src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51480 3793c3a11378 parent 51479 33db4b7189af child 51481 ef949192e5d6 permissions -rw-r--r--
move connected to HOL image; used to show intermediate value theorem
     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   Complex_Main

    12   "~~/src/HOL/Library/Countable_Set"

    13   "~~/src/HOL/Library/Glbs"

    14   "~~/src/HOL/Library/FuncSet"

    15   Linear_Algebra

    16   Norm_Arith

    17 begin

    18

    19 lemma dist_0_norm:

    20   fixes x :: "'a::real_normed_vector"

    21   shows "dist 0 x = norm x"

    22 unfolding dist_norm by simp

    23

    24 lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"

    25   using dist_triangle[of y z x] by (simp add: dist_commute)

    26

    27 (* LEGACY *)

    28 lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"

    29   by (rule LIMSEQ_subseq_LIMSEQ)

    30

    31 lemmas real_isGlb_unique = isGlb_unique[where 'a=real]

    32

    33 lemma countable_PiE:

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

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

    36

    37 subsection {* Topological Basis *}

    38

    39 context topological_space

    40 begin

    41

    42 definition "topological_basis B =

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

    44

    45 lemma topological_basis:

    46   "topological_basis B = (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"

    47   unfolding topological_basis_def

    48   apply safe

    49      apply fastforce

    50     apply fastforce

    51    apply (erule_tac x="x" in allE)

    52    apply simp

    53    apply (rule_tac x="{x}" in exI)

    54   apply auto

    55   done

    56

    57 lemma topological_basis_iff:

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

    59   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'))"

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

    61 proof safe

    62   fix O' and x::'a

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

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

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

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

    67 next

    68   assume H: ?rhs

    69   show "topological_basis B" using assms unfolding topological_basis_def

    70   proof safe

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

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

    73       by (force intro: bchoice simp: Bex_def)

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

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

    76   qed

    77 qed

    78

    79 lemma topological_basisI:

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

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

    82   shows "topological_basis B"

    83   using assms by (subst topological_basis_iff) auto

    84

    85 lemma topological_basisE:

    86   fixes O'

    87   assumes "topological_basis B"

    88   assumes "open O'"

    89   assumes "x \<in> O'"

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

    91 proof atomize_elim

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

    93   with topological_basis_iff assms

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

    95 qed

    96

    97 lemma topological_basis_open:

    98   assumes "topological_basis B"

    99   assumes "X \<in> B"

   100   shows "open X"

   101   using assms

   102   by (simp add: topological_basis_def)

   103

   104 lemma topological_basis_imp_subbasis:

   105   assumes B: "topological_basis B" shows "open = generate_topology B"

   106 proof (intro ext iffI)

   107   fix S :: "'a set" assume "open S"

   108   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"

   109     unfolding topological_basis_def by blast

   110   then show "generate_topology B S"

   111     by (auto intro: generate_topology.intros dest: topological_basis_open)

   112 next

   113   fix S :: "'a set" assume "generate_topology B S" then show "open S"

   114     by induct (auto dest: topological_basis_open[OF B])

   115 qed

   116

   117 lemma basis_dense:

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

   119   assumes "topological_basis B"

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

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

   122 proof (intro allI impI)

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

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

   125   guess B' . note B' = this

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

   127 qed

   128

   129 end

   130

   131 lemma topological_basis_prod:

   132   assumes A: "topological_basis A" and B: "topological_basis B"

   133   shows "topological_basis ((\<lambda>(a, b). a \<times> b)  (A \<times> B))"

   134   unfolding topological_basis_def

   135 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])

   136   fix S :: "('a \<times> 'b) set" assume "open S"

   137   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"

   138   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])

   139     fix x y assume "(x, y) \<in> S"

   140     from open_prod_elim[OF open S this]

   141     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"

   142       by (metis mem_Sigma_iff)

   143     moreover from topological_basisE[OF A a] guess A0 .

   144     moreover from topological_basisE[OF B b] guess B0 .

   145     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"

   146       by (intro UN_I[of "(A0, B0)"]) auto

   147   qed auto

   148 qed (metis A B topological_basis_open open_Times)

   149

   150 subsection {* Countable Basis *}

   151

   152 locale countable_basis =

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

   154   assumes is_basis: "topological_basis B"

   155   assumes countable_basis: "countable B"

   156 begin

   157

   158 lemma open_countable_basis_ex:

   159   assumes "open X"

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

   161   using assms countable_basis is_basis unfolding topological_basis_def by blast

   162

   163 lemma open_countable_basisE:

   164   assumes "open X"

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

   166   using assms open_countable_basis_ex by (atomize_elim) simp

   167

   168 lemma countable_dense_exists:

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

   170 proof -

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

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

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

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

   175 qed

   176

   177 lemma countable_dense_setE:

   178   obtains D :: "'a set"

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

   180   using countable_dense_exists by blast

   181

   182 end

   183

   184 lemma (in first_countable_topology) first_countable_basisE:

   185   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

   186     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"

   187   using first_countable_basis[of x]

   188   apply atomize_elim

   189   apply (elim exE)

   190   apply (rule_tac x="range A" in exI)

   191   apply auto

   192   done

   193

   194 lemma (in first_countable_topology) first_countable_basis_Int_stableE:

   195   obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

   196     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"

   197     "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"

   198 proof atomize_elim

   199   from first_countable_basisE[of x] guess A' . note A' = this

   200   def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n)  N))  (Collect finite::nat set set)"

   201   thus "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>

   202         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"

   203   proof (safe intro!: exI[where x=A])

   204     show "countable A" unfolding A_def by (intro countable_image countable_Collect_finite)

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

   206     thus "x \<in> a" "open a" using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)

   207   next

   208     let ?int = "\<lambda>N. \<Inter>from_nat_into A'  N"

   209     fix a b assume "a \<in> A" "b \<in> A"

   210     then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)" by (auto simp: A_def)

   211     thus "a \<inter> b \<in> A" by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])

   212   next

   213     fix S assume "open S" "x \<in> S" then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast

   214     thus "\<exists>a\<in>A. a \<subseteq> S" using a A'

   215       by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])

   216   qed

   217 qed

   218

   219 lemma (in topological_space) first_countableI:

   220   assumes "countable A" and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

   221    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"

   222   shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

   223 proof (safe intro!: exI[of _ "from_nat_into A"])

   224   have "A \<noteq> {}" using 2[of UNIV] by auto

   225   { fix i show "x \<in> from_nat_into A i" "open (from_nat_into A i)"

   226       using range_from_nat_into_subset[OF A \<noteq> {}] 1 by auto }

   227   { fix S assume "open S" "x\<in>S" from 2[OF this] show "\<exists>i. from_nat_into A i \<subseteq> S"

   228       using subset_range_from_nat_into[OF countable A] by auto }

   229 qed

   230

   231 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

   232 proof

   233   fix x :: "'a \<times> 'b"

   234   from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this

   235   from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this

   236   show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

   237   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b)  (A \<times> B)"], safe)

   238     fix a b assume x: "a \<in> A" "b \<in> B"

   239     with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"

   240       unfolding mem_Times_iff by (auto intro: open_Times)

   241   next

   242     fix S assume "open S" "x \<in> S"

   243     from open_prod_elim[OF this] guess a' b' .

   244     moreover with A(4)[of a'] B(4)[of b']

   245     obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto

   246     ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b)  (A \<times> B). a \<subseteq> S"

   247       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])

   248   qed (simp add: A B)

   249 qed

   250

   251 class second_countable_topology = topological_space +

   252   assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"

   253 begin

   254

   255 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"

   256 proof -

   257   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast

   258   let ?B = "Inter  {b. finite b \<and> b \<subseteq> B }"

   259

   260   show ?thesis

   261   proof (intro exI conjI)

   262     show "countable ?B"

   263       by (intro countable_image countable_Collect_finite_subset B)

   264     { fix S assume "open S"

   265       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"

   266         unfolding B

   267       proof induct

   268         case UNIV show ?case by (intro exI[of _ "{{}}"]) simp

   269       next

   270         case (Int a b)

   271         then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"

   272           and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"

   273           by blast

   274         show ?case

   275           unfolding x y Int_UN_distrib2

   276           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))

   277       next

   278         case (UN K)

   279         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto

   280         then guess k unfolding bchoice_iff ..

   281         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"

   282           by (intro exI[of _ "UNION K k"]) auto

   283       next

   284         case (Basis S) then show ?case

   285           by (intro exI[of _ "{{S}}"]) auto

   286       qed

   287       then have "(\<exists>B'\<subseteq>Inter  {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"

   288         unfolding subset_image_iff by blast }

   289     then show "topological_basis ?B"

   290       unfolding topological_space_class.topological_basis_def

   291       by (safe intro!: topological_space_class.open_Inter)

   292          (simp_all add: B generate_topology.Basis subset_eq)

   293   qed

   294 qed

   295

   296 end

   297

   298 sublocale second_countable_topology <

   299   countable_basis "SOME B. countable B \<and> topological_basis B"

   300   using someI_ex[OF ex_countable_basis]

   301   by unfold_locales safe

   302

   303 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   304 proof

   305   obtain A :: "'a set set" where "countable A" "topological_basis A"

   306     using ex_countable_basis by auto

   307   moreover

   308   obtain B :: "'b set set" where "countable B" "topological_basis B"

   309     using ex_countable_basis by auto

   310   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"

   311     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b)  (A \<times> B)"] topological_basis_prod

   312       topological_basis_imp_subbasis)

   313 qed

   314

   315 instance second_countable_topology \<subseteq> first_countable_topology

   316 proof

   317   fix x :: 'a

   318   def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"

   319   then have B: "countable B" "topological_basis B"

   320     using countable_basis is_basis

   321     by (auto simp: countable_basis is_basis)

   322   then show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"

   323     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])

   324        (fastforce simp: topological_space_class.topological_basis_def)+

   325 qed

   326

   327 subsection {* Polish spaces *}

   328

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

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

   331

   332 class polish_space = complete_space + second_countable_topology

   333

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

   335

   336 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))"

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

   338   morphisms "openin" "topology"

   339   unfolding istopology_def by blast

   340

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

   342   using openin[of U] by blast

   343

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

   345   using topology_inverse[unfolded mem_Collect_eq] .

   346

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

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

   349

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

   351 proof-

   352   { assume "T1=T2"

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

   354   moreover

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

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

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

   358     hence "T1 = T2" unfolding openin_inverse .

   359   }

   360   ultimately show ?thesis by blast

   361 qed

   362

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

   364

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

   366

   367 subsubsection {* Main properties of open sets *}

   368

   369 lemma openin_clauses:

   370   fixes U :: "'a topology"

   371   shows "openin U {}"

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

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

   374   using openin[of U] unfolding istopology_def mem_Collect_eq

   375   by fast+

   376

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

   378   unfolding topspace_def by blast

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

   380

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

   382   using openin_clauses by simp

   383

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

   385   using openin_clauses by simp

   386

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

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

   389

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

   391

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

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

   394 proof

   395   assume ?lhs

   396   then show ?rhs by auto

   397 next

   398   assume H: ?rhs

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

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

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

   402   finally show "openin U S" .

   403 qed

   404

   405

   406 subsubsection {* Closed sets *}

   407

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

   409

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

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

   412 lemma closedin_topspace[intro,simp]:

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

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

   415   by (auto simp add: Diff_Un closedin_def)

   416

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

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

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

   420

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

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

   423

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

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

   426   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   427   apply (metis openin_subset subset_eq)

   428   done

   429

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

   431   by (simp add: openin_closedin_eq)

   432

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

   434 proof-

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

   436     by (auto simp add: topspace_def openin_subset)

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

   438 qed

   439

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

   441 proof-

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

   443     by (auto simp add: topspace_def )

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

   445 qed

   446

   447 subsubsection {* Subspace topology *}

   448

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

   450

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

   452   (is "istopology ?L")

   453 proof-

   454   have "?L {}" by blast

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

   456     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

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

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

   459   moreover

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

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

   462       apply (rule set_eqI)

   463       apply (simp add: Ball_def image_iff)

   464       by metis

   465     from K[unfolded th0 subset_image_iff]

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

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

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

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

   470   ultimately show ?thesis

   471     unfolding subset_eq mem_Collect_eq istopology_def by blast

   472 qed

   473

   474 lemma openin_subtopology:

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

   476   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   477   by auto

   478

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

   480   by (auto simp add: topspace_def openin_subtopology)

   481

   482 lemma closedin_subtopology:

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

   484   unfolding closedin_def topspace_subtopology

   485   apply (simp add: openin_subtopology)

   486   apply (rule iffI)

   487   apply clarify

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

   489   by auto

   490

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

   492   unfolding openin_subtopology

   493   apply (rule iffI, clarify)

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

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

   496   apply auto

   497   done

   498

   499 lemma subtopology_superset:

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

   501   shows "subtopology U V = U"

   502 proof-

   503   {fix S

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

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

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

   507     moreover

   508     {assume S: "openin U S"

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

   510         using openin_subset[OF S] UV by auto}

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

   512   then show ?thesis unfolding topology_eq openin_subtopology by blast

   513 qed

   514

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

   516   by (simp add: subtopology_superset)

   517

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

   519   by (simp add: subtopology_superset)

   520

   521 subsubsection {* The standard Euclidean topology *}

   522

   523 definition

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

   525   "euclidean = topology open"

   526

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

   528   unfolding euclidean_def

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

   530   apply (rule topology_inverse[symmetric])

   531   apply (auto simp add: istopology_def)

   532   done

   533

   534 lemma topspace_euclidean: "topspace euclidean = UNIV"

   535   apply (simp add: topspace_def)

   536   apply (rule set_eqI)

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

   538

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

   540   by (simp add: topspace_euclidean topspace_subtopology)

   541

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

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

   544

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

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

   547

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

   549

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

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

   552

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

   554   by (auto simp add: openin_open)

   555

   556 lemma open_openin_trans[trans]:

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

   558   by (metis Int_absorb1  openin_open_Int)

   559

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

   561   by (auto simp add: openin_open)

   562

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

   564   by (simp add: closedin_subtopology closed_closedin Int_ac)

   565

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

   567   by (metis closedin_closed)

   568

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

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

   571   apply auto

   572   apply (frule closedin_closed_Int[of T S])

   573   by simp

   574

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

   576   by (auto simp add: closedin_closed)

   577

   578 lemma openin_euclidean_subtopology_iff:

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

   580   shows "openin (subtopology euclidean U) S

   581   \<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")

   582 proof

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

   584 next

   585   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}"

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

   587     unfolding T_def

   588     apply clarsimp

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

   590     apply (clarsimp simp add: less_diff_eq)

   591     apply (erule rev_bexI)

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

   593     apply (erule le_less_trans [OF dist_triangle])

   594     done

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

   596     unfolding T_def

   597     apply auto

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

   599     apply auto

   600     done

   601   from 1 2 show ?lhs

   602     unfolding openin_open open_dist by fast

   603 qed

   604

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

   606

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

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

   609   unfolding open_openin openin_open by blast

   610

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

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

   613

   614 lemma closedin_trans[trans]:

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

   616            closedin (subtopology euclidean U) T

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

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

   619

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

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

   622

   623

   624 subsection {* Open and closed balls *}

   625

   626 definition

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

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

   629

   630 definition

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

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

   633

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

   635   by (simp add: ball_def)

   636

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

   638   by (simp add: cball_def)

   639

   640 lemma mem_ball_0:

   641   fixes x :: "'a::real_normed_vector"

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

   643   by (simp add: dist_norm)

   644

   645 lemma mem_cball_0:

   646   fixes x :: "'a::real_normed_vector"

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

   648   by (simp add: dist_norm)

   649

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

   651   by simp

   652

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

   654   by simp

   655

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

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

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

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

   660   by (simp add: set_eq_iff) arith

   661

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

   663   by (simp add: set_eq_iff)

   664

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

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

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

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

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

   670

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

   672   unfolding open_dist ball_def mem_Collect_eq Ball_def

   673   unfolding dist_commute

   674   apply clarify

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

   676   using dist_triangle_alt[where z=x]

   677   apply (clarsimp simp add: diff_less_iff)

   678   apply atomize

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

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

   681   by arith

   682

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

   684   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   685

   686 lemma openE[elim?]:

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

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

   689   using assms unfolding open_contains_ball by auto

   690

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

   692   by (metis open_contains_ball subset_eq centre_in_ball)

   693

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

   695   unfolding mem_ball set_eq_iff

   696   apply (simp add: not_less)

   697   by (metis zero_le_dist order_trans dist_self)

   698

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

   700

   701 lemma euclidean_dist_l2:

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

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

   704   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   706

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

   708

   709 lemma rational_boxes:

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

   711   assumes "0 < e"

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

   713 proof -

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

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

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

   717   proof

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

   719   qed

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

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

   722   proof

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

   724   qed

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

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

   727   show ?thesis

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

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

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

   731       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   733     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

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

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

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

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

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

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

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

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

   742         by (rule power_strict_mono) auto

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

   744         by (simp add: power_divide)

   745     qed auto

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

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

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

   749 qed

   750

   751 lemma open_UNION_box:

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

   753   assumes "open M"

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

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

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

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

   758 proof safe

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

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

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

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

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

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

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

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

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

   768 qed (auto simp: I_def)

   769

   770 subsection{* Connectedness *}

   771

   772 lemma connected_local:

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

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

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

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

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

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

   779                  ~(e2 = {}))"

   780 unfolding connected_def openin_open by (safe, blast+)

   781

   782 lemma exists_diff:

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

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

   785 proof-

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

   787   moreover

   788   {fix S assume H: "P S"

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

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

   791   ultimately show ?thesis by metis

   792 qed

   793

   794 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   797 proof-

   798   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> {})"

   799     unfolding connected_def openin_open closedin_closed

   800     apply (subst exists_diff) by blast

   801   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> {})"

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

   803

   804   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'))"

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

   806     unfolding connected_def openin_open closedin_closed by auto

   807   {fix e2

   808     {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)"

   809         by auto}

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

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

   812   then show ?thesis unfolding th0 th1 by simp

   813 qed

   814

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

   816   by (simp add: connected_def)

   817

   818

   819 subsection{* Limit points *}

   820

   821 definition (in topological_space)

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

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

   824

   825 lemma islimptI:

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

   827   shows "x islimpt S"

   828   using assms unfolding islimpt_def by auto

   829

   830 lemma islimptE:

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

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

   833   using assms unfolding islimpt_def by auto

   834

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

   836   unfolding islimpt_def eventually_at_topological by auto

   837

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

   839   unfolding islimpt_def by fast

   840

   841 lemma islimpt_approachable:

   842   fixes x :: "'a::metric_space"

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

   844   unfolding islimpt_iff_eventually eventually_at by fast

   845

   846 lemma islimpt_approachable_le:

   847   fixes x :: "'a::metric_space"

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

   849   unfolding islimpt_approachable

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

   851     THEN arg_cong [where f=Not]]

   852   by (simp add: Bex_def conj_commute conj_left_commute)

   853

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

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

   856

   857 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

   858   unfolding islimpt_def by blast

   859

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

   861

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

   863   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   864

   865 lemma perfect_choose_dist:

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

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

   868 using islimpt_UNIV [of x]

   869 by (simp add: islimpt_approachable)

   870

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

   872   unfolding closed_def

   873   apply (subst open_subopen)

   874   apply (simp add: islimpt_def subset_eq)

   875   by (metis ComplE ComplI)

   876

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

   878   unfolding islimpt_def by auto

   879

   880 lemma finite_set_avoid:

   881   fixes a :: "'a::metric_space"

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

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

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

   885 next

   886   case (2 x F)

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

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

   889   moreover

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

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

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

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

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

   895   ultimately show ?case by blast

   896 qed

   897

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

   899   by (simp add: islimpt_iff_eventually eventually_conj_iff)

   900

   901 lemma discrete_imp_closed:

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

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

   904   shows "closed S"

   905 proof-

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

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

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

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

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

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

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

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

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

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

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

   917 qed

   918

   919

   920 subsection {* Interior of a Set *}

   921

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

   923

   924 lemma interiorI [intro?]:

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

   926   shows "x \<in> interior S"

   927   using assms unfolding interior_def by fast

   928

   929 lemma interiorE [elim?]:

   930   assumes "x \<in> interior S"

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

   932   using assms unfolding interior_def by fast

   933

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

   935   by (simp add: interior_def open_Union)

   936

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

   938   by (auto simp add: interior_def)

   939

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

   941   by (auto simp add: interior_def)

   942

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

   944   by (intro equalityI interior_subset interior_maximal subset_refl)

   945

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

   947   by (metis open_interior interior_open)

   948

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

   950   by (metis interior_maximal interior_subset subset_trans)

   951

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

   953   using open_empty by (rule interior_open)

   954

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

   956   using open_UNIV by (rule interior_open)

   957

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

   959   using open_interior by (rule interior_open)

   960

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

   962   by (auto simp add: interior_def)

   963

   964 lemma interior_unique:

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

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

   967   shows "interior S = T"

   968   by (intro equalityI assms interior_subset open_interior interior_maximal)

   969

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

   971   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

   972     Int_lower2 interior_maximal interior_subset open_Int open_interior)

   973

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

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

   976   by (simp add: open_subset_interior)

   977

   978 lemma interior_limit_point [intro]:

   979   fixes x :: "'a::perfect_space"

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

   981   using x islimpt_UNIV [of x]

   982   unfolding interior_def islimpt_def

   983   apply (clarsimp, rename_tac T T')

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

   985   apply (auto simp add: open_Int)

   986   done

   987

   988 lemma interior_closed_Un_empty_interior:

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

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

   991 proof

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

   993     by (rule interior_mono, rule Un_upper1)

   994 next

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

   996   proof

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

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

   999     show "x \<in> interior S"

  1000     proof (rule ccontr)

  1001       assume "x \<notin> interior S"

  1002       with x \<in> R open R obtain y where "y \<in> R - S"

  1003         unfolding interior_def by fast

  1004       from open R closed S have "open (R - S)" by (rule open_Diff)

  1005       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

  1006       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

  1007       show "False" unfolding interior_def by fast

  1008     qed

  1009   qed

  1010 qed

  1011

  1012 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1013 proof (rule interior_unique)

  1014   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1015     by (intro Sigma_mono interior_subset)

  1016   show "open (interior A \<times> interior B)"

  1017     by (intro open_Times open_interior)

  1018   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

  1019   proof (safe)

  1020     fix x y assume "(x, y) \<in> T"

  1021     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1022       using open T unfolding open_prod_def by fast

  1023     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1024       using T \<subseteq> A \<times> B by auto

  1025     thus "x \<in> interior A" and "y \<in> interior B"

  1026       by (auto intro: interiorI)

  1027   qed

  1028 qed

  1029

  1030

  1031 subsection {* Closure of a Set *}

  1032

  1033 definition "closure S = S \<union> {x | x. x islimpt S}"

  1034

  1035 lemma interior_closure: "interior S = - (closure (- S))"

  1036   unfolding interior_def closure_def islimpt_def by auto

  1037

  1038 lemma closure_interior: "closure S = - interior (- S)"

  1039   unfolding interior_closure by simp

  1040

  1041 lemma closed_closure[simp, intro]: "closed (closure S)"

  1042   unfolding closure_interior by (simp add: closed_Compl)

  1043

  1044 lemma closure_subset: "S \<subseteq> closure S"

  1045   unfolding closure_def by simp

  1046

  1047 lemma closure_hull: "closure S = closed hull S"

  1048   unfolding hull_def closure_interior interior_def by auto

  1049

  1050 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1051   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1052

  1053 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1054   unfolding closure_eq .

  1055

  1056 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1057   unfolding closure_hull by (rule hull_hull)

  1058

  1059 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1060   unfolding closure_hull by (rule hull_mono)

  1061

  1062 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1063   unfolding closure_hull by (rule hull_minimal)

  1064

  1065 lemma closure_unique:

  1066   assumes "S \<subseteq> T" and "closed T"

  1067   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1068   shows "closure S = T"

  1069   using assms unfolding closure_hull by (rule hull_unique)

  1070

  1071 lemma closure_empty [simp]: "closure {} = {}"

  1072   using closed_empty by (rule closure_closed)

  1073

  1074 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1075   using closed_UNIV by (rule closure_closed)

  1076

  1077 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1078   unfolding closure_interior by simp

  1079

  1080 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1081   using closure_empty closure_subset[of S]

  1082   by blast

  1083

  1084 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1085   using closure_eq[of S] closure_subset[of S]

  1086   by simp

  1087

  1088 lemma open_inter_closure_eq_empty:

  1089   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1090   using open_subset_interior[of S "- T"]

  1091   using interior_subset[of "- T"]

  1092   unfolding closure_interior

  1093   by auto

  1094

  1095 lemma open_inter_closure_subset:

  1096   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1097 proof

  1098   fix x

  1099   assume as: "open S" "x \<in> S \<inter> closure T"

  1100   { assume *:"x islimpt T"

  1101     have "x islimpt (S \<inter> T)"

  1102     proof (rule islimptI)

  1103       fix A

  1104       assume "x \<in> A" "open A"

  1105       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1106         by (simp_all add: open_Int)

  1107       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1108         by (rule islimptE)

  1109       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1110         by simp_all

  1111       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1112     qed

  1113   }

  1114   then show "x \<in> closure (S \<inter> T)" using as

  1115     unfolding closure_def

  1116     by blast

  1117 qed

  1118

  1119 lemma closure_complement: "closure (- S) = - interior S"

  1120   unfolding closure_interior by simp

  1121

  1122 lemma interior_complement: "interior (- S) = - closure S"

  1123   unfolding closure_interior by simp

  1124

  1125 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1126 proof (rule closure_unique)

  1127   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1128     by (intro Sigma_mono closure_subset)

  1129   show "closed (closure A \<times> closure B)"

  1130     by (intro closed_Times closed_closure)

  1131   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

  1132     apply (simp add: closed_def open_prod_def, clarify)

  1133     apply (rule ccontr)

  1134     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1135     apply (simp add: closure_interior interior_def)

  1136     apply (drule_tac x=C in spec)

  1137     apply (drule_tac x=D in spec)

  1138     apply auto

  1139     done

  1140 qed

  1141

  1142

  1143 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1144   unfolding closure_def using islimpt_punctured by blast

  1145

  1146

  1147 subsection {* Frontier (aka boundary) *}

  1148

  1149 definition "frontier S = closure S - interior S"

  1150

  1151 lemma frontier_closed: "closed(frontier S)"

  1152   by (simp add: frontier_def closed_Diff)

  1153

  1154 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1155   by (auto simp add: frontier_def interior_closure)

  1156

  1157 lemma frontier_straddle:

  1158   fixes a :: "'a::metric_space"

  1159   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))"

  1160   unfolding frontier_def closure_interior

  1161   by (auto simp add: mem_interior subset_eq ball_def)

  1162

  1163 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1164   by (metis frontier_def closure_closed Diff_subset)

  1165

  1166 lemma frontier_empty[simp]: "frontier {} = {}"

  1167   by (simp add: frontier_def)

  1168

  1169 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1170 proof-

  1171   { assume "frontier S \<subseteq> S"

  1172     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

  1173     hence "closed S" using closure_subset_eq by auto

  1174   }

  1175   thus ?thesis using frontier_subset_closed[of S] ..

  1176 qed

  1177

  1178 lemma frontier_complement: "frontier(- S) = frontier S"

  1179   by (auto simp add: frontier_def closure_complement interior_complement)

  1180

  1181 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1182   using frontier_complement frontier_subset_eq[of "- S"]

  1183   unfolding open_closed by auto

  1184

  1185 subsection {* Filters and the eventually true'' quantifier *}

  1186

  1187 definition

  1188   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1189     (infixr "indirection" 70) where

  1190   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1191

  1192 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1193

  1194 lemma trivial_limit_within:

  1195   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1196 proof

  1197   assume "trivial_limit (at a within S)"

  1198   thus "\<not> a islimpt S"

  1199     unfolding trivial_limit_def

  1200     unfolding eventually_within eventually_at_topological

  1201     unfolding islimpt_def

  1202     apply (clarsimp simp add: set_eq_iff)

  1203     apply (rename_tac T, rule_tac x=T in exI)

  1204     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1205     done

  1206 next

  1207   assume "\<not> a islimpt S"

  1208   thus "trivial_limit (at a within S)"

  1209     unfolding trivial_limit_def

  1210     unfolding eventually_within eventually_at_topological

  1211     unfolding islimpt_def

  1212     apply clarsimp

  1213     apply (rule_tac x=T in exI)

  1214     apply auto

  1215     done

  1216 qed

  1217

  1218 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1219   using trivial_limit_within [of a UNIV] by simp

  1220

  1221 lemma trivial_limit_at:

  1222   fixes a :: "'a::perfect_space"

  1223   shows "\<not> trivial_limit (at a)"

  1224   by (rule at_neq_bot)

  1225

  1226 lemma trivial_limit_at_infinity:

  1227   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1228   unfolding trivial_limit_def eventually_at_infinity

  1229   apply clarsimp

  1230   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1231    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1232   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1233   apply (drule_tac x=UNIV in spec, simp)

  1234   done

  1235

  1236 lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"

  1237   using islimpt_in_closure by (metis trivial_limit_within)

  1238

  1239 text {* Some property holds "sufficiently close" to the limit point. *}

  1240

  1241 lemma eventually_at: (* FIXME: this replaces Metric_Spaces.eventually_at *)

  1242   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1243 unfolding eventually_at dist_nz by auto

  1244

  1245 lemma eventually_within: (* FIXME: this replaces Topological_Spaces.eventually_within *)

  1246   "eventually P (at a within S) \<longleftrightarrow>

  1247         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1248   by (rule eventually_within_less)

  1249

  1250 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

  1251   unfolding trivial_limit_def

  1252   by (auto elim: eventually_rev_mp)

  1253

  1254 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1255   by simp

  1256

  1257 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1258   by (simp add: filter_eq_iff)

  1259

  1260 text{* Combining theorems for "eventually" *}

  1261

  1262 lemma eventually_rev_mono:

  1263   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1264 using eventually_mono [of P Q] by fast

  1265

  1266 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

  1267   by (simp add: eventually_False)

  1268

  1269

  1270 subsection {* Limits *}

  1271

  1272 lemma Lim:

  1273  "(f ---> l) net \<longleftrightarrow>

  1274         trivial_limit net \<or>

  1275         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1276   unfolding tendsto_iff trivial_limit_eq by auto

  1277

  1278 text{* Show that they yield usual definitions in the various cases. *}

  1279

  1280 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1281            (\<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)"

  1282   by (auto simp add: tendsto_iff eventually_within_le)

  1283

  1284 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1285         (\<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)"

  1286   by (auto simp add: tendsto_iff eventually_within)

  1287

  1288 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1289         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1290   by (auto simp add: tendsto_iff eventually_at)

  1291

  1292 lemma Lim_at_infinity:

  1293   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1294   by (auto simp add: tendsto_iff eventually_at_infinity)

  1295

  1296 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1297   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1298

  1299 text{* The expected monotonicity property. *}

  1300

  1301 lemma Lim_within_empty: "(f ---> l) (net within {})"

  1302   unfolding tendsto_def Limits.eventually_within by simp

  1303

  1304 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"

  1305   unfolding tendsto_def Topological_Spaces.eventually_within

  1306   by (auto elim!: eventually_elim1)

  1307

  1308 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"

  1309   shows "(f ---> l) (net within (S \<union> T))"

  1310   using assms unfolding tendsto_def Limits.eventually_within

  1311   apply clarify

  1312   apply (drule spec, drule (1) mp, drule (1) mp)

  1313   apply (drule spec, drule (1) mp, drule (1) mp)

  1314   apply (auto elim: eventually_elim2)

  1315   done

  1316

  1317 lemma Lim_Un_univ:

  1318  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV

  1319         ==> (f ---> l) net"

  1320   by (metis Lim_Un within_UNIV)

  1321

  1322 text{* Interrelations between restricted and unrestricted limits. *}

  1323

  1324 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"

  1325   (* FIXME: rename *)

  1326   unfolding tendsto_def Limits.eventually_within

  1327   apply (clarify, drule spec, drule (1) mp, drule (1) mp)

  1328   by (auto elim!: eventually_elim1)

  1329

  1330 lemma eventually_within_interior:

  1331   assumes "x \<in> interior S"

  1332   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1333 proof-

  1334   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1335   { assume "?lhs"

  1336     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1337       unfolding Limits.eventually_within eventually_at_topological

  1338       by auto

  1339     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1340       by auto

  1341     then have "?rhs"

  1342       unfolding eventually_at_topological by auto

  1343   } moreover

  1344   { assume "?rhs" hence "?lhs"

  1345       unfolding Limits.eventually_within

  1346       by (auto elim: eventually_elim1)

  1347   } ultimately

  1348   show "?thesis" ..

  1349 qed

  1350

  1351 lemma at_within_interior:

  1352   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1353   by (simp add: filter_eq_iff eventually_within_interior)

  1354

  1355 lemma at_within_open:

  1356   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"

  1357   by (simp only: at_within_interior interior_open)

  1358

  1359 lemma Lim_within_open:

  1360   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

  1361   assumes"a \<in> S" "open S"

  1362   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

  1363   using assms by (simp only: at_within_open)

  1364

  1365 lemma Lim_within_LIMSEQ:

  1366   fixes a :: "'a::metric_space"

  1367   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1368   shows "(X ---> L) (at a within T)"

  1369   using assms unfolding tendsto_def [where l=L]

  1370   by (simp add: sequentially_imp_eventually_within)

  1371

  1372 lemma Lim_right_bound:

  1373   fixes f :: "real \<Rightarrow> real"

  1374   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"

  1375   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1376   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1377 proof cases

  1378   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1379 next

  1380   assume [simp]: "{x<..} \<inter> I \<noteq> {}"

  1381   show ?thesis

  1382   proof (rule Lim_within_LIMSEQ, safe)

  1383     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"

  1384

  1385     show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"

  1386     proof (rule LIMSEQ_I, rule ccontr)

  1387       fix r :: real assume "0 < r"

  1388       with cInf_close[of "f  ({x<..} \<inter> I)" r]

  1389       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto

  1390       from x < y have "0 < y - x" by auto

  1391       from S(2)[THEN LIMSEQ_D, OF this]

  1392       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto

  1393

  1394       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"

  1395       moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1396         using S bnd by (intro cInf_lower[where z=K]) auto

  1397       ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1398         by (auto simp: not_less field_simps)

  1399       with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y

  1400       show False by auto

  1401     qed

  1402   qed

  1403 qed

  1404

  1405 text{* Another limit point characterization. *}

  1406

  1407 lemma islimpt_sequential:

  1408   fixes x :: "'a::first_countable_topology"

  1409   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1410     (is "?lhs = ?rhs")

  1411 proof

  1412   assume ?lhs

  1413   from countable_basis_at_decseq[of x] guess A . note A = this

  1414   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1415   { fix n

  1416     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1417       unfolding islimpt_def using A(1,2)[of n] by auto

  1418     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1419       unfolding f_def by (rule someI_ex)

  1420     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }

  1421   then have "\<forall>n. f n \<in> S - {x}" by auto

  1422   moreover have "(\<lambda>n. f n) ----> x"

  1423   proof (rule topological_tendstoI)

  1424     fix S assume "open S" "x \<in> S"

  1425     from A(3)[OF this] \<And>n. f n \<in> A n

  1426     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)

  1427   qed

  1428   ultimately show ?rhs by fast

  1429 next

  1430   assume ?rhs

  1431   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto

  1432   show ?lhs

  1433     unfolding islimpt_def

  1434   proof safe

  1435     fix T assume "open T" "x \<in> T"

  1436     from lim[THEN topological_tendstoD, OF this] f

  1437     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1438       unfolding eventually_sequentially by auto

  1439   qed

  1440 qed

  1441

  1442 lemma Lim_inv: (* TODO: delete *)

  1443   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1444   assumes "(f ---> l) A" and "l \<noteq> 0"

  1445   shows "((inverse o f) ---> inverse l) A"

  1446   unfolding o_def using assms by (rule tendsto_inverse)

  1447

  1448 lemma Lim_null:

  1449   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1450   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1451   by (simp add: Lim dist_norm)

  1452

  1453 lemma Lim_null_comparison:

  1454   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1455   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1456   shows "(f ---> 0) net"

  1457 proof (rule metric_tendsto_imp_tendsto)

  1458   show "(g ---> 0) net" by fact

  1459   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1460     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1461 qed

  1462

  1463 lemma Lim_transform_bound:

  1464   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1465   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1466   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1467   shows "(f ---> 0) net"

  1468   using assms(1) tendsto_norm_zero [OF assms(2)]

  1469   by (rule Lim_null_comparison)

  1470

  1471 text{* Deducing things about the limit from the elements. *}

  1472

  1473 lemma Lim_in_closed_set:

  1474   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1475   shows "l \<in> S"

  1476 proof (rule ccontr)

  1477   assume "l \<notin> S"

  1478   with closed S have "open (- S)" "l \<in> - S"

  1479     by (simp_all add: open_Compl)

  1480   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1481     by (rule topological_tendstoD)

  1482   with assms(2) have "eventually (\<lambda>x. False) net"

  1483     by (rule eventually_elim2) simp

  1484   with assms(3) show "False"

  1485     by (simp add: eventually_False)

  1486 qed

  1487

  1488 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1489

  1490 lemma Lim_dist_ubound:

  1491   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1492   shows "dist a l <= e"

  1493 proof-

  1494   have "dist a l \<in> {..e}"

  1495   proof (rule Lim_in_closed_set)

  1496     show "closed {..e}" by simp

  1497     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1498     show "\<not> trivial_limit net" by fact

  1499     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1500   qed

  1501   thus ?thesis by simp

  1502 qed

  1503

  1504 lemma Lim_norm_ubound:

  1505   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1506   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1507   shows "norm(l) <= e"

  1508 proof-

  1509   have "norm l \<in> {..e}"

  1510   proof (rule Lim_in_closed_set)

  1511     show "closed {..e}" by simp

  1512     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1513     show "\<not> trivial_limit net" by fact

  1514     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1515   qed

  1516   thus ?thesis by simp

  1517 qed

  1518

  1519 lemma Lim_norm_lbound:

  1520   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1521   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1522   shows "e \<le> norm l"

  1523 proof-

  1524   have "norm l \<in> {e..}"

  1525   proof (rule Lim_in_closed_set)

  1526     show "closed {e..}" by simp

  1527     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1528     show "\<not> trivial_limit net" by fact

  1529     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1530   qed

  1531   thus ?thesis by simp

  1532 qed

  1533

  1534 text{* Limit under bilinear function *}

  1535

  1536 lemma Lim_bilinear:

  1537   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1538   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1539 using bounded_bilinear h (f ---> l) net (g ---> m) net

  1540 by (rule bounded_bilinear.tendsto)

  1541

  1542 text{* These are special for limits out of the same vector space. *}

  1543

  1544 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1545   unfolding id_def by (rule tendsto_ident_at_within)

  1546

  1547 lemma Lim_at_id: "(id ---> a) (at a)"

  1548   unfolding id_def by (rule tendsto_ident_at)

  1549

  1550 lemma Lim_at_zero:

  1551   fixes a :: "'a::real_normed_vector"

  1552   fixes l :: "'b::topological_space"

  1553   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1554   using LIM_offset_zero LIM_offset_zero_cancel ..

  1555

  1556 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1557

  1558 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where

  1559   "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  1560

  1561 lemma netlimit_within:

  1562   "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  1563   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  1564

  1565 lemma netlimit_at:

  1566   fixes a :: "'a::{perfect_space,t2_space}"

  1567   shows "netlimit (at a) = a"

  1568   using netlimit_within [of a UNIV] by simp

  1569

  1570 lemma lim_within_interior:

  1571   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1572   by (simp add: at_within_interior)

  1573

  1574 lemma netlimit_within_interior:

  1575   fixes x :: "'a::{t2_space,perfect_space}"

  1576   assumes "x \<in> interior S"

  1577   shows "netlimit (at x within S) = x"

  1578 using assms by (simp add: at_within_interior netlimit_at)

  1579

  1580 text{* Transformation of limit. *}

  1581

  1582 lemma Lim_transform:

  1583   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1584   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1585   shows "(g ---> l) net"

  1586   using tendsto_diff [OF assms(2) assms(1)] by simp

  1587

  1588 lemma Lim_transform_eventually:

  1589   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1590   apply (rule topological_tendstoI)

  1591   apply (drule (2) topological_tendstoD)

  1592   apply (erule (1) eventually_elim2, simp)

  1593   done

  1594

  1595 lemma Lim_transform_within:

  1596   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1597   and "(f ---> l) (at x within S)"

  1598   shows "(g ---> l) (at x within S)"

  1599 proof (rule Lim_transform_eventually)

  1600   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1601     unfolding eventually_within

  1602     using assms(1,2) by auto

  1603   show "(f ---> l) (at x within S)" by fact

  1604 qed

  1605

  1606 lemma Lim_transform_at:

  1607   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1608   and "(f ---> l) (at x)"

  1609   shows "(g ---> l) (at x)"

  1610 proof (rule Lim_transform_eventually)

  1611   show "eventually (\<lambda>x. f x = g x) (at x)"

  1612     unfolding eventually_at

  1613     using assms(1,2) by auto

  1614   show "(f ---> l) (at x)" by fact

  1615 qed

  1616

  1617 text{* Common case assuming being away from some crucial point like 0. *}

  1618

  1619 lemma Lim_transform_away_within:

  1620   fixes a b :: "'a::t1_space"

  1621   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1622   and "(f ---> l) (at a within S)"

  1623   shows "(g ---> l) (at a within S)"

  1624 proof (rule Lim_transform_eventually)

  1625   show "(f ---> l) (at a within S)" by fact

  1626   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1627     unfolding Limits.eventually_within eventually_at_topological

  1628     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1629 qed

  1630

  1631 lemma Lim_transform_away_at:

  1632   fixes a b :: "'a::t1_space"

  1633   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1634   and fl: "(f ---> l) (at a)"

  1635   shows "(g ---> l) (at a)"

  1636   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl

  1637   by simp

  1638

  1639 text{* Alternatively, within an open set. *}

  1640

  1641 lemma Lim_transform_within_open:

  1642   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1643   and "(f ---> l) (at a)"

  1644   shows "(g ---> l) (at a)"

  1645 proof (rule Lim_transform_eventually)

  1646   show "eventually (\<lambda>x. f x = g x) (at a)"

  1647     unfolding eventually_at_topological

  1648     using assms(1,2,3) by auto

  1649   show "(f ---> l) (at a)" by fact

  1650 qed

  1651

  1652 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1653

  1654 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1655

  1656 lemma Lim_cong_within(*[cong add]*):

  1657   assumes "a = b" "x = y" "S = T"

  1658   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1659   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1660   unfolding tendsto_def Limits.eventually_within eventually_at_topological

  1661   using assms by simp

  1662

  1663 lemma Lim_cong_at(*[cong add]*):

  1664   assumes "a = b" "x = y"

  1665   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1666   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1667   unfolding tendsto_def eventually_at_topological

  1668   using assms by simp

  1669

  1670 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1671

  1672 lemma closure_sequential:

  1673   fixes l :: "'a::first_countable_topology"

  1674   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1675 proof

  1676   assume "?lhs" moreover

  1677   { assume "l \<in> S"

  1678     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1679   } moreover

  1680   { assume "l islimpt S"

  1681     hence "?rhs" unfolding islimpt_sequential by auto

  1682   } ultimately

  1683   show "?rhs" unfolding closure_def by auto

  1684 next

  1685   assume "?rhs"

  1686   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

  1687 qed

  1688

  1689 lemma closed_sequential_limits:

  1690   fixes S :: "'a::first_countable_topology set"

  1691   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1692   unfolding closed_limpt

  1693   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]

  1694   by metis

  1695

  1696 lemma closure_approachable:

  1697   fixes S :: "'a::metric_space set"

  1698   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1699   apply (auto simp add: closure_def islimpt_approachable)

  1700   by (metis dist_self)

  1701

  1702 lemma closed_approachable:

  1703   fixes S :: "'a::metric_space set"

  1704   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1705   by (metis closure_closed closure_approachable)

  1706

  1707 lemma closure_contains_Inf:

  1708   fixes S :: "real set"

  1709   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1710   shows "Inf S \<in> closure S"

  1711   unfolding closure_approachable

  1712 proof safe

  1713   have *: "\<forall>x\<in>S. Inf S \<le> x"

  1714     using cInf_lower_EX[of _ S] assms by metis

  1715

  1716   fix e :: real assume "0 < e"

  1717   then obtain x where x: "x \<in> S" "x < Inf S + e"

  1718     using cInf_close S \<noteq> {} by auto

  1719   moreover then have "x > Inf S - e" using * by auto

  1720   ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)

  1721   then show "\<exists>x\<in>S. dist x (Inf S) < e"

  1722     using x by (auto simp: dist_norm)

  1723 qed

  1724

  1725 lemma closed_contains_Inf:

  1726   fixes S :: "real set"

  1727   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1728     and "closed S"

  1729   shows "Inf S \<in> S"

  1730   by (metis closure_contains_Inf closure_closed assms)

  1731

  1732

  1733 lemma not_trivial_limit_within_ball:

  1734   "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  1735   (is "?lhs = ?rhs")

  1736 proof -

  1737   { assume "?lhs"

  1738     { fix e :: real

  1739       assume "e>0"

  1740       then obtain y where "y:(S-{x}) & dist y x < e"

  1741         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1742         by auto

  1743       then have "y : (S Int ball x e - {x})"

  1744         unfolding ball_def by (simp add: dist_commute)

  1745       then have "S Int ball x e - {x} ~= {}" by blast

  1746     } then have "?rhs" by auto

  1747   }

  1748   moreover

  1749   { assume "?rhs"

  1750     { fix e :: real

  1751       assume "e>0"

  1752       then obtain y where "y : (S Int ball x e - {x})" using ?rhs by blast

  1753       then have "y:(S-{x}) & dist y x < e"

  1754         unfolding ball_def by (simp add: dist_commute)

  1755       then have "EX y:(S-{x}). dist y x < e" by auto

  1756     }

  1757     then have "?lhs"

  1758       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto

  1759   }

  1760   ultimately show ?thesis by auto

  1761 qed

  1762

  1763 subsection {* Infimum Distance *}

  1764

  1765 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1766

  1767 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1768   by (simp add: infdist_def)

  1769

  1770 lemma infdist_nonneg:

  1771   shows "0 \<le> infdist x A"

  1772   using assms by (auto simp add: infdist_def intro: cInf_greatest)

  1773

  1774 lemma infdist_le:

  1775   assumes "a \<in> A"

  1776   assumes "d = dist x a"

  1777   shows "infdist x A \<le> d"

  1778   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

  1779

  1780 lemma infdist_zero[simp]:

  1781   assumes "a \<in> A" shows "infdist a A = 0"

  1782 proof -

  1783   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  1784   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  1785 qed

  1786

  1787 lemma infdist_triangle:

  1788   shows "infdist x A \<le> infdist y A + dist x y"

  1789 proof cases

  1790   assume "A = {}" thus ?thesis by (simp add: infdist_def)

  1791 next

  1792   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto

  1793   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1794   proof (rule cInf_greatest)

  1795     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp

  1796     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1797     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto

  1798     show "infdist x A \<le> d"

  1799       unfolding infdist_notempty[OF A \<noteq> {}]

  1800     proof (rule cInf_lower2)

  1801       show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto

  1802       show "dist x a \<le> d" unfolding d by (rule dist_triangle)

  1803       fix d assume "d \<in> {dist x a |a. a \<in> A}"

  1804       then obtain a where "a \<in> A" "d = dist x a" by auto

  1805       thus "infdist x A \<le> d" by (rule infdist_le)

  1806     qed

  1807   qed

  1808   also have "\<dots> = dist x y + infdist y A"

  1809   proof (rule cInf_eq, safe)

  1810     fix a assume "a \<in> A"

  1811     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)

  1812   next

  1813     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1814     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  1815       by (intro cInf_greatest) (auto simp: field_simps)

  1816     thus "i \<le> dist x y + infdist y A" by simp

  1817   qed

  1818   finally show ?thesis by simp

  1819 qed

  1820

  1821 lemma in_closure_iff_infdist_zero:

  1822   assumes "A \<noteq> {}"

  1823   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1824 proof

  1825   assume "x \<in> closure A"

  1826   show "infdist x A = 0"

  1827   proof (rule ccontr)

  1828     assume "infdist x A \<noteq> 0"

  1829     with infdist_nonneg[of x A] have "infdist x A > 0" by auto

  1830     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto

  1831       by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  1832         eucl_less_not_refl euclidean_trans(2) infdist_le)

  1833     hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  1834     thus False using x \<in> closure A by simp

  1835   qed

  1836 next

  1837   assume x: "infdist x A = 0"

  1838   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)

  1839   show "x \<in> closure A" unfolding closure_approachable

  1840   proof (safe, rule ccontr)

  1841     fix e::real assume "0 < e"

  1842     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1843     hence "infdist x A \<ge> e" using a \<in> A

  1844       unfolding infdist_def

  1845       by (force simp: dist_commute intro: cInf_greatest)

  1846     with x 0 < e show False by auto

  1847   qed

  1848 qed

  1849

  1850 lemma in_closed_iff_infdist_zero:

  1851   assumes "closed A" "A \<noteq> {}"

  1852   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1853 proof -

  1854   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1855     by (rule in_closure_iff_infdist_zero) fact

  1856   with assms show ?thesis by simp

  1857 qed

  1858

  1859 lemma tendsto_infdist [tendsto_intros]:

  1860   assumes f: "(f ---> l) F"

  1861   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  1862 proof (rule tendstoI)

  1863   fix e ::real assume "0 < e"

  1864   from tendstoD[OF f this]

  1865   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1866   proof (eventually_elim)

  1867     fix x

  1868     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1869     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1870       by (simp add: dist_commute dist_real_def)

  1871     also assume "dist (f x) l < e"

  1872     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1873   qed

  1874 qed

  1875

  1876 text{* Some other lemmas about sequences. *}

  1877

  1878 lemma sequentially_offset:

  1879   assumes "eventually (\<lambda>i. P i) sequentially"

  1880   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1881   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1882

  1883 lemma seq_offset:

  1884   assumes "(f ---> l) sequentially"

  1885   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1886   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1887

  1888 lemma seq_offset_neg:

  1889   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1890   apply (rule topological_tendstoI)

  1891   apply (drule (2) topological_tendstoD)

  1892   apply (simp only: eventually_sequentially)

  1893   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1894   apply metis

  1895   by arith

  1896

  1897 lemma seq_offset_rev:

  1898   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1899   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1900

  1901 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1902   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1903

  1904 subsection {* More properties of closed balls *}

  1905

  1906 lemma closed_cball: "closed (cball x e)"

  1907 unfolding cball_def closed_def

  1908 unfolding Collect_neg_eq [symmetric] not_le

  1909 apply (clarsimp simp add: open_dist, rename_tac y)

  1910 apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1911 apply (rename_tac x')

  1912 apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1913 apply simp

  1914 done

  1915

  1916 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1917 proof-

  1918   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1919     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1920   } moreover

  1921   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1922     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto

  1923   } ultimately

  1924   show ?thesis unfolding open_contains_ball by auto

  1925 qed

  1926

  1927 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1928   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1929

  1930 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1931   apply (simp add: interior_def, safe)

  1932   apply (force simp add: open_contains_cball)

  1933   apply (rule_tac x="ball x e" in exI)

  1934   apply (simp add: subset_trans [OF ball_subset_cball])

  1935   done

  1936

  1937 lemma islimpt_ball:

  1938   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1939   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1940 proof

  1941   assume "?lhs"

  1942   { assume "e \<le> 0"

  1943     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1944     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1945   }

  1946   hence "e > 0" by (metis not_less)

  1947   moreover

  1948   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

  1949   ultimately show "?rhs" by auto

  1950 next

  1951   assume "?rhs" hence "e>0"  by auto

  1952   { fix d::real assume "d>0"

  1953     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1954     proof(cases "d \<le> dist x y")

  1955       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1956       proof(cases "x=y")

  1957         case True hence False using d \<le> dist x y d>0 by auto

  1958         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1959       next

  1960         case False

  1961

  1962         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1963               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1964           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1965         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1966           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1967           unfolding scaleR_minus_left scaleR_one

  1968           by (auto simp add: norm_minus_commute)

  1969         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1970           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1971           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1972         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1973         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1974

  1975         moreover

  1976

  1977         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  1978           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  1979         moreover

  1980         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  1981           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  1982           unfolding dist_norm by auto

  1983         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

  1984       qed

  1985     next

  1986       case False hence "d > dist x y" by auto

  1987       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1988       proof(cases "x=y")

  1989         case True

  1990         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  1991           using perfect_choose_dist[of "min e d" y]

  1992           using d > 0 e>0 by auto

  1993         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1994           unfolding x = y

  1995           using z \<noteq> y **

  1996           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  1997       next

  1998         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1999           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  2000       qed

  2001     qed  }

  2002   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  2003 qed

  2004

  2005 lemma closure_ball_lemma:

  2006   fixes x y :: "'a::real_normed_vector"

  2007   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  2008 proof (rule islimptI)

  2009   fix T assume "y \<in> T" "open T"

  2010   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2011     unfolding open_dist by fast

  2012   (* choose point between x and y, within distance r of y. *)

  2013   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2014   def z \<equiv> "y + scaleR k (x - y)"

  2015   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2016     unfolding z_def by (simp add: algebra_simps)

  2017   have "dist z y < r"

  2018     unfolding z_def k_def using 0 < r

  2019     by (simp add: dist_norm min_def)

  2020   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2021   have "dist x z < dist x y"

  2022     unfolding z_def2 dist_norm

  2023     apply (simp add: norm_minus_commute)

  2024     apply (simp only: dist_norm [symmetric])

  2025     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2026     apply (rule mult_strict_right_mono)

  2027     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2028     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2029     done

  2030   hence "z \<in> ball x (dist x y)" by simp

  2031   have "z \<noteq> y"

  2032     unfolding z_def k_def using x \<noteq> y 0 < r

  2033     by (simp add: min_def)

  2034   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2035     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2036     by fast

  2037 qed

  2038

  2039 lemma closure_ball:

  2040   fixes x :: "'a::real_normed_vector"

  2041   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2042 apply (rule equalityI)

  2043 apply (rule closure_minimal)

  2044 apply (rule ball_subset_cball)

  2045 apply (rule closed_cball)

  2046 apply (rule subsetI, rename_tac y)

  2047 apply (simp add: le_less [where 'a=real])

  2048 apply (erule disjE)

  2049 apply (rule subsetD [OF closure_subset], simp)

  2050 apply (simp add: closure_def)

  2051 apply clarify

  2052 apply (rule closure_ball_lemma)

  2053 apply (simp add: zero_less_dist_iff)

  2054 done

  2055

  2056 (* In a trivial vector space, this fails for e = 0. *)

  2057 lemma interior_cball:

  2058   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2059   shows "interior (cball x e) = ball x e"

  2060 proof(cases "e\<ge>0")

  2061   case False note cs = this

  2062   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  2063   { fix y assume "y \<in> cball x e"

  2064     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  2065   hence "cball x e = {}" by auto

  2066   hence "interior (cball x e) = {}" using interior_empty by auto

  2067   ultimately show ?thesis by blast

  2068 next

  2069   case True note cs = this

  2070   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  2071   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2072     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  2073

  2074     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2075       using perfect_choose_dist [of d] by auto

  2076     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  2077     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  2078

  2079     hence "y \<in> ball x e" proof(cases "x = y")

  2080       case True

  2081       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

  2082       thus "y \<in> ball x e" using x = y  by simp

  2083     next

  2084       case False

  2085       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  2086         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2087       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

  2088       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2089       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

  2090         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2091

  2092       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)"

  2093         by (auto simp add: dist_norm algebra_simps)

  2094       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2095         by (auto simp add: algebra_simps)

  2096       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2097         using ** by auto

  2098       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

  2099       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2100       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  2101     qed  }

  2102   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  2103   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  2104 qed

  2105

  2106 lemma frontier_ball:

  2107   fixes a :: "'a::real_normed_vector"

  2108   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  2109   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2110   apply (simp add: set_eq_iff)

  2111   by arith

  2112

  2113 lemma frontier_cball:

  2114   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2115   shows "frontier(cball a e) = {x. dist a x = e}"

  2116   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2117   apply (simp add: set_eq_iff)

  2118   by arith

  2119

  2120 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  2121   apply (simp add: set_eq_iff not_le)

  2122   by (metis zero_le_dist dist_self order_less_le_trans)

  2123 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

  2124

  2125 lemma cball_eq_sing:

  2126   fixes x :: "'a::{metric_space,perfect_space}"

  2127   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2128 proof (rule linorder_cases)

  2129   assume e: "0 < e"

  2130   obtain a where "a \<noteq> x" "dist a x < e"

  2131     using perfect_choose_dist [OF e] by auto

  2132   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  2133   with e show ?thesis by (auto simp add: set_eq_iff)

  2134 qed auto

  2135

  2136 lemma cball_sing:

  2137   fixes x :: "'a::metric_space"

  2138   shows "e = 0 ==> cball x e = {x}"

  2139   by (auto simp add: set_eq_iff)

  2140

  2141

  2142 subsection {* Boundedness *}

  2143

  2144   (* FIXME: This has to be unified with BSEQ!! *)

  2145 definition (in metric_space)

  2146   bounded :: "'a set \<Rightarrow> bool" where

  2147   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2148

  2149 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2150   unfolding bounded_def subset_eq by auto

  2151

  2152 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2153 unfolding bounded_def

  2154 apply safe

  2155 apply (rule_tac x="dist a x + e" in exI, clarify)

  2156 apply (drule (1) bspec)

  2157 apply (erule order_trans [OF dist_triangle add_left_mono])

  2158 apply auto

  2159 done

  2160

  2161 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2162 unfolding bounded_any_center [where a=0]

  2163 by (simp add: dist_norm)

  2164

  2165 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

  2166   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

  2167   using assms by auto

  2168

  2169 lemma bounded_empty [simp]: "bounded {}"

  2170   by (simp add: bounded_def)

  2171

  2172 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2173   by (metis bounded_def subset_eq)

  2174

  2175 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2176   by (metis bounded_subset interior_subset)

  2177

  2178 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

  2179 proof-

  2180   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

  2181   { fix y assume "y \<in> closure S"

  2182     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2183       unfolding closure_sequential by auto

  2184     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2185     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2186       by (rule eventually_mono, simp add: f(1))

  2187     have "dist x y \<le> a"

  2188       apply (rule Lim_dist_ubound [of sequentially f])

  2189       apply (rule trivial_limit_sequentially)

  2190       apply (rule f(2))

  2191       apply fact

  2192       done

  2193   }

  2194   thus ?thesis unfolding bounded_def by auto

  2195 qed

  2196

  2197 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2198   apply (simp add: bounded_def)

  2199   apply (rule_tac x=x in exI)

  2200   apply (rule_tac x=e in exI)

  2201   apply auto

  2202   done

  2203

  2204 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2205   by (metis ball_subset_cball bounded_cball bounded_subset)

  2206

  2207 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2208   apply (auto simp add: bounded_def)

  2209   apply (rename_tac x y r s)

  2210   apply (rule_tac x=x in exI)

  2211   apply (rule_tac x="max r (dist x y + s)" in exI)

  2212   apply (rule ballI, rename_tac z, safe)

  2213   apply (drule (1) bspec, simp)

  2214   apply (drule (1) bspec)

  2215   apply (rule min_max.le_supI2)

  2216   apply (erule order_trans [OF dist_triangle add_left_mono])

  2217   done

  2218

  2219 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2220   by (induct rule: finite_induct[of F], auto)

  2221

  2222 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2223   by (induct set: finite, auto)

  2224

  2225 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2226 proof -

  2227   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp

  2228   hence "bounded {x}" unfolding bounded_def by fast

  2229   thus ?thesis by (metis insert_is_Un bounded_Un)

  2230 qed

  2231

  2232 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2233   by (induct set: finite, simp_all)

  2234

  2235 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2236   apply (simp add: bounded_iff)

  2237   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2238   by metis arith

  2239

  2240 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"

  2241   unfolding Bseq_def bounded_pos by auto

  2242

  2243 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2244   by (metis Int_lower1 Int_lower2 bounded_subset)

  2245

  2246 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2247 apply (metis Diff_subset bounded_subset)

  2248 done

  2249

  2250 lemma not_bounded_UNIV[simp, intro]:

  2251   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2252 proof(auto simp add: bounded_pos not_le)

  2253   obtain x :: 'a where "x \<noteq> 0"

  2254     using perfect_choose_dist [OF zero_less_one] by fast

  2255   fix b::real  assume b: "b >0"

  2256   have b1: "b +1 \<ge> 0" using b by simp

  2257   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2258     by (simp add: norm_sgn)

  2259   then show "\<exists>x::'a. b < norm x" ..

  2260 qed

  2261

  2262 lemma bounded_linear_image:

  2263   assumes "bounded S" "bounded_linear f"

  2264   shows "bounded(f  S)"

  2265 proof-

  2266   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2267   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)

  2268   { fix x assume "x\<in>S"

  2269     hence "norm x \<le> b" using b by auto

  2270     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

  2271       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2272   }

  2273   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

  2274     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

  2275 qed

  2276

  2277 lemma bounded_scaling:

  2278   fixes S :: "'a::real_normed_vector set"

  2279   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2280   apply (rule bounded_linear_image, assumption)

  2281   apply (rule bounded_linear_scaleR_right)

  2282   done

  2283

  2284 lemma bounded_translation:

  2285   fixes S :: "'a::real_normed_vector set"

  2286   assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"

  2287 proof-

  2288   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2289   { fix x assume "x\<in>S"

  2290     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

  2291   }

  2292   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

  2293     by (auto intro!: exI[of _ "b + norm a"])

  2294 qed

  2295

  2296

  2297 text{* Some theorems on sups and infs using the notion "bounded". *}

  2298

  2299 lemma bounded_real:

  2300   fixes S :: "real set"

  2301   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2302   by (simp add: bounded_iff)

  2303

  2304 lemma bounded_has_Sup:

  2305   fixes S :: "real set"

  2306   assumes "bounded S" "S \<noteq> {}"

  2307   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  2308 proof

  2309   fix x assume "x\<in>S"

  2310   thus "x \<le> Sup S"

  2311     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

  2312 next

  2313   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  2314     by (metis cSup_least)

  2315 qed

  2316

  2317 lemma Sup_insert:

  2318   fixes S :: "real set"

  2319   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2320   apply (subst cSup_insert_If)

  2321   apply (rule bounded_has_Sup(1)[of S, rule_format])

  2322   apply (auto simp: sup_max)

  2323   done

  2324

  2325 lemma Sup_insert_finite:

  2326   fixes S :: "real set"

  2327   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2328   apply (rule Sup_insert)

  2329   apply (rule finite_imp_bounded)

  2330   by simp

  2331

  2332 lemma bounded_has_Inf:

  2333   fixes S :: "real set"

  2334   assumes "bounded S"  "S \<noteq> {}"

  2335   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  2336 proof

  2337   fix x assume "x\<in>S"

  2338   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

  2339   thus "x \<ge> Inf S" using x\<in>S

  2340     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

  2341 next

  2342   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  2343     by (metis cInf_greatest)

  2344 qed

  2345

  2346 lemma Inf_insert:

  2347   fixes S :: "real set"

  2348   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2349   apply (subst cInf_insert_if)

  2350   apply (rule bounded_has_Inf(1)[of S, rule_format])

  2351   apply (auto simp: inf_min)

  2352   done

  2353

  2354 lemma Inf_insert_finite:

  2355   fixes S :: "real set"

  2356   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2357   by (rule Inf_insert, rule finite_imp_bounded, simp)

  2358

  2359 subsection {* Compactness *}

  2360

  2361 subsubsection {* Bolzano-Weierstrass property *}

  2362

  2363 lemma heine_borel_imp_bolzano_weierstrass:

  2364   assumes "compact s" "infinite t"  "t \<subseteq> s"

  2365   shows "\<exists>x \<in> s. x islimpt t"

  2366 proof(rule ccontr)

  2367   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2368   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

  2369     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

  2370   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2371     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2372   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2373   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2374     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

  2375     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  }

  2376   hence "inj_on f t" unfolding inj_on_def by simp

  2377   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2378   moreover

  2379   { fix x assume "x\<in>t" "f x \<notin> g"

  2380     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2381     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2382     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2383     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2384   hence "f  t \<subseteq> g" by auto

  2385   ultimately show False using g(2) using finite_subset by auto

  2386 qed

  2387

  2388 lemma acc_point_range_imp_convergent_subsequence:

  2389   fixes l :: "'a :: first_countable_topology"

  2390   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2391   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2392 proof -

  2393   from countable_basis_at_decseq[of l] guess A . note A = this

  2394

  2395   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2396   { fix n i

  2397     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2398       using l A by auto

  2399     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2400       unfolding ex_in_conv by (intro notI) simp

  2401     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2402       by auto

  2403     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2404       by (auto simp: not_le)

  2405     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2406       unfolding s_def by (auto intro: someI2_ex) }

  2407   note s = this

  2408   def r \<equiv> "nat_rec (s 0 0) s"

  2409   have "subseq r"

  2410     by (auto simp: r_def s subseq_Suc_iff)

  2411   moreover

  2412   have "(\<lambda>n. f (r n)) ----> l"

  2413   proof (rule topological_tendstoI)

  2414     fix S assume "open S" "l \<in> S"

  2415     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2416     moreover

  2417     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"

  2418         by (cases i) (simp_all add: r_def s) }

  2419     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2420     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2421       by eventually_elim auto

  2422   qed

  2423   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2424     by (auto simp: convergent_def comp_def)

  2425 qed

  2426

  2427 lemma sequence_infinite_lemma:

  2428   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2429   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2430   shows "infinite (range f)"

  2431 proof

  2432   assume "finite (range f)"

  2433   hence "closed (range f)" by (rule finite_imp_closed)

  2434   hence "open (- range f)" by (rule open_Compl)

  2435   from assms(1) have "l \<in> - range f" by auto

  2436   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2437     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2438   thus False unfolding eventually_sequentially by auto

  2439 qed

  2440

  2441 lemma closure_insert:

  2442   fixes x :: "'a::t1_space"

  2443   shows "closure (insert x s) = insert x (closure s)"

  2444 apply (rule closure_unique)

  2445 apply (rule insert_mono [OF closure_subset])

  2446 apply (rule closed_insert [OF closed_closure])

  2447 apply (simp add: closure_minimal)

  2448 done

  2449

  2450 lemma islimpt_insert:

  2451   fixes x :: "'a::t1_space"

  2452   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2453 proof

  2454   assume *: "x islimpt (insert a s)"

  2455   show "x islimpt s"

  2456   proof (rule islimptI)

  2457     fix t assume t: "x \<in> t" "open t"

  2458     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2459     proof (cases "x = a")

  2460       case True

  2461       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2462         using * t by (rule islimptE)

  2463       with x = a show ?thesis by auto

  2464     next

  2465       case False

  2466       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2467         by (simp_all add: open_Diff)

  2468       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2469         using * t' by (rule islimptE)

  2470       thus ?thesis by auto

  2471     qed

  2472   qed

  2473 next

  2474   assume "x islimpt s" thus "x islimpt (insert a s)"

  2475     by (rule islimpt_subset) auto

  2476 qed

  2477

  2478 lemma islimpt_finite:

  2479   fixes x :: "'a::t1_space"

  2480   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2481 by (induct set: finite, simp_all add: islimpt_insert)

  2482

  2483 lemma islimpt_union_finite:

  2484   fixes x :: "'a::t1_space"

  2485   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2486 by (simp add: islimpt_Un islimpt_finite)

  2487

  2488 lemma islimpt_eq_acc_point:

  2489   fixes l :: "'a :: t1_space"

  2490   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2491 proof (safe intro!: islimptI)

  2492   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2493   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2494     by (auto intro: finite_imp_closed)

  2495   then show False

  2496     by (rule islimptE) auto

  2497 next

  2498   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2499   then have "infinite (T \<inter> S - {l})" by auto

  2500   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2501     unfolding ex_in_conv by (intro notI) simp

  2502   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2503     by auto

  2504 qed

  2505

  2506 lemma islimpt_range_imp_convergent_subsequence:

  2507   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2508   assumes l: "l islimpt (range f)"

  2509   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2510   using l unfolding islimpt_eq_acc_point

  2511   by (rule acc_point_range_imp_convergent_subsequence)

  2512

  2513 lemma sequence_unique_limpt:

  2514   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2515   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  2516   shows "l' = l"

  2517 proof (rule ccontr)

  2518   assume "l' \<noteq> l"

  2519   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2520     using hausdorff [OF l' \<noteq> l] by auto

  2521   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2522     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2523   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2524     unfolding eventually_sequentially by auto

  2525

  2526   have "UNIV = {..<N} \<union> {N..}" by auto

  2527   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  2528   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  2529   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  2530   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2531     using l' \<in> s open s by (rule islimptE)

  2532   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  2533   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  2534   with s \<inter> t = {} show False by simp

  2535 qed

  2536

  2537 lemma bolzano_weierstrass_imp_closed:

  2538   fixes s :: "'a::{first_countable_topology, t2_space} set"

  2539   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2540   shows "closed s"

  2541 proof-

  2542   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2543     hence "l \<in> s"

  2544     proof(cases "\<forall>n. x n \<noteq> l")

  2545       case False thus "l\<in>s" using as(1) by auto

  2546     next

  2547       case True note cas = this

  2548       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  2549       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  2550       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  2551     qed  }

  2552   thus ?thesis unfolding closed_sequential_limits by fast

  2553 qed

  2554

  2555 lemma compact_imp_closed:

  2556   fixes s :: "'a::t2_space set"

  2557   assumes "compact s" shows "closed s"

  2558 unfolding closed_def

  2559 proof (rule openI)

  2560   fix y assume "y \<in> - s"

  2561   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"

  2562   note compact s

  2563   moreover have "\<forall>u\<in>?C. open u" by simp

  2564   moreover have "s \<subseteq> \<Union>?C"

  2565   proof

  2566     fix x assume "x \<in> s"

  2567     with y \<in> - s have "x \<noteq> y" by clarsimp

  2568     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"

  2569       by (rule hausdorff)

  2570     with x \<in> s show "x \<in> \<Union>?C"

  2571       unfolding eventually_nhds by auto

  2572   qed

  2573   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"

  2574     by (rule compactE)

  2575   from D \<subseteq> ?C have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto

  2576   with finite D have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"

  2577     by (simp add: eventually_Ball_finite)

  2578   with s \<subseteq> \<Union>D have "eventually (\<lambda>y. y \<notin> s) (nhds y)"

  2579     by (auto elim!: eventually_mono [rotated])

  2580   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"

  2581     by (simp add: eventually_nhds subset_eq)

  2582 qed

  2583

  2584 lemma compact_imp_bounded:

  2585   assumes "compact U" shows "bounded U"

  2586 proof -

  2587   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto

  2588   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2589     by (elim compactE_image)

  2590   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2591     by (simp add: bounded_UN)

  2592   thus "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2593     by (rule bounded_subset)

  2594 qed

  2595

  2596 text{* In particular, some common special cases. *}

  2597

  2598 lemma compact_union [intro]:

  2599   assumes "compact s" "compact t" shows " compact (s \<union> t)"

  2600 proof (rule compactI)

  2601   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  2602   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  2603     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2604   moreover from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  2605     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2606   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  2607     by (auto intro!: exI[of _ "s' \<union> t'"])

  2608 qed

  2609

  2610 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  2611   by (induct set: finite) auto

  2612

  2613 lemma compact_UN [intro]:

  2614   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  2615   unfolding SUP_def by (rule compact_Union) auto

  2616

  2617 lemma compact_inter_closed [intro]:

  2618   assumes "compact s" and "closed t"

  2619   shows "compact (s \<inter> t)"

  2620 proof (rule compactI)

  2621   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"

  2622   from C closed t have "\<forall>c\<in>C \<union> {-t}. open c" by auto

  2623   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto

  2624   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"

  2625     using compact s unfolding compact_eq_heine_borel by auto

  2626   then guess D ..

  2627   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"

  2628     by (intro exI[of _ "D - {-t}"]) auto

  2629 qed

  2630

  2631 lemma closed_inter_compact [intro]:

  2632   assumes "closed s" and "compact t"

  2633   shows "compact (s \<inter> t)"

  2634   using compact_inter_closed [of t s] assms

  2635   by (simp add: Int_commute)

  2636

  2637 lemma compact_inter [intro]:

  2638   fixes s t :: "'a :: t2_space set"

  2639   assumes "compact s" and "compact t"

  2640   shows "compact (s \<inter> t)"

  2641   using assms by (intro compact_inter_closed compact_imp_closed)

  2642

  2643 lemma compact_sing [simp]: "compact {a}"

  2644   unfolding compact_eq_heine_borel by auto

  2645

  2646 lemma compact_insert [simp]:

  2647   assumes "compact s" shows "compact (insert x s)"

  2648 proof -

  2649   have "compact ({x} \<union> s)"

  2650     using compact_sing assms by (rule compact_union)

  2651   thus ?thesis by simp

  2652 qed

  2653

  2654 lemma finite_imp_compact:

  2655   shows "finite s \<Longrightarrow> compact s"

  2656   by (induct set: finite) simp_all

  2657

  2658 lemma open_delete:

  2659   fixes s :: "'a::t1_space set"

  2660   shows "open s \<Longrightarrow> open (s - {x})"

  2661   by (simp add: open_Diff)

  2662

  2663 text{* Finite intersection property *}

  2664

  2665 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  2666   by (auto simp: inj_on_def)

  2667

  2668 lemma compact_fip:

  2669   "compact U \<longleftrightarrow>

  2670     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"

  2671   (is "_ \<longleftrightarrow> ?R")

  2672 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  2673   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  2674     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  2675   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>uminusA"

  2676     by auto

  2677   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  2678     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  2679   with fi[THEN spec, of B] show False

  2680     by (auto dest: finite_imageD intro: inj_setminus)

  2681 next

  2682   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2683   from cover have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  2684     by auto

  2685   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>uminusB = {}"

  2686     by (metis subset_image_iff)

  2687   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2688     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  2689 qed

  2690

  2691 lemma compact_imp_fip:

  2692   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>

  2693     s \<inter> (\<Inter> f) \<noteq> {}"

  2694   unfolding compact_fip by auto

  2695

  2696 text{*Compactness expressed with filters*}

  2697

  2698 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2699

  2700 lemma eventually_filter_from_subbase:

  2701   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2702     (is "_ \<longleftrightarrow> ?R P")

  2703   unfolding filter_from_subbase_def

  2704 proof (rule eventually_Abs_filter is_filter.intro)+

  2705   show "?R (\<lambda>x. True)"

  2706     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  2707 next

  2708   fix P Q assume "?R P" then guess X ..

  2709   moreover assume "?R Q" then guess Y ..

  2710   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  2711     by (intro exI[of _ "X \<union> Y"]) auto

  2712 next

  2713   fix P Q

  2714   assume "?R P" then guess X ..

  2715   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  2716   ultimately show "?R Q"

  2717     by (intro exI[of _ X]) auto

  2718 qed

  2719

  2720 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  2721   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  2722

  2723 lemma filter_from_subbase_not_bot:

  2724   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  2725   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  2726

  2727 lemma closure_iff_nhds_not_empty:

  2728   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  2729 proof safe

  2730   assume x: "x \<in> closure X"

  2731   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  2732   then have "x \<notin> closure (-S)"

  2733     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  2734   with x have "x \<in> closure X - closure (-S)"

  2735     by auto

  2736   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  2737     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  2738   finally have "X \<inter> S \<noteq> {}" by auto

  2739   then show False using X \<inter> A = {} S \<subseteq> A by auto

  2740 next

  2741   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  2742   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  2743   show "x \<in> closure X"

  2744     by (simp add: closure_subset open_Compl)

  2745 qed

  2746

  2747 lemma compact_filter:

  2748   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"

  2749 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  2750   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  2751   from F have "U \<noteq> {}"

  2752     by (auto simp: eventually_False)

  2753

  2754   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  2755   then have "\<forall>z\<in>Z. closed z"

  2756     by auto

  2757   moreover

  2758   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  2759     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  2760   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  2761   proof (intro allI impI)

  2762     fix B assume "finite B" "B \<subseteq> Z"

  2763     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  2764       by (auto intro!: eventually_Ball_finite)

  2765     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  2766       by eventually_elim auto

  2767     with F show "U \<inter> \<Inter>B \<noteq> {}"

  2768       by (intro notI) (simp add: eventually_False)

  2769   qed

  2770   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  2771     using compact U unfolding compact_fip by blast

  2772   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto

  2773

  2774   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  2775     unfolding eventually_inf eventually_nhds

  2776   proof safe

  2777     fix P Q R S

  2778     assume "eventually R F" "open S" "x \<in> S"

  2779     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  2780     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  2781     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  2782     ultimately show False by (auto simp: set_eq_iff)

  2783   qed

  2784   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  2785     by (metis eventually_bot)

  2786 next

  2787   fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  2788

  2789   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  2790   then have inj_P': "\<And>A. inj_on P' A"

  2791     by (auto intro!: inj_onI simp: fun_eq_iff)

  2792   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  2793   have "F \<noteq> bot"

  2794     unfolding F_def

  2795   proof (safe intro!: filter_from_subbase_not_bot)

  2796     fix X assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  2797     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  2798       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  2799     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto

  2800     with B show False by (auto simp: P'_def fun_eq_iff)

  2801   qed

  2802   moreover have "eventually (\<lambda>x. x \<in> U) F"

  2803     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  2804   moreover assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"

  2805   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  2806     by auto

  2807

  2808   { fix V assume "V \<in> A"

  2809     then have V: "eventually (\<lambda>x. x \<in> V) F"

  2810       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  2811     have "x \<in> closure V"

  2812       unfolding closure_iff_nhds_not_empty

  2813     proof (intro impI allI)

  2814       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"

  2815       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)

  2816       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  2817         by (auto simp: eventually_inf)

  2818       with x show "V \<inter> A \<noteq> {}"

  2819         by (auto simp del: Int_iff simp add: trivial_limit_def)

  2820     qed

  2821     then have "x \<in> V"

  2822       using V \<in> A A(1) by simp }

  2823   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  2824   with U \<inter> \<Inter>A = {} show False by auto

  2825 qed

  2826

  2827 definition "countably_compact U \<longleftrightarrow>

  2828     (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"

  2829

  2830 lemma countably_compactE:

  2831   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  2832   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  2833   using assms unfolding countably_compact_def by metis

  2834

  2835 lemma countably_compactI:

  2836   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"

  2837   shows "countably_compact s"

  2838   using assms unfolding countably_compact_def by metis

  2839

  2840 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  2841   by (auto simp: compact_eq_heine_borel countably_compact_def)

  2842

  2843 lemma countably_compact_imp_compact:

  2844   assumes "countably_compact U"

  2845   assumes ccover: "countable B" "\<forall>b\<in>B. open b"

  2846   assumes basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  2847   shows "compact U"

  2848   using countably_compact U unfolding compact_eq_heine_borel countably_compact_def

  2849 proof safe

  2850   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2851   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  2852

  2853   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  2854   ultimately have "countable C" "\<forall>a\<in>C. open a"

  2855     unfolding C_def using ccover by auto

  2856   moreover

  2857   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  2858   proof safe

  2859     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"

  2860     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast

  2861     with a \<in> A show "x \<in> \<Union>C" unfolding C_def

  2862       by auto

  2863   qed

  2864   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  2865   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  2866     using * by metis

  2867   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  2868     by (auto simp: C_def)

  2869   then guess f unfolding bchoice_iff Bex_def ..

  2870   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2871     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  2872 qed

  2873

  2874 lemma countably_compact_imp_compact_second_countable:

  2875   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2876 proof (rule countably_compact_imp_compact)

  2877   fix T and x :: 'a assume "open T" "x \<in> T"

  2878   from topological_basisE[OF is_basis this] guess b .

  2879   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto

  2880 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  2881

  2882 lemma countably_compact_eq_compact:

  2883   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2884   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  2885

  2886 subsubsection{* Sequential compactness *}

  2887

  2888 definition

  2889   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where

  2890   "seq_compact S \<longleftrightarrow>

  2891    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  2892        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  2893

  2894 lemma seq_compact_imp_countably_compact:

  2895   fixes U :: "'a :: first_countable_topology set"

  2896   assumes "seq_compact U"

  2897   shows "countably_compact U"

  2898 proof (safe intro!: countably_compactI)

  2899   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  2900   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  2901     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  2902   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2903   proof cases

  2904     assume "finite A" with A show ?thesis by auto

  2905   next

  2906     assume "infinite A"

  2907     then have "A \<noteq> {}" by auto

  2908     show ?thesis

  2909     proof (rule ccontr)

  2910       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  2911       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto

  2912       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis

  2913       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  2914       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  2915         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  2916       then have "range X \<subseteq> U" by auto

  2917       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto

  2918       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  2919       obtain n where "x \<in> from_nat_into A n" by auto

  2920       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  2921       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  2922         unfolding tendsto_def by (auto simp: comp_def)

  2923       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  2924         by (auto simp: eventually_sequentially)

  2925       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  2926         by auto

  2927       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  2928         by (auto intro!: exI[of _ "max n N"])

  2929       ultimately show False

  2930         by auto

  2931     qed

  2932   qed

  2933 qed

  2934

  2935 lemma compact_imp_seq_compact:

  2936   fixes U :: "'a :: first_countable_topology set"

  2937   assumes "compact U" shows "seq_compact U"

  2938   unfolding seq_compact_def

  2939 proof safe

  2940   fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"

  2941   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  2942     by (auto simp: eventually_filtermap)

  2943   moreover have "filtermap X sequentially \<noteq> bot"

  2944     by (simp add: trivial_limit_def eventually_filtermap)

  2945   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  2946     using compact U by (auto simp: compact_filter)

  2947

  2948   from countable_basis_at_decseq[of x] guess A . note A = this

  2949   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  2950   { fix n i

  2951     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  2952     proof (rule ccontr)

  2953       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  2954       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto

  2955       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  2956         by (auto simp: eventually_filtermap eventually_sequentially)

  2957       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  2958         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  2959       ultimately have "eventually (\<lambda>x. False) ?F"

  2960         by (auto simp add: eventually_inf)

  2961       with x show False

  2962         by (simp add: eventually_False)

  2963     qed

  2964     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  2965       unfolding s_def by (auto intro: someI2_ex) }

  2966   note s = this

  2967   def r \<equiv> "nat_rec (s 0 0) s"

  2968   have "subseq r"

  2969     by (auto simp: r_def s subseq_Suc_iff)

  2970   moreover

  2971   have "(\<lambda>n. X (r n)) ----> x"

  2972   proof (rule topological_tendstoI)

  2973     fix S assume "open S" "x \<in> S"

  2974     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2975     moreover

  2976     { fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"

  2977         by (cases i) (simp_all add: r_def s) }

  2978     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2979     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  2980       by eventually_elim auto

  2981   qed

  2982   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  2983     using x \<in> U by (auto simp: convergent_def comp_def)

  2984 qed

  2985

  2986 lemma seq_compactI:

  2987   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  2988   shows "seq_compact S"

  2989   unfolding seq_compact_def using assms by fast

  2990

  2991 lemma seq_compactE:

  2992   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  2993   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  2994   using assms unfolding seq_compact_def by fast

  2995

  2996 lemma countably_compact_imp_acc_point:

  2997   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"

  2998   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  2999 proof (rule ccontr)

  3000   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3001   note countably_compact s

  3002   moreover have "\<forall>t\<in>C. open t"

  3003     by (auto simp: C_def)

  3004   moreover

  3005   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3006   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3007   have "s \<subseteq> \<Union>C"

  3008     using t \<subseteq> s

  3009     unfolding C_def Union_image_eq

  3010     apply (safe dest!: s)

  3011     apply (rule_tac a="U \<inter> t" in UN_I)

  3012     apply (auto intro!: interiorI simp add: finite_subset)

  3013     done

  3014   moreover

  3015   from countable t have "countable C"

  3016     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3017   ultimately guess D by (rule countably_compactE)

  3018   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and

  3019     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3020     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3021   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3022     using interior_subset by blast

  3023   moreover have "finite (\<Union>E)"

  3024     using E by auto

  3025   ultimately show False using infinite t by (auto simp: finite_subset)

  3026 qed

  3027

  3028 lemma countable_acc_point_imp_seq_compact:

  3029   fixes s :: "'a::first_countable_topology set"

  3030   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3031   shows "seq_compact s"

  3032 proof -

  3033   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3034     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3035     proof (cases "finite (range f)")

  3036       case True

  3037       obtain l where "infinite {n. f n = f l}"

  3038         using pigeonhole_infinite[OF _ True] by auto

  3039       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3040         using infinite_enumerate by blast

  3041       hence "subseq r \<and> (f \<circ> r) ----> f l"

  3042         by (simp add: fr tendsto_const o_def)

  3043       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3044         by auto

  3045     next

  3046       case False

  3047       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto

  3048       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3049       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3050         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3051       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3052     qed

  3053   }

  3054   thus ?thesis unfolding seq_compact_def by auto

  3055 qed

  3056

  3057 lemma seq_compact_eq_countably_compact:

  3058   fixes U :: "'a :: first_countable_topology set"

  3059   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3060   using

  3061     countable_acc_point_imp_seq_compact

  3062     countably_compact_imp_acc_point

  3063     seq_compact_imp_countably_compact

  3064   by metis

  3065

  3066 lemma seq_compact_eq_acc_point:

  3067   fixes s :: "'a :: first_countable_topology set"

  3068   shows "seq_compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"

  3069   using

  3070     countable_acc_point_imp_seq_compact[of s]

  3071     countably_compact_imp_acc_point[of s]

  3072     seq_compact_imp_countably_compact[of s]

  3073   by metis

  3074

  3075 lemma seq_compact_eq_compact:

  3076   fixes U :: "'a :: second_countable_topology set"

  3077   shows "seq_compact U \<longleftrightarrow> compact U"

  3078   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3079

  3080 lemma bolzano_weierstrass_imp_seq_compact:

  3081   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3082   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3083   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3084

  3085 subsubsection{* Total boundedness *}

  3086

  3087 lemma cauchy_def:

  3088   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  3089 unfolding Cauchy_def by metis

  3090

  3091 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  3092   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3093 declare helper_1.simps[simp del]

  3094

  3095 lemma seq_compact_imp_totally_bounded:

  3096   assumes "seq_compact s"

  3097   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3098 proof(rule, rule, rule ccontr)

  3099   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)"

  3100   def x \<equiv> "helper_1 s e"

  3101   { fix n

  3102     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3103     proof(induct_tac rule:nat_less_induct)

  3104       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3105       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3106       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

  3107       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  3108       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  3109         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  3110       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  3111     qed }

  3112   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  3113   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3114   from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto

  3115   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

  3116   show False

  3117     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3118     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3119     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  3120 qed

  3121

  3122 subsubsection{* Heine-Borel theorem *}

  3123

  3124 lemma seq_compact_imp_heine_borel:

  3125   fixes s :: "'a :: metric_space set"

  3126   assumes "seq_compact s" shows "compact s"

  3127 proof -

  3128   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3129   guess f unfolding choice_iff' .. note f = this

  3130   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3131   have "countably_compact s"

  3132     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3133   then show "compact s"

  3134   proof (rule countably_compact_imp_compact)

  3135     show "countable K"

  3136       unfolding K_def using f

  3137       by (auto intro: countable_finite countable_subset countable_rat

  3138                intro!: countable_image countable_SIGMA countable_UN)

  3139     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3140   next

  3141     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3142     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto

  3143     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto

  3144     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto

  3145     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3146       unfolding Union_image_eq by auto

  3147     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K" by (auto simp: K_def)

  3148     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3149     proof (rule bexI[rotated], safe)

  3150       fix y assume "y \<in> ball k r"

  3151       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3152         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3153       with ball x e \<subseteq> T show "y \<in> T" by auto

  3154     qed (rule x \<in> ball k r)

  3155   qed

  3156 qed

  3157

  3158 lemma compact_eq_seq_compact_metric:

  3159   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3160   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3161

  3162 lemma compact_def:

  3163   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3164    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"

  3165   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3166

  3167 subsubsection {* Complete the chain of compactness variants *}

  3168

  3169 lemma compact_eq_bolzano_weierstrass:

  3170   fixes s :: "'a::metric_space set"

  3171   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3172 proof

  3173   assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3174 next

  3175   assume ?rhs thus ?lhs

  3176     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3177 qed

  3178

  3179 lemma bolzano_weierstrass_imp_bounded:

  3180   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3181   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3182

  3183 text {*

  3184   A metric space (or topological vector space) is said to have the

  3185   Heine-Borel property if every closed and bounded subset is compact.

  3186 *}

  3187

  3188 class heine_borel = metric_space +

  3189   assumes bounded_imp_convergent_subsequence:

  3190     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3191

  3192 lemma bounded_closed_imp_seq_compact:

  3193   fixes s::"'a::heine_borel set"

  3194   assumes "bounded s" and "closed s" shows "seq_compact s"

  3195 proof (unfold seq_compact_def, clarify)

  3196   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3197   with bounded s have "bounded (range f)" by (auto intro: bounded_subset)

  3198   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3199     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3200   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  3201   have "l \<in> s" using closed s fr l

  3202     unfolding closed_sequential_limits by blast

  3203   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3204     using l \<in> s r l by blast

  3205 qed

  3206

  3207 lemma compact_eq_bounded_closed:

  3208   fixes s :: "'a::heine_borel set"

  3209   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3210 proof

  3211   assume ?lhs thus ?rhs

  3212     using compact_imp_closed compact_imp_bounded by blast

  3213 next

  3214   assume ?rhs thus ?lhs

  3215     using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto

  3216 qed

  3217

  3218 (* TODO: is this lemma necessary? *)

  3219 lemma bounded_increasing_convergent:

  3220   fixes s :: "nat \<Rightarrow> real"

  3221   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3222   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3223   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3224

  3225 instance real :: heine_borel

  3226 proof

  3227   fix f :: "nat \<Rightarrow> real"

  3228   assume f: "bounded (range f)"

  3229   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3230     unfolding comp_def by (metis seq_monosub)

  3231   moreover

  3232   then have "Bseq (f \<circ> r)"

  3233     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3234   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3235     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3236 qed

  3237

  3238 lemma compact_lemma:

  3239   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3240   assumes "bounded (range f)"

  3241   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  3242         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3243 proof safe

  3244   fix d :: "'a set" assume d: "d \<subseteq> Basis"

  3245   with finite_Basis have "finite d" by (blast intro: finite_subset)

  3246   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3247       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3248   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  3249   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto

  3250     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)" using bounded (range f)

  3251       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3252     obtain l1::"'a" and r1 where r1:"subseq r1" and

  3253       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3254       using insert(3) using insert(4) by auto

  3255     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f" by simp

  3256     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3257       by (metis (lifting) bounded_subset f' image_subsetI s')

  3258     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3259       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)

  3260     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  3261       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3262     moreover

  3263     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3264     { fix e::real assume "e>0"

  3265       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast

  3266       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)

  3267       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3268         by (rule eventually_subseq)

  3269       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3270         using N1' N2

  3271         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3272     }

  3273     ultimately show ?case by auto

  3274   qed

  3275 qed

  3276

  3277 instance euclidean_space \<subseteq> heine_borel

  3278 proof

  3279   fix f :: "nat \<Rightarrow> 'a"

  3280   assume f: "bounded (range f)"

  3281   then obtain l::'a and r where r: "subseq r"

  3282     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3283     using compact_lemma [OF f] by blast

  3284   { fix e::real assume "e>0"

  3285     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)

  3286     with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"

  3287       by simp

  3288     moreover

  3289     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3290       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3291         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  3292       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3293         apply(rule setsum_strict_mono) using n by auto

  3294       finally have "dist (f (r n)) l < e"

  3295         by auto

  3296     }

  3297     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3298       by (rule eventually_elim1)

  3299   }

  3300   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  3301   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  3302 qed

  3303

  3304 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3305 unfolding bounded_def

  3306 apply clarify

  3307 apply (rule_tac x="a" in exI)

  3308 apply (rule_tac x="e" in exI)

  3309 apply clarsimp

  3310 apply (drule (1) bspec)

  3311 apply (simp add: dist_Pair_Pair)

  3312 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3313 done

  3314

  3315 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3316 unfolding bounded_def

  3317 apply clarify

  3318 apply (rule_tac x="b" in exI)

  3319 apply (rule_tac x="e" in exI)

  3320 apply clarsimp

  3321 apply (drule (1) bspec)

  3322 apply (simp add: dist_Pair_Pair)

  3323 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3324 done

  3325

  3326 instance prod :: (heine_borel, heine_borel) heine_borel

  3327 proof

  3328   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  3329   assume f: "bounded (range f)"

  3330   from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)

  3331   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  3332     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  3333   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  3334     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  3335   obtain l2 r2 where r2: "subseq r2"

  3336     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3337     using bounded_imp_convergent_subsequence [OF s2]

  3338     unfolding o_def by fast

  3339   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3340     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  3341   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3342     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3343   have r: "subseq (r1 \<circ> r2)"

  3344     using r1 r2 unfolding subseq_def by simp

  3345   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3346     using l r by fast

  3347 qed

  3348

  3349 subsubsection{* Completeness *}

  3350

  3351 definition complete :: "'a::metric_space set \<Rightarrow> bool" where

  3352   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  3353

  3354 lemma compact_imp_complete: assumes "compact s" shows "complete s"

  3355 proof-

  3356   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3357     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  3358       using assms unfolding compact_def by blast

  3359

  3360     note lr' = seq_suble [OF lr(2)]

  3361

  3362     { fix e::real assume "e>0"

  3363       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

  3364       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

  3365       { fix n::nat assume n:"n \<ge> max N M"

  3366         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  3367         moreover have "r n \<ge> N" using lr'[of n] n by auto

  3368         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  3369         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)  }

  3370       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  3371     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  3372   thus ?thesis unfolding complete_def by auto

  3373 qed

  3374

  3375 lemma nat_approx_posE:

  3376   fixes e::real

  3377   assumes "0 < e"

  3378   obtains n::nat where "1 / (Suc n) < e"

  3379 proof atomize_elim

  3380   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3381     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3382   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3383     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3384   also have "\<dots> = e" by simp

  3385   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3386 qed

  3387

  3388 lemma compact_eq_totally_bounded:

  3389   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3390     (is "_ \<longleftrightarrow> ?rhs")

  3391 proof

  3392   assume assms: "?rhs"

  3393   then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"

  3394     by (auto simp: choice_iff')

  3395

  3396   show "compact s"

  3397   proof cases

  3398     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3399   next

  3400     assume "s \<noteq> {}"

  3401     show ?thesis

  3402       unfolding compact_def

  3403     proof safe

  3404       fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3405

  3406       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  3407       then have [simp]: "\<And>n. 0 < e n" by auto

  3408       def B \<equiv> "\<lambda>n U. SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3409       { fix n U assume "infinite {n. f n \<in> U}"

  3410         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  3411           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  3412         then guess a ..

  3413         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3414           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  3415         from someI_ex[OF this]

  3416         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  3417           unfolding B_def by auto }

  3418       note B = this

  3419

  3420       def F \<equiv> "nat_rec (B 0 UNIV) B"

  3421       { fix n have "infinite {i. f i \<in> F n}"

  3422           by (induct n) (auto simp: F_def B) }

  3423       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  3424         using B by (simp add: F_def)

  3425       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  3426         using decseq_SucI[of F] by (auto simp: decseq_def)

  3427

  3428       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  3429       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  3430         fix k i

  3431         have "infinite ({n. f n \<in> F k} - {.. i})"

  3432           using infinite {n. f n \<in> F k} by auto

  3433         from infinite_imp_nonempty[OF this]

  3434         show "\<exists>x>i. f x \<in> F k"

  3435           by (simp add: set_eq_iff not_le conj_commute)

  3436       qed

  3437

  3438       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  3439       have "subseq t"

  3440         unfolding subseq_Suc_iff by (simp add: t_def sel)

  3441       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  3442         using f by auto

  3443       moreover

  3444       { fix n have "(f \<circ> t) n \<in> F n"

  3445           by (cases n) (simp_all add: t_def sel) }

  3446       note t = this

  3447

  3448       have "Cauchy (f \<circ> t)"

  3449       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  3450         fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  3451         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  3452           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  3453         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  3454           by (auto simp: subset_eq)

  3455         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  3456         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  3457           by (simp add: dist_commute)

  3458       qed

  3459

  3460       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3461         using assms unfolding complete_def by blast

  3462     qed

  3463   qed

  3464 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  3465

  3466 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  3467 proof-

  3468   { assume ?rhs

  3469     { fix e::real

  3470       assume "e>0"

  3471       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  3472         by (erule_tac x="e/2" in allE) auto

  3473       { fix n m

  3474         assume nm:"N \<le> m \<and> N \<le> n"

  3475         hence "dist (s m) (s n) < e" using N

  3476           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  3477           by blast

  3478       }

  3479       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  3480         by blast

  3481     }

  3482     hence ?lhs

  3483       unfolding cauchy_def

  3484       by blast

  3485   }

  3486   thus ?thesis

  3487     unfolding cauchy_def

  3488     using dist_triangle_half_l

  3489     by blast

  3490 qed

  3491

  3492 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  3493 proof-

  3494   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

  3495   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  3496   moreover

  3497   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  3498   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  3499     unfolding bounded_any_center [where a="s N"] by auto

  3500   ultimately show "?thesis"

  3501     unfolding bounded_any_center [where a="s N"]

  3502     apply(rule_tac x="max a 1" in exI) apply auto

  3503     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  3504 qed

  3505

  3506 instance heine_borel < complete_space

  3507 proof

  3508   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3509   hence "bounded (range f)"

  3510     by (rule cauchy_imp_bounded)

  3511   hence "compact (closure (range f))"

  3512     unfolding compact_eq_bounded_closed by auto

  3513   hence "complete (closure (range f))"

  3514     by (rule compact_imp_complete)

  3515   moreover have "\<forall>n. f n \<in> closure (range f)"

  3516     using closure_subset [of "range f"] by auto

  3517   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  3518     using Cauchy f unfolding complete_def by auto

  3519   then show "convergent f"

  3520     unfolding convergent_def by auto

  3521 qed

  3522

  3523 instance euclidean_space \<subseteq> banach ..

  3524

  3525 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  3526 proof(simp add: complete_def, rule, rule)

  3527   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3528   hence "convergent f" by (rule Cauchy_convergent)

  3529   thus "\<exists>l. f ----> l" unfolding convergent_def .

  3530 qed

  3531

  3532 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  3533 proof -

  3534   { fix x assume "x islimpt s"

  3535     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  3536       unfolding islimpt_sequential by auto

  3537     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  3538       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  3539     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  3540   }

  3541   thus "closed s" unfolding closed_limpt by auto

  3542 qed

  3543

  3544 lemma complete_eq_closed:

  3545   fixes s :: "'a::complete_space set"

  3546   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  3547 proof

  3548   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  3549 next

  3550   assume ?rhs

  3551   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  3552     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  3553     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  }

  3554   thus ?lhs unfolding complete_def by auto

  3555 qed

  3556

  3557 lemma convergent_eq_cauchy:

  3558   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  3559   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  3560   unfolding Cauchy_convergent_iff convergent_def ..

  3561

  3562 lemma convergent_imp_bounded:

  3563   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  3564   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  3565   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  3566

  3567 lemma compact_cball[simp]:

  3568   fixes x :: "'a::heine_borel"

  3569   shows "compact(cball x e)"

  3570   using compact_eq_bounded_closed bounded_cball closed_cball

  3571   by blast

  3572

  3573 lemma compact_frontier_bounded[intro]:

  3574   fixes s :: "'a::heine_borel set"

  3575   shows "bounded s ==> compact(frontier s)"

  3576   unfolding frontier_def

  3577   using compact_eq_bounded_closed

  3578   by blast

  3579

  3580 lemma compact_frontier[intro]:

  3581   fixes s :: "'a::heine_borel set"

  3582   shows "compact s ==> compact (frontier s)"

  3583   using compact_eq_bounded_closed compact_frontier_bounded

  3584   by blast

  3585

  3586 lemma frontier_subset_compact:

  3587   fixes s :: "'a::heine_borel set"

  3588   shows "compact s ==> frontier s \<subseteq> s"

  3589   using frontier_subset_closed compact_eq_bounded_closed

  3590   by blast

  3591

  3592 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3593

  3594 lemma bounded_closed_nest:

  3595   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3596   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3597   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3598 proof-

  3599   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

  3600   from assms(4,1) have *:"seq_compact (s 0)" using bounded_closed_imp_seq_compact[of "s 0"] by auto

  3601

  3602   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3603     unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3604

  3605   { fix n::nat

  3606     { fix e::real assume "e>0"

  3607       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3608       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3609       moreover

  3610       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto

  3611       hence "(x \<circ> r) (max N n) \<in> s n"

  3612         using x apply(erule_tac x=n in allE)

  3613         using x apply(erule_tac x="r (max N n)" in allE)

  3614         using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto

  3615       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3616     }

  3617     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3618   }

  3619   thus ?thesis by auto

  3620 qed

  3621

  3622 text {* Decreasing case does not even need compactness, just completeness. *}

  3623

  3624 lemma decreasing_closed_nest:

  3625   assumes "\<forall>n. closed(s n)"

  3626           "\<forall>n. (s n \<noteq> {})"

  3627           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3628           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3629   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3630 proof-

  3631   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3632   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3633   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3634   { fix e::real assume "e>0"

  3635     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3636     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3637       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3638       hence "dist (t m) (t n) < e" using N by auto

  3639     }

  3640     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3641   }

  3642   hence  "Cauchy t" unfolding cauchy_def by auto

  3643   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3644   { fix n::nat

  3645     { fix e::real assume "e>0"

  3646       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3647       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

  3648       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3649     }

  3650     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3651   }

  3652   then show ?thesis by auto

  3653 qed

  3654

  3655 text {* Strengthen it to the intersection actually being a singleton. *}

  3656

  3657 lemma decreasing_closed_nest_sing:

  3658   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3659   assumes "\<forall>n. closed(s n)"

  3660           "\<forall>n. s n \<noteq> {}"

  3661           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3662           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3663   shows "\<exists>a. \<Inter>(range s) = {a}"

  3664 proof-

  3665   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3666   { fix b assume b:"b \<in> \<Inter>(range s)"

  3667     { fix e::real assume "e>0"

  3668       hence "dist a b < e" using assms(4 )using b using a by blast

  3669     }

  3670     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3671   }

  3672   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3673   thus ?thesis ..

  3674 qed

  3675

  3676 text{* Cauchy-type criteria for uniform convergence. *}

  3677

  3678 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows

  3679  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3680   (\<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")

  3681 proof(rule)

  3682   assume ?lhs

  3683   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

  3684   { fix e::real assume "e>0"

  3685     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

  3686     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3687       hence "dist (s m x) (s n x) < e"

  3688         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3689         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3690         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3691     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  }

  3692   thus ?rhs by auto

  3693 next

  3694   assume ?rhs

  3695   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

  3696   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3697     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3698   { fix e::real assume "e>0"

  3699     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"

  3700       using ?rhs[THEN spec[where x="e/2"]] by auto

  3701     { fix x assume "P x"

  3702       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3703         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3704       fix n::nat assume "n\<ge>N"

  3705       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]]

  3706         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)  }

  3707     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3708   thus ?lhs by auto

  3709 qed

  3710

  3711 lemma uniformly_cauchy_imp_uniformly_convergent:

  3712   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  3713   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"

  3714           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3715   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3716 proof-

  3717   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"

  3718     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3719   moreover

  3720   { fix x assume "P x"

  3721     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3722       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3723   ultimately show ?thesis by auto

  3724 qed

  3725

  3726

  3727 subsection {* Continuity *}

  3728

  3729 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3730

  3731 lemma continuous_within_eps_delta:

  3732   "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)"

  3733   unfolding continuous_within and Lim_within

  3734   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3735

  3736 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3737                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3738   using continuous_within_eps_delta [of x UNIV f] by simp

  3739

  3740 text{* Versions in terms of open balls. *}

  3741

  3742 lemma continuous_within_ball:

  3743  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3744                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3745 proof

  3746   assume ?lhs

  3747   { fix e::real assume "e>0"

  3748     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"

  3749       using ?lhs[unfolded continuous_within Lim_within] by auto

  3750     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3751       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3752         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3753     }

  3754     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)  }

  3755   thus ?rhs by auto

  3756 next

  3757   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3758     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3759 qed

  3760

  3761 lemma continuous_at_ball:

  3762   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3763 proof

  3764   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3765     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)

  3766     unfolding dist_nz[THEN sym] by auto

  3767 next

  3768   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3769     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)

  3770 qed

  3771

  3772 text{* Define setwise continuity in terms of limits within the set. *}

  3773

  3774 lemma continuous_on_iff:

  3775   "continuous_on s f \<longleftrightarrow>

  3776     (\<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)"

  3777 unfolding continuous_on_def Lim_within

  3778 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3779 apply (rename_tac y, case_tac "y = x", simp)

  3780 apply (simp add: dist_nz)

  3781 done

  3782

  3783 definition

  3784   uniformly_continuous_on ::

  3785     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3786 where

  3787   "uniformly_continuous_on s f \<longleftrightarrow>

  3788     (\<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)"

  3789

  3790 text{* Some simple consequential lemmas. *}

  3791

  3792 lemma uniformly_continuous_imp_continuous:

  3793  " uniformly_continuous_on s f ==> continuous_on s f"

  3794   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3795

  3796 lemma continuous_at_imp_continuous_within:

  3797  "continuous (at x) f ==> continuous (at x within s) f"

  3798   unfolding continuous_within continuous_at using Lim_at_within by auto

  3799

  3800 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3801   by simp

  3802

  3803 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3804

  3805 lemma continuous_on_eq_continuous_at:

  3806   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"

  3807   by (auto simp add: continuous_on continuous_at Lim_within_open)

  3808

  3809 lemma continuous_within_subset:

  3810  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3811              ==> continuous (at x within t) f"

  3812   unfolding continuous_within by(metis Lim_within_subset)

  3813

  3814 lemma continuous_on_subset:

  3815   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"

  3816   unfolding continuous_on by (metis subset_eq Lim_within_subset)

  3817

  3818 lemma continuous_on_interior:

  3819   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3820   by (erule interiorE, drule (1) continuous_on_subset,

  3821     simp add: continuous_on_eq_continuous_at)

  3822

  3823 lemma continuous_on_eq:

  3824   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3825   unfolding continuous_on_def tendsto_def Limits.eventually_within

  3826   by simp

  3827

  3828 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3829

  3830 lemma continuous_within_sequentially:

  3831   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3832   shows "continuous (at a within s) f \<longleftrightarrow>

  3833                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3834                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3835 proof

  3836   assume ?lhs

  3837   { 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"

  3838     fix T::"'b set" assume "open T" and "f a \<in> T"

  3839     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"

  3840       unfolding continuous_within tendsto_def eventually_within by auto

  3841     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3842       using x(2) d>0 by simp

  3843     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3844     proof eventually_elim

  3845       case (elim n) thus ?case

  3846         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3847     qed

  3848   }

  3849   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3850 next

  3851   assume ?rhs thus ?lhs

  3852     unfolding continuous_within tendsto_def [where l="f a"]

  3853     by (simp add: sequentially_imp_eventually_within)

  3854 qed

  3855

  3856 lemma continuous_at_sequentially:

  3857   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3858   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  3859                   --> ((f o x) ---> f a) sequentially)"

  3860   using continuous_within_sequentially[of a UNIV f] by simp

  3861

  3862 lemma continuous_on_sequentially:

  3863   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3864   shows "continuous_on s f \<longleftrightarrow>

  3865     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  3866                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  3867 proof

  3868   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  3869 next

  3870   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  3871 qed

  3872

  3873 lemma uniformly_continuous_on_sequentially:

  3874   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  3875                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  3876                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  3877 proof

  3878   assume ?lhs

  3879   { 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"

  3880     { fix e::real assume "e>0"

  3881       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"

  3882         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  3883       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

  3884       { fix n assume "n\<ge>N"

  3885         hence "dist (f (x n)) (f (y n)) < e"

  3886           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

  3887           unfolding dist_commute by simp  }

  3888       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  3889     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  3890   thus ?rhs by auto

  3891 next

  3892   assume ?rhs

  3893   { assume "\<not> ?lhs"

  3894     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

  3895     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"

  3896       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

  3897       by (auto simp add: dist_commute)

  3898     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  3899     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  3900     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"

  3901       unfolding x_def and y_def using fa by auto

  3902     { fix e::real assume "e>0"

  3903       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

  3904       { fix n::nat assume "n\<ge>N"

  3905         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  3906         also have "\<dots> < e" using N by auto

  3907         finally have "inverse (real n + 1) < e" by auto

  3908         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  3909       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  3910     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

  3911     hence False using fxy and e>0 by auto  }

  3912   thus ?lhs unfolding uniformly_continuous_on_def by blast

  3913 qed

  3914

  3915 text{* The usual transformation theorems. *}

  3916

  3917 lemma continuous_transform_within:

  3918   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3919   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  3920           "continuous (at x within s) f"

  3921   shows "continuous (at x within s) g"

  3922 unfolding continuous_within

  3923 proof (rule Lim_transform_within)

  3924   show "0 < d" by fact

  3925   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  3926     using assms(3) by auto

  3927   have "f x = g x"

  3928     using assms(1,2,3) by auto

  3929   thus "(f ---> g x) (at x within s)"

  3930     using assms(4) unfolding continuous_within by simp

  3931 qed

  3932

  3933 lemma continuous_transform_at:

  3934   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3935   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  3936           "continuous (at x) f"

  3937   shows "continuous (at x) g"

  3938   using continuous_transform_within [of d x UNIV f g] assms by simp

  3939

  3940 subsubsection {* Structural rules for pointwise continuity *}

  3941

  3942 lemmas continuous_within_id = continuous_ident

  3943

  3944 lemmas continuous_at_id = isCont_ident

  3945

  3946 lemma continuous_infdist[continuous_intros]:

  3947   assumes "continuous F f"

  3948   shows "continuous F (\<lambda>x. infdist (f x) A)"

  3949   using assms unfolding continuous_def by (rule tendsto_infdist)

  3950

  3951 lemma continuous_infnorm[continuous_intros]:

  3952   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  3953   unfolding continuous_def by (rule tendsto_infnorm)

  3954

  3955 lemma continuous_inner[continuous_intros]:

  3956   assumes "continuous F f" and "continuous F g"

  3957   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  3958   using assms unfolding continuous_def by (rule tendsto_inner)

  3959

  3960 lemmas continuous_at_inverse = isCont_inverse

  3961

  3962 subsubsection {* Structural rules for setwise continuity *}

  3963

  3964 lemma continuous_on_infnorm[continuous_on_intros]:

  3965   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  3966   unfolding continuous_on by (fast intro: tendsto_infnorm)

  3967

  3968 lemma continuous_on_inner[continuous_on_intros]:

  3969   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  3970   assumes "continuous_on s f" and "continuous_on s g"

  3971   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  3972   using bounded_bilinear_inner assms

  3973   by (rule bounded_bilinear.continuous_on)

  3974

  3975 subsubsection {* Structural rules for uniform continuity *}

  3976

  3977 lemma uniformly_continuous_on_id[continuous_on_intros]:

  3978   shows "uniformly_continuous_on s (\<lambda>x. x)"

  3979   unfolding uniformly_continuous_on_def by auto

  3980

  3981 lemma uniformly_continuous_on_const[continuous_on_intros]:

  3982   shows "uniformly_continuous_on s (\<lambda>x. c)"

  3983   unfolding uniformly_continuous_on_def by simp

  3984

  3985 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  3986   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  3987   assumes "uniformly_continuous_on s f"

  3988   assumes "uniformly_continuous_on s g"

  3989   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  3990 proof -

  3991   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  3992       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  3993       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  3994       by arith

  3995   } note le = this

  3996   { fix x y

  3997     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  3998     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  3999     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4000       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4001         simp add: le)

  4002   }

  4003   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  4004     unfolding dist_real_def by simp

  4005 qed

  4006

  4007 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  4008   assumes "uniformly_continuous_on s f"

  4009   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4010   unfolding norm_conv_dist using assms

  4011   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4012

  4013 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  4014   assumes "uniformly_continuous_on s g"

  4015   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4016   using assms unfolding uniformly_continuous_on_sequentially

  4017   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4018   by (auto intro: tendsto_zero)

  4019

  4020 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  4021   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4022   assumes "uniformly_continuous_on s f"

  4023   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4024   using bounded_linear_scaleR_right assms

  4025   by (rule bounded_linear.uniformly_continuous_on)

  4026

  4027 lemma dist_minus:

  4028   fixes x y :: "'a::real_normed_vector"

  4029   shows "dist (- x) (- y) = dist x y"

  4030   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4031

  4032 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  4033   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4034   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4035   unfolding uniformly_continuous_on_def dist_minus .

  4036

  4037 lemma uniformly_continuous_on_add[continuous_on_intros]:

  4038   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4039   assumes "uniformly_continuous_on s f"

  4040   assumes "uniformly_continuous_on s g"

  4041   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4042   using assms unfolding uniformly_continuous_on_sequentially

  4043   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4044   by (auto intro: tendsto_add_zero)

  4045

  4046 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  4047   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4048   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4049   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4050   unfolding ab_diff_minus using assms

  4051   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4052

  4053 text{* Continuity of all kinds is preserved under composition. *}

  4054

  4055 lemmas continuous_at_compose = isCont_o

  4056

  4057 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  4058   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4059   shows "uniformly_continuous_on s (g o f)"

  4060 proof-

  4061   { fix e::real assume "e>0"

  4062     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

  4063     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

  4064     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  }

  4065   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4066 qed

  4067

  4068 text{* Continuity in terms of open preimages. *}

  4069

  4070 lemma continuous_at_open:

  4071   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)))"

  4072 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  4073 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4074

  4075 lemma continuous_imp_tendsto:

  4076   assumes "continuous (at x0) f" and "x ----> x0"

  4077   shows "(f \<circ> x) ----> (f x0)"

  4078 proof (rule topological_tendstoI)

  4079   fix S

  4080   assume "open S" "f x0 \<in> S"

  4081   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  4082      using assms continuous_at_open by metis

  4083   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  4084     using assms T_def by (auto simp: tendsto_def)

  4085   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  4086     using T_def by (auto elim!: eventually_elim1)

  4087 qed

  4088

  4089 lemma continuous_on_open:

  4090   shows "continuous_on s f \<longleftrightarrow>

  4091         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4092             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4093 proof (safe)

  4094   fix t :: "'b set"

  4095   assume 1: "continuous_on s f"

  4096   assume 2: "openin (subtopology euclidean (f  s)) t"

  4097   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  4098     unfolding openin_open by auto

  4099   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  4100   have "open U" unfolding U_def by (simp add: open_Union)

  4101   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  4102   proof (intro ballI iffI)

  4103     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  4104       unfolding U_def t by auto

  4105   next

  4106     fix x assume "x \<in> s" and "f x \<in> t"

  4107     hence "x \<in> s" and "f x \<in> B"

  4108       unfolding t by auto

  4109     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  4110       unfolding t continuous_on_topological by metis

  4111     then show "x \<in> U"

  4112       unfolding U_def by auto

  4113   qed

  4114   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  4115   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4116     unfolding openin_open by fast

  4117 next

  4118   assume "?rhs" show "continuous_on s f"

  4119   unfolding continuous_on_topological

  4120   proof (clarify)

  4121     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  4122     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  4123       unfolding openin_open using open B by auto

  4124     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  4125       using ?rhs by fast

  4126     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  4127       unfolding openin_open using x \<in> s and f x \<in> B by auto

  4128   qed

  4129 qed

  4130

  4131 text {* Similarly in terms of closed sets. *}

  4132

  4133 lemma continuous_on_closed:

  4134   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")

  4135 proof

  4136   assume ?lhs

  4137   { fix t

  4138     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4139     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  4140     assume as:"closedin (subtopology euclidean (f  s)) t"

  4141     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  4142     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"]]

  4143       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  4144   thus ?rhs by auto

  4145 next

  4146   assume ?rhs

  4147   { fix t

  4148     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4149     assume as:"openin (subtopology euclidean (f  s)) t"

  4150     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  4151       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  4152   thus ?lhs unfolding continuous_on_open by auto

  4153 qed

  4154

  4155 text {* Half-global and completely global cases. *}

  4156

  4157 lemma continuous_open_in_preimage:

  4158   assumes "continuous_on s f"  "open t"

  4159   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4160 proof-

  4161   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

  4162   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4163     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4164   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4165 qed

  4166

  4167 lemma continuous_closed_in_preimage:

  4168   assumes "continuous_on s f"  "closed t"

  4169   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4170 proof-

  4171   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

  4172   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4173     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4174   thus ?thesis

  4175     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4176 qed

  4177

  4178 lemma continuous_open_preimage:

  4179   assumes "continuous_on s f" "open s" "open t"

  4180   shows "open {x \<in> s. f x \<in> t}"

  4181 proof-

  4182   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4183     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4184   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4185 qed

  4186

  4187 lemma continuous_closed_preimage:

  4188   assumes "continuous_on s f" "closed s" "closed t"

  4189   shows "closed {x \<in> s. f x \<in> t}"

  4190 proof-

  4191   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4192     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4193   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4194 qed

  4195

  4196 lemma continuous_open_preimage_univ:

  4197   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4198   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4199

  4200 lemma continuous_closed_preimage_univ:

  4201   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4202   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4203

  4204 lemma continuous_open_vimage:

  4205   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4206   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4207

  4208 lemma continuous_closed_vimage:

  4209   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4210   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4211

  4212 lemma interior_image_subset:

  4213   assumes "\<forall>x. continuous (at x) f" "inj f"

  4214   shows "interior (f  s) \<subseteq> f  (interior s)"

  4215 proof

  4216   fix x assume "x \<in> interior (f  s)"

  4217   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4218   hence "x \<in> f  s" by auto

  4219   then obtain y where y: "y \<in> s" "x = f y" by auto

  4220   have "open (vimage f T)"

  4221     using assms(1) open T by (rule continuous_open_vimage)

  4222   moreover have "y \<in> vimage f T"

  4223     using x = f y x \<in> T by simp

  4224   moreover have "vimage f T \<subseteq> s"

  4225     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4226   ultimately have "y \<in> interior s" ..

  4227   with x = f y show "x \<in> f  interior s" ..

  4228 qed

  4229

  4230 text {* Equality of continuous functions on closure and related results. *}

  4231

  4232 lemma continuous_closed_in_preimage_constant:

  4233   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4234   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4235   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4236

  4237 lemma continuous_closed_preimage_constant:

  4238   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4239   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4240   using continuous_closed_preimage[of s f "{a}"] by auto

  4241

  4242 lemma continuous_constant_on_closure:

  4243   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4244   assumes "continuous_on (closure s) f"

  4245           "\<forall>x \<in> s. f x = a"

  4246   shows "\<forall>x \<in> (closure s). f x = a"

  4247     using continuous_closed_preimage_constant[of "closure s" f a]

  4248     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4249

  4250 lemma image_closure_subset:

  4251   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4252   shows "f  (closure s) \<subseteq> t"

  4253 proof-

  4254   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4255   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4256     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4257   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4258     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4259   thus ?thesis by auto

  4260 qed

  4261

  4262 lemma continuous_on_closure_norm_le:

  4263   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4264   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4265   shows "norm(f x) \<le> b"

  4266 proof-

  4267   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4268   show ?thesis

  4269     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4270     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4271 qed

  4272

  4273 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4274

  4275 lemma continuous_within_avoid:

  4276   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4277   assumes "continuous (at x within s) f" and "f x \<noteq> a"

  4278   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4279 proof-

  4280   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  4281     using t1_space [OF f x \<noteq> a] by fast

  4282   have "(f ---> f x) (at x within s)"

  4283     using assms(1) by (simp add: continuous_within)

  4284   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  4285     using open U and f x \<in> U

  4286     unfolding tendsto_def by fast

  4287   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  4288     using a \<notin> U by (fast elim: eventually_mono [rotated])

  4289   thus ?thesis

  4290     unfolding Limits.eventually_within Metric_Spaces.eventually_at

  4291     by (rule ex_forward, cut_tac f x \<noteq> a, auto simp: dist_commute)

  4292 qed

  4293

  4294 lemma continuous_at_avoid:

  4295   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4296   assumes "continuous (at x) f" and "f x \<noteq> a"

  4297   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4298   using assms continuous_within_avoid[of x UNIV f a] by simp

  4299

  4300 lemma continuous_on_avoid:

  4301   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4302   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4303   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4304 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(3) by auto

  4305

  4306 lemma continuous_on_open_avoid:

  4307   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4308   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4309   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4310 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(4) by auto

  4311

  4312 text {* Proving a function is constant by proving open-ness of level set. *}

  4313

  4314 lemma continuous_levelset_open_in_cases:

  4315   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4316   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4317         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4318         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4319 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4320

  4321 lemma continuous_levelset_open_in:

  4322   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4323   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4324         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4325         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4326 using continuous_levelset_open_in_cases[of s f ]

  4327 by meson

  4328

  4329 lemma continuous_levelset_open:

  4330   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4331   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4332   shows "\<forall>x \<in> s. f x = a"

  4333 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4334

  4335 text {* Some arithmetical combinations (more to prove). *}

  4336

  4337 lemma open_scaling[intro]:

  4338   fixes s :: "'a::real_normed_vector set"

  4339   assumes "c \<noteq> 0"  "open s"

  4340   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4341 proof-

  4342   { fix x assume "x \<in> s"

  4343     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

  4344     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4345     moreover

  4346     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4347       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4348         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4349           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4350       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  }

  4351     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  }

  4352   thus ?thesis unfolding open_dist by auto

  4353 qed

  4354

  4355 lemma minus_image_eq_vimage:

  4356   fixes A :: "'a::ab_group_add set"

  4357   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4358   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4359

  4360 lemma open_negations:

  4361   fixes s :: "'a::real_normed_vector set"

  4362   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4363   unfolding scaleR_minus1_left [symmetric]

  4364   by (rule open_scaling, auto)

  4365

  4366 lemma open_translation:

  4367   fixes s :: "'a::real_normed_vector set"

  4368   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4369 proof-

  4370   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4371       by (intro continuous_diff continuous_at_id continuous_const) }

  4372   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4373   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4374 qed

  4375

  4376 lemma open_affinity:

  4377   fixes s :: "'a::real_normed_vector set"

  4378   assumes "open s"  "c \<noteq> 0"

  4379   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4380 proof-

  4381   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4382   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4383   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4384 qed

  4385

  4386 lemma interior_translation:

  4387   fixes s :: "'a::real_normed_vector set"

  4388   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4389 proof (rule set_eqI, rule)

  4390   fix x assume "x \<in> interior (op + a  s)"

  4391   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4392   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

  4393   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

  4394 next

  4395   fix x assume "x \<in> op + a  interior s"

  4396   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

  4397   { fix z have *:"a + y - z = y + a - z" by auto

  4398     assume "z\<in>ball x e"

  4399     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

  4400     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4401   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4402   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4403 qed

  4404

  4405 text {* Topological properties of linear functions. *}

  4406

  4407 lemma linear_lim_0:

  4408   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4409 proof-

  4410   interpret f: bounded_linear f by fact

  4411   have "(f ---> f 0) (at 0)"

  4412     using tendsto_ident_at by (rule f.tendsto)

  4413   thus ?thesis unfolding f.zero .

  4414 qed

  4415

  4416 lemma linear_continuous_at:

  4417   assumes "bounded_linear f"  shows "continuous (at a) f"

  4418   unfolding continuous_at using assms

  4419   apply (rule bounded_linear.tendsto)

  4420   apply (rule tendsto_ident_at)

  4421   done

  4422

  4423 lemma linear_continuous_within:

  4424   shows "bounded_linear f ==> continuous (at x within s) f"

  4425   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4426

  4427 lemma linear_continuous_on:

  4428   shows "bounded_linear f ==> continuous_on s f"

  4429   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4430

  4431 text {* Also bilinear functions, in composition form. *}

  4432

  4433 lemma bilinear_continuous_at_compose:

  4434   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4435         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4436   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4437

  4438 lemma bilinear_continuous_within_compose:

  4439   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4440         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4441   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4442

  4443 lemma bilinear_continuous_on_compose:

  4444   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4445              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4446   unfolding continuous_on_def

  4447   by (fast elim: bounded_bilinear.tendsto)

  4448

  4449 text {* Preservation of compactness and connectedness under continuous function. *}

  4450

  4451 lemma compact_eq_openin_cover:

  4452   "compact S \<longleftrightarrow>

  4453     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4454       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  4455 proof safe

  4456   fix C

  4457   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  4458   hence "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"

  4459     unfolding openin_open by force+

  4460   with compact S obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"

  4461     by (rule compactE)

  4462   hence "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"

  4463     by auto

  4464   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4465 next

  4466   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4467         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  4468   show "compact S"

  4469   proof (rule compactI)

  4470     fix C

  4471     let ?C = "image (\<lambda>T. S \<inter> T) C"

  4472     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  4473     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  4474       unfolding openin_open by auto

  4475     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  4476       by metis

  4477     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  4478     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  4479     proof (intro conjI)

  4480       from D \<subseteq> ?C show "?D \<subseteq> C"

  4481         by (fast intro: inv_into_into)

  4482       from finite D show "finite ?D"

  4483         by (rule finite_imageI)

  4484       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  4485         apply (rule subset_trans)

  4486         apply clarsimp

  4487         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  4488         apply (erule rev_bexI, fast)

  4489         done

  4490     qed

  4491     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4492   qed

  4493 qed

  4494

  4495 lemma connected_continuous_image:

  4496   assumes "continuous_on s f"  "connected s"

  4497   shows "connected(f  s)"

  4498 proof-

  4499   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4500     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4501       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4502       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4503       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4504     hence False using as(1,2)

  4505       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4506   thus ?thesis unfolding connected_clopen by auto

  4507 qed

  4508

  4509 text {* Continuity implies uniform continuity on a compact domain. *}

  4510

  4511 lemma compact_uniformly_continuous:

  4512   assumes f: "continuous_on s f" and s: "compact s"

  4513   shows "uniformly_continuous_on s f"

  4514   unfolding uniformly_continuous_on_def

  4515 proof (cases, safe)

  4516   fix e :: real assume "0 < e" "s \<noteq> {}"

  4517   def [simp]: R \<equiv> "{(y, d). y \<in> s \<and> 0 < d \<and> ball y d \<inter> s \<subseteq> {x \<in> s. f x \<in> ball (f y) (e/2) } }"

  4518   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  4519   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  4520   proof safe

  4521     fix y assume "y \<in> s"

  4522     from continuous_open_in_preimage[OF f open_ball]

  4523     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  4524       unfolding openin_subtopology open_openin by metis

  4525     then obtain d where "ball y d \<subseteq> T" "0 < d"

  4526       using 0 < e y \<in> s by (auto elim!: openE)

  4527     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  4528       by (intro UN_I[of "(y, d)"]) auto

  4529   qed auto

  4530   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  4531     by (rule compactE_image)

  4532   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  4533     by (subst Min_gr_iff) auto

  4534   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  4535   proof (rule, safe)

  4536     fix x x' assume in_s: "x' \<in> s" "x \<in> s"

  4537     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  4538       by blast

  4539     moreover assume "dist x x' < Min (sndD) / 2"

  4540     ultimately have "dist y x' < d"

  4541       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  4542     with D x in_s show  "dist (f x) (f x') < e"

  4543       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  4544   qed (insert D, auto)

  4545 qed auto

  4546

  4547 text{* Continuity of inverse function on compact domain. *}

  4548

  4549 lemma continuous_on_inv:

  4550   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  4551   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4552   shows "continuous_on (f  s) g"

  4553 unfolding continuous_on_topological

  4554 proof (clarsimp simp add: assms(3))

  4555   fix x :: 'a and B :: "'a set"

  4556   assume "x \<in> s" and "open B" and "x \<in> B"

  4557   have 1: "\<forall>x\<in>s. f x \<in> f  (s - B) \<longleftrightarrow> x \<in> s - B"

  4558     using assms(3) by (auto, metis)

  4559   have "continuous_on (s - B) f"

  4560     using continuous_on s f Diff_subset

  4561     by (rule continuous_on_subset)

  4562   moreover have "compact (s - B)"

  4563     using open B and compact s

  4564     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)

  4565   ultimately have "compact (f  (s - B))"

  4566     by (rule compact_continuous_image)

  4567   hence "closed (f  (s - B))"

  4568     by (rule compact_imp_closed)

  4569   hence "open (- f  (s - B))"

  4570     by (rule open_Compl)

  4571   moreover have "f x \<in> - f  (s - B)"

  4572     using x \<in> s and x \<in> B by (simp add: 1)

  4573   moreover have "\<forall>y\<in>s. f y \<in> - f  (s - B) \<longrightarrow> y \<in> B"

  4574     by (simp add: 1)

  4575   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"

  4576     by fast

  4577 qed

  4578

  4579 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4580

  4581 lemma continuous_uniform_limit:

  4582   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4583   assumes "\<not> trivial_limit F"

  4584   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4585   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4586   shows "continuous_on s g"

  4587 proof-

  4588   { fix x and e::real assume "x\<in>s" "e>0"

  4589     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4590       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4591     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4592     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4593       using assms(1) by blast

  4594     have "e / 3 > 0" using e>0 by auto

  4595     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"

  4596       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4597     { fix y assume "y \<in> s" and "dist y x < d"

  4598       hence "dist (f n y) (f n x) < e / 3"

  4599         by (rule d [rule_format])

  4600       hence "dist (f n y) (g x) < 2 * e / 3"

  4601         using dist_triangle [of "f n y" "g x" "f n x"]

  4602         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4603         by auto

  4604       hence "dist (g y) (g x) < e"

  4605         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4606         using dist_triangle3 [of "g y" "g x" "f n y"]

  4607         by auto }

  4608     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4609       using d>0 by auto }

  4610   thus ?thesis unfolding continuous_on_iff by auto

  4611 qed

  4612

  4613

  4614 subsection {* Topological stuff lifted from and dropped to R *}

  4615

  4616 lemma open_real:

  4617   fixes s :: "real set" shows

  4618  "open s \<longleftrightarrow>

  4619         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4620   unfolding open_dist dist_norm by simp

  4621

  4622 lemma islimpt_approachable_real:

  4623   fixes s :: "real set"

  4624   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4625   unfolding islimpt_approachable dist_norm by simp

  4626

  4627 lemma closed_real:

  4628   fixes s :: "real set"

  4629   shows "closed s \<longleftrightarrow>

  4630         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4631             --> x \<in> s)"

  4632   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4633

  4634 lemma continuous_at_real_range:

  4635   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4636   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4637         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4638   unfolding continuous_at unfolding Lim_at

  4639   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4640   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

  4641   apply(erule_tac x=e in allE) by auto

  4642

  4643 lemma continuous_on_real_range:

  4644   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4645   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))"

  4646   unfolding continuous_on_iff dist_norm by simp

  4647

  4648 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4649

  4650 lemma distance_attains_sup:

  4651   assumes "compact s" "s \<noteq> {}"

  4652   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  4653 proof (rule continuous_attains_sup [OF assms])

  4654   { fix x assume "x\<in>s"

  4655     have "(dist a ---> dist a x) (at x within s)"

  4656       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  4657   }

  4658   thus "continuous_on s (dist a)"

  4659     unfolding continuous_on ..

  4660 qed

  4661

  4662 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4663

  4664 lemma distance_attains_inf:

  4665   fixes a :: "'a::heine_borel"

  4666   assumes "closed s"  "s \<noteq> {}"

  4667   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  4668 proof-

  4669   from assms(2) obtain b where "b \<in> s" by auto

  4670   let ?B = "s \<inter> cball a (dist b a)"

  4671   have "?B \<noteq> {}" using b \<in> s by (auto simp add: dist_commute)

  4672   moreover have "continuous_on ?B (dist a)"

  4673     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  4674   moreover have "compact ?B"

  4675     by (intro closed_inter_compact closed s compact_cball)

  4676   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  4677     by (metis continuous_attains_inf)

  4678   thus ?thesis by fastforce

  4679 qed

  4680

  4681

  4682 subsection {* Pasted sets *}

  4683

  4684 lemma bounded_Times:

  4685   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4686 proof-

  4687   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4688     using assms [unfolded bounded_def] by auto

  4689   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4690     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4691   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4692 qed

  4693

  4694 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4695 by (induct x) simp

  4696

  4697 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  4698 unfolding seq_compact_def

  4699 apply clarify

  4700 apply (drule_tac x="fst \<circ> f" in spec)

  4701 apply (drule mp, simp add: mem_Times_iff)

  4702 apply (clarify, rename_tac l1 r1)

  4703 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4704 apply (drule mp, simp add: mem_Times_iff)

  4705 apply (clarify, rename_tac l2 r2)

  4706 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4707 apply (rule_tac x="r1 \<circ> r2" in exI)

  4708 apply (rule conjI, simp add: subseq_def)

  4709 apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  4710 apply (drule (1) tendsto_Pair) back

  4711 apply (simp add: o_def)

  4712 done

  4713

  4714 lemma compact_Times:

  4715   assumes "compact s" "compact t"

  4716   shows "compact (s \<times> t)"

  4717 proof (rule compactI)

  4718   fix C assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  4719   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  4720   proof

  4721     fix x assume "x \<in> s"

  4722     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")

  4723     proof

  4724       fix y assume "y \<in> t"

  4725       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  4726       then show "?P y" by (auto elim!: open_prod_elim)

  4727     qed

  4728     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  4729       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"

  4730       by metis

  4731     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  4732     from compactE_image[OF compact t this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  4733       by auto

  4734     moreover with c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  4735       by (fastforce simp: subset_eq)

  4736     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  4737       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>aD"] conjI) (auto intro!: open_INT)

  4738   qed

  4739   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  4740     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"

  4741     unfolding subset_eq UN_iff by metis

  4742   moreover from compactE_image[OF compact s a] obtain e where e: "e \<subseteq> s" "finite e"

  4743     and s: "s \<subseteq> (\<Union>x\<in>e. a x)" by auto

  4744   moreover

  4745   { from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)" by auto

  4746     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)" using d e \<subseteq> s by (intro UN_mono) auto

  4747     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" . }

  4748   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  4749     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  4750 qed

  4751

  4752 text{* Hence some useful properties follow quite easily. *}

  4753

  4754 lemma compact_scaling:

  4755   fixes s :: "'a::real_normed_vector set"

  4756   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4757 proof-

  4758   let ?f = "\<lambda>x. scaleR c x"

  4759   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4760   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4761     using linear_continuous_at[OF *] assms by auto

  4762 qed

  4763

  4764 lemma compact_negations:

  4765   fixes s :: "'a::real_normed_vector set"

  4766   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4767   using compact_scaling [OF assms, of "- 1"] by auto

  4768

  4769 lemma compact_sums:

  4770   fixes s t :: "'a::real_normed_vector set"

  4771   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4772 proof-

  4773   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4774     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4775   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4776     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4777   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4778 qed

  4779

  4780 lemma compact_differences:

  4781   fixes s t :: "'a::real_normed_vector set"

  4782   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4783 proof-

  4784   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4785     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4786   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4787 qed

  4788

  4789 lemma compact_translation:

  4790   fixes s :: "'a::real_normed_vector set"

  4791   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4792 proof-

  4793   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4794   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4795 qed

  4796

  4797 lemma compact_affinity:

  4798   fixes s :: "'a::real_normed_vector set"

  4799   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4800 proof-

  4801   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4802   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4803 qed

  4804

  4805 text {* Hence we get the following. *}

  4806

  4807 lemma compact_sup_maxdistance:

  4808   fixes s :: "'a::metric_space set"

  4809   assumes "compact s"  "s \<noteq> {}"

  4810   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  4811 proof-

  4812   have "compact (s \<times> s)" using compact s by (intro compact_Times)

  4813   moreover have "s \<times> s \<noteq> {}" using s \<noteq> {} by auto

  4814   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  4815     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  4816   ultimately show ?thesis

  4817     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  4818 qed

  4819

  4820 text {* We can state this in terms of diameter of a set. *}

  4821

  4822 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

  4823

  4824 lemma diameter_bounded_bound:

  4825   fixes s :: "'a :: metric_space set"

  4826   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  4827   shows "dist x y \<le> diameter s"

  4828 proof -

  4829   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4830   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  4831     unfolding bounded_def by auto

  4832   have "dist x y \<le> Sup ?D"

  4833   proof (rule cSup_upper, safe)

  4834     fix a b assume "a \<in> s" "b \<in> s"

  4835     with z[of a] z[of b] dist_triangle[of a b z]

  4836     show "dist a b \<le> 2 * d"

  4837       by (simp add: dist_commute)

  4838   qed (insert s, auto)

  4839   with x \<in> s show ?thesis

  4840     by (auto simp add: diameter_def)

  4841 qed

  4842

  4843 lemma diameter_lower_bounded:

  4844   fixes s :: "'a :: metric_space set"

  4845   assumes s: "bounded s" and d: "0 < d" "d < diameter s"

  4846   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  4847 proof (rule ccontr)

  4848   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4849   assume contr: "\<not> ?thesis"

  4850   moreover

  4851   from d have "s \<noteq> {}"

  4852     by (auto simp: diameter_def)

  4853   then have "?D \<noteq> {}" by auto

  4854   ultimately have "Sup ?D \<le> d"

  4855     by (intro cSup_least) (auto simp: not_less)

  4856   with d < diameter s s \<noteq> {} show False

  4857     by (auto simp: diameter_def)

  4858 qed

  4859

  4860 lemma diameter_bounded:

  4861   assumes "bounded s"

  4862   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  4863         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  4864   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  4865   by auto

  4866

  4867 lemma diameter_compact_attained:

  4868   assumes "compact s"  "s \<noteq> {}"

  4869   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  4870 proof -

  4871   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4872   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  4873     using compact_sup_maxdistance[OF assms] by auto

  4874   hence "diameter s \<le> dist x y"

  4875     unfolding diameter_def by clarsimp (rule cSup_least, fast+)

  4876   thus ?thesis

  4877     by (metis b diameter_bounded_bound order_antisym xys)

  4878 qed

  4879

  4880 text {* Related results with closure as the conclusion. *}

  4881

  4882 lemma closed_scaling:

  4883   fixes s :: "'a::real_normed_vector set"

  4884   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4885 proof(cases "s={}")

  4886   case True thus ?thesis by auto

  4887 next

  4888   case False

  4889   show ?thesis

  4890   proof(cases "c=0")

  4891     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4892     case True thus ?thesis apply auto unfolding * by auto

  4893   next

  4894     case False

  4895     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4896       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4897           using as(1)[THEN spec[where x=n]]

  4898           using c\<noteq>0 by auto

  4899       }

  4900       moreover

  4901       { fix e::real assume "e>0"

  4902         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4903         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4904           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4905         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4906           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4907           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4908       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4909       ultimately have "l \<in> scaleR c  s"

  4910         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"]]

  4911         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4912     thus ?thesis unfolding closed_sequential_limits by fast

  4913   qed

  4914 qed

  4915

  4916 lemma closed_negations:

  4917   fixes s :: "'a::real_normed_vector set"

  4918   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4919   using closed_scaling[OF assms, of "- 1"] by simp

  4920

  4921 lemma compact_closed_sums:

  4922   fixes s :: "'a::real_normed_vector set"

  4923   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4924 proof-

  4925   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4926   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4927     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"

  4928       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4929     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4930       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4931     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4932       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto

  4933     hence "l - l' \<in> t"

  4934       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4935       using f(3) by auto

  4936     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

  4937   }

  4938   thus ?thesis unfolding closed_sequential_limits by fast

  4939 qed

  4940

  4941 lemma closed_compact_sums:

  4942   fixes s t :: "'a::real_normed_vector set"

  4943   assumes "closed s"  "compact t"

  4944   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4945 proof-

  4946   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

  4947     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4948   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4949 qed

  4950

  4951 lemma compact_closed_differences:

  4952   fixes s t :: "'a::real_normed_vector set"

  4953   assumes "compact s"  "closed t"

  4954   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4955 proof-

  4956   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4957     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4958   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4959 qed

  4960

  4961 lemma closed_compact_differences:

  4962   fixes s t :: "'a::real_normed_vector set"

  4963   assumes "closed s" "compact t"

  4964   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4965 proof-

  4966   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  4967     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4968  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  4969 qed

  4970

  4971 lemma closed_translation:

  4972   fixes a :: "'a::real_normed_vector"

  4973   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  4974 proof-

  4975   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  4976   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  4977 qed

  4978

  4979 lemma translation_Compl:

  4980   fixes a :: "'a::ab_group_add"

  4981   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  4982   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  4983

  4984 lemma translation_UNIV:

  4985   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  4986   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  4987

  4988 lemma translation_diff:

  4989   fixes a :: "'a::ab_group_add"

  4990   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  4991   by auto

  4992

  4993 lemma closure_translation:

  4994   fixes a :: "'a::real_normed_vector"

  4995   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  4996 proof-

  4997   have *:"op + a  (- s) = - op + a  s"

  4998     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  4999   show ?thesis unfolding closure_interior translation_Compl

  5000     using interior_translation[of a "- s"] unfolding * by auto

  5001 qed

  5002

  5003 lemma frontier_translation:

  5004   fixes a :: "'a::real_normed_vector"

  5005   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5006   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  5007

  5008

  5009 subsection {* Separation between points and sets *}

  5010

  5011 lemma separate_point_closed:

  5012   fixes s :: "'a::heine_borel set"

  5013   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  5014 proof(cases "s = {}")

  5015   case True

  5016   thus ?thesis by(auto intro!: exI[where x=1])

  5017 next

  5018   case False

  5019   assume "closed s" "a \<notin> s"

  5020   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

  5021   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5022 qed

  5023

  5024 lemma separate_compact_closed:

  5025   fixes s t :: "'a::heine_borel set"

  5026   assumes "compact s" and t: "closed t" "s \<inter> t = {}"

  5027   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5028 proof cases

  5029   assume "s \<noteq> {} \<and> t \<noteq> {}"

  5030   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  5031   let ?inf = "\<lambda>x. infdist x t"

  5032   have "continuous_on s ?inf"

  5033     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

  5034   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  5035     using continuous_attains_inf[OF compact s s \<noteq> {}] by auto

  5036   then have "0 < ?inf x"

  5037     using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  5038   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  5039     using x by (auto intro: order_trans infdist_le)

  5040   ultimately show ?thesis

  5041     by auto

  5042 qed (auto intro!: exI[of _ 1])

  5043

  5044 lemma separate_closed_compact:

  5045   fixes s t :: "'a::heine_borel set"

  5046   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5047   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5048 proof-

  5049   have *:"t \<inter> s = {}" using assms(3) by auto

  5050   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5051     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5052     by (auto simp add: dist_commute)

  5053 qed

  5054

  5055

  5056 subsection {* Intervals *}

  5057

  5058 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5059   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and

  5060   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5061   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5062

  5063 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5064   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5065   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  5066   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5067

  5068 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5069  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and

  5070  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  5071 proof-

  5072   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  5073     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto

  5074     hence "a\<bullet>i < b\<bullet>i" by auto

  5075     hence False using as by auto  }

  5076   moreover

  5077   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  5078     let ?x = "(1/2) *\<^sub>R (a + b)"

  5079     { fix i :: 'a assume i:"i\<in>Basis"

  5080       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5081       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  5082         by (auto simp: inner_add_left) }

  5083     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5084   ultimately show ?th1 by blast

  5085

  5086   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  5087     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto

  5088     hence "a\<bullet>i \<le> b\<bullet>i" by auto

  5089     hence False using as by auto  }

  5090   moreover

  5091   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  5092     let ?x = "(1/2) *\<^sub>R (a + b)"

  5093     { fix i :: 'a assume i:"i\<in>Basis"

  5094       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5095       hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"

  5096         by (auto simp: inner_add_left) }

  5097     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5098   ultimately show ?th2 by blast

  5099 qed

  5100

  5101 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5102   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and

  5103   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  5104   unfolding interval_eq_empty[of a b] by fastforce+

  5105

  5106 lemma interval_sing:

  5107   fixes a :: "'a::ordered_euclidean_space"

  5108   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5109   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5110   by (auto intro: euclidean_eqI simp: ex_in_conv)

  5111

  5112 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5113  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and

  5114  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5115  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and

  5116  "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  5117   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5118   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5119

  5120 lemma interval_open_subset_closed:

  5121   fixes a :: "'a::ordered_euclidean_space"

  5122   shows "{a<..<b} \<subseteq> {a .. b}"

  5123   unfolding subset_eq [unfolded Ball_def] mem_interval

  5124   by (fast intro: less_imp_le)

  5125

  5126 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5127  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and

  5128  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and

  5129  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and

  5130  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)

  5131 proof-

  5132   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5133   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)

  5134   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5135     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5136     fix i :: 'a assume i:"i\<in>Basis"

  5137     (** TODO combine the following two parts as done in the HOL_light version. **)

  5138     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5139       assume as2: "a\<bullet>i > c\<bullet>i"

  5140       { fix j :: 'a assume j:"j\<in>Basis"

  5141         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  5142           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i

  5143           by (auto simp add: as2)  }

  5144       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto

  5145       moreover

  5146       have "?x\<notin>{a .. b}"

  5147         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5148         using as(2)[THEN bspec[where x=i]] and as2 i

  5149         by auto

  5150       ultimately have False using as by auto  }

  5151     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto

  5152     moreover

  5153     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5154       assume as2: "b\<bullet>i < d\<bullet>i"

  5155       { fix j :: 'a assume "j\<in>Basis"

  5156         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  5157           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]

  5158           by (auto simp add: as2) }

  5159       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5160       moreover

  5161       have "?x\<notin>{a .. b}"

  5162         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5163         using as(2)[THEN bspec[where x=i]] and as2 using i

  5164         by auto

  5165       ultimately have False using as by auto  }

  5166     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto

  5167     ultimately

  5168     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  5169   } note part1 = this

  5170   show ?th3

  5171     unfolding subset_eq and Ball_def and mem_interval

  5172     apply(rule,rule,rule,rule)

  5173     apply(rule part1)

  5174     unfolding subset_eq and Ball_def and mem_interval

  5175     prefer 4

  5176     apply auto

  5177     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+

  5178   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5179     fix i :: 'a assume i:"i\<in>Basis"

  5180     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5181     hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this

  5182   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5183     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5184     apply auto by(erule_tac x=xa in allE, simp)+

  5185 qed

  5186

  5187 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5188  "{a .. b} \<inter> {c .. d} =  {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"

  5189   unfolding set_eq_iff and Int_iff and mem_interval by auto

  5190

  5191 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5192   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and

  5193   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and

  5194   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and

  5195   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)

  5196 proof-

  5197   let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"

  5198   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  5199       (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"

  5200     by blast

  5201   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

  5202   show ?th1 unfolding * by (intro **) auto

  5203   show ?th2 unfolding * by (intro **) auto

  5204   show ?th3 unfolding * by (intro **) auto

  5205   show ?th4 unfolding * by (intro **) auto

  5206 qed

  5207

  5208 (* Moved interval_open_subset_closed a bit upwards *)

  5209

  5210 lemma open_interval[intro]:

  5211   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5212 proof-

  5213   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"

  5214     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

  5215       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

  5216   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"

  5217     by (auto simp add: eucl_less [where 'a='a])

  5218   finally show "open {a<..<b}" .

  5219 qed

  5220

  5221 lemma closed_interval[intro]:

  5222   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5223 proof-

  5224   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"

  5225     by (intro closed_INT ballI continuous_closed_vimage allI

  5226       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

  5227   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"

  5228     by (auto simp add: eucl_le [where 'a='a])

  5229   finally show "closed {a .. b}" .

  5230 qed

  5231

  5232 lemma interior_closed_interval [intro]:

  5233   fixes a b :: "'a::ordered_euclidean_space"

  5234   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5235 proof(rule subset_antisym)

  5236   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5237     by (rule interior_maximal)

  5238 next

  5239   { fix x assume "x \<in> interior {a..b}"

  5240     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5241     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

  5242     { fix i :: 'a assume i:"i\<in>Basis"

  5243       have "dist (x - (e / 2) *\<^sub>R i) x < e"

  5244            "dist (x + (e / 2) *\<^sub>R i) x < e"

  5245         unfolding dist_norm apply auto

  5246         unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto

  5247       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"

  5248                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"

  5249         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]

  5250         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]

  5251         unfolding mem_interval using i by blast+

  5252       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"

  5253         using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }

  5254     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5255   thus "?L \<subseteq> ?R" ..

  5256 qed

  5257

  5258 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5259 proof-

  5260   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"

  5261   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  5262     { fix i :: 'a assume "i\<in>Basis"

  5263       hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }

  5264     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5265     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5266   thus ?thesis unfolding interval and bounded_iff by auto

  5267 qed

  5268

  5269 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5270  "bounded {a .. b} \<and> bounded {a<..<b}"

  5271   using bounded_closed_interval[of a b]

  5272   using interval_open_subset_closed[of a b]

  5273   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5274   by simp

  5275

  5276 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5277  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5278   using bounded_interval[of a b] by auto

  5279

  5280 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5281   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]

  5282   by (auto simp: compact_eq_seq_compact_metric)

  5283

  5284 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5285   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5286 proof-

  5287   { fix i :: 'a assume "i\<in>Basis"

  5288     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"

  5289       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }

  5290   thus ?thesis unfolding mem_interval by auto

  5291 qed

  5292

  5293 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5294   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5295   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5296 proof-

  5297   { fix i :: 'a assume i:"i\<in>Basis"

  5298     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp

  5299     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5300       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5301       using x unfolding mem_interval using i apply simp

  5302       using y unfolding mem_interval using i apply simp

  5303       done

  5304     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto

  5305     moreover {

  5306     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp

  5307     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5308       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5309       using x unfolding mem_interval using i apply simp

  5310       using y unfolding mem_interval using i apply simp

  5311       done

  5312     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto

  5313     } ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }

  5314   thus ?thesis unfolding mem_interval by auto

  5315 qed

  5316

  5317 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5318   assumes "{a<..<b} \<noteq> {}"

  5319   shows "closure {a<..<b} = {a .. b}"

  5320 proof-

  5321   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5322   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5323   { fix x assume as:"x \<in> {a .. b}"

  5324     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5325     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5326       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5327       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5328         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5329         by (auto simp add: algebra_simps)

  5330       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

  5331       hence False using fn unfolding f_def using xc by auto  }

  5332     moreover

  5333     { assume "\<not> (f ---> x) sequentially"

  5334       { fix e::real assume "e>0"

  5335         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

  5336         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5337         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)

  5338         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5339       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5340         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5341       hence "(f ---> x) sequentially" unfolding f_def

  5342         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]

  5343         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  }

  5344     ultimately have "x \<in> closure {a<..<b}"

  5345       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5346   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5347 qed

  5348

  5349 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5350   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5351 proof-

  5352   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5353   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"

  5354   { fix x assume "x\<in>s"

  5355     fix i :: 'a assume i:"i\<in>Basis"

  5356     hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF x\<in>s]

  5357       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }

  5358   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5359 qed

  5360

  5361 lemma bounded_subset_open_interval:

  5362   fixes s :: "('a::ordered_euclidean_space) set"

  5363   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5364   by (auto dest!: bounded_subset_open_interval_symmetric)

  5365

  5366 lemma bounded_subset_closed_interval_symmetric:

  5367   fixes s :: "('a::ordered_euclidean_space) set"

  5368   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5369 proof-

  5370   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5371   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5372 qed

  5373

  5374 lemma bounded_subset_closed_interval:

  5375   fixes s :: "('a::ordered_euclidean_space) set"

  5376   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5377   using bounded_subset_closed_interval_symmetric[of s] by auto

  5378

  5379 lemma frontier_closed_interval:

  5380   fixes a b :: "'a::ordered_euclidean_space"

  5381   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5382   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5383

  5384 lemma frontier_open_interval:

  5385   fixes a b :: "'a::ordered_euclidean_space"

  5386   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5387 proof(cases "{a<..<b} = {}")

  5388   case True thus ?thesis using frontier_empty by auto

  5389 next

  5390   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5391 qed

  5392

  5393 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5394   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5395   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5396

  5397

  5398 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5399

  5400 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5401   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"

  5402 proof-

  5403   { fix i :: 'a assume i:"i\<in>Basis"

  5404     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"

  5405     { assume "x\<bullet>i > b\<bullet>i"

  5406       then obtain y where "y \<bullet> i \<le> b \<bullet> i"  "y \<noteq> x"  "dist y x < x\<bullet>i - b\<bullet>i"

  5407         using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto

  5408       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i

  5409         by auto }

  5410     hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto  }

  5411   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5412 qed

  5413

  5414 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5415   shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"

  5416 proof-

  5417   { fix i :: 'a assume i:"i\<in>Basis"

  5418     fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"

  5419     { assume "a\<bullet>i > x\<bullet>i"

  5420       then obtain y where "a \<bullet> i \<le> y \<bullet> i"  "y \<noteq> x"  "dist y x < a\<bullet>i - x\<bullet>i"

  5421         using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto

  5422       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }

  5423     hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto  }

  5424   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5425 qed

  5426

  5427 lemma open_box: "open (box a b)"

  5428 proof -

  5429   have "open (\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i})"

  5430     by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)

  5431   also have "(\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i}) = box a b"

  5432     by (auto simp add: box_def inner_commute)

  5433   finally show ?thesis .

  5434 qed

  5435

  5436 instance euclidean_space \<subseteq> second_countable_topology

  5437 proof

  5438   def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"

  5439   then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp

  5440   def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"

  5441   then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp

  5442   def B \<equiv> "(\<lambda>f. box (a f) (b f))  (Basis \<rightarrow>\<^isub>E (\<rat> \<times> \<rat>))"

  5443

  5444   have "Ball B open" by (simp add: B_def open_box)

  5445   moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"

  5446   proof safe

  5447     fix A::"'a set" assume "open A"

  5448     show "\<exists>B'\<subseteq>B. \<Union>B' = A"

  5449       apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])

  5450       apply (subst (3) open_UNION_box[OF open A])

  5451       apply (auto simp add: a b B_def)

  5452       done

  5453   qed

  5454   ultimately

  5455   have "topological_basis B" unfolding topological_basis_def by blast

  5456   moreover

  5457   have "countable B" unfolding B_def

  5458     by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)

  5459   ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"

  5460     by (blast intro: topological_basis_imp_subbasis)

  5461 qed

  5462

  5463 instance euclidean_space \<subseteq> polish_space ..

  5464

  5465 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5466

  5467 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5468   (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"

  5469

  5470 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5471   "is_interval {a<..<b}" (is ?th2) proof -

  5472   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5473     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5474

  5475 lemma is_interval_empty:

  5476  "is_interval {}"

  5477   unfolding is_interval_def

  5478   by simp

  5479

  5480 lemma is_interval_univ:

  5481  "is_interval UNIV"

  5482   unfolding is_interval_def

  5483   by simp

  5484

  5485

  5486 subsection {* Closure of halfspaces and hyperplanes *}

  5487

  5488 lemma isCont_open_vimage:

  5489   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5490 proof -

  5491   from assms(1) have "continuous_on UNIV f"

  5492     unfolding isCont_def continuous_on_def within_UNIV by simp

  5493   hence "open {x \<in> UNIV. f x \<in> s}"

  5494     using open_UNIV open s by (rule continuous_open_preimage)

  5495   thus "open (f - s)"

  5496     by (simp add: vimage_def)

  5497 qed

  5498

  5499 lemma isCont_closed_vimage:

  5500   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5501   using assms unfolding closed_def vimage_Compl [symmetric]

  5502   by (rule isCont_open_vimage)

  5503

  5504 lemma open_Collect_less:

  5505   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5506   assumes f: "\<And>x. isCont f x"

  5507   assumes g: "\<And>x. isCont g x"

  5508   shows "open {x. f x < g x}"

  5509 proof -

  5510   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5511     using isCont_diff [OF g f] open_real_greaterThan

  5512     by (rule isCont_open_vimage)

  5513   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5514     by auto

  5515   finally show ?thesis .

  5516 qed

  5517

  5518 lemma closed_Collect_le:

  5519   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5520   assumes f: "\<And>x. isCont f x"

  5521   assumes g: "\<And>x. isCont g x"

  5522   shows "closed {x. f x \<le> g x}"

  5523 proof -

  5524   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5525     using isCont_diff [OF g f] closed_real_atLeast

  5526     by (rule isCont_closed_vimage)

  5527   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5528     by auto

  5529   finally show ?thesis .

  5530 qed

  5531

  5532 lemma closed_Collect_eq:

  5533   fixes f g :: "'a::t2_space \<Rightarrow> 'b::t2_space"

  5534   assumes f: "\<And>x. isCont f x"

  5535   assumes g: "\<And>x. isCont g x"

  5536   shows "closed {x. f x = g x}"

  5537 proof -

  5538   have "open {(x::'b, y::'b). x \<noteq> y}"

  5539     unfolding open_prod_def by (auto dest!: hausdorff)

  5540   hence "closed {(x::'b, y::'b). x = y}"

  5541     unfolding closed_def split_def Collect_neg_eq .

  5542   with isCont_Pair [OF f g]

  5543   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5544     by (rule isCont_closed_vimage)

  5545   also have "\<dots> = {x. f x = g x}" by auto

  5546   finally show ?thesis .

  5547 qed

  5548

  5549 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5550   unfolding continuous_at by (intro tendsto_intros)

  5551

  5552 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"

  5553   by (simp add: closed_Collect_le)

  5554

  5555 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

  5556   by (simp add: closed_Collect_le)

  5557

  5558 lemma closed_hyperplane: "closed {x. inner a x = b}"

  5559   by (simp add: closed_Collect_eq)

  5560

  5561 lemma closed_halfspace_component_le:

  5562   shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

  5563   by (simp add: closed_Collect_le)

  5564

  5565 lemma closed_halfspace_component_ge:

  5566   shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

  5567   by (simp add: closed_Collect_le)

  5568

  5569 text {* Openness of halfspaces. *}

  5570

  5571 lemma open_halfspace_lt: "open {x. inner a x < b}"

  5572   by (simp add: open_Collect_less)

  5573

  5574 lemma open_halfspace_gt: "open {x. inner a x > b}"

  5575   by (simp add: open_Collect_less)

  5576

  5577 lemma open_halfspace_component_lt:

  5578   shows "open {x::'a::euclidean_space. x\<bullet>i < a}"

  5579   by (simp add: open_Collect_less)

  5580

  5581 lemma open_halfspace_component_gt:

  5582   shows "open {x::'a::euclidean_space. x\<bullet>i > a}"

  5583   by (simp add: open_Collect_less)

  5584

  5585 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

  5586

  5587 lemma eucl_lessThan_eq_halfspaces:

  5588   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5589   shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"

  5590  by (auto simp: eucl_less[where 'a='a])

  5591

  5592 lemma eucl_greaterThan_eq_halfspaces:

  5593   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5594   shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"

  5595  by (auto simp: eucl_less[where 'a='a])

  5596

  5597 lemma eucl_atMost_eq_halfspaces:

  5598   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5599   shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"

  5600  by (auto simp: eucl_le[where 'a='a])

  5601

  5602 lemma eucl_atLeast_eq_halfspaces:

  5603   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5604   shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"

  5605  by (auto simp: eucl_le[where 'a='a])

  5606

  5607 lemma open_eucl_lessThan[simp, intro]:

  5608   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5609   shows "open {..< a}"

  5610   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

  5611

  5612 lemma open_eucl_greaterThan[simp, intro]:

  5613   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5614   shows "open {a <..}"

  5615   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

  5616

  5617 lemma closed_eucl_atMost[simp, intro]:

  5618   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5619   shows "closed {.. a}"

  5620   unfolding eucl_atMost_eq_halfspaces

  5621   by (simp add: closed_INT closed_Collect_le)

  5622

  5623 lemma closed_eucl_atLeast[simp, intro]:

  5624   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5625   shows "closed {a ..}"

  5626   unfolding eucl_atLeast_eq_halfspaces

  5627   by (simp add: closed_INT closed_Collect_le)

  5628

  5629 text {* This gives a simple derivation of limit component bounds. *}

  5630

  5631 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5632   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"

  5633   shows "l\<bullet>i \<le> b"

  5634   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

  5635

  5636 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5637   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"

  5638   shows "b \<le> l\<bullet>i"

  5639   by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

  5640

  5641 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5642   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"

  5643   shows "l\<bullet>i = b"

  5644   using ev[unfolded order_eq_iff eventually_conj_iff]

  5645   using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto

  5646

  5647 text{* Limits relative to a union.                                               *}

  5648

  5649 lemma eventually_within_Un:

  5650   "eventually P (net within (s \<union> t)) \<longleftrightarrow>

  5651     eventually P (net within s) \<and> eventually P (net within t)"

  5652   unfolding Limits.eventually_within

  5653   by (auto elim!: eventually_rev_mp)

  5654

  5655 lemma Lim_within_union:

  5656  "(f ---> l) (net within (s \<union> t)) \<longleftrightarrow>

  5657   (f ---> l) (net within s) \<and> (f ---> l) (net within t)"

  5658   unfolding tendsto_def

  5659   by (auto simp add: eventually_within_Un)

  5660

  5661 lemma Lim_topological:

  5662  "(f ---> l) net \<longleftrightarrow>

  5663         trivial_limit net \<or>

  5664         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5665   unfolding tendsto_def trivial_limit_eq by auto

  5666

  5667 lemma continuous_on_union:

  5668   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t f"

  5669   shows "continuous_on (s \<union> t) f"

  5670   using assms unfolding continuous_on Lim_within_union

  5671   unfolding Lim_topological trivial_limit_within closed_limpt by auto

  5672

  5673 lemma continuous_on_cases:

  5674   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"

  5675           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"

  5676   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

  5677 proof-

  5678   let ?h = "(\<lambda>x. if P x then f x else g x)"

  5679   have "\<forall>x\<in>s. f x = (if P x then f x else g x)" using assms(5) by auto

  5680   hence "continuous_on s ?h" using continuous_on_eq[of s f ?h] using assms(3) by auto

  5681   moreover

  5682   have "\<forall>x\<in>t. g x = (if P x then f x else g x)" using assms(5) by auto

  5683   hence "continuous_on t ?h" using continuous_on_eq[of t g ?h] using assms(4) by auto

  5684   ultimately show ?thesis using continuous_on_union[OF assms(1,2), of ?h] by auto

  5685 qed

  5686

  5687

  5688 text{* Some more convenient intermediate-value theorem formulations.             *}

  5689

  5690 lemma connected_ivt_hyperplane:

  5691   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5692   shows "\<exists>z \<in> s. inner a z = b"

  5693 proof(rule ccontr)

  5694   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5695   let ?A = "{x. inner a x < b}"

  5696   let ?B = "{x. inner a x > b}"

  5697   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5698   moreover have "?A \<inter> ?B = {}" by auto

  5699   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5700   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

  5701 qed

  5702

  5703 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5704  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s.  z\<bullet>k = a)"

  5705   using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute)

  5706

  5707

  5708 subsection {* Homeomorphisms *}

  5709

  5710 definition "homeomorphism s t f g \<equiv>

  5711      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5712      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5713

  5714 definition

  5715   homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"

  5716     (infixr "homeomorphic" 60) where

  5717   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  5718

  5719 lemma homeomorphic_refl: "s homeomorphic s"

  5720   unfolding homeomorphic_def

  5721   unfolding homeomorphism_def

  5722   using continuous_on_id

  5723   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5724   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5725   by blast

  5726

  5727 lemma homeomorphic_sym:

  5728  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  5729 unfolding homeomorphic_def

  5730 unfolding homeomorphism_def

  5731 by blast

  5732

  5733 lemma homeomorphic_trans:

  5734   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
<