src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author wenzelm Sun Nov 02 17:09:04 2014 +0100 (2014-11-02) changeset 58877 262572d90bc6 parent 58759 e55fe82f3803 child 59587 8ea7b22525cb permissions -rw-r--r--
     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 section {* 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/FuncSet"

    14   Linear_Algebra

    15   Norm_Arith

    16 begin

    17

    18 lemma dist_0_norm:

    19   fixes x :: "'a::real_normed_vector"

    20   shows "dist 0 x = norm x"

    21 unfolding dist_norm by simp

    22

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

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

    25

    26 (* LEGACY *)

    27 lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"

    28   by (rule LIMSEQ_subseq_LIMSEQ)

    29

    30 lemma countable_PiE:

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

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

    33

    34 lemma Lim_within_open:

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

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

    37   by (fact tendsto_within_open)

    38

    39 lemma continuous_on_union:

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

    41   by (fact continuous_on_closed_Un)

    42

    43 lemma continuous_on_cases:

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

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

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

    47   by (rule continuous_on_If) auto

    48

    49

    50 subsection {* Topological Basis *}

    51

    52 context topological_space

    53 begin

    54

    55 definition "topological_basis B \<longleftrightarrow>

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

    57

    58 lemma topological_basis:

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

    60   unfolding topological_basis_def

    61   apply safe

    62      apply fastforce

    63     apply fastforce

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

    65    apply simp

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

    67   apply auto

    68   done

    69

    70 lemma topological_basis_iff:

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

    72   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'))"

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

    74 proof safe

    75   fix O' and x::'a

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

    77   then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)

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

    79   then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto

    80 next

    81   assume H: ?rhs

    82   show "topological_basis B"

    83     using assms unfolding topological_basis_def

    84   proof safe

    85     fix O' :: "'a set"

    86     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     then show "\<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     and "\<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     and "open O'"

   104     and "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'"

   108     by (simp add: topological_basis_def)

   109   with topological_basis_iff assms

   110   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"

   111     using assms by (simp add: Bex_def)

   112 qed

   113

   114 lemma topological_basis_open:

   115   assumes "topological_basis B"

   116     and "X \<in> B"

   117   shows "open X"

   118   using assms by (simp add: topological_basis_def)

   119

   120 lemma topological_basis_imp_subbasis:

   121   assumes B: "topological_basis B"

   122   shows "open = generate_topology B"

   123 proof (intro ext iffI)

   124   fix S :: "'a set"

   125   assume "open S"

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

   127     unfolding topological_basis_def by blast

   128   then show "generate_topology B S"

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

   130 next

   131   fix S :: "'a set"

   132   assume "generate_topology B S"

   133   then show "open S"

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

   135 qed

   136

   137 lemma basis_dense:

   138   fixes B :: "'a set set"

   139     and f :: "'a set \<Rightarrow> 'a"

   140   assumes "topological_basis B"

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

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

   143 proof (intro allI impI)

   144   fix X :: "'a set"

   145   assume "open X" and "X \<noteq> {}"

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

   147   obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .

   148   then show "\<exists>B'\<in>B. f B' \<in> X"

   149     by (auto intro!: choosefrom_basis)

   150 qed

   151

   152 end

   153

   154 lemma topological_basis_prod:

   155   assumes A: "topological_basis A"

   156     and B: "topological_basis B"

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

   158   unfolding topological_basis_def

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

   160   fix S :: "('a \<times> 'b) set"

   161   assume "open S"

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

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

   164     fix x y

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

   166     from open_prod_elim[OF open S this]

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

   168       by (metis mem_Sigma_iff)

   169     moreover

   170     from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"

   171       by (rule topological_basisE)

   172     moreover

   173     from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"

   174       by (rule topological_basisE)

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

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

   177   qed auto

   178 qed (metis A B topological_basis_open open_Times)

   179

   180

   181 subsection {* Countable Basis *}

   182

   183 locale countable_basis =

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

   185   assumes is_basis: "topological_basis B"

   186     and countable_basis: "countable B"

   187 begin

   188

   189 lemma open_countable_basis_ex:

   190   assumes "open X"

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

   192   using assms countable_basis is_basis

   193   unfolding topological_basis_def by blast

   194

   195 lemma open_countable_basisE:

   196   assumes "open X"

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

   198   using assms open_countable_basis_ex

   199   by (atomize_elim) simp

   200

   201 lemma countable_dense_exists:

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

   203 proof -

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

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

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

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

   208 qed

   209

   210 lemma countable_dense_setE:

   211   obtains D :: "'a set"

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

   213   using countable_dense_exists by blast

   214

   215 end

   216

   217 lemma (in first_countable_topology) first_countable_basisE:

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

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

   220   using first_countable_basis[of x]

   221   apply atomize_elim

   222   apply (elim exE)

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

   224   apply auto

   225   done

   226

   227 lemma (in first_countable_topology) first_countable_basis_Int_stableE:

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

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

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

   231 proof atomize_elim

   232   obtain A' where A':

   233     "countable A'"

   234     "\<And>a. a \<in> A' \<Longrightarrow> x \<in> a"

   235     "\<And>a. a \<in> A' \<Longrightarrow> open a"

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

   237     by (rule first_countable_basisE) blast

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

   239   then show "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>

   240         (\<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)"

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

   242     show "countable A"

   243       unfolding A_def by (intro countable_image countable_Collect_finite)

   244     fix a

   245     assume "a \<in> A"

   246     then show "x \<in> a" "open a"

   247       using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)

   248   next

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

   250     fix a b

   251     assume "a \<in> A" "b \<in> A"

   252     then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"

   253       by (auto simp: A_def)

   254     then show "a \<inter> b \<in> A"

   255       by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])

   256   next

   257     fix S

   258     assume "open S" "x \<in> S"

   259     then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast

   260     then show "\<exists>a\<in>A. a \<subseteq> S" using a A'

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

   262   qed

   263 qed

   264

   265 lemma (in topological_space) first_countableI:

   266   assumes "countable A"

   267     and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"

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

   269   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))"

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

   271   fix i

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

   273   show "x \<in> from_nat_into A i" "open (from_nat_into A i)"

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

   275 next

   276   fix S

   277   assume "open S" "x\<in>S" from 2[OF this]

   278   show "\<exists>i. from_nat_into A i \<subseteq> S"

   279     using subset_range_from_nat_into[OF countable A] by auto

   280 qed

   281

   282 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

   283 proof

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

   285   obtain A where A:

   286       "countable A"

   287       "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"

   288       "\<And>a. a \<in> A \<Longrightarrow> open a"

   289       "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"

   290     by (rule first_countable_basisE[of "fst x"]) blast

   291   obtain B where B:

   292       "countable B"

   293       "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"

   294       "\<And>a. a \<in> B \<Longrightarrow> open a"

   295       "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"

   296     by (rule first_countable_basisE[of "snd x"]) blast

   297   show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.

   298     (\<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))"

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

   300     fix a b

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

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

   303       unfolding mem_Times_iff

   304       by (auto intro: open_Times)

   305   next

   306     fix S

   307     assume "open S" "x \<in> S"

   308     then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"

   309       by (rule open_prod_elim)

   310     moreover

   311     from a'b' A(4)[of a'] B(4)[of b']

   312     obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"

   313       by auto

   314     ultimately

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

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

   317   qed (simp add: A B)

   318 qed

   319

   320 class second_countable_topology = topological_space +

   321   assumes ex_countable_subbasis:

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

   323 begin

   324

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

   326 proof -

   327   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"

   328     by blast

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

   330

   331   show ?thesis

   332   proof (intro exI conjI)

   333     show "countable ?B"

   334       by (intro countable_image countable_Collect_finite_subset B)

   335     {

   336       fix S

   337       assume "open S"

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

   339         unfolding B

   340       proof induct

   341         case UNIV

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

   343       next

   344         case (Int a b)

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

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

   347           by blast

   348         show ?case

   349           unfolding x y Int_UN_distrib2

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

   351       next

   352         case (UN K)

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

   354         then obtain k where

   355             "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"

   356           unfolding bchoice_iff ..

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

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

   359       next

   360         case (Basis S)

   361         then show ?case

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

   363       qed

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

   365         unfolding subset_image_iff by blast }

   366     then show "topological_basis ?B"

   367       unfolding topological_space_class.topological_basis_def

   368       by (safe intro!: topological_space_class.open_Inter)

   369          (simp_all add: B generate_topology.Basis subset_eq)

   370   qed

   371 qed

   372

   373 end

   374

   375 sublocale second_countable_topology <

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

   377   using someI_ex[OF ex_countable_basis]

   378   by unfold_locales safe

   379

   380 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   381 proof

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

   383     using ex_countable_basis by auto

   384   moreover

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

   386     using ex_countable_basis by auto

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

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

   389       topological_basis_imp_subbasis)

   390 qed

   391

   392 instance second_countable_topology \<subseteq> first_countable_topology

   393 proof

   394   fix x :: 'a

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

   396   then have B: "countable B" "topological_basis B"

   397     using countable_basis is_basis

   398     by (auto simp: countable_basis is_basis)

   399   then show "\<exists>A::nat \<Rightarrow> 'a set.

   400     (\<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))"

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

   402        (fastforce simp: topological_space_class.topological_basis_def)+

   403 qed

   404

   405

   406 subsection {* Polish spaces *}

   407

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

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

   410

   411 class polish_space = complete_space + second_countable_topology

   412

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

   414

   415 definition "istopology L \<longleftrightarrow>

   416   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))"

   417

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

   419   morphisms "openin" "topology"

   420   unfolding istopology_def by blast

   421

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

   423   using openin[of U] by blast

   424

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

   426   using topology_inverse[unfolded mem_Collect_eq] .

   427

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

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

   430

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

   432 proof

   433   assume "T1 = T2"

   434   then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp

   435 next

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

   437   then have "openin T1 = openin T2" by (simp add: fun_eq_iff)

   438   then have "topology (openin T1) = topology (openin T2)" by simp

   439   then show "T1 = T2" unfolding openin_inverse .

   440 qed

   441

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

   443

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

   445

   446 subsubsection {* Main properties of open sets *}

   447

   448 lemma openin_clauses:

   449   fixes U :: "'a topology"

   450   shows

   451     "openin U {}"

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

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

   454   using openin[of U] unfolding istopology_def mem_Collect_eq by fast+

   455

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

   457   unfolding topspace_def by blast

   458

   459 lemma openin_empty[simp]: "openin U {}"

   460   by (simp add: openin_clauses)

   461

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

   463   using openin_clauses by simp

   464

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

   466   using openin_clauses by simp

   467

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

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

   470

   471 lemma openin_topspace[intro, simp]: "openin U (topspace U)"

   472   by (simp add: openin_Union topspace_def)

   473

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

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

   476 proof

   477   assume ?lhs

   478   then show ?rhs by auto

   479 next

   480   assume H: ?rhs

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

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

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

   484   finally show "openin U S" .

   485 qed

   486

   487

   488 subsubsection {* Closed sets *}

   489

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

   491

   492 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"

   493   by (metis closedin_def)

   494

   495 lemma closedin_empty[simp]: "closedin U {}"

   496   by (simp add: closedin_def)

   497

   498 lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"

   499   by (simp add: closedin_def)

   500

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

   502   by (auto simp add: Diff_Un closedin_def)

   503

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

   505   by auto

   506

   507 lemma closedin_Inter[intro]:

   508   assumes Ke: "K \<noteq> {}"

   509     and Kc: "\<forall>S \<in>K. closedin U S"

   510   shows "closedin U (\<Inter> K)"

   511   using Ke Kc unfolding closedin_def Diff_Inter by auto

   512

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

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

   515

   516 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"

   517   by blast

   518

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

   520   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   521   apply (metis openin_subset subset_eq)

   522   done

   523

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

   525   by (simp add: openin_closedin_eq)

   526

   527 lemma openin_diff[intro]:

   528   assumes oS: "openin U S"

   529     and cT: "closedin U T"

   530   shows "openin U (S - T)"

   531 proof -

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

   533     by (auto simp add: topspace_def openin_subset)

   534   then show ?thesis using oS cT

   535     by (auto simp add: closedin_def)

   536 qed

   537

   538 lemma closedin_diff[intro]:

   539   assumes oS: "closedin U S"

   540     and cT: "openin U T"

   541   shows "closedin U (S - T)"

   542 proof -

   543   have "S - T = S \<inter> (topspace U - T)"

   544     using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)

   545   then show ?thesis

   546     using oS cT by (auto simp add: openin_closedin_eq)

   547 qed

   548

   549

   550 subsubsection {* Subspace topology *}

   551

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

   553

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

   555   (is "istopology ?L")

   556 proof -

   557   have "?L {}" by blast

   558   {

   559     fix A B

   560     assume A: "?L A" and B: "?L B"

   561     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"

   562       by blast

   563     have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"

   564       using Sa Sb by blast+

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

   566   }

   567   moreover

   568   {

   569     fix K

   570     assume K: "K \<subseteq> Collect ?L"

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

   572       by blast

   573     from K[unfolded th0 subset_image_iff]

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

   575       by blast

   576     have "\<Union>K = (\<Union>Sk) \<inter> V"

   577       using Sk by auto

   578     moreover have "openin U (\<Union> Sk)"

   579       using Sk by (auto simp add: subset_eq)

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

   581   }

   582   ultimately show ?thesis

   583     unfolding subset_eq mem_Collect_eq istopology_def by blast

   584 qed

   585

   586 lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"

   587   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   588   by auto

   589

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

   591   by (auto simp add: topspace_def openin_subtopology)

   592

   593 lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"

   594   unfolding closedin_def topspace_subtopology

   595   by (auto simp add: openin_subtopology)

   596

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

   598   unfolding openin_subtopology

   599   by auto (metis IntD1 in_mono openin_subset)

   600

   601 lemma subtopology_superset:

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

   603   shows "subtopology U V = U"

   604 proof -

   605   {

   606     fix S

   607     {

   608       fix T

   609       assume T: "openin U T" "S = T \<inter> V"

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

   611         by blast

   612       have "openin U S"

   613         unfolding eq using T by blast

   614     }

   615     moreover

   616     {

   617       assume S: "openin U S"

   618       then have "\<exists>T. openin U T \<and> S = T \<inter> V"

   619         using openin_subset[OF S] UV by auto

   620     }

   621     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"

   622       by blast

   623   }

   624   then show ?thesis

   625     unfolding topology_eq openin_subtopology by blast

   626 qed

   627

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

   629   by (simp add: subtopology_superset)

   630

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

   632   by (simp add: subtopology_superset)

   633

   634

   635 subsubsection {* The standard Euclidean topology *}

   636

   637 definition euclidean :: "'a::topological_space topology"

   638   where "euclidean = topology open"

   639

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

   641   unfolding euclidean_def

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

   643   apply (rule topology_inverse[symmetric])

   644   apply (auto simp add: istopology_def)

   645   done

   646

   647 lemma topspace_euclidean: "topspace euclidean = UNIV"

   648   apply (simp add: topspace_def)

   649   apply (rule set_eqI)

   650   apply (auto simp add: open_openin[symmetric])

   651   done

   652

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

   654   by (simp add: topspace_euclidean topspace_subtopology)

   655

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

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

   658

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

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

   661

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

   663

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

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

   666

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

   668   by (auto simp add: openin_open)

   669

   670 lemma open_openin_trans[trans]:

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

   672   by (metis Int_absorb1  openin_open_Int)

   673

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

   675   by (auto simp add: openin_open)

   676

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

   678   by (simp add: closedin_subtopology closed_closedin Int_ac)

   679

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

   681   by (metis closedin_closed)

   682

   683 lemma closed_closedin_trans:

   684   "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"

   685   by (metis closedin_closed inf.absorb2)

   686

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

   688   by (auto simp add: closedin_closed)

   689

   690 lemma openin_euclidean_subtopology_iff:

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

   692   shows "openin (subtopology euclidean U) S \<longleftrightarrow>

   693     S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"

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

   695 proof

   696   assume ?lhs

   697   then show ?rhs

   698     unfolding openin_open open_dist by blast

   699 next

   700   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}"

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

   702     unfolding T_def

   703     apply clarsimp

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

   705     apply (clarsimp simp add: less_diff_eq)

   706     by (metis dist_commute dist_triangle_lt)

   707   assume ?rhs then have 2: "S = U \<inter> T"

   708     unfolding T_def

   709     by auto (metis dist_self)

   710   from 1 2 show ?lhs

   711     unfolding openin_open open_dist by fast

   712 qed

   713

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

   715

   716 lemma openin_trans[trans]:

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

   718     openin (subtopology euclidean U) S"

   719   unfolding open_openin openin_open by blast

   720

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

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

   723

   724 lemma closedin_trans[trans]:

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

   726     closedin (subtopology euclidean U) S"

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

   728

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

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

   731

   732

   733 subsection {* Open and closed balls *}

   734

   735 definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"

   736   where "ball x e = {y. dist x y < e}"

   737

   738 definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"

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

   740

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

   742   by (simp add: ball_def)

   743

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

   745   by (simp add: cball_def)

   746

   747 lemma mem_ball_0:

   748   fixes x :: "'a::real_normed_vector"

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

   750   by (simp add: dist_norm)

   751

   752 lemma mem_cball_0:

   753   fixes x :: "'a::real_normed_vector"

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

   755   by (simp add: dist_norm)

   756

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

   758   by simp

   759

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

   761   by simp

   762

   763 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e"

   764   by (simp add: subset_eq)

   765

   766 lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"

   767   by (simp add: subset_eq)

   768

   769 lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"

   770   by (simp add: subset_eq)

   771

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

   773   by (simp add: set_eq_iff) arith

   774

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

   776   by (simp add: set_eq_iff)

   777

   778 lemma diff_less_iff:

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

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

   781   "a - b < c \<longleftrightarrow> a < c + b" "a - b > c \<longleftrightarrow> a > c + b"

   782   by arith+

   783

   784 lemma diff_le_iff:

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

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

   787   "a - b \<le> c \<longleftrightarrow> a \<le> c + b"

   788   "a - b \<ge> c \<longleftrightarrow> a \<ge> c + b"

   789   by arith+

   790

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

   792 proof -

   793   have "open (dist x - {..<e})"

   794     by (intro open_vimage open_lessThan continuous_intros)

   795   also have "dist x - {..<e} = ball x e"

   796     by auto

   797   finally show ?thesis .

   798 qed

   799

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

   801   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   802

   803 lemma openE[elim?]:

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

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

   806   using assms unfolding open_contains_ball by auto

   807

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

   809   by (metis open_contains_ball subset_eq centre_in_ball)

   810

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

   812   unfolding mem_ball set_eq_iff

   813   apply (simp add: not_less)

   814   apply (metis zero_le_dist order_trans dist_self)

   815   done

   816

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

   818

   819 lemma euclidean_dist_l2:

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

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

   822   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   824

   825

   826 subsection {* Boxes *}

   827

   828 abbreviation One :: "'a::euclidean_space"

   829   where "One \<equiv> \<Sum>Basis"

   830

   831 definition (in euclidean_space) eucl_less (infix "<e" 50)

   832   where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"

   833

   834 definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"

   835 definition "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"

   836

   837 lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"

   838   and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"

   839   and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"

   840     "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"

   841   by (auto simp: box_eucl_less eucl_less_def cbox_def)

   842

   843 lemma mem_box_real[simp]:

   844   "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"

   845   "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"

   846   by (auto simp: mem_box)

   847

   848 lemma box_real[simp]:

   849   fixes a b:: real

   850   shows "box a b = {a <..< b}" "cbox a b = {a .. b}"

   851   by auto

   852

   853 lemma box_Int_box:

   854   fixes a :: "'a::euclidean_space"

   855   shows "box a b \<inter> box c d =

   856     box (\<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)"

   857   unfolding set_eq_iff and Int_iff and mem_box by auto

   858

   859 lemma rational_boxes:

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

   861   assumes "e > 0"

   862   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"

   863 proof -

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

   865   then have e: "e' > 0"

   866     using assms by (auto simp: DIM_positive)

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

   868   proof

   869     fix i

   870     from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e

   871     show "?th i" by auto

   872   qed

   873   from choice[OF this] obtain a where

   874     a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..

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

   876   proof

   877     fix i

   878     from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e

   879     show "?th i" by auto

   880   qed

   881   from choice[OF this] obtain b where

   882     b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..

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

   884   show ?thesis

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

   886     fix y :: 'a

   887     assume *: "y \<in> box ?a ?b"

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

   889       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   891     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   892       fix i :: "'a"

   893       assume i: "i \<in> Basis"

   894       have "a i < y\<bullet>i \<and> y\<bullet>i < b i"

   895         using * i by (auto simp: box_def)

   896       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"

   897         using a by auto

   898       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"

   899         using b by auto

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

   901         by auto

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

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

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

   905         by (rule power_strict_mono) auto

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

   907         by (simp add: power_divide)

   908     qed auto

   909     also have "\<dots> = e"

   910       using 0 < e by (simp add: real_eq_of_nat)

   911     finally show "y \<in> ball x e"

   912       by (auto simp: ball_def)

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

   914 qed

   915

   916 lemma open_UNION_box:

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

   918   assumes "open M"

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

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

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

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

   923 proof -

   924   {

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

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

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

   928     moreover obtain a b where ab:

   929       "x \<in> box a b"

   930       "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"

   931       "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"

   932       "box a b \<subseteq> ball x e"

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

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

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

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

   937   }

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

   939 qed

   940

   941 lemma box_eq_empty:

   942   fixes a :: "'a::euclidean_space"

   943   shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)

   944     and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

   945 proof -

   946   {

   947     fix i x

   948     assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"

   949     then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"

   950       unfolding mem_box by (auto simp: box_def)

   951     then have "a\<bullet>i < b\<bullet>i" by auto

   952     then have False using as by auto

   953   }

   954   moreover

   955   {

   956     assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

   957     let ?x = "(1/2) *\<^sub>R (a + b)"

   958     {

   959       fix i :: 'a

   960       assume i: "i \<in> Basis"

   961       have "a\<bullet>i < b\<bullet>i"

   962         using as[THEN bspec[where x=i]] i by auto

   963       then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

   964         by (auto simp: inner_add_left)

   965     }

   966     then have "box a b \<noteq> {}"

   967       using mem_box(1)[of "?x" a b] by auto

   968   }

   969   ultimately show ?th1 by blast

   970

   971   {

   972     fix i x

   973     assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"

   974     then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

   975       unfolding mem_box by auto

   976     then have "a\<bullet>i \<le> b\<bullet>i" by auto

   977     then have False using as by auto

   978   }

   979   moreover

   980   {

   981     assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

   982     let ?x = "(1/2) *\<^sub>R (a + b)"

   983     {

   984       fix i :: 'a

   985       assume i:"i \<in> Basis"

   986       have "a\<bullet>i \<le> b\<bullet>i"

   987         using as[THEN bspec[where x=i]] i by auto

   988       then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"

   989         by (auto simp: inner_add_left)

   990     }

   991     then have "cbox a b \<noteq> {}"

   992       using mem_box(2)[of "?x" a b] by auto

   993   }

   994   ultimately show ?th2 by blast

   995 qed

   996

   997 lemma box_ne_empty:

   998   fixes a :: "'a::euclidean_space"

   999   shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"

  1000   and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  1001   unfolding box_eq_empty[of a b] by fastforce+

  1002

  1003 lemma

  1004   fixes a :: "'a::euclidean_space"

  1005   shows cbox_sing: "cbox a a = {a}"

  1006     and box_sing: "box a a = {}"

  1007   unfolding set_eq_iff mem_box eq_iff [symmetric]

  1008   by (auto intro!: euclidean_eqI[where 'a='a])

  1009      (metis all_not_in_conv nonempty_Basis)

  1010

  1011 lemma subset_box_imp:

  1012   fixes a :: "'a::euclidean_space"

  1013   shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"

  1014     and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"

  1015     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"

  1016      and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"

  1017   unfolding subset_eq[unfolded Ball_def] unfolding mem_box

  1018   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  1019

  1020 lemma box_subset_cbox:

  1021   fixes a :: "'a::euclidean_space"

  1022   shows "box a b \<subseteq> cbox a b"

  1023   unfolding subset_eq [unfolded Ball_def] mem_box

  1024   by (fast intro: less_imp_le)

  1025

  1026 lemma subset_box:

  1027   fixes a :: "'a::euclidean_space"

  1028   shows "cbox c d \<subseteq> cbox 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)

  1029     and "cbox c d \<subseteq> box 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)

  1030     and "box c d \<subseteq> cbox 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)

  1031     and "box c d \<subseteq> box 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)

  1032 proof -

  1033   show ?th1

  1034     unfolding subset_eq and Ball_def and mem_box

  1035     by (auto intro: order_trans)

  1036   show ?th2

  1037     unfolding subset_eq and Ball_def and mem_box

  1038     by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)

  1039   {

  1040     assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  1041     then have "box c d \<noteq> {}"

  1042       unfolding box_eq_empty by auto

  1043     fix i :: 'a

  1044     assume i: "i \<in> Basis"

  1045     (** TODO combine the following two parts as done in the HOL_light version. **)

  1046     {

  1047       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"

  1048       assume as2: "a\<bullet>i > c\<bullet>i"

  1049       {

  1050         fix j :: 'a

  1051         assume j: "j \<in> Basis"

  1052         then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  1053           apply (cases "j = i")

  1054           using as(2)[THEN bspec[where x=j]] i

  1055           apply (auto simp add: as2)

  1056           done

  1057       }

  1058       then have "?x\<in>box c d"

  1059         using i unfolding mem_box by auto

  1060       moreover

  1061       have "?x \<notin> cbox a b"

  1062         unfolding mem_box

  1063         apply auto

  1064         apply (rule_tac x=i in bexI)

  1065         using as(2)[THEN bspec[where x=i]] and as2 i

  1066         apply auto

  1067         done

  1068       ultimately have False using as by auto

  1069     }

  1070     then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto

  1071     moreover

  1072     {

  1073       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"

  1074       assume as2: "b\<bullet>i < d\<bullet>i"

  1075       {

  1076         fix j :: 'a

  1077         assume "j\<in>Basis"

  1078         then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  1079           apply (cases "j = i")

  1080           using as(2)[THEN bspec[where x=j]]

  1081           apply (auto simp add: as2)

  1082           done

  1083       }

  1084       then have "?x\<in>box c d"

  1085         unfolding mem_box by auto

  1086       moreover

  1087       have "?x\<notin>cbox a b"

  1088         unfolding mem_box

  1089         apply auto

  1090         apply (rule_tac x=i in bexI)

  1091         using as(2)[THEN bspec[where x=i]] and as2 using i

  1092         apply auto

  1093         done

  1094       ultimately have False using as by auto

  1095     }

  1096     then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto

  1097     ultimately

  1098     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  1099   } note part1 = this

  1100   show ?th3

  1101     unfolding subset_eq and Ball_def and mem_box

  1102     apply (rule, rule, rule, rule)

  1103     apply (rule part1)

  1104     unfolding subset_eq and Ball_def and mem_box

  1105     prefer 4

  1106     apply auto

  1107     apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+

  1108     done

  1109   {

  1110     assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  1111     fix i :: 'a

  1112     assume i:"i\<in>Basis"

  1113     from as(1) have "box c d \<subseteq> cbox a b"

  1114       using box_subset_cbox[of a b] by auto

  1115     then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"

  1116       using part1 and as(2) using i by auto

  1117   } note * = this

  1118   show ?th4

  1119     unfolding subset_eq and Ball_def and mem_box

  1120     apply (rule, rule, rule, rule)

  1121     apply (rule *)

  1122     unfolding subset_eq and Ball_def and mem_box

  1123     prefer 4

  1124     apply auto

  1125     apply (erule_tac x=xa in allE, simp)+

  1126     done

  1127 qed

  1128

  1129 lemma inter_interval:

  1130   fixes a :: "'a::euclidean_space"

  1131   shows "cbox a b \<inter> cbox c d =

  1132     cbox (\<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)"

  1133   unfolding set_eq_iff and Int_iff and mem_box

  1134   by auto

  1135

  1136 lemma disjoint_interval:

  1137   fixes a::"'a::euclidean_space"

  1138   shows "cbox a b \<inter> cbox 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)

  1139     and "cbox a b \<inter> box 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)

  1140     and "box a b \<inter> cbox 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)

  1141     and "box a b \<inter> box 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)

  1142 proof -

  1143   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"

  1144   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  1145       (\<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)"

  1146     by blast

  1147   note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)

  1148   show ?th1 unfolding * by (intro **) auto

  1149   show ?th2 unfolding * by (intro **) auto

  1150   show ?th3 unfolding * by (intro **) auto

  1151   show ?th4 unfolding * by (intro **) auto

  1152 qed

  1153

  1154 lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"

  1155 proof -

  1156   {

  1157     fix x b :: 'a

  1158     assume [simp]: "b \<in> Basis"

  1159     have "\<bar>x \<bullet> b\<bar> \<le> real (natceiling \<bar>x \<bullet> b\<bar>)"

  1160       by (rule real_natceiling_ge)

  1161     also have "\<dots> \<le> real (natceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)Basis)))"

  1162       by (auto intro!: natceiling_mono)

  1163     also have "\<dots> < real (natceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)Basis)) + 1)"

  1164       by simp

  1165     finally have "\<bar>x \<bullet> b\<bar> < real (natceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)Basis)) + 1)" . }

  1166   then have "\<And>x::'a. \<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n"

  1167     by auto

  1168   moreover have "\<And>x b::'a. \<And>n::nat.  \<bar>x \<bullet> b\<bar> < real n \<longleftrightarrow> - real n < x \<bullet> b \<and> x \<bullet> b < real n"

  1169     by auto

  1170   ultimately show ?thesis

  1171     by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)

  1172 qed

  1173

  1174 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  1175

  1176 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  1177   (\<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)"

  1178

  1179 lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)

  1180   and is_interval_box: "is_interval (box a b)" (is ?th2)

  1181   unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff

  1182   by (meson order_trans le_less_trans less_le_trans less_trans)+

  1183

  1184 lemma is_interval_empty:

  1185  "is_interval {}"

  1186   unfolding is_interval_def

  1187   by simp

  1188

  1189 lemma is_interval_univ:

  1190  "is_interval UNIV"

  1191   unfolding is_interval_def

  1192   by simp

  1193

  1194 lemma mem_is_intervalI:

  1195   assumes "is_interval s"

  1196   assumes "a \<in> s" "b \<in> s"

  1197   assumes "\<And>i. i \<in> Basis \<Longrightarrow> 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"

  1198   shows "x \<in> s"

  1199   by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

  1200

  1201 lemma interval_subst:

  1202   fixes S::"'a::euclidean_space set"

  1203   assumes "is_interval S"

  1204   assumes "x \<in> S" "y j \<in> S"

  1205   assumes "j \<in> Basis"

  1206   shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"

  1207   by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

  1208

  1209 lemma mem_box_componentwiseI:

  1210   fixes S::"'a::euclidean_space set"

  1211   assumes "is_interval S"

  1212   assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i)  S)"

  1213   shows "x \<in> S"

  1214 proof -

  1215   from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"

  1216     by auto

  1217   with finite_Basis obtain s and bs::"'a list" where

  1218     s: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i = s i \<bullet> i" "\<And>i. i \<in> Basis \<Longrightarrow> s i \<in> S" and

  1219     bs: "set bs = Basis" "distinct bs"

  1220     by (metis finite_distinct_list)

  1221   from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast

  1222   def y \<equiv> "rec_list

  1223     (s j)

  1224     (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"

  1225   have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"

  1226     using bs by (auto simp add: s(1)[symmetric] euclidean_representation)

  1227   also have [symmetric]: "y bs = \<dots>"

  1228     using bs(2) bs(1)[THEN equalityD1]

  1229     by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])

  1230   also have "y bs \<in> S"

  1231     using bs(1)[THEN equalityD1]

  1232     apply (induct bs)

  1233     apply (auto simp: y_def j)

  1234     apply (rule interval_subst[OF assms(1)])

  1235     apply (auto simp: s)

  1236     done

  1237   finally show ?thesis .

  1238 qed

  1239

  1240

  1241 subsection{* Connectedness *}

  1242

  1243 lemma connected_local:

  1244  "connected S \<longleftrightarrow>

  1245   \<not> (\<exists>e1 e2.

  1246       openin (subtopology euclidean S) e1 \<and>

  1247       openin (subtopology euclidean S) e2 \<and>

  1248       S \<subseteq> e1 \<union> e2 \<and>

  1249       e1 \<inter> e2 = {} \<and>

  1250       e1 \<noteq> {} \<and>

  1251       e2 \<noteq> {})"

  1252   unfolding connected_def openin_open

  1253   by blast

  1254

  1255 lemma exists_diff:

  1256   fixes P :: "'a set \<Rightarrow> bool"

  1257   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1258 proof -

  1259   {

  1260     assume "?lhs"

  1261     then have ?rhs by blast

  1262   }

  1263   moreover

  1264   {

  1265     fix S

  1266     assume H: "P S"

  1267     have "S = - (- S)" by auto

  1268     with H have "P (- (- S))" by metis

  1269   }

  1270   ultimately show ?thesis by metis

  1271 qed

  1272

  1273 lemma connected_clopen: "connected S \<longleftrightarrow>

  1274   (\<forall>T. openin (subtopology euclidean S) T \<and>

  1275      closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1276 proof -

  1277   have "\<not> connected S \<longleftrightarrow>

  1278     (\<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> {})"

  1279     unfolding connected_def openin_open closedin_closed

  1280     by (metis double_complement)

  1281   then have th0: "connected S \<longleftrightarrow>

  1282     \<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> {})"

  1283     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

  1284     apply (simp add: closed_def)

  1285     apply metis

  1286     done

  1287   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'))"

  1288     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

  1289     unfolding connected_def openin_open closedin_closed by auto

  1290   {

  1291     fix e2

  1292     {

  1293       fix e1

  1294       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)"

  1295         by auto

  1296     }

  1297     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1298       by metis

  1299   }

  1300   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1301     by blast

  1302   then show ?thesis

  1303     unfolding th0 th1 by simp

  1304 qed

  1305

  1306

  1307 subsection{* Limit points *}

  1308

  1309 definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)

  1310   where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"

  1311

  1312 lemma islimptI:

  1313   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1314   shows "x islimpt S"

  1315   using assms unfolding islimpt_def by auto

  1316

  1317 lemma islimptE:

  1318   assumes "x islimpt S" and "x \<in> T" and "open T"

  1319   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

  1320   using assms unfolding islimpt_def by auto

  1321

  1322 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

  1323   unfolding islimpt_def eventually_at_topological by auto

  1324

  1325 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"

  1326   unfolding islimpt_def by fast

  1327

  1328 lemma islimpt_approachable:

  1329   fixes x :: "'a::metric_space"

  1330   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  1331   unfolding islimpt_iff_eventually eventually_at by fast

  1332

  1333 lemma islimpt_approachable_le:

  1334   fixes x :: "'a::metric_space"

  1335   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"

  1336   unfolding islimpt_approachable

  1337   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

  1338     THEN arg_cong [where f=Not]]

  1339   by (simp add: Bex_def conj_commute conj_left_commute)

  1340

  1341 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  1342   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

  1343

  1344 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

  1345   unfolding islimpt_def by blast

  1346

  1347 text {* A perfect space has no isolated points. *}

  1348

  1349 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  1350   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

  1351

  1352 lemma perfect_choose_dist:

  1353   fixes x :: "'a::{perfect_space, metric_space}"

  1354   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

  1355   using islimpt_UNIV [of x]

  1356   by (simp add: islimpt_approachable)

  1357

  1358 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  1359   unfolding closed_def

  1360   apply (subst open_subopen)

  1361   apply (simp add: islimpt_def subset_eq)

  1362   apply (metis ComplE ComplI)

  1363   done

  1364

  1365 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  1366   unfolding islimpt_def by auto

  1367

  1368 lemma finite_set_avoid:

  1369   fixes a :: "'a::metric_space"

  1370   assumes fS: "finite S"

  1371   shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"

  1372 proof (induct rule: finite_induct[OF fS])

  1373   case 1

  1374   then show ?case by (auto intro: zero_less_one)

  1375 next

  1376   case (2 x F)

  1377   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"

  1378     by blast

  1379   show ?case

  1380   proof (cases "x = a")

  1381     case True

  1382     then show ?thesis using d by auto

  1383   next

  1384     case False

  1385     let ?d = "min d (dist a x)"

  1386     have dp: "?d > 0"

  1387       using False d(1) using dist_nz by auto

  1388     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"

  1389       by auto

  1390     with dp False show ?thesis

  1391       by (auto intro!: exI[where x="?d"])

  1392   qed

  1393 qed

  1394

  1395 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  1396   by (simp add: islimpt_iff_eventually eventually_conj_iff)

  1397

  1398 lemma discrete_imp_closed:

  1399   fixes S :: "'a::metric_space set"

  1400   assumes e: "0 < e"

  1401     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  1402   shows "closed S"

  1403 proof -

  1404   {

  1405     fix x

  1406     assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

  1407     from e have e2: "e/2 > 0" by arith

  1408     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"

  1409       by blast

  1410     let ?m = "min (e/2) (dist x y) "

  1411     from e2 y(2) have mp: "?m > 0"

  1412       by (simp add: dist_nz[symmetric])

  1413     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"

  1414       by blast

  1415     have th: "dist z y < e" using z y

  1416       by (intro dist_triangle_lt [where z=x], simp)

  1417     from d[rule_format, OF y(1) z(1) th] y z

  1418     have False by (auto simp add: dist_commute)}

  1419   then show ?thesis

  1420     by (metis islimpt_approachable closed_limpt [where 'a='a])

  1421 qed

  1422

  1423

  1424 subsection {* Interior of a Set *}

  1425

  1426 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

  1427

  1428 lemma interiorI [intro?]:

  1429   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  1430   shows "x \<in> interior S"

  1431   using assms unfolding interior_def by fast

  1432

  1433 lemma interiorE [elim?]:

  1434   assumes "x \<in> interior S"

  1435   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  1436   using assms unfolding interior_def by fast

  1437

  1438 lemma open_interior [simp, intro]: "open (interior S)"

  1439   by (simp add: interior_def open_Union)

  1440

  1441 lemma interior_subset: "interior S \<subseteq> S"

  1442   by (auto simp add: interior_def)

  1443

  1444 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  1445   by (auto simp add: interior_def)

  1446

  1447 lemma interior_open: "open S \<Longrightarrow> interior S = S"

  1448   by (intro equalityI interior_subset interior_maximal subset_refl)

  1449

  1450 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  1451   by (metis open_interior interior_open)

  1452

  1453 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  1454   by (metis interior_maximal interior_subset subset_trans)

  1455

  1456 lemma interior_empty [simp]: "interior {} = {}"

  1457   using open_empty by (rule interior_open)

  1458

  1459 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  1460   using open_UNIV by (rule interior_open)

  1461

  1462 lemma interior_interior [simp]: "interior (interior S) = interior S"

  1463   using open_interior by (rule interior_open)

  1464

  1465 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  1466   by (auto simp add: interior_def)

  1467

  1468 lemma interior_unique:

  1469   assumes "T \<subseteq> S" and "open T"

  1470   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

  1471   shows "interior S = T"

  1472   by (intro equalityI assms interior_subset open_interior interior_maximal)

  1473

  1474 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

  1475   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1476     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1477

  1478 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1479   using open_contains_ball_eq [where S="interior S"]

  1480   by (simp add: open_subset_interior)

  1481

  1482 lemma interior_limit_point [intro]:

  1483   fixes x :: "'a::perfect_space"

  1484   assumes x: "x \<in> interior S"

  1485   shows "x islimpt S"

  1486   using x islimpt_UNIV [of x]

  1487   unfolding interior_def islimpt_def

  1488   apply (clarsimp, rename_tac T T')

  1489   apply (drule_tac x="T \<inter> T'" in spec)

  1490   apply (auto simp add: open_Int)

  1491   done

  1492

  1493 lemma interior_closed_Un_empty_interior:

  1494   assumes cS: "closed S"

  1495     and iT: "interior T = {}"

  1496   shows "interior (S \<union> T) = interior S"

  1497 proof

  1498   show "interior S \<subseteq> interior (S \<union> T)"

  1499     by (rule interior_mono) (rule Un_upper1)

  1500   show "interior (S \<union> T) \<subseteq> interior S"

  1501   proof

  1502     fix x

  1503     assume "x \<in> interior (S \<union> T)"

  1504     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1505     show "x \<in> interior S"

  1506     proof (rule ccontr)

  1507       assume "x \<notin> interior S"

  1508       with x \<in> R open R obtain y where "y \<in> R - S"

  1509         unfolding interior_def by fast

  1510       from open R closed S have "open (R - S)"

  1511         by (rule open_Diff)

  1512       from R \<subseteq> S \<union> T have "R - S \<subseteq> T"

  1513         by fast

  1514       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {} show False

  1515         unfolding interior_def by fast

  1516     qed

  1517   qed

  1518 qed

  1519

  1520 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1521 proof (rule interior_unique)

  1522   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1523     by (intro Sigma_mono interior_subset)

  1524   show "open (interior A \<times> interior B)"

  1525     by (intro open_Times open_interior)

  1526   fix T

  1527   assume "T \<subseteq> A \<times> B" and "open T"

  1528   then show "T \<subseteq> interior A \<times> interior B"

  1529   proof safe

  1530     fix x y

  1531     assume "(x, y) \<in> T"

  1532     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1533       using open T unfolding open_prod_def by fast

  1534     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1535       using T \<subseteq> A \<times> B by auto

  1536     then show "x \<in> interior A" and "y \<in> interior B"

  1537       by (auto intro: interiorI)

  1538   qed

  1539 qed

  1540

  1541

  1542 subsection {* Closure of a Set *}

  1543

  1544 definition "closure S = S \<union> {x | x. x islimpt S}"

  1545

  1546 lemma interior_closure: "interior S = - (closure (- S))"

  1547   unfolding interior_def closure_def islimpt_def by auto

  1548

  1549 lemma closure_interior: "closure S = - interior (- S)"

  1550   unfolding interior_closure by simp

  1551

  1552 lemma closed_closure[simp, intro]: "closed (closure S)"

  1553   unfolding closure_interior by (simp add: closed_Compl)

  1554

  1555 lemma closure_subset: "S \<subseteq> closure S"

  1556   unfolding closure_def by simp

  1557

  1558 lemma closure_hull: "closure S = closed hull S"

  1559   unfolding hull_def closure_interior interior_def by auto

  1560

  1561 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1562   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1563

  1564 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1565   unfolding closure_eq .

  1566

  1567 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1568   unfolding closure_hull by (rule hull_hull)

  1569

  1570 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1571   unfolding closure_hull by (rule hull_mono)

  1572

  1573 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1574   unfolding closure_hull by (rule hull_minimal)

  1575

  1576 lemma closure_unique:

  1577   assumes "S \<subseteq> T"

  1578     and "closed T"

  1579     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1580   shows "closure S = T"

  1581   using assms unfolding closure_hull by (rule hull_unique)

  1582

  1583 lemma closure_empty [simp]: "closure {} = {}"

  1584   using closed_empty by (rule closure_closed)

  1585

  1586 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1587   using closed_UNIV by (rule closure_closed)

  1588

  1589 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1590   unfolding closure_interior by simp

  1591

  1592 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1593   using closure_empty closure_subset[of S]

  1594   by blast

  1595

  1596 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1597   using closure_eq[of S] closure_subset[of S]

  1598   by simp

  1599

  1600 lemma open_inter_closure_eq_empty:

  1601   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1602   using open_subset_interior[of S "- T"]

  1603   using interior_subset[of "- T"]

  1604   unfolding closure_interior

  1605   by auto

  1606

  1607 lemma open_inter_closure_subset:

  1608   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1609 proof

  1610   fix x

  1611   assume as: "open S" "x \<in> S \<inter> closure T"

  1612   {

  1613     assume *: "x islimpt T"

  1614     have "x islimpt (S \<inter> T)"

  1615     proof (rule islimptI)

  1616       fix A

  1617       assume "x \<in> A" "open A"

  1618       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1619         by (simp_all add: open_Int)

  1620       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1621         by (rule islimptE)

  1622       then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1623         by simp_all

  1624       then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1625     qed

  1626   }

  1627   then show "x \<in> closure (S \<inter> T)" using as

  1628     unfolding closure_def

  1629     by blast

  1630 qed

  1631

  1632 lemma closure_complement: "closure (- S) = - interior S"

  1633   unfolding closure_interior by simp

  1634

  1635 lemma interior_complement: "interior (- S) = - closure S"

  1636   unfolding closure_interior by simp

  1637

  1638 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1639 proof (rule closure_unique)

  1640   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1641     by (intro Sigma_mono closure_subset)

  1642   show "closed (closure A \<times> closure B)"

  1643     by (intro closed_Times closed_closure)

  1644   fix T

  1645   assume "A \<times> B \<subseteq> T" and "closed T"

  1646   then show "closure A \<times> closure B \<subseteq> T"

  1647     apply (simp add: closed_def open_prod_def, clarify)

  1648     apply (rule ccontr)

  1649     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1650     apply (simp add: closure_interior interior_def)

  1651     apply (drule_tac x=C in spec)

  1652     apply (drule_tac x=D in spec)

  1653     apply auto

  1654     done

  1655 qed

  1656

  1657 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1658   unfolding closure_def using islimpt_punctured by blast

  1659

  1660

  1661 subsection {* Frontier (aka boundary) *}

  1662

  1663 definition "frontier S = closure S - interior S"

  1664

  1665 lemma frontier_closed: "closed (frontier S)"

  1666   by (simp add: frontier_def closed_Diff)

  1667

  1668 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1669   by (auto simp add: frontier_def interior_closure)

  1670

  1671 lemma frontier_straddle:

  1672   fixes a :: "'a::metric_space"

  1673   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))"

  1674   unfolding frontier_def closure_interior

  1675   by (auto simp add: mem_interior subset_eq ball_def)

  1676

  1677 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1678   by (metis frontier_def closure_closed Diff_subset)

  1679

  1680 lemma frontier_empty[simp]: "frontier {} = {}"

  1681   by (simp add: frontier_def)

  1682

  1683 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1684 proof -

  1685   {

  1686     assume "frontier S \<subseteq> S"

  1687     then have "closure S \<subseteq> S"

  1688       using interior_subset unfolding frontier_def by auto

  1689     then have "closed S"

  1690       using closure_subset_eq by auto

  1691   }

  1692   then show ?thesis using frontier_subset_closed[of S] ..

  1693 qed

  1694

  1695 lemma frontier_complement: "frontier (- S) = frontier S"

  1696   by (auto simp add: frontier_def closure_complement interior_complement)

  1697

  1698 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1699   using frontier_complement frontier_subset_eq[of "- S"]

  1700   unfolding open_closed by auto

  1701

  1702

  1703 subsection {* Filters and the eventually true'' quantifier *}

  1704

  1705 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1706     (infixr "indirection" 70)

  1707   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1708

  1709 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1710

  1711 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1712 proof

  1713   assume "trivial_limit (at a within S)"

  1714   then show "\<not> a islimpt S"

  1715     unfolding trivial_limit_def

  1716     unfolding eventually_at_topological

  1717     unfolding islimpt_def

  1718     apply (clarsimp simp add: set_eq_iff)

  1719     apply (rename_tac T, rule_tac x=T in exI)

  1720     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1721     done

  1722 next

  1723   assume "\<not> a islimpt S"

  1724   then show "trivial_limit (at a within S)"

  1725     unfolding trivial_limit_def eventually_at_topological islimpt_def

  1726     by metis

  1727 qed

  1728

  1729 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1730   using trivial_limit_within [of a UNIV] by simp

  1731

  1732 lemma trivial_limit_at:

  1733   fixes a :: "'a::perfect_space"

  1734   shows "\<not> trivial_limit (at a)"

  1735   by (rule at_neq_bot)

  1736

  1737 lemma trivial_limit_at_infinity:

  1738   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1739   unfolding trivial_limit_def eventually_at_infinity

  1740   apply clarsimp

  1741   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1742    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1743   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1744   apply (drule_tac x=UNIV in spec, simp)

  1745   done

  1746

  1747 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"

  1748   using islimpt_in_closure

  1749   by (metis trivial_limit_within)

  1750

  1751 text {* Some property holds "sufficiently close" to the limit point. *}

  1752

  1753 lemma eventually_at2:

  1754   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1755   unfolding eventually_at dist_nz by auto

  1756

  1757 lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"

  1758   unfolding trivial_limit_def

  1759   by (auto elim: eventually_rev_mp)

  1760

  1761 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1762   by simp

  1763

  1764 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1765   by (simp add: filter_eq_iff)

  1766

  1767 text{* Combining theorems for "eventually" *}

  1768

  1769 lemma eventually_rev_mono:

  1770   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1771   using eventually_mono [of P Q] by fast

  1772

  1773 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"

  1774   by (simp add: eventually_False)

  1775

  1776

  1777 subsection {* Limits *}

  1778

  1779 lemma Lim:

  1780   "(f ---> l) net \<longleftrightarrow>

  1781         trivial_limit net \<or>

  1782         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1783   unfolding tendsto_iff trivial_limit_eq by auto

  1784

  1785 text{* Show that they yield usual definitions in the various cases. *}

  1786

  1787 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1788     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"

  1789   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1790

  1791 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1792     (\<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)"

  1793   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1794

  1795 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1796     (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"

  1797   by (auto simp add: tendsto_iff eventually_at2)

  1798

  1799 lemma Lim_at_infinity:

  1800   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"

  1801   by (auto simp add: tendsto_iff eventually_at_infinity)

  1802

  1803 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1804   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1805

  1806 text{* The expected monotonicity property. *}

  1807

  1808 lemma Lim_Un:

  1809   assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1810   shows "(f ---> l) (at x within (S \<union> T))"

  1811   using assms unfolding at_within_union by (rule filterlim_sup)

  1812

  1813 lemma Lim_Un_univ:

  1814   "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>

  1815     S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"

  1816   by (metis Lim_Un)

  1817

  1818 text{* Interrelations between restricted and unrestricted limits. *}

  1819

  1820 lemma Lim_at_within: (* FIXME: rename *)

  1821   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1822   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1823

  1824 lemma eventually_within_interior:

  1825   assumes "x \<in> interior S"

  1826   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"

  1827   (is "?lhs = ?rhs")

  1828 proof

  1829   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1830   {

  1831     assume "?lhs"

  1832     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1833       unfolding eventually_at_topological

  1834       by auto

  1835     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"

  1836       by auto

  1837     then show "?rhs"

  1838       unfolding eventually_at_topological by auto

  1839   next

  1840     assume "?rhs"

  1841     then show "?lhs"

  1842       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1843   }

  1844 qed

  1845

  1846 lemma at_within_interior:

  1847   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1848   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1849

  1850 lemma Lim_within_LIMSEQ:

  1851   fixes a :: "'a::first_countable_topology"

  1852   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1853   shows "(X ---> L) (at a within T)"

  1854   using assms unfolding tendsto_def [where l=L]

  1855   by (simp add: sequentially_imp_eventually_within)

  1856

  1857 lemma Lim_right_bound:

  1858   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1859     'b::{linorder_topology, conditionally_complete_linorder}"

  1860   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"

  1861     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1862   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1863 proof (cases "{x<..} \<inter> I = {}")

  1864   case True

  1865   then show ?thesis by simp

  1866 next

  1867   case False

  1868   show ?thesis

  1869   proof (rule order_tendstoI)

  1870     fix a

  1871     assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1872     {

  1873       fix y

  1874       assume "y \<in> {x<..} \<inter> I"

  1875       with False bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1876         by (auto intro!: cInf_lower bdd_belowI2 simp del: Inf_image_eq)

  1877       with a have "a < f y"

  1878         by (blast intro: less_le_trans)

  1879     }

  1880     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1881       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1882   next

  1883     fix a

  1884     assume "Inf (f  ({x<..} \<inter> I)) < a"

  1885     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"

  1886       by auto

  1887     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1888       unfolding eventually_at_right[OF x < y] by (metis less_imp_le le_less_trans mono)

  1889     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1890       unfolding eventually_at_filter by eventually_elim simp

  1891   qed

  1892 qed

  1893

  1894 text{* Another limit point characterization. *}

  1895

  1896 lemma islimpt_sequential:

  1897   fixes x :: "'a::first_countable_topology"

  1898   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1899     (is "?lhs = ?rhs")

  1900 proof

  1901   assume ?lhs

  1902   from countable_basis_at_decseq[of x] obtain A where A:

  1903       "\<And>i. open (A i)"

  1904       "\<And>i. x \<in> A i"

  1905       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  1906     by blast

  1907   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1908   {

  1909     fix n

  1910     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1911       unfolding islimpt_def using A(1,2)[of n] by auto

  1912     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1913       unfolding f_def by (rule someI_ex)

  1914     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto

  1915   }

  1916   then have "\<forall>n. f n \<in> S - {x}" by auto

  1917   moreover have "(\<lambda>n. f n) ----> x"

  1918   proof (rule topological_tendstoI)

  1919     fix S

  1920     assume "open S" "x \<in> S"

  1921     from A(3)[OF this] \<And>n. f n \<in> A n

  1922     show "eventually (\<lambda>x. f x \<in> S) sequentially"

  1923       by (auto elim!: eventually_elim1)

  1924   qed

  1925   ultimately show ?rhs by fast

  1926 next

  1927   assume ?rhs

  1928   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"

  1929     by auto

  1930   show ?lhs

  1931     unfolding islimpt_def

  1932   proof safe

  1933     fix T

  1934     assume "open T" "x \<in> T"

  1935     from lim[THEN topological_tendstoD, OF this] f

  1936     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1937       unfolding eventually_sequentially by auto

  1938   qed

  1939 qed

  1940

  1941 lemma Lim_null:

  1942   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1943   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1944   by (simp add: Lim dist_norm)

  1945

  1946 lemma Lim_null_comparison:

  1947   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1948   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1949   shows "(f ---> 0) net"

  1950   using assms(2)

  1951 proof (rule metric_tendsto_imp_tendsto)

  1952   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1953     using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

  1954 qed

  1955

  1956 lemma Lim_transform_bound:

  1957   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1958     and g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1959   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"

  1960     and "(g ---> 0) net"

  1961   shows "(f ---> 0) net"

  1962   using assms(1) tendsto_norm_zero [OF assms(2)]

  1963   by (rule Lim_null_comparison)

  1964

  1965 text{* Deducing things about the limit from the elements. *}

  1966

  1967 lemma Lim_in_closed_set:

  1968   assumes "closed S"

  1969     and "eventually (\<lambda>x. f(x) \<in> S) net"

  1970     and "\<not> trivial_limit net" "(f ---> l) net"

  1971   shows "l \<in> S"

  1972 proof (rule ccontr)

  1973   assume "l \<notin> S"

  1974   with closed S have "open (- S)" "l \<in> - S"

  1975     by (simp_all add: open_Compl)

  1976   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1977     by (rule topological_tendstoD)

  1978   with assms(2) have "eventually (\<lambda>x. False) net"

  1979     by (rule eventually_elim2) simp

  1980   with assms(3) show "False"

  1981     by (simp add: eventually_False)

  1982 qed

  1983

  1984 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1985

  1986 lemma Lim_dist_ubound:

  1987   assumes "\<not>(trivial_limit net)"

  1988     and "(f ---> l) net"

  1989     and "eventually (\<lambda>x. dist a (f x) \<le> e) net"

  1990   shows "dist a l \<le> e"

  1991   using assms by (fast intro: tendsto_le tendsto_intros)

  1992

  1993 lemma Lim_norm_ubound:

  1994   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1995   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"

  1996   shows "norm(l) \<le> e"

  1997   using assms by (fast intro: tendsto_le tendsto_intros)

  1998

  1999 lemma Lim_norm_lbound:

  2000   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  2001   assumes "\<not> trivial_limit net"

  2002     and "(f ---> l) net"

  2003     and "eventually (\<lambda>x. e \<le> norm (f x)) net"

  2004   shows "e \<le> norm l"

  2005   using assms by (fast intro: tendsto_le tendsto_intros)

  2006

  2007 text{* Limit under bilinear function *}

  2008

  2009 lemma Lim_bilinear:

  2010   assumes "(f ---> l) net"

  2011     and "(g ---> m) net"

  2012     and "bounded_bilinear h"

  2013   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  2014   using bounded_bilinear h (f ---> l) net (g ---> m) net

  2015   by (rule bounded_bilinear.tendsto)

  2016

  2017 text{* These are special for limits out of the same vector space. *}

  2018

  2019 lemma Lim_within_id: "(id ---> a) (at a within s)"

  2020   unfolding id_def by (rule tendsto_ident_at)

  2021

  2022 lemma Lim_at_id: "(id ---> a) (at a)"

  2023   unfolding id_def by (rule tendsto_ident_at)

  2024

  2025 lemma Lim_at_zero:

  2026   fixes a :: "'a::real_normed_vector"

  2027     and l :: "'b::topological_space"

  2028   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"

  2029   using LIM_offset_zero LIM_offset_zero_cancel ..

  2030

  2031 text{* It's also sometimes useful to extract the limit point from the filter. *}

  2032

  2033 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  2034   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  2035

  2036 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  2037   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  2038

  2039 lemma netlimit_at:

  2040   fixes a :: "'a::{perfect_space,t2_space}"

  2041   shows "netlimit (at a) = a"

  2042   using netlimit_within [of a UNIV] by simp

  2043

  2044 lemma lim_within_interior:

  2045   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  2046   by (metis at_within_interior)

  2047

  2048 lemma netlimit_within_interior:

  2049   fixes x :: "'a::{t2_space,perfect_space}"

  2050   assumes "x \<in> interior S"

  2051   shows "netlimit (at x within S) = x"

  2052   using assms by (metis at_within_interior netlimit_at)

  2053

  2054 text{* Transformation of limit. *}

  2055

  2056 lemma Lim_transform:

  2057   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  2058   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  2059   shows "(g ---> l) net"

  2060   using tendsto_diff [OF assms(2) assms(1)] by simp

  2061

  2062 lemma Lim_transform_eventually:

  2063   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  2064   apply (rule topological_tendstoI)

  2065   apply (drule (2) topological_tendstoD)

  2066   apply (erule (1) eventually_elim2, simp)

  2067   done

  2068

  2069 lemma Lim_transform_within:

  2070   assumes "0 < d"

  2071     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2072     and "(f ---> l) (at x within S)"

  2073   shows "(g ---> l) (at x within S)"

  2074 proof (rule Lim_transform_eventually)

  2075   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  2076     using assms(1,2) by (auto simp: dist_nz eventually_at)

  2077   show "(f ---> l) (at x within S)" by fact

  2078 qed

  2079

  2080 lemma Lim_transform_at:

  2081   assumes "0 < d"

  2082     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2083     and "(f ---> l) (at x)"

  2084   shows "(g ---> l) (at x)"

  2085   using _ assms(3)

  2086 proof (rule Lim_transform_eventually)

  2087   show "eventually (\<lambda>x. f x = g x) (at x)"

  2088     unfolding eventually_at2

  2089     using assms(1,2) by auto

  2090 qed

  2091

  2092 text{* Common case assuming being away from some crucial point like 0. *}

  2093

  2094 lemma Lim_transform_away_within:

  2095   fixes a b :: "'a::t1_space"

  2096   assumes "a \<noteq> b"

  2097     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2098     and "(f ---> l) (at a within S)"

  2099   shows "(g ---> l) (at a within S)"

  2100 proof (rule Lim_transform_eventually)

  2101   show "(f ---> l) (at a within S)" by fact

  2102   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  2103     unfolding eventually_at_topological

  2104     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  2105 qed

  2106

  2107 lemma Lim_transform_away_at:

  2108   fixes a b :: "'a::t1_space"

  2109   assumes ab: "a\<noteq>b"

  2110     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2111     and fl: "(f ---> l) (at a)"

  2112   shows "(g ---> l) (at a)"

  2113   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  2114

  2115 text{* Alternatively, within an open set. *}

  2116

  2117 lemma Lim_transform_within_open:

  2118   assumes "open S" and "a \<in> S"

  2119     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  2120     and "(f ---> l) (at a)"

  2121   shows "(g ---> l) (at a)"

  2122 proof (rule Lim_transform_eventually)

  2123   show "eventually (\<lambda>x. f x = g x) (at a)"

  2124     unfolding eventually_at_topological

  2125     using assms(1,2,3) by auto

  2126   show "(f ---> l) (at a)" by fact

  2127 qed

  2128

  2129 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  2130

  2131 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  2132

  2133 lemma Lim_cong_within(*[cong add]*):

  2134   assumes "a = b"

  2135     and "x = y"

  2136     and "S = T"

  2137     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  2138   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  2139   unfolding tendsto_def eventually_at_topological

  2140   using assms by simp

  2141

  2142 lemma Lim_cong_at(*[cong add]*):

  2143   assumes "a = b" "x = y"

  2144     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  2145   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  2146   unfolding tendsto_def eventually_at_topological

  2147   using assms by simp

  2148

  2149 text{* Useful lemmas on closure and set of possible sequential limits.*}

  2150

  2151 lemma closure_sequential:

  2152   fixes l :: "'a::first_countable_topology"

  2153   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"

  2154   (is "?lhs = ?rhs")

  2155 proof

  2156   assume "?lhs"

  2157   moreover

  2158   {

  2159     assume "l \<in> S"

  2160     then have "?rhs" using tendsto_const[of l sequentially] by auto

  2161   }

  2162   moreover

  2163   {

  2164     assume "l islimpt S"

  2165     then have "?rhs" unfolding islimpt_sequential by auto

  2166   }

  2167   ultimately show "?rhs"

  2168     unfolding closure_def by auto

  2169 next

  2170   assume "?rhs"

  2171   then show "?lhs" unfolding closure_def islimpt_sequential by auto

  2172 qed

  2173

  2174 lemma closed_sequential_limits:

  2175   fixes S :: "'a::first_countable_topology set"

  2176   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  2177 by (metis closure_sequential closure_subset_eq subset_iff)

  2178

  2179 lemma closure_approachable:

  2180   fixes S :: "'a::metric_space set"

  2181   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  2182   apply (auto simp add: closure_def islimpt_approachable)

  2183   apply (metis dist_self)

  2184   done

  2185

  2186 lemma closed_approachable:

  2187   fixes S :: "'a::metric_space set"

  2188   shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  2189   by (metis closure_closed closure_approachable)

  2190

  2191 lemma closure_contains_Inf:

  2192   fixes S :: "real set"

  2193   assumes "S \<noteq> {}" "bdd_below S"

  2194   shows "Inf S \<in> closure S"

  2195 proof -

  2196   have *: "\<forall>x\<in>S. Inf S \<le> x"

  2197     using cInf_lower[of _ S] assms by metis

  2198   {

  2199     fix e :: real

  2200     assume "e > 0"

  2201     then have "Inf S < Inf S + e" by simp

  2202     with assms obtain x where "x \<in> S" "x < Inf S + e"

  2203       by (subst (asm) cInf_less_iff) auto

  2204     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  2205       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  2206   }

  2207   then show ?thesis unfolding closure_approachable by auto

  2208 qed

  2209

  2210 lemma closed_contains_Inf:

  2211   fixes S :: "real set"

  2212   shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"

  2213   by (metis closure_contains_Inf closure_closed assms)

  2214

  2215 lemma not_trivial_limit_within_ball:

  2216   "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  2217   (is "?lhs = ?rhs")

  2218 proof -

  2219   {

  2220     assume "?lhs"

  2221     {

  2222       fix e :: real

  2223       assume "e > 0"

  2224       then obtain y where "y \<in> S - {x}" and "dist y x < e"

  2225         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2226         by auto

  2227       then have "y \<in> S \<inter> ball x e - {x}"

  2228         unfolding ball_def by (simp add: dist_commute)

  2229       then have "S \<inter> ball x e - {x} \<noteq> {}" by blast

  2230     }

  2231     then have "?rhs" by auto

  2232   }

  2233   moreover

  2234   {

  2235     assume "?rhs"

  2236     {

  2237       fix e :: real

  2238       assume "e > 0"

  2239       then obtain y where "y \<in> S \<inter> ball x e - {x}"

  2240         using ?rhs by blast

  2241       then have "y \<in> S - {x}" and "dist y x < e"

  2242         unfolding ball_def by (simp_all add: dist_commute)

  2243       then have "\<exists>y \<in> S - {x}. dist y x < e"

  2244         by auto

  2245     }

  2246     then have "?lhs"

  2247       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2248       by auto

  2249   }

  2250   ultimately show ?thesis by auto

  2251 qed

  2252

  2253

  2254 subsection {* Infimum Distance *}

  2255

  2256 definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"

  2257

  2258 lemma bdd_below_infdist[intro, simp]: "bdd_below (dist xA)"

  2259   by (auto intro!: zero_le_dist)

  2260

  2261 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"

  2262   by (simp add: infdist_def)

  2263

  2264 lemma infdist_nonneg: "0 \<le> infdist x A"

  2265   by (auto simp add: infdist_def intro: cINF_greatest)

  2266

  2267 lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"

  2268   by (auto intro: cINF_lower simp add: infdist_def)

  2269

  2270 lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"

  2271   by (auto intro!: cINF_lower2 simp add: infdist_def)

  2272

  2273 lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"

  2274   by (auto intro!: antisym infdist_nonneg infdist_le2)

  2275

  2276 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  2277 proof (cases "A = {}")

  2278   case True

  2279   then show ?thesis by (simp add: infdist_def)

  2280 next

  2281   case False

  2282   then obtain a where "a \<in> A" by auto

  2283   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  2284   proof (rule cInf_greatest)

  2285     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  2286       by simp

  2287     fix d

  2288     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  2289     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  2290       by auto

  2291     show "infdist x A \<le> d"

  2292       unfolding infdist_notempty[OF A \<noteq> {}]

  2293     proof (rule cINF_lower2)

  2294       show "a \<in> A" by fact

  2295       show "dist x a \<le> d"

  2296         unfolding d by (rule dist_triangle)

  2297     qed simp

  2298   qed

  2299   also have "\<dots> = dist x y + infdist y A"

  2300   proof (rule cInf_eq, safe)

  2301     fix a

  2302     assume "a \<in> A"

  2303     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  2304       by (auto intro: infdist_le)

  2305   next

  2306     fix i

  2307     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  2308     then have "i - dist x y \<le> infdist y A"

  2309       unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  2310       by (intro cINF_greatest) (auto simp: field_simps)

  2311     then show "i \<le> dist x y + infdist y A"

  2312       by simp

  2313   qed

  2314   finally show ?thesis by simp

  2315 qed

  2316

  2317 lemma in_closure_iff_infdist_zero:

  2318   assumes "A \<noteq> {}"

  2319   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2320 proof

  2321   assume "x \<in> closure A"

  2322   show "infdist x A = 0"

  2323   proof (rule ccontr)

  2324     assume "infdist x A \<noteq> 0"

  2325     with infdist_nonneg[of x A] have "infdist x A > 0"

  2326       by auto

  2327     then have "ball x (infdist x A) \<inter> closure A = {}"

  2328       apply auto

  2329       apply (metis x \<in> closure A closure_approachable dist_commute infdist_le not_less)

  2330       done

  2331     then have "x \<notin> closure A"

  2332       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  2333     then show False using x \<in> closure A by simp

  2334   qed

  2335 next

  2336   assume x: "infdist x A = 0"

  2337   then obtain a where "a \<in> A"

  2338     by atomize_elim (metis all_not_in_conv assms)

  2339   show "x \<in> closure A"

  2340     unfolding closure_approachable

  2341     apply safe

  2342   proof (rule ccontr)

  2343     fix e :: real

  2344     assume "e > 0"

  2345     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  2346     then have "infdist x A \<ge> e" using a \<in> A

  2347       unfolding infdist_def

  2348       by (force simp: dist_commute intro: cINF_greatest)

  2349     with x e > 0 show False by auto

  2350   qed

  2351 qed

  2352

  2353 lemma in_closed_iff_infdist_zero:

  2354   assumes "closed A" "A \<noteq> {}"

  2355   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  2356 proof -

  2357   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2358     by (rule in_closure_iff_infdist_zero) fact

  2359   with assms show ?thesis by simp

  2360 qed

  2361

  2362 lemma tendsto_infdist [tendsto_intros]:

  2363   assumes f: "(f ---> l) F"

  2364   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  2365 proof (rule tendstoI)

  2366   fix e ::real

  2367   assume "e > 0"

  2368   from tendstoD[OF f this]

  2369   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  2370   proof (eventually_elim)

  2371     fix x

  2372     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  2373     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  2374       by (simp add: dist_commute dist_real_def)

  2375     also assume "dist (f x) l < e"

  2376     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  2377   qed

  2378 qed

  2379

  2380 text{* Some other lemmas about sequences. *}

  2381

  2382 lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)

  2383   assumes "eventually (\<lambda>i. P i) sequentially"

  2384   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  2385   using assms by (rule eventually_sequentially_seg [THEN iffD2])

  2386

  2387 lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)

  2388   "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  2389   apply (erule filterlim_compose)

  2390   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)

  2391   apply arith

  2392   done

  2393

  2394 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  2395   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

  2396

  2397 subsection {* More properties of closed balls *}

  2398

  2399 lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *)

  2400   assumes "closed s" and "continuous_on UNIV f"

  2401   shows "closed (vimage f s)"

  2402   using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]

  2403   by simp

  2404

  2405 lemma closed_cball: "closed (cball x e)"

  2406 proof -

  2407   have "closed (dist x - {..e})"

  2408     by (intro closed_vimage closed_atMost continuous_intros)

  2409   also have "dist x - {..e} = cball x e"

  2410     by auto

  2411   finally show ?thesis .

  2412 qed

  2413

  2414 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  2415 proof -

  2416   {

  2417     fix x and e::real

  2418     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  2419     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  2420   }

  2421   moreover

  2422   {

  2423     fix x and e::real

  2424     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  2425     then have "\<exists>d>0. ball x d \<subseteq> S"

  2426       unfolding subset_eq

  2427       apply(rule_tac x="e/2" in exI)

  2428       apply auto

  2429       done

  2430   }

  2431   ultimately show ?thesis

  2432     unfolding open_contains_ball by auto

  2433 qed

  2434

  2435 lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  2436   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  2437

  2438 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  2439   apply (simp add: interior_def, safe)

  2440   apply (force simp add: open_contains_cball)

  2441   apply (rule_tac x="ball x e" in exI)

  2442   apply (simp add: subset_trans [OF ball_subset_cball])

  2443   done

  2444

  2445 lemma islimpt_ball:

  2446   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  2447   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"

  2448   (is "?lhs = ?rhs")

  2449 proof

  2450   assume "?lhs"

  2451   {

  2452     assume "e \<le> 0"

  2453     then have *:"ball x e = {}"

  2454       using ball_eq_empty[of x e] by auto

  2455     have False using ?lhs

  2456       unfolding * using islimpt_EMPTY[of y] by auto

  2457   }

  2458   then have "e > 0" by (metis not_less)

  2459   moreover

  2460   have "y \<in> cball x e"

  2461     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  2462       ball_subset_cball[of x e] ?lhs

  2463     unfolding closed_limpt by auto

  2464   ultimately show "?rhs" by auto

  2465 next

  2466   assume "?rhs"

  2467   then have "e > 0" by auto

  2468   {

  2469     fix d :: real

  2470     assume "d > 0"

  2471     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2472     proof (cases "d \<le> dist x y")

  2473       case True

  2474       then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2475       proof (cases "x = y")

  2476         case True

  2477         then have False

  2478           using d \<le> dist x y d>0 by auto

  2479         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2480           by auto

  2481       next

  2482         case False

  2483         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

  2484           norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2485           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

  2486           by auto

  2487         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  2488           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

  2489           unfolding scaleR_minus_left scaleR_one

  2490           by (auto simp add: norm_minus_commute)

  2491         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  2492           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  2493           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm]

  2494           by auto

  2495         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs

  2496           by (auto simp add: dist_norm)

  2497         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0

  2498           by auto

  2499         moreover

  2500         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2501           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff

  2502           by (auto simp add: dist_commute)

  2503         moreover

  2504         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

  2505           unfolding dist_norm

  2506           apply simp

  2507           unfolding norm_minus_cancel

  2508           using d > 0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2509           unfolding dist_norm

  2510           apply auto

  2511           done

  2512         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2513           apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

  2514           apply auto

  2515           done

  2516       qed

  2517     next

  2518       case False

  2519       then have "d > dist x y" by auto

  2520       show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

  2521       proof (cases "x = y")

  2522         case True

  2523         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2524           using perfect_choose_dist[of "min e d" y]

  2525           using d > 0 e>0 by auto

  2526         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2527           unfolding x = y

  2528           using z \<noteq> y **

  2529           apply (rule_tac x=z in bexI)

  2530           apply (auto simp add: dist_commute)

  2531           done

  2532       next

  2533         case False

  2534         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2535           using d>0 d > dist x y ?rhs

  2536           apply (rule_tac x=x in bexI)

  2537           apply auto

  2538           done

  2539       qed

  2540     qed

  2541   }

  2542   then show "?lhs"

  2543     unfolding mem_cball islimpt_approachable mem_ball by auto

  2544 qed

  2545

  2546 lemma closure_ball_lemma:

  2547   fixes x y :: "'a::real_normed_vector"

  2548   assumes "x \<noteq> y"

  2549   shows "y islimpt ball x (dist x y)"

  2550 proof (rule islimptI)

  2551   fix T

  2552   assume "y \<in> T" "open T"

  2553   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2554     unfolding open_dist by fast

  2555   (* choose point between x and y, within distance r of y. *)

  2556   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2557   def z \<equiv> "y + scaleR k (x - y)"

  2558   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2559     unfolding z_def by (simp add: algebra_simps)

  2560   have "dist z y < r"

  2561     unfolding z_def k_def using 0 < r

  2562     by (simp add: dist_norm min_def)

  2563   then have "z \<in> T"

  2564     using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2565   have "dist x z < dist x y"

  2566     unfolding z_def2 dist_norm

  2567     apply (simp add: norm_minus_commute)

  2568     apply (simp only: dist_norm [symmetric])

  2569     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2570     apply (rule mult_strict_right_mono)

  2571     apply (simp add: k_def zero_less_dist_iff 0 < r x \<noteq> y)

  2572     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2573     done

  2574   then have "z \<in> ball x (dist x y)"

  2575     by simp

  2576   have "z \<noteq> y"

  2577     unfolding z_def k_def using x \<noteq> y 0 < r

  2578     by (simp add: min_def)

  2579   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2580     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2581     by fast

  2582 qed

  2583

  2584 lemma closure_ball:

  2585   fixes x :: "'a::real_normed_vector"

  2586   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2587   apply (rule equalityI)

  2588   apply (rule closure_minimal)

  2589   apply (rule ball_subset_cball)

  2590   apply (rule closed_cball)

  2591   apply (rule subsetI, rename_tac y)

  2592   apply (simp add: le_less [where 'a=real])

  2593   apply (erule disjE)

  2594   apply (rule subsetD [OF closure_subset], simp)

  2595   apply (simp add: closure_def)

  2596   apply clarify

  2597   apply (rule closure_ball_lemma)

  2598   apply (simp add: zero_less_dist_iff)

  2599   done

  2600

  2601 (* In a trivial vector space, this fails for e = 0. *)

  2602 lemma interior_cball:

  2603   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2604   shows "interior (cball x e) = ball x e"

  2605 proof (cases "e \<ge> 0")

  2606   case False note cs = this

  2607   from cs have "ball x e = {}"

  2608     using ball_empty[of e x] by auto

  2609   moreover

  2610   {

  2611     fix y

  2612     assume "y \<in> cball x e"

  2613     then have False

  2614       unfolding mem_cball using dist_nz[of x y] cs by auto

  2615   }

  2616   then have "cball x e = {}" by auto

  2617   then have "interior (cball x e) = {}"

  2618     using interior_empty by auto

  2619   ultimately show ?thesis by blast

  2620 next

  2621   case True note cs = this

  2622   have "ball x e \<subseteq> cball x e"

  2623     using ball_subset_cball by auto

  2624   moreover

  2625   {

  2626     fix S y

  2627     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2628     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

  2629       unfolding open_dist by blast

  2630     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2631       using perfect_choose_dist [of d] by auto

  2632     have "xa \<in> S"

  2633       using d[THEN spec[where x = xa]]

  2634       using xa by (auto simp add: dist_commute)

  2635     then have xa_cball: "xa \<in> cball x e"

  2636       using as(1) by auto

  2637     then have "y \<in> ball x e"

  2638     proof (cases "x = y")

  2639       case True

  2640       then have "e > 0"

  2641         using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2642         by (auto simp add: dist_commute)

  2643       then show "y \<in> ball x e"

  2644         using x = y  by simp

  2645     next

  2646       case False

  2647       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

  2648         unfolding dist_norm

  2649         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2650       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2651         using d as(1)[unfolded subset_eq] by blast

  2652       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2653       hence **:"d / (2 * norm (y - x)) > 0"

  2654         unfolding zero_less_norm_iff[symmetric] using d>0 by auto

  2655       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

  2656         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2657         by (auto simp add: dist_norm algebra_simps)

  2658       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2659         by (auto simp add: algebra_simps)

  2660       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2661         using ** by auto

  2662       also have "\<dots> = (dist y x) + d/2"

  2663         using ** by (auto simp add: distrib_right dist_norm)

  2664       finally have "e \<ge> dist x y +d/2"

  2665         using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2666       then show "y \<in> ball x e"

  2667         unfolding mem_ball using d>0 by auto

  2668     qed

  2669   }

  2670   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

  2671     by auto

  2672   ultimately show ?thesis

  2673     using interior_unique[of "ball x e" "cball x e"]

  2674     using open_ball[of x e]

  2675     by auto

  2676 qed

  2677

  2678 lemma frontier_ball:

  2679   fixes a :: "'a::real_normed_vector"

  2680   shows "0 < e \<Longrightarrow> frontier(ball a e) = {x. dist a x = e}"

  2681   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2682   apply (simp add: set_eq_iff)

  2683   apply arith

  2684   done

  2685

  2686 lemma frontier_cball:

  2687   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2688   shows "frontier (cball a e) = {x. dist a x = e}"

  2689   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2690   apply (simp add: set_eq_iff)

  2691   apply arith

  2692   done

  2693

  2694 lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"

  2695   apply (simp add: set_eq_iff not_le)

  2696   apply (metis zero_le_dist dist_self order_less_le_trans)

  2697   done

  2698

  2699 lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"

  2700   by (simp add: cball_eq_empty)

  2701

  2702 lemma cball_eq_sing:

  2703   fixes x :: "'a::{metric_space,perfect_space}"

  2704   shows "cball x e = {x} \<longleftrightarrow> e = 0"

  2705 proof (rule linorder_cases)

  2706   assume e: "0 < e"

  2707   obtain a where "a \<noteq> x" "dist a x < e"

  2708     using perfect_choose_dist [OF e] by auto

  2709   then have "a \<noteq> x" "dist x a \<le> e"

  2710     by (auto simp add: dist_commute)

  2711   with e show ?thesis by (auto simp add: set_eq_iff)

  2712 qed auto

  2713

  2714 lemma cball_sing:

  2715   fixes x :: "'a::metric_space"

  2716   shows "e = 0 \<Longrightarrow> cball x e = {x}"

  2717   by (auto simp add: set_eq_iff)

  2718

  2719

  2720 subsection {* Boundedness *}

  2721

  2722   (* FIXME: This has to be unified with BSEQ!! *)

  2723 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2724   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2725

  2726 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2727   unfolding bounded_def subset_eq by auto

  2728

  2729 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2730   unfolding bounded_def

  2731   by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)

  2732

  2733 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2734   unfolding bounded_any_center [where a=0]

  2735   by (simp add: dist_norm)

  2736

  2737 lemma bounded_realI:

  2738   assumes "\<forall>x\<in>s. abs (x::real) \<le> B"

  2739   shows "bounded s"

  2740   unfolding bounded_def dist_real_def

  2741   by (metis abs_minus_commute assms diff_0_right)

  2742

  2743 lemma bounded_empty [simp]: "bounded {}"

  2744   by (simp add: bounded_def)

  2745

  2746 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"

  2747   by (metis bounded_def subset_eq)

  2748

  2749 lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"

  2750   by (metis bounded_subset interior_subset)

  2751

  2752 lemma bounded_closure[intro]:

  2753   assumes "bounded S"

  2754   shows "bounded (closure S)"

  2755 proof -

  2756   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2757     unfolding bounded_def by auto

  2758   {

  2759     fix y

  2760     assume "y \<in> closure S"

  2761     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2762       unfolding closure_sequential by auto

  2763     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2764     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2765       by (rule eventually_mono, simp add: f(1))

  2766     have "dist x y \<le> a"

  2767       apply (rule Lim_dist_ubound [of sequentially f])

  2768       apply (rule trivial_limit_sequentially)

  2769       apply (rule f(2))

  2770       apply fact

  2771       done

  2772   }

  2773   then show ?thesis

  2774     unfolding bounded_def by auto

  2775 qed

  2776

  2777 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2778   apply (simp add: bounded_def)

  2779   apply (rule_tac x=x in exI)

  2780   apply (rule_tac x=e in exI)

  2781   apply auto

  2782   done

  2783

  2784 lemma bounded_ball[simp,intro]: "bounded (ball x e)"

  2785   by (metis ball_subset_cball bounded_cball bounded_subset)

  2786

  2787 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2788   apply (auto simp add: bounded_def)

  2789   by (metis Un_iff add_le_cancel_left dist_triangle le_max_iff_disj max.order_iff)

  2790

  2791 lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"

  2792   by (induct rule: finite_induct[of F]) auto

  2793

  2794 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2795   by (induct set: finite) auto

  2796

  2797 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2798 proof -

  2799   have "\<forall>y\<in>{x}. dist x y \<le> 0"

  2800     by simp

  2801   then have "bounded {x}"

  2802     unfolding bounded_def by fast

  2803   then show ?thesis

  2804     by (metis insert_is_Un bounded_Un)

  2805 qed

  2806

  2807 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2808   by (induct set: finite) simp_all

  2809

  2810 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"

  2811   apply (simp add: bounded_iff)

  2812   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")

  2813   apply metis

  2814   apply arith

  2815   done

  2816

  2817 lemma Bseq_eq_bounded:

  2818   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  2819   shows "Bseq f \<longleftrightarrow> bounded (range f)"

  2820   unfolding Bseq_def bounded_pos by auto

  2821

  2822 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2823   by (metis Int_lower1 Int_lower2 bounded_subset)

  2824

  2825 lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"

  2826   by (metis Diff_subset bounded_subset)

  2827

  2828 lemma not_bounded_UNIV[simp, intro]:

  2829   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2830 proof (auto simp add: bounded_pos not_le)

  2831   obtain x :: 'a where "x \<noteq> 0"

  2832     using perfect_choose_dist [OF zero_less_one] by fast

  2833   fix b :: real

  2834   assume b: "b >0"

  2835   have b1: "b +1 \<ge> 0"

  2836     using b by simp

  2837   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2838     by (simp add: norm_sgn)

  2839   then show "\<exists>x::'a. b < norm x" ..

  2840 qed

  2841

  2842 lemma bounded_linear_image:

  2843   assumes "bounded S"

  2844     and "bounded_linear f"

  2845   shows "bounded (f  S)"

  2846 proof -

  2847   from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2848     unfolding bounded_pos by auto

  2849   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"

  2850     using bounded_linear.pos_bounded by (auto simp add: ac_simps)

  2851   {

  2852     fix x

  2853     assume "x \<in> S"

  2854     then have "norm x \<le> b"

  2855       using b by auto

  2856     then have "norm (f x) \<le> B * b"

  2857       using B(2)

  2858       apply (erule_tac x=x in allE)

  2859       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2860       done

  2861   }

  2862   then show ?thesis

  2863     unfolding bounded_pos

  2864     apply (rule_tac x="b*B" in exI)

  2865     using b B by (auto simp add: mult.commute)

  2866 qed

  2867

  2868 lemma bounded_scaling:

  2869   fixes S :: "'a::real_normed_vector set"

  2870   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2871   apply (rule bounded_linear_image)

  2872   apply assumption

  2873   apply (rule bounded_linear_scaleR_right)

  2874   done

  2875

  2876 lemma bounded_translation:

  2877   fixes S :: "'a::real_normed_vector set"

  2878   assumes "bounded S"

  2879   shows "bounded ((\<lambda>x. a + x)  S)"

  2880 proof -

  2881   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2882     unfolding bounded_pos by auto

  2883   {

  2884     fix x

  2885     assume "x \<in> S"

  2886     then have "norm (a + x) \<le> b + norm a"

  2887       using norm_triangle_ineq[of a x] b by auto

  2888   }

  2889   then show ?thesis

  2890     unfolding bounded_pos

  2891     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2892     by (auto intro!: exI[of _ "b + norm a"])

  2893 qed

  2894

  2895

  2896 text{* Some theorems on sups and infs using the notion "bounded". *}

  2897

  2898 lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"

  2899   by (simp add: bounded_iff)

  2900

  2901 lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"

  2902   by (auto simp: bounded_def bdd_above_def dist_real_def)

  2903      (metis abs_le_D1 abs_minus_commute diff_le_eq)

  2904

  2905 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"

  2906   by (auto simp: bounded_def bdd_below_def dist_real_def)

  2907      (metis abs_le_D1 add.commute diff_le_eq)

  2908

  2909 (* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)

  2910

  2911 lemma bounded_has_Sup:

  2912   fixes S :: "real set"

  2913   assumes "bounded S"

  2914     and "S \<noteq> {}"

  2915   shows "\<forall>x\<in>S. x \<le> Sup S"

  2916     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2917 proof

  2918   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2919     using assms by (metis cSup_least)

  2920 qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

  2921

  2922 lemma Sup_insert:

  2923   fixes S :: "real set"

  2924   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2925   by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)

  2926

  2927 lemma Sup_insert_finite:

  2928   fixes S :: "real set"

  2929   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2930   apply (rule Sup_insert)

  2931   apply (rule finite_imp_bounded)

  2932   apply simp

  2933   done

  2934

  2935 lemma bounded_has_Inf:

  2936   fixes S :: "real set"

  2937   assumes "bounded S"

  2938     and "S \<noteq> {}"

  2939   shows "\<forall>x\<in>S. x \<ge> Inf S"

  2940     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2941 proof

  2942   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2943     using assms by (metis cInf_greatest)

  2944 qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

  2945

  2946 lemma Inf_insert:

  2947   fixes S :: "real set"

  2948   shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2949   by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)

  2950

  2951 lemma Inf_insert_finite:

  2952   fixes S :: "real set"

  2953   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2954   apply (rule Inf_insert)

  2955   apply (rule finite_imp_bounded)

  2956   apply simp

  2957   done

  2958

  2959 subsection {* Compactness *}

  2960

  2961 subsubsection {* Bolzano-Weierstrass property *}

  2962

  2963 lemma heine_borel_imp_bolzano_weierstrass:

  2964   assumes "compact s"

  2965     and "infinite t"

  2966     and "t \<subseteq> s"

  2967   shows "\<exists>x \<in> s. x islimpt t"

  2968 proof (rule ccontr)

  2969   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2970   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)"

  2971     unfolding islimpt_def

  2972     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]

  2973     by auto

  2974   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2975     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2976     using f by auto

  2977   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"

  2978     by auto

  2979   {

  2980     fix x y

  2981     assume "x \<in> t" "y \<in> t" "f x = f y"

  2982     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2983       using f[THEN bspec[where x=x]] and t \<subseteq> s by auto

  2984     then have "x = y"

  2985       using f x = f y and f[THEN bspec[where x=y]] and y \<in> t and t \<subseteq> s

  2986       by auto

  2987   }

  2988   then have "inj_on f t"

  2989     unfolding inj_on_def by simp

  2990   then have "infinite (f  t)"

  2991     using assms(2) using finite_imageD by auto

  2992   moreover

  2993   {

  2994     fix x

  2995     assume "x \<in> t" "f x \<notin> g"

  2996     from g(3) assms(3) x \<in> t obtain h where "h \<in> g" and "x \<in> h"

  2997       by auto

  2998     then obtain y where "y \<in> s" "h = f y"

  2999       using g'[THEN bspec[where x=h]] by auto

  3000     then have "y = x"

  3001       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y]

  3002       by auto

  3003     then have False

  3004       using f x \<notin> g h \<in> g unfolding h = f y

  3005       by auto

  3006   }

  3007   then have "f  t \<subseteq> g" by auto

  3008   ultimately show False

  3009     using g(2) using finite_subset by auto

  3010 qed

  3011

  3012 lemma acc_point_range_imp_convergent_subsequence:

  3013   fixes l :: "'a :: first_countable_topology"

  3014   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  3015   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3016 proof -

  3017   from countable_basis_at_decseq[of l]

  3018   obtain A where A:

  3019       "\<And>i. open (A i)"

  3020       "\<And>i. l \<in> A i"

  3021       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3022     by blast

  3023   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  3024   {

  3025     fix n i

  3026     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  3027       using l A by auto

  3028     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  3029       unfolding ex_in_conv by (intro notI) simp

  3030     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  3031       by auto

  3032     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  3033       by (auto simp: not_le)

  3034     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  3035       unfolding s_def by (auto intro: someI2_ex)

  3036   }

  3037   note s = this

  3038   def r \<equiv> "rec_nat (s 0 0) s"

  3039   have "subseq r"

  3040     by (auto simp: r_def s subseq_Suc_iff)

  3041   moreover

  3042   have "(\<lambda>n. f (r n)) ----> l"

  3043   proof (rule topological_tendstoI)

  3044     fix S

  3045     assume "open S" "l \<in> S"

  3046     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3047       by auto

  3048     moreover

  3049     {

  3050       fix i

  3051       assume "Suc 0 \<le> i"

  3052       then have "f (r i) \<in> A i"

  3053         by (cases i) (simp_all add: r_def s)

  3054     }

  3055     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  3056       by (auto simp: eventually_sequentially)

  3057     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  3058       by eventually_elim auto

  3059   qed

  3060   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3061     by (auto simp: convergent_def comp_def)

  3062 qed

  3063

  3064 lemma sequence_infinite_lemma:

  3065   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  3066   assumes "\<forall>n. f n \<noteq> l"

  3067     and "(f ---> l) sequentially"

  3068   shows "infinite (range f)"

  3069 proof

  3070   assume "finite (range f)"

  3071   then have "closed (range f)"

  3072     by (rule finite_imp_closed)

  3073   then have "open (- range f)"

  3074     by (rule open_Compl)

  3075   from assms(1) have "l \<in> - range f"

  3076     by auto

  3077   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  3078     using open (- range f) l \<in> - range f

  3079     by (rule topological_tendstoD)

  3080   then show False

  3081     unfolding eventually_sequentially

  3082     by auto

  3083 qed

  3084

  3085 lemma closure_insert:

  3086   fixes x :: "'a::t1_space"

  3087   shows "closure (insert x s) = insert x (closure s)"

  3088   apply (rule closure_unique)

  3089   apply (rule insert_mono [OF closure_subset])

  3090   apply (rule closed_insert [OF closed_closure])

  3091   apply (simp add: closure_minimal)

  3092   done

  3093

  3094 lemma islimpt_insert:

  3095   fixes x :: "'a::t1_space"

  3096   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  3097 proof

  3098   assume *: "x islimpt (insert a s)"

  3099   show "x islimpt s"

  3100   proof (rule islimptI)

  3101     fix t

  3102     assume t: "x \<in> t" "open t"

  3103     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  3104     proof (cases "x = a")

  3105       case True

  3106       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  3107         using * t by (rule islimptE)

  3108       with x = a show ?thesis by auto

  3109     next

  3110       case False

  3111       with t have t': "x \<in> t - {a}" "open (t - {a})"

  3112         by (simp_all add: open_Diff)

  3113       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  3114         using * t' by (rule islimptE)

  3115       then show ?thesis by auto

  3116     qed

  3117   qed

  3118 next

  3119   assume "x islimpt s"

  3120   then show "x islimpt (insert a s)"

  3121     by (rule islimpt_subset) auto

  3122 qed

  3123

  3124 lemma islimpt_finite:

  3125   fixes x :: "'a::t1_space"

  3126   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  3127   by (induct set: finite) (simp_all add: islimpt_insert)

  3128

  3129 lemma islimpt_union_finite:

  3130   fixes x :: "'a::t1_space"

  3131   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3132   by (simp add: islimpt_Un islimpt_finite)

  3133

  3134 lemma islimpt_eq_acc_point:

  3135   fixes l :: "'a :: t1_space"

  3136   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  3137 proof (safe intro!: islimptI)

  3138   fix U

  3139   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  3140   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  3141     by (auto intro: finite_imp_closed)

  3142   then show False

  3143     by (rule islimptE) auto

  3144 next

  3145   fix T

  3146   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  3147   then have "infinite (T \<inter> S - {l})"

  3148     by auto

  3149   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  3150     unfolding ex_in_conv by (intro notI) simp

  3151   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  3152     by auto

  3153 qed

  3154

  3155 lemma islimpt_range_imp_convergent_subsequence:

  3156   fixes l :: "'a :: {t1_space, first_countable_topology}"

  3157   assumes l: "l islimpt (range f)"

  3158   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3159   using l unfolding islimpt_eq_acc_point

  3160   by (rule acc_point_range_imp_convergent_subsequence)

  3161

  3162 lemma sequence_unique_limpt:

  3163   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3164   assumes "(f ---> l) sequentially"

  3165     and "l' islimpt (range f)"

  3166   shows "l' = l"

  3167 proof (rule ccontr)

  3168   assume "l' \<noteq> l"

  3169   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3170     using hausdorff [OF l' \<noteq> l] by auto

  3171   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3172     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3173   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3174     unfolding eventually_sequentially by auto

  3175

  3176   have "UNIV = {..<N} \<union> {N..}"

  3177     by auto

  3178   then have "l' islimpt (f  ({..<N} \<union> {N..}))"

  3179     using assms(2) by simp

  3180   then have "l' islimpt (f  {..<N} \<union> f  {N..})"

  3181     by (simp add: image_Un)

  3182   then have "l' islimpt (f  {N..})"

  3183     by (simp add: islimpt_union_finite)

  3184   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3185     using l' \<in> s open s by (rule islimptE)

  3186   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"

  3187     by auto

  3188   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t"

  3189     by simp

  3190   with s \<inter> t = {} show False

  3191     by simp

  3192 qed

  3193

  3194 lemma bolzano_weierstrass_imp_closed:

  3195   fixes s :: "'a::{first_countable_topology,t2_space} set"

  3196   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3197   shows "closed s"

  3198 proof -

  3199   {

  3200     fix x l

  3201     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3202     then have "l \<in> s"

  3203     proof (cases "\<forall>n. x n \<noteq> l")

  3204       case False

  3205       then show "l\<in>s" using as(1) by auto

  3206     next

  3207       case True note cas = this

  3208       with as(2) have "infinite (range x)"

  3209         using sequence_infinite_lemma[of x l] by auto

  3210       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  3211         using assms[THEN spec[where x="range x"]] as(1) by auto

  3212       then show "l\<in>s" using sequence_unique_limpt[of x l l']

  3213         using as cas by auto

  3214     qed

  3215   }

  3216   then show ?thesis

  3217     unfolding closed_sequential_limits by fast

  3218 qed

  3219

  3220 lemma compact_imp_bounded:

  3221   assumes "compact U"

  3222   shows "bounded U"

  3223 proof -

  3224   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  3225     using assms by auto

  3226   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  3227     by (rule compactE_image)

  3228   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  3229     by (simp add: bounded_UN)

  3230   then show "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  3231     by (rule bounded_subset)

  3232 qed

  3233

  3234 text{* In particular, some common special cases. *}

  3235

  3236 lemma compact_union [intro]:

  3237   assumes "compact s"

  3238     and "compact t"

  3239   shows " compact (s \<union> t)"

  3240 proof (rule compactI)

  3241   fix f

  3242   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  3243   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  3244     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3245   moreover

  3246   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  3247     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3248   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  3249     by (auto intro!: exI[of _ "s' \<union> t'"])

  3250 qed

  3251

  3252 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3253   by (induct set: finite) auto

  3254

  3255 lemma compact_UN [intro]:

  3256   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3257   unfolding SUP_def by (rule compact_Union) auto

  3258

  3259 lemma closed_inter_compact [intro]:

  3260   assumes "closed s"

  3261     and "compact t"

  3262   shows "compact (s \<inter> t)"

  3263   using compact_inter_closed [of t s] assms

  3264   by (simp add: Int_commute)

  3265

  3266 lemma compact_inter [intro]:

  3267   fixes s t :: "'a :: t2_space set"

  3268   assumes "compact s"

  3269     and "compact t"

  3270   shows "compact (s \<inter> t)"

  3271   using assms by (intro compact_inter_closed compact_imp_closed)

  3272

  3273 lemma compact_sing [simp]: "compact {a}"

  3274   unfolding compact_eq_heine_borel by auto

  3275

  3276 lemma compact_insert [simp]:

  3277   assumes "compact s"

  3278   shows "compact (insert x s)"

  3279 proof -

  3280   have "compact ({x} \<union> s)"

  3281     using compact_sing assms by (rule compact_union)

  3282   then show ?thesis by simp

  3283 qed

  3284

  3285 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  3286   by (induct set: finite) simp_all

  3287

  3288 lemma open_delete:

  3289   fixes s :: "'a::t1_space set"

  3290   shows "open s \<Longrightarrow> open (s - {x})"

  3291   by (simp add: open_Diff)

  3292

  3293 text{*Compactness expressed with filters*}

  3294

  3295 lemma closure_iff_nhds_not_empty:

  3296   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  3297 proof safe

  3298   assume x: "x \<in> closure X"

  3299   fix S A

  3300   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  3301   then have "x \<notin> closure (-S)"

  3302     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  3303   with x have "x \<in> closure X - closure (-S)"

  3304     by auto

  3305   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  3306     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  3307   finally have "X \<inter> S \<noteq> {}" by auto

  3308   then show False using X \<inter> A = {} S \<subseteq> A by auto

  3309 next

  3310   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  3311   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  3312   show "x \<in> closure X"

  3313     by (simp add: closure_subset open_Compl)

  3314 qed

  3315

  3316 lemma compact_filter:

  3317   "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))"

  3318 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  3319   fix F

  3320   assume "compact U"

  3321   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  3322   then have "U \<noteq> {}"

  3323     by (auto simp: eventually_False)

  3324

  3325   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  3326   then have "\<forall>z\<in>Z. closed z"

  3327     by auto

  3328   moreover

  3329   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  3330     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  3331   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  3332   proof (intro allI impI)

  3333     fix B assume "finite B" "B \<subseteq> Z"

  3334     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  3335       by (auto intro!: eventually_Ball_finite)

  3336     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  3337       by eventually_elim auto

  3338     with F show "U \<inter> \<Inter>B \<noteq> {}"

  3339       by (intro notI) (simp add: eventually_False)

  3340   qed

  3341   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  3342     using compact U unfolding compact_fip by blast

  3343   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"

  3344     by auto

  3345

  3346   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  3347     unfolding eventually_inf eventually_nhds

  3348   proof safe

  3349     fix P Q R S

  3350     assume "eventually R F" "open S" "x \<in> S"

  3351     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  3352     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  3353     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  3354     ultimately show False by (auto simp: set_eq_iff)

  3355   qed

  3356   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  3357     by (metis eventually_bot)

  3358 next

  3359   fix A

  3360   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  3361   def F \<equiv> "INF a:insert U A. principal a"

  3362   have "F \<noteq> bot"

  3363     unfolding F_def

  3364   proof (rule INF_filter_not_bot)

  3365     fix X assume "X \<subseteq> insert U A" "finite X"

  3366     moreover with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"

  3367       by auto

  3368     ultimately show "(INF a:X. principal a) \<noteq> bot"

  3369       by (auto simp add: INF_principal_finite principal_eq_bot_iff)

  3370   qed

  3371   moreover

  3372   have "F \<le> principal U"

  3373     unfolding F_def by auto

  3374   then have "eventually (\<lambda>x. x \<in> U) F"

  3375     by (auto simp: le_filter_def eventually_principal)

  3376   moreover

  3377   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)"

  3378   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  3379     by auto

  3380

  3381   { fix V assume "V \<in> A"

  3382     then have "F \<le> principal V"

  3383       unfolding F_def by (intro INF_lower2[of V]) auto

  3384     then have V: "eventually (\<lambda>x. x \<in> V) F"

  3385       by (auto simp: le_filter_def eventually_principal)

  3386     have "x \<in> closure V"

  3387       unfolding closure_iff_nhds_not_empty

  3388     proof (intro impI allI)

  3389       fix S A

  3390       assume "open S" "x \<in> S" "S \<subseteq> A"

  3391       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"

  3392         by (auto simp: eventually_nhds)

  3393       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  3394         by (auto simp: eventually_inf)

  3395       with x show "V \<inter> A \<noteq> {}"

  3396         by (auto simp del: Int_iff simp add: trivial_limit_def)

  3397     qed

  3398     then have "x \<in> V"

  3399       using V \<in> A A(1) by simp

  3400   }

  3401   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  3402   with U \<inter> \<Inter>A = {} show False by auto

  3403 qed

  3404

  3405 definition "countably_compact U \<longleftrightarrow>

  3406     (\<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))"

  3407

  3408 lemma countably_compactE:

  3409   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  3410   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  3411   using assms unfolding countably_compact_def by metis

  3412

  3413 lemma countably_compactI:

  3414   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')"

  3415   shows "countably_compact s"

  3416   using assms unfolding countably_compact_def by metis

  3417

  3418 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  3419   by (auto simp: compact_eq_heine_borel countably_compact_def)

  3420

  3421 lemma countably_compact_imp_compact:

  3422   assumes "countably_compact U"

  3423     and ccover: "countable B" "\<forall>b\<in>B. open b"

  3424     and 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"

  3425   shows "compact U"

  3426   using countably_compact U

  3427   unfolding compact_eq_heine_borel countably_compact_def

  3428 proof safe

  3429   fix A

  3430   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3431   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)"

  3432

  3433   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  3434   ultimately have "countable C" "\<forall>a\<in>C. open a"

  3435     unfolding C_def using ccover by auto

  3436   moreover

  3437   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  3438   proof safe

  3439     fix x a

  3440     assume "x \<in> U" "x \<in> a" "a \<in> A"

  3441     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"

  3442       by blast

  3443     with a \<in> A show "x \<in> \<Union>C"

  3444       unfolding C_def by auto

  3445   qed

  3446   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  3447   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  3448     using * by metis

  3449   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  3450     by (auto simp: C_def)

  3451   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"

  3452     unfolding bchoice_iff Bex_def ..

  3453   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3454     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  3455 qed

  3456

  3457 lemma countably_compact_imp_compact_second_countable:

  3458   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3459 proof (rule countably_compact_imp_compact)

  3460   fix T and x :: 'a

  3461   assume "open T" "x \<in> T"

  3462   from topological_basisE[OF is_basis this] obtain b where

  3463     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .

  3464   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  3465     by blast

  3466 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  3467

  3468 lemma countably_compact_eq_compact:

  3469   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3470   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  3471

  3472 subsubsection{* Sequential compactness *}

  3473

  3474 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"

  3475   where "seq_compact S \<longleftrightarrow>

  3476     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"

  3477

  3478 lemma seq_compactI:

  3479   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3480   shows "seq_compact S"

  3481   unfolding seq_compact_def using assms by fast

  3482

  3483 lemma seq_compactE:

  3484   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3485   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3486   using assms unfolding seq_compact_def by fast

  3487

  3488 lemma closed_sequentially: (* TODO: move upwards *)

  3489   assumes "closed s" and "\<forall>n. f n \<in> s" and "f ----> l"

  3490   shows "l \<in> s"

  3491 proof (rule ccontr)

  3492   assume "l \<notin> s"

  3493   with closed s and f ----> l have "eventually (\<lambda>n. f n \<in> - s) sequentially"

  3494     by (fast intro: topological_tendstoD)

  3495   with \<forall>n. f n \<in> s show "False"

  3496     by simp

  3497 qed

  3498

  3499 lemma seq_compact_inter_closed:

  3500   assumes "seq_compact s" and "closed t"

  3501   shows "seq_compact (s \<inter> t)"

  3502 proof (rule seq_compactI)

  3503   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"

  3504   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  3505     by simp_all

  3506   from seq_compact s and \<forall>n. f n \<in> s

  3507   obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) ----> l"

  3508     by (rule seq_compactE)

  3509   from \<forall>n. f n \<in> t have "\<forall>n. (f \<circ> r) n \<in> t"

  3510     by simp

  3511   from closed t and this and l have "l \<in> t"

  3512     by (rule closed_sequentially)

  3513   with l \<in> s and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3514     by fast

  3515 qed

  3516

  3517 lemma seq_compact_closed_subset:

  3518   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"

  3519   shows "seq_compact s"

  3520   using assms seq_compact_inter_closed [of t s] by (simp add: Int_absorb1)

  3521

  3522 lemma seq_compact_imp_countably_compact:

  3523   fixes U :: "'a :: first_countable_topology set"

  3524   assumes "seq_compact U"

  3525   shows "countably_compact U"

  3526 proof (safe intro!: countably_compactI)

  3527   fix A

  3528   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  3529   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  3530     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  3531   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3532   proof cases

  3533     assume "finite A"

  3534     with A show ?thesis by auto

  3535   next

  3536     assume "infinite A"

  3537     then have "A \<noteq> {}" by auto

  3538     show ?thesis

  3539     proof (rule ccontr)

  3540       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  3541       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"

  3542         by auto

  3543       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"

  3544         by metis

  3545       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  3546       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  3547         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  3548       then have "range X \<subseteq> U"

  3549         by auto

  3550       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"

  3551         by auto

  3552       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  3553       obtain n where "x \<in> from_nat_into A n" by auto

  3554       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  3555       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  3556         unfolding tendsto_def by (auto simp: comp_def)

  3557       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  3558         by (auto simp: eventually_sequentially)

  3559       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  3560         by auto

  3561       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  3562         by (auto intro!: exI[of _ "max n N"])

  3563       ultimately show False

  3564         by auto

  3565     qed

  3566   qed

  3567 qed

  3568

  3569 lemma compact_imp_seq_compact:

  3570   fixes U :: "'a :: first_countable_topology set"

  3571   assumes "compact U"

  3572   shows "seq_compact U"

  3573   unfolding seq_compact_def

  3574 proof safe

  3575   fix X :: "nat \<Rightarrow> 'a"

  3576   assume "\<forall>n. X n \<in> U"

  3577   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  3578     by (auto simp: eventually_filtermap)

  3579   moreover

  3580   have "filtermap X sequentially \<noteq> bot"

  3581     by (simp add: trivial_limit_def eventually_filtermap)

  3582   ultimately

  3583   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  3584     using compact U by (auto simp: compact_filter)

  3585

  3586   from countable_basis_at_decseq[of x]

  3587   obtain A where A:

  3588       "\<And>i. open (A i)"

  3589       "\<And>i. x \<in> A i"

  3590       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3591     by blast

  3592   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3593   {

  3594     fix n i

  3595     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3596     proof (rule ccontr)

  3597       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3598       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"

  3599         by auto

  3600       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3601         by (auto simp: eventually_filtermap eventually_sequentially)

  3602       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3603         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3604       ultimately have "eventually (\<lambda>x. False) ?F"

  3605         by (auto simp add: eventually_inf)

  3606       with x show False

  3607         by (simp add: eventually_False)

  3608     qed

  3609     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3610       unfolding s_def by (auto intro: someI2_ex)

  3611   }

  3612   note s = this

  3613   def r \<equiv> "rec_nat (s 0 0) s"

  3614   have "subseq r"

  3615     by (auto simp: r_def s subseq_Suc_iff)

  3616   moreover

  3617   have "(\<lambda>n. X (r n)) ----> x"

  3618   proof (rule topological_tendstoI)

  3619     fix S

  3620     assume "open S" "x \<in> S"

  3621     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3622       by auto

  3623     moreover

  3624     {

  3625       fix i

  3626       assume "Suc 0 \<le> i"

  3627       then have "X (r i) \<in> A i"

  3628         by (cases i) (simp_all add: r_def s)

  3629     }

  3630     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3631       by (auto simp: eventually_sequentially)

  3632     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3633       by eventually_elim auto

  3634   qed

  3635   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3636     using x \<in> U by (auto simp: convergent_def comp_def)

  3637 qed

  3638

  3639 lemma countably_compact_imp_acc_point:

  3640   assumes "countably_compact s"

  3641     and "countable t"

  3642     and "infinite t"

  3643     and "t \<subseteq> s"

  3644   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3645 proof (rule ccontr)

  3646   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3647   note countably_compact s

  3648   moreover have "\<forall>t\<in>C. open t"

  3649     by (auto simp: C_def)

  3650   moreover

  3651   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3652   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3653   have "s \<subseteq> \<Union>C"

  3654     using t \<subseteq> s

  3655     unfolding C_def Union_image_eq

  3656     apply (safe dest!: s)

  3657     apply (rule_tac a="U \<inter> t" in UN_I)

  3658     apply (auto intro!: interiorI simp add: finite_subset)

  3659     done

  3660   moreover

  3661   from countable t have "countable C"

  3662     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3663   ultimately

  3664   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"

  3665     by (rule countably_compactE)

  3666   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"

  3667     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3668     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3669   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3670     using interior_subset by blast

  3671   moreover have "finite (\<Union>E)"

  3672     using E by auto

  3673   ultimately show False using infinite t

  3674     by (auto simp: finite_subset)

  3675 qed

  3676

  3677 lemma countable_acc_point_imp_seq_compact:

  3678   fixes s :: "'a::first_countable_topology set"

  3679   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>

  3680     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3681   shows "seq_compact s"

  3682 proof -

  3683   {

  3684     fix f :: "nat \<Rightarrow> 'a"

  3685     assume f: "\<forall>n. f n \<in> s"

  3686     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3687     proof (cases "finite (range f)")

  3688       case True

  3689       obtain l where "infinite {n. f n = f l}"

  3690         using pigeonhole_infinite[OF _ True] by auto

  3691       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3692         using infinite_enumerate by blast

  3693       then have "subseq r \<and> (f \<circ> r) ----> f l"

  3694         by (simp add: fr o_def)

  3695       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3696         by auto

  3697     next

  3698       case False

  3699       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"

  3700         by auto

  3701       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3702       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3703         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3704       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3705     qed

  3706   }

  3707   then show ?thesis

  3708     unfolding seq_compact_def by auto

  3709 qed

  3710

  3711 lemma seq_compact_eq_countably_compact:

  3712   fixes U :: "'a :: first_countable_topology set"

  3713   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3714   using

  3715     countable_acc_point_imp_seq_compact

  3716     countably_compact_imp_acc_point

  3717     seq_compact_imp_countably_compact

  3718   by metis

  3719

  3720 lemma seq_compact_eq_acc_point:

  3721   fixes s :: "'a :: first_countable_topology set"

  3722   shows "seq_compact s \<longleftrightarrow>

  3723     (\<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)))"

  3724   using

  3725     countable_acc_point_imp_seq_compact[of s]

  3726     countably_compact_imp_acc_point[of s]

  3727     seq_compact_imp_countably_compact[of s]

  3728   by metis

  3729

  3730 lemma seq_compact_eq_compact:

  3731   fixes U :: "'a :: second_countable_topology set"

  3732   shows "seq_compact U \<longleftrightarrow> compact U"

  3733   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3734

  3735 lemma bolzano_weierstrass_imp_seq_compact:

  3736   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3737   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3738   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3739

  3740 subsubsection{* Totally bounded *}

  3741

  3742 lemma cauchy_def: "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  3743   unfolding Cauchy_def by metis

  3744

  3745 lemma seq_compact_imp_totally_bounded:

  3746   assumes "seq_compact s"

  3747   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"

  3748 proof -

  3749   { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"

  3750     let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"

  3751     have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"

  3752     proof (rule dependent_wellorder_choice)

  3753       fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"

  3754       then have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3755         using *[of "x  {0 ..< n}"] by (auto simp: subset_eq)

  3756       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3757         unfolding subset_eq by auto

  3758       show "\<exists>r. ?Q x n r"

  3759         using z by auto

  3760     qed simp

  3761     then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"

  3762       by blast

  3763     then obtain l r where "l \<in> s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3764       using assms by (metis seq_compact_def)

  3765     from this(3) have "Cauchy (x \<circ> r)"

  3766       using LIMSEQ_imp_Cauchy by auto

  3767     then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"

  3768       unfolding cauchy_def using e > 0 by blast

  3769     then have False

  3770       using x[of "r N" "r (N+1)"] r by (auto simp: subseq_def) }

  3771   then show ?thesis

  3772     by metis

  3773 qed

  3774

  3775 subsubsection{* Heine-Borel theorem *}

  3776

  3777 lemma seq_compact_imp_heine_borel:

  3778   fixes s :: "'a :: metric_space set"

  3779   assumes "seq_compact s"

  3780   shows "compact s"

  3781 proof -

  3782   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3783   obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"

  3784     unfolding choice_iff' ..

  3785   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3786   have "countably_compact s"

  3787     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3788   then show "compact s"

  3789   proof (rule countably_compact_imp_compact)

  3790     show "countable K"

  3791       unfolding K_def using f

  3792       by (auto intro: countable_finite countable_subset countable_rat

  3793                intro!: countable_image countable_SIGMA countable_UN)

  3794     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3795   next

  3796     fix T x

  3797     assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3798     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"

  3799       by auto

  3800     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"

  3801       by auto

  3802     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"

  3803       by auto

  3804     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3805       unfolding Union_image_eq by auto

  3806     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K"

  3807       by (auto simp: K_def)

  3808     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3809     proof (rule bexI[rotated], safe)

  3810       fix y

  3811       assume "y \<in> ball k r"

  3812       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3813         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3814       with ball x e \<subseteq> T show "y \<in> T"

  3815         by auto

  3816     next

  3817       show "x \<in> ball k r" by fact

  3818     qed

  3819   qed

  3820 qed

  3821

  3822 lemma compact_eq_seq_compact_metric:

  3823   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3824   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3825

  3826 lemma compact_def:

  3827   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3828    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"

  3829   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3830

  3831 subsubsection {* Complete the chain of compactness variants *}

  3832

  3833 lemma compact_eq_bolzano_weierstrass:

  3834   fixes s :: "'a::metric_space set"

  3835   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"

  3836   (is "?lhs = ?rhs")

  3837 proof

  3838   assume ?lhs

  3839   then show ?rhs

  3840     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3841 next

  3842   assume ?rhs

  3843   then show ?lhs

  3844     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3845 qed

  3846

  3847 lemma bolzano_weierstrass_imp_bounded:

  3848   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3849   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3850

  3851 subsection {* Metric spaces with the Heine-Borel property *}

  3852

  3853 text {*

  3854   A metric space (or topological vector space) is said to have the

  3855   Heine-Borel property if every closed and bounded subset is compact.

  3856 *}

  3857

  3858 class heine_borel = metric_space +

  3859   assumes bounded_imp_convergent_subsequence:

  3860     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3861

  3862 lemma bounded_closed_imp_seq_compact:

  3863   fixes s::"'a::heine_borel set"

  3864   assumes "bounded s"

  3865     and "closed s"

  3866   shows "seq_compact s"

  3867 proof (unfold seq_compact_def, clarify)

  3868   fix f :: "nat \<Rightarrow> 'a"

  3869   assume f: "\<forall>n. f n \<in> s"

  3870   with bounded s have "bounded (range f)"

  3871     by (auto intro: bounded_subset)

  3872   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3873     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3874   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"

  3875     by simp

  3876   have "l \<in> s" using closed s fr l

  3877     by (rule closed_sequentially)

  3878   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3879     using l \<in> s r l by blast

  3880 qed

  3881

  3882 lemma compact_eq_bounded_closed:

  3883   fixes s :: "'a::heine_borel set"

  3884   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"

  3885   (is "?lhs = ?rhs")

  3886 proof

  3887   assume ?lhs

  3888   then show ?rhs

  3889     using compact_imp_closed compact_imp_bounded

  3890     by blast

  3891 next

  3892   assume ?rhs

  3893   then show ?lhs

  3894     using bounded_closed_imp_seq_compact[of s]

  3895     unfolding compact_eq_seq_compact_metric

  3896     by auto

  3897 qed

  3898

  3899 (* TODO: is this lemma necessary? *)

  3900 lemma bounded_increasing_convergent:

  3901   fixes s :: "nat \<Rightarrow> real"

  3902   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3903   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3904   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3905

  3906 instance real :: heine_borel

  3907 proof

  3908   fix f :: "nat \<Rightarrow> real"

  3909   assume f: "bounded (range f)"

  3910   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3911     unfolding comp_def by (metis seq_monosub)

  3912   then have "Bseq (f \<circ> r)"

  3913     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3914   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3915     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3916 qed

  3917

  3918 lemma compact_lemma:

  3919   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3920   assumes "bounded (range f)"

  3921   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.

  3922     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3923 proof safe

  3924   fix d :: "'a set"

  3925   assume d: "d \<subseteq> Basis"

  3926   with finite_Basis have "finite d"

  3927     by (blast intro: finite_subset)

  3928   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3929     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3930   proof (induct d)

  3931     case empty

  3932     then show ?case

  3933       unfolding subseq_def by auto

  3934   next

  3935     case (insert k d)

  3936     have k[intro]: "k \<in> Basis"

  3937       using insert by auto

  3938     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)"

  3939       using bounded (range f)

  3940       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3941     obtain l1::"'a" and r1 where r1: "subseq r1"

  3942       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3943       using insert(3) using insert(4) by auto

  3944     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f"

  3945       by simp

  3946     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3947       by (metis (lifting) bounded_subset f' image_subsetI s')

  3948     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3949       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]

  3950       by (auto simp: o_def)

  3951     def r \<equiv> "r1 \<circ> r2"

  3952     have r:"subseq r"

  3953       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3954     moreover

  3955     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3956     {

  3957       fix e::real

  3958       assume "e > 0"

  3959       from lr1 e > 0 have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3960         by blast

  3961       from lr2 e > 0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3962         by (rule tendstoD)

  3963       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3964         by (rule eventually_subseq)

  3965       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3966         using N1' N2

  3967         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3968     }

  3969     ultimately show ?case by auto

  3970   qed

  3971 qed

  3972

  3973 instance euclidean_space \<subseteq> heine_borel

  3974 proof

  3975   fix f :: "nat \<Rightarrow> 'a"

  3976   assume f: "bounded (range f)"

  3977   then obtain l::'a and r where r: "subseq r"

  3978     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3979     using compact_lemma [OF f] by blast

  3980   {

  3981     fix e::real

  3982     assume "e > 0"

  3983     hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)

  3984     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"

  3985       by simp

  3986     moreover

  3987     {

  3988       fix n

  3989       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3990       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3991         apply (subst euclidean_dist_l2)

  3992         using zero_le_dist

  3993         apply (rule setL2_le_setsum)

  3994         done

  3995       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3996         apply (rule setsum_strict_mono)

  3997         using n

  3998         apply auto

  3999         done

  4000       finally have "dist (f (r n)) l < e"

  4001         by auto

  4002     }

  4003     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  4004       by (rule eventually_elim1)

  4005   }

  4006   then have *: "((f \<circ> r) ---> l) sequentially"

  4007     unfolding o_def tendsto_iff by simp

  4008   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4009     by auto

  4010 qed

  4011

  4012 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  4013   unfolding bounded_def

  4014   by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)

  4015

  4016 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  4017   unfolding bounded_def

  4018   by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)

  4019

  4020 instance prod :: (heine_borel, heine_borel) heine_borel

  4021 proof

  4022   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  4023   assume f: "bounded (range f)"

  4024   then have "bounded (fst  range f)"

  4025     by (rule bounded_fst)

  4026   then have s1: "bounded (range (fst \<circ> f))"

  4027     by (simp add: image_comp)

  4028   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  4029     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  4030   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  4031     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  4032   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  4033     using bounded_imp_convergent_subsequence [OF s2]

  4034     unfolding o_def by fast

  4035   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  4036     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  4037   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  4038     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  4039   have r: "subseq (r1 \<circ> r2)"

  4040     using r1 r2 unfolding subseq_def by simp

  4041   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4042     using l r by fast

  4043 qed

  4044

  4045 subsubsection {* Completeness *}

  4046

  4047 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  4048   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  4049

  4050 lemma completeI:

  4051   assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f ----> l"

  4052   shows "complete s"

  4053   using assms unfolding complete_def by fast

  4054

  4055 lemma completeE:

  4056   assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"

  4057   obtains l where "l \<in> s" and "f ----> l"

  4058   using assms unfolding complete_def by fast

  4059

  4060 lemma compact_imp_complete:

  4061   assumes "compact s"

  4062   shows "complete s"

  4063 proof -

  4064   {

  4065     fix f

  4066     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  4067     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  4068       using assms unfolding compact_def by blast

  4069

  4070     note lr' = seq_suble [OF lr(2)]

  4071     {

  4072       fix e :: real

  4073       assume "e > 0"

  4074       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"

  4075         unfolding cauchy_def

  4076         using e > 0

  4077         apply (erule_tac x="e/2" in allE)

  4078         apply auto

  4079         done

  4080       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  4081       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"

  4082         using e > 0 by auto

  4083       {

  4084         fix n :: nat

  4085         assume n: "n \<ge> max N M"

  4086         have "dist ((f \<circ> r) n) l < e/2"

  4087           using n M by auto

  4088         moreover have "r n \<ge> N"

  4089           using lr'[of n] n by auto

  4090         then have "dist (f n) ((f \<circ> r) n) < e / 2"

  4091           using N and n by auto

  4092         ultimately have "dist (f n) l < e"

  4093           using dist_triangle_half_r[of "f (r n)" "f n" e l]

  4094           by (auto simp add: dist_commute)

  4095       }

  4096       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  4097     }

  4098     then have "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  4099       unfolding LIMSEQ_def by auto

  4100   }

  4101   then show ?thesis unfolding complete_def by auto

  4102 qed

  4103

  4104 lemma nat_approx_posE:

  4105   fixes e::real

  4106   assumes "0 < e"

  4107   obtains n :: nat where "1 / (Suc n) < e"

  4108 proof atomize_elim

  4109   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  4110     by (rule divide_strict_left_mono) (auto simp: 0 < e)

  4111   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  4112     by (rule divide_left_mono) (auto simp: 0 < e)

  4113   also have "\<dots> = e" by simp

  4114   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  4115 qed

  4116

  4117 lemma compact_eq_totally_bounded:

  4118   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"

  4119     (is "_ \<longleftrightarrow> ?rhs")

  4120 proof

  4121   assume assms: "?rhs"

  4122   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)"

  4123     by (auto simp: choice_iff')

  4124

  4125   show "compact s"

  4126   proof cases

  4127     assume "s = {}"

  4128     then show "compact s" by (simp add: compact_def)

  4129   next

  4130     assume "s \<noteq> {}"

  4131     show ?thesis

  4132       unfolding compact_def

  4133     proof safe

  4134       fix f :: "nat \<Rightarrow> 'a"

  4135       assume f: "\<forall>n. f n \<in> s"

  4136

  4137       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  4138       then have [simp]: "\<And>n. 0 < e n" by auto

  4139       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)"

  4140       {

  4141         fix n U

  4142         assume "infinite {n. f n \<in> U}"

  4143         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  4144           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  4145         then obtain a where

  4146           "a \<in> k (e n)"

  4147           "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..

  4148         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  4149           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  4150         from someI_ex[OF this]

  4151         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  4152           unfolding B_def by auto

  4153       }

  4154       note B = this

  4155

  4156       def F \<equiv> "rec_nat (B 0 UNIV) B"

  4157       {

  4158         fix n

  4159         have "infinite {i. f i \<in> F n}"

  4160           by (induct n) (auto simp: F_def B)

  4161       }

  4162       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  4163         using B by (simp add: F_def)

  4164       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  4165         using decseq_SucI[of F] by (auto simp: decseq_def)

  4166

  4167       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  4168       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  4169         fix k i

  4170         have "infinite ({n. f n \<in> F k} - {.. i})"

  4171           using infinite {n. f n \<in> F k} by auto

  4172         from infinite_imp_nonempty[OF this]

  4173         show "\<exists>x>i. f x \<in> F k"

  4174           by (simp add: set_eq_iff not_le conj_commute)

  4175       qed

  4176

  4177       def t \<equiv> "rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  4178       have "subseq t"

  4179         unfolding subseq_Suc_iff by (simp add: t_def sel)

  4180       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  4181         using f by auto

  4182       moreover

  4183       {

  4184         fix n

  4185         have "(f \<circ> t) n \<in> F n"

  4186           by (cases n) (simp_all add: t_def sel)

  4187       }

  4188       note t = this

  4189

  4190       have "Cauchy (f \<circ> t)"

  4191       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  4192         fix r :: real and N n m

  4193         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  4194         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  4195           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  4196         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  4197           by (auto simp: subset_eq)

  4198         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  4199         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  4200           by (simp add: dist_commute)

  4201       qed

  4202

  4203       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  4204         using assms unfolding complete_def by blast

  4205     qed

  4206   qed

  4207 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  4208

  4209 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  4210 proof -

  4211   {

  4212     assume ?rhs

  4213     {

  4214       fix e::real

  4215       assume "e>0"

  4216       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  4217         by (erule_tac x="e/2" in allE) auto

  4218       {

  4219         fix n m

  4220         assume nm:"N \<le> m \<and> N \<le> n"

  4221         then have "dist (s m) (s n) < e" using N

  4222           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  4223           by blast

  4224       }

  4225       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  4226         by blast

  4227     }

  4228     then have ?lhs

  4229       unfolding cauchy_def

  4230       by blast

  4231   }

  4232   then show ?thesis

  4233     unfolding cauchy_def

  4234     using dist_triangle_half_l

  4235     by blast

  4236 qed

  4237

  4238 lemma cauchy_imp_bounded:

  4239   assumes "Cauchy s"

  4240   shows "bounded (range s)"

  4241 proof -

  4242   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  4243     unfolding cauchy_def

  4244     apply (erule_tac x= 1 in allE)

  4245     apply auto

  4246     done

  4247   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  4248   moreover

  4249   have "bounded (s  {0..N})"

  4250     using finite_imp_bounded[of "s  {1..N}"] by auto

  4251   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  4252     unfolding bounded_any_center [where a="s N"] by auto

  4253   ultimately show "?thesis"

  4254     unfolding bounded_any_center [where a="s N"]

  4255     apply (rule_tac x="max a 1" in exI)

  4256     apply auto

  4257     apply (erule_tac x=y in allE)

  4258     apply (erule_tac x=y in ballE)

  4259     apply auto

  4260     done

  4261 qed

  4262

  4263 instance heine_borel < complete_space

  4264 proof

  4265   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4266   then have "bounded (range f)"

  4267     by (rule cauchy_imp_bounded)

  4268   then have "compact (closure (range f))"

  4269     unfolding compact_eq_bounded_closed by auto

  4270   then have "complete (closure (range f))"

  4271     by (rule compact_imp_complete)

  4272   moreover have "\<forall>n. f n \<in> closure (range f)"

  4273     using closure_subset [of "range f"] by auto

  4274   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  4275     using Cauchy f unfolding complete_def by auto

  4276   then show "convergent f"

  4277     unfolding convergent_def by auto

  4278 qed

  4279

  4280 instance euclidean_space \<subseteq> banach ..

  4281

  4282 lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"

  4283 proof (rule completeI)

  4284   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4285   then have "convergent f" by (rule Cauchy_convergent)

  4286   then show "\<exists>l\<in>UNIV. f ----> l" unfolding convergent_def by simp

  4287 qed

  4288

  4289 lemma complete_imp_closed:

  4290   assumes "complete s"

  4291   shows "closed s"

  4292 proof (unfold closed_sequential_limits, clarify)

  4293   fix f x assume "\<forall>n. f n \<in> s" and "f ----> x"

  4294   from f ----> x have "Cauchy f"

  4295     by (rule LIMSEQ_imp_Cauchy)

  4296   with complete s and \<forall>n. f n \<in> s obtain l where "l \<in> s" and "f ----> l"

  4297     by (rule completeE)

  4298   from f ----> x and f ----> l have "x = l"

  4299     by (rule LIMSEQ_unique)

  4300   with l \<in> s show "x \<in> s"

  4301     by simp

  4302 qed

  4303

  4304 lemma complete_inter_closed:

  4305   assumes "complete s" and "closed t"

  4306   shows "complete (s \<inter> t)"

  4307 proof (rule completeI)

  4308   fix f assume "\<forall>n. f n \<in> s \<inter> t" and "Cauchy f"

  4309   then have "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  4310     by simp_all

  4311   from complete s obtain l where "l \<in> s" and "f ----> l"

  4312     using \<forall>n. f n \<in> s and Cauchy f by (rule completeE)

  4313   from closed t and \<forall>n. f n \<in> t and f ----> l have "l \<in> t"

  4314     by (rule closed_sequentially)

  4315   with l \<in> s and f ----> l show "\<exists>l\<in>s \<inter> t. f ----> l"

  4316     by fast

  4317 qed

  4318

  4319 lemma complete_closed_subset:

  4320   assumes "closed s" and "s \<subseteq> t" and "complete t"

  4321   shows "complete s"

  4322   using assms complete_inter_closed [of t s] by (simp add: Int_absorb1)

  4323

  4324 lemma complete_eq_closed:

  4325   fixes s :: "('a::complete_space) set"

  4326   shows "complete s \<longleftrightarrow> closed s"

  4327 proof

  4328   assume "closed s" then show "complete s"

  4329     using subset_UNIV complete_UNIV by (rule complete_closed_subset)

  4330 next

  4331   assume "complete s" then show "closed s"

  4332     by (rule complete_imp_closed)

  4333 qed

  4334

  4335 lemma convergent_eq_cauchy:

  4336   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  4337   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  4338   unfolding Cauchy_convergent_iff convergent_def ..

  4339

  4340 lemma convergent_imp_bounded:

  4341   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  4342   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  4343   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  4344

  4345 lemma compact_cball[simp]:

  4346   fixes x :: "'a::heine_borel"

  4347   shows "compact (cball x e)"

  4348   using compact_eq_bounded_closed bounded_cball closed_cball

  4349   by blast

  4350

  4351 lemma compact_frontier_bounded[intro]:

  4352   fixes s :: "'a::heine_borel set"

  4353   shows "bounded s \<Longrightarrow> compact (frontier s)"

  4354   unfolding frontier_def

  4355   using compact_eq_bounded_closed

  4356   by blast

  4357

  4358 lemma compact_frontier[intro]:

  4359   fixes s :: "'a::heine_borel set"

  4360   shows "compact s \<Longrightarrow> compact (frontier s)"

  4361   using compact_eq_bounded_closed compact_frontier_bounded

  4362   by blast

  4363

  4364 lemma frontier_subset_compact:

  4365   fixes s :: "'a::heine_borel set"

  4366   shows "compact s \<Longrightarrow> frontier s \<subseteq> s"

  4367   using frontier_subset_closed compact_eq_bounded_closed

  4368   by blast

  4369

  4370 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  4371

  4372 lemma bounded_closed_nest:

  4373   fixes s :: "nat \<Rightarrow> ('a::heine_borel) set"

  4374   assumes "\<forall>n. closed (s n)"

  4375     and "\<forall>n. s n \<noteq> {}"

  4376     and "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4377     and "bounded (s 0)"

  4378   shows "\<exists>a. \<forall>n. a \<in> s n"

  4379 proof -

  4380   from assms(2) obtain x where x: "\<forall>n. x n \<in> s n"

  4381     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4382   from assms(4,1) have "seq_compact (s 0)"

  4383     by (simp add: bounded_closed_imp_seq_compact)

  4384   then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) ----> l"

  4385     using x and assms(3) unfolding seq_compact_def by blast

  4386   have "\<forall>n. l \<in> s n"

  4387   proof

  4388     fix n :: nat

  4389     have "closed (s n)"

  4390       using assms(1) by simp

  4391     moreover have "\<forall>i. (x \<circ> r) i \<in> s i"

  4392       using x and assms(3) and lr(2) [THEN seq_suble] by auto

  4393     then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"

  4394       using assms(3) by (fast intro!: le_add2)

  4395     moreover have "(\<lambda>i. (x \<circ> r) (i + n)) ----> l"

  4396       using lr(3) by (rule LIMSEQ_ignore_initial_segment)

  4397     ultimately show "l \<in> s n"

  4398       by (rule closed_sequentially)

  4399   qed

  4400   then show ?thesis ..

  4401 qed

  4402

  4403 text {* Decreasing case does not even need compactness, just completeness. *}

  4404

  4405 lemma decreasing_closed_nest:

  4406   fixes s :: "nat \<Rightarrow> ('a::complete_space) set"

  4407   assumes

  4408     "\<forall>n. closed (s n)"

  4409     "\<forall>n. s n \<noteq> {}"

  4410     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4411     "\<forall>e>0. \<exists>n. \<forall>x\<in>s n. \<forall>y\<in>s n. dist x y < e"

  4412   shows "\<exists>a. \<forall>n. a \<in> s n"

  4413 proof -

  4414   have "\<forall>n. \<exists>x. x \<in> s n"

  4415     using assms(2) by auto

  4416   then have "\<exists>t. \<forall>n. t n \<in> s n"

  4417     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4418   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  4419   {

  4420     fix e :: real

  4421     assume "e > 0"

  4422     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"

  4423       using assms(4) by auto

  4424     {

  4425       fix m n :: nat

  4426       assume "N \<le> m \<and> N \<le> n"

  4427       then have "t m \<in> s N" "t n \<in> s N"

  4428         using assms(3) t unfolding  subset_eq t by blast+

  4429       then have "dist (t m) (t n) < e"

  4430         using N by auto

  4431     }

  4432     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  4433       by auto

  4434   }

  4435   then have "Cauchy t"

  4436     unfolding cauchy_def by auto

  4437   then obtain l where l:"(t ---> l) sequentially"

  4438     using complete_UNIV unfolding complete_def by auto

  4439   {

  4440     fix n :: nat

  4441     {

  4442       fix e :: real

  4443       assume "e > 0"

  4444       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  4445         using l[unfolded LIMSEQ_def] by auto

  4446       have "t (max n N) \<in> s n"

  4447         using assms(3)

  4448         unfolding subset_eq

  4449         apply (erule_tac x=n in allE)

  4450         apply (erule_tac x="max n N" in allE)

  4451         using t

  4452         apply auto

  4453         done

  4454       then have "\<exists>y\<in>s n. dist y l < e"

  4455         apply (rule_tac x="t (max n N)" in bexI)

  4456         using N

  4457         apply auto

  4458         done

  4459     }

  4460     then have "l \<in> s n"

  4461       using closed_approachable[of "s n" l] assms(1) by auto

  4462   }

  4463   then show ?thesis by auto

  4464 qed

  4465

  4466 text {* Strengthen it to the intersection actually being a singleton. *}

  4467

  4468 lemma decreasing_closed_nest_sing:

  4469   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  4470   assumes

  4471     "\<forall>n. closed(s n)"

  4472     "\<forall>n. s n \<noteq> {}"

  4473     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4474     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  4475   shows "\<exists>a. \<Inter>(range s) = {a}"

  4476 proof -

  4477   obtain a where a: "\<forall>n. a \<in> s n"

  4478     using decreasing_closed_nest[of s] using assms by auto

  4479   {

  4480     fix b

  4481     assume b: "b \<in> \<Inter>(range s)"

  4482     {

  4483       fix e :: real

  4484       assume "e > 0"

  4485       then have "dist a b < e"

  4486         using assms(4) and b and a by blast

  4487     }

  4488     then have "dist a b = 0"

  4489       by (metis dist_eq_0_iff dist_nz less_le)

  4490   }

  4491   with a have "\<Inter>(range s) = {a}"

  4492     unfolding image_def by auto

  4493   then show ?thesis ..

  4494 qed

  4495

  4496 text{* Cauchy-type criteria for uniform convergence. *}

  4497

  4498 lemma uniformly_convergent_eq_cauchy:

  4499   fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"

  4500   shows

  4501     "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e) \<longleftrightarrow>

  4502       (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  \<longrightarrow> dist (s m x) (s n x) < e)"

  4503   (is "?lhs = ?rhs")

  4504 proof

  4505   assume ?lhs

  4506   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"

  4507     by auto

  4508   {

  4509     fix e :: real

  4510     assume "e > 0"

  4511     then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"

  4512       using l[THEN spec[where x="e/2"]] by auto

  4513     {

  4514       fix n m :: nat and x :: "'b"

  4515       assume "N \<le> m \<and> N \<le> n \<and> P x"

  4516       then have "dist (s m x) (s n x) < e"

  4517         using N[THEN spec[where x=m], THEN spec[where x=x]]

  4518         using N[THEN spec[where x=n], THEN spec[where x=x]]

  4519         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

  4520     }

  4521     then have "\<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

  4522   }

  4523   then show ?rhs by auto

  4524 next

  4525   assume ?rhs

  4526   then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"

  4527     unfolding cauchy_def

  4528     apply auto

  4529     apply (erule_tac x=e in allE)

  4530     apply auto

  4531     done

  4532   then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"

  4533     unfolding convergent_eq_cauchy[symmetric]

  4534     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]

  4535     by auto

  4536   {

  4537     fix e :: real

  4538     assume "e > 0"

  4539     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"

  4540       using ?rhs[THEN spec[where x="e/2"]] by auto

  4541     {

  4542       fix x

  4543       assume "P x"

  4544       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  4545         using l[THEN spec[where x=x], unfolded LIMSEQ_def] and e > 0

  4546         by (auto elim!: allE[where x="e/2"])

  4547       fix n :: nat

  4548       assume "n \<ge> N"

  4549       then have "dist(s n x)(l x) < e"

  4550         using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  4551         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

  4552         by (auto simp add: dist_commute)

  4553     }

  4554     then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4555       by auto

  4556   }

  4557   then show ?lhs by auto

  4558 qed

  4559

  4560 lemma uniformly_cauchy_imp_uniformly_convergent:

  4561   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  4562   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"

  4563     and "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n \<longrightarrow> dist(s n x)(l x) < e)"

  4564   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4565 proof -

  4566   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"

  4567     using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto

  4568   moreover

  4569   {

  4570     fix x

  4571     assume "P x"

  4572     then have "l x = l' x"

  4573       using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  4574       using l and assms(2) unfolding LIMSEQ_def by blast

  4575   }

  4576   ultimately show ?thesis by auto

  4577 qed

  4578

  4579

  4580 subsection {* Continuity *}

  4581

  4582 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  4583

  4584 lemma continuous_within_eps_delta:

  4585   "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)"

  4586   unfolding continuous_within and Lim_within

  4587   apply auto

  4588   apply (metis dist_nz dist_self)

  4589   apply blast

  4590   done

  4591

  4592 lemma continuous_at_eps_delta:

  4593   "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  4594   using continuous_within_eps_delta [of x UNIV f] by simp

  4595

  4596 lemma continuous_at_right_real_increasing:

  4597   fixes f :: "real \<Rightarrow> real"

  4598   assumes nondecF: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"

  4599   shows "continuous (at_right a) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f (a + d) - f a < e)"

  4600   apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)

  4601   apply (intro all_cong ex_cong)

  4602   apply safe

  4603   apply (erule_tac x="a + d" in allE)

  4604   apply simp

  4605   apply (simp add: nondecF field_simps)

  4606   apply (drule nondecF)

  4607   apply simp

  4608   done

  4609

  4610 lemma continuous_at_left_real_increasing:

  4611   assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"

  4612   shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"

  4613   apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)

  4614   apply (intro all_cong ex_cong)

  4615   apply safe

  4616   apply (erule_tac x="a - d" in allE)

  4617   apply simp

  4618   apply (simp add: nondecF field_simps)

  4619   apply (cut_tac x="a - d" and y="x" in nondecF)

  4620   apply simp_all

  4621   done

  4622

  4623 text{* Versions in terms of open balls. *}

  4624

  4625 lemma continuous_within_ball:

  4626   "continuous (at x within s) f \<longleftrightarrow>

  4627     (\<forall>e > 0. \<exists>d > 0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e)"

  4628   (is "?lhs = ?rhs")

  4629 proof

  4630   assume ?lhs

  4631   {

  4632     fix e :: real

  4633     assume "e > 0"

  4634     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"

  4635       using ?lhs[unfolded continuous_within Lim_within] by auto

  4636     {

  4637       fix y

  4638       assume "y \<in> f  (ball x d \<inter> s)"

  4639       then have "y \<in> ball (f x) e"

  4640         using d(2)

  4641         unfolding dist_nz[symmetric]

  4642         apply (auto simp add: dist_commute)

  4643         apply (erule_tac x=xa in ballE)

  4644         apply auto

  4645         using e > 0

  4646         apply auto

  4647         done

  4648     }

  4649     then have "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e"

  4650       using d > 0

  4651       unfolding subset_eq ball_def by (auto simp add: dist_commute)

  4652   }

  4653   then show ?rhs by auto

  4654 next

  4655   assume ?rhs

  4656   then show ?lhs

  4657     unfolding continuous_within Lim_within ball_def subset_eq

  4658     apply (auto simp add: dist_commute)

  4659     apply (erule_tac x=e in allE)

  4660     apply auto

  4661     done

  4662 qed

  4663

  4664 lemma continuous_at_ball:

  4665   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  4666 proof

  4667   assume ?lhs

  4668   then show ?rhs

  4669     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4670     apply auto

  4671     apply (erule_tac x=e in allE)

  4672     apply auto

  4673     apply (rule_tac x=d in exI)

  4674     apply auto

  4675     apply (erule_tac x=xa in allE)

  4676     apply (auto simp add: dist_commute dist_nz)

  4677     unfolding dist_nz[symmetric]

  4678     apply auto

  4679     done

  4680 next

  4681   assume ?rhs

  4682   then show ?lhs

  4683     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4684     apply auto

  4685     apply (erule_tac x=e in allE)

  4686     apply auto

  4687     apply (rule_tac x=d in exI)

  4688     apply auto

  4689     apply (erule_tac x="f xa" in allE)

  4690     apply (auto simp add: dist_commute dist_nz)

  4691     done

  4692 qed

  4693

  4694 text{* Define setwise continuity in terms of limits within the set. *}

  4695

  4696 lemma continuous_on_iff:

  4697   "continuous_on s f \<longleftrightarrow>

  4698     (\<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)"

  4699   unfolding continuous_on_def Lim_within

  4700   by (metis dist_pos_lt dist_self)

  4701

  4702 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  4703   where "uniformly_continuous_on s f \<longleftrightarrow>

  4704     (\<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)"

  4705

  4706 text{* Some simple consequential lemmas. *}

  4707

  4708 lemma uniformly_continuous_imp_continuous:

  4709   "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"

  4710   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  4711

  4712 lemma continuous_at_imp_continuous_within:

  4713   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"

  4714   unfolding continuous_within continuous_at using Lim_at_within by auto

  4715

  4716 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  4717   by simp

  4718

  4719 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4720

  4721 lemma continuous_within_subset:

  4722   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"

  4723   unfolding continuous_within by(metis tendsto_within_subset)

  4724

  4725 lemma continuous_on_interior:

  4726   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4727   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)

  4728

  4729 lemma continuous_on_eq:

  4730   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4731   unfolding continuous_on_def tendsto_def eventually_at_topological

  4732   by simp

  4733

  4734 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4735

  4736 lemma continuous_within_sequentially:

  4737   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4738   shows "continuous (at a within s) f \<longleftrightarrow>

  4739     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4740          \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"

  4741   (is "?lhs = ?rhs")

  4742 proof

  4743   assume ?lhs

  4744   {

  4745     fix x :: "nat \<Rightarrow> 'a"

  4746     assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4747     fix T :: "'b set"

  4748     assume "open T" and "f a \<in> T"

  4749     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"

  4750       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  4751     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4752       using x(2) d>0 by simp

  4753     then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4754     proof eventually_elim

  4755       case (elim n)

  4756       then show ?case

  4757         using d x(1) f a \<in> T unfolding dist_nz[symmetric] by auto

  4758     qed

  4759   }

  4760   then show ?rhs

  4761     unfolding tendsto_iff tendsto_def by simp

  4762 next

  4763   assume ?rhs

  4764   then show ?lhs

  4765     unfolding continuous_within tendsto_def [where l="f a"]

  4766     by (simp add: sequentially_imp_eventually_within)

  4767 qed

  4768

  4769 lemma continuous_at_sequentially:

  4770   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4771   shows "continuous (at a) f \<longleftrightarrow>

  4772     (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"

  4773   using continuous_within_sequentially[of a UNIV f] by simp

  4774

  4775 lemma continuous_on_sequentially:

  4776   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4777   shows "continuous_on s f \<longleftrightarrow>

  4778     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4779       --> ((f \<circ> x) ---> f a) sequentially)"

  4780   (is "?lhs = ?rhs")

  4781 proof

  4782   assume ?rhs

  4783   then show ?lhs

  4784     using continuous_within_sequentially[of _ s f]

  4785     unfolding continuous_on_eq_continuous_within

  4786     by auto

  4787 next

  4788   assume ?lhs

  4789   then show ?rhs

  4790     unfolding continuous_on_eq_continuous_within

  4791     using continuous_within_sequentially[of _ s f]

  4792     by auto

  4793 qed

  4794

  4795 lemma uniformly_continuous_on_sequentially:

  4796   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4797                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4798                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4799 proof

  4800   assume ?lhs

  4801   {

  4802     fix x y

  4803     assume x: "\<forall>n. x n \<in> s"

  4804       and y: "\<forall>n. y n \<in> s"

  4805       and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4806     {

  4807       fix e :: real

  4808       assume "e > 0"

  4809       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"

  4810         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4811       obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"

  4812         using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4813       {

  4814         fix n

  4815         assume "n\<ge>N"

  4816         then have "dist (f (x n)) (f (y n)) < e"

  4817           using N[THEN spec[where x=n]]

  4818           using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

  4819           using x and y

  4820           unfolding dist_commute

  4821           by simp

  4822       }

  4823       then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4824         by auto

  4825     }

  4826     then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"

  4827       unfolding LIMSEQ_def and dist_real_def by auto

  4828   }

  4829   then show ?rhs by auto

  4830 next

  4831   assume ?rhs

  4832   {

  4833     assume "\<not> ?lhs"

  4834     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"

  4835       unfolding uniformly_continuous_on_def by auto

  4836     then obtain fa where fa:

  4837       "\<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"

  4838       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"]

  4839       unfolding Bex_def

  4840       by (auto simp add: dist_commute)

  4841     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4842     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4843     have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"

  4844       and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"

  4845       and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4846       unfolding x_def and y_def using fa

  4847       by auto

  4848     {

  4849       fix e :: real

  4850       assume "e > 0"

  4851       then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"

  4852         unfolding real_arch_inv[of e] by auto

  4853       {

  4854         fix n :: nat

  4855         assume "n \<ge> N"

  4856         then have "inverse (real n + 1) < inverse (real N)"

  4857           using real_of_nat_ge_zero and N\<noteq>0 by auto

  4858         also have "\<dots> < e" using N by auto

  4859         finally have "inverse (real n + 1) < e" by auto

  4860         then have "dist (x n) (y n) < e"

  4861           using xy0[THEN spec[where x=n]] by auto

  4862       }

  4863       then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto

  4864     }

  4865     then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4866       using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn

  4867       unfolding LIMSEQ_def dist_real_def by auto

  4868     then have False using fxy and e>0 by auto

  4869   }

  4870   then show ?lhs

  4871     unfolding uniformly_continuous_on_def by blast

  4872 qed

  4873

  4874 text{* The usual transformation theorems. *}

  4875

  4876 lemma continuous_transform_within:

  4877   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4878   assumes "0 < d"

  4879     and "x \<in> s"

  4880     and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4881     and "continuous (at x within s) f"

  4882   shows "continuous (at x within s) g"

  4883   unfolding continuous_within

  4884 proof (rule Lim_transform_within)

  4885   show "0 < d" by fact

  4886   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4887     using assms(3) by auto

  4888   have "f x = g x"

  4889     using assms(1,2,3) by auto

  4890   then show "(f ---> g x) (at x within s)"

  4891     using assms(4) unfolding continuous_within by simp

  4892 qed

  4893

  4894 lemma continuous_transform_at:

  4895   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4896   assumes "0 < d"

  4897     and "\<forall>x'. dist x' x < d --> f x' = g x'"

  4898     and "continuous (at x) f"

  4899   shows "continuous (at x) g"

  4900   using continuous_transform_within [of d x UNIV f g] assms by simp

  4901

  4902

  4903 subsubsection {* Structural rules for pointwise continuity *}

  4904

  4905 lemmas continuous_within_id = continuous_ident

  4906

  4907 lemmas continuous_at_id = isCont_ident

  4908

  4909 lemma continuous_infdist[continuous_intros]:

  4910   assumes "continuous F f"

  4911   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4912   using assms unfolding continuous_def by (rule tendsto_infdist)

  4913

  4914 lemma continuous_infnorm[continuous_intros]:

  4915   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4916   unfolding continuous_def by (rule tendsto_infnorm)

  4917

  4918 lemma continuous_inner[continuous_intros]:

  4919   assumes "continuous F f"

  4920     and "continuous F g"

  4921   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4922   using assms unfolding continuous_def by (rule tendsto_inner)

  4923

  4924 lemmas continuous_at_inverse = isCont_inverse

  4925

  4926 subsubsection {* Structural rules for setwise continuity *}

  4927

  4928 lemma continuous_on_infnorm[continuous_intros]:

  4929   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4930   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4931

  4932 lemma continuous_on_inner[continuous_intros]:

  4933   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4934   assumes "continuous_on s f"

  4935     and "continuous_on s g"

  4936   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4937   using bounded_bilinear_inner assms

  4938   by (rule bounded_bilinear.continuous_on)

  4939

  4940 subsubsection {* Structural rules for uniform continuity *}

  4941

  4942 lemma uniformly_continuous_on_id[continuous_intros]:

  4943   "uniformly_continuous_on s (\<lambda>x. x)"

  4944   unfolding uniformly_continuous_on_def by auto

  4945

  4946 lemma uniformly_continuous_on_const[continuous_intros]:

  4947   "uniformly_continuous_on s (\<lambda>x. c)"

  4948   unfolding uniformly_continuous_on_def by simp

  4949

  4950 lemma uniformly_continuous_on_dist[continuous_intros]:

  4951   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4952   assumes "uniformly_continuous_on s f"

  4953     and "uniformly_continuous_on s g"

  4954   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4955 proof -

  4956   {

  4957     fix a b c d :: 'b

  4958     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4959       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4960       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4961       by arith

  4962   } note le = this

  4963   {

  4964     fix x y

  4965     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4966     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4967     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4968       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4969         simp add: le)

  4970   }

  4971   then show ?thesis

  4972     using assms unfolding uniformly_continuous_on_sequentially

  4973     unfolding dist_real_def by simp

  4974 qed

  4975

  4976 lemma uniformly_continuous_on_norm[continuous_intros]:

  4977   assumes "uniformly_continuous_on s f"

  4978   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4979   unfolding norm_conv_dist using assms

  4980   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4981

  4982 lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:

  4983   assumes "uniformly_continuous_on s g"

  4984   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4985   using assms unfolding uniformly_continuous_on_sequentially

  4986   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4987   by (auto intro: tendsto_zero)

  4988

  4989 lemma uniformly_continuous_on_cmul[continuous_intros]:

  4990   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4991   assumes "uniformly_continuous_on s f"

  4992   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4993   using bounded_linear_scaleR_right assms

  4994   by (rule bounded_linear.uniformly_continuous_on)

  4995

  4996 lemma dist_minus:

  4997   fixes x y :: "'a::real_normed_vector"

  4998   shows "dist (- x) (- y) = dist x y"

  4999   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  5000

  5001 lemma uniformly_continuous_on_minus[continuous_intros]:

  5002   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5003   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  5004   unfolding uniformly_continuous_on_def dist_minus .

  5005

  5006 lemma uniformly_continuous_on_add[continuous_intros]:

  5007   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5008   assumes "uniformly_continuous_on s f"

  5009     and "uniformly_continuous_on s g"

  5010   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  5011   using assms

  5012   unfolding uniformly_continuous_on_sequentially

  5013   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  5014   by (auto intro: tendsto_add_zero)

  5015

  5016 lemma uniformly_continuous_on_diff[continuous_intros]:

  5017   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5018   assumes "uniformly_continuous_on s f"

  5019     and "uniformly_continuous_on s g"

  5020   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  5021   using assms uniformly_continuous_on_add [of s f "- g"]

  5022     by (simp add: fun_Compl_def uniformly_continuous_on_minus)

  5023

  5024 text{* Continuity of all kinds is preserved under composition. *}

  5025

  5026 lemmas continuous_at_compose = isCont_o

  5027

  5028 lemma uniformly_continuous_on_compose[continuous_intros]:

  5029   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  5030   shows "uniformly_continuous_on s (g \<circ> f)"

  5031 proof -

  5032   {

  5033     fix e :: real

  5034     assume "e > 0"

  5035     then obtain d where "d > 0"

  5036       and d: "\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5037       using assms(2) unfolding uniformly_continuous_on_def by auto

  5038     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"

  5039       using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto

  5040     then have "\<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"

  5041       using d>0 using d by auto

  5042   }

  5043   then show ?thesis

  5044     using assms unfolding uniformly_continuous_on_def by auto

  5045 qed

  5046

  5047 text{* Continuity in terms of open preimages. *}

  5048

  5049 lemma continuous_at_open:

  5050   "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)))"

  5051   unfolding continuous_within_topological [of x UNIV f]

  5052   unfolding imp_conjL

  5053   by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  5054

  5055 lemma continuous_imp_tendsto:

  5056   assumes "continuous (at x0) f"

  5057     and "x ----> x0"

  5058   shows "(f \<circ> x) ----> (f x0)"

  5059 proof (rule topological_tendstoI)

  5060   fix S

  5061   assume "open S" "f x0 \<in> S"

  5062   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  5063      using assms continuous_at_open by metis

  5064   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  5065     using assms T_def by (auto simp: tendsto_def)

  5066   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  5067     using T_def by (auto elim!: eventually_elim1)

  5068 qed

  5069

  5070 lemma continuous_on_open:

  5071   "continuous_on s f \<longleftrightarrow>

  5072     (\<forall>t. openin (subtopology euclidean (f  s)) t \<longrightarrow>

  5073       openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5074   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  5075   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5076

  5077 text {* Similarly in terms of closed sets. *}

  5078

  5079 lemma continuous_on_closed:

  5080   "continuous_on s f \<longleftrightarrow>

  5081     (\<forall>t. closedin (subtopology euclidean (f  s)) t \<longrightarrow>

  5082       closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5083   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  5084   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5085

  5086 text {* Half-global and completely global cases. *}

  5087

  5088 lemma continuous_open_in_preimage:

  5089   assumes "continuous_on s f"  "open t"

  5090   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5091 proof -

  5092   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5093     by auto

  5094   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5095     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  5096   then show ?thesis

  5097     using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]]

  5098     using * by auto

  5099 qed

  5100

  5101 lemma continuous_closed_in_preimage:

  5102   assumes "continuous_on s f" and "closed t"

  5103   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5104 proof -

  5105   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5106     by auto

  5107   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5108     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute

  5109     by auto

  5110   then show ?thesis

  5111     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]]

  5112     using * by auto

  5113 qed

  5114

  5115 lemma continuous_open_preimage:

  5116   assumes "continuous_on s f"

  5117     and "open s"

  5118     and "open t"

  5119   shows "open {x \<in> s. f x \<in> t}"

  5120 proof-

  5121   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5122     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  5123   then show ?thesis

  5124     using open_Int[of s T, OF assms(2)] by auto

  5125 qed

  5126

  5127 lemma continuous_closed_preimage:

  5128   assumes "continuous_on s f"

  5129     and "closed s"

  5130     and "closed t"

  5131   shows "closed {x \<in> s. f x \<in> t}"

  5132 proof-

  5133   obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5134     using continuous_closed_in_preimage[OF assms(1,3)]

  5135     unfolding closedin_closed by auto

  5136   then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

  5137 qed

  5138

  5139 lemma continuous_open_preimage_univ:

  5140   "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  5141   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  5142

  5143 lemma continuous_closed_preimage_univ:

  5144   "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s \<Longrightarrow> closed {x. f x \<in> s}"

  5145   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  5146

  5147 lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  5148   unfolding vimage_def by (rule continuous_open_preimage_univ)

  5149

  5150 lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  5151   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  5152

  5153 lemma interior_image_subset:

  5154   assumes "\<forall>x. continuous (at x) f"

  5155     and "inj f"

  5156   shows "interior (f  s) \<subseteq> f  (interior s)"

  5157 proof

  5158   fix x assume "x \<in> interior (f  s)"

  5159   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  5160   then have "x \<in> f  s" by auto

  5161   then obtain y where y: "y \<in> s" "x = f y" by auto

  5162   have "open (vimage f T)"

  5163     using assms(1) open T by (rule continuous_open_vimage)

  5164   moreover have "y \<in> vimage f T"

  5165     using x = f y x \<in> T by simp

  5166   moreover have "vimage f T \<subseteq> s"

  5167     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  5168   ultimately have "y \<in> interior s" ..

  5169   with x = f y show "x \<in> f  interior s" ..

  5170 qed

  5171

  5172 text {* Equality of continuous functions on closure and related results. *}

  5173

  5174 lemma continuous_closed_in_preimage_constant:

  5175   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5176   shows "continuous_on s f \<Longrightarrow> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  5177   using continuous_closed_in_preimage[of s f "{a}"] by auto

  5178

  5179 lemma continuous_closed_preimage_constant:

  5180   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5181   shows "continuous_on s f \<Longrightarrow> closed s \<Longrightarrow> closed {x \<in> s. f x = a}"

  5182   using continuous_closed_preimage[of s f "{a}"] by auto

  5183

  5184 lemma continuous_constant_on_closure:

  5185   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5186   assumes "continuous_on (closure s) f"

  5187     and "\<forall>x \<in> s. f x = a"

  5188   shows "\<forall>x \<in> (closure s). f x = a"

  5189     using continuous_closed_preimage_constant[of "closure s" f a]

  5190       assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset

  5191     unfolding subset_eq

  5192     by auto

  5193

  5194 lemma image_closure_subset:

  5195   assumes "continuous_on (closure s) f"

  5196     and "closed t"

  5197     and "(f  s) \<subseteq> t"

  5198   shows "f  (closure s) \<subseteq> t"

  5199 proof -

  5200   have "s \<subseteq> {x \<in> closure s. f x \<in> t}"

  5201     using assms(3) closure_subset by auto

  5202   moreover have "closed {x \<in> closure s. f x \<in> t}"

  5203     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  5204   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  5205     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  5206   then show ?thesis by auto

  5207 qed

  5208

  5209 lemma continuous_on_closure_norm_le:

  5210   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5211   assumes "continuous_on (closure s) f"

  5212     and "\<forall>y \<in> s. norm(f y) \<le> b"

  5213     and "x \<in> (closure s)"

  5214   shows "norm (f x) \<le> b"

  5215 proof -

  5216   have *: "f  s \<subseteq> cball 0 b"

  5217     using assms(2)[unfolded mem_cball_0[symmetric]] by auto

  5218   show ?thesis

  5219     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  5220     unfolding subset_eq

  5221     apply (erule_tac x="f x" in ballE)

  5222     apply (auto simp add: dist_norm)

  5223     done

  5224 qed

  5225

  5226 text {* Making a continuous function avoid some value in a neighbourhood. *}

  5227

  5228 lemma continuous_within_avoid:

  5229   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5230   assumes "continuous (at x within s) f"

  5231     and "f x \<noteq> a"

  5232   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  5233 proof -

  5234   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  5235     using t1_space [OF f x \<noteq> a] by fast

  5236   have "(f ---> f x) (at x within s)"

  5237     using assms(1) by (simp add: continuous_within)

  5238   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  5239     using open U and f x \<in> U

  5240     unfolding tendsto_def by fast

  5241   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  5242     using a \<notin> U by (fast elim: eventually_mono [rotated])

  5243   then show ?thesis

  5244     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  5245 qed

  5246

  5247 lemma continuous_at_avoid:

  5248   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5249   assumes "continuous (at x) f"

  5250     and "f x \<noteq> a"

  5251   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5252   using assms continuous_within_avoid[of x UNIV f a] by simp

  5253

  5254 lemma continuous_on_avoid:

  5255   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5256   assumes "continuous_on s f"

  5257     and "x \<in> s"

  5258     and "f x \<noteq> a"

  5259   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  5260   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  5261     OF assms(2)] continuous_within_avoid[of x s f a]

  5262   using assms(3)

  5263   by auto

  5264

  5265 lemma continuous_on_open_avoid:

  5266   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5267   assumes "continuous_on s f"

  5268     and "open s"

  5269     and "x \<in> s"

  5270     and "f x \<noteq> a"

  5271   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5272   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  5273   using continuous_at_avoid[of x f a] assms(4)

  5274   by auto

  5275

  5276 text {* Proving a function is constant by proving open-ness of level set. *}

  5277

  5278 lemma continuous_levelset_open_in_cases:

  5279   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5280   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5281         openin (subtopology euclidean s) {x \<in> s. f x = a}

  5282         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  5283   unfolding connected_clopen

  5284   using continuous_closed_in_preimage_constant by auto

  5285

  5286 lemma continuous_levelset_open_in:

  5287   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5288   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5289         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  5290         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"

  5291   using continuous_levelset_open_in_cases[of s f ]

  5292   by meson

  5293

  5294 lemma continuous_levelset_open:

  5295   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5296   assumes "connected s"

  5297     and "continuous_on s f"

  5298     and "open {x \<in> s. f x = a}"

  5299     and "\<exists>x \<in> s.  f x = a"

  5300   shows "\<forall>x \<in> s. f x = a"

  5301   using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

  5302   using assms (3,4)

  5303   by fast

  5304

  5305 text {* Some arithmetical combinations (more to prove). *}

  5306

  5307 lemma open_scaling[intro]:

  5308   fixes s :: "'a::real_normed_vector set"

  5309   assumes "c \<noteq> 0"

  5310     and "open s"

  5311   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  5312 proof -

  5313   {

  5314     fix x

  5315     assume "x \<in> s"

  5316     then obtain e where "e>0"

  5317       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  5318       by auto

  5319     have "e * abs c > 0"

  5320       using assms(1)[unfolded zero_less_abs_iff[symmetric]] e>0 by auto

  5321     moreover

  5322     {

  5323       fix y

  5324       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  5325       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  5326         unfolding dist_norm

  5327         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  5328           assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

  5329       then have "y \<in> op *\<^sub>R c  s"

  5330         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]

  5331         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  5332         using assms(1)

  5333         unfolding dist_norm scaleR_scaleR

  5334         by auto

  5335     }

  5336     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s"

  5337       apply (rule_tac x="e * abs c" in exI)

  5338       apply auto

  5339       done

  5340   }

  5341   then show ?thesis unfolding open_dist by auto

  5342 qed

  5343

  5344 lemma minus_image_eq_vimage:

  5345   fixes A :: "'a::ab_group_add set"

  5346   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  5347   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  5348

  5349 lemma open_negations:

  5350   fixes s :: "'a::real_normed_vector set"

  5351   shows "open s \<Longrightarrow> open ((\<lambda>x. - x)  s)"

  5352   using open_scaling [of "- 1" s] by simp

  5353

  5354 lemma open_translation:

  5355   fixes s :: "'a::real_normed_vector set"

  5356   assumes "open s"

  5357   shows "open((\<lambda>x. a + x)  s)"

  5358 proof -

  5359   {

  5360     fix x

  5361     have "continuous (at x) (\<lambda>x. x - a)"

  5362       by (intro continuous_diff continuous_at_id continuous_const)

  5363   }

  5364   moreover have "{x. x - a \<in> s} = op + a  s"

  5365     by force

  5366   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]

  5367     using assms by auto

  5368 qed

  5369

  5370 lemma open_affinity:

  5371   fixes s :: "'a::real_normed_vector set"

  5372   assumes "open s"  "c \<noteq> 0"

  5373   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5374 proof -

  5375   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  5376     unfolding o_def ..

  5377   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s"

  5378     by auto

  5379   then show ?thesis

  5380     using assms open_translation[of "op *\<^sub>R c  s" a]

  5381     unfolding *

  5382     by auto

  5383 qed

  5384

  5385 lemma interior_translation:

  5386   fixes s :: "'a::real_normed_vector set"

  5387   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  5388 proof (rule set_eqI, rule)

  5389   fix x

  5390   assume "x \<in> interior (op + a  s)"

  5391   then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a  s"

  5392     unfolding mem_interior by auto

  5393   then have "ball (x - a) e \<subseteq> s"

  5394     unfolding subset_eq Ball_def mem_ball dist_norm

  5395     apply auto

  5396     apply (erule_tac x="a + xa" in allE)

  5397     unfolding ab_group_add_class.diff_diff_eq[symmetric]

  5398     apply auto

  5399     done

  5400   then show "x \<in> op + a  interior s"

  5401     unfolding image_iff

  5402     apply (rule_tac x="x - a" in bexI)

  5403     unfolding mem_interior

  5404     using e > 0

  5405     apply auto

  5406     done

  5407 next

  5408   fix x

  5409   assume "x \<in> op + a  interior s"

  5410   then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"

  5411     unfolding image_iff Bex_def mem_interior by auto

  5412   {

  5413     fix z

  5414     have *: "a + y - z = y + a - z" by auto

  5415     assume "z \<in> ball x e"

  5416     then have "z - a \<in> s"

  5417       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  5418       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  5419       by auto

  5420     then have "z \<in> op + a  s"

  5421       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  5422   }

  5423   then have "ball x e \<subseteq> op + a  s"

  5424     unfolding subset_eq by auto

  5425   then show "x \<in> interior (op + a  s)"

  5426     unfolding mem_interior using e > 0 by auto

  5427 qed

  5428

  5429 text {* Topological properties of linear functions. *}

  5430

  5431 lemma linear_lim_0:

  5432   assumes "bounded_linear f"

  5433   shows "(f ---> 0) (at (0))"

  5434 proof -

  5435   interpret f: bounded_linear f by fact

  5436   have "(f ---> f 0) (at 0)"

  5437     using tendsto_ident_at by (rule f.tendsto)

  5438   then show ?thesis unfolding f.zero .

  5439 qed

  5440

  5441 lemma linear_continuous_at:

  5442   assumes "bounded_linear f"

  5443   shows "continuous (at a) f"

  5444   unfolding continuous_at using assms

  5445   apply (rule bounded_linear.tendsto)

  5446   apply (rule tendsto_ident_at)

  5447   done

  5448

  5449 lemma linear_continuous_within:

  5450   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"

  5451   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  5452

  5453 lemma linear_continuous_on:

  5454   "bounded_linear f \<Longrightarrow> continuous_on s f"

  5455   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  5456

  5457 text {* Also bilinear functions, in composition form. *}

  5458

  5459 lemma bilinear_continuous_at_compose:

  5460   "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5461     continuous (at x) (\<lambda>x. h (f x) (g x))"

  5462   unfolding continuous_at

  5463   using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

  5464   by auto

  5465

  5466 lemma bilinear_continuous_within_compose:

  5467   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5468     continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  5469   unfolding continuous_within

  5470   using Lim_bilinear[of f "f x"]

  5471   by auto

  5472

  5473 lemma bilinear_continuous_on_compose:

  5474   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5475     continuous_on s (\<lambda>x. h (f x) (g x))"

  5476   unfolding continuous_on_def

  5477   by (fast elim: bounded_bilinear.tendsto)

  5478

  5479 text {* Preservation of compactness and connectedness under continuous function. *}

  5480

  5481 lemma compact_eq_openin_cover:

  5482   "compact S \<longleftrightarrow>

  5483     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5484       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  5485 proof safe

  5486   fix C

  5487   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  5488   then have "\<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}"

  5489     unfolding openin_open by force+

  5490   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"

  5491     by (rule compactE)

  5492   then have "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)"

  5493     by auto

  5494   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5495 next

  5496   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5497         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  5498   show "compact S"

  5499   proof (rule compactI)

  5500     fix C

  5501     let ?C = "image (\<lambda>T. S \<inter> T) C"

  5502     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  5503     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  5504       unfolding openin_open by auto

  5505     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  5506       by metis

  5507     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  5508     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  5509     proof (intro conjI)

  5510       from D \<subseteq> ?C show "?D \<subseteq> C"

  5511         by (fast intro: inv_into_into)

  5512       from finite D show "finite ?D"

  5513         by (rule finite_imageI)

  5514       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  5515         apply (rule subset_trans)

  5516         apply clarsimp

  5517         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  5518         apply (erule rev_bexI, fast)

  5519         done

  5520     qed

  5521     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5522   qed

  5523 qed

  5524

  5525 lemma connected_continuous_image:

  5526   assumes "continuous_on s f"

  5527     and "connected s"

  5528   shows "connected(f  s)"

  5529 proof -

  5530   {

  5531     fix T

  5532     assume as:

  5533       "T \<noteq> {}"

  5534       "T \<noteq> f  s"

  5535       "openin (subtopology euclidean (f  s)) T"

  5536       "closedin (subtopology euclidean (f  s)) T"

  5537     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  5538       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  5539       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  5540       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  5541     then have False using as(1,2)

  5542       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

  5543   }

  5544   then show ?thesis

  5545     unfolding connected_clopen by auto

  5546 qed

  5547

  5548 text {* Continuity implies uniform continuity on a compact domain. *}

  5549

  5550 lemma compact_uniformly_continuous:

  5551   assumes f: "continuous_on s f"

  5552     and s: "compact s"

  5553   shows "uniformly_continuous_on s f"

  5554   unfolding uniformly_continuous_on_def

  5555 proof (cases, safe)

  5556   fix e :: real

  5557   assume "0 < e" "s \<noteq> {}"

  5558   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) } }"

  5559   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  5560   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  5561   proof safe

  5562     fix y

  5563     assume "y \<in> s"

  5564     from continuous_open_in_preimage[OF f open_ball]

  5565     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  5566       unfolding openin_subtopology open_openin by metis

  5567     then obtain d where "ball y d \<subseteq> T" "0 < d"

  5568       using 0 < e y \<in> s by (auto elim!: openE)

  5569     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  5570       by (intro UN_I[of "(y, d)"]) auto

  5571   qed auto

  5572   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  5573     by (rule compactE_image)

  5574   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  5575     by (subst Min_gr_iff) auto

  5576   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  5577   proof (rule, safe)

  5578     fix x x'

  5579     assume in_s: "x' \<in> s" "x \<in> s"

  5580     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  5581       by blast

  5582     moreover assume "dist x x' < Min (sndD) / 2"

  5583     ultimately have "dist y x' < d"

  5584       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  5585     with D x in_s show  "dist (f x) (f x') < e"

  5586       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  5587   qed (insert D, auto)

  5588 qed auto

  5589

  5590 text {* A uniformly convergent limit of continuous functions is continuous. *}

  5591

  5592 lemma continuous_uniform_limit:

  5593   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  5594   assumes "\<not> trivial_limit F"

  5595     and "eventually (\<lambda>n. continuous_on s (f n)) F"

  5596     and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  5597   shows "continuous_on s g"

  5598 proof -

  5599   {

  5600     fix x and e :: real

  5601     assume "x\<in>s" "e>0"

  5602     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  5603       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  5604     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  5605     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  5606       using assms(1) by blast

  5607     have "e / 3 > 0" using e>0 by auto

  5608     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"

  5609       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  5610     {

  5611       fix y

  5612       assume "y \<in> s" and "dist y x < d"

  5613       then have "dist (f n y) (f n x) < e / 3"

  5614         by (rule d [rule_format])

  5615       then have "dist (f n y) (g x) < 2 * e / 3"

  5616         using dist_triangle [of "f n y" "g x" "f n x"]

  5617         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  5618         by auto

  5619       then have "dist (g y) (g x) < e"

  5620         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  5621         using dist_triangle3 [of "g y" "g x" "f n y"]

  5622         by auto

  5623     }

  5624     then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5625       using d>0 by auto

  5626   }

  5627   then show ?thesis

  5628     unfolding continuous_on_iff by auto

  5629 qed

  5630

  5631

  5632 subsection {* Topological stuff lifted from and dropped to R *}

  5633

  5634 lemma open_real:

  5635   fixes s :: "real set"

  5636   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"

  5637   unfolding open_dist dist_norm by simp

  5638

  5639 lemma islimpt_approachable_real:

  5640   fixes s :: "real set"

  5641   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  5642   unfolding islimpt_approachable dist_norm by simp

  5643

  5644 lemma closed_real:

  5645   fixes s :: "real set"

  5646   shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e) \<longrightarrow> x \<in> s)"

  5647   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5648

  5649 lemma continuous_at_real_range:

  5650   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5651   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5652   unfolding continuous_at

  5653   unfolding Lim_at

  5654   unfolding dist_nz[symmetric]

  5655   unfolding dist_norm

  5656   apply auto

  5657   apply (erule_tac x=e in allE)

  5658   apply auto

  5659   apply (rule_tac x=d in exI)

  5660   apply auto

  5661   apply (erule_tac x=x' in allE)

  5662   apply auto

  5663   apply (erule_tac x=e in allE)

  5664   apply auto

  5665   done

  5666

  5667 lemma continuous_on_real_range:

  5668   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5669   shows "continuous_on s f \<longleftrightarrow>

  5670     (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> abs(f x' - f x) < e))"

  5671   unfolding continuous_on_iff dist_norm by simp

  5672

  5673 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5674

  5675 lemma distance_attains_sup:

  5676   assumes "compact s" "s \<noteq> {}"

  5677   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  5678 proof (rule continuous_attains_sup [OF assms])

  5679   {

  5680     fix x

  5681     assume "x\<in>s"

  5682     have "(dist a ---> dist a x) (at x within s)"

  5683       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  5684   }

  5685   then show "continuous_on s (dist a)"

  5686     unfolding continuous_on ..

  5687 qed

  5688

  5689 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5690

  5691 lemma distance_attains_inf:

  5692   fixes a :: "'a::heine_borel"

  5693   assumes "closed s"

  5694     and "s \<noteq> {}"

  5695   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  5696 proof -

  5697   from assms(2) obtain b where "b \<in> s" by auto

  5698   let ?B = "s \<inter> cball a (dist b a)"

  5699   have "?B \<noteq> {}" using b \<in> s

  5700     by (auto simp add: dist_commute)

  5701   moreover have "continuous_on ?B (dist a)"

  5702     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  5703   moreover have "compact ?B"

  5704     by (intro closed_inter_compact closed s compact_cball)

  5705   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  5706     by (metis continuous_attains_inf)

  5707   then show ?thesis by fastforce

  5708 qed

  5709

  5710

  5711 subsection {* Pasted sets *}

  5712

  5713 lemma bounded_Times:

  5714   assumes "bounded s" "bounded t"

  5715   shows "bounded (s \<times> t)"

  5716 proof -

  5717   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5718     using assms [unfolded bounded_def] by auto

  5719   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  5720     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5721   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5722 qed

  5723

  5724 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5725   by (induct x) simp

  5726

  5727 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5728   unfolding seq_compact_def

  5729   apply clarify

  5730   apply (drule_tac x="fst \<circ> f" in spec)

  5731   apply (drule mp, simp add: mem_Times_iff)

  5732   apply (clarify, rename_tac l1 r1)

  5733   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5734   apply (drule mp, simp add: mem_Times_iff)

  5735   apply (clarify, rename_tac l2 r2)

  5736   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5737   apply (rule_tac x="r1 \<circ> r2" in exI)

  5738   apply (rule conjI, simp add: subseq_def)

  5739   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  5740   apply (drule (1) tendsto_Pair) back

  5741   apply (simp add: o_def)

  5742   done

  5743

  5744 lemma compact_Times:

  5745   assumes "compact s" "compact t"

  5746   shows "compact (s \<times> t)"

  5747 proof (rule compactI)

  5748   fix C

  5749   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  5750   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)"

  5751   proof

  5752     fix x

  5753     assume "x \<in> s"

  5754     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")

  5755     proof

  5756       fix y

  5757       assume "y \<in> t"

  5758       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  5759       then show "?P y" by (auto elim!: open_prod_elim)

  5760     qed

  5761     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  5762       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"

  5763       by metis

  5764     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  5765     from compactE_image[OF compact t this] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  5766       by auto

  5767     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  5768       by (fastforce simp: subset_eq)

  5769     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  5770       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  5771   qed

  5772   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  5773     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"

  5774     unfolding subset_eq UN_iff by metis

  5775   moreover

  5776   from compactE_image[OF compact s a]

  5777   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  5778     by auto

  5779   moreover

  5780   {

  5781     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  5782       by auto

  5783     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  5784       using d e \<subseteq> s by (intro UN_mono) auto

  5785     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  5786   }

  5787   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  5788     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  5789 qed

  5790

  5791 text{* Hence some useful properties follow quite easily. *}

  5792

  5793 lemma compact_scaling:

  5794   fixes s :: "'a::real_normed_vector set"

  5795   assumes "compact s"

  5796   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5797 proof -

  5798   let ?f = "\<lambda>x. scaleR c x"

  5799   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5800   show ?thesis

  5801     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5802     using linear_continuous_at[OF *] assms

  5803     by auto

  5804 qed

  5805

  5806 lemma compact_negations:

  5807   fixes s :: "'a::real_normed_vector set"

  5808   assumes "compact s"

  5809   shows "compact ((\<lambda>x. - x)  s)"

  5810   using compact_scaling [OF assms, of "- 1"] by auto

  5811

  5812 lemma compact_sums:

  5813   fixes s t :: "'a::real_normed_vector set"

  5814   assumes "compact s"

  5815     and "compact t"

  5816   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5817 proof -

  5818   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5819     apply auto

  5820     unfolding image_iff

  5821     apply (rule_tac x="(xa, y)" in bexI)

  5822     apply auto

  5823     done

  5824   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5825     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5826   then show ?thesis

  5827     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5828 qed

  5829

  5830 lemma compact_differences:

  5831   fixes s t :: "'a::real_normed_vector set"

  5832   assumes "compact s"

  5833     and "compact t"

  5834   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5835 proof-

  5836   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5837     apply auto

  5838     apply (rule_tac x= xa in exI)

  5839     apply auto

  5840     done

  5841   then show ?thesis

  5842     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5843 qed

  5844

  5845 lemma compact_translation:

  5846   fixes s :: "'a::real_normed_vector set"

  5847   assumes "compact s"

  5848   shows "compact ((\<lambda>x. a + x)  s)"

  5849 proof -

  5850   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  5851     by auto

  5852   then show ?thesis

  5853     using compact_sums[OF assms compact_sing[of a]] by auto

  5854 qed

  5855

  5856 lemma compact_affinity:

  5857   fixes s :: "'a::real_normed_vector set"

  5858   assumes "compact s"

  5859   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5860 proof -

  5861   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  5862     by auto

  5863   then show ?thesis

  5864     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5865 qed

  5866

  5867 text {* Hence we get the following. *}

  5868

  5869 lemma compact_sup_maxdistance:

  5870   fixes s :: "'a::metric_space set"

  5871   assumes "compact s"

  5872     and "s \<noteq> {}"

  5873   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5874 proof -

  5875   have "compact (s \<times> s)"

  5876     using compact s by (intro compact_Times)

  5877   moreover have "s \<times> s \<noteq> {}"

  5878     using s \<noteq> {} by auto

  5879   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  5880     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  5881   ultimately show ?thesis

  5882     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  5883 qed

  5884

  5885 text {* We can state this in terms of diameter of a set. *}

  5886

  5887 definition diameter :: "'a::metric_space set \<Rightarrow> real" where

  5888   "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"

  5889

  5890 lemma diameter_bounded_bound:

  5891   fixes s :: "'a :: metric_space set"

  5892   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  5893   shows "dist x y \<le> diameter s"

  5894 proof -

  5895   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  5896     unfolding bounded_def by auto

  5897   have "bdd_above (split dist  (s\<times>s))"

  5898   proof (intro bdd_aboveI, safe)

  5899     fix a b

  5900     assume "a \<in> s" "b \<in> s"

  5901     with z[of a] z[of b] dist_triangle[of a b z]

  5902     show "dist a b \<le> 2 * d"

  5903       by (simp add: dist_commute)

  5904   qed

  5905   moreover have "(x,y) \<in> s\<times>s" using s by auto

  5906   ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"

  5907     by (rule cSUP_upper2) simp

  5908   with x \<in> s show ?thesis

  5909     by (auto simp add: diameter_def)

  5910 qed

  5911

  5912 lemma diameter_lower_bounded:

  5913   fixes s :: "'a :: metric_space set"

  5914   assumes s: "bounded s"

  5915     and d: "0 < d" "d < diameter s"

  5916   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  5917 proof (rule ccontr)

  5918   assume contr: "\<not> ?thesis"

  5919   moreover have "s \<noteq> {}"

  5920     using d by (auto simp add: diameter_def)

  5921   ultimately have "diameter s \<le> d"

  5922     by (auto simp: not_less diameter_def intro!: cSUP_least)

  5923   with d < diameter s show False by auto

  5924 qed

  5925

  5926 lemma diameter_bounded:

  5927   assumes "bounded s"

  5928   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  5929     and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  5930   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  5931   by auto

  5932

  5933 lemma diameter_compact_attained:

  5934   assumes "compact s"

  5935     and "s \<noteq> {}"

  5936   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  5937 proof -

  5938   have b: "bounded s" using assms(1)

  5939     by (rule compact_imp_bounded)

  5940   then obtain x y where xys: "x\<in>s" "y\<in>s"

  5941     and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5942     using compact_sup_maxdistance[OF assms] by auto

  5943   then have "diameter s \<le> dist x y"

  5944     unfolding diameter_def

  5945     apply clarsimp

  5946     apply (rule cSUP_least)

  5947     apply fast+

  5948     done

  5949   then show ?thesis

  5950     by (metis b diameter_bounded_bound order_antisym xys)

  5951 qed

  5952

  5953 text {* Related results with closure as the conclusion. *}

  5954

  5955 lemma closed_scaling:

  5956   fixes s :: "'a::real_normed_vector set"

  5957   assumes "closed s"

  5958   shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5959 proof (cases "c = 0")

  5960   case True then show ?thesis

  5961     by (auto simp add: image_constant_conv)

  5962 next

  5963   case False

  5964   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) - s)"

  5965     by (simp add: continuous_closed_vimage)

  5966   also have "(\<lambda>x. inverse c *\<^sub>R x) - s = (\<lambda>x. c *\<^sub>R x)  s"

  5967     using c \<noteq> 0 by (auto elim: image_eqI [rotated])

  5968   finally show ?thesis .

  5969 qed

  5970

  5971 lemma closed_negations:

  5972   fixes s :: "'a::real_normed_vector set"

  5973   assumes "closed s"

  5974   shows "closed ((\<lambda>x. -x)  s)"

  5975   using closed_scaling[OF assms, of "- 1"] by simp

  5976

  5977 lemma compact_closed_sums:

  5978   fixes s :: "'a::real_normed_vector set"

  5979   assumes "compact s" and "closed t"

  5980   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5981 proof -

  5982   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  5983   {

  5984     fix x l

  5985     assume as: "\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  5986     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"

  5987       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  5988     obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  5989       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  5990     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  5991       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

  5992       unfolding o_def

  5993       by auto

  5994     then have "l - l' \<in> t"

  5995       using assms(2)[unfolded closed_sequential_limits,

  5996         THEN spec[where x="\<lambda> n. snd (f (r n))"],

  5997         THEN spec[where x="l - l'"]]

  5998       using f(3)

  5999       by auto

  6000     then have "l \<in> ?S"

  6001       using l' \<in> s

  6002       apply auto

  6003       apply (rule_tac x=l' in exI)

  6004       apply (rule_tac x="l - l'" in exI)

  6005       apply auto

  6006       done

  6007   }

  6008   then show ?thesis

  6009     unfolding closed_sequential_limits by fast

  6010 qed

  6011

  6012 lemma closed_compact_sums:

  6013   fixes s t :: "'a::real_normed_vector set"

  6014   assumes "closed s"

  6015     and "compact t"

  6016   shows "closed {x + y | x y. x \<in> s \<and> y \&l`