src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author wenzelm Tue Aug 13 16:25:47 2013 +0200 (2013-08-13) changeset 53015 a1119cf551e8 parent 52625 b74bf6c0e5a1 child 53255 addd7b9b2bff permissions -rw-r--r--
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
     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 lemma Lim_within_open:

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

    39   shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

    40   by (fact tendsto_within_open)

    41

    42 lemma continuous_on_union:

    43   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"

    44   by (fact continuous_on_closed_Un)

    45

    46 lemma continuous_on_cases:

    47   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>

    48     \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>

    49     continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"

    50   by (rule continuous_on_If) auto

    51

    52 subsection {* Topological Basis *}

    53

    54 context topological_space

    55 begin

    56

    57 definition "topological_basis B =

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

    59

    60 lemma topological_basis:

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

    62   unfolding topological_basis_def

    63   apply safe

    64      apply fastforce

    65     apply fastforce

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

    67    apply simp

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

    69   apply auto

    70   done

    71

    72 lemma topological_basis_iff:

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

    74   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'))"

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

    76 proof safe

    77   fix O' and x::'a

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

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

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

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

    82 next

    83   assume H: ?rhs

    84   show "topological_basis B" using assms unfolding topological_basis_def

    85   proof safe

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

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

    88       by (force intro: bchoice simp: Bex_def)

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

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

    91   qed

    92 qed

    93

    94 lemma topological_basisI:

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

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

    97   shows "topological_basis B"

    98   using assms by (subst topological_basis_iff) auto

    99

   100 lemma topological_basisE:

   101   fixes O'

   102   assumes "topological_basis B"

   103   assumes "open O'"

   104   assumes "x \<in> O'"

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

   106 proof atomize_elim

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

   108   with topological_basis_iff assms

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

   110 qed

   111

   112 lemma topological_basis_open:

   113   assumes "topological_basis B"

   114   assumes "X \<in> B"

   115   shows "open X"

   116   using assms

   117   by (simp add: topological_basis_def)

   118

   119 lemma topological_basis_imp_subbasis:

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

   121 proof (intro ext iffI)

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

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

   124     unfolding topological_basis_def by blast

   125   then show "generate_topology B S"

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

   127 next

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

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

   130 qed

   131

   132 lemma basis_dense:

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

   134   assumes "topological_basis B"

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

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

   137 proof (intro allI impI)

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

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

   140   guess B' . note B' = this

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

   142 qed

   143

   144 end

   145

   146 lemma topological_basis_prod:

   147   assumes A: "topological_basis A" and B: "topological_basis B"

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

   149   unfolding topological_basis_def

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

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

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

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

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

   155     from open_prod_elim[OF open S this]

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

   157       by (metis mem_Sigma_iff)

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

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

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

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

   162   qed auto

   163 qed (metis A B topological_basis_open open_Times)

   164

   165 subsection {* Countable Basis *}

   166

   167 locale countable_basis =

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

   169   assumes is_basis: "topological_basis B"

   170   assumes countable_basis: "countable B"

   171 begin

   172

   173 lemma open_countable_basis_ex:

   174   assumes "open X"

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

   176   using assms countable_basis is_basis unfolding topological_basis_def by blast

   177

   178 lemma open_countable_basisE:

   179   assumes "open X"

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

   181   using assms open_countable_basis_ex by (atomize_elim) simp

   182

   183 lemma countable_dense_exists:

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

   185 proof -

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

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

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

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

   190 qed

   191

   192 lemma countable_dense_setE:

   193   obtains D :: "'a set"

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

   195   using countable_dense_exists by blast

   196

   197 end

   198

   199 lemma (in first_countable_topology) first_countable_basisE:

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

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

   202   using first_countable_basis[of x]

   203   apply atomize_elim

   204   apply (elim exE)

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

   206   apply auto

   207   done

   208

   209 lemma (in first_countable_topology) first_countable_basis_Int_stableE:

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

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

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

   213 proof atomize_elim

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

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

   216   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>

   217         (\<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)"

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

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

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

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

   222   next

   223     let ?int = "\<lambda>N. \<Inter>(from_nat_into A'  N)"

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

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

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

   227   next

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

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

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

   231   qed

   232 qed

   233

   234 lemma (in topological_space) first_countableI:

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

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

   237   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))"

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

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

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

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

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

   243       using subset_range_from_nat_into[OF countable A] by auto }

   244 qed

   245

   246 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

   247 proof

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

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

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

   251   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))"

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

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

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

   255       unfolding mem_Times_iff by (auto intro: open_Times)

   256   next

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

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

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

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

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

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

   263   qed (simp add: A B)

   264 qed

   265

   266 class second_countable_topology = topological_space +

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

   268 begin

   269

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

   271 proof -

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

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

   274

   275   show ?thesis

   276   proof (intro exI conjI)

   277     show "countable ?B"

   278       by (intro countable_image countable_Collect_finite_subset B)

   279     { fix S assume "open S"

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

   281         unfolding B

   282       proof induct

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

   284       next

   285         case (Int a b)

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

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

   288           by blast

   289         show ?case

   290           unfolding x y Int_UN_distrib2

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

   292       next

   293         case (UN K)

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

   295         then guess k unfolding bchoice_iff ..

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

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

   298       next

   299         case (Basis S) then show ?case

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

   301       qed

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

   303         unfolding subset_image_iff by blast }

   304     then show "topological_basis ?B"

   305       unfolding topological_space_class.topological_basis_def

   306       by (safe intro!: topological_space_class.open_Inter)

   307          (simp_all add: B generate_topology.Basis subset_eq)

   308   qed

   309 qed

   310

   311 end

   312

   313 sublocale second_countable_topology <

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

   315   using someI_ex[OF ex_countable_basis]

   316   by unfold_locales safe

   317

   318 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   319 proof

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

   321     using ex_countable_basis by auto

   322   moreover

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

   324     using ex_countable_basis by auto

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

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

   327       topological_basis_imp_subbasis)

   328 qed

   329

   330 instance second_countable_topology \<subseteq> first_countable_topology

   331 proof

   332   fix x :: 'a

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

   334   then have B: "countable B" "topological_basis B"

   335     using countable_basis is_basis

   336     by (auto simp: countable_basis is_basis)

   337   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))"

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

   339        (fastforce simp: topological_space_class.topological_basis_def)+

   340 qed

   341

   342 subsection {* Polish spaces *}

   343

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

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

   346

   347 class polish_space = complete_space + second_countable_topology

   348

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

   350

   351 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))"

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

   353   morphisms "openin" "topology"

   354   unfolding istopology_def by blast

   355

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

   357   using openin[of U] by blast

   358

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

   360   using topology_inverse[unfolded mem_Collect_eq] .

   361

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

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

   364

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

   366 proof-

   367   { assume "T1=T2"

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

   369   moreover

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

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

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

   373     hence "T1 = T2" unfolding openin_inverse .

   374   }

   375   ultimately show ?thesis by blast

   376 qed

   377

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

   379

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

   381

   382 subsubsection {* Main properties of open sets *}

   383

   384 lemma openin_clauses:

   385   fixes U :: "'a topology"

   386   shows "openin U {}"

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

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

   389   using openin[of U] unfolding istopology_def mem_Collect_eq

   390   by fast+

   391

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

   393   unfolding topspace_def by blast

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

   395

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

   397   using openin_clauses by simp

   398

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

   400   using openin_clauses by simp

   401

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

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

   404

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

   406

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

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

   409 proof

   410   assume ?lhs

   411   then show ?rhs by auto

   412 next

   413   assume H: ?rhs

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

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

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

   417   finally show "openin U S" .

   418 qed

   419

   420

   421 subsubsection {* Closed sets *}

   422

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

   424

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

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

   427 lemma closedin_topspace[intro,simp]:

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

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

   430   by (auto simp add: Diff_Un closedin_def)

   431

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

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

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

   435

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

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

   438

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

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

   441   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   442   apply (metis openin_subset subset_eq)

   443   done

   444

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

   446   by (simp add: openin_closedin_eq)

   447

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

   449 proof-

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

   451     by (auto simp add: topspace_def openin_subset)

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

   453 qed

   454

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

   456 proof-

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

   458     by (auto simp add: topspace_def )

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

   460 qed

   461

   462 subsubsection {* Subspace topology *}

   463

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

   465

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

   467   (is "istopology ?L")

   468 proof-

   469   have "?L {}" by blast

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

   471     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

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

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

   474   moreover

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

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

   477       apply (rule set_eqI)

   478       apply (simp add: Ball_def image_iff)

   479       by metis

   480     from K[unfolded th0 subset_image_iff]

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

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

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

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

   485   ultimately show ?thesis

   486     unfolding subset_eq mem_Collect_eq istopology_def by blast

   487 qed

   488

   489 lemma openin_subtopology:

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

   491   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   492   by auto

   493

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

   495   by (auto simp add: topspace_def openin_subtopology)

   496

   497 lemma closedin_subtopology:

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

   499   unfolding closedin_def topspace_subtopology

   500   apply (simp add: openin_subtopology)

   501   apply (rule iffI)

   502   apply clarify

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

   504   by auto

   505

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

   507   unfolding openin_subtopology

   508   apply (rule iffI, clarify)

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

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

   511   apply auto

   512   done

   513

   514 lemma subtopology_superset:

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

   516   shows "subtopology U V = U"

   517 proof-

   518   {fix S

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

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

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

   522     moreover

   523     {assume S: "openin U S"

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

   525         using openin_subset[OF S] UV by auto}

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

   527   then show ?thesis unfolding topology_eq openin_subtopology by blast

   528 qed

   529

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

   531   by (simp add: subtopology_superset)

   532

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

   534   by (simp add: subtopology_superset)

   535

   536 subsubsection {* The standard Euclidean topology *}

   537

   538 definition

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

   540   "euclidean = topology open"

   541

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

   543   unfolding euclidean_def

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

   545   apply (rule topology_inverse[symmetric])

   546   apply (auto simp add: istopology_def)

   547   done

   548

   549 lemma topspace_euclidean: "topspace euclidean = UNIV"

   550   apply (simp add: topspace_def)

   551   apply (rule set_eqI)

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

   553

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

   555   by (simp add: topspace_euclidean topspace_subtopology)

   556

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

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

   559

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

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

   562

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

   564

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

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

   567

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

   569   by (auto simp add: openin_open)

   570

   571 lemma open_openin_trans[trans]:

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

   573   by (metis Int_absorb1  openin_open_Int)

   574

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

   576   by (auto simp add: openin_open)

   577

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

   579   by (simp add: closedin_subtopology closed_closedin Int_ac)

   580

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

   582   by (metis closedin_closed)

   583

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

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

   586   apply auto

   587   apply (frule closedin_closed_Int[of T S])

   588   apply simp

   589   done

   590

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

   592   by (auto simp add: closedin_closed)

   593

   594 lemma openin_euclidean_subtopology_iff:

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

   596   shows "openin (subtopology euclidean U) S

   597   \<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")

   598 proof

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

   600 next

   601   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}"

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

   603     unfolding T_def

   604     apply clarsimp

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

   606     apply (clarsimp simp add: less_diff_eq)

   607     apply (erule rev_bexI)

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

   609     apply (erule le_less_trans [OF dist_triangle])

   610     done

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

   612     unfolding T_def

   613     apply auto

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

   615     apply auto

   616     done

   617   from 1 2 show ?lhs

   618     unfolding openin_open open_dist by fast

   619 qed

   620

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

   622

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

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

   625   unfolding open_openin openin_open by blast

   626

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

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

   629

   630 lemma closedin_trans[trans]:

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

   632            closedin (subtopology euclidean U) T

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

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

   635

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

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

   638

   639

   640 subsection {* Open and closed balls *}

   641

   642 definition

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

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

   645

   646 definition

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

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

   649

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

   651   by (simp add: ball_def)

   652

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

   654   by (simp add: cball_def)

   655

   656 lemma mem_ball_0:

   657   fixes x :: "'a::real_normed_vector"

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

   659   by (simp add: dist_norm)

   660

   661 lemma mem_cball_0:

   662   fixes x :: "'a::real_normed_vector"

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

   664   by (simp add: dist_norm)

   665

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

   667   by simp

   668

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

   670   by simp

   671

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

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

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

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

   676   by (simp add: set_eq_iff) arith

   677

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

   679   by (simp add: set_eq_iff)

   680

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

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

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

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

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

   686

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

   688   unfolding open_dist ball_def mem_Collect_eq Ball_def

   689   unfolding dist_commute

   690   apply clarify

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

   692   using dist_triangle_alt[where z=x]

   693   apply (clarsimp simp add: diff_less_iff)

   694   apply atomize

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

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

   697   apply arith

   698   done

   699

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

   701   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   702

   703 lemma openE[elim?]:

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

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

   706   using assms unfolding open_contains_ball by auto

   707

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

   709   by (metis open_contains_ball subset_eq centre_in_ball)

   710

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

   712   unfolding mem_ball set_eq_iff

   713   apply (simp add: not_less)

   714   apply (metis zero_le_dist order_trans dist_self)

   715   done

   716

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

   718

   719 lemma euclidean_dist_l2:

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

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

   722   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   724

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

   726

   727 lemma rational_boxes:

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

   729   assumes "0 < e"

   730   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"

   731 proof -

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

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

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

   735   proof

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

   737   qed

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

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

   740   proof

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

   742   qed

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

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

   745   show ?thesis

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

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

   748     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"

   749       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   751     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

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

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

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

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

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

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

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

   759       then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"

   760         by (rule power_strict_mono) auto

   761       then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"

   762         by (simp add: power_divide)

   763     qed auto

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

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

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

   767 qed

   768

   769 lemma open_UNION_box:

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

   771   assumes "open M"

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

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

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

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

   776 proof -

   777   {

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

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

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

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

   782       "\<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"

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

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

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

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

   787   }

   788   then show ?thesis by (auto simp: I_def)

   789 qed

   790

   791

   792 subsection{* Connectedness *}

   793

   794 lemma connected_local:

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

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

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

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

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

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

   801                  ~(e2 = {}))"

   802 unfolding connected_def openin_open by (safe, blast+)

   803

   804 lemma exists_diff:

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

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

   807 proof-

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

   809   moreover

   810   {fix S assume H: "P S"

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

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

   813   ultimately show ?thesis by metis

   814 qed

   815

   816 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   819 proof-

   820   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> {})"

   821     unfolding connected_def openin_open closedin_closed

   822     apply (subst exists_diff)

   823     apply blast

   824     done

   825   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> {})"

   826     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

   827     apply (simp add: closed_def)

   828     apply metis

   829     done

   830

   831   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'))"

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

   833     unfolding connected_def openin_open closedin_closed by auto

   834   {fix e2

   835     {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)"

   836         by auto}

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

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

   839   then show ?thesis unfolding th0 th1 by simp

   840 qed

   841

   842 lemma connected_empty[simp, intro]: "connected {}"  (* FIXME duplicate? *)

   843   by simp

   844

   845

   846 subsection{* Limit points *}

   847

   848 definition (in topological_space)

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

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

   851

   852 lemma islimptI:

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

   854   shows "x islimpt S"

   855   using assms unfolding islimpt_def by auto

   856

   857 lemma islimptE:

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

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

   860   using assms unfolding islimpt_def by auto

   861

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

   863   unfolding islimpt_def eventually_at_topological by auto

   864

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

   866   unfolding islimpt_def by fast

   867

   868 lemma islimpt_approachable:

   869   fixes x :: "'a::metric_space"

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

   871   unfolding islimpt_iff_eventually eventually_at by fast

   872

   873 lemma islimpt_approachable_le:

   874   fixes x :: "'a::metric_space"

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

   876   unfolding islimpt_approachable

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

   878     THEN arg_cong [where f=Not]]

   879   by (simp add: Bex_def conj_commute conj_left_commute)

   880

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

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

   883

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

   885   unfolding islimpt_def by blast

   886

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

   888

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

   890   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   891

   892 lemma perfect_choose_dist:

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

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

   895 using islimpt_UNIV [of x]

   896 by (simp add: islimpt_approachable)

   897

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

   899   unfolding closed_def

   900   apply (subst open_subopen)

   901   apply (simp add: islimpt_def subset_eq)

   902   apply (metis ComplE ComplI)

   903   done

   904

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

   906   unfolding islimpt_def by auto

   907

   908 lemma finite_set_avoid:

   909   fixes a :: "'a::metric_space"

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

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

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

   913 next

   914   case (2 x F)

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

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

   917   moreover

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

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

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

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

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

   923   ultimately show ?case by blast

   924 qed

   925

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

   927   by (simp add: islimpt_iff_eventually eventually_conj_iff)

   928

   929 lemma discrete_imp_closed:

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

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

   932   shows "closed S"

   933 proof-

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

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

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

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

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

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

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

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

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

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

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

   945 qed

   946

   947

   948 subsection {* Interior of a Set *}

   949

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

   951

   952 lemma interiorI [intro?]:

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

   954   shows "x \<in> interior S"

   955   using assms unfolding interior_def by fast

   956

   957 lemma interiorE [elim?]:

   958   assumes "x \<in> interior S"

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

   960   using assms unfolding interior_def by fast

   961

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

   963   by (simp add: interior_def open_Union)

   964

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

   966   by (auto simp add: interior_def)

   967

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

   969   by (auto simp add: interior_def)

   970

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

   972   by (intro equalityI interior_subset interior_maximal subset_refl)

   973

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

   975   by (metis open_interior interior_open)

   976

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

   978   by (metis interior_maximal interior_subset subset_trans)

   979

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

   981   using open_empty by (rule interior_open)

   982

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

   984   using open_UNIV by (rule interior_open)

   985

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

   987   using open_interior by (rule interior_open)

   988

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

   990   by (auto simp add: interior_def)

   991

   992 lemma interior_unique:

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

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

   995   shows "interior S = T"

   996   by (intro equalityI assms interior_subset open_interior interior_maximal)

   997

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

   999   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1000     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1001

  1002 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1003   using open_contains_ball_eq [where S="interior S"]

  1004   by (simp add: open_subset_interior)

  1005

  1006 lemma interior_limit_point [intro]:

  1007   fixes x :: "'a::perfect_space"

  1008   assumes x: "x \<in> interior S" shows "x islimpt S"

  1009   using x islimpt_UNIV [of x]

  1010   unfolding interior_def islimpt_def

  1011   apply (clarsimp, rename_tac T T')

  1012   apply (drule_tac x="T \<inter> T'" in spec)

  1013   apply (auto simp add: open_Int)

  1014   done

  1015

  1016 lemma interior_closed_Un_empty_interior:

  1017   assumes cS: "closed S" and iT: "interior T = {}"

  1018   shows "interior (S \<union> T) = interior S"

  1019 proof

  1020   show "interior S \<subseteq> interior (S \<union> T)"

  1021     by (rule interior_mono, rule Un_upper1)

  1022 next

  1023   show "interior (S \<union> T) \<subseteq> interior S"

  1024   proof

  1025     fix x assume "x \<in> interior (S \<union> T)"

  1026     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1027     show "x \<in> interior S"

  1028     proof (rule ccontr)

  1029       assume "x \<notin> interior S"

  1030       with x \<in> R open R obtain y where "y \<in> R - S"

  1031         unfolding interior_def by fast

  1032       from open R closed S have "open (R - S)" by (rule open_Diff)

  1033       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

  1034       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

  1035       show "False" unfolding interior_def by fast

  1036     qed

  1037   qed

  1038 qed

  1039

  1040 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1041 proof (rule interior_unique)

  1042   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1043     by (intro Sigma_mono interior_subset)

  1044   show "open (interior A \<times> interior B)"

  1045     by (intro open_Times open_interior)

  1046   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

  1047   proof (safe)

  1048     fix x y assume "(x, y) \<in> T"

  1049     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1050       using open T unfolding open_prod_def by fast

  1051     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1052       using T \<subseteq> A \<times> B by auto

  1053     thus "x \<in> interior A" and "y \<in> interior B"

  1054       by (auto intro: interiorI)

  1055   qed

  1056 qed

  1057

  1058

  1059 subsection {* Closure of a Set *}

  1060

  1061 definition "closure S = S \<union> {x | x. x islimpt S}"

  1062

  1063 lemma interior_closure: "interior S = - (closure (- S))"

  1064   unfolding interior_def closure_def islimpt_def by auto

  1065

  1066 lemma closure_interior: "closure S = - interior (- S)"

  1067   unfolding interior_closure by simp

  1068

  1069 lemma closed_closure[simp, intro]: "closed (closure S)"

  1070   unfolding closure_interior by (simp add: closed_Compl)

  1071

  1072 lemma closure_subset: "S \<subseteq> closure S"

  1073   unfolding closure_def by simp

  1074

  1075 lemma closure_hull: "closure S = closed hull S"

  1076   unfolding hull_def closure_interior interior_def by auto

  1077

  1078 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1079   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1080

  1081 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1082   unfolding closure_eq .

  1083

  1084 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1085   unfolding closure_hull by (rule hull_hull)

  1086

  1087 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1088   unfolding closure_hull by (rule hull_mono)

  1089

  1090 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1091   unfolding closure_hull by (rule hull_minimal)

  1092

  1093 lemma closure_unique:

  1094   assumes "S \<subseteq> T" and "closed T"

  1095   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1096   shows "closure S = T"

  1097   using assms unfolding closure_hull by (rule hull_unique)

  1098

  1099 lemma closure_empty [simp]: "closure {} = {}"

  1100   using closed_empty by (rule closure_closed)

  1101

  1102 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1103   using closed_UNIV by (rule closure_closed)

  1104

  1105 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1106   unfolding closure_interior by simp

  1107

  1108 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1109   using closure_empty closure_subset[of S]

  1110   by blast

  1111

  1112 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1113   using closure_eq[of S] closure_subset[of S]

  1114   by simp

  1115

  1116 lemma open_inter_closure_eq_empty:

  1117   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1118   using open_subset_interior[of S "- T"]

  1119   using interior_subset[of "- T"]

  1120   unfolding closure_interior

  1121   by auto

  1122

  1123 lemma open_inter_closure_subset:

  1124   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1125 proof

  1126   fix x

  1127   assume as: "open S" "x \<in> S \<inter> closure T"

  1128   { assume *:"x islimpt T"

  1129     have "x islimpt (S \<inter> T)"

  1130     proof (rule islimptI)

  1131       fix A

  1132       assume "x \<in> A" "open A"

  1133       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1134         by (simp_all add: open_Int)

  1135       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1136         by (rule islimptE)

  1137       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1138         by simp_all

  1139       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1140     qed

  1141   }

  1142   then show "x \<in> closure (S \<inter> T)" using as

  1143     unfolding closure_def

  1144     by blast

  1145 qed

  1146

  1147 lemma closure_complement: "closure (- S) = - interior S"

  1148   unfolding closure_interior by simp

  1149

  1150 lemma interior_complement: "interior (- S) = - closure S"

  1151   unfolding closure_interior by simp

  1152

  1153 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1154 proof (rule closure_unique)

  1155   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1156     by (intro Sigma_mono closure_subset)

  1157   show "closed (closure A \<times> closure B)"

  1158     by (intro closed_Times closed_closure)

  1159   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

  1160     apply (simp add: closed_def open_prod_def, clarify)

  1161     apply (rule ccontr)

  1162     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1163     apply (simp add: closure_interior interior_def)

  1164     apply (drule_tac x=C in spec)

  1165     apply (drule_tac x=D in spec)

  1166     apply auto

  1167     done

  1168 qed

  1169

  1170

  1171 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1172   unfolding closure_def using islimpt_punctured by blast

  1173

  1174

  1175 subsection {* Frontier (aka boundary) *}

  1176

  1177 definition "frontier S = closure S - interior S"

  1178

  1179 lemma frontier_closed: "closed(frontier S)"

  1180   by (simp add: frontier_def closed_Diff)

  1181

  1182 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1183   by (auto simp add: frontier_def interior_closure)

  1184

  1185 lemma frontier_straddle:

  1186   fixes a :: "'a::metric_space"

  1187   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))"

  1188   unfolding frontier_def closure_interior

  1189   by (auto simp add: mem_interior subset_eq ball_def)

  1190

  1191 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1192   by (metis frontier_def closure_closed Diff_subset)

  1193

  1194 lemma frontier_empty[simp]: "frontier {} = {}"

  1195   by (simp add: frontier_def)

  1196

  1197 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1198 proof-

  1199   { assume "frontier S \<subseteq> S"

  1200     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

  1201     hence "closed S" using closure_subset_eq by auto

  1202   }

  1203   thus ?thesis using frontier_subset_closed[of S] ..

  1204 qed

  1205

  1206 lemma frontier_complement: "frontier(- S) = frontier S"

  1207   by (auto simp add: frontier_def closure_complement interior_complement)

  1208

  1209 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1210   using frontier_complement frontier_subset_eq[of "- S"]

  1211   unfolding open_closed by auto

  1212

  1213 subsection {* Filters and the eventually true'' quantifier *}

  1214

  1215 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1216     (infixr "indirection" 70)

  1217   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1218

  1219 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1220

  1221 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1222 proof

  1223   assume "trivial_limit (at a within S)"

  1224   thus "\<not> a islimpt S"

  1225     unfolding trivial_limit_def

  1226     unfolding eventually_at_topological

  1227     unfolding islimpt_def

  1228     apply (clarsimp simp add: set_eq_iff)

  1229     apply (rename_tac T, rule_tac x=T in exI)

  1230     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1231     done

  1232 next

  1233   assume "\<not> a islimpt S"

  1234   thus "trivial_limit (at a within S)"

  1235     unfolding trivial_limit_def

  1236     unfolding eventually_at_topological

  1237     unfolding islimpt_def

  1238     apply clarsimp

  1239     apply (rule_tac x=T in exI)

  1240     apply auto

  1241     done

  1242 qed

  1243

  1244 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1245   using trivial_limit_within [of a UNIV] by simp

  1246

  1247 lemma trivial_limit_at:

  1248   fixes a :: "'a::perfect_space"

  1249   shows "\<not> trivial_limit (at a)"

  1250   by (rule at_neq_bot)

  1251

  1252 lemma trivial_limit_at_infinity:

  1253   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1254   unfolding trivial_limit_def eventually_at_infinity

  1255   apply clarsimp

  1256   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1257    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1258   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1259   apply (drule_tac x=UNIV in spec, simp)

  1260   done

  1261

  1262 lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"

  1263   using islimpt_in_closure by (metis trivial_limit_within)

  1264

  1265 text {* Some property holds "sufficiently close" to the limit point. *}

  1266

  1267 lemma eventually_at2:

  1268   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1269 unfolding eventually_at dist_nz by auto

  1270

  1271 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

  1272   unfolding trivial_limit_def

  1273   by (auto elim: eventually_rev_mp)

  1274

  1275 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1276   by simp

  1277

  1278 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1279   by (simp add: filter_eq_iff)

  1280

  1281 text{* Combining theorems for "eventually" *}

  1282

  1283 lemma eventually_rev_mono:

  1284   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1285 using eventually_mono [of P Q] by fast

  1286

  1287 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

  1288   by (simp add: eventually_False)

  1289

  1290

  1291 subsection {* Limits *}

  1292

  1293 lemma Lim:

  1294  "(f ---> l) net \<longleftrightarrow>

  1295         trivial_limit net \<or>

  1296         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1297   unfolding tendsto_iff trivial_limit_eq by auto

  1298

  1299 text{* Show that they yield usual definitions in the various cases. *}

  1300

  1301 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1302            (\<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)"

  1303   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1304

  1305 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1306         (\<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)"

  1307   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1308

  1309 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1310         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1311   by (auto simp add: tendsto_iff eventually_at2)

  1312

  1313 lemma Lim_at_infinity:

  1314   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1315   by (auto simp add: tendsto_iff eventually_at_infinity)

  1316

  1317 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1318   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1319

  1320 text{* The expected monotonicity property. *}

  1321

  1322 lemma Lim_within_empty: "(f ---> l) (at x within {})"

  1323   unfolding tendsto_def eventually_at_filter by simp

  1324

  1325 lemma Lim_Un: assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1326   shows "(f ---> l) (at x within (S \<union> T))"

  1327   using assms unfolding tendsto_def eventually_at_filter

  1328   apply clarify

  1329   apply (drule spec, drule (1) mp, drule (1) mp)

  1330   apply (drule spec, drule (1) mp, drule (1) mp)

  1331   apply (auto elim: eventually_elim2)

  1332   done

  1333

  1334 lemma Lim_Un_univ:

  1335  "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = UNIV

  1336         ==> (f ---> l) (at x)"

  1337   by (metis Lim_Un)

  1338

  1339 text{* Interrelations between restricted and unrestricted limits. *}

  1340

  1341

  1342 lemma Lim_at_within: (* FIXME: rename *)

  1343   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1344   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1345

  1346 lemma eventually_within_interior:

  1347   assumes "x \<in> interior S"

  1348   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1349 proof -

  1350   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1351   { assume "?lhs"

  1352     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1353       unfolding eventually_at_topological

  1354       by auto

  1355     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1356       by auto

  1357     then have "?rhs"

  1358       unfolding eventually_at_topological by auto

  1359   }

  1360   moreover

  1361   { assume "?rhs" hence "?lhs"

  1362       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1363   }

  1364   ultimately show "?thesis" ..

  1365 qed

  1366

  1367 lemma at_within_interior:

  1368   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1369   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1370

  1371 lemma Lim_within_LIMSEQ:

  1372   fixes a :: "'a::metric_space"

  1373   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1374   shows "(X ---> L) (at a within T)"

  1375   using assms unfolding tendsto_def [where l=L]

  1376   by (simp add: sequentially_imp_eventually_within)

  1377

  1378 lemma Lim_right_bound:

  1379   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1380     'b::{linorder_topology, conditionally_complete_linorder}"

  1381   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"

  1382   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1383   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1384 proof cases

  1385   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1386 next

  1387   assume e: "{x<..} \<inter> I \<noteq> {}"

  1388   show ?thesis

  1389   proof (rule order_tendstoI)

  1390     fix a assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1391     { fix y assume "y \<in> {x<..} \<inter> I"

  1392       with e bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1393         by (auto intro: cInf_lower)

  1394       with a have "a < f y" by (blast intro: less_le_trans) }

  1395     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1396       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1397   next

  1398     fix a assume "Inf (f  ({x<..} \<inter> I)) < a"

  1399     from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a" by auto

  1400     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1401       unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

  1402     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1403       unfolding eventually_at_filter by eventually_elim simp

  1404   qed

  1405 qed

  1406

  1407 text{* Another limit point characterization. *}

  1408

  1409 lemma islimpt_sequential:

  1410   fixes x :: "'a::first_countable_topology"

  1411   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1412     (is "?lhs = ?rhs")

  1413 proof

  1414   assume ?lhs

  1415   from countable_basis_at_decseq[of x] guess A . note A = this

  1416   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1417   { fix n

  1418     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1419       unfolding islimpt_def using A(1,2)[of n] by auto

  1420     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1421       unfolding f_def by (rule someI_ex)

  1422     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }

  1423   then have "\<forall>n. f n \<in> S - {x}" by auto

  1424   moreover have "(\<lambda>n. f n) ----> x"

  1425   proof (rule topological_tendstoI)

  1426     fix S assume "open S" "x \<in> S"

  1427     from A(3)[OF this] \<And>n. f n \<in> A n

  1428     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)

  1429   qed

  1430   ultimately show ?rhs by fast

  1431 next

  1432   assume ?rhs

  1433   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto

  1434   show ?lhs

  1435     unfolding islimpt_def

  1436   proof safe

  1437     fix T assume "open T" "x \<in> T"

  1438     from lim[THEN topological_tendstoD, OF this] f

  1439     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1440       unfolding eventually_sequentially by auto

  1441   qed

  1442 qed

  1443

  1444 lemma Lim_inv: (* TODO: delete *)

  1445   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1446   assumes "(f ---> l) A" and "l \<noteq> 0"

  1447   shows "((inverse o f) ---> inverse l) A"

  1448   unfolding o_def using assms by (rule tendsto_inverse)

  1449

  1450 lemma Lim_null:

  1451   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1452   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1453   by (simp add: Lim dist_norm)

  1454

  1455 lemma Lim_null_comparison:

  1456   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1457   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1458   shows "(f ---> 0) net"

  1459 proof (rule metric_tendsto_imp_tendsto)

  1460   show "(g ---> 0) net" by fact

  1461   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1462     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1463 qed

  1464

  1465 lemma Lim_transform_bound:

  1466   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1467   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1468   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1469   shows "(f ---> 0) net"

  1470   using assms(1) tendsto_norm_zero [OF assms(2)]

  1471   by (rule Lim_null_comparison)

  1472

  1473 text{* Deducing things about the limit from the elements. *}

  1474

  1475 lemma Lim_in_closed_set:

  1476   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1477   shows "l \<in> S"

  1478 proof (rule ccontr)

  1479   assume "l \<notin> S"

  1480   with closed S have "open (- S)" "l \<in> - S"

  1481     by (simp_all add: open_Compl)

  1482   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1483     by (rule topological_tendstoD)

  1484   with assms(2) have "eventually (\<lambda>x. False) net"

  1485     by (rule eventually_elim2) simp

  1486   with assms(3) show "False"

  1487     by (simp add: eventually_False)

  1488 qed

  1489

  1490 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1491

  1492 lemma Lim_dist_ubound:

  1493   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1494   shows "dist a l <= e"

  1495 proof -

  1496   have "dist a l \<in> {..e}"

  1497   proof (rule Lim_in_closed_set)

  1498     show "closed {..e}" by simp

  1499     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1500     show "\<not> trivial_limit net" by fact

  1501     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1502   qed

  1503   thus ?thesis by simp

  1504 qed

  1505

  1506 lemma Lim_norm_ubound:

  1507   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1508   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1509   shows "norm(l) <= e"

  1510 proof -

  1511   have "norm l \<in> {..e}"

  1512   proof (rule Lim_in_closed_set)

  1513     show "closed {..e}" by simp

  1514     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1515     show "\<not> trivial_limit net" by fact

  1516     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1517   qed

  1518   thus ?thesis by simp

  1519 qed

  1520

  1521 lemma Lim_norm_lbound:

  1522   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1523   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1524   shows "e \<le> norm l"

  1525 proof -

  1526   have "norm l \<in> {e..}"

  1527   proof (rule Lim_in_closed_set)

  1528     show "closed {e..}" by simp

  1529     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1530     show "\<not> trivial_limit net" by fact

  1531     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1532   qed

  1533   thus ?thesis by simp

  1534 qed

  1535

  1536 text{* Limit under bilinear function *}

  1537

  1538 lemma Lim_bilinear:

  1539   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1540   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1541   using bounded_bilinear h (f ---> l) net (g ---> m) net

  1542   by (rule bounded_bilinear.tendsto)

  1543

  1544 text{* These are special for limits out of the same vector space. *}

  1545

  1546 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1547   unfolding id_def by (rule tendsto_ident_at)

  1548

  1549 lemma Lim_at_id: "(id ---> a) (at a)"

  1550   unfolding id_def by (rule tendsto_ident_at)

  1551

  1552 lemma Lim_at_zero:

  1553   fixes a :: "'a::real_normed_vector"

  1554   fixes l :: "'b::topological_space"

  1555   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1556   using LIM_offset_zero LIM_offset_zero_cancel ..

  1557

  1558 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1559

  1560 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  1561   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  1562

  1563 lemma netlimit_within:

  1564   "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  1565   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  1566

  1567 lemma netlimit_at:

  1568   fixes a :: "'a::{perfect_space,t2_space}"

  1569   shows "netlimit (at a) = a"

  1570   using netlimit_within [of a UNIV] by simp

  1571

  1572 lemma lim_within_interior:

  1573   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1574   by (metis at_within_interior)

  1575

  1576 lemma netlimit_within_interior:

  1577   fixes x :: "'a::{t2_space,perfect_space}"

  1578   assumes "x \<in> interior S"

  1579   shows "netlimit (at x within S) = x"

  1580   using assms by (metis at_within_interior netlimit_at)

  1581

  1582 text{* Transformation of limit. *}

  1583

  1584 lemma Lim_transform:

  1585   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1586   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1587   shows "(g ---> l) net"

  1588   using tendsto_diff [OF assms(2) assms(1)] by simp

  1589

  1590 lemma Lim_transform_eventually:

  1591   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1592   apply (rule topological_tendstoI)

  1593   apply (drule (2) topological_tendstoD)

  1594   apply (erule (1) eventually_elim2, simp)

  1595   done

  1596

  1597 lemma Lim_transform_within:

  1598   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1599   and "(f ---> l) (at x within S)"

  1600   shows "(g ---> l) (at x within S)"

  1601 proof (rule Lim_transform_eventually)

  1602   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1603     using assms(1,2) by (auto simp: dist_nz eventually_at)

  1604   show "(f ---> l) (at x within S)" by fact

  1605 qed

  1606

  1607 lemma Lim_transform_at:

  1608   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1609   and "(f ---> l) (at x)"

  1610   shows "(g ---> l) (at x)"

  1611 proof (rule Lim_transform_eventually)

  1612   show "eventually (\<lambda>x. f x = g x) (at x)"

  1613     unfolding eventually_at2

  1614     using assms(1,2) by auto

  1615   show "(f ---> l) (at x)" by fact

  1616 qed

  1617

  1618 text{* Common case assuming being away from some crucial point like 0. *}

  1619

  1620 lemma Lim_transform_away_within:

  1621   fixes a b :: "'a::t1_space"

  1622   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1623   and "(f ---> l) (at a within S)"

  1624   shows "(g ---> l) (at a within S)"

  1625 proof (rule Lim_transform_eventually)

  1626   show "(f ---> l) (at a within S)" by fact

  1627   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1628     unfolding eventually_at_topological

  1629     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1630 qed

  1631

  1632 lemma Lim_transform_away_at:

  1633   fixes a b :: "'a::t1_space"

  1634   assumes ab: "a\<noteq>b"

  1635     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1636     and fl: "(f ---> l) (at a)"

  1637   shows "(g ---> l) (at a)"

  1638   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  1639

  1640 text{* Alternatively, within an open set. *}

  1641

  1642 lemma Lim_transform_within_open:

  1643   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1644   and "(f ---> l) (at a)"

  1645   shows "(g ---> l) (at a)"

  1646 proof (rule Lim_transform_eventually)

  1647   show "eventually (\<lambda>x. f x = g x) (at a)"

  1648     unfolding eventually_at_topological

  1649     using assms(1,2,3) by auto

  1650   show "(f ---> l) (at a)" by fact

  1651 qed

  1652

  1653 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1654

  1655 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1656

  1657 lemma Lim_cong_within(*[cong add]*):

  1658   assumes "a = b" "x = y" "S = T"

  1659   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1660   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1661   unfolding tendsto_def eventually_at_topological

  1662   using assms by simp

  1663

  1664 lemma Lim_cong_at(*[cong add]*):

  1665   assumes "a = b" "x = y"

  1666   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1667   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1668   unfolding tendsto_def eventually_at_topological

  1669   using assms by simp

  1670

  1671 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1672

  1673 lemma closure_sequential:

  1674   fixes l :: "'a::first_countable_topology"

  1675   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1676 proof

  1677   assume "?lhs" moreover

  1678   { assume "l \<in> S"

  1679     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1680   }

  1681   moreover

  1682   { assume "l islimpt S"

  1683     hence "?rhs" unfolding islimpt_sequential by auto

  1684   }

  1685   ultimately show "?rhs"

  1686     unfolding closure_def by auto

  1687 next

  1688   assume "?rhs"

  1689   thus "?lhs" unfolding closure_def islimpt_sequential by auto

  1690 qed

  1691

  1692 lemma closed_sequential_limits:

  1693   fixes S :: "'a::first_countable_topology set"

  1694   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1695   unfolding closed_limpt

  1696   using closure_sequential [where 'a='a] closure_closed [where 'a='a]

  1697     closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1698   by metis

  1699

  1700 lemma closure_approachable:

  1701   fixes S :: "'a::metric_space set"

  1702   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1703   apply (auto simp add: closure_def islimpt_approachable)

  1704   apply (metis dist_self)

  1705   done

  1706

  1707 lemma closed_approachable:

  1708   fixes S :: "'a::metric_space set"

  1709   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1710   by (metis closure_closed closure_approachable)

  1711

  1712 lemma closure_contains_Inf:

  1713   fixes S :: "real set"

  1714   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1715   shows "Inf S \<in> closure S"

  1716 proof -

  1717   have *: "\<forall>x\<in>S. Inf S \<le> x"

  1718     using cInf_lower_EX[of _ S] assms by metis

  1719   {

  1720     fix e :: real assume "0 < e"

  1721     then have "Inf S < Inf S + e" by simp

  1722     with assms obtain x where "x \<in> S" "x < Inf S + e"

  1723       by (subst (asm) cInf_less_iff[of _ B]) auto

  1724     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  1725       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  1726   }

  1727   then show ?thesis unfolding closure_approachable by auto

  1728 qed

  1729

  1730 lemma closed_contains_Inf:

  1731   fixes S :: "real set"

  1732   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1733     and "closed S"

  1734   shows "Inf S \<in> S"

  1735   by (metis closure_contains_Inf closure_closed assms)

  1736

  1737

  1738 lemma not_trivial_limit_within_ball:

  1739   "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  1740   (is "?lhs = ?rhs")

  1741 proof -

  1742   { assume "?lhs"

  1743     { fix e :: real

  1744       assume "e>0"

  1745       then obtain y where "y:(S-{x}) & dist y x < e"

  1746         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1747         by auto

  1748       then have "y : (S Int ball x e - {x})"

  1749         unfolding ball_def by (simp add: dist_commute)

  1750       then have "S Int ball x e - {x} ~= {}" by blast

  1751     }

  1752     then have "?rhs" by auto

  1753   }

  1754   moreover

  1755   { assume "?rhs"

  1756     { fix e :: real

  1757       assume "e>0"

  1758       then obtain y where "y : (S Int ball x e - {x})" using ?rhs by blast

  1759       then have "y:(S-{x}) & dist y x < e"

  1760         unfolding ball_def by (simp add: dist_commute)

  1761       then have "EX y:(S-{x}). dist y x < e" by auto

  1762     }

  1763     then have "?lhs"

  1764       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto

  1765   }

  1766   ultimately show ?thesis by auto

  1767 qed

  1768

  1769

  1770 subsection {* Infimum Distance *}

  1771

  1772 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1773

  1774 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1775   by (simp add: infdist_def)

  1776

  1777 lemma infdist_nonneg: "0 \<le> infdist x A"

  1778   by (auto simp add: infdist_def intro: cInf_greatest)

  1779

  1780 lemma infdist_le:

  1781   assumes "a \<in> A"

  1782     and "d = dist x a"

  1783   shows "infdist x A \<le> d"

  1784   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

  1785

  1786 lemma infdist_zero[simp]:

  1787   assumes "a \<in> A"

  1788   shows "infdist a A = 0"

  1789 proof -

  1790   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  1791   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  1792 qed

  1793

  1794 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  1795 proof cases

  1796   assume "A = {}"

  1797   thus ?thesis by (simp add: infdist_def)

  1798 next

  1799   assume "A \<noteq> {}"

  1800   then obtain a where "a \<in> A" by auto

  1801   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1802   proof (rule cInf_greatest)

  1803     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp

  1804     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1805     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto

  1806     show "infdist x A \<le> d"

  1807       unfolding infdist_notempty[OF A \<noteq> {}]

  1808     proof (rule cInf_lower2)

  1809       show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto

  1810       show "dist x a \<le> d" unfolding d by (rule dist_triangle)

  1811       fix d assume "d \<in> {dist x a |a. a \<in> A}"

  1812       then obtain a where "a \<in> A" "d = dist x a" by auto

  1813       thus "infdist x A \<le> d" by (rule infdist_le)

  1814     qed

  1815   qed

  1816   also have "\<dots> = dist x y + infdist y A"

  1817   proof (rule cInf_eq, safe)

  1818     fix a assume "a \<in> A"

  1819     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)

  1820   next

  1821     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1822     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  1823       by (intro cInf_greatest) (auto simp: field_simps)

  1824     thus "i \<le> dist x y + infdist y A" by simp

  1825   qed

  1826   finally show ?thesis by simp

  1827 qed

  1828

  1829 lemma in_closure_iff_infdist_zero:

  1830   assumes "A \<noteq> {}"

  1831   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1832 proof

  1833   assume "x \<in> closure A"

  1834   show "infdist x A = 0"

  1835   proof (rule ccontr)

  1836     assume "infdist x A \<noteq> 0"

  1837     with infdist_nonneg[of x A] have "infdist x A > 0" by auto

  1838     hence "ball x (infdist x A) \<inter> closure A = {}"

  1839       apply auto

  1840       apply (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  1841         eucl_less_not_refl euclidean_trans(2) infdist_le)

  1842       done

  1843     hence "x \<notin> closure A"

  1844       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  1845     thus False using x \<in> closure A by simp

  1846   qed

  1847 next

  1848   assume x: "infdist x A = 0"

  1849   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)

  1850   show "x \<in> closure A" unfolding closure_approachable

  1851   proof (safe, rule ccontr)

  1852     fix e::real assume "0 < e"

  1853     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1854     hence "infdist x A \<ge> e" using a \<in> A

  1855       unfolding infdist_def

  1856       by (force simp: dist_commute intro: cInf_greatest)

  1857     with x 0 < e show False by auto

  1858   qed

  1859 qed

  1860

  1861 lemma in_closed_iff_infdist_zero:

  1862   assumes "closed A" "A \<noteq> {}"

  1863   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1864 proof -

  1865   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1866     by (rule in_closure_iff_infdist_zero) fact

  1867   with assms show ?thesis by simp

  1868 qed

  1869

  1870 lemma tendsto_infdist [tendsto_intros]:

  1871   assumes f: "(f ---> l) F"

  1872   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  1873 proof (rule tendstoI)

  1874   fix e ::real assume "0 < e"

  1875   from tendstoD[OF f this]

  1876   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1877   proof (eventually_elim)

  1878     fix x

  1879     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1880     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1881       by (simp add: dist_commute dist_real_def)

  1882     also assume "dist (f x) l < e"

  1883     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1884   qed

  1885 qed

  1886

  1887 text{* Some other lemmas about sequences. *}

  1888

  1889 lemma sequentially_offset:

  1890   assumes "eventually (\<lambda>i. P i) sequentially"

  1891   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1892   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1893

  1894 lemma seq_offset:

  1895   assumes "(f ---> l) sequentially"

  1896   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1897   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1898

  1899 lemma seq_offset_neg:

  1900   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1901   apply (rule topological_tendstoI)

  1902   apply (drule (2) topological_tendstoD)

  1903   apply (simp only: eventually_sequentially)

  1904   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1905   apply metis

  1906   apply arith

  1907   done

  1908

  1909 lemma seq_offset_rev:

  1910   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1911   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1912

  1913 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1914   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1915

  1916 subsection {* More properties of closed balls *}

  1917

  1918 lemma closed_cball: "closed (cball x e)"

  1919   unfolding cball_def closed_def

  1920   unfolding Collect_neg_eq [symmetric] not_le

  1921   apply (clarsimp simp add: open_dist, rename_tac y)

  1922   apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1923   apply (rename_tac x')

  1924   apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1925   apply simp

  1926   done

  1927

  1928 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1929 proof -

  1930   {

  1931     fix x and e::real

  1932     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1933     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1934   }

  1935   moreover

  1936   {

  1937     fix x and e::real

  1938     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1939     hence "\<exists>d>0. ball x d \<subseteq> S"

  1940       unfolding subset_eq

  1941       apply(rule_tac x="e/2" in exI)

  1942       apply auto

  1943       done

  1944   }

  1945   ultimately show ?thesis

  1946     unfolding open_contains_ball by auto

  1947 qed

  1948

  1949 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1950   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1951

  1952 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1953   apply (simp add: interior_def, safe)

  1954   apply (force simp add: open_contains_cball)

  1955   apply (rule_tac x="ball x e" in exI)

  1956   apply (simp add: subset_trans [OF ball_subset_cball])

  1957   done

  1958

  1959 lemma islimpt_ball:

  1960   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1961   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1962 proof

  1963   assume "?lhs"

  1964   { assume "e \<le> 0"

  1965     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1966     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1967   }

  1968   hence "e > 0" by (metis not_less)

  1969   moreover

  1970   have "y \<in> cball x e"

  1971     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  1972       ball_subset_cball[of x e] ?lhs

  1973     unfolding closed_limpt by auto

  1974   ultimately show "?rhs" by auto

  1975 next

  1976   assume "?rhs" hence "e>0"  by auto

  1977   { fix d::real assume "d>0"

  1978     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1979     proof(cases "d \<le> dist x y")

  1980       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1981       proof(cases "x=y")

  1982         case True hence False using d \<le> dist x y d>0 by auto

  1983         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1984       next

  1985         case False

  1986

  1987         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1988               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1989           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1990         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1991           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1992           unfolding scaleR_minus_left scaleR_one

  1993           by (auto simp add: norm_minus_commute)

  1994         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1995           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1996           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1997         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1998         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1999

  2000         moreover

  2001

  2002         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2003           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  2004         moreover

  2005         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  2006           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2007           unfolding dist_norm by auto

  2008         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

  2009       qed

  2010     next

  2011       case False hence "d > dist x y" by auto

  2012       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2013       proof(cases "x=y")

  2014         case True

  2015         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2016           using perfect_choose_dist[of "min e d" y]

  2017           using d > 0 e>0 by auto

  2018         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2019           unfolding x = y

  2020           using z \<noteq> y **

  2021           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  2022       next

  2023         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2024           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  2025       qed

  2026     qed  }

  2027   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  2028 qed

  2029

  2030 lemma closure_ball_lemma:

  2031   fixes x y :: "'a::real_normed_vector"

  2032   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  2033 proof (rule islimptI)

  2034   fix T assume "y \<in> T" "open T"

  2035   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2036     unfolding open_dist by fast

  2037   (* choose point between x and y, within distance r of y. *)

  2038   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2039   def z \<equiv> "y + scaleR k (x - y)"

  2040   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2041     unfolding z_def by (simp add: algebra_simps)

  2042   have "dist z y < r"

  2043     unfolding z_def k_def using 0 < r

  2044     by (simp add: dist_norm min_def)

  2045   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2046   have "dist x z < dist x y"

  2047     unfolding z_def2 dist_norm

  2048     apply (simp add: norm_minus_commute)

  2049     apply (simp only: dist_norm [symmetric])

  2050     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2051     apply (rule mult_strict_right_mono)

  2052     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2053     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2054     done

  2055   hence "z \<in> ball x (dist x y)" by simp

  2056   have "z \<noteq> y"

  2057     unfolding z_def k_def using x \<noteq> y 0 < r

  2058     by (simp add: min_def)

  2059   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2060     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2061     by fast

  2062 qed

  2063

  2064 lemma closure_ball:

  2065   fixes x :: "'a::real_normed_vector"

  2066   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2067   apply (rule equalityI)

  2068   apply (rule closure_minimal)

  2069   apply (rule ball_subset_cball)

  2070   apply (rule closed_cball)

  2071   apply (rule subsetI, rename_tac y)

  2072   apply (simp add: le_less [where 'a=real])

  2073   apply (erule disjE)

  2074   apply (rule subsetD [OF closure_subset], simp)

  2075   apply (simp add: closure_def)

  2076   apply clarify

  2077   apply (rule closure_ball_lemma)

  2078   apply (simp add: zero_less_dist_iff)

  2079   done

  2080

  2081 (* In a trivial vector space, this fails for e = 0. *)

  2082 lemma interior_cball:

  2083   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2084   shows "interior (cball x e) = ball x e"

  2085 proof (cases "e\<ge>0")

  2086   case False note cs = this

  2087   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  2088   { fix y assume "y \<in> cball x e"

  2089     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  2090   hence "cball x e = {}" by auto

  2091   hence "interior (cball x e) = {}" using interior_empty by auto

  2092   ultimately show ?thesis by blast

  2093 next

  2094   case True note cs = this

  2095   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  2096   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2097     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  2098

  2099     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2100       using perfect_choose_dist [of d] by auto

  2101     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  2102     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  2103

  2104     hence "y \<in> ball x e" proof(cases "x = y")

  2105       case True

  2106       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2107         by (auto simp add: dist_commute)

  2108       thus "y \<in> ball x e" using x = y  by simp

  2109     next

  2110       case False

  2111       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  2112         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2113       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2114         using d as(1)[unfolded subset_eq] by blast

  2115       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2116       hence **:"d / (2 * norm (y - x)) > 0"

  2117         unfolding zero_less_norm_iff[THEN sym]

  2118         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2119

  2120       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)"

  2121         by (auto simp add: dist_norm algebra_simps)

  2122       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2123         by (auto simp add: algebra_simps)

  2124       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2125         using ** by auto

  2126       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

  2127       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2128       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  2129     qed

  2130   }

  2131   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  2132   ultimately show ?thesis

  2133     using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  2134 qed

  2135

  2136 lemma frontier_ball:

  2137   fixes a :: "'a::real_normed_vector"

  2138   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  2139   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2140   apply (simp add: set_eq_iff)

  2141   apply arith

  2142   done

  2143

  2144 lemma frontier_cball:

  2145   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2146   shows "frontier(cball a e) = {x. dist a x = e}"

  2147   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2148   apply (simp add: set_eq_iff)

  2149   apply arith

  2150   done

  2151

  2152 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  2153   apply (simp add: set_eq_iff not_le)

  2154   apply (metis zero_le_dist dist_self order_less_le_trans)

  2155   done

  2156

  2157 lemma cball_empty: "e < 0 ==> cball x e = {}"

  2158   by (simp add: cball_eq_empty)

  2159

  2160 lemma cball_eq_sing:

  2161   fixes x :: "'a::{metric_space,perfect_space}"

  2162   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2163 proof (rule linorder_cases)

  2164   assume e: "0 < e"

  2165   obtain a where "a \<noteq> x" "dist a x < e"

  2166     using perfect_choose_dist [OF e] by auto

  2167   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  2168   with e show ?thesis by (auto simp add: set_eq_iff)

  2169 qed auto

  2170

  2171 lemma cball_sing:

  2172   fixes x :: "'a::metric_space"

  2173   shows "e = 0 ==> cball x e = {x}"

  2174   by (auto simp add: set_eq_iff)

  2175

  2176

  2177 subsection {* Boundedness *}

  2178

  2179   (* FIXME: This has to be unified with BSEQ!! *)

  2180 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2181   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2182

  2183 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2184   unfolding bounded_def subset_eq by auto

  2185

  2186 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2187   unfolding bounded_def

  2188   apply safe

  2189   apply (rule_tac x="dist a x + e" in exI, clarify)

  2190   apply (drule (1) bspec)

  2191   apply (erule order_trans [OF dist_triangle add_left_mono])

  2192   apply auto

  2193   done

  2194

  2195 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2196   unfolding bounded_any_center [where a=0]

  2197   by (simp add: dist_norm)

  2198

  2199 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

  2200   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

  2201   using assms by auto

  2202

  2203 lemma bounded_empty [simp]: "bounded {}"

  2204   by (simp add: bounded_def)

  2205

  2206 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2207   by (metis bounded_def subset_eq)

  2208

  2209 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2210   by (metis bounded_subset interior_subset)

  2211

  2212 lemma bounded_closure[intro]:

  2213   assumes "bounded S"

  2214   shows "bounded (closure S)"

  2215 proof -

  2216   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2217     unfolding bounded_def by auto

  2218   {

  2219     fix y

  2220     assume "y \<in> closure S"

  2221     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2222       unfolding closure_sequential by auto

  2223     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2224     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2225       by (rule eventually_mono, simp add: f(1))

  2226     have "dist x y \<le> a"

  2227       apply (rule Lim_dist_ubound [of sequentially f])

  2228       apply (rule trivial_limit_sequentially)

  2229       apply (rule f(2))

  2230       apply fact

  2231       done

  2232   }

  2233   thus ?thesis unfolding bounded_def by auto

  2234 qed

  2235

  2236 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2237   apply (simp add: bounded_def)

  2238   apply (rule_tac x=x in exI)

  2239   apply (rule_tac x=e in exI)

  2240   apply auto

  2241   done

  2242

  2243 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2244   by (metis ball_subset_cball bounded_cball bounded_subset)

  2245

  2246 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2247   apply (auto simp add: bounded_def)

  2248   apply (rename_tac x y r s)

  2249   apply (rule_tac x=x in exI)

  2250   apply (rule_tac x="max r (dist x y + s)" in exI)

  2251   apply (rule ballI, rename_tac z, safe)

  2252   apply (drule (1) bspec, simp)

  2253   apply (drule (1) bspec)

  2254   apply (rule min_max.le_supI2)

  2255   apply (erule order_trans [OF dist_triangle add_left_mono])

  2256   done

  2257

  2258 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2259   by (induct rule: finite_induct[of F]) auto

  2260

  2261 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2262   by (induct set: finite) auto

  2263

  2264 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2265 proof -

  2266   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp

  2267   hence "bounded {x}" unfolding bounded_def by fast

  2268   thus ?thesis by (metis insert_is_Un bounded_Un)

  2269 qed

  2270

  2271 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2272   by (induct set: finite) simp_all

  2273

  2274 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2275   apply (simp add: bounded_iff)

  2276   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2277   apply metis

  2278   apply arith

  2279   done

  2280

  2281 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"

  2282   unfolding Bseq_def bounded_pos by auto

  2283

  2284 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2285   by (metis Int_lower1 Int_lower2 bounded_subset)

  2286

  2287 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2288   by (metis Diff_subset bounded_subset)

  2289

  2290 lemma not_bounded_UNIV[simp, intro]:

  2291   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2292 proof(auto simp add: bounded_pos not_le)

  2293   obtain x :: 'a where "x \<noteq> 0"

  2294     using perfect_choose_dist [OF zero_less_one] by fast

  2295   fix b::real  assume b: "b >0"

  2296   have b1: "b +1 \<ge> 0" using b by simp

  2297   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2298     by (simp add: norm_sgn)

  2299   then show "\<exists>x::'a. b < norm x" ..

  2300 qed

  2301

  2302 lemma bounded_linear_image:

  2303   assumes "bounded S" "bounded_linear f"

  2304   shows "bounded(f  S)"

  2305 proof -

  2306   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"

  2307     unfolding bounded_pos by auto

  2308   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"

  2309     using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2310   {

  2311     fix x assume "x\<in>S"

  2312     hence "norm x \<le> b" using b by auto

  2313     hence "norm (f x) \<le> B * b" using B(2)

  2314       apply (erule_tac x=x in allE)

  2315       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2316       done

  2317   }

  2318   thus ?thesis unfolding bounded_pos

  2319     apply (rule_tac x="b*B" in exI)

  2320     using b B mult_pos_pos [of b B]

  2321     apply (auto simp add: mult_commute)

  2322     done

  2323 qed

  2324

  2325 lemma bounded_scaling:

  2326   fixes S :: "'a::real_normed_vector set"

  2327   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2328   apply (rule bounded_linear_image, assumption)

  2329   apply (rule bounded_linear_scaleR_right)

  2330   done

  2331

  2332 lemma bounded_translation:

  2333   fixes S :: "'a::real_normed_vector set"

  2334   assumes "bounded S"

  2335   shows "bounded ((\<lambda>x. a + x)  S)"

  2336 proof-

  2337   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"

  2338     unfolding bounded_pos by auto

  2339   {

  2340     fix x

  2341     assume "x\<in>S"

  2342     hence "norm (a + x) \<le> b + norm a"

  2343       using norm_triangle_ineq[of a x] b by auto

  2344   }

  2345   thus ?thesis

  2346     unfolding bounded_pos

  2347     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2348     by (auto intro!: exI[of _ "b + norm a"])

  2349 qed

  2350

  2351

  2352 text{* Some theorems on sups and infs using the notion "bounded". *}

  2353

  2354 lemma bounded_real:

  2355   fixes S :: "real set"

  2356   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2357   by (simp add: bounded_iff)

  2358

  2359 lemma bounded_has_Sup:

  2360   fixes S :: "real set"

  2361   assumes "bounded S" "S \<noteq> {}"

  2362   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  2363 proof

  2364   fix x assume "x\<in>S"

  2365   thus "x \<le> Sup S"

  2366     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

  2367 next

  2368   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  2369     by (metis cSup_least)

  2370 qed

  2371

  2372 lemma Sup_insert:

  2373   fixes S :: "real set"

  2374   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2375   apply (subst cSup_insert_If)

  2376   apply (rule bounded_has_Sup(1)[of S, rule_format])

  2377   apply (auto simp: sup_max)

  2378   done

  2379

  2380 lemma Sup_insert_finite:

  2381   fixes S :: "real set"

  2382   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2383   apply (rule Sup_insert)

  2384   apply (rule finite_imp_bounded)

  2385   apply simp

  2386   done

  2387

  2388 lemma bounded_has_Inf:

  2389   fixes S :: "real set"

  2390   assumes "bounded S"  "S \<noteq> {}"

  2391   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  2392 proof

  2393   fix x

  2394   assume "x\<in>S"

  2395   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a"

  2396     unfolding bounded_real by auto

  2397   thus "x \<ge> Inf S" using x\<in>S

  2398     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

  2399 next

  2400   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  2401     by (metis cInf_greatest)

  2402 qed

  2403

  2404 lemma Inf_insert:

  2405   fixes S :: "real set"

  2406   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2407   apply (subst cInf_insert_if)

  2408   apply (rule bounded_has_Inf(1)[of S, rule_format])

  2409   apply (auto simp: inf_min)

  2410   done

  2411

  2412 lemma Inf_insert_finite:

  2413   fixes S :: "real set"

  2414   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2415   by (rule Inf_insert, rule finite_imp_bounded, simp)

  2416

  2417 subsection {* Compactness *}

  2418

  2419 subsubsection {* Bolzano-Weierstrass property *}

  2420

  2421 lemma heine_borel_imp_bolzano_weierstrass:

  2422   assumes "compact s" "infinite t"  "t \<subseteq> s"

  2423   shows "\<exists>x \<in> s. x islimpt t"

  2424 proof(rule ccontr)

  2425   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2426   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)"

  2427     unfolding islimpt_def

  2428     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

  2429   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2430     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2431     using f by auto

  2432   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2433   {

  2434     fix x y

  2435     assume "x\<in>t" "y\<in>t" "f x = f y"

  2436     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2437       using f[THEN bspec[where x=x]] and t\<subseteq>s by auto

  2438     hence "x = y"

  2439       using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto

  2440   }

  2441   hence "inj_on f t"

  2442     unfolding inj_on_def by simp

  2443   hence "infinite (f  t)"

  2444     using assms(2) using finite_imageD by auto

  2445   moreover

  2446   {

  2447     fix x

  2448     assume "x\<in>t" "f x \<notin> g"

  2449     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2450     then obtain y where "y\<in>s" "h = f y"

  2451       using g'[THEN bspec[where x=h]] by auto

  2452     hence "y = x"

  2453       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2454     hence False

  2455       using f x \<notin> g h\<in>g unfolding h = f y by auto

  2456   }

  2457   hence "f  t \<subseteq> g" by auto

  2458   ultimately show False

  2459     using g(2) using finite_subset by auto

  2460 qed

  2461

  2462 lemma acc_point_range_imp_convergent_subsequence:

  2463   fixes l :: "'a :: first_countable_topology"

  2464   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2465   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2466 proof -

  2467   from countable_basis_at_decseq[of l] guess A . note A = this

  2468

  2469   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2470   {

  2471     fix n i

  2472     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2473       using l A by auto

  2474     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2475       unfolding ex_in_conv by (intro notI) simp

  2476     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2477       by auto

  2478     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2479       by (auto simp: not_le)

  2480     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2481       unfolding s_def by (auto intro: someI2_ex)

  2482   }

  2483   note s = this

  2484   def r \<equiv> "nat_rec (s 0 0) s"

  2485   have "subseq r"

  2486     by (auto simp: r_def s subseq_Suc_iff)

  2487   moreover

  2488   have "(\<lambda>n. f (r n)) ----> l"

  2489   proof (rule topological_tendstoI)

  2490     fix S assume "open S" "l \<in> S"

  2491     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2492     moreover

  2493     {

  2494       fix i

  2495       assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"

  2496         by (cases i) (simp_all add: r_def s)

  2497     }

  2498     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  2499       by (auto simp: eventually_sequentially)

  2500     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2501       by eventually_elim auto

  2502   qed

  2503   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2504     by (auto simp: convergent_def comp_def)

  2505 qed

  2506

  2507 lemma sequence_infinite_lemma:

  2508   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2509   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2510   shows "infinite (range f)"

  2511 proof

  2512   assume "finite (range f)"

  2513   hence "closed (range f)" by (rule finite_imp_closed)

  2514   hence "open (- range f)" by (rule open_Compl)

  2515   from assms(1) have "l \<in> - range f" by auto

  2516   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2517     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2518   thus False unfolding eventually_sequentially by auto

  2519 qed

  2520

  2521 lemma closure_insert:

  2522   fixes x :: "'a::t1_space"

  2523   shows "closure (insert x s) = insert x (closure s)"

  2524   apply (rule closure_unique)

  2525   apply (rule insert_mono [OF closure_subset])

  2526   apply (rule closed_insert [OF closed_closure])

  2527   apply (simp add: closure_minimal)

  2528   done

  2529

  2530 lemma islimpt_insert:

  2531   fixes x :: "'a::t1_space"

  2532   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2533 proof

  2534   assume *: "x islimpt (insert a s)"

  2535   show "x islimpt s"

  2536   proof (rule islimptI)

  2537     fix t assume t: "x \<in> t" "open t"

  2538     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2539     proof (cases "x = a")

  2540       case True

  2541       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2542         using * t by (rule islimptE)

  2543       with x = a show ?thesis by auto

  2544     next

  2545       case False

  2546       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2547         by (simp_all add: open_Diff)

  2548       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2549         using * t' by (rule islimptE)

  2550       thus ?thesis by auto

  2551     qed

  2552   qed

  2553 next

  2554   assume "x islimpt s" thus "x islimpt (insert a s)"

  2555     by (rule islimpt_subset) auto

  2556 qed

  2557

  2558 lemma islimpt_finite:

  2559   fixes x :: "'a::t1_space"

  2560   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2561   by (induct set: finite) (simp_all add: islimpt_insert)

  2562

  2563 lemma islimpt_union_finite:

  2564   fixes x :: "'a::t1_space"

  2565   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2566   by (simp add: islimpt_Un islimpt_finite)

  2567

  2568 lemma islimpt_eq_acc_point:

  2569   fixes l :: "'a :: t1_space"

  2570   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2571 proof (safe intro!: islimptI)

  2572   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2573   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2574     by (auto intro: finite_imp_closed)

  2575   then show False

  2576     by (rule islimptE) auto

  2577 next

  2578   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2579   then have "infinite (T \<inter> S - {l})" by auto

  2580   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2581     unfolding ex_in_conv by (intro notI) simp

  2582   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2583     by auto

  2584 qed

  2585

  2586 lemma islimpt_range_imp_convergent_subsequence:

  2587   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2588   assumes l: "l islimpt (range f)"

  2589   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2590   using l unfolding islimpt_eq_acc_point

  2591   by (rule acc_point_range_imp_convergent_subsequence)

  2592

  2593 lemma sequence_unique_limpt:

  2594   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2595   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  2596   shows "l' = l"

  2597 proof (rule ccontr)

  2598   assume "l' \<noteq> l"

  2599   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2600     using hausdorff [OF l' \<noteq> l] by auto

  2601   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2602     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2603   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2604     unfolding eventually_sequentially by auto

  2605

  2606   have "UNIV = {..<N} \<union> {N..}" by auto

  2607   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  2608   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  2609   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  2610   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2611     using l' \<in> s open s by (rule islimptE)

  2612   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  2613   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  2614   with s \<inter> t = {} show False by simp

  2615 qed

  2616

  2617 lemma bolzano_weierstrass_imp_closed:

  2618   fixes s :: "'a::{first_countable_topology, t2_space} set"

  2619   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2620   shows "closed s"

  2621 proof -

  2622   {

  2623     fix x l

  2624     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2625     hence "l \<in> s"

  2626     proof (cases "\<forall>n. x n \<noteq> l")

  2627       case False

  2628       thus "l\<in>s" using as(1) by auto

  2629     next

  2630       case True note cas = this

  2631       with as(2) have "infinite (range x)"

  2632         using sequence_infinite_lemma[of x l] by auto

  2633       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  2634         using assms[THEN spec[where x="range x"]] as(1) by auto

  2635       thus "l\<in>s" using sequence_unique_limpt[of x l l']

  2636         using as cas by auto

  2637     qed

  2638   }

  2639   thus ?thesis unfolding closed_sequential_limits by fast

  2640 qed

  2641

  2642 lemma compact_imp_bounded:

  2643   assumes "compact U"

  2644   shows "bounded U"

  2645 proof -

  2646   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  2647     using assms by auto

  2648   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2649     by (rule compactE_image)

  2650   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2651     by (simp add: bounded_UN)

  2652   thus "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2653     by (rule bounded_subset)

  2654 qed

  2655

  2656 text{* In particular, some common special cases. *}

  2657

  2658 lemma compact_union [intro]:

  2659   assumes "compact s" "compact t" shows " compact (s \<union> t)"

  2660 proof (rule compactI)

  2661   fix f

  2662   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  2663   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  2664     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2665   moreover

  2666   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  2667     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2668   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  2669     by (auto intro!: exI[of _ "s' \<union> t'"])

  2670 qed

  2671

  2672 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  2673   by (induct set: finite) auto

  2674

  2675 lemma compact_UN [intro]:

  2676   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  2677   unfolding SUP_def by (rule compact_Union) auto

  2678

  2679 lemma closed_inter_compact [intro]:

  2680   assumes "closed s" and "compact t"

  2681   shows "compact (s \<inter> t)"

  2682   using compact_inter_closed [of t s] assms

  2683   by (simp add: Int_commute)

  2684

  2685 lemma compact_inter [intro]:

  2686   fixes s t :: "'a :: t2_space set"

  2687   assumes "compact s" and "compact t"

  2688   shows "compact (s \<inter> t)"

  2689   using assms by (intro compact_inter_closed compact_imp_closed)

  2690

  2691 lemma compact_sing [simp]: "compact {a}"

  2692   unfolding compact_eq_heine_borel by auto

  2693

  2694 lemma compact_insert [simp]:

  2695   assumes "compact s" shows "compact (insert x s)"

  2696 proof -

  2697   have "compact ({x} \<union> s)"

  2698     using compact_sing assms by (rule compact_union)

  2699   thus ?thesis by simp

  2700 qed

  2701

  2702 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  2703   by (induct set: finite) simp_all

  2704

  2705 lemma open_delete:

  2706   fixes s :: "'a::t1_space set"

  2707   shows "open s \<Longrightarrow> open (s - {x})"

  2708   by (simp add: open_Diff)

  2709

  2710 text{* Finite intersection property *}

  2711

  2712 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  2713   by (auto simp: inj_on_def)

  2714

  2715 lemma compact_fip:

  2716   "compact U \<longleftrightarrow>

  2717     (\<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> {})"

  2718   (is "_ \<longleftrightarrow> ?R")

  2719 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  2720   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  2721     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  2722   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>(uminusA)"

  2723     by auto

  2724   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  2725     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  2726   with fi[THEN spec, of B] show False

  2727     by (auto dest: finite_imageD intro: inj_setminus)

  2728 next

  2729   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2730   from cover have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  2731     by auto

  2732   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>(uminusB) = {}"

  2733     by (metis subset_image_iff)

  2734   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2735     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  2736 qed

  2737

  2738 lemma compact_imp_fip:

  2739   "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>

  2740     s \<inter> (\<Inter> f) \<noteq> {}"

  2741   unfolding compact_fip by auto

  2742

  2743 text{*Compactness expressed with filters*}

  2744

  2745 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2746

  2747 lemma eventually_filter_from_subbase:

  2748   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2749     (is "_ \<longleftrightarrow> ?R P")

  2750   unfolding filter_from_subbase_def

  2751 proof (rule eventually_Abs_filter is_filter.intro)+

  2752   show "?R (\<lambda>x. True)"

  2753     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  2754 next

  2755   fix P Q assume "?R P" then guess X ..

  2756   moreover assume "?R Q" then guess Y ..

  2757   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  2758     by (intro exI[of _ "X \<union> Y"]) auto

  2759 next

  2760   fix P Q

  2761   assume "?R P" then guess X ..

  2762   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  2763   ultimately show "?R Q"

  2764     by (intro exI[of _ X]) auto

  2765 qed

  2766

  2767 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  2768   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  2769

  2770 lemma filter_from_subbase_not_bot:

  2771   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  2772   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  2773

  2774 lemma closure_iff_nhds_not_empty:

  2775   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  2776 proof safe

  2777   assume x: "x \<in> closure X"

  2778   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  2779   then have "x \<notin> closure (-S)"

  2780     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  2781   with x have "x \<in> closure X - closure (-S)"

  2782     by auto

  2783   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  2784     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  2785   finally have "X \<inter> S \<noteq> {}" by auto

  2786   then show False using X \<inter> A = {} S \<subseteq> A by auto

  2787 next

  2788   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  2789   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  2790   show "x \<in> closure X"

  2791     by (simp add: closure_subset open_Compl)

  2792 qed

  2793

  2794 lemma compact_filter:

  2795   "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))"

  2796 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  2797   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  2798   from F have "U \<noteq> {}"

  2799     by (auto simp: eventually_False)

  2800

  2801   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  2802   then have "\<forall>z\<in>Z. closed z"

  2803     by auto

  2804   moreover

  2805   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  2806     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  2807   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  2808   proof (intro allI impI)

  2809     fix B assume "finite B" "B \<subseteq> Z"

  2810     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  2811       by (auto intro!: eventually_Ball_finite)

  2812     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  2813       by eventually_elim auto

  2814     with F show "U \<inter> \<Inter>B \<noteq> {}"

  2815       by (intro notI) (simp add: eventually_False)

  2816   qed

  2817   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  2818     using compact U unfolding compact_fip by blast

  2819   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto

  2820

  2821   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  2822     unfolding eventually_inf eventually_nhds

  2823   proof safe

  2824     fix P Q R S

  2825     assume "eventually R F" "open S" "x \<in> S"

  2826     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  2827     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  2828     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  2829     ultimately show False by (auto simp: set_eq_iff)

  2830   qed

  2831   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  2832     by (metis eventually_bot)

  2833 next

  2834   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 = {}"

  2835

  2836   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  2837   then have inj_P': "\<And>A. inj_on P' A"

  2838     by (auto intro!: inj_onI simp: fun_eq_iff)

  2839   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  2840   have "F \<noteq> bot"

  2841     unfolding F_def

  2842   proof (safe intro!: filter_from_subbase_not_bot)

  2843     fix X assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  2844     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  2845       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  2846     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto

  2847     with B show False by (auto simp: P'_def fun_eq_iff)

  2848   qed

  2849   moreover have "eventually (\<lambda>x. x \<in> U) F"

  2850     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  2851   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)"

  2852   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  2853     by auto

  2854

  2855   { fix V assume "V \<in> A"

  2856     then have V: "eventually (\<lambda>x. x \<in> V) F"

  2857       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  2858     have "x \<in> closure V"

  2859       unfolding closure_iff_nhds_not_empty

  2860     proof (intro impI allI)

  2861       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"

  2862       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)

  2863       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  2864         by (auto simp: eventually_inf)

  2865       with x show "V \<inter> A \<noteq> {}"

  2866         by (auto simp del: Int_iff simp add: trivial_limit_def)

  2867     qed

  2868     then have "x \<in> V"

  2869       using V \<in> A A(1) by simp }

  2870   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  2871   with U \<inter> \<Inter>A = {} show False by auto

  2872 qed

  2873

  2874 definition "countably_compact U \<longleftrightarrow>

  2875     (\<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))"

  2876

  2877 lemma countably_compactE:

  2878   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  2879   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  2880   using assms unfolding countably_compact_def by metis

  2881

  2882 lemma countably_compactI:

  2883   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')"

  2884   shows "countably_compact s"

  2885   using assms unfolding countably_compact_def by metis

  2886

  2887 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  2888   by (auto simp: compact_eq_heine_borel countably_compact_def)

  2889

  2890 lemma countably_compact_imp_compact:

  2891   assumes "countably_compact U"

  2892   assumes ccover: "countable B" "\<forall>b\<in>B. open b"

  2893   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"

  2894   shows "compact U"

  2895   using countably_compact U unfolding compact_eq_heine_borel countably_compact_def

  2896 proof safe

  2897   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2898   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)"

  2899

  2900   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  2901   ultimately have "countable C" "\<forall>a\<in>C. open a"

  2902     unfolding C_def using ccover by auto

  2903   moreover

  2904   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  2905   proof safe

  2906     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"

  2907     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast

  2908     with a \<in> A show "x \<in> \<Union>C" unfolding C_def

  2909       by auto

  2910   qed

  2911   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  2912   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  2913     using * by metis

  2914   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  2915     by (auto simp: C_def)

  2916   then guess f unfolding bchoice_iff Bex_def ..

  2917   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2918     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  2919 qed

  2920

  2921 lemma countably_compact_imp_compact_second_countable:

  2922   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2923 proof (rule countably_compact_imp_compact)

  2924   fix T and x :: 'a assume "open T" "x \<in> T"

  2925   from topological_basisE[OF is_basis this] guess b .

  2926   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto

  2927 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  2928

  2929 lemma countably_compact_eq_compact:

  2930   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2931   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  2932

  2933 subsubsection{* Sequential compactness *}

  2934

  2935 definition

  2936   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where

  2937   "seq_compact S \<longleftrightarrow>

  2938    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  2939        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  2940

  2941 lemma seq_compact_imp_countably_compact:

  2942   fixes U :: "'a :: first_countable_topology set"

  2943   assumes "seq_compact U"

  2944   shows "countably_compact U"

  2945 proof (safe intro!: countably_compactI)

  2946   fix A

  2947   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  2948   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  2949     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  2950   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2951   proof cases

  2952     assume "finite A"

  2953     with A show ?thesis by auto

  2954   next

  2955     assume "infinite A"

  2956     then have "A \<noteq> {}" by auto

  2957     show ?thesis

  2958     proof (rule ccontr)

  2959       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  2960       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto

  2961       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis

  2962       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  2963       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  2964         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  2965       then have "range X \<subseteq> U" by auto

  2966       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto

  2967       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  2968       obtain n where "x \<in> from_nat_into A n" by auto

  2969       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  2970       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  2971         unfolding tendsto_def by (auto simp: comp_def)

  2972       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  2973         by (auto simp: eventually_sequentially)

  2974       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  2975         by auto

  2976       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  2977         by (auto intro!: exI[of _ "max n N"])

  2978       ultimately show False

  2979         by auto

  2980     qed

  2981   qed

  2982 qed

  2983

  2984 lemma compact_imp_seq_compact:

  2985   fixes U :: "'a :: first_countable_topology set"

  2986   assumes "compact U" shows "seq_compact U"

  2987   unfolding seq_compact_def

  2988 proof safe

  2989   fix X :: "nat \<Rightarrow> 'a"

  2990   assume "\<forall>n. X n \<in> U"

  2991   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  2992     by (auto simp: eventually_filtermap)

  2993   moreover

  2994   have "filtermap X sequentially \<noteq> bot"

  2995     by (simp add: trivial_limit_def eventually_filtermap)

  2996   ultimately

  2997   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  2998     using compact U by (auto simp: compact_filter)

  2999

  3000   from countable_basis_at_decseq[of x] guess A . note A = this

  3001   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3002   {

  3003     fix n i

  3004     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3005     proof (rule ccontr)

  3006       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3007       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto

  3008       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3009         by (auto simp: eventually_filtermap eventually_sequentially)

  3010       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3011         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3012       ultimately have "eventually (\<lambda>x. False) ?F"

  3013         by (auto simp add: eventually_inf)

  3014       with x show False

  3015         by (simp add: eventually_False)

  3016     qed

  3017     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3018       unfolding s_def by (auto intro: someI2_ex)

  3019   }

  3020   note s = this

  3021   def r \<equiv> "nat_rec (s 0 0) s"

  3022   have "subseq r"

  3023     by (auto simp: r_def s subseq_Suc_iff)

  3024   moreover

  3025   have "(\<lambda>n. X (r n)) ----> x"

  3026   proof (rule topological_tendstoI)

  3027     fix S

  3028     assume "open S" "x \<in> S"

  3029     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  3030     moreover

  3031     {

  3032       fix i

  3033       assume "Suc 0 \<le> i"

  3034       then have "X (r i) \<in> A i"

  3035         by (cases i) (simp_all add: r_def s)

  3036     }

  3037     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3038       by (auto simp: eventually_sequentially)

  3039     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3040       by eventually_elim auto

  3041   qed

  3042   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3043     using x \<in> U by (auto simp: convergent_def comp_def)

  3044 qed

  3045

  3046 lemma seq_compactI:

  3047   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  3048   shows "seq_compact S"

  3049   unfolding seq_compact_def using assms by fast

  3050

  3051 lemma seq_compactE:

  3052   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3053   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3054   using assms unfolding seq_compact_def by fast

  3055

  3056 lemma countably_compact_imp_acc_point:

  3057   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"

  3058   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3059 proof (rule ccontr)

  3060   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3061   note countably_compact s

  3062   moreover have "\<forall>t\<in>C. open t"

  3063     by (auto simp: C_def)

  3064   moreover

  3065   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3066   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3067   have "s \<subseteq> \<Union>C"

  3068     using t \<subseteq> s

  3069     unfolding C_def Union_image_eq

  3070     apply (safe dest!: s)

  3071     apply (rule_tac a="U \<inter> t" in UN_I)

  3072     apply (auto intro!: interiorI simp add: finite_subset)

  3073     done

  3074   moreover

  3075   from countable t have "countable C"

  3076     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3077   ultimately guess D by (rule countably_compactE)

  3078   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and

  3079     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3080     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3081   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3082     using interior_subset by blast

  3083   moreover have "finite (\<Union>E)"

  3084     using E by auto

  3085   ultimately show False using infinite t by (auto simp: finite_subset)

  3086 qed

  3087

  3088 lemma countable_acc_point_imp_seq_compact:

  3089   fixes s :: "'a::first_countable_topology set"

  3090   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))"

  3091   shows "seq_compact s"

  3092 proof -

  3093   {

  3094     fix f :: "nat \<Rightarrow> 'a"

  3095     assume f: "\<forall>n. f n \<in> s"

  3096     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3097     proof (cases "finite (range f)")

  3098       case True

  3099       obtain l where "infinite {n. f n = f l}"

  3100         using pigeonhole_infinite[OF _ True] by auto

  3101       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3102         using infinite_enumerate by blast

  3103       hence "subseq r \<and> (f \<circ> r) ----> f l"

  3104         by (simp add: fr tendsto_const o_def)

  3105       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3106         by auto

  3107     next

  3108       case False

  3109       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto

  3110       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3111       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3112         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3113       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3114     qed

  3115   }

  3116   thus ?thesis unfolding seq_compact_def by auto

  3117 qed

  3118

  3119 lemma seq_compact_eq_countably_compact:

  3120   fixes U :: "'a :: first_countable_topology set"

  3121   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3122   using

  3123     countable_acc_point_imp_seq_compact

  3124     countably_compact_imp_acc_point

  3125     seq_compact_imp_countably_compact

  3126   by metis

  3127

  3128 lemma seq_compact_eq_acc_point:

  3129   fixes s :: "'a :: first_countable_topology set"

  3130   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)))"

  3131   using

  3132     countable_acc_point_imp_seq_compact[of s]

  3133     countably_compact_imp_acc_point[of s]

  3134     seq_compact_imp_countably_compact[of s]

  3135   by metis

  3136

  3137 lemma seq_compact_eq_compact:

  3138   fixes U :: "'a :: second_countable_topology set"

  3139   shows "seq_compact U \<longleftrightarrow> compact U"

  3140   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3141

  3142 lemma bolzano_weierstrass_imp_seq_compact:

  3143   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3144   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3145   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3146

  3147 subsubsection{* Total boundedness *}

  3148

  3149 lemma cauchy_def:

  3150   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  3151   unfolding Cauchy_def by metis

  3152

  3153 fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  3154   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3155 declare helper_1.simps[simp del]

  3156

  3157 lemma seq_compact_imp_totally_bounded:

  3158   assumes "seq_compact s"

  3159   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3160 proof (rule, rule, rule ccontr)

  3161   fix e::real

  3162   assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3163   def x \<equiv> "helper_1 s e"

  3164   {

  3165     fix n

  3166     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3167     proof (induct n rule: nat_less_induct)

  3168       fix n

  3169       def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3170       assume as: "\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3171       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3172         using assm

  3173         apply simp

  3174         apply (erule_tac x="x  {0 ..< n}" in allE)

  3175         using as

  3176         apply auto

  3177         done

  3178       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3179         unfolding subset_eq by auto

  3180       have "Q (x n)"

  3181         unfolding x_def and helper_1.simps[of s e n]

  3182         apply (rule someI2[where a=z])

  3183         unfolding x_def[symmetric] and Q_def

  3184         using z

  3185         apply auto

  3186         done

  3187       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3188         unfolding Q_def by auto

  3189     qed

  3190   }

  3191   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"

  3192     by blast+

  3193   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3194     using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3195   from this(3) have "Cauchy (x \<circ> r)"

  3196     using LIMSEQ_imp_Cauchy by auto

  3197   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"

  3198     unfolding cauchy_def using e>0 by auto

  3199   show False

  3200     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3201     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3202     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

  3203     by auto

  3204 qed

  3205

  3206 subsubsection{* Heine-Borel theorem *}

  3207

  3208 lemma seq_compact_imp_heine_borel:

  3209   fixes s :: "'a :: metric_space set"

  3210   assumes "seq_compact s" shows "compact s"

  3211 proof -

  3212   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3213   guess f unfolding choice_iff' .. note f = this

  3214   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3215   have "countably_compact s"

  3216     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3217   then show "compact s"

  3218   proof (rule countably_compact_imp_compact)

  3219     show "countable K"

  3220       unfolding K_def using f

  3221       by (auto intro: countable_finite countable_subset countable_rat

  3222                intro!: countable_image countable_SIGMA countable_UN)

  3223     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3224   next

  3225     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3226     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto

  3227     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto

  3228     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto

  3229     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3230       unfolding Union_image_eq by auto

  3231     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K" by (auto simp: K_def)

  3232     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3233     proof (rule bexI[rotated], safe)

  3234       fix y assume "y \<in> ball k r"

  3235       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3236         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3237       with ball x e \<subseteq> T show "y \<in> T" by auto

  3238     qed (rule x \<in> ball k r)

  3239   qed

  3240 qed

  3241

  3242 lemma compact_eq_seq_compact_metric:

  3243   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3244   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3245

  3246 lemma compact_def:

  3247   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3248    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"

  3249   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3250

  3251 subsubsection {* Complete the chain of compactness variants *}

  3252

  3253 lemma compact_eq_bolzano_weierstrass:

  3254   fixes s :: "'a::metric_space set"

  3255   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3256 proof

  3257   assume ?lhs

  3258   thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3259 next

  3260   assume ?rhs

  3261   thus ?lhs

  3262     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3263 qed

  3264

  3265 lemma bolzano_weierstrass_imp_bounded:

  3266   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3267   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3268

  3269 text {*

  3270   A metric space (or topological vector space) is said to have the

  3271   Heine-Borel property if every closed and bounded subset is compact.

  3272 *}

  3273

  3274 class heine_borel = metric_space +

  3275   assumes bounded_imp_convergent_subsequence:

  3276     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3277

  3278 lemma bounded_closed_imp_seq_compact:

  3279   fixes s::"'a::heine_borel set"

  3280   assumes "bounded s" and "closed s" shows "seq_compact s"

  3281 proof (unfold seq_compact_def, clarify)

  3282   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3283   with bounded s have "bounded (range f)" by (auto intro: bounded_subset)

  3284   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3285     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3286   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  3287   have "l \<in> s" using closed s fr l

  3288     unfolding closed_sequential_limits by blast

  3289   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3290     using l \<in> s r l by blast

  3291 qed

  3292

  3293 lemma compact_eq_bounded_closed:

  3294   fixes s :: "'a::heine_borel set"

  3295   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3296 proof

  3297   assume ?lhs

  3298   thus ?rhs

  3299     using compact_imp_closed compact_imp_bounded

  3300     by blast

  3301 next

  3302   assume ?rhs

  3303   thus ?lhs

  3304     using bounded_closed_imp_seq_compact[of s]

  3305     unfolding compact_eq_seq_compact_metric

  3306     by auto

  3307 qed

  3308

  3309 (* TODO: is this lemma necessary? *)

  3310 lemma bounded_increasing_convergent:

  3311   fixes s :: "nat \<Rightarrow> real"

  3312   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3313   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3314   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3315

  3316 instance real :: heine_borel

  3317 proof

  3318   fix f :: "nat \<Rightarrow> real"

  3319   assume f: "bounded (range f)"

  3320   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3321     unfolding comp_def by (metis seq_monosub)

  3322   moreover

  3323   then have "Bseq (f \<circ> r)"

  3324     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3325   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3326     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3327 qed

  3328

  3329 lemma compact_lemma:

  3330   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3331   assumes "bounded (range f)"

  3332   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  3333         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3334 proof safe

  3335   fix d :: "'a set"

  3336   assume d: "d \<subseteq> Basis"

  3337   with finite_Basis have "finite d" by (blast intro: finite_subset)

  3338   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3339     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3340   proof (induct d)

  3341     case empty

  3342     thus ?case unfolding subseq_def by auto

  3343   next

  3344     case (insert k d)

  3345     have k[intro]:"k\<in>Basis" using insert by auto

  3346     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)" using bounded (range f)

  3347       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3348     obtain l1::"'a" and r1 where r1:"subseq r1" and

  3349       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3350       using insert(3) using insert(4) by auto

  3351     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f" by simp

  3352     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3353       by (metis (lifting) bounded_subset f' image_subsetI s')

  3354     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3355       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)

  3356     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  3357       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3358     moreover

  3359     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3360     {

  3361       fix e::real

  3362       assume "e>0"

  3363       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3364         by blast

  3365       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3366         by (rule tendstoD)

  3367       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3368         by (rule eventually_subseq)

  3369       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3370         using N1' N2

  3371         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3372     }

  3373     ultimately show ?case by auto

  3374   qed

  3375 qed

  3376

  3377 instance euclidean_space \<subseteq> heine_borel

  3378 proof

  3379   fix f :: "nat \<Rightarrow> 'a"

  3380   assume f: "bounded (range f)"

  3381   then obtain l::'a and r where r: "subseq r"

  3382     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3383     using compact_lemma [OF f] by blast

  3384   {

  3385     fix e::real

  3386     assume "e>0"

  3387     hence "0 < e / real_of_nat DIM('a)"

  3388       by (auto intro!: divide_pos_pos DIM_positive)

  3389     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"

  3390       by simp

  3391     moreover

  3392     {

  3393       fix n

  3394       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3395       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3396         apply (subst euclidean_dist_l2)

  3397         using zero_le_dist

  3398         by (rule setL2_le_setsum)

  3399       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3400         apply (rule setsum_strict_mono)

  3401         using n

  3402         by auto

  3403       finally have "dist (f (r n)) l < e"

  3404         by auto

  3405     }

  3406     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3407       by (rule eventually_elim1)

  3408   }

  3409   hence *: "((f \<circ> r) ---> l) sequentially"

  3410     unfolding o_def tendsto_iff by simp

  3411   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3412     by auto

  3413 qed

  3414

  3415 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3416   unfolding bounded_def

  3417   apply clarify

  3418   apply (rule_tac x="a" in exI)

  3419   apply (rule_tac x="e" in exI)

  3420   apply clarsimp

  3421   apply (drule (1) bspec)

  3422   apply (simp add: dist_Pair_Pair)

  3423   apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3424   done

  3425

  3426 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3427   unfolding bounded_def

  3428   apply clarify

  3429   apply (rule_tac x="b" in exI)

  3430   apply (rule_tac x="e" in exI)

  3431   apply clarsimp

  3432   apply (drule (1) bspec)

  3433   apply (simp add: dist_Pair_Pair)

  3434   apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3435   done

  3436

  3437 instance prod :: (heine_borel, heine_borel) heine_borel

  3438 proof

  3439   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  3440   assume f: "bounded (range f)"

  3441   from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)

  3442   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  3443     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  3444   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  3445     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  3446   obtain l2 r2 where r2: "subseq r2"

  3447     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3448     using bounded_imp_convergent_subsequence [OF s2]

  3449     unfolding o_def by fast

  3450   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3451     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  3452   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3453     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3454   have r: "subseq (r1 \<circ> r2)"

  3455     using r1 r2 unfolding subseq_def by simp

  3456   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3457     using l r by fast

  3458 qed

  3459

  3460 subsubsection{* Completeness *}

  3461

  3462 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  3463   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  3464

  3465 lemma compact_imp_complete:

  3466   assumes "compact s"

  3467   shows "complete s"

  3468 proof -

  3469   {

  3470     fix f

  3471     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3472     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  3473       using assms unfolding compact_def by blast

  3474

  3475     note lr' = seq_suble [OF lr(2)]

  3476

  3477     {

  3478       fix e::real

  3479       assume "e>0"

  3480       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"

  3481         unfolding cauchy_def

  3482         using e>0 apply (erule_tac x="e/2" in allE)

  3483         apply auto

  3484         done

  3485       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  3486       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using e>0 by auto

  3487       {

  3488         fix n::nat

  3489         assume n:"n \<ge> max N M"

  3490         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  3491         moreover have "r n \<ge> N" using lr'[of n] n by auto

  3492         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  3493         ultimately have "dist (f n) l < e"

  3494           using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)

  3495       }

  3496       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  3497     }

  3498     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  3499       unfolding LIMSEQ_def by auto

  3500   }

  3501   thus ?thesis unfolding complete_def by auto

  3502 qed

  3503

  3504 lemma nat_approx_posE:

  3505   fixes e::real

  3506   assumes "0 < e"

  3507   obtains n::nat where "1 / (Suc n) < e"

  3508 proof atomize_elim

  3509   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3510     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3511   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3512     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3513   also have "\<dots> = e" by simp

  3514   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3515 qed

  3516

  3517 lemma compact_eq_totally_bounded:

  3518   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3519     (is "_ \<longleftrightarrow> ?rhs")

  3520 proof

  3521   assume assms: "?rhs"

  3522   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)"

  3523     by (auto simp: choice_iff')

  3524

  3525   show "compact s"

  3526   proof cases

  3527     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3528   next

  3529     assume "s \<noteq> {}"

  3530     show ?thesis

  3531       unfolding compact_def

  3532     proof safe

  3533       fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3534

  3535       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  3536       then have [simp]: "\<And>n. 0 < e n" by auto

  3537       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)"

  3538       { fix n U assume "infinite {n. f n \<in> U}"

  3539         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  3540           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  3541         then guess a ..

  3542         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3543           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  3544         from someI_ex[OF this]

  3545         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  3546           unfolding B_def by auto }

  3547       note B = this

  3548

  3549       def F \<equiv> "nat_rec (B 0 UNIV) B"

  3550       { fix n have "infinite {i. f i \<in> F n}"

  3551           by (induct n) (auto simp: F_def B) }

  3552       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  3553         using B by (simp add: F_def)

  3554       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  3555         using decseq_SucI[of F] by (auto simp: decseq_def)

  3556

  3557       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  3558       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  3559         fix k i

  3560         have "infinite ({n. f n \<in> F k} - {.. i})"

  3561           using infinite {n. f n \<in> F k} by auto

  3562         from infinite_imp_nonempty[OF this]

  3563         show "\<exists>x>i. f x \<in> F k"

  3564           by (simp add: set_eq_iff not_le conj_commute)

  3565       qed

  3566

  3567       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  3568       have "subseq t"

  3569         unfolding subseq_Suc_iff by (simp add: t_def sel)

  3570       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  3571         using f by auto

  3572       moreover

  3573       { fix n have "(f \<circ> t) n \<in> F n"

  3574           by (cases n) (simp_all add: t_def sel) }

  3575       note t = this

  3576

  3577       have "Cauchy (f \<circ> t)"

  3578       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  3579         fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  3580         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  3581           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  3582         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  3583           by (auto simp: subset_eq)

  3584         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  3585         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  3586           by (simp add: dist_commute)

  3587       qed

  3588

  3589       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3590         using assms unfolding complete_def by blast

  3591     qed

  3592   qed

  3593 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  3594

  3595 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  3596 proof-

  3597   { assume ?rhs

  3598     { fix e::real

  3599       assume "e>0"

  3600       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  3601         by (erule_tac x="e/2" in allE) auto

  3602       { fix n m

  3603         assume nm:"N \<le> m \<and> N \<le> n"

  3604         hence "dist (s m) (s n) < e" using N

  3605           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  3606           by blast

  3607       }

  3608       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  3609         by blast

  3610     }

  3611     hence ?lhs

  3612       unfolding cauchy_def

  3613       by blast

  3614   }

  3615   thus ?thesis

  3616     unfolding cauchy_def

  3617     using dist_triangle_half_l

  3618     by blast

  3619 qed

  3620

  3621 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  3622 proof-

  3623   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  3624     unfolding cauchy_def

  3625     apply (erule_tac x= 1 in allE)

  3626     apply auto

  3627     done

  3628   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  3629   moreover

  3630   have "bounded (s  {0..N})"

  3631     using finite_imp_bounded[of "s  {1..N}"] by auto

  3632   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  3633     unfolding bounded_any_center [where a="s N"] by auto

  3634   ultimately show "?thesis"

  3635     unfolding bounded_any_center [where a="s N"]

  3636     apply (rule_tac x="max a 1" in exI)

  3637     apply auto

  3638     apply (erule_tac x=y in allE)

  3639     apply (erule_tac x=y in ballE)

  3640     apply auto

  3641     done

  3642 qed

  3643

  3644 instance heine_borel < complete_space

  3645 proof

  3646   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3647   hence "bounded (range f)"

  3648     by (rule cauchy_imp_bounded)

  3649   hence "compact (closure (range f))"

  3650     unfolding compact_eq_bounded_closed by auto

  3651   hence "complete (closure (range f))"

  3652     by (rule compact_imp_complete)

  3653   moreover have "\<forall>n. f n \<in> closure (range f)"

  3654     using closure_subset [of "range f"] by auto

  3655   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  3656     using Cauchy f unfolding complete_def by auto

  3657   then show "convergent f"

  3658     unfolding convergent_def by auto

  3659 qed

  3660

  3661 instance euclidean_space \<subseteq> banach ..

  3662

  3663 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  3664 proof(simp add: complete_def, rule, rule)

  3665   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3666   hence "convergent f" by (rule Cauchy_convergent)

  3667   thus "\<exists>l. f ----> l" unfolding convergent_def .

  3668 qed

  3669

  3670 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  3671 proof -

  3672   { fix x assume "x islimpt s"

  3673     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  3674       unfolding islimpt_sequential by auto

  3675     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  3676       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  3677     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  3678   }

  3679   thus "closed s" unfolding closed_limpt by auto

  3680 qed

  3681

  3682 lemma complete_eq_closed:

  3683   fixes s :: "'a::complete_space set"

  3684   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  3685 proof

  3686   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  3687 next

  3688   assume ?rhs

  3689   {

  3690     fix f

  3691     assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  3692     then obtain l where "(f ---> l) sequentially"

  3693       using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  3694     hence "\<exists>l\<in>s. (f ---> l) sequentially"

  3695       using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]

  3696       using as(1) by auto

  3697   }

  3698   thus ?lhs unfolding complete_def by auto

  3699 qed

  3700

  3701 lemma convergent_eq_cauchy:

  3702   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  3703   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  3704   unfolding Cauchy_convergent_iff convergent_def ..

  3705

  3706 lemma convergent_imp_bounded:

  3707   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  3708   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  3709   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  3710

  3711 lemma compact_cball[simp]:

  3712   fixes x :: "'a::heine_borel"

  3713   shows "compact(cball x e)"

  3714   using compact_eq_bounded_closed bounded_cball closed_cball

  3715   by blast

  3716

  3717 lemma compact_frontier_bounded[intro]:

  3718   fixes s :: "'a::heine_borel set"

  3719   shows "bounded s ==> compact(frontier s)"

  3720   unfolding frontier_def

  3721   using compact_eq_bounded_closed

  3722   by blast

  3723

  3724 lemma compact_frontier[intro]:

  3725   fixes s :: "'a::heine_borel set"

  3726   shows "compact s ==> compact (frontier s)"

  3727   using compact_eq_bounded_closed compact_frontier_bounded

  3728   by blast

  3729

  3730 lemma frontier_subset_compact:

  3731   fixes s :: "'a::heine_borel set"

  3732   shows "compact s ==> frontier s \<subseteq> s"

  3733   using frontier_subset_closed compact_eq_bounded_closed

  3734   by blast

  3735

  3736 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3737

  3738 lemma bounded_closed_nest:

  3739   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3740     "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3741   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3742 proof -

  3743   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n"

  3744     using choice[of "\<lambda>n x. x\<in> s n"] by auto

  3745   from assms(4,1) have *:"seq_compact (s 0)"

  3746     using bounded_closed_imp_seq_compact[of "s 0"] by auto

  3747

  3748   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3749     unfolding seq_compact_def

  3750     apply (erule_tac x=x in allE)

  3751     using x using assms(3)

  3752     apply blast

  3753     done

  3754

  3755   { fix n::nat

  3756     { fix e::real assume "e>0"

  3757       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e"

  3758         unfolding LIMSEQ_def by auto

  3759       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3760       moreover

  3761       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto

  3762       hence "(x \<circ> r) (max N n) \<in> s n"

  3763         using x apply (erule_tac x=n in allE)

  3764         using x apply (erule_tac x="r (max N n)" in allE)

  3765         using assms(3) apply (erule_tac x=n in allE)

  3766         apply (erule_tac x="r (max N n)" in allE)

  3767         apply auto

  3768         done

  3769       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3770     }

  3771     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3772   }

  3773   thus ?thesis by auto

  3774 qed

  3775

  3776 text {* Decreasing case does not even need compactness, just completeness. *}

  3777

  3778 lemma decreasing_closed_nest:

  3779   assumes "\<forall>n. closed(s n)"

  3780           "\<forall>n. (s n \<noteq> {})"

  3781           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3782           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3783   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3784 proof-

  3785   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3786   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3787   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3788   { fix e::real assume "e>0"

  3789     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3790     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3791       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3792       hence "dist (t m) (t n) < e" using N by auto

  3793     }

  3794     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3795   }

  3796   hence  "Cauchy t" unfolding cauchy_def by auto

  3797   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3798   { fix n::nat

  3799     { fix e::real assume "e>0"

  3800       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3801       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

  3802       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3803     }

  3804     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3805   }

  3806   then show ?thesis by auto

  3807 qed

  3808

  3809 text {* Strengthen it to the intersection actually being a singleton. *}

  3810

  3811 lemma decreasing_closed_nest_sing:

  3812   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3813   assumes "\<forall>n. closed(s n)"

  3814           "\<forall>n. s n \<noteq> {}"

  3815           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3816           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3817   shows "\<exists>a. \<Inter>(range s) = {a}"

  3818 proof-

  3819   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3820   { fix b assume b:"b \<in> \<Inter>(range s)"

  3821     { fix e::real assume "e>0"

  3822       hence "dist a b < e" using assms(4 )using b using a by blast

  3823     }

  3824     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3825   }

  3826   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3827   thus ?thesis ..

  3828 qed

  3829

  3830 text{* Cauchy-type criteria for uniform convergence. *}

  3831

  3832 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows

  3833  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3834   (\<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")

  3835 proof(rule)

  3836   assume ?lhs

  3837   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

  3838   { fix e::real assume "e>0"

  3839     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

  3840     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3841       hence "dist (s m x) (s n x) < e"

  3842         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3843         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3844         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3845     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  }

  3846   thus ?rhs by auto

  3847 next

  3848   assume ?rhs

  3849   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

  3850   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3851     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3852   { fix e::real assume "e>0"

  3853     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"

  3854       using ?rhs[THEN spec[where x="e/2"]] by auto

  3855     { fix x assume "P x"

  3856       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3857         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3858       fix n::nat assume "n\<ge>N"

  3859       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]]

  3860         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)  }

  3861     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3862   thus ?lhs by auto

  3863 qed

  3864

  3865 lemma uniformly_cauchy_imp_uniformly_convergent:

  3866   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  3867   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"

  3868           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3869   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3870 proof-

  3871   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"

  3872     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3873   moreover

  3874   { fix x assume "P x"

  3875     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3876       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3877   ultimately show ?thesis by auto

  3878 qed

  3879

  3880

  3881 subsection {* Continuity *}

  3882

  3883 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3884

  3885 lemma continuous_within_eps_delta:

  3886   "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)"

  3887   unfolding continuous_within and Lim_within

  3888   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3889

  3890 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3891                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3892   using continuous_within_eps_delta [of x UNIV f] by simp

  3893

  3894 text{* Versions in terms of open balls. *}

  3895

  3896 lemma continuous_within_ball:

  3897  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3898                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3899 proof

  3900   assume ?lhs

  3901   { fix e::real assume "e>0"

  3902     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"

  3903       using ?lhs[unfolded continuous_within Lim_within] by auto

  3904     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3905       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3906         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3907     }

  3908     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)  }

  3909   thus ?rhs by auto

  3910 next

  3911   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3912     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3913 qed

  3914

  3915 lemma continuous_at_ball:

  3916   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3917 proof

  3918   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3919     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)

  3920     unfolding dist_nz[THEN sym] by auto

  3921 next

  3922   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3923     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)

  3924 qed

  3925

  3926 text{* Define setwise continuity in terms of limits within the set. *}

  3927

  3928 lemma continuous_on_iff:

  3929   "continuous_on s f \<longleftrightarrow>

  3930     (\<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)"

  3931 unfolding continuous_on_def Lim_within

  3932 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3933 apply (rename_tac y, case_tac "y = x", simp)

  3934 apply (simp add: dist_nz)

  3935 done

  3936

  3937 definition

  3938   uniformly_continuous_on ::

  3939     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3940 where

  3941   "uniformly_continuous_on s f \<longleftrightarrow>

  3942     (\<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)"

  3943

  3944 text{* Some simple consequential lemmas. *}

  3945

  3946 lemma uniformly_continuous_imp_continuous:

  3947  " uniformly_continuous_on s f ==> continuous_on s f"

  3948   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3949

  3950 lemma continuous_at_imp_continuous_within:

  3951  "continuous (at x) f ==> continuous (at x within s) f"

  3952   unfolding continuous_within continuous_at using Lim_at_within by auto

  3953

  3954 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3955   by simp

  3956

  3957 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3958

  3959 lemma continuous_within_subset:

  3960  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3961              ==> continuous (at x within t) f"

  3962   unfolding continuous_within by(metis tendsto_within_subset)

  3963

  3964 lemma continuous_on_interior:

  3965   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3966   by (erule interiorE, drule (1) continuous_on_subset,

  3967     simp add: continuous_on_eq_continuous_at)

  3968

  3969 lemma continuous_on_eq:

  3970   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3971   unfolding continuous_on_def tendsto_def eventually_at_topological

  3972   by simp

  3973

  3974 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3975

  3976 lemma continuous_within_sequentially:

  3977   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3978   shows "continuous (at a within s) f \<longleftrightarrow>

  3979                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3980                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3981 proof

  3982   assume ?lhs

  3983   { 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"

  3984     fix T::"'b set" assume "open T" and "f a \<in> T"

  3985     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"

  3986       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  3987     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3988       using x(2) d>0 by simp

  3989     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3990     proof eventually_elim

  3991       case (elim n) thus ?case

  3992         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3993     qed

  3994   }

  3995   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3996 next

  3997   assume ?rhs thus ?lhs

  3998     unfolding continuous_within tendsto_def [where l="f a"]

  3999     by (simp add: sequentially_imp_eventually_within)

  4000 qed

  4001

  4002 lemma continuous_at_sequentially:

  4003   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4004   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  4005                   --> ((f o x) ---> f a) sequentially)"

  4006   using continuous_within_sequentially[of a UNIV f] by simp

  4007

  4008 lemma continuous_on_sequentially:

  4009   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4010   shows "continuous_on s f \<longleftrightarrow>

  4011     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4012                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  4013 proof

  4014   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  4015 next

  4016   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  4017 qed

  4018

  4019 lemma uniformly_continuous_on_sequentially:

  4020   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4021                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4022                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4023 proof

  4024   assume ?lhs

  4025   { 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"

  4026     { fix e::real assume "e>0"

  4027       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"

  4028         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4029       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

  4030       { fix n assume "n\<ge>N"

  4031         hence "dist (f (x n)) (f (y n)) < e"

  4032           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

  4033           unfolding dist_commute by simp  }

  4034       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  4035     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  4036   thus ?rhs by auto

  4037 next

  4038   assume ?rhs

  4039   { assume "\<not> ?lhs"

  4040     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

  4041     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"

  4042       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

  4043       by (auto simp add: dist_commute)

  4044     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4045     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4046     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"

  4047       unfolding x_def and y_def using fa by auto

  4048     { fix e::real assume "e>0"

  4049       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

  4050       { fix n::nat assume "n\<ge>N"

  4051         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  4052         also have "\<dots> < e" using N by auto

  4053         finally have "inverse (real n + 1) < e" by auto

  4054         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  4055       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  4056     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

  4057     hence False using fxy and e>0 by auto  }

  4058   thus ?lhs unfolding uniformly_continuous_on_def by blast

  4059 qed

  4060

  4061 text{* The usual transformation theorems. *}

  4062

  4063 lemma continuous_transform_within:

  4064   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4065   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4066           "continuous (at x within s) f"

  4067   shows "continuous (at x within s) g"

  4068 unfolding continuous_within

  4069 proof (rule Lim_transform_within)

  4070   show "0 < d" by fact

  4071   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4072     using assms(3) by auto

  4073   have "f x = g x"

  4074     using assms(1,2,3) by auto

  4075   thus "(f ---> g x) (at x within s)"

  4076     using assms(4) unfolding continuous_within by simp

  4077 qed

  4078

  4079 lemma continuous_transform_at:

  4080   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4081   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  4082           "continuous (at x) f"

  4083   shows "continuous (at x) g"

  4084   using continuous_transform_within [of d x UNIV f g] assms by simp

  4085

  4086 subsubsection {* Structural rules for pointwise continuity *}

  4087

  4088 lemmas continuous_within_id = continuous_ident

  4089

  4090 lemmas continuous_at_id = isCont_ident

  4091

  4092 lemma continuous_infdist[continuous_intros]:

  4093   assumes "continuous F f"

  4094   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4095   using assms unfolding continuous_def by (rule tendsto_infdist)

  4096

  4097 lemma continuous_infnorm[continuous_intros]:

  4098   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4099   unfolding continuous_def by (rule tendsto_infnorm)

  4100

  4101 lemma continuous_inner[continuous_intros]:

  4102   assumes "continuous F f" and "continuous F g"

  4103   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4104   using assms unfolding continuous_def by (rule tendsto_inner)

  4105

  4106 lemmas continuous_at_inverse = isCont_inverse

  4107

  4108 subsubsection {* Structural rules for setwise continuity *}

  4109

  4110 lemma continuous_on_infnorm[continuous_on_intros]:

  4111   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4112   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4113

  4114 lemma continuous_on_inner[continuous_on_intros]:

  4115   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4116   assumes "continuous_on s f" and "continuous_on s g"

  4117   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4118   using bounded_bilinear_inner assms

  4119   by (rule bounded_bilinear.continuous_on)

  4120

  4121 subsubsection {* Structural rules for uniform continuity *}

  4122

  4123 lemma uniformly_continuous_on_id[continuous_on_intros]:

  4124   shows "uniformly_continuous_on s (\<lambda>x. x)"

  4125   unfolding uniformly_continuous_on_def by auto

  4126

  4127 lemma uniformly_continuous_on_const[continuous_on_intros]:

  4128   shows "uniformly_continuous_on s (\<lambda>x. c)"

  4129   unfolding uniformly_continuous_on_def by simp

  4130

  4131 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  4132   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4133   assumes "uniformly_continuous_on s f"

  4134   assumes "uniformly_continuous_on s g"

  4135   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4136 proof -

  4137   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4138       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4139       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4140       by arith

  4141   } note le = this

  4142   { fix x y

  4143     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4144     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4145     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4146       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4147         simp add: le)

  4148   }

  4149   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  4150     unfolding dist_real_def by simp

  4151 qed

  4152

  4153 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  4154   assumes "uniformly_continuous_on s f"

  4155   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4156   unfolding norm_conv_dist using assms

  4157   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4158

  4159 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  4160   assumes "uniformly_continuous_on s g"

  4161   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4162   using assms unfolding uniformly_continuous_on_sequentially

  4163   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4164   by (auto intro: tendsto_zero)

  4165

  4166 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  4167   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4168   assumes "uniformly_continuous_on s f"

  4169   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4170   using bounded_linear_scaleR_right assms

  4171   by (rule bounded_linear.uniformly_continuous_on)

  4172

  4173 lemma dist_minus:

  4174   fixes x y :: "'a::real_normed_vector"

  4175   shows "dist (- x) (- y) = dist x y"

  4176   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4177

  4178 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  4179   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4180   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4181   unfolding uniformly_continuous_on_def dist_minus .

  4182

  4183 lemma uniformly_continuous_on_add[continuous_on_intros]:

  4184   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4185   assumes "uniformly_continuous_on s f"

  4186   assumes "uniformly_continuous_on s g"

  4187   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4188   using assms unfolding uniformly_continuous_on_sequentially

  4189   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4190   by (auto intro: tendsto_add_zero)

  4191

  4192 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  4193   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4194   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4195   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4196   unfolding ab_diff_minus using assms

  4197   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4198

  4199 text{* Continuity of all kinds is preserved under composition. *}

  4200

  4201 lemmas continuous_at_compose = isCont_o

  4202

  4203 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  4204   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4205   shows "uniformly_continuous_on s (g o f)"

  4206 proof-

  4207   { fix e::real assume "e>0"

  4208     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

  4209     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

  4210     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  }

  4211   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4212 qed

  4213

  4214 text{* Continuity in terms of open preimages. *}

  4215

  4216 lemma continuous_at_open:

  4217   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)))"

  4218 unfolding continuous_within_topological [of x UNIV f]

  4219 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4220

  4221 lemma continuous_imp_tendsto:

  4222   assumes "continuous (at x0) f" and "x ----> x0"

  4223   shows "(f \<circ> x) ----> (f x0)"

  4224 proof (rule topological_tendstoI)

  4225   fix S

  4226   assume "open S" "f x0 \<in> S"

  4227   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  4228      using assms continuous_at_open by metis

  4229   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  4230     using assms T_def by (auto simp: tendsto_def)

  4231   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  4232     using T_def by (auto elim!: eventually_elim1)

  4233 qed

  4234

  4235 lemma continuous_on_open:

  4236   "continuous_on s f \<longleftrightarrow>

  4237         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4238             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4239   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  4240   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4241

  4242 text {* Similarly in terms of closed sets. *}

  4243

  4244 lemma continuous_on_closed:

  4245   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")

  4246   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  4247   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4248

  4249 text {* Half-global and completely global cases. *}

  4250

  4251 lemma continuous_open_in_preimage:

  4252   assumes "continuous_on s f"  "open t"

  4253   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4254 proof-

  4255   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

  4256   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4257     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4258   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4259 qed

  4260

  4261 lemma continuous_closed_in_preimage:

  4262   assumes "continuous_on s f"  "closed t"

  4263   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4264 proof-

  4265   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

  4266   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4267     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4268   thus ?thesis

  4269     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4270 qed

  4271

  4272 lemma continuous_open_preimage:

  4273   assumes "continuous_on s f" "open s" "open t"

  4274   shows "open {x \<in> s. f x \<in> t}"

  4275 proof-

  4276   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4277     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4278   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4279 qed

  4280

  4281 lemma continuous_closed_preimage:

  4282   assumes "continuous_on s f" "closed s" "closed t"

  4283   shows "closed {x \<in> s. f x \<in> t}"

  4284 proof-

  4285   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4286     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4287   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4288 qed

  4289

  4290 lemma continuous_open_preimage_univ:

  4291   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4292   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4293

  4294 lemma continuous_closed_preimage_univ:

  4295   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4296   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4297

  4298 lemma continuous_open_vimage:

  4299   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4300   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4301

  4302 lemma continuous_closed_vimage:

  4303   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4304   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4305

  4306 lemma interior_image_subset:

  4307   assumes "\<forall>x. continuous (at x) f" "inj f"

  4308   shows "interior (f  s) \<subseteq> f  (interior s)"

  4309 proof

  4310   fix x assume "x \<in> interior (f  s)"

  4311   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4312   hence "x \<in> f  s" by auto

  4313   then obtain y where y: "y \<in> s" "x = f y" by auto

  4314   have "open (vimage f T)"

  4315     using assms(1) open T by (rule continuous_open_vimage)

  4316   moreover have "y \<in> vimage f T"

  4317     using x = f y x \<in> T by simp

  4318   moreover have "vimage f T \<subseteq> s"

  4319     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4320   ultimately have "y \<in> interior s" ..

  4321   with x = f y show "x \<in> f  interior s" ..

  4322 qed

  4323

  4324 text {* Equality of continuous functions on closure and related results. *}

  4325

  4326 lemma continuous_closed_in_preimage_constant:

  4327   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4328   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4329   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4330

  4331 lemma continuous_closed_preimage_constant:

  4332   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4333   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4334   using continuous_closed_preimage[of s f "{a}"] by auto

  4335

  4336 lemma continuous_constant_on_closure:

  4337   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4338   assumes "continuous_on (closure s) f"

  4339           "\<forall>x \<in> s. f x = a"

  4340   shows "\<forall>x \<in> (closure s). f x = a"

  4341     using continuous_closed_preimage_constant[of "closure s" f a]

  4342     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4343

  4344 lemma image_closure_subset:

  4345   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4346   shows "f  (closure s) \<subseteq> t"

  4347 proof-

  4348   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4349   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4350     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4351   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4352     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4353   thus ?thesis by auto

  4354 qed

  4355

  4356 lemma continuous_on_closure_norm_le:

  4357   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4358   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4359   shows "norm(f x) \<le> b"

  4360 proof-

  4361   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4362   show ?thesis

  4363     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4364     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4365 qed

  4366

  4367 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4368

  4369 lemma continuous_within_avoid:

  4370   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4371   assumes "continuous (at x within s) f" and "f x \<noteq> a"

  4372   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4373 proof-

  4374   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  4375     using t1_space [OF f x \<noteq> a] by fast

  4376   have "(f ---> f x) (at x within s)"

  4377     using assms(1) by (simp add: continuous_within)

  4378   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  4379     using open U and f x \<in> U

  4380     unfolding tendsto_def by fast

  4381   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  4382     using a \<notin> U by (fast elim: eventually_mono [rotated])

  4383   thus ?thesis

  4384     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  4385 qed

  4386

  4387 lemma continuous_at_avoid:

  4388   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4389   assumes "continuous (at x) f" and "f x \<noteq> a"

  4390   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4391   using assms continuous_within_avoid[of x UNIV f a] by simp

  4392

  4393 lemma continuous_on_avoid:

  4394   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4395   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4396   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4397 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

  4398

  4399 lemma continuous_on_open_avoid:

  4400   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4401   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4402   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4403 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

  4404

  4405 text {* Proving a function is constant by proving open-ness of level set. *}

  4406

  4407 lemma continuous_levelset_open_in_cases:

  4408   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4409   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4410         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4411         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4412 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4413

  4414 lemma continuous_levelset_open_in:

  4415   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4416   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4417         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4418         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4419 using continuous_levelset_open_in_cases[of s f ]

  4420 by meson

  4421

  4422 lemma continuous_levelset_open:

  4423   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4424   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4425   shows "\<forall>x \<in> s. f x = a"

  4426 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4427

  4428 text {* Some arithmetical combinations (more to prove). *}

  4429

  4430 lemma open_scaling[intro]:

  4431   fixes s :: "'a::real_normed_vector set"

  4432   assumes "c \<noteq> 0"  "open s"

  4433   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4434 proof-

  4435   { fix x assume "x \<in> s"

  4436     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

  4437     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4438     moreover

  4439     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4440       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4441         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4442           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4443       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  }

  4444     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  }

  4445   thus ?thesis unfolding open_dist by auto

  4446 qed

  4447

  4448 lemma minus_image_eq_vimage:

  4449   fixes A :: "'a::ab_group_add set"

  4450   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4451   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4452

  4453 lemma open_negations:

  4454   fixes s :: "'a::real_normed_vector set"

  4455   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4456   unfolding scaleR_minus1_left [symmetric]

  4457   by (rule open_scaling, auto)

  4458

  4459 lemma open_translation:

  4460   fixes s :: "'a::real_normed_vector set"

  4461   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4462 proof-

  4463   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4464       by (intro continuous_diff continuous_at_id continuous_const) }

  4465   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4466   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4467 qed

  4468

  4469 lemma open_affinity:

  4470   fixes s :: "'a::real_normed_vector set"

  4471   assumes "open s"  "c \<noteq> 0"

  4472   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4473 proof-

  4474   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4475   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4476   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4477 qed

  4478

  4479 lemma interior_translation:

  4480   fixes s :: "'a::real_normed_vector set"

  4481   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4482 proof (rule set_eqI, rule)

  4483   fix x assume "x \<in> interior (op + a  s)"

  4484   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4485   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

  4486   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

  4487 next

  4488   fix x assume "x \<in> op + a  interior s"

  4489   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

  4490   { fix z have *:"a + y - z = y + a - z" by auto

  4491     assume "z\<in>ball x e"

  4492     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

  4493     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4494   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4495   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4496 qed

  4497

  4498 text {* Topological properties of linear functions. *}

  4499

  4500 lemma linear_lim_0:

  4501   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4502 proof-

  4503   interpret f: bounded_linear f by fact

  4504   have "(f ---> f 0) (at 0)"

  4505     using tendsto_ident_at by (rule f.tendsto)

  4506   thus ?thesis unfolding f.zero .

  4507 qed

  4508

  4509 lemma linear_continuous_at:

  4510   assumes "bounded_linear f"  shows "continuous (at a) f"

  4511   unfolding continuous_at using assms

  4512   apply (rule bounded_linear.tendsto)

  4513   apply (rule tendsto_ident_at)

  4514   done

  4515

  4516 lemma linear_continuous_within:

  4517   shows "bounded_linear f ==> continuous (at x within s) f"

  4518   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4519

  4520 lemma linear_continuous_on:

  4521   shows "bounded_linear f ==> continuous_on s f"

  4522   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4523

  4524 text {* Also bilinear functions, in composition form. *}

  4525

  4526 lemma bilinear_continuous_at_compose:

  4527   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4528         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4529   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4530

  4531 lemma bilinear_continuous_within_compose:

  4532   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4533         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4534   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4535

  4536 lemma bilinear_continuous_on_compose:

  4537   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4538              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4539   unfolding continuous_on_def

  4540   by (fast elim: bounded_bilinear.tendsto)

  4541

  4542 text {* Preservation of compactness and connectedness under continuous function. *}

  4543

  4544 lemma compact_eq_openin_cover:

  4545   "compact S \<longleftrightarrow>

  4546     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4547       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  4548 proof safe

  4549   fix C

  4550   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  4551   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}"

  4552     unfolding openin_open by force+

  4553   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"

  4554     by (rule compactE)

  4555   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)"

  4556     by auto

  4557   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4558 next

  4559   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4560         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  4561   show "compact S"

  4562   proof (rule compactI)

  4563     fix C

  4564     let ?C = "image (\<lambda>T. S \<inter> T) C"

  4565     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  4566     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  4567       unfolding openin_open by auto

  4568     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  4569       by metis

  4570     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  4571     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  4572     proof (intro conjI)

  4573       from D \<subseteq> ?C show "?D \<subseteq> C"

  4574         by (fast intro: inv_into_into)

  4575       from finite D show "finite ?D"

  4576         by (rule finite_imageI)

  4577       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  4578         apply (rule subset_trans)

  4579         apply clarsimp

  4580         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  4581         apply (erule rev_bexI, fast)

  4582         done

  4583     qed

  4584     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4585   qed

  4586 qed

  4587

  4588 lemma connected_continuous_image:

  4589   assumes "continuous_on s f"  "connected s"

  4590   shows "connected(f  s)"

  4591 proof-

  4592   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4593     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4594       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4595       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4596       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4597     hence False using as(1,2)

  4598       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4599   thus ?thesis unfolding connected_clopen by auto

  4600 qed

  4601

  4602 text {* Continuity implies uniform continuity on a compact domain. *}

  4603

  4604 lemma compact_uniformly_continuous:

  4605   assumes f: "continuous_on s f" and s: "compact s"

  4606   shows "uniformly_continuous_on s f"

  4607   unfolding uniformly_continuous_on_def

  4608 proof (cases, safe)

  4609   fix e :: real assume "0 < e" "s \<noteq> {}"

  4610   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) } }"

  4611   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  4612   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  4613   proof safe

  4614     fix y assume "y \<in> s"

  4615     from continuous_open_in_preimage[OF f open_ball]

  4616     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  4617       unfolding openin_subtopology open_openin by metis

  4618     then obtain d where "ball y d \<subseteq> T" "0 < d"

  4619       using 0 < e y \<in> s by (auto elim!: openE)

  4620     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  4621       by (intro UN_I[of "(y, d)"]) auto

  4622   qed auto

  4623   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  4624     by (rule compactE_image)

  4625   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  4626     by (subst Min_gr_iff) auto

  4627   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  4628   proof (rule, safe)

  4629     fix x x' assume in_s: "x' \<in> s" "x \<in> s"

  4630     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  4631       by blast

  4632     moreover assume "dist x x' < Min (sndD) / 2"

  4633     ultimately have "dist y x' < d"

  4634       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  4635     with D x in_s show  "dist (f x) (f x') < e"

  4636       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  4637   qed (insert D, auto)

  4638 qed auto

  4639

  4640 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4641

  4642 lemma continuous_uniform_limit:

  4643   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4644   assumes "\<not> trivial_limit F"

  4645   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4646   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4647   shows "continuous_on s g"

  4648 proof-

  4649   { fix x and e::real assume "x\<in>s" "e>0"

  4650     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4651       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4652     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4653     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4654       using assms(1) by blast

  4655     have "e / 3 > 0" using e>0 by auto

  4656     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"

  4657       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4658     { fix y assume "y \<in> s" and "dist y x < d"

  4659       hence "dist (f n y) (f n x) < e / 3"

  4660         by (rule d [rule_format])

  4661       hence "dist (f n y) (g x) < 2 * e / 3"

  4662         using dist_triangle [of "f n y" "g x" "f n x"]

  4663         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4664         by auto

  4665       hence "dist (g y) (g x) < e"

  4666         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4667         using dist_triangle3 [of "g y" "g x" "f n y"]

  4668         by auto }

  4669     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4670       using d>0 by auto }

  4671   thus ?thesis unfolding continuous_on_iff by auto

  4672 qed

  4673

  4674

  4675 subsection {* Topological stuff lifted from and dropped to R *}

  4676

  4677 lemma open_real:

  4678   fixes s :: "real set" shows

  4679  "open s \<longleftrightarrow>

  4680         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4681   unfolding open_dist dist_norm by simp

  4682

  4683 lemma islimpt_approachable_real:

  4684   fixes s :: "real set"

  4685   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4686   unfolding islimpt_approachable dist_norm by simp

  4687

  4688 lemma closed_real:

  4689   fixes s :: "real set"

  4690   shows "closed s \<longleftrightarrow>

  4691         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4692             --> x \<in> s)"

  4693   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4694

  4695 lemma continuous_at_real_range:

  4696   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4697   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4698         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4699   unfolding continuous_at unfolding Lim_at

  4700   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4701   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

  4702   apply(erule_tac x=e in allE) by auto

  4703

  4704 lemma continuous_on_real_range:

  4705   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4706   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))"

  4707   unfolding continuous_on_iff dist_norm by simp

  4708

  4709 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4710

  4711 lemma distance_attains_sup:

  4712   assumes "compact s" "s \<noteq> {}"

  4713   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  4714 proof (rule continuous_attains_sup [OF assms])

  4715   { fix x assume "x\<in>s"

  4716     have "(dist a ---> dist a x) (at x within s)"

  4717       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  4718   }

  4719   thus "continuous_on s (dist a)"

  4720     unfolding continuous_on ..

  4721 qed

  4722

  4723 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4724

  4725 lemma distance_attains_inf:

  4726   fixes a :: "'a::heine_borel"

  4727   assumes "closed s"  "s \<noteq> {}"

  4728   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  4729 proof-

  4730   from assms(2) obtain b where "b \<in> s" by auto

  4731   let ?B = "s \<inter> cball a (dist b a)"

  4732   have "?B \<noteq> {}" using b \<in> s by (auto simp add: dist_commute)

  4733   moreover have "continuous_on ?B (dist a)"

  4734     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  4735   moreover have "compact ?B"

  4736     by (intro closed_inter_compact closed s compact_cball)

  4737   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  4738     by (metis continuous_attains_inf)

  4739   thus ?thesis by fastforce

  4740 qed

  4741

  4742

  4743 subsection {* Pasted sets *}

  4744

  4745 lemma bounded_Times:

  4746   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4747 proof-

  4748   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4749     using assms [unfolded bounded_def] by auto

  4750   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  4751     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4752   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4753 qed

  4754

  4755 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4756 by (induct x) simp

  4757

  4758 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  4759 unfolding seq_compact_def

  4760 apply clarify

  4761 apply (drule_tac x="fst \<circ> f" in spec)

  4762 apply (drule mp, simp add: mem_Times_iff)

  4763 apply (clarify, rename_tac l1 r1)

  4764 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4765 apply (drule mp, simp add: mem_Times_iff)

  4766 apply (clarify, rename_tac l2 r2)

  4767 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4768 apply (rule_tac x="r1 \<circ> r2" in exI)

  4769 apply (rule conjI, simp add: subseq_def)

  4770 apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  4771 apply (drule (1) tendsto_Pair) back

  4772 apply (simp add: o_def)

  4773 done

  4774

  4775 lemma compact_Times:

  4776   assumes "compact s" "compact t"

  4777   shows "compact (s \<times> t)"

  4778 proof (rule compactI)

  4779   fix C assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  4780   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)"

  4781   proof

  4782     fix x assume "x \<in> s"

  4783     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")

  4784     proof

  4785       fix y assume "y \<in> t"

  4786       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  4787       then show "?P y" by (auto elim!: open_prod_elim)

  4788     qed

  4789     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  4790       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"

  4791       by metis

  4792     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  4793     from compactE_image[OF compact t this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  4794       by auto

  4795     moreover with c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  4796       by (fastforce simp: subset_eq)

  4797     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  4798       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  4799   qed

  4800   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  4801     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"

  4802     unfolding subset_eq UN_iff by metis

  4803   moreover from compactE_image[OF compact s a] obtain e where e: "e \<subseteq> s" "finite e"

  4804     and s: "s \<subseteq> (\<Union>x\<in>e. a x)" by auto

  4805   moreover

  4806   { from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)" by auto

  4807     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)" using d e \<subseteq> s by (intro UN_mono) auto

  4808     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" . }

  4809   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  4810     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  4811 qed

  4812

  4813 text{* Hence some useful properties follow quite easily. *}

  4814

  4815 lemma compact_scaling:

  4816   fixes s :: "'a::real_normed_vector set"

  4817   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4818 proof-

  4819   let ?f = "\<lambda>x. scaleR c x"

  4820   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4821   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4822     using linear_continuous_at[OF *] assms by auto

  4823 qed

  4824

  4825 lemma compact_negations:

  4826   fixes s :: "'a::real_normed_vector set"

  4827   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4828   using compact_scaling [OF assms, of "- 1"] by auto

  4829

  4830 lemma compact_sums:

  4831   fixes s t :: "'a::real_normed_vector set"

  4832   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4833 proof-

  4834   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4835     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4836   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4837     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4838   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4839 qed

  4840

  4841 lemma compact_differences:

  4842   fixes s t :: "'a::real_normed_vector set"

  4843   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4844 proof-

  4845   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4846     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4847   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4848 qed

  4849

  4850 lemma compact_translation:

  4851   fixes s :: "'a::real_normed_vector set"

  4852   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4853 proof-

  4854   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4855   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4856 qed

  4857

  4858 lemma compact_affinity:

  4859   fixes s :: "'a::real_normed_vector set"

  4860   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4861 proof-

  4862   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4863   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4864 qed

  4865

  4866 text {* Hence we get the following. *}

  4867

  4868 lemma compact_sup_maxdistance:

  4869   fixes s :: "'a::metric_space set"

  4870   assumes "compact s"  "s \<noteq> {}"

  4871   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  4872 proof-

  4873   have "compact (s \<times> s)" using compact s by (intro compact_Times)

  4874   moreover have "s \<times> s \<noteq> {}" using s \<noteq> {} by auto

  4875   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  4876     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  4877   ultimately show ?thesis

  4878     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  4879 qed

  4880

  4881 text {* We can state this in terms of diameter of a set. *}

  4882

  4883 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

  4884

  4885 lemma diameter_bounded_bound:

  4886   fixes s :: "'a :: metric_space set"

  4887   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  4888   shows "dist x y \<le> diameter s"

  4889 proof -

  4890   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4891   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  4892     unfolding bounded_def by auto

  4893   have "dist x y \<le> Sup ?D"

  4894   proof (rule cSup_upper, safe)

  4895     fix a b assume "a \<in> s" "b \<in> s"

  4896     with z[of a] z[of b] dist_triangle[of a b z]

  4897     show "dist a b \<le> 2 * d"

  4898       by (simp add: dist_commute)

  4899   qed (insert s, auto)

  4900   with x \<in> s show ?thesis

  4901     by (auto simp add: diameter_def)

  4902 qed

  4903

  4904 lemma diameter_lower_bounded:

  4905   fixes s :: "'a :: metric_space set"

  4906   assumes s: "bounded s" and d: "0 < d" "d < diameter s"

  4907   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  4908 proof (rule ccontr)

  4909   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4910   assume contr: "\<not> ?thesis"

  4911   moreover

  4912   from d have "s \<noteq> {}"

  4913     by (auto simp: diameter_def)

  4914   then have "?D \<noteq> {}" by auto

  4915   ultimately have "Sup ?D \<le> d"

  4916     by (intro cSup_least) (auto simp: not_less)

  4917   with d < diameter s s \<noteq> {} show False

  4918     by (auto simp: diameter_def)

  4919 qed

  4920

  4921 lemma diameter_bounded:

  4922   assumes "bounded s"

  4923   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  4924         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  4925   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  4926   by auto

  4927

  4928 lemma diameter_compact_attained:

  4929   assumes "compact s"  "s \<noteq> {}"

  4930   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  4931 proof -

  4932   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4933   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"

  4934     using compact_sup_maxdistance[OF assms] by auto

  4935   hence "diameter s \<le> dist x y"

  4936     unfolding diameter_def by clarsimp (rule cSup_least, fast+)

  4937   thus ?thesis

  4938     by (metis b diameter_bounded_bound order_antisym xys)

  4939 qed

  4940

  4941 text {* Related results with closure as the conclusion. *}

  4942

  4943 lemma closed_scaling:

  4944   fixes s :: "'a::real_normed_vector set"

  4945   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4946 proof(cases "s={}")

  4947   case True thus ?thesis by auto

  4948 next

  4949   case False

  4950   show ?thesis

  4951   proof(cases "c=0")

  4952     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4953     case True thus ?thesis apply auto unfolding * by auto

  4954   next

  4955     case False

  4956     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4957       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4958           using as(1)[THEN spec[where x=n]]

  4959           using c\<noteq>0 by auto

  4960       }

  4961       moreover

  4962       { fix e::real assume "e>0"

  4963         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4964         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4965           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4966         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4967           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4968           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4969       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4970       ultimately have "l \<in> scaleR c  s"

  4971         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"]]

  4972         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4973     thus ?thesis unfolding closed_sequential_limits by fast

  4974   qed

  4975 qed

  4976

  4977 lemma closed_negations:

  4978   fixes s :: "'a::real_normed_vector set"

  4979   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4980   using closed_scaling[OF assms, of "- 1"] by simp

  4981

  4982 lemma compact_closed_sums:

  4983   fixes s :: "'a::real_normed_vector set"

  4984   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4985 proof-

  4986   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4987   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4988     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"

  4989       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4990     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4991       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4992     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4993       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto

  4994     hence "l - l' \<in> t"

  4995       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4996       using f(3) by auto

  4997     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

  4998   }

  4999   thus ?thesis unfolding closed_sequential_limits by fast

  5000 qed

  5001

  5002 lemma closed_compact_sums:

  5003   fixes s t :: "'a::real_normed_vector set"

  5004   assumes "closed s"  "compact t"

  5005   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5006 proof-

  5007   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

  5008     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  5009   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  5010 qed

  5011

  5012 lemma compact_closed_differences:

  5013   fixes s t :: "'a::real_normed_vector set"

  5014   assumes "compact s"  "closed t"

  5015   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5016 proof-

  5017   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  5018     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5019   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  5020 qed

  5021

  5022 lemma closed_compact_differences:

  5023   fixes s t :: "'a::real_normed_vector set"

  5024   assumes "closed s" "compact t"

  5025   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5026 proof-

  5027   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  5028     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5029  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  5030 qed

  5031

  5032 lemma closed_translation:

  5033   fixes a :: "'a::real_normed_vector"

  5034   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  5035 proof-

  5036   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  5037   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  5038 qed

  5039

  5040 lemma translation_Compl:

  5041   fixes a :: "'a::ab_group_add"

  5042   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5043   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  5044

  5045 lemma translation_UNIV:

  5046   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  5047   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  5048

  5049 lemma translation_diff:

  5050   fixes a :: "'a::ab_group_add"

  5051   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5052   by auto

  5053

  5054 lemma closure_translation:

  5055   fixes a :: "'a::real_normed_vector"

  5056   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5057 proof-

  5058   have *:"op + a  (- s) = - op + a  s"

  5059     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  5060   show ?thesis unfolding closure_interior translation_Compl

  5061     using interior_translation[of a "- s"] unfolding * by auto

  5062 qed

  5063

  5064 lemma frontier_translation:

  5065   fixes a :: "'a::real_normed_vector"

  5066   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5067   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  5068

  5069

  5070 subsection {* Separation between points and sets *}

  5071

  5072 lemma separate_point_closed:

  5073   fixes s :: "'a::heine_borel set"

  5074   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  5075 proof(cases "s = {}")

  5076   case True

  5077   thus ?thesis by(auto intro!: exI[where x=1])

  5078 next

  5079   case False

  5080   assume "closed s" "a \<notin> s"

  5081   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

  5082   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5083 qed

  5084

  5085 lemma separate_compact_closed:

  5086   fixes s t :: "'a::heine_borel set"

  5087   assumes "compact s" and t: "closed t" "s \<inter> t = {}"

  5088   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5089 proof cases

  5090   assume "s \<noteq> {} \<and> t \<noteq> {}"

  5091   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  5092   let ?inf = "\<lambda>x. infdist x t"

  5093   have "continuous_on s ?inf"

  5094     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

  5095   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  5096     using continuous_attains_inf[OF compact s s \<noteq> {}] by auto

  5097   then have "0 < ?inf x"

  5098     using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  5099   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  5100     using x by (auto intro: order_trans infdist_le)

  5101   ultimately show ?thesis

  5102     by auto

  5103 qed (auto intro!: exI[of _ 1])

  5104

  5105 lemma separate_closed_compact:

  5106   fixes s t :: "'a::heine_borel set"

  5107   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5108   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5109 proof-

  5110   have *:"t \<inter> s = {}" using assms(3) by auto

  5111   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5112     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5113     by (auto simp add: dist_commute)

  5114 qed

  5115

  5116

  5117 subsection {* Intervals *}

  5118

  5119 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5120   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and

  5121   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5122   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5123

  5124 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5125   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5126   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  5127   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5128

  5129 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5130  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and

  5131  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  5132 proof-

  5133   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  5134     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto

  5135     hence "a\<bullet>i < b\<bullet>i" by auto

  5136     hence False using as by auto  }

  5137   moreover

  5138   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  5139     let ?x = "(1/2) *\<^sub>R (a + b)"

  5140     { fix i :: 'a assume i:"i\<in>Basis"

  5141       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5142       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  5143         by (auto simp: inner_add_left) }

  5144     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5145   ultimately show ?th1 by blast

  5146

  5147   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  5148     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto

  5149     hence "a\<bullet>i \<le> b\<bullet>i" by auto

  5150     hence False using as by auto  }

  5151   moreover

  5152   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  5153     let ?x = "(1/2) *\<^sub>R (a + b)"

  5154     { fix i :: 'a assume i:"i\<in>Basis"

  5155       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5156       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"

  5157         by (auto simp: inner_add_left) }

  5158     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5159   ultimately show ?th2 by blast

  5160 qed

  5161

  5162 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5163   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and

  5164   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  5165   unfolding interval_eq_empty[of a b] by fastforce+

  5166

  5167 lemma interval_sing:

  5168   fixes a :: "'a::ordered_euclidean_space"

  5169   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5170   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5171   by (auto intro: euclidean_eqI simp: ex_in_conv)

  5172

  5173 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5174  "(\<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

  5175  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5176  "(\<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

  5177  "(\<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}"

  5178   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5179   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5180

  5181 lemma interval_open_subset_closed:

  5182   fixes a :: "'a::ordered_euclidean_space"

  5183   shows "{a<..<b} \<subseteq> {a .. b}"

  5184   unfolding subset_eq [unfolded Ball_def] mem_interval

  5185   by (fast intro: less_imp_le)

  5186

  5187 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5188  "{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

  5189  "{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

  5190  "{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

  5191  "{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)

  5192 proof-

  5193   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5194   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)

  5195   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5196     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5197     fix i :: 'a assume i:"i\<in>Basis"

  5198     (** TODO combine the following two parts as done in the HOL_light version. **)

  5199     { 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"

  5200       assume as2: "a\<bullet>i > c\<bullet>i"

  5201       { fix j :: 'a assume j:"j\<in>Basis"

  5202         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  5203           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i

  5204           by (auto simp add: as2)  }

  5205       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto

  5206       moreover

  5207       have "?x\<notin>{a .. b}"

  5208         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5209         using as(2)[THEN bspec[where x=i]] and as2 i

  5210         by auto

  5211       ultimately have False using as by auto  }

  5212     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto

  5213     moreover

  5214     { 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"

  5215       assume as2: "b\<bullet>i < d\<bullet>i"

  5216       { fix j :: 'a assume "j\<in>Basis"

  5217         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  5218           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]

  5219           by (auto simp add: as2) }

  5220       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5221       moreover

  5222       have "?x\<notin>{a .. b}"

  5223         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5224         using as(2)[THEN bspec[where x=i]] and as2 using i

  5225         by auto

  5226       ultimately have False using as by auto  }

  5227     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto

  5228     ultimately

  5229     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  5230   } note part1 = this

  5231   show ?th3

  5232     unfolding subset_eq and Ball_def and mem_interval

  5233     apply(rule,rule,rule,rule)

  5234     apply(rule part1)

  5235     unfolding subset_eq and Ball_def and mem_interval

  5236     prefer 4

  5237     apply auto

  5238     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+

  5239   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5240     fix i :: 'a assume i:"i\<in>Basis"

  5241     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5242     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

  5243   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5244     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5245     apply auto by(erule_tac x=xa in allE, simp)+

  5246 qed

  5247

  5248 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5249  "{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)}"

  5250   unfolding set_eq_iff and Int_iff and mem_interval by auto

  5251

  5252 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5253   "{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

  5254   "{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

  5255   "{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

  5256   "{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)

  5257 proof-

  5258   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"

  5259   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  5260       (\<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)"

  5261     by blast

  5262   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

  5263   show ?th1 unfolding * by (intro **) auto

  5264   show ?th2 unfolding * by (intro **) auto

  5265   show ?th3 unfolding * by (intro **) auto

  5266   show ?th4 unfolding * by (intro **) auto

  5267 qed

  5268

  5269 (* Moved interval_open_subset_closed a bit upwards *)

  5270

  5271 lemma open_interval[intro]:

  5272   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5273 proof-

  5274   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"

  5275     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

  5276       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

  5277   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"

  5278     by (auto simp add: eucl_less [where 'a='a])

  5279   finally show "open {a<..<b}" .

  5280 qed

  5281

  5282 lemma closed_interval[intro]:

  5283   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5284 proof-

  5285   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"

  5286     by (intro closed_INT ballI continuous_closed_vimage allI

  5287       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

  5288   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"

  5289     by (auto simp add: eucl_le [where 'a='a])

  5290   finally show "closed {a .. b}" .

  5291 qed

  5292

  5293 lemma interior_closed_interval [intro]:

  5294   fixes a b :: "'a::ordered_euclidean_space"

  5295   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5296 proof(rule subset_antisym)

  5297   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5298     by (rule interior_maximal)

  5299 next

  5300   { fix x assume "x \<in> interior {a..b}"

  5301     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5302     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

  5303     { fix i :: 'a assume i:"i\<in>Basis"

  5304       have "dist (x - (e / 2) *\<^sub>R i) x < e"

  5305            "dist (x + (e / 2) *\<^sub>R i) x < e"

  5306         unfolding dist_norm apply auto

  5307         unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto

  5308       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"

  5309                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"

  5310         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]

  5311         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]

  5312         unfolding mem_interval using i by blast+

  5313       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"

  5314         using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }

  5315     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5316   thus "?L \<subseteq> ?R" ..

  5317 qed

  5318

  5319 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5320 proof-

  5321   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"

  5322   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  5323     { fix i :: 'a assume "i\<in>Basis"

  5324       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  }

  5325     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5326     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5327   thus ?thesis unfolding interval and bounded_iff by auto

  5328 qed

  5329

  5330 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5331  "bounded {a .. b} \<and> bounded {a<..<b}"

  5332   using bounded_closed_interval[of a b]

  5333   using interval_open_subset_closed[of a b]

  5334   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5335   by simp

  5336

  5337 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5338  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5339   using bounded_interval[of a b] by auto

  5340

  5341 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5342   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]

  5343   by (auto simp: compact_eq_seq_compact_metric)

  5344

  5345 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5346   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5347 proof-

  5348   { fix i :: 'a assume "i\<in>Basis"

  5349     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"

  5350       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }

  5351   thus ?thesis unfolding mem_interval by auto

  5352 qed

  5353

  5354 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5355   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5356   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5357 proof-

  5358   { fix i :: 'a assume i:"i\<in>Basis"

  5359     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp

  5360     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5361       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5362       using x unfolding mem_interval using i apply simp

  5363       using y unfolding mem_interval using i apply simp

  5364       done

  5365     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto

  5366     moreover {

  5367     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp

  5368     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5369       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5370       using x unfolding mem_interval using i apply simp

  5371       using y unfolding mem_interval using i apply simp

  5372       done

  5373     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto

  5374     } 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 }

  5375   thus ?thesis unfolding mem_interval by auto

  5376 qed

  5377

  5378 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5379   assumes "{a<..<b} \<noteq> {}"

  5380   shows "closure {a<..<b} = {a .. b}"

  5381 proof-

  5382   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5383   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5384   { fix x assume as:"x \<in> {a .. b}"

  5385     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5386     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5387       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5388       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5389         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5390         by (auto simp add: algebra_simps)

  5391       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

  5392       hence False using fn unfolding f_def using xc by auto  }

  5393     moreover

  5394     { assume "\<not> (f ---> x) sequentially"

  5395       { fix e::real assume "e>0"

  5396         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

  5397         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5398         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)

  5399         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5400       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5401         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5402       hence "(f ---> x) sequentially" unfolding f_def

  5403         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]

  5404         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  }

  5405     ultimately have "x \<in> closure {a<..<b}"

  5406       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5407   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5408 qed

  5409

  5410 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5411   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5412 proof-

  5413   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5414   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"

  5415   { fix x assume "x\<in>s"

  5416     fix i :: 'a assume i:"i\<in>Basis"

  5417     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]

  5418       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }

  5419   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5420 qed

  5421

  5422 lemma bounded_subset_open_interval:

  5423   fixes s :: "('a::ordered_euclidean_space) set"

  5424   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5425   by (auto dest!: bounded_subset_open_interval_symmetric)

  5426

  5427 lemma bounded_subset_closed_interval_symmetric:

  5428   fixes s :: "('a::ordered_euclidean_space) set"

  5429   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5430 proof-

  5431   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5432   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5433 qed

  5434

  5435 lemma bounded_subset_closed_interval:

  5436   fixes s :: "('a::ordered_euclidean_space) set"

  5437   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5438   using bounded_subset_closed_interval_symmetric[of s] by auto

  5439

  5440 lemma frontier_closed_interval:

  5441   fixes a b :: "'a::ordered_euclidean_space"

  5442   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5443   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5444

  5445 lemma frontier_open_interval:

  5446   fixes a b :: "'a::ordered_euclidean_space"

  5447   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5448 proof(cases "{a<..<b} = {}")

  5449   case True thus ?thesis using frontier_empty by auto

  5450 next

  5451   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5452 qed

  5453

  5454 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5455   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5456   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5457

  5458

  5459 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5460

  5461 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5462   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"

  5463 proof-

  5464   { fix i :: 'a assume i:"i\<in>Basis"

  5465     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"

  5466     { assume "x\<bullet>i > b\<bullet>i"

  5467       then obtain y where "y \<bullet> i \<le> b \<bullet> i"  "y \<noteq> x"  "dist y x < x\<bullet>i - b\<bullet>i"

  5468         using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto

  5469       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i

  5470         by auto }

  5471     hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto  }

  5472   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5473 qed

  5474

  5475 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5476   shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"

  5477 proof-

  5478   { fix i :: 'a assume i:"i\<in>Basis"

  5479     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"

  5480     { assume "a\<bullet>i > x\<bullet>i"

  5481       then obtain y where "a \<bullet> i \<le> y \<bullet> i"  "y \<noteq> x"  "dist y x < a\<bullet>i - x\<bullet>i"

  5482         using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto

  5483       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }

  5484     hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto  }

  5485   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5486 qed

  5487

  5488 lemma open_box: "open (box a b)"

  5489 proof -

  5490   have "open (\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i})"

  5491     by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)

  5492   also have "(\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i}) = box a b"

  5493     by (auto simp add: box_def inner_commute)

  5494   finally show ?thesis .

  5495 qed

  5496

  5497 instance euclidean_space \<subseteq> second_countable_topology

  5498 proof

  5499   def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"

  5500   then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp

  5501   def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"

  5502   then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp

  5503   def B \<equiv> "(\<lambda>f. box (a f) (b f))  (Basis \<rightarrow>\<^sub>E (\<rat> \<times> \<rat>))"

  5504

  5505   have "Ball B open" by (simp add: B_def open_box)

  5506   moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"

  5507   proof safe

  5508     fix A::"'a set" assume "open A"

  5509     show "\<exists>B'\<subseteq>B. \<Union>B' = A"

  5510       apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])

  5511       apply (subst (3) open_UNION_box[OF open A])

  5512       apply (auto simp add: a b B_def)

  5513       done

  5514   qed

  5515   ultimately

  5516   have "topological_basis B" unfolding topological_basis_def by blast

  5517   moreover

  5518   have "countable B" unfolding B_def

  5519     by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)

  5520   ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"

  5521     by (blast intro: topological_basis_imp_subbasis)

  5522 qed

  5523

  5524 instance euclidean_space \<subseteq> polish_space ..

  5525

  5526 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5527

  5528 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5529   (\<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)"

  5530

  5531 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5532   "is_interval {a<..<b}" (is ?th2) proof -

  5533   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5534     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5535

  5536 lemma is_interval_empty:

  5537  "is_interval {}"

  5538   unfolding is_interval_def

  5539   by simp

  5540

  5541 lemma is_interval_univ:

  5542  "is_interval UNIV"

  5543   unfolding is_interval_def

  5544   by simp

  5545

  5546

  5547 subsection {* Closure of halfspaces and hyperplanes *}

  5548

  5549 lemma isCont_open_vimage:

  5550   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5551 proof -

  5552   from assms(1) have "continuous_on UNIV f"

  5553     unfolding isCont_def continuous_on_def by simp

  5554   hence "open {x \<in> UNIV. f x \<in> s}"

  5555     using open_UNIV open s by (rule continuous_open_preimage)

  5556   thus "open (f - s)"

  5557     by (simp add: vimage_def)

  5558 qed

  5559

  5560 lemma isCont_closed_vimage:

  5561   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5562   using assms unfolding closed_def vimage_Compl [symmetric]

  5563   by (rule isCont_open_vimage)

  5564

  5565 lemma open_Collect_less:

  5566   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5567   assumes f: "\<And>x. isCont f x"

  5568   assumes g: "\<And>x. isCont g x"

  5569   shows "open {x. f x < g x}"

  5570 proof -

  5571   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5572     using isCont_diff [OF g f] open_real_greaterThan

  5573     by (rule isCont_open_vimage)

  5574   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5575     by auto

  5576   finally show ?thesis .

  5577 qed

  5578

  5579 lemma closed_Collect_le:

  5580   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5581   assumes f: "\<And>x. isCont f x"

  5582   assumes g: "\<And>x. isCont g x"

  5583   shows "closed {x. f x \<le> g x}"

  5584 proof -

  5585   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5586     using isCont_diff [OF g f] closed_real_atLeast

  5587     by (rule isCont_closed_vimage)

  5588   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5589     by auto

  5590   finally show ?thesis .

  5591 qed

  5592

  5593 lemma closed_Collect_eq:

  5594   fixes f g :: "'a::t2_space \<Rightarrow> 'b::t2_space"

  5595   assumes f: "\<And>x. isCont f x"

  5596   assumes g: "\<And>x. isCont g x"

  5597   shows "closed {x. f x = g x}"

  5598 proof -

  5599   have "open {(x::'b, y::'b). x \<noteq> y}"

  5600     unfolding open_prod_def by (auto dest!: hausdorff)

  5601   hence "closed {(x::'b, y::'b). x = y}"

  5602     unfolding closed_def split_def Collect_neg_eq .

  5603   with isCont_Pair [OF f g]

  5604   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5605     by (rule isCont_closed_vimage)

  5606   also have "\<dots> = {x. f x = g x}" by auto

  5607   finally show ?thesis .

  5608 qed

  5609

  5610 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5611   unfolding continuous_at by (intro tendsto_intros)

  5612

  5613 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"

  5614   by (simp add: closed_Collect_le)

  5615

  5616 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

  5617   by (simp add: closed_Collect_le)

  5618

  5619 lemma closed_hyperplane: "closed {x. inner a x = b}"

  5620   by (simp add: closed_Collect_eq)

  5621

  5622 lemma closed_halfspace_component_le:

  5623   shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

  5624   by (simp add: closed_Collect_le)

  5625

  5626 lemma closed_halfspace_component_ge:

  5627   shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

  5628   by (simp add: closed_Collect_le)

  5629

  5630 text {* Openness of halfspaces. *}

  5631

  5632 lemma open_halfspace_lt: "open {x. inner a x < b}"

  5633   by (simp add: open_Collect_less)

  5634

  5635 lemma open_halfspace_gt: "open {x. inner a x > b}"

  5636   by (simp add: open_Collect_less)

  5637

  5638 lemma open_halfspace_component_lt:

  5639   shows "open {x::'a::euclidean_space. x\<bullet>i < a}"

  5640   by (simp add: open_Collect_less)

  5641

  5642 lemma open_halfspace_component_gt:

  5643   shows "open {x::'a::euclidean_space. x\<bullet>i > a}"

  5644   by (simp add: open_Collect_less)

  5645

  5646 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

  5647

  5648 lemma eucl_lessThan_eq_halfspaces:

  5649   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5650   shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"

  5651  by (auto simp: eucl_less[where 'a='a])

  5652

  5653 lemma eucl_greaterThan_eq_halfspaces:

  5654   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5655   shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"

  5656  by (auto simp: eucl_less[where 'a='a])

  5657

  5658 lemma eucl_atMost_eq_halfspaces:

  5659   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5660   shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"

  5661  by (auto simp: eucl_le[where 'a='a])

  5662

  5663 lemma eucl_atLeast_eq_halfspaces:

  5664   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5665   shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"

  5666  by (auto simp: eucl_le[where 'a='a])

  5667

  5668 lemma open_eucl_lessThan[simp, intro]:

  5669   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5670   shows "open {..< a}"

  5671   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

  5672

  5673 lemma open_eucl_greaterThan[simp, intro]:

  5674   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5675   shows "open {a <..}"

  5676   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

  5677

  5678 lemma closed_eucl_atMost[simp, intro]:

  5679   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5680   shows "closed {.. a}"

  5681   unfolding eucl_atMost_eq_halfspaces

  5682   by (simp add: closed_INT closed_Collect_le)

  5683

  5684 lemma closed_eucl_atLeast[simp, intro]:

  5685   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5686   shows "closed {a ..}"

  5687   unfolding eucl_atLeast_eq_halfspaces

  5688   by (simp add: closed_INT closed_Collect_le)

  5689

  5690 text {* This gives a simple derivation of limit component bounds. *}

  5691

  5692 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5693   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"

  5694   shows "l\<bullet>i \<le> b"

  5695   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

  5696

  5697 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5698   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"

  5699   shows "b \<le> l\<bullet>i"

  5700   by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

  5701

  5702 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5703   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"

  5704   shows "l\<bullet>i = b"

  5705   using ev[unfolded order_eq_iff eventually_conj_iff]

  5706   using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto

  5707

  5708 text{* Limits relative to a union.                                               *}

  5709

  5710 lemma eventually_within_Un:

  5711   "eventually P (at x within (s \<union> t)) \<longleftrightarrow> eventually P (at x within s) \<and> eventually P (at x within t)"

  5712   unfolding eventually_at_filter

  5713   by (auto elim!: eventually_rev_mp)

  5714

  5715 lemma Lim_within_union:

  5716  "(f ---> l) (at x within (s \<union> t)) \<longleftrightarrow>

  5717   (f ---> l) (at x within s) \<and> (f ---> l) (at x within t)"

  5718   unfolding tendsto_def

  5719   by (auto simp add: eventually_within_Un)

  5720

  5721 lemma Lim_topological:

  5722  "(f ---> l) net \<longleftrightarrow>

  5723         trivial_limit net \<or>

  5724         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5725   unfolding tendsto_def trivial_limit_eq by auto

  5726

  5727 text{* Some more convenient intermediate-value theorem formulations.             *}

  5728

  5729 lemma connected_ivt_hyperplane:

  5730   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5731   shows "\<exists>z \<in> s. inner a z = b"

  5732 proof(rule ccontr)

  5733   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5734   let ?A = "{x. inner a x < b}"

  5735   let ?B = "{x. inner a x > b}"

  5736   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5737   moreover have "?A \<inter> ?B = {}" by auto

  5738   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5739   ultimately show False

  5740     using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5)

  5741     by auto

  5742 qed

  5743

  5744 lemma connected_ivt_component:

  5745   fixes x::"'a::euclidean_space"

  5746   shows "connected s \<Longrightarrow>

  5747     x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow>

  5748     x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s.  z\<bullet>k = a)"

  5749   using connected_ivt_hyperplane[of s x y "k::'a" a]

  5750   by (auto simp: inner_commute)

  5751

  5752

  5753 subsection {* Homeomorphisms *}

  5754

  5755 definition "homeomorphism s t f g \<longleftrightarrow>

  5756   (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5757   (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5758

  5759 definition

  5760   homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"

  5761     (infixr "homeomorphic" 60) where

  5762   "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  5763

  5764 lemma homeomorphic_refl: "s homeomorphic s"

  5765   unfolding homeomorphic_def

  5766   unfolding homeomorphism_def

  5767   using continuous_on_id

  5768   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5769   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5770   apply blast

  5771   done

  5772

  5773 lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  5774   unfolding homeomorphic_def

  5775   unfolding homeomorphism_def

  5776   by blast

  5777

  5778 lemma homeomorphic_trans:

  5779   assumes "s homeomorphic t" "t homeomorphic u"

  5780   shows "s homeomorphic u"

  5781 proof-

  5782   obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x"  "f1  s = t"

  5783       "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1  t = s" "continuous_on t g1"

  5784     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto

  5785   obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x"  "f2  t = u" "continuous_on t f2"

  5786       "\<forall>y\<in>u. f2 (g2 y) = y" "g2  u = t" "continuous_on u g2"

  5787     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto

  5788

  5789   {

  5790     fix x

  5791     assume "x\<in>s"

  5792     hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x"

  5793       using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2)

  5794       by auto

  5795   }

  5796   moreover have "(