by auto   1714 hence "ball x (infdist x A) \<inter> closure A = {}" apply auto   1715 by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute   1716 eucl_less_not_refl euclidean_trans(2) infdist_le)   1717 hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)   1718 thus False using x \<in> closure A by simp   1719 qed   1720 next   1721 assume x: "infdist x A = 0"   1722 then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)   1723 show "x \<in> closure A" unfolding closure_approachable   1724 proof (safe, rule ccontr)   1725 fix e::real assume "0 < e"   1726 assume "\<not> (\<exists>y\<in>A. dist y x < e)"   1727 hence "infdist x A \<ge> e" using a \<in> A   1728 unfolding infdist_def   1729 by (force intro: Inf_greatest simp: dist_commute)   1730 with x 0 < e show False by auto   1731 qed   1732 qed   1733   1734 lemma   1735 in_closed_iff_infdist_zero:   1736 assumes "closed A" "A \<noteq> {}"   1737 shows "x \<in> A \<longleftrightarrow> infdist x A = 0"   1738 proof -   1739 have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"   1740 by (rule in_closure_iff_infdist_zero) fact   1741 with assms show ?thesis by simp   1742 qed   1743   1744 lemma tendsto_infdist [tendsto_intros]:   1745 assumes f: "(f ---> l) F"   1746 shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"   1747 proof (rule tendstoI)   1748 fix e ::real assume "0 < e"   1749 from tendstoD[OF f this]   1750 show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"   1751 proof (eventually_elim)   1752 fix x   1753 from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]   1754 have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"   1755 by (simp add: dist_commute dist_real_def)   1756 also assume "dist (f x) l < e"   1757 finally show "dist (infdist (f x) A) (infdist l A) < e" .   1758 qed   1759 qed   1760   1761 text{* Some other lemmas about sequences. *}   1762   1763 lemma sequentially_offset:   1764 assumes "eventually (\<lambda>i. P i) sequentially"   1765 shows "eventually (\<lambda>i. P (i + k)) sequentially"   1766 using assms unfolding eventually_sequentially by (metis trans_le_add1)   1767   1768 lemma seq_offset:   1769 assumes "(f ---> l) sequentially"   1770 shows "((\<lambda>i. f (i + k)) ---> l) sequentially"   1771 using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)   1772   1773 lemma seq_offset_neg:   1774 "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"   1775 apply (rule topological_tendstoI)   1776 apply (drule (2) topological_tendstoD)   1777 apply (simp only: eventually_sequentially)   1778 apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")   1779 apply metis   1780 by arith   1781   1782 lemma seq_offset_rev:   1783 "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"   1784 by (rule LIMSEQ_offset) (* FIXME: redundant *)   1785   1786 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"   1787 using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)   1788   1789 subsection {* More properties of closed balls *}   1790   1791 lemma closed_cball: "closed (cball x e)"   1792 unfolding cball_def closed_def   1793 unfolding Collect_neg_eq [symmetric] not_le   1794 apply (clarsimp simp add: open_dist, rename_tac y)   1795 apply (rule_tac x="dist x y - e" in exI, clarsimp)   1796 apply (rename_tac x')   1797 apply (cut_tac x=x and y=x' and z=y in dist_triangle)   1798 apply simp   1799 done   1800   1801 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"   1802 proof-   1803 { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"   1804 hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)   1805 } moreover   1806 { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"   1807 hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto   1808 } ultimately   1809 show ?thesis unfolding open_contains_ball by auto   1810 qed   1811   1812 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"   1813 by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)   1814   1815 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"   1816 apply (simp add: interior_def, safe)   1817 apply (force simp add: open_contains_cball)   1818 apply (rule_tac x="ball x e" in exI)   1819 apply (simp add: subset_trans [OF ball_subset_cball])   1820 done   1821   1822 lemma islimpt_ball:   1823 fixes x y :: "'a::{real_normed_vector,perfect_space}"   1824 shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")   1825 proof   1826 assume "?lhs"   1827 { assume "e \<le> 0"   1828 hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto   1829 have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto   1830 }   1831 hence "e > 0" by (metis not_less)   1832 moreover   1833 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   1834 ultimately show "?rhs" by auto   1835 next   1836 assume "?rhs" hence "e>0" by auto   1837 { fix d::real assume "d>0"   1838 have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1839 proof(cases "d \<le> dist x y")   1840 case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1841 proof(cases "x=y")   1842 case True hence False using d \<le> dist x y d>0 by auto   1843 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto   1844 next   1845 case False   1846   1847 have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))   1848 = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"   1849 unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto   1850 also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"   1851 using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]   1852 unfolding scaleR_minus_left scaleR_one   1853 by (auto simp add: norm_minus_commute)   1854 also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"   1855 unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]   1856 unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto   1857 also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)   1858 finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto   1859   1860 moreover   1861   1862 have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"   1863 using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)   1864 moreover   1865 have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel   1866 using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]   1867 unfolding dist_norm by auto   1868 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   1869 qed   1870 next   1871 case False hence "d > dist x y" by auto   1872 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1873 proof(cases "x=y")   1874 case True   1875 obtain z where **: "z \<noteq> y" "dist z y < min e d"   1876 using perfect_choose_dist[of "min e d" y]   1877 using d > 0 e>0 by auto   1878 show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1879 unfolding x = y   1880 using z \<noteq> y **   1881 by (rule_tac x=z in bexI, auto simp add: dist_commute)   1882 next   1883 case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"   1884 using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)   1885 qed   1886 qed }   1887 thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto   1888 qed   1889   1890 lemma closure_ball_lemma:   1891 fixes x y :: "'a::real_normed_vector"   1892 assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"   1893 proof (rule islimptI)   1894 fix T assume "y \<in> T" "open T"   1895 then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"   1896 unfolding open_dist by fast   1897 (* choose point between x and y, within distance r of y. *)   1898 def k \<equiv> "min 1 (r / (2 * dist x y))"   1899 def z \<equiv> "y + scaleR k (x - y)"   1900 have z_def2: "z = x + scaleR (1 - k) (y - x)"   1901 unfolding z_def by (simp add: algebra_simps)   1902 have "dist z y < r"   1903 unfolding z_def k_def using 0 < r   1904 by (simp add: dist_norm min_def)   1905 hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp   1906 have "dist x z < dist x y"   1907 unfolding z_def2 dist_norm   1908 apply (simp add: norm_minus_commute)   1909 apply (simp only: dist_norm [symmetric])   1910 apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)   1911 apply (rule mult_strict_right_mono)   1912 apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)   1913 apply (simp add: zero_less_dist_iff x \<noteq> y)   1914 done   1915 hence "z \<in> ball x (dist x y)" by simp   1916 have "z \<noteq> y"   1917 unfolding z_def k_def using x \<noteq> y 0 < r   1918 by (simp add: min_def)   1919 show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"   1920 using z \<in> ball x (dist x y) z \<in> T z \<noteq> y   1921 by fast   1922 qed   1923   1924 lemma closure_ball:   1925 fixes x :: "'a::real_normed_vector"   1926 shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"   1927 apply (rule equalityI)   1928 apply (rule closure_minimal)   1929 apply (rule ball_subset_cball)   1930 apply (rule closed_cball)   1931 apply (rule subsetI, rename_tac y)   1932 apply (simp add: le_less [where 'a=real])   1933 apply (erule disjE)   1934 apply (rule subsetD [OF closure_subset], simp)   1935 apply (simp add: closure_def)   1936 apply clarify   1937 apply (rule closure_ball_lemma)   1938 apply (simp add: zero_less_dist_iff)   1939 done   1940   1941 (* In a trivial vector space, this fails for e = 0. *)   1942 lemma interior_cball:   1943 fixes x :: "'a::{real_normed_vector, perfect_space}"   1944 shows "interior (cball x e) = ball x e"   1945 proof(cases "e\<ge>0")   1946 case False note cs = this   1947 from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover   1948 { fix y assume "y \<in> cball x e"   1949 hence False unfolding mem_cball using dist_nz[of x y] cs by auto }   1950 hence "cball x e = {}" by auto   1951 hence "interior (cball x e) = {}" using interior_empty by auto   1952 ultimately show ?thesis by blast   1953 next   1954 case True note cs = this   1955 have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover   1956 { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"   1957 then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast   1958   1959 then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"   1960 using perfect_choose_dist [of d] by auto   1961 have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)   1962 hence xa_cball:"xa \<in> cball x e" using as(1) by auto   1963   1964 hence "y \<in> ball x e" proof(cases "x = y")   1965 case True   1966 hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)   1967 thus "y \<in> ball x e" using x = y  by simp   1968 next   1969 case False   1970 have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm   1971 using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto   1972 hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast   1973 have "y - x \<noteq> 0" using x \<noteq> y by auto   1974 hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]   1975 using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto   1976   1977 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)"   1978 by (auto simp add: dist_norm algebra_simps)   1979 also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"   1980 by (auto simp add: algebra_simps)   1981 also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"   1982 using ** by auto   1983 also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)   1984 finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)   1985 thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto   1986 qed }   1987 hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto   1988 ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto   1989 qed   1990   1991 lemma frontier_ball:   1992 fixes a :: "'a::real_normed_vector"   1993 shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"   1994 apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)   1995 apply (simp add: set_eq_iff)   1996 by arith   1997   1998 lemma frontier_cball:   1999 fixes a :: "'a::{real_normed_vector, perfect_space}"   2000 shows "frontier(cball a e) = {x. dist a x = e}"   2001 apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)   2002 apply (simp add: set_eq_iff)   2003 by arith   2004   2005 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"   2006 apply (simp add: set_eq_iff not_le)   2007 by (metis zero_le_dist dist_self order_less_le_trans)   2008 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)   2009   2010 lemma cball_eq_sing:   2011 fixes x :: "'a::{metric_space,perfect_space}"   2012 shows "(cball x e = {x}) \<longleftrightarrow> e = 0"   2013 proof (rule linorder_cases)   2014 assume e: "0 < e"   2015 obtain a where "a \<noteq> x" "dist a x < e"   2016 using perfect_choose_dist [OF e] by auto   2017 hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)   2018 with e show ?thesis by (auto simp add: set_eq_iff)   2019 qed auto   2020   2021 lemma cball_sing:   2022 fixes x :: "'a::metric_space"   2023 shows "e = 0 ==> cball x e = {x}"   2024 by (auto simp add: set_eq_iff)   2025   2026   2027 subsection {* Boundedness *}   2028   2029 (* FIXME: This has to be unified with BSEQ!! *)   2030 definition (in metric_space)   2031 bounded :: "'a set \<Rightarrow> bool" where   2032 "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"   2033   2034 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"   2035 unfolding bounded_def   2036 apply safe   2037 apply (rule_tac x="dist a x + e" in exI, clarify)   2038 apply (drule (1) bspec)   2039 apply (erule order_trans [OF dist_triangle add_left_mono])   2040 apply auto   2041 done   2042   2043 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"   2044 unfolding bounded_any_center [where a=0]   2045 by (simp add: dist_norm)   2046   2047 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"   2048 unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)   2049 using assms by auto   2050   2051 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)   2052 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"   2053 by (metis bounded_def subset_eq)   2054   2055 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"   2056 by (metis bounded_subset interior_subset)   2057   2058 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"   2059 proof-   2060 from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto   2061 { fix y assume "y \<in> closure S"   2062 then obtain f where f: "\<forall>n. f n \<in> S" "(f ---> y) sequentially"   2063 unfolding closure_sequential by auto   2064 have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp   2065 hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"   2066 by (rule eventually_mono, simp add: f(1))   2067 have "dist x y \<le> a"   2068 apply (rule Lim_dist_ubound [of sequentially f])   2069 apply (rule trivial_limit_sequentially)   2070 apply (rule f(2))   2071 apply fact   2072 done   2073 }   2074 thus ?thesis unfolding bounded_def by auto   2075 qed   2076   2077 lemma bounded_cball[simp,intro]: "bounded (cball x e)"   2078 apply (simp add: bounded_def)   2079 apply (rule_tac x=x in exI)   2080 apply (rule_tac x=e in exI)   2081 apply auto   2082 done   2083   2084 lemma bounded_ball[simp,intro]: "bounded(ball x e)"   2085 by (metis ball_subset_cball bounded_cball bounded_subset)   2086   2087 lemma finite_imp_bounded[intro]:   2088 fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"   2089 proof-   2090 { fix a and F :: "'a set" assume as:"bounded F"   2091 then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto   2092 hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto   2093 hence "bounded (insert a F)" unfolding bounded_def by (intro exI)   2094 }   2095 thus ?thesis using finite_induct[of S bounded] using bounded_empty assms by auto   2096 qed   2097   2098 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"   2099 apply (auto simp add: bounded_def)   2100 apply (rename_tac x y r s)   2101 apply (rule_tac x=x in exI)   2102 apply (rule_tac x="max r (dist x y + s)" in exI)   2103 apply (rule ballI, rename_tac z, safe)   2104 apply (drule (1) bspec, simp)   2105 apply (drule (1) bspec)   2106 apply (rule min_max.le_supI2)   2107 apply (erule order_trans [OF dist_triangle add_left_mono])   2108 done   2109   2110 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"   2111 by (induct rule: finite_induct[of F], auto)   2112   2113 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"   2114 apply (simp add: bounded_iff)   2115 apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")   2116 by metis arith   2117   2118 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"   2119 by (metis Int_lower1 Int_lower2 bounded_subset)   2120   2121 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"   2122 apply (metis Diff_subset bounded_subset)   2123 done   2124   2125 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"   2126 by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)   2127   2128 lemma not_bounded_UNIV[simp, intro]:   2129 "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"   2130 proof(auto simp add: bounded_pos not_le)   2131 obtain x :: 'a where "x \<noteq> 0"   2132 using perfect_choose_dist [OF zero_less_one] by fast   2133 fix b::real assume b: "b >0"   2134 have b1: "b +1 \<ge> 0" using b by simp   2135 with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"   2136 by (simp add: norm_sgn)   2137 then show "\<exists>x::'a. b < norm x" ..   2138 qed   2139   2140 lemma bounded_linear_image:   2141 assumes "bounded S" "bounded_linear f"   2142 shows "bounded(f  S)"   2143 proof-   2144 from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto   2145 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)   2146 { fix x assume "x\<in>S"   2147 hence "norm x \<le> b" using b by auto   2148 hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)   2149 by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)   2150 }   2151 thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)   2152 using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)   2153 qed   2154   2155 lemma bounded_scaling:   2156 fixes S :: "'a::real_normed_vector set"   2157 shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"   2158 apply (rule bounded_linear_image, assumption)   2159 apply (rule bounded_linear_scaleR_right)   2160 done   2161   2162 lemma bounded_translation:   2163 fixes S :: "'a::real_normed_vector set"   2164 assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"   2165 proof-   2166 from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto   2167 { fix x assume "x\<in>S"   2168 hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto   2169 }   2170 thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]   2171 by (auto intro!: exI[of _ "b + norm a"])   2172 qed   2173   2174   2175 text{* Some theorems on sups and infs using the notion "bounded". *}   2176   2177 lemma bounded_real:   2178 fixes S :: "real set"   2179 shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"   2180 by (simp add: bounded_iff)   2181   2182 lemma bounded_has_Sup:   2183 fixes S :: "real set"   2184 assumes "bounded S" "S \<noteq> {}"   2185 shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"   2186 proof   2187 fix x assume "x\<in>S"   2188 thus "x \<le> Sup S"   2189 by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)   2190 next   2191 show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms   2192 by (metis SupInf.Sup_least)   2193 qed   2194   2195 lemma Sup_insert:   2196 fixes S :: "real set"   2197 shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"   2198 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)   2199   2200 lemma Sup_insert_finite:   2201 fixes S :: "real set"   2202 shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"   2203 apply (rule Sup_insert)   2204 apply (rule finite_imp_bounded)   2205 by simp   2206   2207 lemma bounded_has_Inf:   2208 fixes S :: "real set"   2209 assumes "bounded S" "S \<noteq> {}"   2210 shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"   2211 proof   2212 fix x assume "x\<in>S"   2213 from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto   2214 thus "x \<ge> Inf S" using x\<in>S   2215 by (metis Inf_lower_EX abs_le_D2 minus_le_iff)   2216 next   2217 show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms   2218 by (metis SupInf.Inf_greatest)   2219 qed   2220   2221 lemma Inf_insert:   2222 fixes S :: "real set"   2223 shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"   2224 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)   2225 lemma Inf_insert_finite:   2226 fixes S :: "real set"   2227 shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"   2228 by (rule Inf_insert, rule finite_imp_bounded, simp)   2229   2230 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)   2231 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"   2232 apply (frule isGlb_isLb)   2233 apply (frule_tac x = y in isGlb_isLb)   2234 apply (blast intro!: order_antisym dest!: isGlb_le_isLb)   2235 done   2236   2237   2238 subsection {* Equivalent versions of compactness *}   2239   2240 subsubsection{* Sequential compactness *}   2241   2242 definition   2243 compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)   2244 "compact S \<longleftrightarrow>   2245 (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>   2246 (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"   2247   2248 lemma compactI:   2249 assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"   2250 shows "compact S"   2251 unfolding compact_def using assms by fast   2252   2253 lemma compactE:   2254 assumes "compact S" "\<forall>n. f n \<in> S"   2255 obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"   2256 using assms unfolding compact_def by fast   2257   2258 text {*   2259 A metric space (or topological vector space) is said to have the   2260 Heine-Borel property if every closed and bounded subset is compact.   2261 *}   2262   2263 class heine_borel = metric_space +   2264 assumes bounded_imp_convergent_subsequence:   2265 "bounded s \<Longrightarrow> \<forall>n. f n \<in> s   2266 \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2267   2268 lemma bounded_closed_imp_compact:   2269 fixes s::"'a::heine_borel set"   2270 assumes "bounded s" and "closed s" shows "compact s"   2271 proof (unfold compact_def, clarify)   2272 fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"   2273 obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"   2274 using bounded_imp_convergent_subsequence [OF bounded s \<forall>n. f n \<in> s] by auto   2275 from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp   2276 have "l \<in> s" using closed s fr l   2277 unfolding closed_sequential_limits by blast   2278 show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2279 using l \<in> s r l by blast   2280 qed   2281   2282 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"   2283 proof(induct n)   2284 show "0 \<le> r 0" by auto   2285 next   2286 fix n assume "n \<le> r n"   2287 moreover have "r n < r (Suc n)"   2288 using assms [unfolded subseq_def] by auto   2289 ultimately show "Suc n \<le> r (Suc n)" by auto   2290 qed   2291   2292 lemma eventually_subseq:   2293 assumes r: "subseq r"   2294 shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"   2295 unfolding eventually_sequentially   2296 by (metis subseq_bigger [OF r] le_trans)   2297   2298 lemma lim_subseq:   2299 "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"   2300 unfolding tendsto_def eventually_sequentially o_def   2301 by (metis subseq_bigger le_trans)   2302   2303 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"   2304 unfolding Ex1_def   2305 apply (rule_tac x="nat_rec e f" in exI)   2306 apply (rule conjI)+   2307 apply (rule def_nat_rec_0, simp)   2308 apply (rule allI, rule def_nat_rec_Suc, simp)   2309 apply (rule allI, rule impI, rule ext)   2310 apply (erule conjE)   2311 apply (induct_tac x)   2312 apply simp   2313 apply (erule_tac x="n" in allE)   2314 apply (simp)   2315 done   2316   2317 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"   2318 assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"   2319 shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"   2320 proof-   2321 have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto   2322 then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto   2323 { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"   2324 { fix n::nat   2325 obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto   2326 have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto   2327 with n have "s N \<le> t - e" using e>0 by auto   2328 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 }   2329 hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto   2330 hence False using isLub_le_isUb[OF t, of "t - e"] and e>0 by auto }   2331 thus ?thesis by blast   2332 qed   2333   2334 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"   2335 assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"   2336 shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"   2337 using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]   2338 unfolding monoseq_def incseq_def   2339 apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]   2340 unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto   2341   2342 (* TODO: merge this lemma with the ones above *)   2343 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"   2344 assumes "bounded {s n| n::nat. True}" "\<forall>n. (s n) \<le>(s(Suc n))"   2345 shows "\<exists>l. (s ---> l) sequentially"   2346 proof-   2347 obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff] by auto   2348 { fix m::nat   2349 have "\<And> n. n\<ge>m \<longrightarrow> (s m) \<le> (s n)"   2350 apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)   2351 apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq) }   2352 hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto   2353 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]   2354 unfolding monoseq_def by auto   2355 thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="l" in exI)   2356 unfolding dist_norm by auto   2357 qed   2358   2359 lemma compact_real_lemma:   2360 assumes "\<forall>n::nat. abs(s n) \<le> b"   2361 shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"   2362 proof-   2363 obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"   2364 using seq_monosub[of s] by auto   2365 thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms   2366 unfolding tendsto_iff dist_norm eventually_sequentially by auto   2367 qed   2368   2369 instance real :: heine_borel   2370 proof   2371 fix s :: "real set" and f :: "nat \<Rightarrow> real"   2372 assume s: "bounded s" and f: "\<forall>n. f n \<in> s"   2373 then obtain b where b: "\<forall>n. abs (f n) \<le> b"   2374 unfolding bounded_iff by auto   2375 obtain l :: real and r :: "nat \<Rightarrow> nat" where   2376 r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"   2377 using compact_real_lemma [OF b] by auto   2378 thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"   2379 by auto   2380 qed   2381   2382 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x$$ i)  s)"

  2383   apply (erule bounded_linear_image)

  2384   apply (rule bounded_linear_euclidean_component)

  2385   done

  2386

  2387 lemma compact_lemma:

  2388   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  2389   assumes "bounded s" and "\<forall>n. f n \<in> s"

  2390   shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>

  2391         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2392 proof

  2393   fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"

  2394   have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto

  2395   hence "\<exists>l::'a. \<exists>r. subseq r \<and>

  2396       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2397   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  2398   next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto

  2399     have s': "bounded ((\<lambda>x. x $$k)  s)" using bounded s by (rule bounded_component)   2400 obtain l1::"'a" and r1 where r1:"subseq r1" and   2401 lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n)$$ i) (l1 $$i) < e) sequentially"   2402 using insert(3) using insert(4) by auto   2403 have f': "\<forall>n. f (r1 n)$$ k \<in> (\<lambda>x. x $$k)  s" using \<forall>n. f n \<in> s by simp   2404 obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i))$$ k) ---> l2) sequentially"

  2405       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

  2406     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  2407       using r1 and r2 unfolding r_def o_def subseq_def by auto

  2408     moreover

  2409     def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"   2410 { fix e::real assume "e>0"   2411 from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n)$$ i) (l1 $$i) < e) sequentially" by blast   2412 from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n))$$ k) l2 < e) sequentially" by (rule tendstoD)

  2413       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$i) (l1$$ i) < e) sequentially"

  2414         by (rule eventually_subseq)

  2415       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$i) (l$$ i) < e) sequentially"

  2416         using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def

  2417         using insert.prems by auto

  2418     }

  2419     ultimately show ?case by auto

  2420   qed

  2421   thus "\<exists>l::'a. \<exists>r. subseq r \<and>

  2422       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$i) (l$$ i) < e) sequentially)"

  2423     apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe

  2424     apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe

  2425     apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)

  2426     apply(erule_tac x=i in ballE)

  2427   proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a

  2428     assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"

  2429     hence *:"i\<ge>DIM('a)" by auto

  2430     thus "dist (f (r n) $$i) (l$$ i) < e" using e by auto

  2431   qed

  2432 qed

  2433

  2434 instance euclidean_space \<subseteq> heine_borel

  2435 proof

  2436   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"

  2437   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2438   then obtain l::'a and r where r: "subseq r"

  2439     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$i) (l$$ i) < e) sequentially"

  2440     using compact_lemma [OF s f] by blast

  2441   let ?d = "{..<DIM('a)}"

  2442   { fix e::real assume "e>0"

  2443     hence "0 < e / (real_of_nat (card ?d))"

  2444       using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto

  2445     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$i) (l$$ i) < e / (real_of_nat (card ?d))) sequentially"

  2446       by simp

  2447     moreover

  2448     { fix n assume n: "\<forall>i. dist (f (r n) $$i) (l$$ i) < e / (real_of_nat (card ?d))"

  2449       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$i) (l$$ i))"

  2450         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  2451       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"

  2452         apply(rule setsum_strict_mono) using n by auto

  2453       finally have "dist (f (r n)) l < e" unfolding setsum_constant

  2454         using DIM_positive[where 'a='a] by auto

  2455     }

  2456     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  2457       by (rule eventually_elim1)

  2458   }

  2459   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  2460   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  2461 qed

  2462

  2463 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  2464 unfolding bounded_def

  2465 apply clarify

  2466 apply (rule_tac x="a" in exI)

  2467 apply (rule_tac x="e" in exI)

  2468 apply clarsimp

  2469 apply (drule (1) bspec)

  2470 apply (simp add: dist_Pair_Pair)

  2471 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  2472 done

  2473

  2474 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  2475 unfolding bounded_def

  2476 apply clarify

  2477 apply (rule_tac x="b" in exI)

  2478 apply (rule_tac x="e" in exI)

  2479 apply clarsimp

  2480 apply (drule (1) bspec)

  2481 apply (simp add: dist_Pair_Pair)

  2482 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  2483 done

  2484

  2485 instance prod :: (heine_borel, heine_borel) heine_borel

  2486 proof

  2487   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"

  2488   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  2489   from s have s1: "bounded (fst  s)" by (rule bounded_fst)

  2490   from f have f1: "\<forall>n. fst (f n) \<in> fst  s" by simp

  2491   obtain l1 r1 where r1: "subseq r1"

  2492     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"

  2493     using bounded_imp_convergent_subsequence [OF s1 f1]

  2494     unfolding o_def by fast

  2495   from s have s2: "bounded (snd  s)" by (rule bounded_snd)

  2496   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd  s" by simp

  2497   obtain l2 r2 where r2: "subseq r2"

  2498     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  2499     using bounded_imp_convergent_subsequence [OF s2 f2]

  2500     unfolding o_def by fast

  2501   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  2502     using lim_subseq [OF r2 l1] unfolding o_def .

  2503   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  2504     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  2505   have r: "subseq (r1 \<circ> r2)"

  2506     using r1 r2 unfolding subseq_def by simp

  2507   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2508     using l r by fast

  2509 qed

  2510

  2511 subsubsection{* Completeness *}

  2512

  2513 lemma cauchy_def:

  2514   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  2515 unfolding Cauchy_def by blast

  2516

  2517 definition

  2518   complete :: "'a::metric_space set \<Rightarrow> bool" where

  2519   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f

  2520                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"

  2521

  2522 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  2523 proof-

  2524   { assume ?rhs

  2525     { fix e::real

  2526       assume "e>0"

  2527       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  2528         by (erule_tac x="e/2" in allE) auto

  2529       { fix n m

  2530         assume nm:"N \<le> m \<and> N \<le> n"

  2531         hence "dist (s m) (s n) < e" using N

  2532           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  2533           by blast

  2534       }

  2535       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  2536         by blast

  2537     }

  2538     hence ?lhs

  2539       unfolding cauchy_def

  2540       by blast

  2541   }

  2542   thus ?thesis

  2543     unfolding cauchy_def

  2544     using dist_triangle_half_l

  2545     by blast

  2546 qed

  2547

  2548 lemma convergent_imp_cauchy:

  2549  "(s ---> l) sequentially ==> Cauchy s"

  2550 proof(simp only: cauchy_def, rule, rule)

  2551   fix e::real assume "e>0" "(s ---> l) sequentially"

  2552   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

  2553   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

  2554 qed

  2555

  2556 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  2557 proof-

  2558   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

  2559   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  2560   moreover

  2561   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  2562   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  2563     unfolding bounded_any_center [where a="s N"] by auto

  2564   ultimately show "?thesis"

  2565     unfolding bounded_any_center [where a="s N"]

  2566     apply(rule_tac x="max a 1" in exI) apply auto

  2567     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  2568 qed

  2569

  2570 lemma compact_imp_complete: assumes "compact s" shows "complete s"

  2571 proof-

  2572   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  2573     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

  2574

  2575     note lr' = subseq_bigger [OF lr(2)]

  2576

  2577     { fix e::real assume "e>0"

  2578       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

  2579       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

  2580       { fix n::nat assume n:"n \<ge> max N M"

  2581         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  2582         moreover have "r n \<ge> N" using lr'[of n] n by auto

  2583         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  2584         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)  }

  2585       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  2586     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  2587   thus ?thesis unfolding complete_def by auto

  2588 qed

  2589

  2590 instance heine_borel < complete_space

  2591 proof

  2592   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2593   hence "bounded (range f)"

  2594     by (rule cauchy_imp_bounded)

  2595   hence "compact (closure (range f))"

  2596     using bounded_closed_imp_compact [of "closure (range f)"] by auto

  2597   hence "complete (closure (range f))"

  2598     by (rule compact_imp_complete)

  2599   moreover have "\<forall>n. f n \<in> closure (range f)"

  2600     using closure_subset [of "range f"] by auto

  2601   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  2602     using Cauchy f unfolding complete_def by auto

  2603   then show "convergent f"

  2604     unfolding convergent_def by auto

  2605 qed

  2606

  2607 instance euclidean_space \<subseteq> banach ..

  2608

  2609 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  2610 proof(simp add: complete_def, rule, rule)

  2611   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  2612   hence "convergent f" by (rule Cauchy_convergent)

  2613   thus "\<exists>l. f ----> l" unfolding convergent_def .

  2614 qed

  2615

  2616 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  2617 proof -

  2618   { fix x assume "x islimpt s"

  2619     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  2620       unfolding islimpt_sequential by auto

  2621     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  2622       using complete s[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto

  2623     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  2624   }

  2625   thus "closed s" unfolding closed_limpt by auto

  2626 qed

  2627

  2628 lemma complete_eq_closed:

  2629   fixes s :: "'a::complete_space set"

  2630   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  2631 proof

  2632   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  2633 next

  2634   assume ?rhs

  2635   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  2636     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  2637     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  }

  2638   thus ?lhs unfolding complete_def by auto

  2639 qed

  2640

  2641 lemma convergent_eq_cauchy:

  2642   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  2643   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  2644   unfolding Cauchy_convergent_iff convergent_def ..

  2645

  2646 lemma convergent_imp_bounded:

  2647   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  2648   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  2649   by (intro cauchy_imp_bounded convergent_imp_cauchy)

  2650

  2651 subsubsection{* Total boundedness *}

  2652

  2653 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  2654   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  2655 declare helper_1.simps[simp del]

  2656

  2657 lemma compact_imp_totally_bounded:

  2658   assumes "compact s"

  2659   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  2660 proof(rule, rule, rule ccontr)

  2661   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)"

  2662   def x \<equiv> "helper_1 s e"

  2663   { fix n

  2664     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  2665     proof(induct_tac rule:nat_less_induct)

  2666       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  2667       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  2668       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

  2669       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  2670       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  2671         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  2672       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  2673     qed }

  2674   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  2675   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

  2676   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto

  2677   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

  2678   show False

  2679     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  2680     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  2681     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  2682 qed

  2683

  2684 subsubsection{* Heine-Borel theorem *}

  2685

  2686 text {* Following Burkill \& Burkill vol. 2. *}

  2687

  2688 lemma heine_borel_lemma: fixes s::"'a::metric_space set"

  2689   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"

  2690   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"

  2691 proof(rule ccontr)

  2692   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"

  2693   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto

  2694   { fix n::nat

  2695     have "1 / real (n + 1) > 0" by auto

  2696     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 }

  2697   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

  2698   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)"

  2699     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

  2700

  2701   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"

  2702     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto

  2703

  2704   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto

  2705   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"

  2706     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto

  2707

  2708   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"

  2709     using lr[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] by auto

  2710

  2711   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and e>0 by auto

  2712   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"

  2713     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2

  2714     using subseq_bigger[OF r, of "N1 + N2"] by auto

  2715

  2716   def x \<equiv> "(f (r (N1 + N2)))"

  2717   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def

  2718     using f[THEN spec[where x="r (N1 + N2)"]] using b\<in>t by auto

  2719   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto

  2720   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto

  2721

  2722   have "dist x l < e/2" using N1 unfolding x_def o_def by auto

  2723   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)

  2724

  2725   thus False using e and y\<notin>b by auto

  2726 qed

  2727

  2728 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  2729                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"

  2730 proof clarify

  2731   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"

  2732   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

  2733   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto

  2734   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

  2735   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast

  2736

  2737   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

  2738   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" by auto

  2739

  2740   have "finite (bb  k)" using k(1) by auto

  2741   moreover

  2742   { fix x assume "x\<in>s"

  2743     hence "x\<in>\<Union>(\<lambda>x. ball x e)  k" using k(3)  unfolding subset_eq by auto

  2744     hence "\<exists>X\<in>bb  k. x \<in> X" using bb k(2) by blast

  2745     hence "x \<in> \<Union>(bb  k)" using  Union_iff[of x "bb  k"] by auto

  2746   }

  2747   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

  2748 qed

  2749

  2750 subsubsection {* Bolzano-Weierstrass property *}

  2751

  2752 lemma heine_borel_imp_bolzano_weierstrass:

  2753   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'))"

  2754           "infinite t"  "t \<subseteq> s"

  2755   shows "\<exists>x \<in> s. x islimpt t"

  2756 proof(rule ccontr)

  2757   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2758   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

  2759     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

  2760   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2761     using assms(1)[THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2762   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2763   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2764     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

  2765     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  }

  2766   hence "inj_on f t" unfolding inj_on_def by simp

  2767   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2768   moreover

  2769   { fix x assume "x\<in>t" "f x \<notin> g"

  2770     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2771     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2772     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2773     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2774   hence "f  t \<subseteq> g" by auto

  2775   ultimately show False using g(2) using finite_subset by auto

  2776 qed

  2777

  2778 subsubsection {* Complete the chain of compactness variants *}

  2779

  2780 lemma islimpt_range_imp_convergent_subsequence:

  2781   fixes f :: "nat \<Rightarrow> 'a::metric_space"

  2782   assumes "l islimpt (range f)"

  2783   shows "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2784 proof (intro exI conjI)

  2785   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2786     using assms unfolding islimpt_def

  2787     by (drule_tac x="ball l e" in spec)

  2788        (auto simp add: zero_less_dist_iff dist_commute)

  2789

  2790   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"

  2791   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"

  2792     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])

  2793   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"

  2794     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])

  2795   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"

  2796     unfolding t_def by (simp add: Least_le)

  2797   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"

  2798     unfolding t_def by (drule not_less_Least) simp

  2799   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"

  2800     apply (rule t_le)

  2801     apply (erule f_t_neq)

  2802     apply (erule (1) less_le_trans [OF f_t_closer])

  2803     done

  2804   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"

  2805     by (drule f_t_closer) auto

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

  2807     apply (subst less_le)

  2808     apply (rule conjI)

  2809     apply (rule t_antimono)

  2810     apply (erule f_t_neq)

  2811     apply (erule f_t_closer [THEN less_imp_le])

  2812     apply (rule t_dist_f_neq [symmetric])

  2813     apply (erule f_t_neq)

  2814     done

  2815   have dist_f_t_less':

  2816     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"

  2817     apply (simp add: le_less)

  2818     apply (erule disjE)

  2819     apply (rule less_trans)

  2820     apply (erule f_t_closer)

  2821     apply (rule le_less_trans)

  2822     apply (erule less_tD)

  2823     apply (erule f_t_neq)

  2824     apply (erule f_t_closer)

  2825     apply (erule subst)

  2826     apply (erule f_t_closer)

  2827     done

  2828

  2829   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"

  2830   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"

  2831     unfolding r_def by simp_all

  2832   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"

  2833     by (induct_tac n) (simp_all add: r_simps f_t_neq)

  2834

  2835   show "subseq r"

  2836     unfolding subseq_Suc_iff

  2837     apply (rule allI)

  2838     apply (case_tac n)

  2839     apply (simp_all add: r_simps)

  2840     apply (rule t_less, rule zero_less_one)

  2841     apply (rule t_less, rule f_r_neq)

  2842     done

  2843   show "((f \<circ> r) ---> l) sequentially"

  2844     unfolding LIMSEQ_def o_def

  2845     apply (clarify, rename_tac e, rule_tac x="t e" in exI, clarify)

  2846     apply (drule le_trans, rule seq_suble [OF subseq r])

  2847     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)

  2848     done

  2849 qed

  2850

  2851 lemma finite_range_imp_infinite_repeats:

  2852   fixes f :: "nat \<Rightarrow> 'a"

  2853   assumes "finite (range f)"

  2854   shows "\<exists>k. infinite {n. f n = k}"

  2855 proof -

  2856   { fix A :: "'a set" assume "finite A"

  2857     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"

  2858     proof (induct)

  2859       case empty thus ?case by simp

  2860     next

  2861       case (insert x A)

  2862      show ?case

  2863       proof (cases "finite {n. f n = x}")

  2864         case True

  2865         with infinite {n. f n \<in> insert x A}

  2866         have "infinite {n. f n \<in> A}" by simp

  2867         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)

  2868       next

  2869         case False thus "\<exists>k. infinite {n. f n = k}" ..

  2870       qed

  2871     qed

  2872   } note H = this

  2873   from assms show "\<exists>k. infinite {n. f n = k}"

  2874     by (rule H) simp

  2875 qed

  2876

  2877 lemma bolzano_weierstrass_imp_compact:

  2878   fixes s :: "'a::metric_space set"

  2879   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2880   shows "compact s"

  2881 proof -

  2882   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  2883     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2884     proof (cases "finite (range f)")

  2885       case True

  2886       hence "\<exists>l. infinite {n. f n = l}"

  2887         by (rule finite_range_imp_infinite_repeats)

  2888       then obtain l where "infinite {n. f n = l}" ..

  2889       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"

  2890         by (rule infinite_enumerate)

  2891       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto

  2892       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2893         unfolding o_def by (simp add: fr tendsto_const)

  2894       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2895         by - (rule exI)

  2896       from f have "\<forall>n. f (r n) \<in> s" by simp

  2897       hence "l \<in> s" by (simp add: fr)

  2898       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2899         by (rule rev_bexI) fact

  2900     next

  2901       case False

  2902       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto

  2903       then obtain l where "l \<in> s" "l islimpt (range f)" ..

  2904       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2905         using l islimpt (range f)

  2906         by (rule islimpt_range_imp_convergent_subsequence)

  2907       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  2908     qed

  2909   }

  2910   thus ?thesis unfolding compact_def by auto

  2911 qed

  2912

  2913 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where

  2914   "helper_2 beyond 0 = beyond 0" |

  2915   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"

  2916

  2917 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"

  2918   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2919   shows "bounded s"

  2920 proof(rule ccontr)

  2921   assume "\<not> bounded s"

  2922   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"

  2923     unfolding bounded_any_center [where a=undefined]

  2924     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto

  2925   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"

  2926     unfolding linorder_not_le by auto

  2927   def x \<equiv> "helper_2 beyond"

  2928

  2929   { fix m n ::nat assume "m<n"

  2930     hence "dist undefined (x m) + 1 < dist undefined (x n)"

  2931     proof(induct n)

  2932       case 0 thus ?case by auto

  2933     next

  2934       case (Suc n)

  2935       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"

  2936         unfolding x_def and helper_2.simps

  2937         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto

  2938       thus ?case proof(cases "m < n")

  2939         case True thus ?thesis using Suc and * by auto

  2940       next

  2941         case False hence "m = n" using Suc(2) by auto

  2942         thus ?thesis using * by auto

  2943       qed

  2944     qed  } note * = this

  2945   { fix m n ::nat assume "m\<noteq>n"

  2946     have "1 < dist (x m) (x n)"

  2947     proof(cases "m<n")

  2948       case True

  2949       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto

  2950       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith

  2951     next

  2952       case False hence "n<m" using m\<noteq>n by auto

  2953       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto

  2954       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith

  2955     qed  } note ** = this

  2956   { fix a b assume "x a = x b" "a \<noteq> b"

  2957     hence False using **[of a b] by auto  }

  2958   hence "inj x" unfolding inj_on_def by auto

  2959   moreover

  2960   { fix n::nat

  2961     have "x n \<in> s"

  2962     proof(cases "n = 0")

  2963       case True thus ?thesis unfolding x_def using beyond by auto

  2964     next

  2965       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto

  2966       thus ?thesis unfolding x_def using beyond by auto

  2967     qed  }

  2968   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

  2969

  2970   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto

  2971   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

  2972   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"]]

  2973     unfolding dist_nz by auto

  2974   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto

  2975 qed

  2976

  2977 lemma sequence_infinite_lemma:

  2978   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2979   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2980   shows "infinite (range f)"

  2981 proof

  2982   assume "finite (range f)"

  2983   hence "closed (range f)" by (rule finite_imp_closed)

  2984   hence "open (- range f)" by (rule open_Compl)

  2985   from assms(1) have "l \<in> - range f" by auto

  2986   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2987     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2988   thus False unfolding eventually_sequentially by auto

  2989 qed

  2990

  2991 lemma closure_insert:

  2992   fixes x :: "'a::t1_space"

  2993   shows "closure (insert x s) = insert x (closure s)"

  2994 apply (rule closure_unique)

  2995 apply (rule insert_mono [OF closure_subset])

  2996 apply (rule closed_insert [OF closed_closure])

  2997 apply (simp add: closure_minimal)

  2998 done

  2999

  3000 lemma islimpt_insert:

  3001   fixes x :: "'a::t1_space"

  3002   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  3003 proof

  3004   assume *: "x islimpt (insert a s)"

  3005   show "x islimpt s"

  3006   proof (rule islimptI)

  3007     fix t assume t: "x \<in> t" "open t"

  3008     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  3009     proof (cases "x = a")

  3010       case True

  3011       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  3012         using * t by (rule islimptE)

  3013       with x = a show ?thesis by auto

  3014     next

  3015       case False

  3016       with t have t': "x \<in> t - {a}" "open (t - {a})"

  3017         by (simp_all add: open_Diff)

  3018       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  3019         using * t' by (rule islimptE)

  3020       thus ?thesis by auto

  3021     qed

  3022   qed

  3023 next

  3024   assume "x islimpt s" thus "x islimpt (insert a s)"

  3025     by (rule islimpt_subset) auto

  3026 qed

  3027

  3028 lemma islimpt_union_finite:

  3029   fixes x :: "'a::t1_space"

  3030   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3031 by (induct set: finite, simp_all add: islimpt_insert)

  3032

  3033 lemma sequence_unique_limpt:

  3034   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3035   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  3036   shows "l' = l"

  3037 proof (rule ccontr)

  3038   assume "l' \<noteq> l"

  3039   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3040     using hausdorff [OF l' \<noteq> l] by auto

  3041   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3042     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3043   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3044     unfolding eventually_sequentially by auto

  3045

  3046   have "UNIV = {..<N} \<union> {N..}" by auto

  3047   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  3048   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  3049   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  3050   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3051     using l' \<in> s open s by (rule islimptE)

  3052   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  3053   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  3054   with s \<inter> t = {} show False by simp

  3055 qed

  3056

  3057 lemma bolzano_weierstrass_imp_closed:

  3058   fixes s :: "'a::metric_space set" (* TODO: can this be generalized? *)

  3059   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3060   shows "closed s"

  3061 proof-

  3062   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3063     hence "l \<in> s"

  3064     proof(cases "\<forall>n. x n \<noteq> l")

  3065       case False thus "l\<in>s" using as(1) by auto

  3066     next

  3067       case True note cas = this

  3068       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  3069       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  3070       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  3071     qed  }

  3072   thus ?thesis unfolding closed_sequential_limits by fast

  3073 qed

  3074

  3075 text {* Hence express everything as an equivalence. *}

  3076

  3077 lemma compact_eq_heine_borel:

  3078   fixes s :: "'a::metric_space set"

  3079   shows "compact s \<longleftrightarrow>

  3080            (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

  3081                --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))" (is "?lhs = ?rhs")

  3082 proof

  3083   assume ?lhs thus ?rhs by (rule compact_imp_heine_borel)

  3084 next

  3085   assume ?rhs

  3086   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"

  3087     by (blast intro: heine_borel_imp_bolzano_weierstrass[of s])

  3088   thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3089 qed

  3090

  3091 lemma compact_eq_bolzano_weierstrass:

  3092   fixes s :: "'a::metric_space set"

  3093   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3094 proof

  3095   assume ?lhs thus ?rhs unfolding compact_eq_heine_borel using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3096 next

  3097   assume ?rhs thus ?lhs by (rule bolzano_weierstrass_imp_compact)

  3098 qed

  3099

  3100 lemma nat_approx_posE:

  3101   fixes e::real

  3102   assumes "0 < e"

  3103   obtains n::nat where "1 / (Suc n) < e"

  3104 proof atomize_elim

  3105   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3106     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3107   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3108     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3109   also have "\<dots> = e" by simp

  3110   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3111 qed

  3112

  3113 lemma compact_eq_totally_bounded:

  3114   shows "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3115 proof (safe intro!: compact_imp_complete)

  3116   fix e::real

  3117   def f \<equiv> "(\<lambda>x::'a. ball x e)  UNIV"

  3118   assume "0 < e" "compact s"

  3119   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3120     by (simp add: compact_eq_heine_borel)

  3121   moreover

  3122   have d0: "\<And>x::'a. dist x x < e" using 0 < e by simp

  3123   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f" by (auto simp: f_def intro!: d0)

  3124   ultimately have "(\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')" ..

  3125   then guess K .. note K = this

  3126   have "\<forall>K'\<in>K. \<exists>k. K' = ball k e" using K by (auto simp: f_def)

  3127   then obtain k where "\<And>K'. K' \<in> K \<Longrightarrow> K' = ball (k K') e" unfolding bchoice_iff by blast

  3128   thus "\<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" using K

  3129     by (intro exI[where x="k  K"]) (auto simp: f_def)

  3130 next

  3131   assume assms: "complete s" "\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k"

  3132   show "compact s"

  3133   proof cases

  3134     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3135   next

  3136     assume "s \<noteq> {}"

  3137     show ?thesis

  3138       unfolding compact_def

  3139     proof safe

  3140       fix f::"nat \<Rightarrow> _" assume "\<forall>n. f n \<in> s" hence f: "\<And>n. f n \<in> s" by simp

  3141       from assms have "\<forall>e. \<exists>k. e>0 \<longrightarrow> finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))" by simp

  3142       then obtain K where

  3143         K: "\<And>e. e > 0 \<Longrightarrow> finite (K e) \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3144         unfolding choice_iff by blast

  3145       {

  3146         fix e::real and f' have f': "\<And>n::nat. (f o f') n \<in> s" using f by auto

  3147         assume "e > 0"

  3148         from K[OF this] have K: "finite (K e)" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3149           by simp_all

  3150         have "\<exists>k\<in>(K e). \<exists>r. subseq r \<and> (\<forall>i. (f o f' o r) i \<in> ball k e)"

  3151         proof (rule ccontr)

  3152           from K have "finite (K e)" "K e \<noteq> {}" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3153             using s \<noteq> {}

  3154             by auto

  3155           moreover

  3156           assume "\<not> (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f \<circ> f' o r) i \<in> ball k e))"

  3157           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

  3158           ultimately

  3159           show False using f'

  3160           proof (induct arbitrary: s f f' rule: finite_ne_induct)

  3161             case (singleton x)

  3162             have "\<exists>i. (f \<circ> f' o id) i \<notin> ball x e" by (rule singleton) (auto simp: subseq_def)

  3163             thus ?case using singleton by (auto simp: ball_def)

  3164           next

  3165             case (insert x A)

  3166             show ?case

  3167             proof cases

  3168               have inf_ms: "infinite ((f o f') - s)" using insert by (simp add: vimage_def)

  3169               have "infinite ((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A)))"

  3170                 using insert by (intro infinite_super[OF _ inf_ms]) auto

  3171               also have "((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A))) =

  3172                 {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

  3173               finally have "infinite \<dots>" .

  3174               moreover assume "finite {m. (f o f') m \<in> ball x e}"

  3175               ultimately have inf: "infinite {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by blast

  3176               hence "A \<noteq> {}" by auto then obtain k where "k \<in> A" by auto

  3177               def r \<equiv> "enumerate {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3178               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3179                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3180               hence "subseq r" by (simp add: subseq_def)

  3181               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3182                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3183               show False

  3184               proof (rule insert)

  3185                 show "\<Union>(\<lambda>x. ball x e)  A \<subseteq> \<Union>(\<lambda>x. ball x e)  A" by simp

  3186                 fix k s assume "k \<in> A" "subseq s"

  3187                 thus "\<exists>i. (f o f' o r o s) i \<notin> ball k e" using subseq r

  3188                   by (subst (2) o_assoc[symmetric]) (intro insert(6) subseq_o, simp_all)

  3189               next

  3190                 fix n show "(f \<circ> f' o r) n \<in> \<Union>(\<lambda>x. ball x e)  A" using r_in_set by auto

  3191               qed

  3192             next

  3193               assume inf: "infinite {m. (f o f') m \<in> ball x e}"

  3194               def r \<equiv> "enumerate {m. (f o f') m \<in> ball x e}"

  3195               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3196                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3197               hence "subseq r" by (simp add: subseq_def)

  3198               from insert(6)[OF insertI1 this] obtain i where "(f o f') (r i) \<notin> ball x e" by auto

  3199               moreover

  3200               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> ball x e}"

  3201                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3202               hence "(f o f') (r i) \<in> ball x e" by simp

  3203               ultimately show False by simp

  3204             qed

  3205           qed

  3206         qed

  3207       }

  3208       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

  3209       let ?e = "\<lambda>n. 1 / real (Suc n)"

  3210       let ?P = "\<lambda>n s. \<exists>k\<in>K (?e n). (\<forall>i. (f o s) i \<in> ball k (?e n))"

  3211       interpret subseqs ?P using ex by unfold_locales force

  3212       from complete s have limI: "\<And>f. (\<And>n. f n \<in> s) \<Longrightarrow> Cauchy f \<Longrightarrow> (\<exists>l\<in>s. f ----> l)"

  3213         by (simp add: complete_def)

  3214       have "\<exists>l\<in>s. (f o diagseq) ----> l"

  3215       proof (intro limI metric_CauchyI)

  3216         fix e::real assume "0 < e" hence "0 < e / 2" by auto

  3217         from nat_approx_posE[OF this] guess n . note n = this

  3218         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) n) < e"

  3219         proof (rule exI[where x="Suc n"], safe)

  3220           fix m mm assume "Suc n \<le> m" "Suc n \<le> mm"

  3221           let ?e = "1 / real (Suc n)"

  3222           from reducer_reduces[of n] obtain k where

  3223             "k\<in>K ?e"  "\<And>i. (f o seqseq (Suc n)) i \<in> ball k ?e"

  3224             unfolding seqseq_reducer by auto

  3225           moreover

  3226           note diagseq_sub[OF Suc n \<le> m] diagseq_sub[OF Suc n \<le> mm]

  3227           ultimately have "{(f o diagseq) m, (f o diagseq) mm} \<subseteq> ball k ?e" by auto

  3228           also have "\<dots> \<subseteq> ball k (e / 2)" using n by (intro subset_ball) simp

  3229           finally

  3230           have "dist k ((f \<circ> diagseq) m) + dist k ((f \<circ> diagseq) mm) < e / 2 + e /2"

  3231             by (intro add_strict_mono) auto

  3232           hence "dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k < e"

  3233             by (simp add: dist_commute)

  3234           moreover have "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) \<le>

  3235             dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k"

  3236             by (rule dist_triangle2)

  3237           ultimately show "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) < e"

  3238             by simp

  3239         qed

  3240       next

  3241         fix n show "(f o diagseq) n \<in> s" using f by simp

  3242       qed

  3243       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l" using subseq_diagseq by auto

  3244     qed

  3245   qed

  3246 qed

  3247

  3248 lemma compact_eq_bounded_closed:

  3249   fixes s :: "'a::heine_borel set"

  3250   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3251 proof

  3252   assume ?lhs thus ?rhs unfolding compact_eq_bolzano_weierstrass using bolzano_weierstrass_imp_bounded bolzano_weierstrass_imp_closed by auto

  3253 next

  3254   assume ?rhs thus ?lhs using bounded_closed_imp_compact by auto

  3255 qed

  3256

  3257 lemma compact_imp_bounded:

  3258   fixes s :: "'a::metric_space set"

  3259   shows "compact s ==> bounded s"

  3260 proof -

  3261   assume "compact s"

  3262   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')"

  3263     by (rule compact_imp_heine_borel)

  3264   hence "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t)"

  3265     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3266   thus "bounded s"

  3267     by (rule bolzano_weierstrass_imp_bounded)

  3268 qed

  3269

  3270 lemma compact_imp_closed:

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

  3272   shows "compact s ==> closed 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 "closed s"

  3280     by (rule bolzano_weierstrass_imp_closed)

  3281 qed

  3282

  3283 text{* In particular, some common special cases. *}

  3284

  3285 lemma compact_empty[simp]:

  3286  "compact {}"

  3287   unfolding compact_def

  3288   by simp

  3289

  3290 lemma compact_union [intro]:

  3291   assumes "compact s" and "compact t"

  3292   shows "compact (s \<union> t)"

  3293 proof (rule compactI)

  3294   fix f :: "nat \<Rightarrow> 'a"

  3295   assume "\<forall>n. f n \<in> s \<union> t"

  3296   hence "infinite {n. f n \<in> s \<union> t}" by simp

  3297   hence "infinite {n. f n \<in> s} \<or> infinite {n. f n \<in> t}" by simp

  3298   thus "\<exists>l\<in>s \<union> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3299   proof

  3300     assume "infinite {n. f n \<in> s}"

  3301     from infinite_enumerate [OF this]

  3302     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> s" by auto

  3303     obtain r l where "l \<in> s" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3304       using compact s \<forall>n. (f \<circ> q) n \<in> s by (rule compactE)

  3305     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3306       using subseq q by (simp_all add: subseq_o o_assoc)

  3307     thus ?thesis by auto

  3308   next

  3309     assume "infinite {n. f n \<in> t}"

  3310     from infinite_enumerate [OF this]

  3311     obtain q where "subseq q" "\<forall>n. (f \<circ> q) n \<in> t" by auto

  3312     obtain r l where "l \<in> t" "subseq r" "((f \<circ> q \<circ> r) ---> l) sequentially"

  3313       using compact t \<forall>n. (f \<circ> q) n \<in> t by (rule compactE)

  3314     hence "l \<in> s \<union> t" "subseq (q \<circ> r)" "((f \<circ> (q \<circ> r)) ---> l) sequentially"

  3315       using subseq q by (simp_all add: subseq_o o_assoc)

  3316     thus ?thesis by auto

  3317   qed

  3318 qed

  3319

  3320 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3321   by (induct set: finite) auto

  3322

  3323 lemma compact_UN [intro]:

  3324   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3325   unfolding SUP_def by (rule compact_Union) auto

  3326

  3327 lemma compact_inter_closed [intro]:

  3328   assumes "compact s" and "closed t"

  3329   shows "compact (s \<inter> t)"

  3330 proof (rule compactI)

  3331   fix f :: "nat \<Rightarrow> 'a"

  3332   assume "\<forall>n. f n \<in> s \<inter> t"

  3333   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t" by simp_all

  3334   obtain l r where "l \<in> s" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3335     using compact s \<forall>n. f n \<in> s by (rule compactE)

  3336   moreover

  3337   from closed t \<forall>n. f n \<in> t ((f \<circ> r) ---> l) sequentially have "l \<in> t"

  3338     unfolding closed_sequential_limits o_def by fast

  3339   ultimately show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3340     by auto

  3341 qed

  3342

  3343 lemma closed_inter_compact [intro]:

  3344   assumes "closed s" and "compact t"

  3345   shows "compact (s \<inter> t)"

  3346   using compact_inter_closed [of t s] assms

  3347   by (simp add: Int_commute)

  3348

  3349 lemma compact_inter [intro]:

  3350   assumes "compact s" and "compact t"

  3351   shows "compact (s \<inter> t)"

  3352   using assms by (intro compact_inter_closed compact_imp_closed)

  3353

  3354 lemma compact_sing [simp]: "compact {a}"

  3355   unfolding compact_def o_def subseq_def

  3356   by (auto simp add: tendsto_const)

  3357

  3358 lemma compact_insert [simp]:

  3359   assumes "compact s" shows "compact (insert x s)"

  3360 proof -

  3361   have "compact ({x} \<union> s)"

  3362     using compact_sing assms by (rule compact_union)

  3363   thus ?thesis by simp

  3364 qed

  3365

  3366 lemma finite_imp_compact:

  3367   shows "finite s \<Longrightarrow> compact s"

  3368   by (induct set: finite) simp_all

  3369

  3370 lemma compact_cball[simp]:

  3371   fixes x :: "'a::heine_borel"

  3372   shows "compact(cball x e)"

  3373   using compact_eq_bounded_closed bounded_cball closed_cball

  3374   by blast

  3375

  3376 lemma compact_frontier_bounded[intro]:

  3377   fixes s :: "'a::heine_borel set"

  3378   shows "bounded s ==> compact(frontier s)"

  3379   unfolding frontier_def

  3380   using compact_eq_bounded_closed

  3381   by blast

  3382

  3383 lemma compact_frontier[intro]:

  3384   fixes s :: "'a::heine_borel set"

  3385   shows "compact s ==> compact (frontier s)"

  3386   using compact_eq_bounded_closed compact_frontier_bounded

  3387   by blast

  3388

  3389 lemma frontier_subset_compact:

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

  3391   shows "compact s ==> frontier s \<subseteq> s"

  3392   using frontier_subset_closed compact_eq_bounded_closed

  3393   by blast

  3394

  3395 lemma open_delete:

  3396   fixes s :: "'a::t1_space set"

  3397   shows "open s \<Longrightarrow> open (s - {x})"

  3398   by (simp add: open_Diff)

  3399

  3400 text{* Finite intersection property. I could make it an equivalence in fact. *}

  3401

  3402 lemma compact_imp_fip:

  3403   assumes "compact s"  "\<forall>t \<in> f. closed t"

  3404         "\<forall>f'. finite f' \<and> f' \<subseteq> f --> (s \<inter> (\<Inter> f') \<noteq> {})"

  3405   shows "s \<inter> (\<Inter> f) \<noteq> {}"

  3406 proof

  3407   assume as:"s \<inter> (\<Inter> f) = {}"

  3408   hence "s \<subseteq> \<Union> uminus  f" by auto

  3409   moreover have "Ball (uminus  f) open" using open_Diff closed_Diff using assms(2) by auto

  3410   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

  3411   hence "finite (uminus  f') \<and> uminus  f' \<subseteq> f" by(auto simp add: Diff_Diff_Int)

  3412   hence "s \<inter> \<Inter>uminus  f' \<noteq> {}" using assms(3)[THEN spec[where x="uminus  f'"]] by auto

  3413   thus False using f'(3) unfolding subset_eq and Union_iff by blast

  3414 qed

  3415

  3416

  3417 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3418

  3419 lemma bounded_closed_nest:

  3420   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3421   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3422   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3423 proof-

  3424   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

  3425   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto

  3426

  3427   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3428     unfolding compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3429

  3430   { fix n::nat

  3431     { fix e::real assume "e>0"

  3432       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3433       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3434       moreover

  3435       have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto

  3436       hence "(x \<circ> r) (max N n) \<in> s n"

  3437         using x apply(erule_tac x=n in allE)

  3438         using x apply(erule_tac x="r (max N n)" in allE)

  3439         using assms(3) apply(erule_tac x=n in allE)apply( erule_tac x="r (max N n)" in allE) by auto

  3440       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3441     }

  3442     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3443   }

  3444   thus ?thesis by auto

  3445 qed

  3446

  3447 text {* Decreasing case does not even need compactness, just completeness. *}

  3448

  3449 lemma decreasing_closed_nest:

  3450   assumes "\<forall>n. closed(s n)"

  3451           "\<forall>n. (s n \<noteq> {})"

  3452           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3453           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3454   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3455 proof-

  3456   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3457   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3458   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3459   { fix e::real assume "e>0"

  3460     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3461     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3462       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3463       hence "dist (t m) (t n) < e" using N by auto

  3464     }

  3465     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3466   }

  3467   hence  "Cauchy t" unfolding cauchy_def by auto

  3468   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3469   { fix n::nat

  3470     { fix e::real assume "e>0"

  3471       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3472       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

  3473       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3474     }

  3475     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3476   }

  3477   then show ?thesis by auto

  3478 qed

  3479

  3480 text {* Strengthen it to the intersection actually being a singleton. *}

  3481

  3482 lemma decreasing_closed_nest_sing:

  3483   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3484   assumes "\<forall>n. closed(s n)"

  3485           "\<forall>n. s n \<noteq> {}"

  3486           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3487           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3488   shows "\<exists>a. \<Inter>(range s) = {a}"

  3489 proof-

  3490   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3491   { fix b assume b:"b \<in> \<Inter>(range s)"

  3492     { fix e::real assume "e>0"

  3493       hence "dist a b < e" using assms(4 )using b using a by blast

  3494     }

  3495     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3496   }

  3497   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3498   thus ?thesis ..

  3499 qed

  3500

  3501 text{* Cauchy-type criteria for uniform convergence. *}

  3502

  3503 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows

  3504  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3505   (\<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")

  3506 proof(rule)

  3507   assume ?lhs

  3508   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

  3509   { fix e::real assume "e>0"

  3510     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

  3511     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3512       hence "dist (s m x) (s n x) < e"

  3513         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3514         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3515         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3516     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  }

  3517   thus ?rhs by auto

  3518 next

  3519   assume ?rhs

  3520   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

  3521   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3522     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3523   { fix e::real assume "e>0"

  3524     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"

  3525       using ?rhs[THEN spec[where x="e/2"]] by auto

  3526     { fix x assume "P x"

  3527       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3528         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3529       fix n::nat assume "n\<ge>N"

  3530       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]]

  3531         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)  }

  3532     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3533   thus ?lhs by auto

  3534 qed

  3535

  3536 lemma uniformly_cauchy_imp_uniformly_convergent:

  3537   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"

  3538   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"

  3539           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3540   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3541 proof-

  3542   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"

  3543     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3544   moreover

  3545   { fix x assume "P x"

  3546     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3547       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3548   ultimately show ?thesis by auto

  3549 qed

  3550

  3551

  3552 subsection {* Continuity *}

  3553

  3554 text {* Define continuity over a net to take in restrictions of the set. *}

  3555

  3556 definition

  3557   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3558   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"

  3559

  3560 lemma continuous_trivial_limit:

  3561  "trivial_limit net ==> continuous net f"

  3562   unfolding continuous_def tendsto_def trivial_limit_eq by auto

  3563

  3564 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"

  3565   unfolding continuous_def

  3566   unfolding tendsto_def

  3567   using netlimit_within[of x s]

  3568   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)

  3569

  3570 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"

  3571   using continuous_within [of x UNIV f] by simp

  3572

  3573 lemma continuous_at_within:

  3574   assumes "continuous (at x) f"  shows "continuous (at x within s) f"

  3575   using assms unfolding continuous_at continuous_within

  3576   by (rule Lim_at_within)

  3577

  3578 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3579

  3580 lemma continuous_within_eps_delta:

  3581   "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)"

  3582   unfolding continuous_within and Lim_within

  3583   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3584

  3585 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3586                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3587   using continuous_within_eps_delta [of x UNIV f] by simp

  3588

  3589 text{* Versions in terms of open balls. *}

  3590

  3591 lemma continuous_within_ball:

  3592  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3593                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3594 proof

  3595   assume ?lhs

  3596   { fix e::real assume "e>0"

  3597     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"

  3598       using ?lhs[unfolded continuous_within Lim_within] by auto

  3599     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3600       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3601         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3602     }

  3603     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)  }

  3604   thus ?rhs by auto

  3605 next

  3606   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3607     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3608 qed

  3609

  3610 lemma continuous_at_ball:

  3611   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3612 proof

  3613   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3614     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)

  3615     unfolding dist_nz[THEN sym] by auto

  3616 next

  3617   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3618     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)

  3619 qed

  3620

  3621 text{* Define setwise continuity in terms of limits within the set. *}

  3622

  3623 definition

  3624   continuous_on ::

  3625     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3626 where

  3627   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"

  3628

  3629 lemma continuous_on_topological:

  3630   "continuous_on s f \<longleftrightarrow>

  3631     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  3632       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  3633 unfolding continuous_on_def tendsto_def

  3634 unfolding Limits.eventually_within eventually_at_topological

  3635 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  3636

  3637 lemma continuous_on_iff:

  3638   "continuous_on s f \<longleftrightarrow>

  3639     (\<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)"

  3640 unfolding continuous_on_def Lim_within

  3641 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3642 apply (rename_tac y, case_tac "y = x", simp)

  3643 apply (simp add: dist_nz)

  3644 done

  3645

  3646 definition

  3647   uniformly_continuous_on ::

  3648     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3649 where

  3650   "uniformly_continuous_on s f \<longleftrightarrow>

  3651     (\<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)"

  3652

  3653 text{* Some simple consequential lemmas. *}

  3654

  3655 lemma uniformly_continuous_imp_continuous:

  3656  " uniformly_continuous_on s f ==> continuous_on s f"

  3657   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3658

  3659 lemma continuous_at_imp_continuous_within:

  3660  "continuous (at x) f ==> continuous (at x within s) f"

  3661   unfolding continuous_within continuous_at using Lim_at_within by auto

  3662

  3663 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3664 unfolding tendsto_def by (simp add: trivial_limit_eq)

  3665

  3666 lemma continuous_at_imp_continuous_on:

  3667   assumes "\<forall>x\<in>s. continuous (at x) f"

  3668   shows "continuous_on s f"

  3669 unfolding continuous_on_def

  3670 proof

  3671   fix x assume "x \<in> s"

  3672   with assms have *: "(f ---> f (netlimit (at x))) (at x)"

  3673     unfolding continuous_def by simp

  3674   have "(f ---> f x) (at x)"

  3675   proof (cases "trivial_limit (at x)")

  3676     case True thus ?thesis

  3677       by (rule Lim_trivial_limit)

  3678   next

  3679     case False

  3680     hence 1: "netlimit (at x) = x"

  3681       using netlimit_within [of x UNIV] by simp

  3682     with * show ?thesis by simp

  3683   qed

  3684   thus "(f ---> f x) (at x within s)"

  3685     by (rule Lim_at_within)

  3686 qed

  3687

  3688 lemma continuous_on_eq_continuous_within:

  3689   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"

  3690 unfolding continuous_on_def continuous_def

  3691 apply (rule ball_cong [OF refl])

  3692 apply (case_tac "trivial_limit (at x within s)")

  3693 apply (simp add: Lim_trivial_limit)

  3694 apply (simp add: netlimit_within)

  3695 done

  3696

  3697 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3698

  3699 lemma continuous_on_eq_continuous_at:

  3700   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"

  3701   by (auto simp add: continuous_on continuous_at Lim_within_open)

  3702

  3703 lemma continuous_within_subset:

  3704  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3705              ==> continuous (at x within t) f"

  3706   unfolding continuous_within by(metis Lim_within_subset)

  3707

  3708 lemma continuous_on_subset:

  3709   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"

  3710   unfolding continuous_on by (metis subset_eq Lim_within_subset)

  3711

  3712 lemma continuous_on_interior:

  3713   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3714   by (erule interiorE, drule (1) continuous_on_subset,

  3715     simp add: continuous_on_eq_continuous_at)

  3716

  3717 lemma continuous_on_eq:

  3718   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3719   unfolding continuous_on_def tendsto_def Limits.eventually_within

  3720   by simp

  3721

  3722 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3723

  3724 lemma continuous_within_sequentially:

  3725   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3726   shows "continuous (at a within s) f \<longleftrightarrow>

  3727                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3728                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3729 proof

  3730   assume ?lhs

  3731   { 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"

  3732     fix T::"'b set" assume "open T" and "f a \<in> T"

  3733     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"

  3734       unfolding continuous_within tendsto_def eventually_within by auto

  3735     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3736       using x(2) d>0 by simp

  3737     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3738     proof eventually_elim

  3739       case (elim n) thus ?case

  3740         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3741     qed

  3742   }

  3743   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3744 next

  3745   assume ?rhs thus ?lhs

  3746     unfolding continuous_within tendsto_def [where l="f a"]

  3747     by (simp add: sequentially_imp_eventually_within)

  3748 qed

  3749

  3750 lemma continuous_at_sequentially:

  3751   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3752   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  3753                   --> ((f o x) ---> f a) sequentially)"

  3754   using continuous_within_sequentially[of a UNIV f] by simp

  3755

  3756 lemma continuous_on_sequentially:

  3757   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3758   shows "continuous_on s f \<longleftrightarrow>

  3759     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  3760                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  3761 proof

  3762   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  3763 next

  3764   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  3765 qed

  3766

  3767 lemma uniformly_continuous_on_sequentially:

  3768   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  3769                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  3770                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  3771 proof

  3772   assume ?lhs

  3773   { 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"

  3774     { fix e::real assume "e>0"

  3775       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"

  3776         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  3777       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

  3778       { fix n assume "n\<ge>N"

  3779         hence "dist (f (x n)) (f (y n)) < e"

  3780           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

  3781           unfolding dist_commute by simp  }

  3782       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  3783     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  3784   thus ?rhs by auto

  3785 next

  3786   assume ?rhs

  3787   { assume "\<not> ?lhs"

  3788     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

  3789     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"

  3790       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

  3791       by (auto simp add: dist_commute)

  3792     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  3793     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  3794     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"

  3795       unfolding x_def and y_def using fa by auto

  3796     { fix e::real assume "e>0"

  3797       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

  3798       { fix n::nat assume "n\<ge>N"

  3799         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  3800         also have "\<dots> < e" using N by auto

  3801         finally have "inverse (real n + 1) < e" by auto

  3802         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  3803       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  3804     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

  3805     hence False using fxy and e>0 by auto  }

  3806   thus ?lhs unfolding uniformly_continuous_on_def by blast

  3807 qed

  3808

  3809 text{* The usual transformation theorems. *}

  3810

  3811 lemma continuous_transform_within:

  3812   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3813   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  3814           "continuous (at x within s) f"

  3815   shows "continuous (at x within s) g"

  3816 unfolding continuous_within

  3817 proof (rule Lim_transform_within)

  3818   show "0 < d" by fact

  3819   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  3820     using assms(3) by auto

  3821   have "f x = g x"

  3822     using assms(1,2,3) by auto

  3823   thus "(f ---> g x) (at x within s)"

  3824     using assms(4) unfolding continuous_within by simp

  3825 qed

  3826

  3827 lemma continuous_transform_at:

  3828   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3829   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  3830           "continuous (at x) f"

  3831   shows "continuous (at x) g"

  3832   using continuous_transform_within [of d x UNIV f g] assms by simp

  3833

  3834 subsubsection {* Structural rules for pointwise continuity *}

  3835

  3836 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"

  3837   unfolding continuous_within by (rule tendsto_ident_at_within)

  3838

  3839 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"

  3840   unfolding continuous_at by (rule tendsto_ident_at)

  3841

  3842 lemma continuous_const: "continuous F (\<lambda>x. c)"

  3843   unfolding continuous_def by (rule tendsto_const)

  3844

  3845 lemma continuous_dist:

  3846   assumes "continuous F f" and "continuous F g"

  3847   shows "continuous F (\<lambda>x. dist (f x) (g x))"

  3848   using assms unfolding continuous_def by (rule tendsto_dist)

  3849

  3850 lemma continuous_infdist:

  3851   assumes "continuous F f"

  3852   shows "continuous F (\<lambda>x. infdist (f x) A)"

  3853   using assms unfolding continuous_def by (rule tendsto_infdist)

  3854

  3855 lemma continuous_norm:

  3856   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"

  3857   unfolding continuous_def by (rule tendsto_norm)

  3858

  3859 lemma continuous_infnorm:

  3860   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  3861   unfolding continuous_def by (rule tendsto_infnorm)

  3862

  3863 lemma continuous_add:

  3864   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3865   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"

  3866   unfolding continuous_def by (rule tendsto_add)

  3867

  3868 lemma continuous_minus:

  3869   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3870   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"

  3871   unfolding continuous_def by (rule tendsto_minus)

  3872

  3873 lemma continuous_diff:

  3874   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3875   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"

  3876   unfolding continuous_def by (rule tendsto_diff)

  3877

  3878 lemma continuous_scaleR:

  3879   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  3880   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"

  3881   unfolding continuous_def by (rule tendsto_scaleR)

  3882

  3883 lemma continuous_mult:

  3884   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"

  3885   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"

  3886   unfolding continuous_def by (rule tendsto_mult)

  3887

  3888 lemma continuous_inner:

  3889   assumes "continuous F f" and "continuous F g"

  3890   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  3891   using assms unfolding continuous_def by (rule tendsto_inner)

  3892

  3893 lemma continuous_euclidean_component:

  3894   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $$i)"   3895 unfolding continuous_def by (rule tendsto_euclidean_component)   3896   3897 lemma continuous_inverse:   3898 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3899 assumes "continuous F f" and "f (netlimit F) \<noteq> 0"   3900 shows "continuous F (\<lambda>x. inverse (f x))"   3901 using assms unfolding continuous_def by (rule tendsto_inverse)   3902   3903 lemma continuous_at_within_inverse:   3904 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3905 assumes "continuous (at a within s) f" and "f a \<noteq> 0"   3906 shows "continuous (at a within s) (\<lambda>x. inverse (f x))"   3907 using assms unfolding continuous_within by (rule tendsto_inverse)   3908   3909 lemma continuous_at_inverse:   3910 fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"   3911 assumes "continuous (at a) f" and "f a \<noteq> 0"   3912 shows "continuous (at a) (\<lambda>x. inverse (f x))"   3913 using assms unfolding continuous_at by (rule tendsto_inverse)   3914   3915 lemmas continuous_intros = continuous_at_id continuous_within_id   3916 continuous_const continuous_dist continuous_norm continuous_infnorm   3917 continuous_add continuous_minus continuous_diff   3918 continuous_scaleR continuous_mult   3919 continuous_inner continuous_euclidean_component   3920 continuous_at_inverse continuous_at_within_inverse   3921   3922 subsubsection {* Structural rules for setwise continuity *}   3923   3924 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"   3925 unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)   3926   3927 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"   3928 unfolding continuous_on_def by (auto intro: tendsto_intros)   3929   3930 lemma continuous_on_norm:   3931 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"   3932 unfolding continuous_on_def by (fast intro: tendsto_norm)   3933   3934 lemma continuous_on_infnorm:   3935 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"   3936 unfolding continuous_on by (fast intro: tendsto_infnorm)   3937   3938 lemma continuous_on_minus:   3939 fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3940 shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"   3941 unfolding continuous_on_def by (auto intro: tendsto_intros)   3942   3943 lemma continuous_on_add:   3944 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3945 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3946 \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"   3947 unfolding continuous_on_def by (auto intro: tendsto_intros)   3948   3949 lemma continuous_on_diff:   3950 fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3951 shows "continuous_on s f \<Longrightarrow> continuous_on s g   3952 \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"   3953 unfolding continuous_on_def by (auto intro: tendsto_intros)   3954   3955 lemma (in bounded_linear) continuous_on:   3956 "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"   3957 unfolding continuous_on_def by (fast intro: tendsto)   3958   3959 lemma (in bounded_bilinear) continuous_on:   3960 "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"   3961 unfolding continuous_on_def by (fast intro: tendsto)   3962   3963 lemma continuous_on_scaleR:   3964 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"   3965 assumes "continuous_on s f" and "continuous_on s g"   3966 shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"   3967 using bounded_bilinear_scaleR assms   3968 by (rule bounded_bilinear.continuous_on)   3969   3970 lemma continuous_on_mult:   3971 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"   3972 assumes "continuous_on s f" and "continuous_on s g"   3973 shows "continuous_on s (\<lambda>x. f x * g x)"   3974 using bounded_bilinear_mult assms   3975 by (rule bounded_bilinear.continuous_on)   3976   3977 lemma continuous_on_inner:   3978 fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"   3979 assumes "continuous_on s f" and "continuous_on s g"   3980 shows "continuous_on s (\<lambda>x. inner (f x) (g x))"   3981 using bounded_bilinear_inner assms   3982 by (rule bounded_bilinear.continuous_on)   3983   3984 lemma continuous_on_euclidean_component:   3985 "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x$$ i)"

  3986   using bounded_linear_euclidean_component

  3987   by (rule bounded_linear.continuous_on)

  3988

  3989 lemma continuous_on_inverse:

  3990   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"

  3991   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

  3992   shows "continuous_on s (\<lambda>x. inverse (f x))"

  3993   using assms unfolding continuous_on by (fast intro: tendsto_inverse)

  3994

  3995 subsubsection {* Structural rules for uniform continuity *}

  3996

  3997 lemma uniformly_continuous_on_id:

  3998   shows "uniformly_continuous_on s (\<lambda>x. x)"

  3999   unfolding uniformly_continuous_on_def by auto

  4000

  4001 lemma uniformly_continuous_on_const:

  4002   shows "uniformly_continuous_on s (\<lambda>x. c)"

  4003   unfolding uniformly_continuous_on_def by simp

  4004

  4005 lemma uniformly_continuous_on_dist:

  4006   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4007   assumes "uniformly_continuous_on s f"

  4008   assumes "uniformly_continuous_on s g"

  4009   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4010 proof -

  4011   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4012       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4013       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4014       by arith

  4015   } note le = this

  4016   { fix x y

  4017     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4018     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4019     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4020       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4021         simp add: le)

  4022   }

  4023   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  4024     unfolding dist_real_def by simp

  4025 qed

  4026

  4027 lemma uniformly_continuous_on_norm:

  4028   assumes "uniformly_continuous_on s f"

  4029   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4030   unfolding norm_conv_dist using assms

  4031   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4032

  4033 lemma (in bounded_linear) uniformly_continuous_on:

  4034   assumes "uniformly_continuous_on s g"

  4035   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4036   using assms unfolding uniformly_continuous_on_sequentially

  4037   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4038   by (auto intro: tendsto_zero)

  4039

  4040 lemma uniformly_continuous_on_cmul:

  4041   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4042   assumes "uniformly_continuous_on s f"

  4043   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4044   using bounded_linear_scaleR_right assms

  4045   by (rule bounded_linear.uniformly_continuous_on)

  4046

  4047 lemma dist_minus:

  4048   fixes x y :: "'a::real_normed_vector"

  4049   shows "dist (- x) (- y) = dist x y"

  4050   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4051

  4052 lemma uniformly_continuous_on_minus:

  4053   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4054   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4055   unfolding uniformly_continuous_on_def dist_minus .

  4056

  4057 lemma uniformly_continuous_on_add:

  4058   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4059   assumes "uniformly_continuous_on s f"

  4060   assumes "uniformly_continuous_on s g"

  4061   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4062   using assms unfolding uniformly_continuous_on_sequentially

  4063   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4064   by (auto intro: tendsto_add_zero)

  4065

  4066 lemma uniformly_continuous_on_diff:

  4067   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4068   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4069   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4070   unfolding ab_diff_minus using assms

  4071   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4072

  4073 text{* Continuity of all kinds is preserved under composition. *}

  4074

  4075 lemma continuous_within_topological:

  4076   "continuous (at x within s) f \<longleftrightarrow>

  4077     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  4078       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  4079 unfolding continuous_within

  4080 unfolding tendsto_def Limits.eventually_within eventually_at_topological

  4081 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  4082

  4083 lemma continuous_within_compose:

  4084   assumes "continuous (at x within s) f"

  4085   assumes "continuous (at (f x) within f  s) g"

  4086   shows "continuous (at x within s) (g o f)"

  4087 using assms unfolding continuous_within_topological by simp metis

  4088

  4089 lemma continuous_at_compose:

  4090   assumes "continuous (at x) f" and "continuous (at (f x)) g"

  4091   shows "continuous (at x) (g o f)"

  4092 proof-

  4093   have "continuous (at (f x) within range f) g" using assms(2)

  4094     using continuous_within_subset[of "f x" UNIV g "range f"] by simp

  4095   thus ?thesis using assms(1)

  4096     using continuous_within_compose[of x UNIV f g] by simp

  4097 qed

  4098

  4099 lemma continuous_on_compose:

  4100   "continuous_on s f \<Longrightarrow> continuous_on (f  s) g \<Longrightarrow> continuous_on s (g o f)"

  4101   unfolding continuous_on_topological by simp metis

  4102

  4103 lemma uniformly_continuous_on_compose:

  4104   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4105   shows "uniformly_continuous_on s (g o f)"

  4106 proof-

  4107   { fix e::real assume "e>0"

  4108     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

  4109     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

  4110     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  }

  4111   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4112 qed

  4113

  4114 lemmas continuous_on_intros = continuous_on_id continuous_on_const

  4115   continuous_on_compose continuous_on_norm continuous_on_infnorm

  4116   continuous_on_add continuous_on_minus continuous_on_diff

  4117   continuous_on_scaleR continuous_on_mult continuous_on_inverse

  4118   continuous_on_inner continuous_on_euclidean_component

  4119   uniformly_continuous_on_id uniformly_continuous_on_const

  4120   uniformly_continuous_on_dist uniformly_continuous_on_norm

  4121   uniformly_continuous_on_compose uniformly_continuous_on_add

  4122   uniformly_continuous_on_minus uniformly_continuous_on_diff

  4123   uniformly_continuous_on_cmul

  4124

  4125 text{* Continuity in terms of open preimages. *}

  4126

  4127 lemma continuous_at_open:

  4128   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)))"

  4129 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  4130 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4131

  4132 lemma continuous_on_open:

  4133   shows "continuous_on s f \<longleftrightarrow>

  4134         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4135             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4136 proof (safe)

  4137   fix t :: "'b set"

  4138   assume 1: "continuous_on s f"

  4139   assume 2: "openin (subtopology euclidean (f  s)) t"

  4140   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  4141     unfolding openin_open by auto

  4142   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  4143   have "open U" unfolding U_def by (simp add: open_Union)

  4144   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  4145   proof (intro ballI iffI)

  4146     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  4147       unfolding U_def t by auto

  4148   next

  4149     fix x assume "x \<in> s" and "f x \<in> t"

  4150     hence "x \<in> s" and "f x \<in> B"

  4151       unfolding t by auto

  4152     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  4153       unfolding t continuous_on_topological by metis

  4154     then show "x \<in> U"

  4155       unfolding U_def by auto

  4156   qed

  4157   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  4158   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4159     unfolding openin_open by fast

  4160 next

  4161   assume "?rhs" show "continuous_on s f"

  4162   unfolding continuous_on_topological

  4163   proof (clarify)

  4164     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  4165     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  4166       unfolding openin_open using open B by auto

  4167     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  4168       using ?rhs by fast

  4169     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  4170       unfolding openin_open using x \<in> s and f x \<in> B by auto

  4171   qed

  4172 qed

  4173

  4174 text {* Similarly in terms of closed sets. *}

  4175

  4176 lemma continuous_on_closed:

  4177   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")

  4178 proof

  4179   assume ?lhs

  4180   { fix t

  4181     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4182     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  4183     assume as:"closedin (subtopology euclidean (f  s)) t"

  4184     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  4185     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"]]

  4186       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  4187   thus ?rhs by auto

  4188 next

  4189   assume ?rhs

  4190   { fix t

  4191     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4192     assume as:"openin (subtopology euclidean (f  s)) t"

  4193     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  4194       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  4195   thus ?lhs unfolding continuous_on_open by auto

  4196 qed

  4197

  4198 text {* Half-global and completely global cases. *}

  4199

  4200 lemma continuous_open_in_preimage:

  4201   assumes "continuous_on s f"  "open t"

  4202   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4203 proof-

  4204   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

  4205   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4206     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4207   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4208 qed

  4209

  4210 lemma continuous_closed_in_preimage:

  4211   assumes "continuous_on s f"  "closed t"

  4212   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4213 proof-

  4214   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

  4215   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4216     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4217   thus ?thesis

  4218     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4219 qed

  4220

  4221 lemma continuous_open_preimage:

  4222   assumes "continuous_on s f" "open s" "open t"

  4223   shows "open {x \<in> s. f x \<in> t}"

  4224 proof-

  4225   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4226     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4227   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4228 qed

  4229

  4230 lemma continuous_closed_preimage:

  4231   assumes "continuous_on s f" "closed s" "closed t"

  4232   shows "closed {x \<in> s. f x \<in> t}"

  4233 proof-

  4234   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4235     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4236   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4237 qed

  4238

  4239 lemma continuous_open_preimage_univ:

  4240   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4241   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4242

  4243 lemma continuous_closed_preimage_univ:

  4244   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4245   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4246

  4247 lemma continuous_open_vimage:

  4248   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4249   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4250

  4251 lemma continuous_closed_vimage:

  4252   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4253   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4254

  4255 lemma interior_image_subset:

  4256   assumes "\<forall>x. continuous (at x) f" "inj f"

  4257   shows "interior (f  s) \<subseteq> f  (interior s)"

  4258 proof

  4259   fix x assume "x \<in> interior (f  s)"

  4260   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4261   hence "x \<in> f  s" by auto

  4262   then obtain y where y: "y \<in> s" "x = f y" by auto

  4263   have "open (vimage f T)"

  4264     using assms(1) open T by (rule continuous_open_vimage)

  4265   moreover have "y \<in> vimage f T"

  4266     using x = f y x \<in> T by simp

  4267   moreover have "vimage f T \<subseteq> s"

  4268     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4269   ultimately have "y \<in> interior s" ..

  4270   with x = f y show "x \<in> f  interior s" ..

  4271 qed

  4272

  4273 text {* Equality of continuous functions on closure and related results. *}

  4274

  4275 lemma continuous_closed_in_preimage_constant:

  4276   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4277   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4278   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4279

  4280 lemma continuous_closed_preimage_constant:

  4281   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4282   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4283   using continuous_closed_preimage[of s f "{a}"] by auto

  4284

  4285 lemma continuous_constant_on_closure:

  4286   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4287   assumes "continuous_on (closure s) f"

  4288           "\<forall>x \<in> s. f x = a"

  4289   shows "\<forall>x \<in> (closure s). f x = a"

  4290     using continuous_closed_preimage_constant[of "closure s" f a]

  4291     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4292

  4293 lemma image_closure_subset:

  4294   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4295   shows "f  (closure s) \<subseteq> t"

  4296 proof-

  4297   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4298   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4299     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4300   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4301     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4302   thus ?thesis by auto

  4303 qed

  4304

  4305 lemma continuous_on_closure_norm_le:

  4306   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4307   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4308   shows "norm(f x) \<le> b"

  4309 proof-

  4310   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4311   show ?thesis

  4312     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4313     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4314 qed

  4315

  4316 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4317

  4318 lemma continuous_within_avoid:

  4319   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4320   assumes "continuous (at x within s) f"  "x \<in> s"  "f x \<noteq> a"

  4321   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4322 proof-

  4323   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"

  4324     using assms(1)[unfolded continuous_within Lim_within, THEN spec[where x="dist (f x) a"]] assms(3)[unfolded dist_nz] by auto

  4325   { fix y assume " y\<in>s"  "dist x y < d"

  4326     hence "f y \<noteq> a" using d[THEN bspec[where x=y]] assms(3)[unfolded dist_nz]

  4327       apply auto unfolding dist_nz[THEN sym] by (auto simp add: dist_commute) }

  4328   thus ?thesis using d>0 by auto

  4329 qed

  4330

  4331 lemma continuous_at_avoid:

  4332   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *)

  4333   assumes "continuous (at x) f" and "f x \<noteq> a"

  4334   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4335   using assms continuous_within_avoid[of x UNIV f a] by simp

  4336

  4337 lemma continuous_on_avoid:

  4338   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4339   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4340   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4341 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

  4342

  4343 lemma continuous_on_open_avoid:

  4344   fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* TODO: generalize *)

  4345   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4346   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4347 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

  4348

  4349 text {* Proving a function is constant by proving open-ness of level set. *}

  4350

  4351 lemma continuous_levelset_open_in_cases:

  4352   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4353   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4354         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4355         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4356 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4357

  4358 lemma continuous_levelset_open_in:

  4359   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4360   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4361         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4362         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4363 using continuous_levelset_open_in_cases[of s f ]

  4364 by meson

  4365

  4366 lemma continuous_levelset_open:

  4367   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4368   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4369   shows "\<forall>x \<in> s. f x = a"

  4370 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4371

  4372 text {* Some arithmetical combinations (more to prove). *}

  4373

  4374 lemma open_scaling[intro]:

  4375   fixes s :: "'a::real_normed_vector set"

  4376   assumes "c \<noteq> 0"  "open s"

  4377   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4378 proof-

  4379   { fix x assume "x \<in> s"

  4380     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

  4381     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4382     moreover

  4383     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4384       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4385         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4386           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4387       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  }

  4388     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  }

  4389   thus ?thesis unfolding open_dist by auto

  4390 qed

  4391

  4392 lemma minus_image_eq_vimage:

  4393   fixes A :: "'a::ab_group_add set"

  4394   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4395   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4396

  4397 lemma open_negations:

  4398   fixes s :: "'a::real_normed_vector set"

  4399   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4400   unfolding scaleR_minus1_left [symmetric]

  4401   by (rule open_scaling, auto)

  4402

  4403 lemma open_translation:

  4404   fixes s :: "'a::real_normed_vector set"

  4405   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4406 proof-

  4407   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4408       by (intro continuous_diff continuous_at_id continuous_const) }

  4409   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4410   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4411 qed

  4412

  4413 lemma open_affinity:

  4414   fixes s :: "'a::real_normed_vector set"

  4415   assumes "open s"  "c \<noteq> 0"

  4416   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4417 proof-

  4418   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4419   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4420   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4421 qed

  4422

  4423 lemma interior_translation:

  4424   fixes s :: "'a::real_normed_vector set"

  4425   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4426 proof (rule set_eqI, rule)

  4427   fix x assume "x \<in> interior (op + a  s)"

  4428   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4429   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

  4430   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

  4431 next

  4432   fix x assume "x \<in> op + a  interior s"

  4433   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

  4434   { fix z have *:"a + y - z = y + a - z" by auto

  4435     assume "z\<in>ball x e"

  4436     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

  4437     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4438   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4439   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4440 qed

  4441

  4442 text {* Topological properties of linear functions. *}

  4443

  4444 lemma linear_lim_0:

  4445   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4446 proof-

  4447   interpret f: bounded_linear f by fact

  4448   have "(f ---> f 0) (at 0)"

  4449     using tendsto_ident_at by (rule f.tendsto)

  4450   thus ?thesis unfolding f.zero .

  4451 qed

  4452

  4453 lemma linear_continuous_at:

  4454   assumes "bounded_linear f"  shows "continuous (at a) f"

  4455   unfolding continuous_at using assms

  4456   apply (rule bounded_linear.tendsto)

  4457   apply (rule tendsto_ident_at)

  4458   done

  4459

  4460 lemma linear_continuous_within:

  4461   shows "bounded_linear f ==> continuous (at x within s) f"

  4462   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4463

  4464 lemma linear_continuous_on:

  4465   shows "bounded_linear f ==> continuous_on s f"

  4466   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4467

  4468 text {* Also bilinear functions, in composition form. *}

  4469

  4470 lemma bilinear_continuous_at_compose:

  4471   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4472         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4473   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4474

  4475 lemma bilinear_continuous_within_compose:

  4476   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4477         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4478   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4479

  4480 lemma bilinear_continuous_on_compose:

  4481   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4482              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4483   unfolding continuous_on_def

  4484   by (fast elim: bounded_bilinear.tendsto)

  4485

  4486 text {* Preservation of compactness and connectedness under continuous function. *}

  4487

  4488 lemma compact_continuous_image:

  4489   assumes "continuous_on s f"  "compact s"

  4490   shows "compact(f  s)"

  4491 proof-

  4492   { fix x assume x:"\<forall>n::nat. x n \<in> f  s"

  4493     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

  4494     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

  4495     { fix e::real assume "e>0"

  4496       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

  4497       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

  4498       { 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  }

  4499       hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto  }

  4500     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  }

  4501   thus ?thesis unfolding compact_def by auto

  4502 qed

  4503

  4504 lemma connected_continuous_image:

  4505   assumes "continuous_on s f"  "connected s"

  4506   shows "connected(f  s)"

  4507 proof-

  4508   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4509     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4510       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4511       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4512       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4513     hence False using as(1,2)

  4514       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4515   thus ?thesis unfolding connected_clopen by auto

  4516 qed

  4517

  4518 text {* Continuity implies uniform continuity on a compact domain. *}

  4519

  4520 lemma compact_uniformly_continuous:

  4521   assumes "continuous_on s f"  "compact s"

  4522   shows "uniformly_continuous_on s f"

  4523 proof-

  4524     { fix x assume x:"x\<in>s"

  4525       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

  4526       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  }

  4527     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

  4528     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)"

  4529       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

  4530

  4531   { fix e::real assume "e>0"

  4532

  4533     { 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  }

  4534     hence "s \<subseteq> \<Union>{ball x (d x (e / 2)) |x. x \<in> s}" unfolding subset_eq by auto

  4535     moreover

  4536     { fix b assume "b\<in>{ball x (d x (e / 2)) |x. x \<in> s}" hence "open b" by auto  }

  4537     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

  4538

  4539     { fix x y assume "x\<in>s" "y\<in>s" and as:"dist y x < ea"

  4540       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

  4541       hence "x\<in>ball z (d z (e / 2))" using ea>0 unfolding subset_eq by auto

  4542       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

  4543         by (auto  simp add: dist_commute)

  4544       moreover have "y\<in>ball z (d z (e / 2))" using as and ea>0 and z[unfolded subset_eq]

  4545         by (auto simp add: dist_commute)

  4546       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

  4547         by (auto  simp add: dist_commute)

  4548       ultimately have "dist (f y) (f x) < e" using dist_triangle_half_r[of "f z" "f x" e "f y"]

  4549         by (auto simp add: dist_commute)  }

  4550     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  }

  4551   thus ?thesis unfolding uniformly_continuous_on_def by auto

  4552 qed

  4553

  4554 text{* Continuity of inverse function on compact domain. *}

  4555

  4556 lemma continuous_on_inv:

  4557   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"

  4558     (* TODO: can this be generalized more? *)

  4559   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4560   shows "continuous_on (f  s) g"

  4561 proof-

  4562   have *:"g  f  s = s" using assms(3) by (auto simp add: image_iff)

  4563   { fix t assume t:"closedin (subtopology euclidean (g  f  s)) t"

  4564     then obtain T where T: "closed T" "t = s \<inter> T" unfolding closedin_closed unfolding * by auto

  4565     have "continuous_on (s \<inter> T) f" using continuous_on_subset[OF assms(1), of "s \<inter> t"]

  4566       unfolding T(2) and Int_left_absorb by auto

  4567     moreover have "compact (s \<inter> T)"

  4568       using assms(2) unfolding compact_eq_bounded_closed

  4569       using bounded_subset[of s "s \<inter> T"] and T(1) by auto

  4570     ultimately have "closed (f  t)" using T(1) unfolding T(2)

  4571       using compact_continuous_image [of "s \<inter> T" f] unfolding compact_eq_bounded_closed by auto

  4572     moreover have "{x \<in> f  s. g x \<in> t} = f  s \<inter> f  t" using assms(3) unfolding T(2) by auto

  4573     ultimately have "closedin (subtopology euclidean (f  s)) {x \<in> f  s. g x \<in> t}"

  4574       unfolding closedin_closed by auto  }

  4575   thus ?thesis unfolding continuous_on_closed by auto

  4576 qed

  4577

  4578 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4579

  4580 lemma continuous_uniform_limit:

  4581   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4582   assumes "\<not> trivial_limit F"

  4583   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4584   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4585   shows "continuous_on s g"

  4586 proof-

  4587   { fix x and e::real assume "x\<in>s" "e>0"

  4588     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4589       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4590     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4591     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4592       using assms(1) by blast

  4593     have "e / 3 > 0" using e>0 by auto

  4594     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"

  4595       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4596     { fix y assume "y \<in> s" and "dist y x < d"

  4597       hence "dist (f n y) (f n x) < e / 3"

  4598         by (rule d [rule_format])

  4599       hence "dist (f n y) (g x) < 2 * e / 3"

  4600         using dist_triangle [of "f n y" "g x" "f n x"]

  4601         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4602         by auto

  4603       hence "dist (g y) (g x) < e"

  4604         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4605         using dist_triangle3 [of "g y" "g x" "f n y"]

  4606         by auto }

  4607     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4608       using d>0 by auto }

  4609   thus ?thesis unfolding continuous_on_iff by auto

  4610 qed

  4611

  4612

  4613 subsection {* Topological stuff lifted from and dropped to R *}

  4614

  4615 lemma open_real:

  4616   fixes s :: "real set" shows

  4617  "open s \<longleftrightarrow>

  4618         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4619   unfolding open_dist dist_norm by simp

  4620

  4621 lemma islimpt_approachable_real:

  4622   fixes s :: "real set"

  4623   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4624   unfolding islimpt_approachable dist_norm by simp

  4625

  4626 lemma closed_real:

  4627   fixes s :: "real set"

  4628   shows "closed s \<longleftrightarrow>

  4629         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4630             --> x \<in> s)"

  4631   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4632

  4633 lemma continuous_at_real_range:

  4634   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4635   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4636         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4637   unfolding continuous_at unfolding Lim_at

  4638   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4639   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

  4640   apply(erule_tac x=e in allE) by auto

  4641

  4642 lemma continuous_on_real_range:

  4643   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4644   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))"

  4645   unfolding continuous_on_iff dist_norm by simp

  4646

  4647 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4648

  4649 lemma compact_attains_sup:

  4650   fixes s :: "real set"

  4651   assumes "compact s"  "s \<noteq> {}"

  4652   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"

  4653 proof-

  4654   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4655   { 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"

  4656     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto

  4657     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto

  4658     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using e>0 by auto  }

  4659   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]

  4660     apply(rule_tac x="Sup s" in bexI) by auto

  4661 qed

  4662

  4663 lemma Inf:

  4664   fixes S :: "real set"

  4665   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"

  4666 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)

  4667

  4668 lemma compact_attains_inf:

  4669   fixes s :: "real set"

  4670   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"

  4671 proof-

  4672   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  4673   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"

  4674       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"

  4675     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto

  4676     moreover

  4677     { fix x assume "x \<in> s"

  4678       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto

  4679       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) x\<in>s unfolding * by auto }

  4680     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto

  4681     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using e>0 by auto  }

  4682   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]

  4683     apply(rule_tac x="Inf s" in bexI) by auto

  4684 qed

  4685

  4686 lemma continuous_attains_sup:

  4687   fixes f :: "'a::metric_space \<Rightarrow> real"

  4688   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4689         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"

  4690   using compact_attains_sup[of "f  s"]

  4691   using compact_continuous_image[of s f] by auto

  4692

  4693 lemma continuous_attains_inf:

  4694   fixes f :: "'a::metric_space \<Rightarrow> real"

  4695   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  4696         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"

  4697   using compact_attains_inf[of "f  s"]

  4698   using compact_continuous_image[of s f] by auto

  4699

  4700 lemma distance_attains_sup:

  4701   assumes "compact s" "s \<noteq> {}"

  4702   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"

  4703 proof (rule continuous_attains_sup [OF assms])

  4704   { fix x assume "x\<in>s"

  4705     have "(dist a ---> dist a x) (at x within s)"

  4706       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  4707   }

  4708   thus "continuous_on s (dist a)"

  4709     unfolding continuous_on ..

  4710 qed

  4711

  4712 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4713

  4714 lemma distance_attains_inf:

  4715   fixes a :: "'a::heine_borel"

  4716   assumes "closed s"  "s \<noteq> {}"

  4717   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"

  4718 proof-

  4719   from assms(2) obtain b where "b\<in>s" by auto

  4720   let ?B = "cball a (dist b a) \<inter> s"

  4721   have "b \<in> ?B" using b\<in>s by (simp add: dist_commute)

  4722   hence "?B \<noteq> {}" by auto

  4723   moreover

  4724   { fix x assume "x\<in>?B"

  4725     fix e::real assume "e>0"

  4726     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"

  4727       from as have "\<bar>dist a x' - dist a x\<bar> < e"

  4728         unfolding abs_less_iff minus_diff_eq

  4729         using dist_triangle2 [of a x' x]

  4730         using dist_triangle [of a x x']

  4731         by arith

  4732     }

  4733     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"

  4734       using e>0 by auto

  4735   }

  4736   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"

  4737     unfolding continuous_on Lim_within dist_norm real_norm_def

  4738     by fast

  4739   moreover have "compact ?B"

  4740     using compact_cball[of a "dist b a"]

  4741     unfolding compact_eq_bounded_closed

  4742     using bounded_Int and closed_Int and assms(1) by auto

  4743   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"

  4744     using continuous_attains_inf[of ?B "dist a"] by fastforce

  4745   thus ?thesis by fastforce

  4746 qed

  4747

  4748

  4749 subsection {* Pasted sets *}

  4750

  4751 lemma bounded_Times:

  4752   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4753 proof-

  4754   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4755     using assms [unfolded bounded_def] by auto

  4756   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4757     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4758   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4759 qed

  4760

  4761 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4762 by (induct x) simp

  4763

  4764 lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"

  4765 unfolding compact_def

  4766 apply clarify

  4767 apply (drule_tac x="fst \<circ> f" in spec)

  4768 apply (drule mp, simp add: mem_Times_iff)

  4769 apply (clarify, rename_tac l1 r1)

  4770 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4771 apply (drule mp, simp add: mem_Times_iff)

  4772 apply (clarify, rename_tac l2 r2)

  4773 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4774 apply (rule_tac x="r1 \<circ> r2" in exI)

  4775 apply (rule conjI, simp add: subseq_def)

  4776 apply (drule_tac r=r2 in lim_subseq [rotated], assumption)

  4777 apply (drule (1) tendsto_Pair) back

  4778 apply (simp add: o_def)

  4779 done

  4780

  4781 text{* Hence some useful properties follow quite easily. *}

  4782

  4783 lemma compact_scaling:

  4784   fixes s :: "'a::real_normed_vector set"

  4785   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4786 proof-

  4787   let ?f = "\<lambda>x. scaleR c x"

  4788   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4789   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4790     using linear_continuous_at[OF *] assms by auto

  4791 qed

  4792

  4793 lemma compact_negations:

  4794   fixes s :: "'a::real_normed_vector set"

  4795   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4796   using compact_scaling [OF assms, of "- 1"] by auto

  4797

  4798 lemma compact_sums:

  4799   fixes s t :: "'a::real_normed_vector set"

  4800   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4801 proof-

  4802   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4803     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4804   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4805     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4806   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4807 qed

  4808

  4809 lemma compact_differences:

  4810   fixes s t :: "'a::real_normed_vector set"

  4811   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4812 proof-

  4813   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4814     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4815   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4816 qed

  4817

  4818 lemma compact_translation:

  4819   fixes s :: "'a::real_normed_vector set"

  4820   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4821 proof-

  4822   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4823   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4824 qed

  4825

  4826 lemma compact_affinity:

  4827   fixes s :: "'a::real_normed_vector set"

  4828   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4829 proof-

  4830   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4831   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4832 qed

  4833

  4834 text {* Hence we get the following. *}

  4835

  4836 lemma compact_sup_maxdistance:

  4837   fixes s :: "'a::real_normed_vector set"

  4838   assumes "compact s"  "s \<noteq> {}"

  4839   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"

  4840 proof-

  4841   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using s \<noteq> {} by auto

  4842   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"

  4843     using compact_differences[OF assms(1) assms(1)]

  4844     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

  4845   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto

  4846   thus ?thesis using x(2)[unfolded x = a - b] by blast

  4847 qed

  4848

  4849 text {* We can state this in terms of diameter of a set. *}

  4850

  4851 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"

  4852   (* TODO: generalize to class metric_space *)

  4853

  4854 lemma diameter_bounded:

  4855   assumes "bounded s"

  4856   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4857         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"

  4858 proof-

  4859   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"

  4860   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto

  4861   { fix x y assume "x \<in> s" "y \<in> s"

  4862     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)  }

  4863   note * = this

  4864   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto

  4865     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

  4866       by simp (blast del: Sup_upper intro!: * Sup_upper) }

  4867   moreover

  4868   { fix d::real assume "d>0" "d < diameter s"

  4869     hence "s\<noteq>{}" unfolding diameter_def by auto

  4870     have "\<exists>d' \<in> ?D. d' > d"

  4871     proof(rule ccontr)

  4872       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"

  4873       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)

  4874       thus False using d < diameter s s\<noteq>{}

  4875         apply (auto simp add: diameter_def)

  4876         apply (drule Sup_real_iff [THEN [2] rev_iffD2])

  4877         apply (auto, force)

  4878         done

  4879     qed

  4880     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }

  4881   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  4882         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto

  4883 qed

  4884

  4885 lemma diameter_bounded_bound:

  4886  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"

  4887   using diameter_bounded by blast

  4888

  4889 lemma diameter_compact_attained:

  4890   fixes s :: "'a::real_normed_vector set"

  4891   assumes "compact s"  "s \<noteq> {}"

  4892   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"

  4893 proof-

  4894   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4895   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

  4896   hence "diameter s \<le> norm (x - y)"

  4897     unfolding diameter_def by clarsimp (rule Sup_least, fast+)

  4898   thus ?thesis

  4899     by (metis b diameter_bounded_bound order_antisym xys)

  4900 qed

  4901

  4902 text {* Related results with closure as the conclusion. *}

  4903

  4904 lemma closed_scaling:

  4905   fixes s :: "'a::real_normed_vector set"

  4906   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4907 proof(cases "s={}")

  4908   case True thus ?thesis by auto

  4909 next

  4910   case False

  4911   show ?thesis

  4912   proof(cases "c=0")

  4913     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4914     case True thus ?thesis apply auto unfolding * by auto

  4915   next

  4916     case False

  4917     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4918       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4919           using as(1)[THEN spec[where x=n]]

  4920           using c\<noteq>0 by auto

  4921       }

  4922       moreover

  4923       { fix e::real assume "e>0"

  4924         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4925         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4926           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4927         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4928           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4929           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4930       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4931       ultimately have "l \<in> scaleR c  s"

  4932         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"]]

  4933         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4934     thus ?thesis unfolding closed_sequential_limits by fast

  4935   qed

  4936 qed

  4937

  4938 lemma closed_negations:

  4939   fixes s :: "'a::real_normed_vector set"

  4940   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4941   using closed_scaling[OF assms, of "- 1"] by simp

  4942

  4943 lemma compact_closed_sums:

  4944   fixes s :: "'a::real_normed_vector set"

  4945   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4946 proof-

  4947   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4948   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4949     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"

  4950       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4951     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4952       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4953     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4954       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto

  4955     hence "l - l' \<in> t"

  4956       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4957       using f(3) by auto

  4958     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

  4959   }

  4960   thus ?thesis unfolding closed_sequential_limits by fast

  4961 qed

  4962

  4963 lemma closed_compact_sums:

  4964   fixes s t :: "'a::real_normed_vector set"

  4965   assumes "closed s"  "compact t"

  4966   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4967 proof-

  4968   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

  4969     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4970   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4971 qed

  4972

  4973 lemma compact_closed_differences:

  4974   fixes s t :: "'a::real_normed_vector set"

  4975   assumes "compact s"  "closed t"

  4976   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4977 proof-

  4978   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4979     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4980   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4981 qed

  4982

  4983 lemma closed_compact_differences:

  4984   fixes s t :: "'a::real_normed_vector set"

  4985   assumes "closed s" "compact t"

  4986   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4987 proof-

  4988   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  4989     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4990  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  4991 qed

  4992

  4993 lemma closed_translation:

  4994   fixes a :: "'a::real_normed_vector"

  4995   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  4996 proof-

  4997   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  4998   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  4999 qed

  5000

  5001 lemma translation_Compl:

  5002   fixes a :: "'a::ab_group_add"

  5003   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5004   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  5005

  5006 lemma translation_UNIV:

  5007   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  5008   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  5009

  5010 lemma translation_diff:

  5011   fixes a :: "'a::ab_group_add"

  5012   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5013   by auto

  5014

  5015 lemma closure_translation:

  5016   fixes a :: "'a::real_normed_vector"

  5017   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5018 proof-

  5019   have *:"op + a  (- s) = - op + a  s"

  5020     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  5021   show ?thesis unfolding closure_interior translation_Compl

  5022     using interior_translation[of a "- s"] unfolding * by auto

  5023 qed

  5024

  5025 lemma frontier_translation:

  5026   fixes a :: "'a::real_normed_vector"

  5027   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5028   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  5029

  5030

  5031 subsection {* Separation between points and sets *}

  5032

  5033 lemma separate_point_closed:

  5034   fixes s :: "'a::heine_borel set"

  5035   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  5036 proof(cases "s = {}")

  5037   case True

  5038   thus ?thesis by(auto intro!: exI[where x=1])

  5039 next

  5040   case False

  5041   assume "closed s" "a \<notin> s"

  5042   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

  5043   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5044 qed

  5045

  5046 lemma separate_compact_closed:

  5047   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5048     (* TODO: does this generalize to heine_borel? *)

  5049   assumes "compact s" and "closed t" and "s \<inter> t = {}"

  5050   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5051 proof-

  5052   have "0 \<notin> {x - y |x y. x \<in> s \<and> y \<in> t}" using assms(3) by auto

  5053   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"

  5054     using separate_point_closed[OF compact_closed_differences[OF assms(1,2)], of 0] by auto

  5055   { fix x y assume "x\<in>s" "y\<in>t"

  5056     hence "x - y \<in> {x - y |x y. x \<in> s \<and> y \<in> t}" by auto

  5057     hence "d \<le> dist (x - y) 0" using d[THEN bspec[where x="x - y"]] using dist_commute

  5058       by (auto  simp add: dist_commute)

  5059     hence "d \<le> dist x y" unfolding dist_norm by auto  }

  5060   thus ?thesis using d>0 by auto

  5061 qed

  5062

  5063 lemma separate_closed_compact:

  5064   fixes s t :: "'a::{heine_borel, real_normed_vector} set"

  5065   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5066   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5067 proof-

  5068   have *:"t \<inter> s = {}" using assms(3) by auto

  5069   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5070     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5071     by (auto simp add: dist_commute)

  5072 qed

  5073

  5074

  5075 subsection {* Intervals *}

  5076

  5077 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5078   "{a <..< b} = {x::'a. \<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i}" and

  5079   "{a .. b} = {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i}"

  5080   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5081

  5082 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5083   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < x$$i \<and> x$$i < b$$i)"

  5084   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> x$$i \<and> x$$i \<le> b$$i)"

  5085   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5086

  5087 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5088  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i))" (is ?th1) and

  5089  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i<DIM('a). b$$i < a$$i))" (is ?th2)

  5090 proof-

  5091   { fix i x assume i:"i<DIM('a)" and as:"b$$i \<le> a$$i" and x:"x\<in>{a <..< b}"

  5092     hence "a $$i < x$$ i \<and> x $$i < b$$ i" unfolding mem_interval by auto

  5093     hence "a$$i < b$$i" by auto

  5094     hence False using as by auto  }

  5095   moreover

  5096   { assume as:"\<forall>i<DIM('a). \<not> (b$$i \<le> a$$i)"

  5097     let ?x = "(1/2) *\<^sub>R (a + b)"

  5098     { fix i assume i:"i<DIM('a)"

  5099       have "a$$i < b$$i" using as[THEN spec[where x=i]] using i by auto

  5100       hence "a$$i < ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i < b$$i"

  5101         unfolding euclidean_simps by auto }

  5102     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5103   ultimately show ?th1 by blast

  5104

  5105   { fix i x assume i:"i<DIM('a)" and as:"b$$i < a$$i" and x:"x\<in>{a .. b}"

  5106     hence "a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i" unfolding mem_interval by auto

  5107     hence "a$$i \<le> b$$i" by auto

  5108     hence False using as by auto  }

  5109   moreover

  5110   { assume as:"\<forall>i<DIM('a). \<not> (b$$i < a$$i)"

  5111     let ?x = "(1/2) *\<^sub>R (a + b)"

  5112     { fix i assume i:"i<DIM('a)"

  5113       have "a$$i \<le> b$$i" using as[THEN spec[where x=i]] by auto

  5114       hence "a$$i \<le> ((1/2) *\<^sub>R (a+b))$$ i" "((1/2) *\<^sub>R (a+b)) $$i \<le> b$$i"

  5115         unfolding euclidean_simps by auto }

  5116     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5117   ultimately show ?th2 by blast

  5118 qed

  5119

  5120 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5121   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i \<le> b$$i)" and

  5122   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"

  5123   unfolding interval_eq_empty[of a b] by fastforce+

  5124

  5125 lemma interval_sing:

  5126   fixes a :: "'a::ordered_euclidean_space"

  5127   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5128   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5129   by (auto simp add: euclidean_eq[where 'a='a] eq_commute

  5130     eucl_less[where 'a='a] eucl_le[where 'a='a])

  5131

  5132 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5133  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  5134  "(\<forall>i<DIM('a). a$$i < c$$i \<and> d$$i < b$$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5135  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  5136  "(\<forall>i<DIM('a). a$$i \<le> c$$i \<and> d$$i \<le> b$$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  5137   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5138   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5139

  5140 lemma interval_open_subset_closed:

  5141   fixes a :: "'a::ordered_euclidean_space"

  5142   shows "{a<..<b} \<subseteq> {a .. b}"

  5143   unfolding subset_eq [unfolded Ball_def] mem_interval

  5144   by (fast intro: less_imp_le)

  5145

  5146 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5147  "{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

  5148  "{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

  5149  "{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

  5150  "{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)

  5151 proof-

  5152   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5153   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)

  5154   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i<DIM('a). c$$i < d$$i"

  5155     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5156     fix i assume i:"i<DIM('a)"

  5157     (** TODO combine the following two parts as done in the HOL_light version. **)

  5158     { let ?x = "(\<chi>\<chi> j. (if j=i then ((min (a$$j) (d$$j))+c$$j)/2 else (c$$j+d$$j)/2))::'a"   5159 assume as2: "a$$i > c$$i"   5160 { fix j assume j:"j<DIM('a)"   5161 hence "c$$ j < ?x $$j \<and> ?x$$ j < d $$j"   5162 apply(cases "j=i") using as(2)[THEN spec[where x=j]] i   5163 by (auto simp add: as2) }   5164 hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto   5165 moreover   5166 have "?x\<notin>{a .. b}"   5167 unfolding mem_interval apply auto apply(rule_tac x=i in exI)   5168 using as(2)[THEN spec[where x=i]] and as2 i   5169 by auto   5170 ultimately have False using as by auto }   5171 hence "a$$i \<le> c$$i" by(rule ccontr)auto   5172 moreover   5173 { let ?x = "(\<chi>\<chi> j. (if j=i then ((max (b$$j) (c$$j))+d$$j)/2 else (c$$j+d$$j)/2))::'a"

  5174       assume as2: "b$$i < d$$i"

  5175       { fix j assume "j<DIM('a)"

  5176         hence "d $$j > ?x$$ j \<and> ?x $$j > c$$ j"

  5177           apply(cases "j=i") using as(2)[THEN spec[where x=j]]

  5178           by (auto simp add: as2)  }

  5179       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5180       moreover

  5181       have "?x\<notin>{a .. b}"

  5182         unfolding mem_interval apply auto apply(rule_tac x=i in exI)

  5183         using as(2)[THEN spec[where x=i]] and as2 using i

  5184         by auto

  5185       ultimately have False using as by auto  }

  5186     hence "b$$i \<ge> d$$i" by(rule ccontr)auto

  5187     ultimately

  5188     have "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" by auto

  5189   } note part1 = this

  5190   show ?th3 unfolding subset_eq and Ball_def and mem_interval

  5191     apply(rule,rule,rule,rule) apply(rule part1) unfolding subset_eq and Ball_def and mem_interval

  5192     prefer 4 apply auto by(erule_tac x=i in allE,erule_tac x=i in allE,fastforce)+

  5193   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i<DIM('a). c$$i < d$$i"

  5194     fix i assume i:"i<DIM('a)"

  5195     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5196     hence "a$$i \<le> c$$i \<and> d$$i \<le> b$$i" using part1 and as(2) using i by auto  } note * = this

  5197   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5198     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5199     apply auto by(erule_tac x=i in allE, simp)+

  5200 qed

  5201

  5202 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5203   "{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

  5204   "{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

  5205   "{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

  5206   "{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)

  5207 proof-

  5208   let ?z = "(\<chi>\<chi> i. ((max (a$$i) (c$$i)) + (min (b$$i) (d$$i))) / 2)::'a"

  5209   note * = set_eq_iff Int_iff empty_iff mem_interval all_conj_distrib[THEN sym] eq_False

  5210   show ?th1 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5211     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5212   show ?th2 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5213     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5214   show ?th3 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5215     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5216   show ?th4 unfolding * apply safe apply(erule_tac x="?z" in allE)

  5217     unfolding not_all apply(erule exE,rule_tac x=x in exI) apply(erule_tac[2-] x=i in allE) by auto

  5218 qed

  5219

  5220 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5221  "{a .. b} \<inter> {c .. d} =  {(\<chi>\<chi> i. max (a$$i) (c$$i)) .. (\<chi>\<chi> i. min (b$$i) (d$$i))}"

  5222   unfolding set_eq_iff and Int_iff and mem_interval

  5223   by auto

  5224

  5225 (* Moved interval_open_subset_closed a bit upwards *)

  5226

  5227 lemma open_interval[intro]:

  5228   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5229 proof-

  5230   have "open (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i})"   5231 by (intro open_INT finite_lessThan ballI continuous_open_vimage allI   5232 linear_continuous_at bounded_linear_euclidean_component   5233 open_real_greaterThanLessThan)   5234 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i<..<b$$i}) = {a<..<b}"

  5235     by (auto simp add: eucl_less [where 'a='a])

  5236   finally show "open {a<..<b}" .

  5237 qed

  5238

  5239 lemma closed_interval[intro]:

  5240   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5241 proof-

  5242   have "closed (\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i})"   5243 by (intro closed_INT ballI continuous_closed_vimage allI   5244 linear_continuous_at bounded_linear_euclidean_component   5245 closed_real_atLeastAtMost)   5246 also have "(\<Inter>i<DIM('a). (\<lambda>x. x$$i) - {a$$i .. b$$i}) = {a .. b}"

  5247     by (auto simp add: eucl_le [where 'a='a])

  5248   finally show "closed {a .. b}" .

  5249 qed

  5250

  5251 lemma interior_closed_interval [intro]:

  5252   fixes a b :: "'a::ordered_euclidean_space"

  5253   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5254 proof(rule subset_antisym)

  5255   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5256     by (rule interior_maximal)

  5257 next

  5258   { fix x assume "x \<in> interior {a..b}"

  5259     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5260     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

  5261     { fix i assume i:"i<DIM('a)"

  5262       have "dist (x - (e / 2) *\<^sub>R basis i) x < e"

  5263            "dist (x + (e / 2) *\<^sub>R basis i) x < e"

  5264         unfolding dist_norm apply auto

  5265         unfolding norm_minus_cancel using norm_basis and e>0 by auto

  5266       hence "a $$i \<le> (x - (e / 2) *\<^sub>R basis i)$$ i"

  5267                      "(x + (e / 2) *\<^sub>R basis i) $$i \<le> b$$ i"

  5268         using e[THEN spec[where x="x - (e/2) *\<^sub>R basis i"]]

  5269         and   e[THEN spec[where x="x + (e/2) *\<^sub>R basis i"]]

  5270         unfolding mem_interval using i by blast+

  5271       hence "a $$i < x$$ i" and "x $$i < b$$ i" unfolding euclidean_simps

  5272         unfolding basis_component using e>0 i by auto  }

  5273     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5274   thus "?L \<subseteq> ?R" ..

  5275 qed

  5276

  5277 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5278 proof-

  5279   let ?b = "\<Sum>i<DIM('a). \<bar>a$$i\<bar> + \<bar>b$$i\<bar>"

  5280   { fix x::"'a" assume x:"\<forall>i<DIM('a). a $$i \<le> x$$ i \<and> x $$i \<le> b$$ i"

  5281     { fix i assume "i<DIM('a)"

  5282       hence "\<bar>x$$i\<bar> \<le> \<bar>a$$i\<bar> + \<bar>b$$i\<bar>" using x[THEN spec[where x=i]] by auto }   5283 hence "(\<Sum>i<DIM('a). \<bar>x$$ i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5284     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5285   thus ?thesis unfolding interval and bounded_iff by auto

  5286 qed

  5287

  5288 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5289  "bounded {a .. b} \<and> bounded {a<..<b}"

  5290   using bounded_closed_interval[of a b]

  5291   using interval_open_subset_closed[of a b]

  5292   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5293   by simp

  5294

  5295 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5296  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5297   using bounded_interval[of a b] by auto

  5298

  5299 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5300   using bounded_closed_imp_compact[of "{a..b}"] using bounded_interval[of a b]

  5301   by auto

  5302

  5303 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5304   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5305 proof-

  5306   { fix i assume "i<DIM('a)"

  5307     hence "a $$i < ((1 / 2) *\<^sub>R (a + b))$$ i \<and> ((1 / 2) *\<^sub>R (a + b)) $$i < b$$ i"

  5308       using assms[unfolded interval_ne_empty, THEN spec[where x=i]]

  5309       unfolding euclidean_simps by auto  }

  5310   thus ?thesis unfolding mem_interval by auto

  5311 qed

  5312

  5313 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5314   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5315   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5316 proof-

  5317   { fix i assume i:"i<DIM('a)"

  5318     have "a $$i = e * a$$i + (1 - e) * a$$i" unfolding left_diff_distrib by simp   5319 also have "\<dots> < e * x$$ i + (1 - e) * y $$i" apply(rule add_less_le_mono)   5320 using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all   5321 using x unfolding mem_interval using i apply simp   5322 using y unfolding mem_interval using i apply simp   5323 done   5324 finally have "a$$ i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i" unfolding euclidean_simps by auto   5325 moreover {   5326 have "b$$ i = e * b$$i + (1 - e) * b$$i" unfolding left_diff_distrib by simp

  5327     also have "\<dots> > e * x $$i + (1 - e) * y$$ i" apply(rule add_less_le_mono)

  5328       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5329       using x unfolding mem_interval using i apply simp

  5330       using y unfolding mem_interval using i apply simp

  5331       done

  5332     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) $$i < b$$ i" unfolding euclidean_simps by auto

  5333     } 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 }

  5334   thus ?thesis unfolding mem_interval by auto

  5335 qed

  5336

  5337 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5338   assumes "{a<..<b} \<noteq> {}"

  5339   shows "closure {a<..<b} = {a .. b}"

  5340 proof-

  5341   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5342   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5343   { fix x assume as:"x \<in> {a .. b}"

  5344     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5345     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5346       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5347       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5348         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5349         by (auto simp add: algebra_simps)

  5350       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

  5351       hence False using fn unfolding f_def using xc by auto  }

  5352     moreover

  5353     { assume "\<not> (f ---> x) sequentially"

  5354       { fix e::real assume "e>0"

  5355         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

  5356         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5357         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)

  5358         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5359       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5360         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5361       hence "(f ---> x) sequentially" unfolding f_def

  5362         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]

  5363         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  }

  5364     ultimately have "x \<in> closure {a<..<b}"

  5365       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5366   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5367 qed

  5368

  5369 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5370   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5371 proof-

  5372   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5373   def a \<equiv> "(\<chi>\<chi> i. b+1)::'a"

  5374   { fix x assume "x\<in>s"

  5375     fix i assume i:"i<DIM('a)"

  5376     hence "(-a)$$i < x$$i" and "x$$i < a$$i" using b[THEN bspec[where x=x], OF x\<in>s]

  5377       and component_le_norm[of x i] unfolding euclidean_simps and a_def by auto  }

  5378   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5379 qed

  5380

  5381 lemma bounded_subset_open_interval:

  5382   fixes s :: "('a::ordered_euclidean_space) set"

  5383   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5384   by (auto dest!: bounded_subset_open_interval_symmetric)

  5385

  5386 lemma bounded_subset_closed_interval_symmetric:

  5387   fixes s :: "('a::ordered_euclidean_space) set"

  5388   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5389 proof-

  5390   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5391   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5392 qed

  5393

  5394 lemma bounded_subset_closed_interval:

  5395   fixes s :: "('a::ordered_euclidean_space) set"

  5396   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5397   using bounded_subset_closed_interval_symmetric[of s] by auto

  5398

  5399 lemma frontier_closed_interval:

  5400   fixes a b :: "'a::ordered_euclidean_space"

  5401   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5402   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5403

  5404 lemma frontier_open_interval:

  5405   fixes a b :: "'a::ordered_euclidean_space"

  5406   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5407 proof(cases "{a<..<b} = {}")

  5408   case True thus ?thesis using frontier_empty by auto

  5409 next

  5410   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5411 qed

  5412

  5413 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5414   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5415   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5416

  5417

  5418 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5419

  5420 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5421   shows "closed {x::'a. \<forall>i<DIM('a). x$$i \<le> b$$i}"

  5422 proof-

  5423   { fix i assume i:"i<DIM('a)"

  5424     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"

  5425     { assume "x$$i > b$$i"

  5426       then obtain y where "y $$i \<le> b$$ i"  "y \<noteq> x"  "dist y x < x$$i - b$$i"

  5427         using x[THEN spec[where x="x$$i - b$$i"]] using i by auto

  5428       hence False using component_le_norm[of "y - x" i] unfolding dist_norm euclidean_simps using i

  5429         by auto   }

  5430     hence "x$$i \<le> b$$i" by(rule ccontr)auto  }

  5431   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5432 qed

  5433

  5434 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5435   shows "closed {x::'a. \<forall>i<DIM('a). a$$i \<le> x$$i}"

  5436 proof-

  5437   { fix i assume i:"i<DIM('a)"

  5438     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"

  5439     { assume "a$$i > x$$i"

  5440       then obtain y where "a $$i \<le> y$$ i"  "y \<noteq> x"  "dist y x < a$$i - x$$i"

  5441         using x[THEN spec[where x="a$$i - x$$i"]] i by auto

  5442       hence False using component_le_norm[of "y - x" i] unfolding dist_norm and euclidean_simps by auto   }

  5443     hence "a$$i \<le> x$$i" by(rule ccontr)auto  }

  5444   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5445 qed

  5446

  5447 instance ordered_euclidean_space \<subseteq> enumerable_basis

  5448 proof

  5449   def to_cube \<equiv> "\<lambda>(a, b). {Chi (real_of_rat \<circ> op ! a)<..<Chi (real_of_rat \<circ> op ! b)}::'a set"

  5450   def enum \<equiv> "\<lambda>n. (to_cube (from_nat n)::'a set)"

  5451   have "Ball (range enum) open" unfolding enum_def

  5452   proof safe

  5453     fix n show "open (to_cube (from_nat n))"

  5454       by (cases "from_nat n::rat list \<times> rat list")

  5455          (simp add: open_interval to_cube_def)

  5456   qed

  5457   moreover have "(\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>range enum. \<Union>B' = x))"

  5458   proof safe

  5459     fix x::"'a set" assume "open x"

  5460     def lists \<equiv> "{(a, b) |a b. to_cube (a, b) \<subseteq> x}"

  5461     from open_UNION[OF open x]

  5462     have "\<Union>(to_cube  lists) = x" unfolding lists_def to_cube_def

  5463      by simp

  5464     moreover have "to_cube  lists \<subseteq> range enum"

  5465     proof

  5466       fix x assume "x \<in> to_cube  lists"

  5467       then obtain l where "l \<in> lists" "x = to_cube l" by auto

  5468       hence "x = enum (to_nat l)" by (simp add: to_cube_def enum_def)

  5469       thus "x \<in> range enum" by simp

  5470     qed

  5471     ultimately

  5472     show "\<exists>B'\<subseteq>range enum. \<Union>B' = x" by blast

  5473   qed

  5474   ultimately

  5475   show "\<exists>f::nat\<Rightarrow>'a set. topological_basis (range f)" unfolding topological_basis_def by blast

  5476 qed

  5477

  5478 instance ordered_euclidean_space \<subseteq> polish_space ..

  5479

  5480 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5481

  5482 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5483   (\<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)"

  5484

  5485 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5486   "is_interval {a<..<b}" (is ?th2) proof -

  5487   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5488     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5489

  5490 lemma is_interval_empty:

  5491  "is_interval {}"

  5492   unfolding is_interval_def

  5493   by simp

  5494

  5495 lemma is_interval_univ:

  5496  "is_interval UNIV"

  5497   unfolding is_interval_def

  5498   by simp

  5499

  5500

  5501 subsection {* Closure of halfspaces and hyperplanes *}

  5502

  5503 lemma isCont_open_vimage:

  5504   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5505 proof -

  5506   from assms(1) have "continuous_on UNIV f"

  5507     unfolding isCont_def continuous_on_def within_UNIV by simp

  5508   hence "open {x \<in> UNIV. f x \<in> s}"

  5509     using open_UNIV open s by (rule continuous_open_preimage)

  5510   thus "open (f - s)"

  5511     by (simp add: vimage_def)

  5512 qed

  5513

  5514 lemma isCont_closed_vimage:

  5515   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5516   using assms unfolding closed_def vimage_Compl [symmetric]

  5517   by (rule isCont_open_vimage)

  5518

  5519 lemma open_Collect_less:

  5520   fixes f g :: "'a::topological_space \<Rightarrow> real"

  5521   assumes f: "\<And>x. isCont f x"

  5522   assumes g: "\<And>x. isCont g x"

  5523   shows "open {x. f x < g x}"

  5524 proof -

  5525   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5526     using isCont_diff [OF g f] open_real_greaterThan

  5527     by (rule isCont_open_vimage)

  5528   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5529     by auto

  5530   finally show ?thesis .

  5531 qed

  5532

  5533 lemma closed_Collect_le:

  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 "closed {x. f x \<le> g x}"

  5538 proof -

  5539   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5540     using isCont_diff [OF g f] closed_real_atLeast

  5541     by (rule isCont_closed_vimage)

  5542   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5543     by auto

  5544   finally show ?thesis .

  5545 qed

  5546

  5547 lemma closed_Collect_eq:

  5548   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  5549   assumes f: "\<And>x. isCont f x"

  5550   assumes g: "\<And>x. isCont g x"

  5551   shows "closed {x. f x = g x}"

  5552 proof -

  5553   have "open {(x::'b, y::'b). x \<noteq> y}"

  5554     unfolding open_prod_def by (auto dest!: hausdorff)

  5555   hence "closed {(x::'b, y::'b). x = y}"

  5556     unfolding closed_def split_def Collect_neg_eq .

  5557   with isCont_Pair [OF f g]

  5558   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5559     by (rule isCont_closed_vimage)

  5560   also have "\<dots> = {x. f x = g x}" by auto

  5561   finally show ?thesis .

  5562 qed

  5563

  5564 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5565   unfolding continuous_at by (intro tendsto_intros)

  5566

  5567 lemma continuous_at_euclidean_component[intro!, simp]: "continuous (at x) (\<lambda>x. x $$i)"   5568 unfolding euclidean_component_def by (rule continuous_at_inner)   5569   5570 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"   5571 by (simp add: closed_Collect_le)   5572   5573 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"   5574 by (simp add: closed_Collect_le)   5575   5576 lemma closed_hyperplane: "closed {x. inner a x = b}"   5577 by (simp add: closed_Collect_eq)   5578   5579 lemma closed_halfspace_component_le:   5580 shows "closed {x::'a::euclidean_space. x$$i \<le> a}"

  5581   by (simp add: closed_Collect_le)

  5582

  5583 lemma closed_halfspace_component_ge:

  5584   shows "closed {x::'a::euclidean_space. x$$i \<ge> a}"   5585 by (simp add: closed_Collect_le)   5586   5587 text {* Openness of halfspaces. *}   5588   5589 lemma open_halfspace_lt: "open {x. inner a x < b}"   5590 by (simp add: open_Collect_less)   5591   5592 lemma open_halfspace_gt: "open {x. inner a x > b}"   5593 by (simp add: open_Collect_less)   5594   5595 lemma open_halfspace_component_lt:   5596 shows "open {x::'a::euclidean_space. x$$i < a}"

  5597   by (simp add: open_Collect_less)

  5598

  5599 lemma open_halfspace_component_gt:

  5600   shows "open {x::'a::euclidean_space. x$$i > a}"   5601 by (simp add: open_Collect_less)   5602   5603 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}   5604   5605 lemma eucl_lessThan_eq_halfspaces:   5606 fixes a :: "'a\<Colon>ordered_euclidean_space"   5607 shows "{..<a} = (\<Inter>i<DIM('a). {x. x$$ i < a $$i})"   5608 by (auto simp: eucl_less[where 'a='a])   5609   5610 lemma eucl_greaterThan_eq_halfspaces:   5611 fixes a :: "'a\<Colon>ordered_euclidean_space"   5612 shows "{a<..} = (\<Inter>i<DIM('a). {x. a$$ i < x $$i})"   5613 by (auto simp: eucl_less[where 'a='a])   5614   5615 lemma eucl_atMost_eq_halfspaces:   5616 fixes a :: "'a\<Colon>ordered_euclidean_space"   5617 shows "{.. a} = (\<Inter>i<DIM('a). {x. x$$ i \<le> a $$i})"   5618 by (auto simp: eucl_le[where 'a='a])   5619   5620 lemma eucl_atLeast_eq_halfspaces:   5621 fixes a :: "'a\<Colon>ordered_euclidean_space"   5622 shows "{a ..} = (\<Inter>i<DIM('a). {x. a$$ i \<le> x $$i})"   5623 by (auto simp: eucl_le[where 'a='a])   5624   5625 lemma open_eucl_lessThan[simp, intro]:   5626 fixes a :: "'a\<Colon>ordered_euclidean_space"   5627 shows "open {..< a}"   5628 by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)   5629   5630 lemma open_eucl_greaterThan[simp, intro]:   5631 fixes a :: "'a\<Colon>ordered_euclidean_space"   5632 shows "open {a <..}"   5633 by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)   5634   5635 lemma closed_eucl_atMost[simp, intro]:   5636 fixes a :: "'a\<Colon>ordered_euclidean_space"   5637 shows "closed {.. a}"   5638 unfolding eucl_atMost_eq_halfspaces   5639 by (simp add: closed_INT closed_Collect_le)   5640   5641 lemma closed_eucl_atLeast[simp, intro]:   5642 fixes a :: "'a\<Colon>ordered_euclidean_space"   5643 shows "closed {a ..}"   5644 unfolding eucl_atLeast_eq_halfspaces   5645 by (simp add: closed_INT closed_Collect_le)   5646   5647 lemma open_vimage_euclidean_component: "open S \<Longrightarrow> open ((\<lambda>x. x$$ i) - S)"

  5648   by (auto intro!: continuous_open_vimage)

  5649

  5650 text {* This gives a simple derivation of limit component bounds. *}

  5651

  5652 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5653   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$$i \<le> b) net"   5654 shows "l$$i \<le> b"

  5655 proof-

  5656   { fix x have "x \<in> {x::'b. inner (basis i) x \<le> b} \<longleftrightarrow> x$$i \<le> b"   5657 unfolding euclidean_component_def by auto } note * = this   5658 show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<le> b}" f net l] unfolding *   5659 using closed_halfspace_le[of "(basis i)::'b" b] and assms(1,2,3) by auto   5660 qed   5661   5662 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"   5663 assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$$i) net"

  5664   shows "b \<le> l$$i"   5665 proof-   5666 { fix x have "x \<in> {x::'b. inner (basis i) x \<ge> b} \<longleftrightarrow> x$$i \<ge> b"

  5667       unfolding euclidean_component_def by auto  } note * = this

  5668   show ?thesis using Lim_in_closed_set[of "{x. inner (basis i) x \<ge> b}" f net l] unfolding *

  5669     using closed_halfspace_ge[of b "(basis i)"] and assms(1,2,3) by auto

  5670 qed

  5671

  5672 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5673   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"   5674 shows "l$$i = b"

  5675   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

  5676 text{* Limits relative to a union.                                               *}

  5677

  5678 lemma eventually_within_Un:

  5679   "eventually P (net within (s \<union> t)) \<longleftrightarrow>

  5680     eventually P (net within s) \<and> eventually P (net within t)"

  5681   unfolding Limits.eventually_within

  5682   by (auto elim!: eventually_rev_mp)

  5683

  5684 lemma Lim_within_union:

  5685  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>

  5686   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"

  5687   unfolding tendsto_def

  5688   by (auto simp add: eventually_within_Un)

  5689

  5690 lemma Lim_topological:

  5691  "(f ---> l) net \<longleftrightarrow>

  5692         trivial_limit net \<or>

  5693         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5694   unfolding tendsto_def trivial_limit_eq by auto

  5695

  5696 lemma continuous_on_union:

  5697   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"

  5698   shows "continuous_on (s \<union> t) f"

  5699   using assms unfolding continuous_on Lim_within_union

  5700   unfolding Lim_topological trivial_limit_within closed_limpt by auto

  5701

  5702 lemma continuous_on_cases:

  5703   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"

  5704           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"

  5705   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  5706 proof-

  5707   let ?h = "(\<lambda>x. if P x then f x else g x)"

  5708   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto

  5709   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto

  5710   moreover

  5711   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto

  5712   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto

  5713   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto

  5714 qed

  5715

  5716

  5717 text{* Some more convenient intermediate-value theorem formulations.             *}

  5718

  5719 lemma connected_ivt_hyperplane:

  5720   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5721   shows "\<exists>z \<in> s. inner a z = b"

  5722 proof(rule ccontr)

  5723   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5724   let ?A = "{x. inner a x < b}"

  5725   let ?B = "{x. inner a x > b}"

  5726   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5727   moreover have "?A \<inter> ?B = {}" by auto

  5728   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5729   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

  5730 qed

  5731

  5732 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5733  "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)"

  5734   using connected_ivt_hyperplane[of s x y "(basis k)::'a" a]

  5735   unfolding euclidean_component_def by auto

  5736

  5737

  5738 subsection {* Homeomorphisms *}

  5739

  5740 definition "homeomorphism s t f g \<equiv>

  5741      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5742      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5743

  5744 definition

  5745   homeomorphic :: "'a::metric_space set \<Rightarrow> 'b::metric_space set \<Rightarrow> bool"

  5746     (infixr "homeomorphic" 60) where

  5747   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  5748

  5749 lemma homeomorphic_refl: "s homeomorphic s"

  5750   unfolding homeomorphic_def

  5751   unfolding homeomorphism_def

  5752   using continuous_on_id

  5753   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5754   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5755   by blast

  5756

  5757 lemma homeomorphic_sym:

  5758  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  5759 unfolding homeomorphic_def

  5760 unfolding homeomorphism_def

  5761 by blast

  5762
`