src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author haftmann Sat May 25 15:44:29 2013 +0200 (2013-05-25) changeset 52141 eff000cab70f parent 51773 9328c6681f3c child 52624 8a7b59a81088 permissions -rw-r--r--
weaker precendence of syntax for big intersection and union on sets
     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   by simp

   589

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

   591   by (auto simp add: closedin_closed)

   592

   593 lemma openin_euclidean_subtopology_iff:

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

   595   shows "openin (subtopology euclidean U) S

   596   \<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")

   597 proof

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

   599 next

   600   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}"

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

   602     unfolding T_def

   603     apply clarsimp

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

   605     apply (clarsimp simp add: less_diff_eq)

   606     apply (erule rev_bexI)

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

   608     apply (erule le_less_trans [OF dist_triangle])

   609     done

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

   611     unfolding T_def

   612     apply auto

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

   614     apply auto

   615     done

   616   from 1 2 show ?lhs

   617     unfolding openin_open open_dist by fast

   618 qed

   619

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

   621

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

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

   624   unfolding open_openin openin_open by blast

   625

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

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

   628

   629 lemma closedin_trans[trans]:

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

   631            closedin (subtopology euclidean U) T

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

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

   634

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

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

   637

   638

   639 subsection {* Open and closed balls *}

   640

   641 definition

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

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

   644

   645 definition

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

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

   648

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

   650   by (simp add: ball_def)

   651

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

   653   by (simp add: cball_def)

   654

   655 lemma mem_ball_0:

   656   fixes x :: "'a::real_normed_vector"

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

   658   by (simp add: dist_norm)

   659

   660 lemma mem_cball_0:

   661   fixes x :: "'a::real_normed_vector"

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

   663   by (simp add: dist_norm)

   664

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

   666   by simp

   667

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

   669   by simp

   670

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

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

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

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

   675   by (simp add: set_eq_iff) arith

   676

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

   678   by (simp add: set_eq_iff)

   679

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

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

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

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

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

   685

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

   687   unfolding open_dist ball_def mem_Collect_eq Ball_def

   688   unfolding dist_commute

   689   apply clarify

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

   691   using dist_triangle_alt[where z=x]

   692   apply (clarsimp simp add: diff_less_iff)

   693   apply atomize

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

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

   696   by arith

   697

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

   699   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   700

   701 lemma openE[elim?]:

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

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

   704   using assms unfolding open_contains_ball by auto

   705

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

   707   by (metis open_contains_ball subset_eq centre_in_ball)

   708

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

   710   unfolding mem_ball set_eq_iff

   711   apply (simp add: not_less)

   712   by (metis zero_le_dist order_trans dist_self)

   713

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

   715

   716 lemma euclidean_dist_l2:

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

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

   719   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   721

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

   723

   724 lemma rational_boxes:

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

   726   assumes "0 < e"

   727   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"

   728 proof -

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

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

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

   732   proof

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

   734   qed

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

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

   737   proof

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

   739   qed

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

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

   742   show ?thesis

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

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

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

   746       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   748     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

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

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

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

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

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

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

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

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

   757         by (rule power_strict_mono) auto

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

   759         by (simp add: power_divide)

   760     qed auto

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

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

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

   764 qed

   765

   766 lemma open_UNION_box:

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

   768   assumes "open M"

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

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

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

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

   773 proof safe

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

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

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

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

   778     "\<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"

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

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

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

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

   783 qed (auto simp: I_def)

   784

   785 subsection{* Connectedness *}

   786

   787 lemma connected_local:

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

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

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

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

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

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

   794                  ~(e2 = {}))"

   795 unfolding connected_def openin_open by (safe, blast+)

   796

   797 lemma exists_diff:

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

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

   800 proof-

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

   802   moreover

   803   {fix S assume H: "P S"

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

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

   806   ultimately show ?thesis by metis

   807 qed

   808

   809 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   812 proof-

   813   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> {})"

   814     unfolding connected_def openin_open closedin_closed

   815     apply (subst exists_diff) by blast

   816   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> {})"

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

   818

   819   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'))"

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

   821     unfolding connected_def openin_open closedin_closed by auto

   822   {fix e2

   823     {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)"

   824         by auto}

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

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

   827   then show ?thesis unfolding th0 th1 by simp

   828 qed

   829

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

   831   by (simp add: connected_def)

   832

   833

   834 subsection{* Limit points *}

   835

   836 definition (in topological_space)

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

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

   839

   840 lemma islimptI:

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

   842   shows "x islimpt S"

   843   using assms unfolding islimpt_def by auto

   844

   845 lemma islimptE:

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

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

   848   using assms unfolding islimpt_def by auto

   849

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

   851   unfolding islimpt_def eventually_at_topological by auto

   852

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

   854   unfolding islimpt_def by fast

   855

   856 lemma islimpt_approachable:

   857   fixes x :: "'a::metric_space"

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

   859   unfolding islimpt_iff_eventually eventually_at by fast

   860

   861 lemma islimpt_approachable_le:

   862   fixes x :: "'a::metric_space"

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

   864   unfolding islimpt_approachable

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

   866     THEN arg_cong [where f=Not]]

   867   by (simp add: Bex_def conj_commute conj_left_commute)

   868

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

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

   871

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

   873   unfolding islimpt_def by blast

   874

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

   876

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

   878   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   879

   880 lemma perfect_choose_dist:

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

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

   883 using islimpt_UNIV [of x]

   884 by (simp add: islimpt_approachable)

   885

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

   887   unfolding closed_def

   888   apply (subst open_subopen)

   889   apply (simp add: islimpt_def subset_eq)

   890   by (metis ComplE ComplI)

   891

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

   893   unfolding islimpt_def by auto

   894

   895 lemma finite_set_avoid:

   896   fixes a :: "'a::metric_space"

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

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

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

   900 next

   901   case (2 x F)

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

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

   904   moreover

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

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

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

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

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

   910   ultimately show ?case by blast

   911 qed

   912

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

   914   by (simp add: islimpt_iff_eventually eventually_conj_iff)

   915

   916 lemma discrete_imp_closed:

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

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

   919   shows "closed S"

   920 proof-

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

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

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

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

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

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

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

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

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

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

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

   932 qed

   933

   934

   935 subsection {* Interior of a Set *}

   936

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

   938

   939 lemma interiorI [intro?]:

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

   941   shows "x \<in> interior S"

   942   using assms unfolding interior_def by fast

   943

   944 lemma interiorE [elim?]:

   945   assumes "x \<in> interior S"

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

   947   using assms unfolding interior_def by fast

   948

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

   950   by (simp add: interior_def open_Union)

   951

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

   953   by (auto simp add: interior_def)

   954

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

   956   by (auto simp add: interior_def)

   957

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

   959   by (intro equalityI interior_subset interior_maximal subset_refl)

   960

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

   962   by (metis open_interior interior_open)

   963

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

   965   by (metis interior_maximal interior_subset subset_trans)

   966

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

   968   using open_empty by (rule interior_open)

   969

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

   971   using open_UNIV by (rule interior_open)

   972

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

   974   using open_interior by (rule interior_open)

   975

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

   977   by (auto simp add: interior_def)

   978

   979 lemma interior_unique:

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

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

   982   shows "interior S = T"

   983   by (intro equalityI assms interior_subset open_interior interior_maximal)

   984

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

   986   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

   987     Int_lower2 interior_maximal interior_subset open_Int open_interior)

   988

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

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

   991   by (simp add: open_subset_interior)

   992

   993 lemma interior_limit_point [intro]:

   994   fixes x :: "'a::perfect_space"

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

   996   using x islimpt_UNIV [of x]

   997   unfolding interior_def islimpt_def

   998   apply (clarsimp, rename_tac T T')

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

  1000   apply (auto simp add: open_Int)

  1001   done

  1002

  1003 lemma interior_closed_Un_empty_interior:

  1004   assumes cS: "closed S" and iT: "interior T = {}"

  1005   shows "interior (S \<union> T) = interior S"

  1006 proof

  1007   show "interior S \<subseteq> interior (S \<union> T)"

  1008     by (rule interior_mono, rule Un_upper1)

  1009 next

  1010   show "interior (S \<union> T) \<subseteq> interior S"

  1011   proof

  1012     fix x assume "x \<in> interior (S \<union> T)"

  1013     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1014     show "x \<in> interior S"

  1015     proof (rule ccontr)

  1016       assume "x \<notin> interior S"

  1017       with x \<in> R open R obtain y where "y \<in> R - S"

  1018         unfolding interior_def by fast

  1019       from open R closed S have "open (R - S)" by (rule open_Diff)

  1020       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

  1021       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

  1022       show "False" unfolding interior_def by fast

  1023     qed

  1024   qed

  1025 qed

  1026

  1027 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1028 proof (rule interior_unique)

  1029   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1030     by (intro Sigma_mono interior_subset)

  1031   show "open (interior A \<times> interior B)"

  1032     by (intro open_Times open_interior)

  1033   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

  1034   proof (safe)

  1035     fix x y assume "(x, y) \<in> T"

  1036     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1037       using open T unfolding open_prod_def by fast

  1038     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1039       using T \<subseteq> A \<times> B by auto

  1040     thus "x \<in> interior A" and "y \<in> interior B"

  1041       by (auto intro: interiorI)

  1042   qed

  1043 qed

  1044

  1045

  1046 subsection {* Closure of a Set *}

  1047

  1048 definition "closure S = S \<union> {x | x. x islimpt S}"

  1049

  1050 lemma interior_closure: "interior S = - (closure (- S))"

  1051   unfolding interior_def closure_def islimpt_def by auto

  1052

  1053 lemma closure_interior: "closure S = - interior (- S)"

  1054   unfolding interior_closure by simp

  1055

  1056 lemma closed_closure[simp, intro]: "closed (closure S)"

  1057   unfolding closure_interior by (simp add: closed_Compl)

  1058

  1059 lemma closure_subset: "S \<subseteq> closure S"

  1060   unfolding closure_def by simp

  1061

  1062 lemma closure_hull: "closure S = closed hull S"

  1063   unfolding hull_def closure_interior interior_def by auto

  1064

  1065 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1066   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1067

  1068 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1069   unfolding closure_eq .

  1070

  1071 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1072   unfolding closure_hull by (rule hull_hull)

  1073

  1074 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1075   unfolding closure_hull by (rule hull_mono)

  1076

  1077 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1078   unfolding closure_hull by (rule hull_minimal)

  1079

  1080 lemma closure_unique:

  1081   assumes "S \<subseteq> T" and "closed T"

  1082   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1083   shows "closure S = T"

  1084   using assms unfolding closure_hull by (rule hull_unique)

  1085

  1086 lemma closure_empty [simp]: "closure {} = {}"

  1087   using closed_empty by (rule closure_closed)

  1088

  1089 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1090   using closed_UNIV by (rule closure_closed)

  1091

  1092 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1093   unfolding closure_interior by simp

  1094

  1095 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1096   using closure_empty closure_subset[of S]

  1097   by blast

  1098

  1099 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1100   using closure_eq[of S] closure_subset[of S]

  1101   by simp

  1102

  1103 lemma open_inter_closure_eq_empty:

  1104   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1105   using open_subset_interior[of S "- T"]

  1106   using interior_subset[of "- T"]

  1107   unfolding closure_interior

  1108   by auto

  1109

  1110 lemma open_inter_closure_subset:

  1111   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1112 proof

  1113   fix x

  1114   assume as: "open S" "x \<in> S \<inter> closure T"

  1115   { assume *:"x islimpt T"

  1116     have "x islimpt (S \<inter> T)"

  1117     proof (rule islimptI)

  1118       fix A

  1119       assume "x \<in> A" "open A"

  1120       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1121         by (simp_all add: open_Int)

  1122       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1123         by (rule islimptE)

  1124       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1125         by simp_all

  1126       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1127     qed

  1128   }

  1129   then show "x \<in> closure (S \<inter> T)" using as

  1130     unfolding closure_def

  1131     by blast

  1132 qed

  1133

  1134 lemma closure_complement: "closure (- S) = - interior S"

  1135   unfolding closure_interior by simp

  1136

  1137 lemma interior_complement: "interior (- S) = - closure S"

  1138   unfolding closure_interior by simp

  1139

  1140 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1141 proof (rule closure_unique)

  1142   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1143     by (intro Sigma_mono closure_subset)

  1144   show "closed (closure A \<times> closure B)"

  1145     by (intro closed_Times closed_closure)

  1146   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

  1147     apply (simp add: closed_def open_prod_def, clarify)

  1148     apply (rule ccontr)

  1149     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1150     apply (simp add: closure_interior interior_def)

  1151     apply (drule_tac x=C in spec)

  1152     apply (drule_tac x=D in spec)

  1153     apply auto

  1154     done

  1155 qed

  1156

  1157

  1158 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1159   unfolding closure_def using islimpt_punctured by blast

  1160

  1161

  1162 subsection {* Frontier (aka boundary) *}

  1163

  1164 definition "frontier S = closure S - interior S"

  1165

  1166 lemma frontier_closed: "closed(frontier S)"

  1167   by (simp add: frontier_def closed_Diff)

  1168

  1169 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1170   by (auto simp add: frontier_def interior_closure)

  1171

  1172 lemma frontier_straddle:

  1173   fixes a :: "'a::metric_space"

  1174   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))"

  1175   unfolding frontier_def closure_interior

  1176   by (auto simp add: mem_interior subset_eq ball_def)

  1177

  1178 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1179   by (metis frontier_def closure_closed Diff_subset)

  1180

  1181 lemma frontier_empty[simp]: "frontier {} = {}"

  1182   by (simp add: frontier_def)

  1183

  1184 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1185 proof-

  1186   { assume "frontier S \<subseteq> S"

  1187     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

  1188     hence "closed S" using closure_subset_eq by auto

  1189   }

  1190   thus ?thesis using frontier_subset_closed[of S] ..

  1191 qed

  1192

  1193 lemma frontier_complement: "frontier(- S) = frontier S"

  1194   by (auto simp add: frontier_def closure_complement interior_complement)

  1195

  1196 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1197   using frontier_complement frontier_subset_eq[of "- S"]

  1198   unfolding open_closed by auto

  1199

  1200 subsection {* Filters and the eventually true'' quantifier *}

  1201

  1202 definition

  1203   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1204     (infixr "indirection" 70) where

  1205   "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1206

  1207 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1208

  1209 lemma trivial_limit_within:

  1210   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1211 proof

  1212   assume "trivial_limit (at a within S)"

  1213   thus "\<not> a islimpt S"

  1214     unfolding trivial_limit_def

  1215     unfolding eventually_at_topological

  1216     unfolding islimpt_def

  1217     apply (clarsimp simp add: set_eq_iff)

  1218     apply (rename_tac T, rule_tac x=T in exI)

  1219     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1220     done

  1221 next

  1222   assume "\<not> a islimpt S"

  1223   thus "trivial_limit (at a within S)"

  1224     unfolding trivial_limit_def

  1225     unfolding eventually_at_topological

  1226     unfolding islimpt_def

  1227     apply clarsimp

  1228     apply (rule_tac x=T in exI)

  1229     apply auto

  1230     done

  1231 qed

  1232

  1233 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1234   using trivial_limit_within [of a UNIV] by simp

  1235

  1236 lemma trivial_limit_at:

  1237   fixes a :: "'a::perfect_space"

  1238   shows "\<not> trivial_limit (at a)"

  1239   by (rule at_neq_bot)

  1240

  1241 lemma trivial_limit_at_infinity:

  1242   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1243   unfolding trivial_limit_def eventually_at_infinity

  1244   apply clarsimp

  1245   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1246    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1247   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1248   apply (drule_tac x=UNIV in spec, simp)

  1249   done

  1250

  1251 lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"

  1252   using islimpt_in_closure by (metis trivial_limit_within)

  1253

  1254 text {* Some property holds "sufficiently close" to the limit point. *}

  1255

  1256 lemma eventually_at2:

  1257   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1258 unfolding eventually_at dist_nz by auto

  1259

  1260 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

  1261   unfolding trivial_limit_def

  1262   by (auto elim: eventually_rev_mp)

  1263

  1264 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1265   by simp

  1266

  1267 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1268   by (simp add: filter_eq_iff)

  1269

  1270 text{* Combining theorems for "eventually" *}

  1271

  1272 lemma eventually_rev_mono:

  1273   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1274 using eventually_mono [of P Q] by fast

  1275

  1276 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

  1277   by (simp add: eventually_False)

  1278

  1279

  1280 subsection {* Limits *}

  1281

  1282 lemma Lim:

  1283  "(f ---> l) net \<longleftrightarrow>

  1284         trivial_limit net \<or>

  1285         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1286   unfolding tendsto_iff trivial_limit_eq by auto

  1287

  1288 text{* Show that they yield usual definitions in the various cases. *}

  1289

  1290 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1291            (\<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)"

  1292   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1293

  1294 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1295         (\<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)"

  1296   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1297

  1298 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1299         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1300   by (auto simp add: tendsto_iff eventually_at2)

  1301

  1302 lemma Lim_at_infinity:

  1303   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1304   by (auto simp add: tendsto_iff eventually_at_infinity)

  1305

  1306 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1307   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1308

  1309 text{* The expected monotonicity property. *}

  1310

  1311 lemma Lim_within_empty: "(f ---> l) (at x within {})"

  1312   unfolding tendsto_def eventually_at_filter by simp

  1313

  1314 lemma Lim_Un: assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1315   shows "(f ---> l) (at x within (S \<union> T))"

  1316   using assms unfolding tendsto_def eventually_at_filter

  1317   apply clarify

  1318   apply (drule spec, drule (1) mp, drule (1) mp)

  1319   apply (drule spec, drule (1) mp, drule (1) mp)

  1320   apply (auto elim: eventually_elim2)

  1321   done

  1322

  1323 lemma Lim_Un_univ:

  1324  "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = UNIV

  1325         ==> (f ---> l) (at x)"

  1326   by (metis Lim_Un)

  1327

  1328 text{* Interrelations between restricted and unrestricted limits. *}

  1329

  1330

  1331 lemma Lim_at_within: (* FIXME: rename *)

  1332   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1333   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1334

  1335 lemma eventually_within_interior:

  1336   assumes "x \<in> interior S"

  1337   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1338 proof-

  1339   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1340   { assume "?lhs"

  1341     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1342       unfolding eventually_at_topological

  1343       by auto

  1344     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1345       by auto

  1346     then have "?rhs"

  1347       unfolding eventually_at_topological by auto

  1348   } moreover

  1349   { assume "?rhs" hence "?lhs"

  1350       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1351   } ultimately

  1352   show "?thesis" ..

  1353 qed

  1354

  1355 lemma at_within_interior:

  1356   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1357   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1358

  1359 lemma Lim_within_LIMSEQ:

  1360   fixes a :: "'a::metric_space"

  1361   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1362   shows "(X ---> L) (at a within T)"

  1363   using assms unfolding tendsto_def [where l=L]

  1364   by (simp add: sequentially_imp_eventually_within)

  1365

  1366 lemma Lim_right_bound:

  1367   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1368     'b::{linorder_topology, conditionally_complete_linorder}"

  1369   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"

  1370   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1371   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1372 proof cases

  1373   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1374 next

  1375   assume e: "{x<..} \<inter> I \<noteq> {}"

  1376   show ?thesis

  1377   proof (rule order_tendstoI)

  1378     fix a assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1379     { fix y assume "y \<in> {x<..} \<inter> I"

  1380       with e bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1381         by (auto intro: cInf_lower)

  1382       with a have "a < f y" by (blast intro: less_le_trans) }

  1383     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1384       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1385   next

  1386     fix a assume "Inf (f  ({x<..} \<inter> I)) < a"

  1387     from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a" by auto

  1388     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1389       unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

  1390     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1391       unfolding eventually_at_filter by eventually_elim simp

  1392   qed

  1393 qed

  1394

  1395 text{* Another limit point characterization. *}

  1396

  1397 lemma islimpt_sequential:

  1398   fixes x :: "'a::first_countable_topology"

  1399   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1400     (is "?lhs = ?rhs")

  1401 proof

  1402   assume ?lhs

  1403   from countable_basis_at_decseq[of x] guess A . note A = this

  1404   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1405   { fix n

  1406     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1407       unfolding islimpt_def using A(1,2)[of n] by auto

  1408     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1409       unfolding f_def by (rule someI_ex)

  1410     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }

  1411   then have "\<forall>n. f n \<in> S - {x}" by auto

  1412   moreover have "(\<lambda>n. f n) ----> x"

  1413   proof (rule topological_tendstoI)

  1414     fix S assume "open S" "x \<in> S"

  1415     from A(3)[OF this] \<And>n. f n \<in> A n

  1416     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)

  1417   qed

  1418   ultimately show ?rhs by fast

  1419 next

  1420   assume ?rhs

  1421   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto

  1422   show ?lhs

  1423     unfolding islimpt_def

  1424   proof safe

  1425     fix T assume "open T" "x \<in> T"

  1426     from lim[THEN topological_tendstoD, OF this] f

  1427     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1428       unfolding eventually_sequentially by auto

  1429   qed

  1430 qed

  1431

  1432 lemma Lim_inv: (* TODO: delete *)

  1433   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1434   assumes "(f ---> l) A" and "l \<noteq> 0"

  1435   shows "((inverse o f) ---> inverse l) A"

  1436   unfolding o_def using assms by (rule tendsto_inverse)

  1437

  1438 lemma Lim_null:

  1439   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1440   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1441   by (simp add: Lim dist_norm)

  1442

  1443 lemma Lim_null_comparison:

  1444   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1445   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1446   shows "(f ---> 0) net"

  1447 proof (rule metric_tendsto_imp_tendsto)

  1448   show "(g ---> 0) net" by fact

  1449   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1450     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1451 qed

  1452

  1453 lemma Lim_transform_bound:

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

  1455   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1456   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1457   shows "(f ---> 0) net"

  1458   using assms(1) tendsto_norm_zero [OF assms(2)]

  1459   by (rule Lim_null_comparison)

  1460

  1461 text{* Deducing things about the limit from the elements. *}

  1462

  1463 lemma Lim_in_closed_set:

  1464   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1465   shows "l \<in> S"

  1466 proof (rule ccontr)

  1467   assume "l \<notin> S"

  1468   with closed S have "open (- S)" "l \<in> - S"

  1469     by (simp_all add: open_Compl)

  1470   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1471     by (rule topological_tendstoD)

  1472   with assms(2) have "eventually (\<lambda>x. False) net"

  1473     by (rule eventually_elim2) simp

  1474   with assms(3) show "False"

  1475     by (simp add: eventually_False)

  1476 qed

  1477

  1478 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1479

  1480 lemma Lim_dist_ubound:

  1481   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1482   shows "dist a l <= e"

  1483 proof-

  1484   have "dist a l \<in> {..e}"

  1485   proof (rule Lim_in_closed_set)

  1486     show "closed {..e}" by simp

  1487     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1488     show "\<not> trivial_limit net" by fact

  1489     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1490   qed

  1491   thus ?thesis by simp

  1492 qed

  1493

  1494 lemma Lim_norm_ubound:

  1495   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1496   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1497   shows "norm(l) <= e"

  1498 proof-

  1499   have "norm l \<in> {..e}"

  1500   proof (rule Lim_in_closed_set)

  1501     show "closed {..e}" by simp

  1502     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1503     show "\<not> trivial_limit net" by fact

  1504     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1505   qed

  1506   thus ?thesis by simp

  1507 qed

  1508

  1509 lemma Lim_norm_lbound:

  1510   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1511   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1512   shows "e \<le> norm l"

  1513 proof-

  1514   have "norm l \<in> {e..}"

  1515   proof (rule Lim_in_closed_set)

  1516     show "closed {e..}" by simp

  1517     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1518     show "\<not> trivial_limit net" by fact

  1519     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1520   qed

  1521   thus ?thesis by simp

  1522 qed

  1523

  1524 text{* Limit under bilinear function *}

  1525

  1526 lemma Lim_bilinear:

  1527   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1528   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1529 using bounded_bilinear h (f ---> l) net (g ---> m) net

  1530 by (rule bounded_bilinear.tendsto)

  1531

  1532 text{* These are special for limits out of the same vector space. *}

  1533

  1534 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1535   unfolding id_def by (rule tendsto_ident_at)

  1536

  1537 lemma Lim_at_id: "(id ---> a) (at a)"

  1538   unfolding id_def by (rule tendsto_ident_at)

  1539

  1540 lemma Lim_at_zero:

  1541   fixes a :: "'a::real_normed_vector"

  1542   fixes l :: "'b::topological_space"

  1543   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1544   using LIM_offset_zero LIM_offset_zero_cancel ..

  1545

  1546 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1547

  1548 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where

  1549   "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  1550

  1551 lemma netlimit_within:

  1552   "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  1553   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  1554

  1555 lemma netlimit_at:

  1556   fixes a :: "'a::{perfect_space,t2_space}"

  1557   shows "netlimit (at a) = a"

  1558   using netlimit_within [of a UNIV] by simp

  1559

  1560 lemma lim_within_interior:

  1561   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1562   by (metis at_within_interior)

  1563

  1564 lemma netlimit_within_interior:

  1565   fixes x :: "'a::{t2_space,perfect_space}"

  1566   assumes "x \<in> interior S"

  1567   shows "netlimit (at x within S) = x"

  1568 using assms by (metis at_within_interior netlimit_at)

  1569

  1570 text{* Transformation of limit. *}

  1571

  1572 lemma Lim_transform:

  1573   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1574   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1575   shows "(g ---> l) net"

  1576   using tendsto_diff [OF assms(2) assms(1)] by simp

  1577

  1578 lemma Lim_transform_eventually:

  1579   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1580   apply (rule topological_tendstoI)

  1581   apply (drule (2) topological_tendstoD)

  1582   apply (erule (1) eventually_elim2, simp)

  1583   done

  1584

  1585 lemma Lim_transform_within:

  1586   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1587   and "(f ---> l) (at x within S)"

  1588   shows "(g ---> l) (at x within S)"

  1589 proof (rule Lim_transform_eventually)

  1590   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1591     using assms(1,2) by (auto simp: dist_nz eventually_at)

  1592   show "(f ---> l) (at x within S)" by fact

  1593 qed

  1594

  1595 lemma Lim_transform_at:

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

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

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

  1599 proof (rule Lim_transform_eventually)

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

  1601     unfolding eventually_at2

  1602     using assms(1,2) by auto

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

  1604 qed

  1605

  1606 text{* Common case assuming being away from some crucial point like 0. *}

  1607

  1608 lemma Lim_transform_away_within:

  1609   fixes a b :: "'a::t1_space"

  1610   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1611   and "(f ---> l) (at a within S)"

  1612   shows "(g ---> l) (at a within S)"

  1613 proof (rule Lim_transform_eventually)

  1614   show "(f ---> l) (at a within S)" by fact

  1615   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1616     unfolding eventually_at_topological

  1617     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1618 qed

  1619

  1620 lemma Lim_transform_away_at:

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

  1622   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1623   and fl: "(f ---> l) (at a)"

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

  1625   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl

  1626   by simp

  1627

  1628 text{* Alternatively, within an open set. *}

  1629

  1630 lemma Lim_transform_within_open:

  1631   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1632   and "(f ---> l) (at a)"

  1633   shows "(g ---> l) (at a)"

  1634 proof (rule Lim_transform_eventually)

  1635   show "eventually (\<lambda>x. f x = g x) (at a)"

  1636     unfolding eventually_at_topological

  1637     using assms(1,2,3) by auto

  1638   show "(f ---> l) (at a)" by fact

  1639 qed

  1640

  1641 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1642

  1643 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1644

  1645 lemma Lim_cong_within(*[cong add]*):

  1646   assumes "a = b" "x = y" "S = T"

  1647   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1648   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1649   unfolding tendsto_def eventually_at_topological

  1650   using assms by simp

  1651

  1652 lemma Lim_cong_at(*[cong add]*):

  1653   assumes "a = b" "x = y"

  1654   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1655   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1656   unfolding tendsto_def eventually_at_topological

  1657   using assms by simp

  1658

  1659 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1660

  1661 lemma closure_sequential:

  1662   fixes l :: "'a::first_countable_topology"

  1663   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1664 proof

  1665   assume "?lhs" moreover

  1666   { assume "l \<in> S"

  1667     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1668   } moreover

  1669   { assume "l islimpt S"

  1670     hence "?rhs" unfolding islimpt_sequential by auto

  1671   } ultimately

  1672   show "?rhs" unfolding closure_def by auto

  1673 next

  1674   assume "?rhs"

  1675   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

  1676 qed

  1677

  1678 lemma closed_sequential_limits:

  1679   fixes S :: "'a::first_countable_topology set"

  1680   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1681   unfolding closed_limpt

  1682   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1683   by metis

  1684

  1685 lemma closure_approachable:

  1686   fixes S :: "'a::metric_space set"

  1687   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1688   apply (auto simp add: closure_def islimpt_approachable)

  1689   by (metis dist_self)

  1690

  1691 lemma closed_approachable:

  1692   fixes S :: "'a::metric_space set"

  1693   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1694   by (metis closure_closed closure_approachable)

  1695

  1696 lemma closure_contains_Inf:

  1697   fixes S :: "real set"

  1698   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1699   shows "Inf S \<in> closure S"

  1700   unfolding closure_approachable

  1701 proof safe

  1702   have *: "\<forall>x\<in>S. Inf S \<le> x"

  1703     using cInf_lower_EX[of _ S] assms by metis

  1704

  1705   fix e :: real assume "0 < e"

  1706   then have "Inf S < Inf S + e" by simp

  1707   with assms obtain x where "x \<in> S" "x < Inf S + e"

  1708     by (subst (asm) cInf_less_iff[of _ B]) auto

  1709   with * show "\<exists>x\<in>S. dist x (Inf S) < e"

  1710     by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  1711 qed

  1712

  1713 lemma closed_contains_Inf:

  1714   fixes S :: "real set"

  1715   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1716     and "closed S"

  1717   shows "Inf S \<in> S"

  1718   by (metis closure_contains_Inf closure_closed assms)

  1719

  1720

  1721 lemma not_trivial_limit_within_ball:

  1722   "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  1723   (is "?lhs = ?rhs")

  1724 proof -

  1725   { assume "?lhs"

  1726     { fix e :: real

  1727       assume "e>0"

  1728       then obtain y where "y:(S-{x}) & dist y x < e"

  1729         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1730         by auto

  1731       then have "y : (S Int ball x e - {x})"

  1732         unfolding ball_def by (simp add: dist_commute)

  1733       then have "S Int ball x e - {x} ~= {}" by blast

  1734     } then have "?rhs" by auto

  1735   }

  1736   moreover

  1737   { assume "?rhs"

  1738     { fix e :: real

  1739       assume "e>0"

  1740       then obtain y where "y : (S Int ball x e - {x})" using ?rhs by blast

  1741       then have "y:(S-{x}) & dist y x < e"

  1742         unfolding ball_def by (simp add: dist_commute)

  1743       then have "EX y:(S-{x}). dist y x < e" by auto

  1744     }

  1745     then have "?lhs"

  1746       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto

  1747   }

  1748   ultimately show ?thesis by auto

  1749 qed

  1750

  1751 subsection {* Infimum Distance *}

  1752

  1753 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1754

  1755 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1756   by (simp add: infdist_def)

  1757

  1758 lemma infdist_nonneg:

  1759   shows "0 \<le> infdist x A"

  1760   using assms by (auto simp add: infdist_def intro: cInf_greatest)

  1761

  1762 lemma infdist_le:

  1763   assumes "a \<in> A"

  1764   assumes "d = dist x a"

  1765   shows "infdist x A \<le> d"

  1766   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

  1767

  1768 lemma infdist_zero[simp]:

  1769   assumes "a \<in> A" shows "infdist a A = 0"

  1770 proof -

  1771   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  1772   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  1773 qed

  1774

  1775 lemma infdist_triangle:

  1776   shows "infdist x A \<le> infdist y A + dist x y"

  1777 proof cases

  1778   assume "A = {}" thus ?thesis by (simp add: infdist_def)

  1779 next

  1780   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto

  1781   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1782   proof (rule cInf_greatest)

  1783     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp

  1784     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1785     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto

  1786     show "infdist x A \<le> d"

  1787       unfolding infdist_notempty[OF A \<noteq> {}]

  1788     proof (rule cInf_lower2)

  1789       show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto

  1790       show "dist x a \<le> d" unfolding d by (rule dist_triangle)

  1791       fix d assume "d \<in> {dist x a |a. a \<in> A}"

  1792       then obtain a where "a \<in> A" "d = dist x a" by auto

  1793       thus "infdist x A \<le> d" by (rule infdist_le)

  1794     qed

  1795   qed

  1796   also have "\<dots> = dist x y + infdist y A"

  1797   proof (rule cInf_eq, safe)

  1798     fix a assume "a \<in> A"

  1799     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)

  1800   next

  1801     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1802     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  1803       by (intro cInf_greatest) (auto simp: field_simps)

  1804     thus "i \<le> dist x y + infdist y A" by simp

  1805   qed

  1806   finally show ?thesis by simp

  1807 qed

  1808

  1809 lemma in_closure_iff_infdist_zero:

  1810   assumes "A \<noteq> {}"

  1811   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1812 proof

  1813   assume "x \<in> closure A"

  1814   show "infdist x A = 0"

  1815   proof (rule ccontr)

  1816     assume "infdist x A \<noteq> 0"

  1817     with infdist_nonneg[of x A] have "infdist x A > 0" by auto

  1818     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto

  1819       by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  1820         eucl_less_not_refl euclidean_trans(2) infdist_le)

  1821     hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  1822     thus False using x \<in> closure A by simp

  1823   qed

  1824 next

  1825   assume x: "infdist x A = 0"

  1826   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)

  1827   show "x \<in> closure A" unfolding closure_approachable

  1828   proof (safe, rule ccontr)

  1829     fix e::real assume "0 < e"

  1830     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1831     hence "infdist x A \<ge> e" using a \<in> A

  1832       unfolding infdist_def

  1833       by (force simp: dist_commute intro: cInf_greatest)

  1834     with x 0 < e show False by auto

  1835   qed

  1836 qed

  1837

  1838 lemma in_closed_iff_infdist_zero:

  1839   assumes "closed A" "A \<noteq> {}"

  1840   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1841 proof -

  1842   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1843     by (rule in_closure_iff_infdist_zero) fact

  1844   with assms show ?thesis by simp

  1845 qed

  1846

  1847 lemma tendsto_infdist [tendsto_intros]:

  1848   assumes f: "(f ---> l) F"

  1849   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  1850 proof (rule tendstoI)

  1851   fix e ::real assume "0 < e"

  1852   from tendstoD[OF f this]

  1853   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1854   proof (eventually_elim)

  1855     fix x

  1856     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1857     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1858       by (simp add: dist_commute dist_real_def)

  1859     also assume "dist (f x) l < e"

  1860     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1861   qed

  1862 qed

  1863

  1864 text{* Some other lemmas about sequences. *}

  1865

  1866 lemma sequentially_offset:

  1867   assumes "eventually (\<lambda>i. P i) sequentially"

  1868   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1869   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1870

  1871 lemma seq_offset:

  1872   assumes "(f ---> l) sequentially"

  1873   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1874   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1875

  1876 lemma seq_offset_neg:

  1877   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1878   apply (rule topological_tendstoI)

  1879   apply (drule (2) topological_tendstoD)

  1880   apply (simp only: eventually_sequentially)

  1881   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1882   apply metis

  1883   by arith

  1884

  1885 lemma seq_offset_rev:

  1886   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1887   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1888

  1889 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1890   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1891

  1892 subsection {* More properties of closed balls *}

  1893

  1894 lemma closed_cball: "closed (cball x e)"

  1895 unfolding cball_def closed_def

  1896 unfolding Collect_neg_eq [symmetric] not_le

  1897 apply (clarsimp simp add: open_dist, rename_tac y)

  1898 apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1899 apply (rename_tac x')

  1900 apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1901 apply simp

  1902 done

  1903

  1904 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1905 proof-

  1906   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1907     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1908   } moreover

  1909   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1910     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto

  1911   } ultimately

  1912   show ?thesis unfolding open_contains_ball by auto

  1913 qed

  1914

  1915 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1916   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1917

  1918 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1919   apply (simp add: interior_def, safe)

  1920   apply (force simp add: open_contains_cball)

  1921   apply (rule_tac x="ball x e" in exI)

  1922   apply (simp add: subset_trans [OF ball_subset_cball])

  1923   done

  1924

  1925 lemma islimpt_ball:

  1926   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1927   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1928 proof

  1929   assume "?lhs"

  1930   { assume "e \<le> 0"

  1931     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1932     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1933   }

  1934   hence "e > 0" by (metis not_less)

  1935   moreover

  1936   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] ?lhs unfolding closed_limpt by auto

  1937   ultimately show "?rhs" by auto

  1938 next

  1939   assume "?rhs" hence "e>0"  by auto

  1940   { fix d::real assume "d>0"

  1941     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1942     proof(cases "d \<le> dist x y")

  1943       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1944       proof(cases "x=y")

  1945         case True hence False using d \<le> dist x y d>0 by auto

  1946         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1947       next

  1948         case False

  1949

  1950         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1951               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1952           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1953         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1954           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1955           unfolding scaleR_minus_left scaleR_one

  1956           by (auto simp add: norm_minus_commute)

  1957         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1958           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1959           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1960         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1961         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1962

  1963         moreover

  1964

  1965         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  1966           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  1967         moreover

  1968         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  1969           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  1970           unfolding dist_norm by auto

  1971         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

  1972       qed

  1973     next

  1974       case False hence "d > dist x y" by auto

  1975       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1976       proof(cases "x=y")

  1977         case True

  1978         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  1979           using perfect_choose_dist[of "min e d" y]

  1980           using d > 0 e>0 by auto

  1981         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1982           unfolding x = y

  1983           using z \<noteq> y **

  1984           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  1985       next

  1986         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1987           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  1988       qed

  1989     qed  }

  1990   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  1991 qed

  1992

  1993 lemma closure_ball_lemma:

  1994   fixes x y :: "'a::real_normed_vector"

  1995   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  1996 proof (rule islimptI)

  1997   fix T assume "y \<in> T" "open T"

  1998   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  1999     unfolding open_dist by fast

  2000   (* choose point between x and y, within distance r of y. *)

  2001   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2002   def z \<equiv> "y + scaleR k (x - y)"

  2003   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2004     unfolding z_def by (simp add: algebra_simps)

  2005   have "dist z y < r"

  2006     unfolding z_def k_def using 0 < r

  2007     by (simp add: dist_norm min_def)

  2008   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2009   have "dist x z < dist x y"

  2010     unfolding z_def2 dist_norm

  2011     apply (simp add: norm_minus_commute)

  2012     apply (simp only: dist_norm [symmetric])

  2013     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2014     apply (rule mult_strict_right_mono)

  2015     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2016     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2017     done

  2018   hence "z \<in> ball x (dist x y)" by simp

  2019   have "z \<noteq> y"

  2020     unfolding z_def k_def using x \<noteq> y 0 < r

  2021     by (simp add: min_def)

  2022   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2023     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2024     by fast

  2025 qed

  2026

  2027 lemma closure_ball:

  2028   fixes x :: "'a::real_normed_vector"

  2029   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2030 apply (rule equalityI)

  2031 apply (rule closure_minimal)

  2032 apply (rule ball_subset_cball)

  2033 apply (rule closed_cball)

  2034 apply (rule subsetI, rename_tac y)

  2035 apply (simp add: le_less [where 'a=real])

  2036 apply (erule disjE)

  2037 apply (rule subsetD [OF closure_subset], simp)

  2038 apply (simp add: closure_def)

  2039 apply clarify

  2040 apply (rule closure_ball_lemma)

  2041 apply (simp add: zero_less_dist_iff)

  2042 done

  2043

  2044 (* In a trivial vector space, this fails for e = 0. *)

  2045 lemma interior_cball:

  2046   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2047   shows "interior (cball x e) = ball x e"

  2048 proof(cases "e\<ge>0")

  2049   case False note cs = this

  2050   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  2051   { fix y assume "y \<in> cball x e"

  2052     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  2053   hence "cball x e = {}" by auto

  2054   hence "interior (cball x e) = {}" using interior_empty by auto

  2055   ultimately show ?thesis by blast

  2056 next

  2057   case True note cs = this

  2058   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  2059   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2060     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  2061

  2062     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2063       using perfect_choose_dist [of d] by auto

  2064     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  2065     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  2066

  2067     hence "y \<in> ball x e" proof(cases "x = y")

  2068       case True

  2069       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

  2070       thus "y \<in> ball x e" using x = y  by simp

  2071     next

  2072       case False

  2073       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  2074         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2075       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

  2076       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2077       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

  2078         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2079

  2080       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)"

  2081         by (auto simp add: dist_norm algebra_simps)

  2082       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2083         by (auto simp add: algebra_simps)

  2084       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2085         using ** by auto

  2086       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

  2087       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2088       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  2089     qed  }

  2090   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  2091   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  2092 qed

  2093

  2094 lemma frontier_ball:

  2095   fixes a :: "'a::real_normed_vector"

  2096   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  2097   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2098   apply (simp add: set_eq_iff)

  2099   by arith

  2100

  2101 lemma frontier_cball:

  2102   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2103   shows "frontier(cball a e) = {x. dist a x = e}"

  2104   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2105   apply (simp add: set_eq_iff)

  2106   by arith

  2107

  2108 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  2109   apply (simp add: set_eq_iff not_le)

  2110   by (metis zero_le_dist dist_self order_less_le_trans)

  2111 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

  2112

  2113 lemma cball_eq_sing:

  2114   fixes x :: "'a::{metric_space,perfect_space}"

  2115   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2116 proof (rule linorder_cases)

  2117   assume e: "0 < e"

  2118   obtain a where "a \<noteq> x" "dist a x < e"

  2119     using perfect_choose_dist [OF e] by auto

  2120   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  2121   with e show ?thesis by (auto simp add: set_eq_iff)

  2122 qed auto

  2123

  2124 lemma cball_sing:

  2125   fixes x :: "'a::metric_space"

  2126   shows "e = 0 ==> cball x e = {x}"

  2127   by (auto simp add: set_eq_iff)

  2128

  2129

  2130 subsection {* Boundedness *}

  2131

  2132   (* FIXME: This has to be unified with BSEQ!! *)

  2133 definition (in metric_space)

  2134   bounded :: "'a set \<Rightarrow> bool" where

  2135   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2136

  2137 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2138   unfolding bounded_def subset_eq by auto

  2139

  2140 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2141 unfolding bounded_def

  2142 apply safe

  2143 apply (rule_tac x="dist a x + e" in exI, clarify)

  2144 apply (drule (1) bspec)

  2145 apply (erule order_trans [OF dist_triangle add_left_mono])

  2146 apply auto

  2147 done

  2148

  2149 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2150 unfolding bounded_any_center [where a=0]

  2151 by (simp add: dist_norm)

  2152

  2153 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

  2154   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

  2155   using assms by auto

  2156

  2157 lemma bounded_empty [simp]: "bounded {}"

  2158   by (simp add: bounded_def)

  2159

  2160 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2161   by (metis bounded_def subset_eq)

  2162

  2163 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2164   by (metis bounded_subset interior_subset)

  2165

  2166 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

  2167 proof-

  2168   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

  2169   { fix y assume "y \<in> closure S"

  2170     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2171       unfolding closure_sequential by auto

  2172     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2173     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2174       by (rule eventually_mono, simp add: f(1))

  2175     have "dist x y \<le> a"

  2176       apply (rule Lim_dist_ubound [of sequentially f])

  2177       apply (rule trivial_limit_sequentially)

  2178       apply (rule f(2))

  2179       apply fact

  2180       done

  2181   }

  2182   thus ?thesis unfolding bounded_def by auto

  2183 qed

  2184

  2185 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2186   apply (simp add: bounded_def)

  2187   apply (rule_tac x=x in exI)

  2188   apply (rule_tac x=e in exI)

  2189   apply auto

  2190   done

  2191

  2192 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2193   by (metis ball_subset_cball bounded_cball bounded_subset)

  2194

  2195 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2196   apply (auto simp add: bounded_def)

  2197   apply (rename_tac x y r s)

  2198   apply (rule_tac x=x in exI)

  2199   apply (rule_tac x="max r (dist x y + s)" in exI)

  2200   apply (rule ballI, rename_tac z, safe)

  2201   apply (drule (1) bspec, simp)

  2202   apply (drule (1) bspec)

  2203   apply (rule min_max.le_supI2)

  2204   apply (erule order_trans [OF dist_triangle add_left_mono])

  2205   done

  2206

  2207 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2208   by (induct rule: finite_induct[of F], auto)

  2209

  2210 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2211   by (induct set: finite, auto)

  2212

  2213 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2214 proof -

  2215   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp

  2216   hence "bounded {x}" unfolding bounded_def by fast

  2217   thus ?thesis by (metis insert_is_Un bounded_Un)

  2218 qed

  2219

  2220 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2221   by (induct set: finite, simp_all)

  2222

  2223 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2224   apply (simp add: bounded_iff)

  2225   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2226   by metis arith

  2227

  2228 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"

  2229   unfolding Bseq_def bounded_pos by auto

  2230

  2231 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2232   by (metis Int_lower1 Int_lower2 bounded_subset)

  2233

  2234 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2235 apply (metis Diff_subset bounded_subset)

  2236 done

  2237

  2238 lemma not_bounded_UNIV[simp, intro]:

  2239   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2240 proof(auto simp add: bounded_pos not_le)

  2241   obtain x :: 'a where "x \<noteq> 0"

  2242     using perfect_choose_dist [OF zero_less_one] by fast

  2243   fix b::real  assume b: "b >0"

  2244   have b1: "b +1 \<ge> 0" using b by simp

  2245   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2246     by (simp add: norm_sgn)

  2247   then show "\<exists>x::'a. b < norm x" ..

  2248 qed

  2249

  2250 lemma bounded_linear_image:

  2251   assumes "bounded S" "bounded_linear f"

  2252   shows "bounded(f  S)"

  2253 proof-

  2254   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2255   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2256   { fix x assume "x\<in>S"

  2257     hence "norm x \<le> b" using b by auto

  2258     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

  2259       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2260   }

  2261   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

  2262     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

  2263 qed

  2264

  2265 lemma bounded_scaling:

  2266   fixes S :: "'a::real_normed_vector set"

  2267   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2268   apply (rule bounded_linear_image, assumption)

  2269   apply (rule bounded_linear_scaleR_right)

  2270   done

  2271

  2272 lemma bounded_translation:

  2273   fixes S :: "'a::real_normed_vector set"

  2274   assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"

  2275 proof-

  2276   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2277   { fix x assume "x\<in>S"

  2278     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

  2279   }

  2280   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

  2281     by (auto intro!: exI[of _ "b + norm a"])

  2282 qed

  2283

  2284

  2285 text{* Some theorems on sups and infs using the notion "bounded". *}

  2286

  2287 lemma bounded_real:

  2288   fixes S :: "real set"

  2289   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2290   by (simp add: bounded_iff)

  2291

  2292 lemma bounded_has_Sup:

  2293   fixes S :: "real set"

  2294   assumes "bounded S" "S \<noteq> {}"

  2295   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  2296 proof

  2297   fix x assume "x\<in>S"

  2298   thus "x \<le> Sup S"

  2299     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

  2300 next

  2301   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  2302     by (metis cSup_least)

  2303 qed

  2304

  2305 lemma Sup_insert:

  2306   fixes S :: "real set"

  2307   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2308   apply (subst cSup_insert_If)

  2309   apply (rule bounded_has_Sup(1)[of S, rule_format])

  2310   apply (auto simp: sup_max)

  2311   done

  2312

  2313 lemma Sup_insert_finite:

  2314   fixes S :: "real set"

  2315   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2316   apply (rule Sup_insert)

  2317   apply (rule finite_imp_bounded)

  2318   by simp

  2319

  2320 lemma bounded_has_Inf:

  2321   fixes S :: "real set"

  2322   assumes "bounded S"  "S \<noteq> {}"

  2323   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  2324 proof

  2325   fix x assume "x\<in>S"

  2326   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

  2327   thus "x \<ge> Inf S" using x\<in>S

  2328     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

  2329 next

  2330   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  2331     by (metis cInf_greatest)

  2332 qed

  2333

  2334 lemma Inf_insert:

  2335   fixes S :: "real set"

  2336   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2337   apply (subst cInf_insert_if)

  2338   apply (rule bounded_has_Inf(1)[of S, rule_format])

  2339   apply (auto simp: inf_min)

  2340   done

  2341

  2342 lemma Inf_insert_finite:

  2343   fixes S :: "real set"

  2344   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2345   by (rule Inf_insert, rule finite_imp_bounded, simp)

  2346

  2347 subsection {* Compactness *}

  2348

  2349 subsubsection {* Bolzano-Weierstrass property *}

  2350

  2351 lemma heine_borel_imp_bolzano_weierstrass:

  2352   assumes "compact s" "infinite t"  "t \<subseteq> s"

  2353   shows "\<exists>x \<in> s. x islimpt t"

  2354 proof(rule ccontr)

  2355   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2356   then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def

  2357     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

  2358   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2359     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2360   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2361   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2362     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and t\<subseteq>s by auto

  2363     hence "x = y" using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto  }

  2364   hence "inj_on f t" unfolding inj_on_def by simp

  2365   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2366   moreover

  2367   { fix x assume "x\<in>t" "f x \<notin> g"

  2368     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2369     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2370     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2371     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2372   hence "f  t \<subseteq> g" by auto

  2373   ultimately show False using g(2) using finite_subset by auto

  2374 qed

  2375

  2376 lemma acc_point_range_imp_convergent_subsequence:

  2377   fixes l :: "'a :: first_countable_topology"

  2378   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2379   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2380 proof -

  2381   from countable_basis_at_decseq[of l] guess A . note A = this

  2382

  2383   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2384   { fix n i

  2385     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2386       using l A by auto

  2387     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2388       unfolding ex_in_conv by (intro notI) simp

  2389     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2390       by auto

  2391     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2392       by (auto simp: not_le)

  2393     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2394       unfolding s_def by (auto intro: someI2_ex) }

  2395   note s = this

  2396   def r \<equiv> "nat_rec (s 0 0) s"

  2397   have "subseq r"

  2398     by (auto simp: r_def s subseq_Suc_iff)

  2399   moreover

  2400   have "(\<lambda>n. f (r n)) ----> l"

  2401   proof (rule topological_tendstoI)

  2402     fix S assume "open S" "l \<in> S"

  2403     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2404     moreover

  2405     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"

  2406         by (cases i) (simp_all add: r_def s) }

  2407     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2408     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2409       by eventually_elim auto

  2410   qed

  2411   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2412     by (auto simp: convergent_def comp_def)

  2413 qed

  2414

  2415 lemma sequence_infinite_lemma:

  2416   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2417   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2418   shows "infinite (range f)"

  2419 proof

  2420   assume "finite (range f)"

  2421   hence "closed (range f)" by (rule finite_imp_closed)

  2422   hence "open (- range f)" by (rule open_Compl)

  2423   from assms(1) have "l \<in> - range f" by auto

  2424   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2425     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2426   thus False unfolding eventually_sequentially by auto

  2427 qed

  2428

  2429 lemma closure_insert:

  2430   fixes x :: "'a::t1_space"

  2431   shows "closure (insert x s) = insert x (closure s)"

  2432 apply (rule closure_unique)

  2433 apply (rule insert_mono [OF closure_subset])

  2434 apply (rule closed_insert [OF closed_closure])

  2435 apply (simp add: closure_minimal)

  2436 done

  2437

  2438 lemma islimpt_insert:

  2439   fixes x :: "'a::t1_space"

  2440   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2441 proof

  2442   assume *: "x islimpt (insert a s)"

  2443   show "x islimpt s"

  2444   proof (rule islimptI)

  2445     fix t assume t: "x \<in> t" "open t"

  2446     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2447     proof (cases "x = a")

  2448       case True

  2449       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2450         using * t by (rule islimptE)

  2451       with x = a show ?thesis by auto

  2452     next

  2453       case False

  2454       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2455         by (simp_all add: open_Diff)

  2456       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2457         using * t' by (rule islimptE)

  2458       thus ?thesis by auto

  2459     qed

  2460   qed

  2461 next

  2462   assume "x islimpt s" thus "x islimpt (insert a s)"

  2463     by (rule islimpt_subset) auto

  2464 qed

  2465

  2466 lemma islimpt_finite:

  2467   fixes x :: "'a::t1_space"

  2468   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2469 by (induct set: finite, simp_all add: islimpt_insert)

  2470

  2471 lemma islimpt_union_finite:

  2472   fixes x :: "'a::t1_space"

  2473   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2474 by (simp add: islimpt_Un islimpt_finite)

  2475

  2476 lemma islimpt_eq_acc_point:

  2477   fixes l :: "'a :: t1_space"

  2478   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2479 proof (safe intro!: islimptI)

  2480   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2481   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2482     by (auto intro: finite_imp_closed)

  2483   then show False

  2484     by (rule islimptE) auto

  2485 next

  2486   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2487   then have "infinite (T \<inter> S - {l})" by auto

  2488   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2489     unfolding ex_in_conv by (intro notI) simp

  2490   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2491     by auto

  2492 qed

  2493

  2494 lemma islimpt_range_imp_convergent_subsequence:

  2495   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2496   assumes l: "l islimpt (range f)"

  2497   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2498   using l unfolding islimpt_eq_acc_point

  2499   by (rule acc_point_range_imp_convergent_subsequence)

  2500

  2501 lemma sequence_unique_limpt:

  2502   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2503   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  2504   shows "l' = l"

  2505 proof (rule ccontr)

  2506   assume "l' \<noteq> l"

  2507   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2508     using hausdorff [OF l' \<noteq> l] by auto

  2509   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2510     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2511   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2512     unfolding eventually_sequentially by auto

  2513

  2514   have "UNIV = {..<N} \<union> {N..}" by auto

  2515   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  2516   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  2517   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  2518   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2519     using l' \<in> s open s by (rule islimptE)

  2520   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  2521   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  2522   with s \<inter> t = {} show False by simp

  2523 qed

  2524

  2525 lemma bolzano_weierstrass_imp_closed:

  2526   fixes s :: "'a::{first_countable_topology, t2_space} set"

  2527   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2528   shows "closed s"

  2529 proof-

  2530   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2531     hence "l \<in> s"

  2532     proof(cases "\<forall>n. x n \<noteq> l")

  2533       case False thus "l\<in>s" using as(1) by auto

  2534     next

  2535       case True note cas = this

  2536       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  2537       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  2538       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  2539     qed  }

  2540   thus ?thesis unfolding closed_sequential_limits by fast

  2541 qed

  2542

  2543 lemma compact_imp_bounded:

  2544   assumes "compact U" shows "bounded U"

  2545 proof -

  2546   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto

  2547   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2548     by (elim compactE_image)

  2549   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2550     by (simp add: bounded_UN)

  2551   thus "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2552     by (rule bounded_subset)

  2553 qed

  2554

  2555 text{* In particular, some common special cases. *}

  2556

  2557 lemma compact_union [intro]:

  2558   assumes "compact s" "compact t" shows " compact (s \<union> t)"

  2559 proof (rule compactI)

  2560   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  2561   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  2562     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2563   moreover from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  2564     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2565   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  2566     by (auto intro!: exI[of _ "s' \<union> t'"])

  2567 qed

  2568

  2569 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  2570   by (induct set: finite) auto

  2571

  2572 lemma compact_UN [intro]:

  2573   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  2574   unfolding SUP_def by (rule compact_Union) auto

  2575

  2576 lemma closed_inter_compact [intro]:

  2577   assumes "closed s" and "compact t"

  2578   shows "compact (s \<inter> t)"

  2579   using compact_inter_closed [of t s] assms

  2580   by (simp add: Int_commute)

  2581

  2582 lemma compact_inter [intro]:

  2583   fixes s t :: "'a :: t2_space set"

  2584   assumes "compact s" and "compact t"

  2585   shows "compact (s \<inter> t)"

  2586   using assms by (intro compact_inter_closed compact_imp_closed)

  2587

  2588 lemma compact_sing [simp]: "compact {a}"

  2589   unfolding compact_eq_heine_borel by auto

  2590

  2591 lemma compact_insert [simp]:

  2592   assumes "compact s" shows "compact (insert x s)"

  2593 proof -

  2594   have "compact ({x} \<union> s)"

  2595     using compact_sing assms by (rule compact_union)

  2596   thus ?thesis by simp

  2597 qed

  2598

  2599 lemma finite_imp_compact:

  2600   shows "finite s \<Longrightarrow> compact s"

  2601   by (induct set: finite) simp_all

  2602

  2603 lemma open_delete:

  2604   fixes s :: "'a::t1_space set"

  2605   shows "open s \<Longrightarrow> open (s - {x})"

  2606   by (simp add: open_Diff)

  2607

  2608 text{* Finite intersection property *}

  2609

  2610 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  2611   by (auto simp: inj_on_def)

  2612

  2613 lemma compact_fip:

  2614   "compact U \<longleftrightarrow>

  2615     (\<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> {})"

  2616   (is "_ \<longleftrightarrow> ?R")

  2617 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  2618   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  2619     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  2620   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>(uminusA)"

  2621     by auto

  2622   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  2623     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  2624   with fi[THEN spec, of B] show False

  2625     by (auto dest: finite_imageD intro: inj_setminus)

  2626 next

  2627   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2628   from cover have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  2629     by auto

  2630   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>(uminusB) = {}"

  2631     by (metis subset_image_iff)

  2632   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2633     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  2634 qed

  2635

  2636 lemma compact_imp_fip:

  2637   "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>

  2638     s \<inter> (\<Inter> f) \<noteq> {}"

  2639   unfolding compact_fip by auto

  2640

  2641 text{*Compactness expressed with filters*}

  2642

  2643 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2644

  2645 lemma eventually_filter_from_subbase:

  2646   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2647     (is "_ \<longleftrightarrow> ?R P")

  2648   unfolding filter_from_subbase_def

  2649 proof (rule eventually_Abs_filter is_filter.intro)+

  2650   show "?R (\<lambda>x. True)"

  2651     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  2652 next

  2653   fix P Q assume "?R P" then guess X ..

  2654   moreover assume "?R Q" then guess Y ..

  2655   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  2656     by (intro exI[of _ "X \<union> Y"]) auto

  2657 next

  2658   fix P Q

  2659   assume "?R P" then guess X ..

  2660   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  2661   ultimately show "?R Q"

  2662     by (intro exI[of _ X]) auto

  2663 qed

  2664

  2665 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  2666   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  2667

  2668 lemma filter_from_subbase_not_bot:

  2669   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  2670   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  2671

  2672 lemma closure_iff_nhds_not_empty:

  2673   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  2674 proof safe

  2675   assume x: "x \<in> closure X"

  2676   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  2677   then have "x \<notin> closure (-S)"

  2678     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  2679   with x have "x \<in> closure X - closure (-S)"

  2680     by auto

  2681   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  2682     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  2683   finally have "X \<inter> S \<noteq> {}" by auto

  2684   then show False using X \<inter> A = {} S \<subseteq> A by auto

  2685 next

  2686   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  2687   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  2688   show "x \<in> closure X"

  2689     by (simp add: closure_subset open_Compl)

  2690 qed

  2691

  2692 lemma compact_filter:

  2693   "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))"

  2694 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  2695   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  2696   from F have "U \<noteq> {}"

  2697     by (auto simp: eventually_False)

  2698

  2699   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  2700   then have "\<forall>z\<in>Z. closed z"

  2701     by auto

  2702   moreover

  2703   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  2704     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  2705   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  2706   proof (intro allI impI)

  2707     fix B assume "finite B" "B \<subseteq> Z"

  2708     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  2709       by (auto intro!: eventually_Ball_finite)

  2710     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  2711       by eventually_elim auto

  2712     with F show "U \<inter> \<Inter>B \<noteq> {}"

  2713       by (intro notI) (simp add: eventually_False)

  2714   qed

  2715   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  2716     using compact U unfolding compact_fip by blast

  2717   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto

  2718

  2719   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  2720     unfolding eventually_inf eventually_nhds

  2721   proof safe

  2722     fix P Q R S

  2723     assume "eventually R F" "open S" "x \<in> S"

  2724     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  2725     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  2726     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  2727     ultimately show False by (auto simp: set_eq_iff)

  2728   qed

  2729   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  2730     by (metis eventually_bot)

  2731 next

  2732   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 = {}"

  2733

  2734   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  2735   then have inj_P': "\<And>A. inj_on P' A"

  2736     by (auto intro!: inj_onI simp: fun_eq_iff)

  2737   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  2738   have "F \<noteq> bot"

  2739     unfolding F_def

  2740   proof (safe intro!: filter_from_subbase_not_bot)

  2741     fix X assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  2742     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  2743       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  2744     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto

  2745     with B show False by (auto simp: P'_def fun_eq_iff)

  2746   qed

  2747   moreover have "eventually (\<lambda>x. x \<in> U) F"

  2748     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  2749   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)"

  2750   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  2751     by auto

  2752

  2753   { fix V assume "V \<in> A"

  2754     then have V: "eventually (\<lambda>x. x \<in> V) F"

  2755       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  2756     have "x \<in> closure V"

  2757       unfolding closure_iff_nhds_not_empty

  2758     proof (intro impI allI)

  2759       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"

  2760       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)

  2761       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  2762         by (auto simp: eventually_inf)

  2763       with x show "V \<inter> A \<noteq> {}"

  2764         by (auto simp del: Int_iff simp add: trivial_limit_def)

  2765     qed

  2766     then have "x \<in> V"

  2767       using V \<in> A A(1) by simp }

  2768   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  2769   with U \<inter> \<Inter>A = {} show False by auto

  2770 qed

  2771

  2772 definition "countably_compact U \<longleftrightarrow>

  2773     (\<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))"

  2774

  2775 lemma countably_compactE:

  2776   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  2777   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  2778   using assms unfolding countably_compact_def by metis

  2779

  2780 lemma countably_compactI:

  2781   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')"

  2782   shows "countably_compact s"

  2783   using assms unfolding countably_compact_def by metis

  2784

  2785 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  2786   by (auto simp: compact_eq_heine_borel countably_compact_def)

  2787

  2788 lemma countably_compact_imp_compact:

  2789   assumes "countably_compact U"

  2790   assumes ccover: "countable B" "\<forall>b\<in>B. open b"

  2791   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"

  2792   shows "compact U"

  2793   using countably_compact U unfolding compact_eq_heine_borel countably_compact_def

  2794 proof safe

  2795   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2796   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)"

  2797

  2798   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  2799   ultimately have "countable C" "\<forall>a\<in>C. open a"

  2800     unfolding C_def using ccover by auto

  2801   moreover

  2802   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  2803   proof safe

  2804     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"

  2805     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast

  2806     with a \<in> A show "x \<in> \<Union>C" unfolding C_def

  2807       by auto

  2808   qed

  2809   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  2810   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  2811     using * by metis

  2812   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  2813     by (auto simp: C_def)

  2814   then guess f unfolding bchoice_iff Bex_def ..

  2815   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2816     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  2817 qed

  2818

  2819 lemma countably_compact_imp_compact_second_countable:

  2820   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2821 proof (rule countably_compact_imp_compact)

  2822   fix T and x :: 'a assume "open T" "x \<in> T"

  2823   from topological_basisE[OF is_basis this] guess b .

  2824   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto

  2825 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  2826

  2827 lemma countably_compact_eq_compact:

  2828   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2829   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  2830

  2831 subsubsection{* Sequential compactness *}

  2832

  2833 definition

  2834   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where

  2835   "seq_compact S \<longleftrightarrow>

  2836    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  2837        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  2838

  2839 lemma seq_compact_imp_countably_compact:

  2840   fixes U :: "'a :: first_countable_topology set"

  2841   assumes "seq_compact U"

  2842   shows "countably_compact U"

  2843 proof (safe intro!: countably_compactI)

  2844   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  2845   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  2846     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  2847   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2848   proof cases

  2849     assume "finite A" with A show ?thesis by auto

  2850   next

  2851     assume "infinite A"

  2852     then have "A \<noteq> {}" by auto

  2853     show ?thesis

  2854     proof (rule ccontr)

  2855       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  2856       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto

  2857       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis

  2858       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  2859       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  2860         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  2861       then have "range X \<subseteq> U" by auto

  2862       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto

  2863       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  2864       obtain n where "x \<in> from_nat_into A n" by auto

  2865       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  2866       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  2867         unfolding tendsto_def by (auto simp: comp_def)

  2868       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  2869         by (auto simp: eventually_sequentially)

  2870       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  2871         by auto

  2872       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  2873         by (auto intro!: exI[of _ "max n N"])

  2874       ultimately show False

  2875         by auto

  2876     qed

  2877   qed

  2878 qed

  2879

  2880 lemma compact_imp_seq_compact:

  2881   fixes U :: "'a :: first_countable_topology set"

  2882   assumes "compact U" shows "seq_compact U"

  2883   unfolding seq_compact_def

  2884 proof safe

  2885   fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"

  2886   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  2887     by (auto simp: eventually_filtermap)

  2888   moreover have "filtermap X sequentially \<noteq> bot"

  2889     by (simp add: trivial_limit_def eventually_filtermap)

  2890   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  2891     using compact U by (auto simp: compact_filter)

  2892

  2893   from countable_basis_at_decseq[of x] guess A . note A = this

  2894   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  2895   { fix n i

  2896     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  2897     proof (rule ccontr)

  2898       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  2899       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto

  2900       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  2901         by (auto simp: eventually_filtermap eventually_sequentially)

  2902       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  2903         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  2904       ultimately have "eventually (\<lambda>x. False) ?F"

  2905         by (auto simp add: eventually_inf)

  2906       with x show False

  2907         by (simp add: eventually_False)

  2908     qed

  2909     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  2910       unfolding s_def by (auto intro: someI2_ex) }

  2911   note s = this

  2912   def r \<equiv> "nat_rec (s 0 0) s"

  2913   have "subseq r"

  2914     by (auto simp: r_def s subseq_Suc_iff)

  2915   moreover

  2916   have "(\<lambda>n. X (r n)) ----> x"

  2917   proof (rule topological_tendstoI)

  2918     fix S assume "open S" "x \<in> S"

  2919     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2920     moreover

  2921     { fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"

  2922         by (cases i) (simp_all add: r_def s) }

  2923     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2924     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  2925       by eventually_elim auto

  2926   qed

  2927   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  2928     using x \<in> U by (auto simp: convergent_def comp_def)

  2929 qed

  2930

  2931 lemma seq_compactI:

  2932   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  2933   shows "seq_compact S"

  2934   unfolding seq_compact_def using assms by fast

  2935

  2936 lemma seq_compactE:

  2937   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  2938   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  2939   using assms unfolding seq_compact_def by fast

  2940

  2941 lemma countably_compact_imp_acc_point:

  2942   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"

  2943   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  2944 proof (rule ccontr)

  2945   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  2946   note countably_compact s

  2947   moreover have "\<forall>t\<in>C. open t"

  2948     by (auto simp: C_def)

  2949   moreover

  2950   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  2951   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  2952   have "s \<subseteq> \<Union>C"

  2953     using t \<subseteq> s

  2954     unfolding C_def Union_image_eq

  2955     apply (safe dest!: s)

  2956     apply (rule_tac a="U \<inter> t" in UN_I)

  2957     apply (auto intro!: interiorI simp add: finite_subset)

  2958     done

  2959   moreover

  2960   from countable t have "countable C"

  2961     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  2962   ultimately guess D by (rule countably_compactE)

  2963   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and

  2964     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  2965     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  2966   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  2967     using interior_subset by blast

  2968   moreover have "finite (\<Union>E)"

  2969     using E by auto

  2970   ultimately show False using infinite t by (auto simp: finite_subset)

  2971 qed

  2972

  2973 lemma countable_acc_point_imp_seq_compact:

  2974   fixes s :: "'a::first_countable_topology set"

  2975   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))"

  2976   shows "seq_compact s"

  2977 proof -

  2978   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  2979     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2980     proof (cases "finite (range f)")

  2981       case True

  2982       obtain l where "infinite {n. f n = f l}"

  2983         using pigeonhole_infinite[OF _ True] by auto

  2984       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  2985         using infinite_enumerate by blast

  2986       hence "subseq r \<and> (f \<circ> r) ----> f l"

  2987         by (simp add: fr tendsto_const o_def)

  2988       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2989         by auto

  2990     next

  2991       case False

  2992       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto

  2993       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  2994       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  2995         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  2996       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  2997     qed

  2998   }

  2999   thus ?thesis unfolding seq_compact_def by auto

  3000 qed

  3001

  3002 lemma seq_compact_eq_countably_compact:

  3003   fixes U :: "'a :: first_countable_topology set"

  3004   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3005   using

  3006     countable_acc_point_imp_seq_compact

  3007     countably_compact_imp_acc_point

  3008     seq_compact_imp_countably_compact

  3009   by metis

  3010

  3011 lemma seq_compact_eq_acc_point:

  3012   fixes s :: "'a :: first_countable_topology set"

  3013   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)))"

  3014   using

  3015     countable_acc_point_imp_seq_compact[of s]

  3016     countably_compact_imp_acc_point[of s]

  3017     seq_compact_imp_countably_compact[of s]

  3018   by metis

  3019

  3020 lemma seq_compact_eq_compact:

  3021   fixes U :: "'a :: second_countable_topology set"

  3022   shows "seq_compact U \<longleftrightarrow> compact U"

  3023   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3024

  3025 lemma bolzano_weierstrass_imp_seq_compact:

  3026   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3027   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3028   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3029

  3030 subsubsection{* Total boundedness *}

  3031

  3032 lemma cauchy_def:

  3033   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  3034 unfolding Cauchy_def by metis

  3035

  3036 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  3037   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3038 declare helper_1.simps[simp del]

  3039

  3040 lemma seq_compact_imp_totally_bounded:

  3041   assumes "seq_compact s"

  3042   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3043 proof(rule, rule, rule ccontr)

  3044   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3045   def x \<equiv> "helper_1 s e"

  3046   { fix n

  3047     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3048     proof(induct_tac rule:nat_less_induct)

  3049       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3050       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3051       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x  {0 ..< n}" in allE) using as by auto

  3052       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  3053       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  3054         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  3055       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  3056     qed }

  3057   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  3058   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3059   from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto

  3060   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using e>0 by auto

  3061   show False

  3062     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3063     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3064     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  3065 qed

  3066

  3067 subsubsection{* Heine-Borel theorem *}

  3068

  3069 lemma seq_compact_imp_heine_borel:

  3070   fixes s :: "'a :: metric_space set"

  3071   assumes "seq_compact s" shows "compact s"

  3072 proof -

  3073   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3074   guess f unfolding choice_iff' .. note f = this

  3075   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3076   have "countably_compact s"

  3077     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3078   then show "compact s"

  3079   proof (rule countably_compact_imp_compact)

  3080     show "countable K"

  3081       unfolding K_def using f

  3082       by (auto intro: countable_finite countable_subset countable_rat

  3083                intro!: countable_image countable_SIGMA countable_UN)

  3084     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3085   next

  3086     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3087     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto

  3088     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto

  3089     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto

  3090     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3091       unfolding Union_image_eq by auto

  3092     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K" by (auto simp: K_def)

  3093     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3094     proof (rule bexI[rotated], safe)

  3095       fix y assume "y \<in> ball k r"

  3096       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3097         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3098       with ball x e \<subseteq> T show "y \<in> T" by auto

  3099     qed (rule x \<in> ball k r)

  3100   qed

  3101 qed

  3102

  3103 lemma compact_eq_seq_compact_metric:

  3104   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3105   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3106

  3107 lemma compact_def:

  3108   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3109    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"

  3110   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3111

  3112 subsubsection {* Complete the chain of compactness variants *}

  3113

  3114 lemma compact_eq_bolzano_weierstrass:

  3115   fixes s :: "'a::metric_space set"

  3116   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3117 proof

  3118   assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3119 next

  3120   assume ?rhs thus ?lhs

  3121     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3122 qed

  3123

  3124 lemma bolzano_weierstrass_imp_bounded:

  3125   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3126   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3127

  3128 text {*

  3129   A metric space (or topological vector space) is said to have the

  3130   Heine-Borel property if every closed and bounded subset is compact.

  3131 *}

  3132

  3133 class heine_borel = metric_space +

  3134   assumes bounded_imp_convergent_subsequence:

  3135     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3136

  3137 lemma bounded_closed_imp_seq_compact:

  3138   fixes s::"'a::heine_borel set"

  3139   assumes "bounded s" and "closed s" shows "seq_compact s"

  3140 proof (unfold seq_compact_def, clarify)

  3141   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3142   with bounded s have "bounded (range f)" by (auto intro: bounded_subset)

  3143   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3144     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3145   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  3146   have "l \<in> s" using closed s fr l

  3147     unfolding closed_sequential_limits by blast

  3148   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3149     using l \<in> s r l by blast

  3150 qed

  3151

  3152 lemma compact_eq_bounded_closed:

  3153   fixes s :: "'a::heine_borel set"

  3154   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3155 proof

  3156   assume ?lhs thus ?rhs

  3157     using compact_imp_closed compact_imp_bounded by blast

  3158 next

  3159   assume ?rhs thus ?lhs

  3160     using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto

  3161 qed

  3162

  3163 (* TODO: is this lemma necessary? *)

  3164 lemma bounded_increasing_convergent:

  3165   fixes s :: "nat \<Rightarrow> real"

  3166   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3167   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3168   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3169

  3170 instance real :: heine_borel

  3171 proof

  3172   fix f :: "nat \<Rightarrow> real"

  3173   assume f: "bounded (range f)"

  3174   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3175     unfolding comp_def by (metis seq_monosub)

  3176   moreover

  3177   then have "Bseq (f \<circ> r)"

  3178     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3179   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3180     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3181 qed

  3182

  3183 lemma compact_lemma:

  3184   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3185   assumes "bounded (range f)"

  3186   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  3187         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3188 proof safe

  3189   fix d :: "'a set" assume d: "d \<subseteq> Basis"

  3190   with finite_Basis have "finite d" by (blast intro: finite_subset)

  3191   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3192       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3193   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  3194   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto

  3195     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)" using bounded (range f)

  3196       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3197     obtain l1::"'a" and r1 where r1:"subseq r1" and

  3198       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3199       using insert(3) using insert(4) by auto

  3200     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f" by simp

  3201     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3202       by (metis (lifting) bounded_subset f' image_subsetI s')

  3203     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3204       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)

  3205     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  3206       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3207     moreover

  3208     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3209     { fix e::real assume "e>0"

  3210       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast

  3211       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)

  3212       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3213         by (rule eventually_subseq)

  3214       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3215         using N1' N2

  3216         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3217     }

  3218     ultimately show ?case by auto

  3219   qed

  3220 qed

  3221

  3222 instance euclidean_space \<subseteq> heine_borel

  3223 proof

  3224   fix f :: "nat \<Rightarrow> 'a"

  3225   assume f: "bounded (range f)"

  3226   then obtain l::'a and r where r: "subseq r"

  3227     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3228     using compact_lemma [OF f] by blast

  3229   { fix e::real assume "e>0"

  3230     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)

  3231     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"

  3232       by simp

  3233     moreover

  3234     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3235       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3236         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  3237       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3238         apply(rule setsum_strict_mono) using n by auto

  3239       finally have "dist (f (r n)) l < e"

  3240         by auto

  3241     }

  3242     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3243       by (rule eventually_elim1)

  3244   }

  3245   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  3246   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  3247 qed

  3248

  3249 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3250 unfolding bounded_def

  3251 apply clarify

  3252 apply (rule_tac x="a" in exI)

  3253 apply (rule_tac x="e" in exI)

  3254 apply clarsimp

  3255 apply (drule (1) bspec)

  3256 apply (simp add: dist_Pair_Pair)

  3257 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3258 done

  3259

  3260 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3261 unfolding bounded_def

  3262 apply clarify

  3263 apply (rule_tac x="b" in exI)

  3264 apply (rule_tac x="e" in exI)

  3265 apply clarsimp

  3266 apply (drule (1) bspec)

  3267 apply (simp add: dist_Pair_Pair)

  3268 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3269 done

  3270

  3271 instance prod :: (heine_borel, heine_borel) heine_borel

  3272 proof

  3273   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  3274   assume f: "bounded (range f)"

  3275   from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)

  3276   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  3277     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  3278   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  3279     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  3280   obtain l2 r2 where r2: "subseq r2"

  3281     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3282     using bounded_imp_convergent_subsequence [OF s2]

  3283     unfolding o_def by fast

  3284   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3285     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  3286   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3287     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3288   have r: "subseq (r1 \<circ> r2)"

  3289     using r1 r2 unfolding subseq_def by simp

  3290   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3291     using l r by fast

  3292 qed

  3293

  3294 subsubsection{* Completeness *}

  3295

  3296 definition complete :: "'a::metric_space set \<Rightarrow> bool" where

  3297   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  3298

  3299 lemma compact_imp_complete: assumes "compact s" shows "complete s"

  3300 proof-

  3301   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3302     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  3303       using assms unfolding compact_def by blast

  3304

  3305     note lr' = seq_suble [OF lr(2)]

  3306

  3307     { fix e::real assume "e>0"

  3308       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using e>0 apply (erule_tac x="e/2" in allE) by auto

  3309       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using e>0 by auto

  3310       { fix n::nat assume n:"n \<ge> max N M"

  3311         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  3312         moreover have "r n \<ge> N" using lr'[of n] n by auto

  3313         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  3314         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }

  3315       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  3316     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  3317   thus ?thesis unfolding complete_def by auto

  3318 qed

  3319

  3320 lemma nat_approx_posE:

  3321   fixes e::real

  3322   assumes "0 < e"

  3323   obtains n::nat where "1 / (Suc n) < e"

  3324 proof atomize_elim

  3325   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3326     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3327   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3328     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3329   also have "\<dots> = e" by simp

  3330   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3331 qed

  3332

  3333 lemma compact_eq_totally_bounded:

  3334   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3335     (is "_ \<longleftrightarrow> ?rhs")

  3336 proof

  3337   assume assms: "?rhs"

  3338   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)"

  3339     by (auto simp: choice_iff')

  3340

  3341   show "compact s"

  3342   proof cases

  3343     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3344   next

  3345     assume "s \<noteq> {}"

  3346     show ?thesis

  3347       unfolding compact_def

  3348     proof safe

  3349       fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3350

  3351       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  3352       then have [simp]: "\<And>n. 0 < e n" by auto

  3353       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)"

  3354       { fix n U assume "infinite {n. f n \<in> U}"

  3355         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  3356           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  3357         then guess a ..

  3358         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3359           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  3360         from someI_ex[OF this]

  3361         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  3362           unfolding B_def by auto }

  3363       note B = this

  3364

  3365       def F \<equiv> "nat_rec (B 0 UNIV) B"

  3366       { fix n have "infinite {i. f i \<in> F n}"

  3367           by (induct n) (auto simp: F_def B) }

  3368       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  3369         using B by (simp add: F_def)

  3370       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  3371         using decseq_SucI[of F] by (auto simp: decseq_def)

  3372

  3373       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  3374       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  3375         fix k i

  3376         have "infinite ({n. f n \<in> F k} - {.. i})"

  3377           using infinite {n. f n \<in> F k} by auto

  3378         from infinite_imp_nonempty[OF this]

  3379         show "\<exists>x>i. f x \<in> F k"

  3380           by (simp add: set_eq_iff not_le conj_commute)

  3381       qed

  3382

  3383       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  3384       have "subseq t"

  3385         unfolding subseq_Suc_iff by (simp add: t_def sel)

  3386       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  3387         using f by auto

  3388       moreover

  3389       { fix n have "(f \<circ> t) n \<in> F n"

  3390           by (cases n) (simp_all add: t_def sel) }

  3391       note t = this

  3392

  3393       have "Cauchy (f \<circ> t)"

  3394       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  3395         fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  3396         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  3397           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  3398         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  3399           by (auto simp: subset_eq)

  3400         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  3401         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  3402           by (simp add: dist_commute)

  3403       qed

  3404

  3405       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3406         using assms unfolding complete_def by blast

  3407     qed

  3408   qed

  3409 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  3410

  3411 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  3412 proof-

  3413   { assume ?rhs

  3414     { fix e::real

  3415       assume "e>0"

  3416       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  3417         by (erule_tac x="e/2" in allE) auto

  3418       { fix n m

  3419         assume nm:"N \<le> m \<and> N \<le> n"

  3420         hence "dist (s m) (s n) < e" using N

  3421           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  3422           by blast

  3423       }

  3424       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  3425         by blast

  3426     }

  3427     hence ?lhs

  3428       unfolding cauchy_def

  3429       by blast

  3430   }

  3431   thus ?thesis

  3432     unfolding cauchy_def

  3433     using dist_triangle_half_l

  3434     by blast

  3435 qed

  3436

  3437 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  3438 proof-

  3439   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto

  3440   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  3441   moreover

  3442   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  3443   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  3444     unfolding bounded_any_center [where a="s N"] by auto

  3445   ultimately show "?thesis"

  3446     unfolding bounded_any_center [where a="s N"]

  3447     apply(rule_tac x="max a 1" in exI) apply auto

  3448     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  3449 qed

  3450

  3451 instance heine_borel < complete_space

  3452 proof

  3453   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3454   hence "bounded (range f)"

  3455     by (rule cauchy_imp_bounded)

  3456   hence "compact (closure (range f))"

  3457     unfolding compact_eq_bounded_closed by auto

  3458   hence "complete (closure (range f))"

  3459     by (rule compact_imp_complete)

  3460   moreover have "\<forall>n. f n \<in> closure (range f)"

  3461     using closure_subset [of "range f"] by auto

  3462   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  3463     using Cauchy f unfolding complete_def by auto

  3464   then show "convergent f"

  3465     unfolding convergent_def by auto

  3466 qed

  3467

  3468 instance euclidean_space \<subseteq> banach ..

  3469

  3470 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  3471 proof(simp add: complete_def, rule, rule)

  3472   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3473   hence "convergent f" by (rule Cauchy_convergent)

  3474   thus "\<exists>l. f ----> l" unfolding convergent_def .

  3475 qed

  3476

  3477 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  3478 proof -

  3479   { fix x assume "x islimpt s"

  3480     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  3481       unfolding islimpt_sequential by auto

  3482     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  3483       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  3484     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  3485   }

  3486   thus "closed s" unfolding closed_limpt by auto

  3487 qed

  3488

  3489 lemma complete_eq_closed:

  3490   fixes s :: "'a::complete_space set"

  3491   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  3492 proof

  3493   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  3494 next

  3495   assume ?rhs

  3496   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  3497     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  3498     hence "\<exists>l\<in>s. (f ---> l) sequentially" using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }

  3499   thus ?lhs unfolding complete_def by auto

  3500 qed

  3501

  3502 lemma convergent_eq_cauchy:

  3503   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  3504   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  3505   unfolding Cauchy_convergent_iff convergent_def ..

  3506

  3507 lemma convergent_imp_bounded:

  3508   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  3509   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  3510   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  3511

  3512 lemma compact_cball[simp]:

  3513   fixes x :: "'a::heine_borel"

  3514   shows "compact(cball x e)"

  3515   using compact_eq_bounded_closed bounded_cball closed_cball

  3516   by blast

  3517

  3518 lemma compact_frontier_bounded[intro]:

  3519   fixes s :: "'a::heine_borel set"

  3520   shows "bounded s ==> compact(frontier s)"

  3521   unfolding frontier_def

  3522   using compact_eq_bounded_closed

  3523   by blast

  3524

  3525 lemma compact_frontier[intro]:

  3526   fixes s :: "'a::heine_borel set"

  3527   shows "compact s ==> compact (frontier s)"

  3528   using compact_eq_bounded_closed compact_frontier_bounded

  3529   by blast

  3530

  3531 lemma frontier_subset_compact:

  3532   fixes s :: "'a::heine_borel set"

  3533   shows "compact s ==> frontier s \<subseteq> s"

  3534   using frontier_subset_closed compact_eq_bounded_closed

  3535   by blast

  3536

  3537 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3538

  3539 lemma bounded_closed_nest:

  3540   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3541   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3542   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3543 proof-

  3544   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto

  3545   from assms(4,1) have *:"seq_compact (s 0)" using bounded_closed_imp_seq_compact[of "s 0"] by auto

  3546

  3547   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3548     unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3549

  3550   { fix n::nat

  3551     { fix e::real assume "e>0"

  3552       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3553       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3554       moreover

  3555       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto

  3556       hence "(x \<circ> r) (max N n) \<in> s n"

  3557         using x apply(erule_tac x=n in allE)

  3558         using x apply(erule_tac x="r (max N n)" in allE)

  3559         using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto

  3560       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3561     }

  3562     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3563   }

  3564   thus ?thesis by auto

  3565 qed

  3566

  3567 text {* Decreasing case does not even need compactness, just completeness. *}

  3568

  3569 lemma decreasing_closed_nest:

  3570   assumes "\<forall>n. closed(s n)"

  3571           "\<forall>n. (s n \<noteq> {})"

  3572           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3573           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3574   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3575 proof-

  3576   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3577   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3578   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3579   { fix e::real assume "e>0"

  3580     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3581     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3582       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3583       hence "dist (t m) (t n) < e" using N by auto

  3584     }

  3585     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3586   }

  3587   hence  "Cauchy t" unfolding cauchy_def by auto

  3588   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3589   { fix n::nat

  3590     { fix e::real assume "e>0"

  3591       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3592       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

  3593       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3594     }

  3595     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3596   }

  3597   then show ?thesis by auto

  3598 qed

  3599

  3600 text {* Strengthen it to the intersection actually being a singleton. *}

  3601

  3602 lemma decreasing_closed_nest_sing:

  3603   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3604   assumes "\<forall>n. closed(s n)"

  3605           "\<forall>n. s n \<noteq> {}"

  3606           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3607           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3608   shows "\<exists>a. \<Inter>(range s) = {a}"

  3609 proof-

  3610   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3611   { fix b assume b:"b \<in> \<Inter>(range s)"

  3612     { fix e::real assume "e>0"

  3613       hence "dist a b < e" using assms(4 )using b using a by blast

  3614     }

  3615     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3616   }

  3617   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3618   thus ?thesis ..

  3619 qed

  3620

  3621 text{* Cauchy-type criteria for uniform convergence. *}

  3622

  3623 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows

  3624  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3625   (\<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")

  3626 proof(rule)

  3627   assume ?lhs

  3628   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

  3629   { fix e::real assume "e>0"

  3630     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

  3631     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3632       hence "dist (s m x) (s n x) < e"

  3633         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3634         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3635         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3636     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  }

  3637   thus ?rhs by auto

  3638 next

  3639   assume ?rhs

  3640   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

  3641   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3642     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3643   { fix e::real assume "e>0"

  3644     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"

  3645       using ?rhs[THEN spec[where x="e/2"]] by auto

  3646     { fix x assume "P x"

  3647       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3648         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3649       fix n::nat assume "n\<ge>N"

  3650       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]]

  3651         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)  }

  3652     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3653   thus ?lhs by auto

  3654 qed

  3655

  3656 lemma uniformly_cauchy_imp_uniformly_convergent:

  3657   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  3658   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"

  3659           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3660   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3661 proof-

  3662   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"

  3663     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3664   moreover

  3665   { fix x assume "P x"

  3666     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3667       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3668   ultimately show ?thesis by auto

  3669 qed

  3670

  3671

  3672 subsection {* Continuity *}

  3673

  3674 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3675

  3676 lemma continuous_within_eps_delta:

  3677   "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)"

  3678   unfolding continuous_within and Lim_within

  3679   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3680

  3681 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3682                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3683   using continuous_within_eps_delta [of x UNIV f] by simp

  3684

  3685 text{* Versions in terms of open balls. *}

  3686

  3687 lemma continuous_within_ball:

  3688  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3689                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3690 proof

  3691   assume ?lhs

  3692   { fix e::real assume "e>0"

  3693     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"

  3694       using ?lhs[unfolded continuous_within Lim_within] by auto

  3695     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3696       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3697         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3698     }

  3699     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)  }

  3700   thus ?rhs by auto

  3701 next

  3702   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3703     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3704 qed

  3705

  3706 lemma continuous_at_ball:

  3707   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3708 proof

  3709   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3710     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)

  3711     unfolding dist_nz[THEN sym] by auto

  3712 next

  3713   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3714     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)

  3715 qed

  3716

  3717 text{* Define setwise continuity in terms of limits within the set. *}

  3718

  3719 lemma continuous_on_iff:

  3720   "continuous_on s f \<longleftrightarrow>

  3721     (\<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)"

  3722 unfolding continuous_on_def Lim_within

  3723 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3724 apply (rename_tac y, case_tac "y = x", simp)

  3725 apply (simp add: dist_nz)

  3726 done

  3727

  3728 definition

  3729   uniformly_continuous_on ::

  3730     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3731 where

  3732   "uniformly_continuous_on s f \<longleftrightarrow>

  3733     (\<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)"

  3734

  3735 text{* Some simple consequential lemmas. *}

  3736

  3737 lemma uniformly_continuous_imp_continuous:

  3738  " uniformly_continuous_on s f ==> continuous_on s f"

  3739   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3740

  3741 lemma continuous_at_imp_continuous_within:

  3742  "continuous (at x) f ==> continuous (at x within s) f"

  3743   unfolding continuous_within continuous_at using Lim_at_within by auto

  3744

  3745 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3746   by simp

  3747

  3748 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  3749

  3750 lemma continuous_within_subset:

  3751  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  3752              ==> continuous (at x within t) f"

  3753   unfolding continuous_within by(metis tendsto_within_subset)

  3754

  3755 lemma continuous_on_interior:

  3756   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  3757   by (erule interiorE, drule (1) continuous_on_subset,

  3758     simp add: continuous_on_eq_continuous_at)

  3759

  3760 lemma continuous_on_eq:

  3761   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  3762   unfolding continuous_on_def tendsto_def eventually_at_topological

  3763   by simp

  3764

  3765 text {* Characterization of various kinds of continuity in terms of sequences. *}

  3766

  3767 lemma continuous_within_sequentially:

  3768   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3769   shows "continuous (at a within s) f \<longleftrightarrow>

  3770                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  3771                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  3772 proof

  3773   assume ?lhs

  3774   { 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"

  3775     fix T::"'b set" assume "open T" and "f a \<in> T"

  3776     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"

  3777       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  3778     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  3779       using x(2) d>0 by simp

  3780     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  3781     proof eventually_elim

  3782       case (elim n) thus ?case

  3783         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  3784     qed

  3785   }

  3786   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  3787 next

  3788   assume ?rhs thus ?lhs

  3789     unfolding continuous_within tendsto_def [where l="f a"]

  3790     by (simp add: sequentially_imp_eventually_within)

  3791 qed

  3792

  3793 lemma continuous_at_sequentially:

  3794   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3795   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  3796                   --> ((f o x) ---> f a) sequentially)"

  3797   using continuous_within_sequentially[of a UNIV f] by simp

  3798

  3799 lemma continuous_on_sequentially:

  3800   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3801   shows "continuous_on s f \<longleftrightarrow>

  3802     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  3803                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  3804 proof

  3805   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  3806 next

  3807   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  3808 qed

  3809

  3810 lemma uniformly_continuous_on_sequentially:

  3811   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  3812                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  3813                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  3814 proof

  3815   assume ?lhs

  3816   { 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"

  3817     { fix e::real assume "e>0"

  3818       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"

  3819         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  3820       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

  3821       { fix n assume "n\<ge>N"

  3822         hence "dist (f (x n)) (f (y n)) < e"

  3823           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

  3824           unfolding dist_commute by simp  }

  3825       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  3826     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  3827   thus ?rhs by auto

  3828 next

  3829   assume ?rhs

  3830   { assume "\<not> ?lhs"

  3831     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

  3832     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"

  3833       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

  3834       by (auto simp add: dist_commute)

  3835     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  3836     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  3837     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"

  3838       unfolding x_def and y_def using fa by auto

  3839     { fix e::real assume "e>0"

  3840       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

  3841       { fix n::nat assume "n\<ge>N"

  3842         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  3843         also have "\<dots> < e" using N by auto

  3844         finally have "inverse (real n + 1) < e" by auto

  3845         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  3846       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  3847     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

  3848     hence False using fxy and e>0 by auto  }

  3849   thus ?lhs unfolding uniformly_continuous_on_def by blast

  3850 qed

  3851

  3852 text{* The usual transformation theorems. *}

  3853

  3854 lemma continuous_transform_within:

  3855   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3856   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  3857           "continuous (at x within s) f"

  3858   shows "continuous (at x within s) g"

  3859 unfolding continuous_within

  3860 proof (rule Lim_transform_within)

  3861   show "0 < d" by fact

  3862   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  3863     using assms(3) by auto

  3864   have "f x = g x"

  3865     using assms(1,2,3) by auto

  3866   thus "(f ---> g x) (at x within s)"

  3867     using assms(4) unfolding continuous_within by simp

  3868 qed

  3869

  3870 lemma continuous_transform_at:

  3871   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  3872   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  3873           "continuous (at x) f"

  3874   shows "continuous (at x) g"

  3875   using continuous_transform_within [of d x UNIV f g] assms by simp

  3876

  3877 subsubsection {* Structural rules for pointwise continuity *}

  3878

  3879 lemmas continuous_within_id = continuous_ident

  3880

  3881 lemmas continuous_at_id = isCont_ident

  3882

  3883 lemma continuous_infdist[continuous_intros]:

  3884   assumes "continuous F f"

  3885   shows "continuous F (\<lambda>x. infdist (f x) A)"

  3886   using assms unfolding continuous_def by (rule tendsto_infdist)

  3887

  3888 lemma continuous_infnorm[continuous_intros]:

  3889   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  3890   unfolding continuous_def by (rule tendsto_infnorm)

  3891

  3892 lemma continuous_inner[continuous_intros]:

  3893   assumes "continuous F f" and "continuous F g"

  3894   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  3895   using assms unfolding continuous_def by (rule tendsto_inner)

  3896

  3897 lemmas continuous_at_inverse = isCont_inverse

  3898

  3899 subsubsection {* Structural rules for setwise continuity *}

  3900

  3901 lemma continuous_on_infnorm[continuous_on_intros]:

  3902   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  3903   unfolding continuous_on by (fast intro: tendsto_infnorm)

  3904

  3905 lemma continuous_on_inner[continuous_on_intros]:

  3906   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  3907   assumes "continuous_on s f" and "continuous_on s g"

  3908   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  3909   using bounded_bilinear_inner assms

  3910   by (rule bounded_bilinear.continuous_on)

  3911

  3912 subsubsection {* Structural rules for uniform continuity *}

  3913

  3914 lemma uniformly_continuous_on_id[continuous_on_intros]:

  3915   shows "uniformly_continuous_on s (\<lambda>x. x)"

  3916   unfolding uniformly_continuous_on_def by auto

  3917

  3918 lemma uniformly_continuous_on_const[continuous_on_intros]:

  3919   shows "uniformly_continuous_on s (\<lambda>x. c)"

  3920   unfolding uniformly_continuous_on_def by simp

  3921

  3922 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  3923   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  3924   assumes "uniformly_continuous_on s f"

  3925   assumes "uniformly_continuous_on s g"

  3926   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  3927 proof -

  3928   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  3929       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  3930       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  3931       by arith

  3932   } note le = this

  3933   { fix x y

  3934     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  3935     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  3936     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  3937       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  3938         simp add: le)

  3939   }

  3940   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  3941     unfolding dist_real_def by simp

  3942 qed

  3943

  3944 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  3945   assumes "uniformly_continuous_on s f"

  3946   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  3947   unfolding norm_conv_dist using assms

  3948   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  3949

  3950 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  3951   assumes "uniformly_continuous_on s g"

  3952   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  3953   using assms unfolding uniformly_continuous_on_sequentially

  3954   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  3955   by (auto intro: tendsto_zero)

  3956

  3957 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  3958   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3959   assumes "uniformly_continuous_on s f"

  3960   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  3961   using bounded_linear_scaleR_right assms

  3962   by (rule bounded_linear.uniformly_continuous_on)

  3963

  3964 lemma dist_minus:

  3965   fixes x y :: "'a::real_normed_vector"

  3966   shows "dist (- x) (- y) = dist x y"

  3967   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  3968

  3969 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  3970   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3971   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  3972   unfolding uniformly_continuous_on_def dist_minus .

  3973

  3974 lemma uniformly_continuous_on_add[continuous_on_intros]:

  3975   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3976   assumes "uniformly_continuous_on s f"

  3977   assumes "uniformly_continuous_on s g"

  3978   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  3979   using assms unfolding uniformly_continuous_on_sequentially

  3980   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  3981   by (auto intro: tendsto_add_zero)

  3982

  3983 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  3984   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  3985   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  3986   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  3987   unfolding ab_diff_minus using assms

  3988   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  3989

  3990 text{* Continuity of all kinds is preserved under composition. *}

  3991

  3992 lemmas continuous_at_compose = isCont_o

  3993

  3994 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  3995   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  3996   shows "uniformly_continuous_on s (g o f)"

  3997 proof-

  3998   { fix e::real assume "e>0"

  3999     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

  4000     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

  4001     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  }

  4002   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4003 qed

  4004

  4005 text{* Continuity in terms of open preimages. *}

  4006

  4007 lemma continuous_at_open:

  4008   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)))"

  4009 unfolding continuous_within_topological [of x UNIV f]

  4010 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4011

  4012 lemma continuous_imp_tendsto:

  4013   assumes "continuous (at x0) f" and "x ----> x0"

  4014   shows "(f \<circ> x) ----> (f x0)"

  4015 proof (rule topological_tendstoI)

  4016   fix S

  4017   assume "open S" "f x0 \<in> S"

  4018   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  4019      using assms continuous_at_open by metis

  4020   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  4021     using assms T_def by (auto simp: tendsto_def)

  4022   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  4023     using T_def by (auto elim!: eventually_elim1)

  4024 qed

  4025

  4026 lemma continuous_on_open:

  4027   "continuous_on s f \<longleftrightarrow>

  4028         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4029             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4030   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  4031   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4032

  4033 text {* Similarly in terms of closed sets. *}

  4034

  4035 lemma continuous_on_closed:

  4036   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")

  4037   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  4038   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4039

  4040 text {* Half-global and completely global cases. *}

  4041

  4042 lemma continuous_open_in_preimage:

  4043   assumes "continuous_on s f"  "open t"

  4044   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4045 proof-

  4046   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

  4047   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4048     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4049   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4050 qed

  4051

  4052 lemma continuous_closed_in_preimage:

  4053   assumes "continuous_on s f"  "closed t"

  4054   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4055 proof-

  4056   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

  4057   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4058     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4059   thus ?thesis

  4060     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4061 qed

  4062

  4063 lemma continuous_open_preimage:

  4064   assumes "continuous_on s f" "open s" "open t"

  4065   shows "open {x \<in> s. f x \<in> t}"

  4066 proof-

  4067   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4068     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4069   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4070 qed

  4071

  4072 lemma continuous_closed_preimage:

  4073   assumes "continuous_on s f" "closed s" "closed t"

  4074   shows "closed {x \<in> s. f x \<in> t}"

  4075 proof-

  4076   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4077     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4078   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4079 qed

  4080

  4081 lemma continuous_open_preimage_univ:

  4082   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4083   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4084

  4085 lemma continuous_closed_preimage_univ:

  4086   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4087   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4088

  4089 lemma continuous_open_vimage:

  4090   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4091   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4092

  4093 lemma continuous_closed_vimage:

  4094   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4095   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4096

  4097 lemma interior_image_subset:

  4098   assumes "\<forall>x. continuous (at x) f" "inj f"

  4099   shows "interior (f  s) \<subseteq> f  (interior s)"

  4100 proof

  4101   fix x assume "x \<in> interior (f  s)"

  4102   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4103   hence "x \<in> f  s" by auto

  4104   then obtain y where y: "y \<in> s" "x = f y" by auto

  4105   have "open (vimage f T)"

  4106     using assms(1) open T by (rule continuous_open_vimage)

  4107   moreover have "y \<in> vimage f T"

  4108     using x = f y x \<in> T by simp

  4109   moreover have "vimage f T \<subseteq> s"

  4110     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4111   ultimately have "y \<in> interior s" ..

  4112   with x = f y show "x \<in> f  interior s" ..

  4113 qed

  4114

  4115 text {* Equality of continuous functions on closure and related results. *}

  4116

  4117 lemma continuous_closed_in_preimage_constant:

  4118   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4119   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4120   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4121

  4122 lemma continuous_closed_preimage_constant:

  4123   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4124   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4125   using continuous_closed_preimage[of s f "{a}"] by auto

  4126

  4127 lemma continuous_constant_on_closure:

  4128   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4129   assumes "continuous_on (closure s) f"

  4130           "\<forall>x \<in> s. f x = a"

  4131   shows "\<forall>x \<in> (closure s). f x = a"

  4132     using continuous_closed_preimage_constant[of "closure s" f a]

  4133     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4134

  4135 lemma image_closure_subset:

  4136   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4137   shows "f  (closure s) \<subseteq> t"

  4138 proof-

  4139   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4140   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4141     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4142   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4143     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4144   thus ?thesis by auto

  4145 qed

  4146

  4147 lemma continuous_on_closure_norm_le:

  4148   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4149   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4150   shows "norm(f x) \<le> b"

  4151 proof-

  4152   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4153   show ?thesis

  4154     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4155     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4156 qed

  4157

  4158 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4159

  4160 lemma continuous_within_avoid:

  4161   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4162   assumes "continuous (at x within s) f" and "f x \<noteq> a"

  4163   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4164 proof-

  4165   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  4166     using t1_space [OF f x \<noteq> a] by fast

  4167   have "(f ---> f x) (at x within s)"

  4168     using assms(1) by (simp add: continuous_within)

  4169   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  4170     using open U and f x \<in> U

  4171     unfolding tendsto_def by fast

  4172   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  4173     using a \<notin> U by (fast elim: eventually_mono [rotated])

  4174   thus ?thesis

  4175     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  4176 qed

  4177

  4178 lemma continuous_at_avoid:

  4179   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4180   assumes "continuous (at x) f" and "f x \<noteq> a"

  4181   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4182   using assms continuous_within_avoid[of x UNIV f a] by simp

  4183

  4184 lemma continuous_on_avoid:

  4185   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4186   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4187   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4188 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

  4189

  4190 lemma continuous_on_open_avoid:

  4191   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4192   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4193   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4194 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

  4195

  4196 text {* Proving a function is constant by proving open-ness of level set. *}

  4197

  4198 lemma continuous_levelset_open_in_cases:

  4199   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4200   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4201         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4202         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4203 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4204

  4205 lemma continuous_levelset_open_in:

  4206   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4207   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4208         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4209         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4210 using continuous_levelset_open_in_cases[of s f ]

  4211 by meson

  4212

  4213 lemma continuous_levelset_open:

  4214   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4215   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4216   shows "\<forall>x \<in> s. f x = a"

  4217 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4218

  4219 text {* Some arithmetical combinations (more to prove). *}

  4220

  4221 lemma open_scaling[intro]:

  4222   fixes s :: "'a::real_normed_vector set"

  4223   assumes "c \<noteq> 0"  "open s"

  4224   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4225 proof-

  4226   { fix x assume "x \<in> s"

  4227     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

  4228     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4229     moreover

  4230     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4231       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4232         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4233           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4234       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  }

  4235     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  }

  4236   thus ?thesis unfolding open_dist by auto

  4237 qed

  4238

  4239 lemma minus_image_eq_vimage:

  4240   fixes A :: "'a::ab_group_add set"

  4241   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4242   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4243

  4244 lemma open_negations:

  4245   fixes s :: "'a::real_normed_vector set"

  4246   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4247   unfolding scaleR_minus1_left [symmetric]

  4248   by (rule open_scaling, auto)

  4249

  4250 lemma open_translation:

  4251   fixes s :: "'a::real_normed_vector set"

  4252   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4253 proof-

  4254   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4255       by (intro continuous_diff continuous_at_id continuous_const) }

  4256   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4257   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4258 qed

  4259

  4260 lemma open_affinity:

  4261   fixes s :: "'a::real_normed_vector set"

  4262   assumes "open s"  "c \<noteq> 0"

  4263   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4264 proof-

  4265   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4266   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4267   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4268 qed

  4269

  4270 lemma interior_translation:

  4271   fixes s :: "'a::real_normed_vector set"

  4272   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4273 proof (rule set_eqI, rule)

  4274   fix x assume "x \<in> interior (op + a  s)"

  4275   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4276   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

  4277   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

  4278 next

  4279   fix x assume "x \<in> op + a  interior s"

  4280   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

  4281   { fix z have *:"a + y - z = y + a - z" by auto

  4282     assume "z\<in>ball x e"

  4283     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

  4284     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4285   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4286   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4287 qed

  4288

  4289 text {* Topological properties of linear functions. *}

  4290

  4291 lemma linear_lim_0:

  4292   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4293 proof-

  4294   interpret f: bounded_linear f by fact

  4295   have "(f ---> f 0) (at 0)"

  4296     using tendsto_ident_at by (rule f.tendsto)

  4297   thus ?thesis unfolding f.zero .

  4298 qed

  4299

  4300 lemma linear_continuous_at:

  4301   assumes "bounded_linear f"  shows "continuous (at a) f"

  4302   unfolding continuous_at using assms

  4303   apply (rule bounded_linear.tendsto)

  4304   apply (rule tendsto_ident_at)

  4305   done

  4306

  4307 lemma linear_continuous_within:

  4308   shows "bounded_linear f ==> continuous (at x within s) f"

  4309   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4310

  4311 lemma linear_continuous_on:

  4312   shows "bounded_linear f ==> continuous_on s f"

  4313   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4314

  4315 text {* Also bilinear functions, in composition form. *}

  4316

  4317 lemma bilinear_continuous_at_compose:

  4318   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4319         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4320   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4321

  4322 lemma bilinear_continuous_within_compose:

  4323   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4324         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4325   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4326

  4327 lemma bilinear_continuous_on_compose:

  4328   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4329              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4330   unfolding continuous_on_def

  4331   by (fast elim: bounded_bilinear.tendsto)

  4332

  4333 text {* Preservation of compactness and connectedness under continuous function. *}

  4334

  4335 lemma compact_eq_openin_cover:

  4336   "compact S \<longleftrightarrow>

  4337     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4338       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  4339 proof safe

  4340   fix C

  4341   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  4342   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}"

  4343     unfolding openin_open by force+

  4344   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"

  4345     by (rule compactE)

  4346   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)"

  4347     by auto

  4348   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4349 next

  4350   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4351         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  4352   show "compact S"

  4353   proof (rule compactI)

  4354     fix C

  4355     let ?C = "image (\<lambda>T. S \<inter> T) C"

  4356     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  4357     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  4358       unfolding openin_open by auto

  4359     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  4360       by metis

  4361     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  4362     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  4363     proof (intro conjI)

  4364       from D \<subseteq> ?C show "?D \<subseteq> C"

  4365         by (fast intro: inv_into_into)

  4366       from finite D show "finite ?D"

  4367         by (rule finite_imageI)

  4368       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  4369         apply (rule subset_trans)

  4370         apply clarsimp

  4371         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  4372         apply (erule rev_bexI, fast)

  4373         done

  4374     qed

  4375     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4376   qed

  4377 qed

  4378

  4379 lemma connected_continuous_image:

  4380   assumes "continuous_on s f"  "connected s"

  4381   shows "connected(f  s)"

  4382 proof-

  4383   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4384     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4385       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4386       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4387       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4388     hence False using as(1,2)

  4389       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4390   thus ?thesis unfolding connected_clopen by auto

  4391 qed

  4392

  4393 text {* Continuity implies uniform continuity on a compact domain. *}

  4394

  4395 lemma compact_uniformly_continuous:

  4396   assumes f: "continuous_on s f" and s: "compact s"

  4397   shows "uniformly_continuous_on s f"

  4398   unfolding uniformly_continuous_on_def

  4399 proof (cases, safe)

  4400   fix e :: real assume "0 < e" "s \<noteq> {}"

  4401   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) } }"

  4402   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  4403   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  4404   proof safe

  4405     fix y assume "y \<in> s"

  4406     from continuous_open_in_preimage[OF f open_ball]

  4407     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  4408       unfolding openin_subtopology open_openin by metis

  4409     then obtain d where "ball y d \<subseteq> T" "0 < d"

  4410       using 0 < e y \<in> s by (auto elim!: openE)

  4411     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  4412       by (intro UN_I[of "(y, d)"]) auto

  4413   qed auto

  4414   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  4415     by (rule compactE_image)

  4416   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  4417     by (subst Min_gr_iff) auto

  4418   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  4419   proof (rule, safe)

  4420     fix x x' assume in_s: "x' \<in> s" "x \<in> s"

  4421     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  4422       by blast

  4423     moreover assume "dist x x' < Min (sndD) / 2"

  4424     ultimately have "dist y x' < d"

  4425       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  4426     with D x in_s show  "dist (f x) (f x') < e"

  4427       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  4428   qed (insert D, auto)

  4429 qed auto

  4430

  4431 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4432

  4433 lemma continuous_uniform_limit:

  4434   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4435   assumes "\<not> trivial_limit F"

  4436   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4437   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4438   shows "continuous_on s g"

  4439 proof-

  4440   { fix x and e::real assume "x\<in>s" "e>0"

  4441     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4442       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4443     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4444     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4445       using assms(1) by blast

  4446     have "e / 3 > 0" using e>0 by auto

  4447     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"

  4448       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4449     { fix y assume "y \<in> s" and "dist y x < d"

  4450       hence "dist (f n y) (f n x) < e / 3"

  4451         by (rule d [rule_format])

  4452       hence "dist (f n y) (g x) < 2 * e / 3"

  4453         using dist_triangle [of "f n y" "g x" "f n x"]

  4454         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4455         by auto

  4456       hence "dist (g y) (g x) < e"

  4457         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4458         using dist_triangle3 [of "g y" "g x" "f n y"]

  4459         by auto }

  4460     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4461       using d>0 by auto }

  4462   thus ?thesis unfolding continuous_on_iff by auto

  4463 qed

  4464

  4465

  4466 subsection {* Topological stuff lifted from and dropped to R *}

  4467

  4468 lemma open_real:

  4469   fixes s :: "real set" shows

  4470  "open s \<longleftrightarrow>

  4471         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4472   unfolding open_dist dist_norm by simp

  4473

  4474 lemma islimpt_approachable_real:

  4475   fixes s :: "real set"

  4476   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4477   unfolding islimpt_approachable dist_norm by simp

  4478

  4479 lemma closed_real:

  4480   fixes s :: "real set"

  4481   shows "closed s \<longleftrightarrow>

  4482         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4483             --> x \<in> s)"

  4484   unfolding closed_limpt islimpt_approachable dist_norm by simp

  4485

  4486 lemma continuous_at_real_range:

  4487   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4488   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  4489         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  4490   unfolding continuous_at unfolding Lim_at

  4491   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  4492   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

  4493   apply(erule_tac x=e in allE) by auto

  4494

  4495 lemma continuous_on_real_range:

  4496   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  4497   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))"

  4498   unfolding continuous_on_iff dist_norm by simp

  4499

  4500 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  4501

  4502 lemma distance_attains_sup:

  4503   assumes "compact s" "s \<noteq> {}"

  4504   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  4505 proof (rule continuous_attains_sup [OF assms])

  4506   { fix x assume "x\<in>s"

  4507     have "(dist a ---> dist a x) (at x within s)"

  4508       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  4509   }

  4510   thus "continuous_on s (dist a)"

  4511     unfolding continuous_on ..

  4512 qed

  4513

  4514 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  4515

  4516 lemma distance_attains_inf:

  4517   fixes a :: "'a::heine_borel"

  4518   assumes "closed s"  "s \<noteq> {}"

  4519   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  4520 proof-

  4521   from assms(2) obtain b where "b \<in> s" by auto

  4522   let ?B = "s \<inter> cball a (dist b a)"

  4523   have "?B \<noteq> {}" using b \<in> s by (auto simp add: dist_commute)

  4524   moreover have "continuous_on ?B (dist a)"

  4525     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  4526   moreover have "compact ?B"

  4527     by (intro closed_inter_compact closed s compact_cball)

  4528   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  4529     by (metis continuous_attains_inf)

  4530   thus ?thesis by fastforce

  4531 qed

  4532

  4533

  4534 subsection {* Pasted sets *}

  4535

  4536 lemma bounded_Times:

  4537   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  4538 proof-

  4539   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  4540     using assms [unfolded bounded_def] by auto

  4541   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  4542     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  4543   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  4544 qed

  4545

  4546 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  4547 by (induct x) simp

  4548

  4549 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  4550 unfolding seq_compact_def

  4551 apply clarify

  4552 apply (drule_tac x="fst \<circ> f" in spec)

  4553 apply (drule mp, simp add: mem_Times_iff)

  4554 apply (clarify, rename_tac l1 r1)

  4555 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  4556 apply (drule mp, simp add: mem_Times_iff)

  4557 apply (clarify, rename_tac l2 r2)

  4558 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  4559 apply (rule_tac x="r1 \<circ> r2" in exI)

  4560 apply (rule conjI, simp add: subseq_def)

  4561 apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  4562 apply (drule (1) tendsto_Pair) back

  4563 apply (simp add: o_def)

  4564 done

  4565

  4566 lemma compact_Times:

  4567   assumes "compact s" "compact t"

  4568   shows "compact (s \<times> t)"

  4569 proof (rule compactI)

  4570   fix C assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  4571   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)"

  4572   proof

  4573     fix x assume "x \<in> s"

  4574     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")

  4575     proof

  4576       fix y assume "y \<in> t"

  4577       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  4578       then show "?P y" by (auto elim!: open_prod_elim)

  4579     qed

  4580     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  4581       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"

  4582       by metis

  4583     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  4584     from compactE_image[OF compact t this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  4585       by auto

  4586     moreover with c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  4587       by (fastforce simp: subset_eq)

  4588     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  4589       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  4590   qed

  4591   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  4592     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"

  4593     unfolding subset_eq UN_iff by metis

  4594   moreover from compactE_image[OF compact s a] obtain e where e: "e \<subseteq> s" "finite e"

  4595     and s: "s \<subseteq> (\<Union>x\<in>e. a x)" by auto

  4596   moreover

  4597   { from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)" by auto

  4598     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)" using d e \<subseteq> s by (intro UN_mono) auto

  4599     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" . }

  4600   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  4601     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  4602 qed

  4603

  4604 text{* Hence some useful properties follow quite easily. *}

  4605

  4606 lemma compact_scaling:

  4607   fixes s :: "'a::real_normed_vector set"

  4608   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  4609 proof-

  4610   let ?f = "\<lambda>x. scaleR c x"

  4611   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  4612   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  4613     using linear_continuous_at[OF *] assms by auto

  4614 qed

  4615

  4616 lemma compact_negations:

  4617   fixes s :: "'a::real_normed_vector set"

  4618   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  4619   using compact_scaling [OF assms, of "- 1"] by auto

  4620

  4621 lemma compact_sums:

  4622   fixes s t :: "'a::real_normed_vector set"

  4623   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  4624 proof-

  4625   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  4626     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  4627   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  4628     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  4629   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  4630 qed

  4631

  4632 lemma compact_differences:

  4633   fixes s t :: "'a::real_normed_vector set"

  4634   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  4635 proof-

  4636   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  4637     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4638   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  4639 qed

  4640

  4641 lemma compact_translation:

  4642   fixes s :: "'a::real_normed_vector set"

  4643   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  4644 proof-

  4645   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  4646   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  4647 qed

  4648

  4649 lemma compact_affinity:

  4650   fixes s :: "'a::real_normed_vector set"

  4651   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4652 proof-

  4653   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  4654   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  4655 qed

  4656

  4657 text {* Hence we get the following. *}

  4658

  4659 lemma compact_sup_maxdistance:

  4660   fixes s :: "'a::metric_space set"

  4661   assumes "compact s"  "s \<noteq> {}"

  4662   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  4663 proof-

  4664   have "compact (s \<times> s)" using compact s by (intro compact_Times)

  4665   moreover have "s \<times> s \<noteq> {}" using s \<noteq> {} by auto

  4666   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  4667     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  4668   ultimately show ?thesis

  4669     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  4670 qed

  4671

  4672 text {* We can state this in terms of diameter of a set. *}

  4673

  4674 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

  4675

  4676 lemma diameter_bounded_bound:

  4677   fixes s :: "'a :: metric_space set"

  4678   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  4679   shows "dist x y \<le> diameter s"

  4680 proof -

  4681   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4682   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  4683     unfolding bounded_def by auto

  4684   have "dist x y \<le> Sup ?D"

  4685   proof (rule cSup_upper, safe)

  4686     fix a b assume "a \<in> s" "b \<in> s"

  4687     with z[of a] z[of b] dist_triangle[of a b z]

  4688     show "dist a b \<le> 2 * d"

  4689       by (simp add: dist_commute)

  4690   qed (insert s, auto)

  4691   with x \<in> s show ?thesis

  4692     by (auto simp add: diameter_def)

  4693 qed

  4694

  4695 lemma diameter_lower_bounded:

  4696   fixes s :: "'a :: metric_space set"

  4697   assumes s: "bounded s" and d: "0 < d" "d < diameter s"

  4698   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  4699 proof (rule ccontr)

  4700   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  4701   assume contr: "\<not> ?thesis"

  4702   moreover

  4703   from d have "s \<noteq> {}"

  4704     by (auto simp: diameter_def)

  4705   then have "?D \<noteq> {}" by auto

  4706   ultimately have "Sup ?D \<le> d"

  4707     by (intro cSup_least) (auto simp: not_less)

  4708   with d < diameter s s \<noteq> {} show False

  4709     by (auto simp: diameter_def)

  4710 qed

  4711

  4712 lemma diameter_bounded:

  4713   assumes "bounded s"

  4714   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  4715         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  4716   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  4717   by auto

  4718

  4719 lemma diameter_compact_attained:

  4720   assumes "compact s"  "s \<noteq> {}"

  4721   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  4722 proof -

  4723   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  4724   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"

  4725     using compact_sup_maxdistance[OF assms] by auto

  4726   hence "diameter s \<le> dist x y"

  4727     unfolding diameter_def by clarsimp (rule cSup_least, fast+)

  4728   thus ?thesis

  4729     by (metis b diameter_bounded_bound order_antisym xys)

  4730 qed

  4731

  4732 text {* Related results with closure as the conclusion. *}

  4733

  4734 lemma closed_scaling:

  4735   fixes s :: "'a::real_normed_vector set"

  4736   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  4737 proof(cases "s={}")

  4738   case True thus ?thesis by auto

  4739 next

  4740   case False

  4741   show ?thesis

  4742   proof(cases "c=0")

  4743     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  4744     case True thus ?thesis apply auto unfolding * by auto

  4745   next

  4746     case False

  4747     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  4748       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  4749           using as(1)[THEN spec[where x=n]]

  4750           using c\<noteq>0 by auto

  4751       }

  4752       moreover

  4753       { fix e::real assume "e>0"

  4754         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  4755         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  4756           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  4757         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  4758           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  4759           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  4760       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  4761       ultimately have "l \<in> scaleR c  s"

  4762         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"]]

  4763         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  4764     thus ?thesis unfolding closed_sequential_limits by fast

  4765   qed

  4766 qed

  4767

  4768 lemma closed_negations:

  4769   fixes s :: "'a::real_normed_vector set"

  4770   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  4771   using closed_scaling[OF assms, of "- 1"] by simp

  4772

  4773 lemma compact_closed_sums:

  4774   fixes s :: "'a::real_normed_vector set"

  4775   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4776 proof-

  4777   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  4778   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  4779     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"

  4780       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  4781     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  4782       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  4783     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  4784       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto

  4785     hence "l - l' \<in> t"

  4786       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  4787       using f(3) by auto

  4788     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

  4789   }

  4790   thus ?thesis unfolding closed_sequential_limits by fast

  4791 qed

  4792

  4793 lemma closed_compact_sums:

  4794   fixes s t :: "'a::real_normed_vector set"

  4795   assumes "closed s"  "compact t"

  4796   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  4797 proof-

  4798   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

  4799     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  4800   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  4801 qed

  4802

  4803 lemma compact_closed_differences:

  4804   fixes s t :: "'a::real_normed_vector set"

  4805   assumes "compact s"  "closed t"

  4806   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4807 proof-

  4808   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  4809     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4810   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  4811 qed

  4812

  4813 lemma closed_compact_differences:

  4814   fixes s t :: "'a::real_normed_vector set"

  4815   assumes "closed s" "compact t"

  4816   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  4817 proof-

  4818   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  4819     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  4820  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  4821 qed

  4822

  4823 lemma closed_translation:

  4824   fixes a :: "'a::real_normed_vector"

  4825   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  4826 proof-

  4827   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  4828   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  4829 qed

  4830

  4831 lemma translation_Compl:

  4832   fixes a :: "'a::ab_group_add"

  4833   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  4834   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  4835

  4836 lemma translation_UNIV:

  4837   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  4838   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  4839

  4840 lemma translation_diff:

  4841   fixes a :: "'a::ab_group_add"

  4842   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  4843   by auto

  4844

  4845 lemma closure_translation:

  4846   fixes a :: "'a::real_normed_vector"

  4847   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  4848 proof-

  4849   have *:"op + a  (- s) = - op + a  s"

  4850     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  4851   show ?thesis unfolding closure_interior translation_Compl

  4852     using interior_translation[of a "- s"] unfolding * by auto

  4853 qed

  4854

  4855 lemma frontier_translation:

  4856   fixes a :: "'a::real_normed_vector"

  4857   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  4858   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  4859

  4860

  4861 subsection {* Separation between points and sets *}

  4862

  4863 lemma separate_point_closed:

  4864   fixes s :: "'a::heine_borel set"

  4865   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  4866 proof(cases "s = {}")

  4867   case True

  4868   thus ?thesis by(auto intro!: exI[where x=1])

  4869 next

  4870   case False

  4871   assume "closed s" "a \<notin> s"

  4872   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

  4873   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  4874 qed

  4875

  4876 lemma separate_compact_closed:

  4877   fixes s t :: "'a::heine_borel set"

  4878   assumes "compact s" and t: "closed t" "s \<inter> t = {}"

  4879   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  4880 proof cases

  4881   assume "s \<noteq> {} \<and> t \<noteq> {}"

  4882   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  4883   let ?inf = "\<lambda>x. infdist x t"

  4884   have "continuous_on s ?inf"

  4885     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

  4886   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  4887     using continuous_attains_inf[OF compact s s \<noteq> {}] by auto

  4888   then have "0 < ?inf x"

  4889     using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  4890   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  4891     using x by (auto intro: order_trans infdist_le)

  4892   ultimately show ?thesis

  4893     by auto

  4894 qed (auto intro!: exI[of _ 1])

  4895

  4896 lemma separate_closed_compact:

  4897   fixes s t :: "'a::heine_borel set"

  4898   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  4899   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  4900 proof-

  4901   have *:"t \<inter> s = {}" using assms(3) by auto

  4902   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  4903     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  4904     by (auto simp add: dist_commute)

  4905 qed

  4906

  4907

  4908 subsection {* Intervals *}

  4909

  4910 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  4911   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and

  4912   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  4913   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  4914

  4915 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4916   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  4917   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  4918   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  4919

  4920 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  4921  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and

  4922  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  4923 proof-

  4924   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  4925     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto

  4926     hence "a\<bullet>i < b\<bullet>i" by auto

  4927     hence False using as by auto  }

  4928   moreover

  4929   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  4930     let ?x = "(1/2) *\<^sub>R (a + b)"

  4931     { fix i :: 'a assume i:"i\<in>Basis"

  4932       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  4933       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  4934         by (auto simp: inner_add_left) }

  4935     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  4936   ultimately show ?th1 by blast

  4937

  4938   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  4939     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto

  4940     hence "a\<bullet>i \<le> b\<bullet>i" by auto

  4941     hence False using as by auto  }

  4942   moreover

  4943   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  4944     let ?x = "(1/2) *\<^sub>R (a + b)"

  4945     { fix i :: 'a assume i:"i\<in>Basis"

  4946       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  4947       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"

  4948         by (auto simp: inner_add_left) }

  4949     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  4950   ultimately show ?th2 by blast

  4951 qed

  4952

  4953 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  4954   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and

  4955   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  4956   unfolding interval_eq_empty[of a b] by fastforce+

  4957

  4958 lemma interval_sing:

  4959   fixes a :: "'a::ordered_euclidean_space"

  4960   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  4961   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  4962   by (auto intro: euclidean_eqI simp: ex_in_conv)

  4963

  4964 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  4965  "(\<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

  4966  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  4967  "(\<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

  4968  "(\<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}"

  4969   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  4970   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  4971

  4972 lemma interval_open_subset_closed:

  4973   fixes a :: "'a::ordered_euclidean_space"

  4974   shows "{a<..<b} \<subseteq> {a .. b}"

  4975   unfolding subset_eq [unfolded Ball_def] mem_interval

  4976   by (fast intro: less_imp_le)

  4977

  4978 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  4979  "{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

  4980  "{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

  4981  "{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

  4982  "{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)

  4983 proof-

  4984   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  4985   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)

  4986   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  4987     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  4988     fix i :: 'a assume i:"i\<in>Basis"

  4989     (** TODO combine the following two parts as done in the HOL_light version. **)

  4990     { 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"

  4991       assume as2: "a\<bullet>i > c\<bullet>i"

  4992       { fix j :: 'a assume j:"j\<in>Basis"

  4993         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  4994           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i

  4995           by (auto simp add: as2)  }

  4996       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto

  4997       moreover

  4998       have "?x\<notin>{a .. b}"

  4999         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5000         using as(2)[THEN bspec[where x=i]] and as2 i

  5001         by auto

  5002       ultimately have False using as by auto  }

  5003     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto

  5004     moreover

  5005     { 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"

  5006       assume as2: "b\<bullet>i < d\<bullet>i"

  5007       { fix j :: 'a assume "j\<in>Basis"

  5008         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  5009           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]

  5010           by (auto simp add: as2) }

  5011       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5012       moreover

  5013       have "?x\<notin>{a .. b}"

  5014         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5015         using as(2)[THEN bspec[where x=i]] and as2 using i

  5016         by auto

  5017       ultimately have False using as by auto  }

  5018     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto

  5019     ultimately

  5020     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  5021   } note part1 = this

  5022   show ?th3

  5023     unfolding subset_eq and Ball_def and mem_interval

  5024     apply(rule,rule,rule,rule)

  5025     apply(rule part1)

  5026     unfolding subset_eq and Ball_def and mem_interval

  5027     prefer 4

  5028     apply auto

  5029     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+

  5030   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5031     fix i :: 'a assume i:"i\<in>Basis"

  5032     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5033     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

  5034   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5035     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5036     apply auto by(erule_tac x=xa in allE, simp)+

  5037 qed

  5038

  5039 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5040  "{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)}"

  5041   unfolding set_eq_iff and Int_iff and mem_interval by auto

  5042

  5043 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5044   "{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

  5045   "{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

  5046   "{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

  5047   "{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)

  5048 proof-

  5049   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"

  5050   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  5051       (\<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)"

  5052     by blast

  5053   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

  5054   show ?th1 unfolding * by (intro **) auto

  5055   show ?th2 unfolding * by (intro **) auto

  5056   show ?th3 unfolding * by (intro **) auto

  5057   show ?th4 unfolding * by (intro **) auto

  5058 qed

  5059

  5060 (* Moved interval_open_subset_closed a bit upwards *)

  5061

  5062 lemma open_interval[intro]:

  5063   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5064 proof-

  5065   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"

  5066     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

  5067       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

  5068   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"

  5069     by (auto simp add: eucl_less [where 'a='a])

  5070   finally show "open {a<..<b}" .

  5071 qed

  5072

  5073 lemma closed_interval[intro]:

  5074   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5075 proof-

  5076   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"

  5077     by (intro closed_INT ballI continuous_closed_vimage allI

  5078       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

  5079   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"

  5080     by (auto simp add: eucl_le [where 'a='a])

  5081   finally show "closed {a .. b}" .

  5082 qed

  5083

  5084 lemma interior_closed_interval [intro]:

  5085   fixes a b :: "'a::ordered_euclidean_space"

  5086   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5087 proof(rule subset_antisym)

  5088   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5089     by (rule interior_maximal)

  5090 next

  5091   { fix x assume "x \<in> interior {a..b}"

  5092     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5093     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

  5094     { fix i :: 'a assume i:"i\<in>Basis"

  5095       have "dist (x - (e / 2) *\<^sub>R i) x < e"

  5096            "dist (x + (e / 2) *\<^sub>R i) x < e"

  5097         unfolding dist_norm apply auto

  5098         unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto

  5099       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"

  5100                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"

  5101         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]

  5102         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]

  5103         unfolding mem_interval using i by blast+

  5104       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"

  5105         using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }

  5106     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5107   thus "?L \<subseteq> ?R" ..

  5108 qed

  5109

  5110 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5111 proof-

  5112   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"

  5113   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  5114     { fix i :: 'a assume "i\<in>Basis"

  5115       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  }

  5116     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5117     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5118   thus ?thesis unfolding interval and bounded_iff by auto

  5119 qed

  5120

  5121 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5122  "bounded {a .. b} \<and> bounded {a<..<b}"

  5123   using bounded_closed_interval[of a b]

  5124   using interval_open_subset_closed[of a b]

  5125   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5126   by simp

  5127

  5128 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5129  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5130   using bounded_interval[of a b] by auto

  5131

  5132 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5133   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]

  5134   by (auto simp: compact_eq_seq_compact_metric)

  5135

  5136 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5137   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5138 proof-

  5139   { fix i :: 'a assume "i\<in>Basis"

  5140     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"

  5141       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }

  5142   thus ?thesis unfolding mem_interval by auto

  5143 qed

  5144

  5145 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5146   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5147   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5148 proof-

  5149   { fix i :: 'a assume i:"i\<in>Basis"

  5150     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp

  5151     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5152       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5153       using x unfolding mem_interval using i apply simp

  5154       using y unfolding mem_interval using i apply simp

  5155       done

  5156     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto

  5157     moreover {

  5158     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp

  5159     also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5160       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5161       using x unfolding mem_interval using i apply simp

  5162       using y unfolding mem_interval using i apply simp

  5163       done

  5164     finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto

  5165     } 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 }

  5166   thus ?thesis unfolding mem_interval by auto

  5167 qed

  5168

  5169 lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"

  5170   assumes "{a<..<b} \<noteq> {}"

  5171   shows "closure {a<..<b} = {a .. b}"

  5172 proof-

  5173   have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto

  5174   let ?c = "(1 / 2) *\<^sub>R (a + b)"

  5175   { fix x assume as:"x \<in> {a .. b}"

  5176     def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"

  5177     { fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"

  5178       have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto

  5179       have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =

  5180         x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"

  5181         by (auto simp add: algebra_simps)

  5182       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

  5183       hence False using fn unfolding f_def using xc by auto  }

  5184     moreover

  5185     { assume "\<not> (f ---> x) sequentially"

  5186       { fix e::real assume "e>0"

  5187         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

  5188         then obtain N::nat where "inverse (real (N + 1)) < e" by auto

  5189         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)

  5190         hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }

  5191       hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"

  5192         unfolding LIMSEQ_def by(auto simp add: dist_norm)

  5193       hence "(f ---> x) sequentially" unfolding f_def

  5194         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]

  5195         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  }

  5196     ultimately have "x \<in> closure {a<..<b}"

  5197       using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }

  5198   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast

  5199 qed

  5200

  5201 lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"

  5202   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"

  5203 proof-

  5204   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto

  5205   def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"

  5206   { fix x assume "x\<in>s"

  5207     fix i :: 'a assume i:"i\<in>Basis"

  5208     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]

  5209       and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }

  5210   thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])

  5211 qed

  5212

  5213 lemma bounded_subset_open_interval:

  5214   fixes s :: "('a::ordered_euclidean_space) set"

  5215   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"

  5216   by (auto dest!: bounded_subset_open_interval_symmetric)

  5217

  5218 lemma bounded_subset_closed_interval_symmetric:

  5219   fixes s :: "('a::ordered_euclidean_space) set"

  5220   assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"

  5221 proof-

  5222   obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto

  5223   thus ?thesis using interval_open_subset_closed[of "-a" a] by auto

  5224 qed

  5225

  5226 lemma bounded_subset_closed_interval:

  5227   fixes s :: "('a::ordered_euclidean_space) set"

  5228   shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"

  5229   using bounded_subset_closed_interval_symmetric[of s] by auto

  5230

  5231 lemma frontier_closed_interval:

  5232   fixes a b :: "'a::ordered_euclidean_space"

  5233   shows "frontier {a .. b} = {a .. b} - {a<..<b}"

  5234   unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

  5235

  5236 lemma frontier_open_interval:

  5237   fixes a b :: "'a::ordered_euclidean_space"

  5238   shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"

  5239 proof(cases "{a<..<b} = {}")

  5240   case True thus ?thesis using frontier_empty by auto

  5241 next

  5242   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto

  5243 qed

  5244

  5245 lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"

  5246   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"

  5247   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

  5248

  5249

  5250 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)

  5251

  5252 lemma closed_interval_left: fixes b::"'a::euclidean_space"

  5253   shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"

  5254 proof-

  5255   { fix i :: 'a assume i:"i\<in>Basis"

  5256     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"

  5257     { assume "x\<bullet>i > b\<bullet>i"

  5258       then obtain y where "y \<bullet> i \<le> b \<bullet> i"  "y \<noteq> x"  "dist y x < x\<bullet>i - b\<bullet>i"

  5259         using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto

  5260       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i

  5261         by auto }

  5262     hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto  }

  5263   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5264 qed

  5265

  5266 lemma closed_interval_right: fixes a::"'a::euclidean_space"

  5267   shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"

  5268 proof-

  5269   { fix i :: 'a assume i:"i\<in>Basis"

  5270     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"

  5271     { assume "a\<bullet>i > x\<bullet>i"

  5272       then obtain y where "a \<bullet> i \<le> y \<bullet> i"  "y \<noteq> x"  "dist y x < a\<bullet>i - x\<bullet>i"

  5273         using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto

  5274       hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }

  5275     hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto  }

  5276   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast

  5277 qed

  5278

  5279 lemma open_box: "open (box a b)"

  5280 proof -

  5281   have "open (\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i})"

  5282     by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)

  5283   also have "(\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i}) = box a b"

  5284     by (auto simp add: box_def inner_commute)

  5285   finally show ?thesis .

  5286 qed

  5287

  5288 instance euclidean_space \<subseteq> second_countable_topology

  5289 proof

  5290   def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"

  5291   then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp

  5292   def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"

  5293   then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp

  5294   def B \<equiv> "(\<lambda>f. box (a f) (b f))  (Basis \<rightarrow>\<^isub>E (\<rat> \<times> \<rat>))"

  5295

  5296   have "Ball B open" by (simp add: B_def open_box)

  5297   moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"

  5298   proof safe

  5299     fix A::"'a set" assume "open A"

  5300     show "\<exists>B'\<subseteq>B. \<Union>B' = A"

  5301       apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])

  5302       apply (subst (3) open_UNION_box[OF open A])

  5303       apply (auto simp add: a b B_def)

  5304       done

  5305   qed

  5306   ultimately

  5307   have "topological_basis B" unfolding topological_basis_def by blast

  5308   moreover

  5309   have "countable B" unfolding B_def

  5310     by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)

  5311   ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"

  5312     by (blast intro: topological_basis_imp_subbasis)

  5313 qed

  5314

  5315 instance euclidean_space \<subseteq> polish_space ..

  5316

  5317 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  5318

  5319 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  5320   (\<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)"

  5321

  5322 lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)

  5323   "is_interval {a<..<b}" (is ?th2) proof -

  5324   show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff

  5325     by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

  5326

  5327 lemma is_interval_empty:

  5328  "is_interval {}"

  5329   unfolding is_interval_def

  5330   by simp

  5331

  5332 lemma is_interval_univ:

  5333  "is_interval UNIV"

  5334   unfolding is_interval_def

  5335   by simp

  5336

  5337

  5338 subsection {* Closure of halfspaces and hyperplanes *}

  5339

  5340 lemma isCont_open_vimage:

  5341   assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"

  5342 proof -

  5343   from assms(1) have "continuous_on UNIV f"

  5344     unfolding isCont_def continuous_on_def by simp

  5345   hence "open {x \<in> UNIV. f x \<in> s}"

  5346     using open_UNIV open s by (rule continuous_open_preimage)

  5347   thus "open (f - s)"

  5348     by (simp add: vimage_def)

  5349 qed

  5350

  5351 lemma isCont_closed_vimage:

  5352   assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"

  5353   using assms unfolding closed_def vimage_Compl [symmetric]

  5354   by (rule isCont_open_vimage)

  5355

  5356 lemma open_Collect_less:

  5357   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5358   assumes f: "\<And>x. isCont f x"

  5359   assumes g: "\<And>x. isCont g x"

  5360   shows "open {x. f x < g x}"

  5361 proof -

  5362   have "open ((\<lambda>x. g x - f x) - {0<..})"

  5363     using isCont_diff [OF g f] open_real_greaterThan

  5364     by (rule isCont_open_vimage)

  5365   also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"

  5366     by auto

  5367   finally show ?thesis .

  5368 qed

  5369

  5370 lemma closed_Collect_le:

  5371   fixes f g :: "'a::t2_space \<Rightarrow> real"

  5372   assumes f: "\<And>x. isCont f x"

  5373   assumes g: "\<And>x. isCont g x"

  5374   shows "closed {x. f x \<le> g x}"

  5375 proof -

  5376   have "closed ((\<lambda>x. g x - f x) - {0..})"

  5377     using isCont_diff [OF g f] closed_real_atLeast

  5378     by (rule isCont_closed_vimage)

  5379   also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"

  5380     by auto

  5381   finally show ?thesis .

  5382 qed

  5383

  5384 lemma closed_Collect_eq:

  5385   fixes f g :: "'a::t2_space \<Rightarrow> 'b::t2_space"

  5386   assumes f: "\<And>x. isCont f x"

  5387   assumes g: "\<And>x. isCont g x"

  5388   shows "closed {x. f x = g x}"

  5389 proof -

  5390   have "open {(x::'b, y::'b). x \<noteq> y}"

  5391     unfolding open_prod_def by (auto dest!: hausdorff)

  5392   hence "closed {(x::'b, y::'b). x = y}"

  5393     unfolding closed_def split_def Collect_neg_eq .

  5394   with isCont_Pair [OF f g]

  5395   have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"

  5396     by (rule isCont_closed_vimage)

  5397   also have "\<dots> = {x. f x = g x}" by auto

  5398   finally show ?thesis .

  5399 qed

  5400

  5401 lemma continuous_at_inner: "continuous (at x) (inner a)"

  5402   unfolding continuous_at by (intro tendsto_intros)

  5403

  5404 lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"

  5405   by (simp add: closed_Collect_le)

  5406

  5407 lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

  5408   by (simp add: closed_Collect_le)

  5409

  5410 lemma closed_hyperplane: "closed {x. inner a x = b}"

  5411   by (simp add: closed_Collect_eq)

  5412

  5413 lemma closed_halfspace_component_le:

  5414   shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

  5415   by (simp add: closed_Collect_le)

  5416

  5417 lemma closed_halfspace_component_ge:

  5418   shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

  5419   by (simp add: closed_Collect_le)

  5420

  5421 text {* Openness of halfspaces. *}

  5422

  5423 lemma open_halfspace_lt: "open {x. inner a x < b}"

  5424   by (simp add: open_Collect_less)

  5425

  5426 lemma open_halfspace_gt: "open {x. inner a x > b}"

  5427   by (simp add: open_Collect_less)

  5428

  5429 lemma open_halfspace_component_lt:

  5430   shows "open {x::'a::euclidean_space. x\<bullet>i < a}"

  5431   by (simp add: open_Collect_less)

  5432

  5433 lemma open_halfspace_component_gt:

  5434   shows "open {x::'a::euclidean_space. x\<bullet>i > a}"

  5435   by (simp add: open_Collect_less)

  5436

  5437 text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

  5438

  5439 lemma eucl_lessThan_eq_halfspaces:

  5440   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5441   shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"

  5442  by (auto simp: eucl_less[where 'a='a])

  5443

  5444 lemma eucl_greaterThan_eq_halfspaces:

  5445   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5446   shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"

  5447  by (auto simp: eucl_less[where 'a='a])

  5448

  5449 lemma eucl_atMost_eq_halfspaces:

  5450   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5451   shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"

  5452  by (auto simp: eucl_le[where 'a='a])

  5453

  5454 lemma eucl_atLeast_eq_halfspaces:

  5455   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5456   shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"

  5457  by (auto simp: eucl_le[where 'a='a])

  5458

  5459 lemma open_eucl_lessThan[simp, intro]:

  5460   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5461   shows "open {..< a}"

  5462   by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

  5463

  5464 lemma open_eucl_greaterThan[simp, intro]:

  5465   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5466   shows "open {a <..}"

  5467   by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

  5468

  5469 lemma closed_eucl_atMost[simp, intro]:

  5470   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5471   shows "closed {.. a}"

  5472   unfolding eucl_atMost_eq_halfspaces

  5473   by (simp add: closed_INT closed_Collect_le)

  5474

  5475 lemma closed_eucl_atLeast[simp, intro]:

  5476   fixes a :: "'a\<Colon>ordered_euclidean_space"

  5477   shows "closed {a ..}"

  5478   unfolding eucl_atLeast_eq_halfspaces

  5479   by (simp add: closed_INT closed_Collect_le)

  5480

  5481 text {* This gives a simple derivation of limit component bounds. *}

  5482

  5483 lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5484   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"

  5485   shows "l\<bullet>i \<le> b"

  5486   by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

  5487

  5488 lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5489   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"

  5490   shows "b \<le> l\<bullet>i"

  5491   by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

  5492

  5493 lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"

  5494   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"

  5495   shows "l\<bullet>i = b"

  5496   using ev[unfolded order_eq_iff eventually_conj_iff]

  5497   using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto

  5498

  5499 text{* Limits relative to a union.                                               *}

  5500

  5501 lemma eventually_within_Un:

  5502   "eventually P (at x within (s \<union> t)) \<longleftrightarrow> eventually P (at x within s) \<and> eventually P (at x within t)"

  5503   unfolding eventually_at_filter

  5504   by (auto elim!: eventually_rev_mp)

  5505

  5506 lemma Lim_within_union:

  5507  "(f ---> l) (at x within (s \<union> t)) \<longleftrightarrow>

  5508   (f ---> l) (at x within s) \<and> (f ---> l) (at x within t)"

  5509   unfolding tendsto_def

  5510   by (auto simp add: eventually_within_Un)

  5511

  5512 lemma Lim_topological:

  5513  "(f ---> l) net \<longleftrightarrow>

  5514         trivial_limit net \<or>

  5515         (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"

  5516   unfolding tendsto_def trivial_limit_eq by auto

  5517

  5518 text{* Some more convenient intermediate-value theorem formulations.             *}

  5519

  5520 lemma connected_ivt_hyperplane:

  5521   assumes "connected s" "x \<in> s" "y \<in> s" "inner a x \<le> b" "b \<le> inner a y"

  5522   shows "\<exists>z \<in> s. inner a z = b"

  5523 proof(rule ccontr)

  5524   assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"

  5525   let ?A = "{x. inner a x < b}"

  5526   let ?B = "{x. inner a x > b}"

  5527   have "open ?A" "open ?B" using open_halfspace_lt and open_halfspace_gt by auto

  5528   moreover have "?A \<inter> ?B = {}" by auto

  5529   moreover have "s \<subseteq> ?A \<union> ?B" using as by auto

  5530   ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto

  5531 qed

  5532

  5533 lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows

  5534  "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s.  z\<bullet>k = a)"

  5535   using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute)

  5536

  5537

  5538 subsection {* Homeomorphisms *}

  5539

  5540 definition "homeomorphism s t f g \<equiv>

  5541      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f  s = t) \<and> continuous_on s f \<and>

  5542      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g  t = s) \<and> continuous_on t g"

  5543

  5544 definition

  5545   homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"

  5546     (infixr "homeomorphic" 60) where

  5547   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

  5548

  5549 lemma homeomorphic_refl: "s homeomorphic s"

  5550   unfolding homeomorphic_def

  5551   unfolding homeomorphism_def

  5552   using continuous_on_id

  5553   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5554   apply(rule_tac x = "(\<lambda>x. x)" in exI)

  5555   by blast

  5556

  5557 lemma homeomorphic_sym:

  5558  "s homeomorphic t \<longleftrightarrow> t homeomorphic s"

  5559 unfolding homeomorphic_def

  5560 unfolding homeomorphism_def

  5561 by blast

  5562

  5563 lemma homeomorphic_trans:

  5564   assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"

  5565 proof-

  5566   obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x"  "f1  s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1  t = s" "continuous_on t g1"

  5567     using assms(1) unfolding homeomorphic_def homeomorphism_def by auto

  5568   obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x"  "f2  t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2  u = t" "continuous_on u g2"

  5569     using assms(2) unfolding homeomorphic_def homeomorphism_def by auto

  5570

  5571   { fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }

  5572   moreover have "(f2 \<circ> f1)  s = u" using fg1(2) fg2(2) by auto

  5573   moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto

  5574   moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }

  5575   moreover have "(g1 \<circ> g2)  u = s" using fg1(5) fg2(5) by auto

  5576   moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto

  5577   ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto

  5578 qed

  5579

  5580 lemma homeomorphic_minimal:

  5581  "s homeomorphic t \<longleftrightarrow>

  5582     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>

  5583            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>

  5584            continuous_on s f \<and> continuous_on t g)"

  5585 unfolding homeomorphic_def homeomorphism_def

  5586 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)

  5587 apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto

  5588 unfolding image_iff

  5589 apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)

  5590 apply auto apply(rule_tac x="g x" in bexI) apply auto

  5591 apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)

  5592 apply auto apply(rule_tac x="f x" in bexI) by auto

  5593

  5594 text {* Relatively weak hypotheses if a set is compact. *}

  5595

  5596 lemma homeomorphism_compact:

  5597   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  5598   assumes "compact s" "continuous_on s f"  "f  s = t"  "inj_on f s"

  5599   shows "\<exists>g. homeomorphism s t f g"

  5600 proof-

  5601   def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = x"

  5602   have g:"\<forall>x\<in>s. g (f x) = x" using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto

  5603   { fix y assume "y\<in>t"

  5604     then obtain x where x:"f x = y" "x\<in>s" using assms(3) by auto

  5605     hence "g (f x) = x" using g by auto

  5606     hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }

  5607   hence g':"\<forall>x\<in>t. f (g x) = x" by auto

  5608   moreover

  5609   { fix x

  5610     have "x\<in>s \<Longrightarrow> x \<in> g  t" using g[THEN bspec[where x=x]] unfolding image_iff using assms(3) by(auto intro!: bexI[where x="f x"])

  5611     moreover

  5612     { assume "x\<in>g  t"

  5613       then obtain y where y:"y\<in>t" "g y = x" by auto

  5614       then obtain x' where x':"x'\<in>s" "f x' = y" using assms(3) by auto

  5615       hence "x \<in> s" unfolding g_def using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"] unfolding y(2)[THEN sym] and g_def by auto }

  5616     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g  t" ..  }

  5617   hence "g  t = s" by auto

  5618   ultimately

  5619   show ?thesis unfolding homeomorphism_def homeomorphic_def

  5620     apply(rule_tac x=g in exI) using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2) by auto

  5621 qed

  5622

  5623 lemma homeomorphic_compact:

  5624   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  5625   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f  s = t) \<Longrightarrow> inj_on f s

  5626           \<Longrightarrow> s homeomorphic t"

  5627   unfolding homeomorphic_def by (metis homeomorphism_compact)

  5628

  5629 text{* Preservation of topological properties.                                   *}

  5630

  5631 lemma homeomorphic_compactness:

  5632  "s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"

  5633 unfolding homeomorphic_def homeomorphism_def

  5634 by (metis compact_continuous_image)

  5635

  5636 text{* Results on translation, scaling etc.                                      *}

  5637

  5638 lemma homeomorphic_scaling:

  5639   fixes s :: "'a::real_normed_vector set"

  5640   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x)  s)"

  5641   unfolding homeomorphic_minimal

  5642   apply(rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)

  5643   apply(rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)

  5644   using assms by (auto simp add: continuous_on_intros)

  5645

  5646 lemma homeomorphic_translation:

  5647   fixes s :: "'a::real_normed_vector set"

  5648   shows "s homeomorphic ((\<lambda>x. a + x)  s)"

  5649   unfolding homeomorphic_minimal

  5650   apply(rule_tac x="\<lambda>x. a + x" in exI)

  5651   apply(rule_tac x="\<lambda>x. -a + x" in exI)

  5652   using continuous_on_add[OF continuous_on_const continuous_on_id] by auto

  5653

  5654 lemma homeomorphic_affinity:

  5655   fixes s :: "'a::real_normed_vector set"

  5656   assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5657 proof-

  5658   have *:"op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  5659   show ?thesis

  5660     using homeomorphic_trans

  5661     using homeomorphic_scaling[OF assms, of s]

  5662     using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x)  s" a] unfolding * by auto

  5663 qed

  5664

  5665 lemma homeomorphic_balls:

  5666   fixes a b ::"'a::real_normed_vector"

  5667   assumes "0 < d"  "0 < e"

  5668   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)

  5669         "(cball a d) homeomorphic (cball b e)" (is ?cth)

  5670 proof-

  5671   show ?th unfolding homeomorphic_minimal

  5672     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)

  5673     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)

  5674     using assms

  5675     apply (auto intro!: continuous_on_intros

  5676                 simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)

  5677     done

  5678 next

  5679   show ?cth unfolding homeomorphic_minimal

  5680     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)

  5681     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)

  5682     using assms

  5683     apply (auto intro!: continuous_on_intros

  5684                 simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)

  5685     done

  5686 qed

  5687

  5688 text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}

  5689

  5690 lemma cauchy_isometric:

  5691   fixes x :: "nat \<Rightarrow> 'a::euclidean_space"

  5692   assumes e:"0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and xs:"\<forall>n::nat. x n \<in> s" and cf:"Cauchy(f o x)"

  5693   shows "Cauchy x"

  5694 proof-

  5695   interpret f: bounded_linear f by fact

  5696   { fix d::real assume "d>0"

  5697     then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"

  5698       using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] and e and mult_pos_pos[of e d] by auto

  5699     { fix n assume "n\<ge>N"

  5700       have "e * norm (x n - x N) \<le> norm (f (x n - x N))"

  5701         using subspace_sub[OF s, of "x n" "x N"] using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]

  5702         using normf[THEN bspec[where x="x n - x N"]] by auto

  5703       also have "norm (f (x n - x N)) < e * d"

  5704         using N \<le> n N unfolding f.diff[THEN sym] by auto

  5705       finally have "norm (x n - x N) < d" using e>0 by simp }

  5706     hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }

  5707   thus ?thesis unfolding cauchy and dist_norm by auto

  5708 qed

  5709

  5710 lemma complete_isometric_image:

  5711   fixes f :: "'a::euclidean_space => 'b::euclidean_space"

  5712   assumes "0 < e" and s:"subspace s" and f:"bounded_linear f" and normf:"\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)" and cs:"complete s"

  5713   shows "complete(f  s)"

  5714 proof-

  5715   { fix g assume as:"\<forall>n::nat. g n \<in> f  s" and cfg:"Cauchy g"

  5716     then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"

  5717       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto

  5718     hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto

  5719     hence "f \<circ> x = g" unfolding fun_eq_iff by auto

  5720     then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"

  5721       using cs[unfolded complete_def, THEN spec[where x="x"]]

  5722       using cauchy_isometric[OF 0<e s f normf] and cfg and x(1) by auto

  5723     hence "\<exists>l\<in>f  s. (g ---> l) sequentially"

  5724       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]

  5725       unfolding f \<circ> x = g by auto  }

  5726   thus ?thesis unfolding complete_def by auto

  5727 qed

  5728

  5729 lemma injective_imp_isometric: fixes f::&quo