src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author hoelzl Mon Jun 30 15:45:21 2014 +0200 (2014-06-30) changeset 57447 87429bdecad5 parent 57418 6ab1c7cb0b8d child 57448 159e45728ceb permissions -rw-r--r--
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
     1 (*  title:      HOL/Library/Topology_Euclidian_Space.thy

     2     Author:     Amine Chaieb, University of Cambridge

     3     Author:     Robert Himmelmann, TU Muenchen

     4     Author:     Brian Huffman, Portland State University

     5 *)

     6

     7 header {* Elementary topology in Euclidean space. *}

     8

     9 theory Topology_Euclidean_Space

    10 imports

    11   Complex_Main

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

    13   "~~/src/HOL/Library/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   { fix x b :: 'a assume [simp]: "b \<in> Basis"

  1157     have "\<bar>x \<bullet> b\<bar> \<le> real (natceiling \<bar>x \<bullet> b\<bar>)"

  1158       by (rule real_natceiling_ge)

  1159     also have "\<dots> \<le> real (natceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)Basis)))"

  1160       by (auto intro!: natceiling_mono)

  1161     also have "\<dots> < real (natceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)Basis)) + 1)"

  1162       by simp

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

  1164   then have "\<And>x::'a. \<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n"

  1165     by auto

  1166   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"

  1167     by auto

  1168   ultimately show ?thesis

  1169     by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)

  1170 qed

  1171

  1172 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  1173

  1174 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  1175   (\<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)"

  1176

  1177 lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)

  1178   and is_interval_box: "is_interval (box a b)" (is ?th2)

  1179   unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff

  1180   by (meson order_trans le_less_trans less_le_trans less_trans)+

  1181

  1182 lemma is_interval_empty:

  1183  "is_interval {}"

  1184   unfolding is_interval_def

  1185   by simp

  1186

  1187 lemma is_interval_univ:

  1188  "is_interval UNIV"

  1189   unfolding is_interval_def

  1190   by simp

  1191

  1192 lemma mem_is_intervalI:

  1193   assumes "is_interval s"

  1194   assumes "a \<in> s" "b \<in> s"

  1195   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"

  1196   shows "x \<in> s"

  1197   by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

  1198

  1199 lemma interval_subst:

  1200   fixes S::"'a::euclidean_space set"

  1201   assumes "is_interval S"

  1202   assumes "x \<in> S" "y j \<in> S"

  1203   assumes "j \<in> Basis"

  1204   shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"

  1205   by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

  1206

  1207 lemma mem_box_componentwiseI:

  1208   fixes S::"'a::euclidean_space set"

  1209   assumes "is_interval S"

  1210   assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i)  S)"

  1211   shows "x \<in> S"

  1212 proof -

  1213   from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"

  1214     by auto

  1215   with finite_Basis obtain s and bs::"'a list" where

  1216     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

  1217     bs: "set bs = Basis" "distinct bs"

  1218     by (metis finite_distinct_list)

  1219   from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast

  1220   def y \<equiv> "rec_list

  1221     (s j)

  1222     (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"

  1223   have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"

  1224     using bs by (auto simp add: s(1)[symmetric] euclidean_representation)

  1225   also have [symmetric]: "y bs = \<dots>"

  1226     using bs(2) bs(1)[THEN equalityD1]

  1227     by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])

  1228   also have "y bs \<in> S"

  1229     using bs(1)[THEN equalityD1]

  1230     apply (induct bs)

  1231     apply (auto simp: y_def j)

  1232     apply (rule interval_subst[OF assms(1)])

  1233     apply (auto simp: s)

  1234     done

  1235   finally show ?thesis .

  1236 qed

  1237

  1238

  1239 subsection{* Connectedness *}

  1240

  1241 lemma connected_local:

  1242  "connected S \<longleftrightarrow>

  1243   \<not> (\<exists>e1 e2.

  1244       openin (subtopology euclidean S) e1 \<and>

  1245       openin (subtopology euclidean S) e2 \<and>

  1246       S \<subseteq> e1 \<union> e2 \<and>

  1247       e1 \<inter> e2 = {} \<and>

  1248       e1 \<noteq> {} \<and>

  1249       e2 \<noteq> {})"

  1250   unfolding connected_def openin_open

  1251   by blast

  1252

  1253 lemma exists_diff:

  1254   fixes P :: "'a set \<Rightarrow> bool"

  1255   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1256 proof -

  1257   {

  1258     assume "?lhs"

  1259     then have ?rhs by blast

  1260   }

  1261   moreover

  1262   {

  1263     fix S

  1264     assume H: "P S"

  1265     have "S = - (- S)" by auto

  1266     with H have "P (- (- S))" by metis

  1267   }

  1268   ultimately show ?thesis by metis

  1269 qed

  1270

  1271 lemma connected_clopen: "connected S \<longleftrightarrow>

  1272   (\<forall>T. openin (subtopology euclidean S) T \<and>

  1273      closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1274 proof -

  1275   have "\<not> connected S \<longleftrightarrow>

  1276     (\<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> {})"

  1277     unfolding connected_def openin_open closedin_closed

  1278     by (metis double_complement)

  1279   then have th0: "connected S \<longleftrightarrow>

  1280     \<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> {})"

  1281     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

  1282     apply (simp add: closed_def)

  1283     apply metis

  1284     done

  1285   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'))"

  1286     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

  1287     unfolding connected_def openin_open closedin_closed by auto

  1288   {

  1289     fix e2

  1290     {

  1291       fix e1

  1292       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)"

  1293         by auto

  1294     }

  1295     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1296       by metis

  1297   }

  1298   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1299     by blast

  1300   then show ?thesis

  1301     unfolding th0 th1 by simp

  1302 qed

  1303

  1304

  1305 subsection{* Limit points *}

  1306

  1307 definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)

  1308   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))"

  1309

  1310 lemma islimptI:

  1311   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1312   shows "x islimpt S"

  1313   using assms unfolding islimpt_def by auto

  1314

  1315 lemma islimptE:

  1316   assumes "x islimpt S" and "x \<in> T" and "open T"

  1317   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

  1318   using assms unfolding islimpt_def by auto

  1319

  1320 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

  1321   unfolding islimpt_def eventually_at_topological by auto

  1322

  1323 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"

  1324   unfolding islimpt_def by fast

  1325

  1326 lemma islimpt_approachable:

  1327   fixes x :: "'a::metric_space"

  1328   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  1329   unfolding islimpt_iff_eventually eventually_at by fast

  1330

  1331 lemma islimpt_approachable_le:

  1332   fixes x :: "'a::metric_space"

  1333   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"

  1334   unfolding islimpt_approachable

  1335   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

  1336     THEN arg_cong [where f=Not]]

  1337   by (simp add: Bex_def conj_commute conj_left_commute)

  1338

  1339 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  1340   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

  1341

  1342 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

  1343   unfolding islimpt_def by blast

  1344

  1345 text {* A perfect space has no isolated points. *}

  1346

  1347 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  1348   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

  1349

  1350 lemma perfect_choose_dist:

  1351   fixes x :: "'a::{perfect_space, metric_space}"

  1352   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

  1353   using islimpt_UNIV [of x]

  1354   by (simp add: islimpt_approachable)

  1355

  1356 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  1357   unfolding closed_def

  1358   apply (subst open_subopen)

  1359   apply (simp add: islimpt_def subset_eq)

  1360   apply (metis ComplE ComplI)

  1361   done

  1362

  1363 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  1364   unfolding islimpt_def by auto

  1365

  1366 lemma finite_set_avoid:

  1367   fixes a :: "'a::metric_space"

  1368   assumes fS: "finite S"

  1369   shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"

  1370 proof (induct rule: finite_induct[OF fS])

  1371   case 1

  1372   then show ?case by (auto intro: zero_less_one)

  1373 next

  1374   case (2 x F)

  1375   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"

  1376     by blast

  1377   show ?case

  1378   proof (cases "x = a")

  1379     case True

  1380     then show ?thesis using d by auto

  1381   next

  1382     case False

  1383     let ?d = "min d (dist a x)"

  1384     have dp: "?d > 0"

  1385       using False d(1) using dist_nz by auto

  1386     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"

  1387       by auto

  1388     with dp False show ?thesis

  1389       by (auto intro!: exI[where x="?d"])

  1390   qed

  1391 qed

  1392

  1393 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  1394   by (simp add: islimpt_iff_eventually eventually_conj_iff)

  1395

  1396 lemma discrete_imp_closed:

  1397   fixes S :: "'a::metric_space set"

  1398   assumes e: "0 < e"

  1399     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  1400   shows "closed S"

  1401 proof -

  1402   {

  1403     fix x

  1404     assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

  1405     from e have e2: "e/2 > 0" by arith

  1406     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"

  1407       by blast

  1408     let ?m = "min (e/2) (dist x y) "

  1409     from e2 y(2) have mp: "?m > 0"

  1410       by (simp add: dist_nz[symmetric])

  1411     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"

  1412       by blast

  1413     have th: "dist z y < e" using z y

  1414       by (intro dist_triangle_lt [where z=x], simp)

  1415     from d[rule_format, OF y(1) z(1) th] y z

  1416     have False by (auto simp add: dist_commute)}

  1417   then show ?thesis

  1418     by (metis islimpt_approachable closed_limpt [where 'a='a])

  1419 qed

  1420

  1421

  1422 subsection {* Interior of a Set *}

  1423

  1424 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

  1425

  1426 lemma interiorI [intro?]:

  1427   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  1428   shows "x \<in> interior S"

  1429   using assms unfolding interior_def by fast

  1430

  1431 lemma interiorE [elim?]:

  1432   assumes "x \<in> interior S"

  1433   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  1434   using assms unfolding interior_def by fast

  1435

  1436 lemma open_interior [simp, intro]: "open (interior S)"

  1437   by (simp add: interior_def open_Union)

  1438

  1439 lemma interior_subset: "interior S \<subseteq> S"

  1440   by (auto simp add: interior_def)

  1441

  1442 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  1443   by (auto simp add: interior_def)

  1444

  1445 lemma interior_open: "open S \<Longrightarrow> interior S = S"

  1446   by (intro equalityI interior_subset interior_maximal subset_refl)

  1447

  1448 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  1449   by (metis open_interior interior_open)

  1450

  1451 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  1452   by (metis interior_maximal interior_subset subset_trans)

  1453

  1454 lemma interior_empty [simp]: "interior {} = {}"

  1455   using open_empty by (rule interior_open)

  1456

  1457 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  1458   using open_UNIV by (rule interior_open)

  1459

  1460 lemma interior_interior [simp]: "interior (interior S) = interior S"

  1461   using open_interior by (rule interior_open)

  1462

  1463 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  1464   by (auto simp add: interior_def)

  1465

  1466 lemma interior_unique:

  1467   assumes "T \<subseteq> S" and "open T"

  1468   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

  1469   shows "interior S = T"

  1470   by (intro equalityI assms interior_subset open_interior interior_maximal)

  1471

  1472 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

  1473   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1474     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1475

  1476 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1477   using open_contains_ball_eq [where S="interior S"]

  1478   by (simp add: open_subset_interior)

  1479

  1480 lemma interior_limit_point [intro]:

  1481   fixes x :: "'a::perfect_space"

  1482   assumes x: "x \<in> interior S"

  1483   shows "x islimpt S"

  1484   using x islimpt_UNIV [of x]

  1485   unfolding interior_def islimpt_def

  1486   apply (clarsimp, rename_tac T T')

  1487   apply (drule_tac x="T \<inter> T'" in spec)

  1488   apply (auto simp add: open_Int)

  1489   done

  1490

  1491 lemma interior_closed_Un_empty_interior:

  1492   assumes cS: "closed S"

  1493     and iT: "interior T = {}"

  1494   shows "interior (S \<union> T) = interior S"

  1495 proof

  1496   show "interior S \<subseteq> interior (S \<union> T)"

  1497     by (rule interior_mono) (rule Un_upper1)

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

  1499   proof

  1500     fix x

  1501     assume "x \<in> interior (S \<union> T)"

  1502     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1503     show "x \<in> interior S"

  1504     proof (rule ccontr)

  1505       assume "x \<notin> interior S"

  1506       with x \<in> R open R obtain y where "y \<in> R - S"

  1507         unfolding interior_def by fast

  1508       from open R closed S have "open (R - S)"

  1509         by (rule open_Diff)

  1510       from R \<subseteq> S \<union> T have "R - S \<subseteq> T"

  1511         by fast

  1512       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {} show False

  1513         unfolding interior_def by fast

  1514     qed

  1515   qed

  1516 qed

  1517

  1518 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1519 proof (rule interior_unique)

  1520   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1521     by (intro Sigma_mono interior_subset)

  1522   show "open (interior A \<times> interior B)"

  1523     by (intro open_Times open_interior)

  1524   fix T

  1525   assume "T \<subseteq> A \<times> B" and "open T"

  1526   then show "T \<subseteq> interior A \<times> interior B"

  1527   proof safe

  1528     fix x y

  1529     assume "(x, y) \<in> T"

  1530     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1531       using open T unfolding open_prod_def by fast

  1532     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1533       using T \<subseteq> A \<times> B by auto

  1534     then show "x \<in> interior A" and "y \<in> interior B"

  1535       by (auto intro: interiorI)

  1536   qed

  1537 qed

  1538

  1539

  1540 subsection {* Closure of a Set *}

  1541

  1542 definition "closure S = S \<union> {x | x. x islimpt S}"

  1543

  1544 lemma interior_closure: "interior S = - (closure (- S))"

  1545   unfolding interior_def closure_def islimpt_def by auto

  1546

  1547 lemma closure_interior: "closure S = - interior (- S)"

  1548   unfolding interior_closure by simp

  1549

  1550 lemma closed_closure[simp, intro]: "closed (closure S)"

  1551   unfolding closure_interior by (simp add: closed_Compl)

  1552

  1553 lemma closure_subset: "S \<subseteq> closure S"

  1554   unfolding closure_def by simp

  1555

  1556 lemma closure_hull: "closure S = closed hull S"

  1557   unfolding hull_def closure_interior interior_def by auto

  1558

  1559 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1560   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1561

  1562 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1563   unfolding closure_eq .

  1564

  1565 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1566   unfolding closure_hull by (rule hull_hull)

  1567

  1568 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1569   unfolding closure_hull by (rule hull_mono)

  1570

  1571 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1572   unfolding closure_hull by (rule hull_minimal)

  1573

  1574 lemma closure_unique:

  1575   assumes "S \<subseteq> T"

  1576     and "closed T"

  1577     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1578   shows "closure S = T"

  1579   using assms unfolding closure_hull by (rule hull_unique)

  1580

  1581 lemma closure_empty [simp]: "closure {} = {}"

  1582   using closed_empty by (rule closure_closed)

  1583

  1584 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1585   using closed_UNIV by (rule closure_closed)

  1586

  1587 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1588   unfolding closure_interior by simp

  1589

  1590 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1591   using closure_empty closure_subset[of S]

  1592   by blast

  1593

  1594 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1595   using closure_eq[of S] closure_subset[of S]

  1596   by simp

  1597

  1598 lemma open_inter_closure_eq_empty:

  1599   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1600   using open_subset_interior[of S "- T"]

  1601   using interior_subset[of "- T"]

  1602   unfolding closure_interior

  1603   by auto

  1604

  1605 lemma open_inter_closure_subset:

  1606   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1607 proof

  1608   fix x

  1609   assume as: "open S" "x \<in> S \<inter> closure T"

  1610   {

  1611     assume *: "x islimpt T"

  1612     have "x islimpt (S \<inter> T)"

  1613     proof (rule islimptI)

  1614       fix A

  1615       assume "x \<in> A" "open A"

  1616       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1617         by (simp_all add: open_Int)

  1618       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1619         by (rule islimptE)

  1620       then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1621         by simp_all

  1622       then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1623     qed

  1624   }

  1625   then show "x \<in> closure (S \<inter> T)" using as

  1626     unfolding closure_def

  1627     by blast

  1628 qed

  1629

  1630 lemma closure_complement: "closure (- S) = - interior S"

  1631   unfolding closure_interior by simp

  1632

  1633 lemma interior_complement: "interior (- S) = - closure S"

  1634   unfolding closure_interior by simp

  1635

  1636 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1637 proof (rule closure_unique)

  1638   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1639     by (intro Sigma_mono closure_subset)

  1640   show "closed (closure A \<times> closure B)"

  1641     by (intro closed_Times closed_closure)

  1642   fix T

  1643   assume "A \<times> B \<subseteq> T" and "closed T"

  1644   then show "closure A \<times> closure B \<subseteq> T"

  1645     apply (simp add: closed_def open_prod_def, clarify)

  1646     apply (rule ccontr)

  1647     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1648     apply (simp add: closure_interior interior_def)

  1649     apply (drule_tac x=C in spec)

  1650     apply (drule_tac x=D in spec)

  1651     apply auto

  1652     done

  1653 qed

  1654

  1655 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1656   unfolding closure_def using islimpt_punctured by blast

  1657

  1658

  1659 subsection {* Frontier (aka boundary) *}

  1660

  1661 definition "frontier S = closure S - interior S"

  1662

  1663 lemma frontier_closed: "closed (frontier S)"

  1664   by (simp add: frontier_def closed_Diff)

  1665

  1666 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1667   by (auto simp add: frontier_def interior_closure)

  1668

  1669 lemma frontier_straddle:

  1670   fixes a :: "'a::metric_space"

  1671   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))"

  1672   unfolding frontier_def closure_interior

  1673   by (auto simp add: mem_interior subset_eq ball_def)

  1674

  1675 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1676   by (metis frontier_def closure_closed Diff_subset)

  1677

  1678 lemma frontier_empty[simp]: "frontier {} = {}"

  1679   by (simp add: frontier_def)

  1680

  1681 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1682 proof-

  1683   {

  1684     assume "frontier S \<subseteq> S"

  1685     then have "closure S \<subseteq> S"

  1686       using interior_subset unfolding frontier_def by auto

  1687     then have "closed S"

  1688       using closure_subset_eq by auto

  1689   }

  1690   then show ?thesis using frontier_subset_closed[of S] ..

  1691 qed

  1692

  1693 lemma frontier_complement: "frontier(- S) = frontier S"

  1694   by (auto simp add: frontier_def closure_complement interior_complement)

  1695

  1696 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1697   using frontier_complement frontier_subset_eq[of "- S"]

  1698   unfolding open_closed by auto

  1699

  1700 subsection {* Filters and the eventually true'' quantifier *}

  1701

  1702 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1703     (infixr "indirection" 70)

  1704   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1705

  1706 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1707

  1708 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1709 proof

  1710   assume "trivial_limit (at a within S)"

  1711   then show "\<not> a islimpt S"

  1712     unfolding trivial_limit_def

  1713     unfolding eventually_at_topological

  1714     unfolding islimpt_def

  1715     apply (clarsimp simp add: set_eq_iff)

  1716     apply (rename_tac T, rule_tac x=T in exI)

  1717     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1718     done

  1719 next

  1720   assume "\<not> a islimpt S"

  1721   then show "trivial_limit (at a within S)"

  1722     unfolding trivial_limit_def eventually_at_topological islimpt_def

  1723     by metis

  1724 qed

  1725

  1726 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1727   using trivial_limit_within [of a UNIV] by simp

  1728

  1729 lemma trivial_limit_at:

  1730   fixes a :: "'a::perfect_space"

  1731   shows "\<not> trivial_limit (at a)"

  1732   by (rule at_neq_bot)

  1733

  1734 lemma trivial_limit_at_infinity:

  1735   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1736   unfolding trivial_limit_def eventually_at_infinity

  1737   apply clarsimp

  1738   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1739    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1740   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1741   apply (drule_tac x=UNIV in spec, simp)

  1742   done

  1743

  1744 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"

  1745   using islimpt_in_closure

  1746   by (metis trivial_limit_within)

  1747

  1748 text {* Some property holds "sufficiently close" to the limit point. *}

  1749

  1750 lemma eventually_at2:

  1751   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1752   unfolding eventually_at dist_nz by auto

  1753

  1754 lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"

  1755   unfolding trivial_limit_def

  1756   by (auto elim: eventually_rev_mp)

  1757

  1758 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1759   by simp

  1760

  1761 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1762   by (simp add: filter_eq_iff)

  1763

  1764 text{* Combining theorems for "eventually" *}

  1765

  1766 lemma eventually_rev_mono:

  1767   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1768   using eventually_mono [of P Q] by fast

  1769

  1770 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"

  1771   by (simp add: eventually_False)

  1772

  1773

  1774 subsection {* Limits *}

  1775

  1776 lemma Lim:

  1777   "(f ---> l) net \<longleftrightarrow>

  1778         trivial_limit net \<or>

  1779         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1780   unfolding tendsto_iff trivial_limit_eq by auto

  1781

  1782 text{* Show that they yield usual definitions in the various cases. *}

  1783

  1784 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1785     (\<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)"

  1786   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1787

  1788 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1789     (\<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)"

  1790   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1791

  1792 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1793     (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"

  1794   by (auto simp add: tendsto_iff eventually_at2)

  1795

  1796 lemma Lim_at_infinity:

  1797   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"

  1798   by (auto simp add: tendsto_iff eventually_at_infinity)

  1799

  1800 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1801   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1802

  1803 text{* The expected monotonicity property. *}

  1804

  1805 lemma Lim_Un:

  1806   assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1807   shows "(f ---> l) (at x within (S \<union> T))"

  1808   using assms unfolding at_within_union by (rule filterlim_sup)

  1809

  1810 lemma Lim_Un_univ:

  1811   "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>

  1812     S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"

  1813   by (metis Lim_Un)

  1814

  1815 text{* Interrelations between restricted and unrestricted limits. *}

  1816

  1817 lemma Lim_at_within: (* FIXME: rename *)

  1818   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1819   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1820

  1821 lemma eventually_within_interior:

  1822   assumes "x \<in> interior S"

  1823   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"

  1824   (is "?lhs = ?rhs")

  1825 proof

  1826   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1827   {

  1828     assume "?lhs"

  1829     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"

  1830       unfolding eventually_at_topological

  1831       by auto

  1832     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"

  1833       by auto

  1834     then show "?rhs"

  1835       unfolding eventually_at_topological by auto

  1836   next

  1837     assume "?rhs"

  1838     then show "?lhs"

  1839       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1840   }

  1841 qed

  1842

  1843 lemma at_within_interior:

  1844   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1845   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1846

  1847 lemma Lim_within_LIMSEQ:

  1848   fixes a :: "'a::first_countable_topology"

  1849   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1850   shows "(X ---> L) (at a within T)"

  1851   using assms unfolding tendsto_def [where l=L]

  1852   by (simp add: sequentially_imp_eventually_within)

  1853

  1854 lemma Lim_right_bound:

  1855   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1856     'b::{linorder_topology, conditionally_complete_linorder}"

  1857   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"

  1858     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1859   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1860 proof (cases "{x<..} \<inter> I = {}")

  1861   case True

  1862   then show ?thesis by simp

  1863 next

  1864   case False

  1865   show ?thesis

  1866   proof (rule order_tendstoI)

  1867     fix a

  1868     assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1869     {

  1870       fix y

  1871       assume "y \<in> {x<..} \<inter> I"

  1872       with False bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1873         by (auto intro!: cInf_lower bdd_belowI2 simp del: Inf_image_eq)

  1874       with a have "a < f y"

  1875         by (blast intro: less_le_trans)

  1876     }

  1877     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1878       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1879   next

  1880     fix a

  1881     assume "Inf (f  ({x<..} \<inter> I)) < a"

  1882     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"

  1883       by auto

  1884     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1885       unfolding eventually_at_right[OF x < y] by (metis less_imp_le le_less_trans mono)

  1886     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1887       unfolding eventually_at_filter by eventually_elim simp

  1888   qed

  1889 qed

  1890

  1891 text{* Another limit point characterization. *}

  1892

  1893 lemma islimpt_sequential:

  1894   fixes x :: "'a::first_countable_topology"

  1895   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1896     (is "?lhs = ?rhs")

  1897 proof

  1898   assume ?lhs

  1899   from countable_basis_at_decseq[of x] obtain A where A:

  1900       "\<And>i. open (A i)"

  1901       "\<And>i. x \<in> A i"

  1902       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  1903     by blast

  1904   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1905   {

  1906     fix n

  1907     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1908       unfolding islimpt_def using A(1,2)[of n] by auto

  1909     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1910       unfolding f_def by (rule someI_ex)

  1911     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto

  1912   }

  1913   then have "\<forall>n. f n \<in> S - {x}" by auto

  1914   moreover have "(\<lambda>n. f n) ----> x"

  1915   proof (rule topological_tendstoI)

  1916     fix S

  1917     assume "open S" "x \<in> S"

  1918     from A(3)[OF this] \<And>n. f n \<in> A n

  1919     show "eventually (\<lambda>x. f x \<in> S) sequentially"

  1920       by (auto elim!: eventually_elim1)

  1921   qed

  1922   ultimately show ?rhs by fast

  1923 next

  1924   assume ?rhs

  1925   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"

  1926     by auto

  1927   show ?lhs

  1928     unfolding islimpt_def

  1929   proof safe

  1930     fix T

  1931     assume "open T" "x \<in> T"

  1932     from lim[THEN topological_tendstoD, OF this] f

  1933     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1934       unfolding eventually_sequentially by auto

  1935   qed

  1936 qed

  1937

  1938 lemma Lim_null:

  1939   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1940   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1941   by (simp add: Lim dist_norm)

  1942

  1943 lemma Lim_null_comparison:

  1944   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1945   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1946   shows "(f ---> 0) net"

  1947   using assms(2)

  1948 proof (rule metric_tendsto_imp_tendsto)

  1949   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1950     using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

  1951 qed

  1952

  1953 lemma Lim_transform_bound:

  1954   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1955     and g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1956   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"

  1957     and "(g ---> 0) net"

  1958   shows "(f ---> 0) net"

  1959   using assms(1) tendsto_norm_zero [OF assms(2)]

  1960   by (rule Lim_null_comparison)

  1961

  1962 text{* Deducing things about the limit from the elements. *}

  1963

  1964 lemma Lim_in_closed_set:

  1965   assumes "closed S"

  1966     and "eventually (\<lambda>x. f(x) \<in> S) net"

  1967     and "\<not> trivial_limit net" "(f ---> l) net"

  1968   shows "l \<in> S"

  1969 proof (rule ccontr)

  1970   assume "l \<notin> S"

  1971   with closed S have "open (- S)" "l \<in> - S"

  1972     by (simp_all add: open_Compl)

  1973   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1974     by (rule topological_tendstoD)

  1975   with assms(2) have "eventually (\<lambda>x. False) net"

  1976     by (rule eventually_elim2) simp

  1977   with assms(3) show "False"

  1978     by (simp add: eventually_False)

  1979 qed

  1980

  1981 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1982

  1983 lemma Lim_dist_ubound:

  1984   assumes "\<not>(trivial_limit net)"

  1985     and "(f ---> l) net"

  1986     and "eventually (\<lambda>x. dist a (f x) \<le> e) net"

  1987   shows "dist a l \<le> e"

  1988   using assms by (fast intro: tendsto_le tendsto_intros)

  1989

  1990 lemma Lim_norm_ubound:

  1991   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1992   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"

  1993   shows "norm(l) \<le> e"

  1994   using assms by (fast intro: tendsto_le tendsto_intros)

  1995

  1996 lemma Lim_norm_lbound:

  1997   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1998   assumes "\<not> trivial_limit net"

  1999     and "(f ---> l) net"

  2000     and "eventually (\<lambda>x. e \<le> norm (f x)) net"

  2001   shows "e \<le> norm l"

  2002   using assms by (fast intro: tendsto_le tendsto_intros)

  2003

  2004 text{* Limit under bilinear function *}

  2005

  2006 lemma Lim_bilinear:

  2007   assumes "(f ---> l) net"

  2008     and "(g ---> m) net"

  2009     and "bounded_bilinear h"

  2010   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  2011   using bounded_bilinear h (f ---> l) net (g ---> m) net

  2012   by (rule bounded_bilinear.tendsto)

  2013

  2014 text{* These are special for limits out of the same vector space. *}

  2015

  2016 lemma Lim_within_id: "(id ---> a) (at a within s)"

  2017   unfolding id_def by (rule tendsto_ident_at)

  2018

  2019 lemma Lim_at_id: "(id ---> a) (at a)"

  2020   unfolding id_def by (rule tendsto_ident_at)

  2021

  2022 lemma Lim_at_zero:

  2023   fixes a :: "'a::real_normed_vector"

  2024     and l :: "'b::topological_space"

  2025   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"

  2026   using LIM_offset_zero LIM_offset_zero_cancel ..

  2027

  2028 text{* It's also sometimes useful to extract the limit point from the filter. *}

  2029

  2030 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  2031   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  2032

  2033 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  2034   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  2035

  2036 lemma netlimit_at:

  2037   fixes a :: "'a::{perfect_space,t2_space}"

  2038   shows "netlimit (at a) = a"

  2039   using netlimit_within [of a UNIV] by simp

  2040

  2041 lemma lim_within_interior:

  2042   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  2043   by (metis at_within_interior)

  2044

  2045 lemma netlimit_within_interior:

  2046   fixes x :: "'a::{t2_space,perfect_space}"

  2047   assumes "x \<in> interior S"

  2048   shows "netlimit (at x within S) = x"

  2049   using assms by (metis at_within_interior netlimit_at)

  2050

  2051 text{* Transformation of limit. *}

  2052

  2053 lemma Lim_transform:

  2054   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  2055   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  2056   shows "(g ---> l) net"

  2057   using tendsto_diff [OF assms(2) assms(1)] by simp

  2058

  2059 lemma Lim_transform_eventually:

  2060   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  2061   apply (rule topological_tendstoI)

  2062   apply (drule (2) topological_tendstoD)

  2063   apply (erule (1) eventually_elim2, simp)

  2064   done

  2065

  2066 lemma Lim_transform_within:

  2067   assumes "0 < d"

  2068     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2069     and "(f ---> l) (at x within S)"

  2070   shows "(g ---> l) (at x within S)"

  2071 proof (rule Lim_transform_eventually)

  2072   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  2073     using assms(1,2) by (auto simp: dist_nz eventually_at)

  2074   show "(f ---> l) (at x within S)" by fact

  2075 qed

  2076

  2077 lemma Lim_transform_at:

  2078   assumes "0 < d"

  2079     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2080     and "(f ---> l) (at x)"

  2081   shows "(g ---> l) (at x)"

  2082   using _ assms(3)

  2083 proof (rule Lim_transform_eventually)

  2084   show "eventually (\<lambda>x. f x = g x) (at x)"

  2085     unfolding eventually_at2

  2086     using assms(1,2) by auto

  2087 qed

  2088

  2089 text{* Common case assuming being away from some crucial point like 0. *}

  2090

  2091 lemma Lim_transform_away_within:

  2092   fixes a b :: "'a::t1_space"

  2093   assumes "a \<noteq> b"

  2094     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2095     and "(f ---> l) (at a within S)"

  2096   shows "(g ---> l) (at a within S)"

  2097 proof (rule Lim_transform_eventually)

  2098   show "(f ---> l) (at a within S)" by fact

  2099   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  2100     unfolding eventually_at_topological

  2101     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  2102 qed

  2103

  2104 lemma Lim_transform_away_at:

  2105   fixes a b :: "'a::t1_space"

  2106   assumes ab: "a\<noteq>b"

  2107     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2108     and fl: "(f ---> l) (at a)"

  2109   shows "(g ---> l) (at a)"

  2110   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  2111

  2112 text{* Alternatively, within an open set. *}

  2113

  2114 lemma Lim_transform_within_open:

  2115   assumes "open S" and "a \<in> S"

  2116     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  2117     and "(f ---> l) (at a)"

  2118   shows "(g ---> l) (at a)"

  2119 proof (rule Lim_transform_eventually)

  2120   show "eventually (\<lambda>x. f x = g x) (at a)"

  2121     unfolding eventually_at_topological

  2122     using assms(1,2,3) by auto

  2123   show "(f ---> l) (at a)" by fact

  2124 qed

  2125

  2126 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  2127

  2128 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  2129

  2130 lemma Lim_cong_within(*[cong add]*):

  2131   assumes "a = b"

  2132     and "x = y"

  2133     and "S = T"

  2134     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  2135   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  2136   unfolding tendsto_def eventually_at_topological

  2137   using assms by simp

  2138

  2139 lemma Lim_cong_at(*[cong add]*):

  2140   assumes "a = b" "x = y"

  2141     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  2142   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  2143   unfolding tendsto_def eventually_at_topological

  2144   using assms by simp

  2145

  2146 text{* Useful lemmas on closure and set of possible sequential limits.*}

  2147

  2148 lemma closure_sequential:

  2149   fixes l :: "'a::first_countable_topology"

  2150   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"

  2151   (is "?lhs = ?rhs")

  2152 proof

  2153   assume "?lhs"

  2154   moreover

  2155   {

  2156     assume "l \<in> S"

  2157     then have "?rhs" using tendsto_const[of l sequentially] by auto

  2158   }

  2159   moreover

  2160   {

  2161     assume "l islimpt S"

  2162     then have "?rhs" unfolding islimpt_sequential by auto

  2163   }

  2164   ultimately show "?rhs"

  2165     unfolding closure_def by auto

  2166 next

  2167   assume "?rhs"

  2168   then show "?lhs" unfolding closure_def islimpt_sequential by auto

  2169 qed

  2170

  2171 lemma closed_sequential_limits:

  2172   fixes S :: "'a::first_countable_topology set"

  2173   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  2174 by (metis closure_sequential closure_subset_eq subset_iff)

  2175

  2176 lemma closure_approachable:

  2177   fixes S :: "'a::metric_space set"

  2178   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  2179   apply (auto simp add: closure_def islimpt_approachable)

  2180   apply (metis dist_self)

  2181   done

  2182

  2183 lemma closed_approachable:

  2184   fixes S :: "'a::metric_space set"

  2185   shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  2186   by (metis closure_closed closure_approachable)

  2187

  2188 lemma closure_contains_Inf:

  2189   fixes S :: "real set"

  2190   assumes "S \<noteq> {}" "bdd_below S"

  2191   shows "Inf S \<in> closure S"

  2192 proof -

  2193   have *: "\<forall>x\<in>S. Inf S \<le> x"

  2194     using cInf_lower[of _ S] assms by metis

  2195   {

  2196     fix e :: real

  2197     assume "e > 0"

  2198     then have "Inf S < Inf S + e" by simp

  2199     with assms obtain x where "x \<in> S" "x < Inf S + e"

  2200       by (subst (asm) cInf_less_iff) auto

  2201     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  2202       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  2203   }

  2204   then show ?thesis unfolding closure_approachable by auto

  2205 qed

  2206

  2207 lemma closed_contains_Inf:

  2208   fixes S :: "real set"

  2209   shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"

  2210   by (metis closure_contains_Inf closure_closed assms)

  2211

  2212 lemma not_trivial_limit_within_ball:

  2213   "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  2214   (is "?lhs = ?rhs")

  2215 proof -

  2216   {

  2217     assume "?lhs"

  2218     {

  2219       fix e :: real

  2220       assume "e > 0"

  2221       then obtain y where "y \<in> S - {x}" and "dist y x < e"

  2222         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2223         by auto

  2224       then have "y \<in> S \<inter> ball x e - {x}"

  2225         unfolding ball_def by (simp add: dist_commute)

  2226       then have "S \<inter> ball x e - {x} \<noteq> {}" by blast

  2227     }

  2228     then have "?rhs" by auto

  2229   }

  2230   moreover

  2231   {

  2232     assume "?rhs"

  2233     {

  2234       fix e :: real

  2235       assume "e > 0"

  2236       then obtain y where "y \<in> S \<inter> ball x e - {x}"

  2237         using ?rhs by blast

  2238       then have "y \<in> S - {x}" and "dist y x < e"

  2239         unfolding ball_def by (simp_all add: dist_commute)

  2240       then have "\<exists>y \<in> S - {x}. dist y x < e"

  2241         by auto

  2242     }

  2243     then have "?lhs"

  2244       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2245       by auto

  2246   }

  2247   ultimately show ?thesis by auto

  2248 qed

  2249

  2250

  2251 subsection {* Infimum Distance *}

  2252

  2253 definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"

  2254

  2255 lemma bdd_below_infdist[intro, simp]: "bdd_below (dist xA)"

  2256   by (auto intro!: zero_le_dist)

  2257

  2258 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"

  2259   by (simp add: infdist_def)

  2260

  2261 lemma infdist_nonneg: "0 \<le> infdist x A"

  2262   by (auto simp add: infdist_def intro: cINF_greatest)

  2263

  2264 lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"

  2265   by (auto intro: cINF_lower simp add: infdist_def)

  2266

  2267 lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"

  2268   by (auto intro!: cINF_lower2 simp add: infdist_def)

  2269

  2270 lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"

  2271   by (auto intro!: antisym infdist_nonneg infdist_le2)

  2272

  2273 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  2274 proof (cases "A = {}")

  2275   case True

  2276   then show ?thesis by (simp add: infdist_def)

  2277 next

  2278   case False

  2279   then obtain a where "a \<in> A" by auto

  2280   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  2281   proof (rule cInf_greatest)

  2282     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  2283       by simp

  2284     fix d

  2285     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  2286     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  2287       by auto

  2288     show "infdist x A \<le> d"

  2289       unfolding infdist_notempty[OF A \<noteq> {}]

  2290     proof (rule cINF_lower2)

  2291       show "a \<in> A" by fact

  2292       show "dist x a \<le> d"

  2293         unfolding d by (rule dist_triangle)

  2294     qed simp

  2295   qed

  2296   also have "\<dots> = dist x y + infdist y A"

  2297   proof (rule cInf_eq, safe)

  2298     fix a

  2299     assume "a \<in> A"

  2300     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  2301       by (auto intro: infdist_le)

  2302   next

  2303     fix i

  2304     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  2305     then have "i - dist x y \<le> infdist y A"

  2306       unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  2307       by (intro cINF_greatest) (auto simp: field_simps)

  2308     then show "i \<le> dist x y + infdist y A"

  2309       by simp

  2310   qed

  2311   finally show ?thesis by simp

  2312 qed

  2313

  2314 lemma in_closure_iff_infdist_zero:

  2315   assumes "A \<noteq> {}"

  2316   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2317 proof

  2318   assume "x \<in> closure A"

  2319   show "infdist x A = 0"

  2320   proof (rule ccontr)

  2321     assume "infdist x A \<noteq> 0"

  2322     with infdist_nonneg[of x A] have "infdist x A > 0"

  2323       by auto

  2324     then have "ball x (infdist x A) \<inter> closure A = {}"

  2325       apply auto

  2326       apply (metis x \<in> closure A closure_approachable dist_commute infdist_le not_less)

  2327       done

  2328     then have "x \<notin> closure A"

  2329       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  2330     then show False using x \<in> closure A by simp

  2331   qed

  2332 next

  2333   assume x: "infdist x A = 0"

  2334   then obtain a where "a \<in> A"

  2335     by atomize_elim (metis all_not_in_conv assms)

  2336   show "x \<in> closure A"

  2337     unfolding closure_approachable

  2338     apply safe

  2339   proof (rule ccontr)

  2340     fix e :: real

  2341     assume "e > 0"

  2342     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  2343     then have "infdist x A \<ge> e" using a \<in> A

  2344       unfolding infdist_def

  2345       by (force simp: dist_commute intro: cINF_greatest)

  2346     with x e > 0 show False by auto

  2347   qed

  2348 qed

  2349

  2350 lemma in_closed_iff_infdist_zero:

  2351   assumes "closed A" "A \<noteq> {}"

  2352   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  2353 proof -

  2354   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2355     by (rule in_closure_iff_infdist_zero) fact

  2356   with assms show ?thesis by simp

  2357 qed

  2358

  2359 lemma tendsto_infdist [tendsto_intros]:

  2360   assumes f: "(f ---> l) F"

  2361   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  2362 proof (rule tendstoI)

  2363   fix e ::real

  2364   assume "e > 0"

  2365   from tendstoD[OF f this]

  2366   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  2367   proof (eventually_elim)

  2368     fix x

  2369     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  2370     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  2371       by (simp add: dist_commute dist_real_def)

  2372     also assume "dist (f x) l < e"

  2373     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  2374   qed

  2375 qed

  2376

  2377 text{* Some other lemmas about sequences. *}

  2378

  2379 lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)

  2380   assumes "eventually (\<lambda>i. P i) sequentially"

  2381   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  2382   using assms by (rule eventually_sequentially_seg [THEN iffD2])

  2383

  2384 lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)

  2385   "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  2386   apply (erule filterlim_compose)

  2387   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)

  2388   apply arith

  2389   done

  2390

  2391 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  2392   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

  2393

  2394 subsection {* More properties of closed balls *}

  2395

  2396 lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *)

  2397   assumes "closed s" and "continuous_on UNIV f"

  2398   shows "closed (vimage f s)"

  2399   using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]

  2400   by simp

  2401

  2402 lemma closed_cball: "closed (cball x e)"

  2403 proof -

  2404   have "closed (dist x - {..e})"

  2405     by (intro closed_vimage closed_atMost continuous_intros)

  2406   also have "dist x - {..e} = cball x e"

  2407     by auto

  2408   finally show ?thesis .

  2409 qed

  2410

  2411 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  2412 proof -

  2413   {

  2414     fix x and e::real

  2415     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  2416     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  2417   }

  2418   moreover

  2419   {

  2420     fix x and e::real

  2421     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  2422     then have "\<exists>d>0. ball x d \<subseteq> S"

  2423       unfolding subset_eq

  2424       apply(rule_tac x="e/2" in exI)

  2425       apply auto

  2426       done

  2427   }

  2428   ultimately show ?thesis

  2429     unfolding open_contains_ball by auto

  2430 qed

  2431

  2432 lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  2433   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  2434

  2435 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  2436   apply (simp add: interior_def, safe)

  2437   apply (force simp add: open_contains_cball)

  2438   apply (rule_tac x="ball x e" in exI)

  2439   apply (simp add: subset_trans [OF ball_subset_cball])

  2440   done

  2441

  2442 lemma islimpt_ball:

  2443   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  2444   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"

  2445   (is "?lhs = ?rhs")

  2446 proof

  2447   assume "?lhs"

  2448   {

  2449     assume "e \<le> 0"

  2450     then have *:"ball x e = {}"

  2451       using ball_eq_empty[of x e] by auto

  2452     have False using ?lhs

  2453       unfolding * using islimpt_EMPTY[of y] by auto

  2454   }

  2455   then have "e > 0" by (metis not_less)

  2456   moreover

  2457   have "y \<in> cball x e"

  2458     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  2459       ball_subset_cball[of x e] ?lhs

  2460     unfolding closed_limpt by auto

  2461   ultimately show "?rhs" by auto

  2462 next

  2463   assume "?rhs"

  2464   then have "e > 0" by auto

  2465   {

  2466     fix d :: real

  2467     assume "d > 0"

  2468     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2469     proof (cases "d \<le> dist x y")

  2470       case True

  2471       then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2472       proof (cases "x = y")

  2473         case True

  2474         then have False

  2475           using d \<le> dist x y d>0 by auto

  2476         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2477           by auto

  2478       next

  2479         case False

  2480         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

  2481           norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2482           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

  2483           by auto

  2484         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  2485           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

  2486           unfolding scaleR_minus_left scaleR_one

  2487           by (auto simp add: norm_minus_commute)

  2488         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  2489           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  2490           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm]

  2491           by auto

  2492         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs

  2493           by (auto simp add: dist_norm)

  2494         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0

  2495           by auto

  2496         moreover

  2497         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2498           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff

  2499           by (auto simp add: dist_commute)

  2500         moreover

  2501         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

  2502           unfolding dist_norm

  2503           apply simp

  2504           unfolding norm_minus_cancel

  2505           using d > 0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2506           unfolding dist_norm

  2507           apply auto

  2508           done

  2509         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2510           apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

  2511           apply auto

  2512           done

  2513       qed

  2514     next

  2515       case False

  2516       then have "d > dist x y" by auto

  2517       show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

  2518       proof (cases "x = y")

  2519         case True

  2520         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2521           using perfect_choose_dist[of "min e d" y]

  2522           using d > 0 e>0 by auto

  2523         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2524           unfolding x = y

  2525           using z \<noteq> y **

  2526           apply (rule_tac x=z in bexI)

  2527           apply (auto simp add: dist_commute)

  2528           done

  2529       next

  2530         case False

  2531         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2532           using d>0 d > dist x y ?rhs

  2533           apply (rule_tac x=x in bexI)

  2534           apply auto

  2535           done

  2536       qed

  2537     qed

  2538   }

  2539   then show "?lhs"

  2540     unfolding mem_cball islimpt_approachable mem_ball by auto

  2541 qed

  2542

  2543 lemma closure_ball_lemma:

  2544   fixes x y :: "'a::real_normed_vector"

  2545   assumes "x \<noteq> y"

  2546   shows "y islimpt ball x (dist x y)"

  2547 proof (rule islimptI)

  2548   fix T

  2549   assume "y \<in> T" "open T"

  2550   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2551     unfolding open_dist by fast

  2552   (* choose point between x and y, within distance r of y. *)

  2553   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2554   def z \<equiv> "y + scaleR k (x - y)"

  2555   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2556     unfolding z_def by (simp add: algebra_simps)

  2557   have "dist z y < r"

  2558     unfolding z_def k_def using 0 < r

  2559     by (simp add: dist_norm min_def)

  2560   then have "z \<in> T"

  2561     using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2562   have "dist x z < dist x y"

  2563     unfolding z_def2 dist_norm

  2564     apply (simp add: norm_minus_commute)

  2565     apply (simp only: dist_norm [symmetric])

  2566     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2567     apply (rule mult_strict_right_mono)

  2568     apply (simp add: k_def zero_less_dist_iff 0 < r x \<noteq> y)

  2569     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2570     done

  2571   then have "z \<in> ball x (dist x y)"

  2572     by simp

  2573   have "z \<noteq> y"

  2574     unfolding z_def k_def using x \<noteq> y 0 < r

  2575     by (simp add: min_def)

  2576   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2577     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2578     by fast

  2579 qed

  2580

  2581 lemma closure_ball:

  2582   fixes x :: "'a::real_normed_vector"

  2583   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2584   apply (rule equalityI)

  2585   apply (rule closure_minimal)

  2586   apply (rule ball_subset_cball)

  2587   apply (rule closed_cball)

  2588   apply (rule subsetI, rename_tac y)

  2589   apply (simp add: le_less [where 'a=real])

  2590   apply (erule disjE)

  2591   apply (rule subsetD [OF closure_subset], simp)

  2592   apply (simp add: closure_def)

  2593   apply clarify

  2594   apply (rule closure_ball_lemma)

  2595   apply (simp add: zero_less_dist_iff)

  2596   done

  2597

  2598 (* In a trivial vector space, this fails for e = 0. *)

  2599 lemma interior_cball:

  2600   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2601   shows "interior (cball x e) = ball x e"

  2602 proof (cases "e \<ge> 0")

  2603   case False note cs = this

  2604   from cs have "ball x e = {}"

  2605     using ball_empty[of e x] by auto

  2606   moreover

  2607   {

  2608     fix y

  2609     assume "y \<in> cball x e"

  2610     then have False

  2611       unfolding mem_cball using dist_nz[of x y] cs by auto

  2612   }

  2613   then have "cball x e = {}" by auto

  2614   then have "interior (cball x e) = {}"

  2615     using interior_empty by auto

  2616   ultimately show ?thesis by blast

  2617 next

  2618   case True note cs = this

  2619   have "ball x e \<subseteq> cball x e"

  2620     using ball_subset_cball by auto

  2621   moreover

  2622   {

  2623     fix S y

  2624     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2625     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

  2626       unfolding open_dist by blast

  2627     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2628       using perfect_choose_dist [of d] by auto

  2629     have "xa \<in> S"

  2630       using d[THEN spec[where x = xa]]

  2631       using xa by (auto simp add: dist_commute)

  2632     then have xa_cball: "xa \<in> cball x e"

  2633       using as(1) by auto

  2634     then have "y \<in> ball x e"

  2635     proof (cases "x = y")

  2636       case True

  2637       then have "e > 0"

  2638         using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2639         by (auto simp add: dist_commute)

  2640       then show "y \<in> ball x e"

  2641         using x = y  by simp

  2642     next

  2643       case False

  2644       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

  2645         unfolding dist_norm

  2646         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2647       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2648         using d as(1)[unfolded subset_eq] by blast

  2649       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2650       hence **:"d / (2 * norm (y - x)) > 0"

  2651         unfolding zero_less_norm_iff[symmetric] using d>0 by auto

  2652       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

  2653         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2654         by (auto simp add: dist_norm algebra_simps)

  2655       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2656         by (auto simp add: algebra_simps)

  2657       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2658         using ** by auto

  2659       also have "\<dots> = (dist y x) + d/2"

  2660         using ** by (auto simp add: distrib_right dist_norm)

  2661       finally have "e \<ge> dist x y +d/2"

  2662         using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2663       then show "y \<in> ball x e"

  2664         unfolding mem_ball using d>0 by auto

  2665     qed

  2666   }

  2667   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

  2668     by auto

  2669   ultimately show ?thesis

  2670     using interior_unique[of "ball x e" "cball x e"]

  2671     using open_ball[of x e]

  2672     by auto

  2673 qed

  2674

  2675 lemma frontier_ball:

  2676   fixes a :: "'a::real_normed_vector"

  2677   shows "0 < e \<Longrightarrow> frontier(ball a e) = {x. dist a x = e}"

  2678   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2679   apply (simp add: set_eq_iff)

  2680   apply arith

  2681   done

  2682

  2683 lemma frontier_cball:

  2684   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2685   shows "frontier (cball a e) = {x. dist a x = e}"

  2686   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2687   apply (simp add: set_eq_iff)

  2688   apply arith

  2689   done

  2690

  2691 lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"

  2692   apply (simp add: set_eq_iff not_le)

  2693   apply (metis zero_le_dist dist_self order_less_le_trans)

  2694   done

  2695

  2696 lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"

  2697   by (simp add: cball_eq_empty)

  2698

  2699 lemma cball_eq_sing:

  2700   fixes x :: "'a::{metric_space,perfect_space}"

  2701   shows "cball x e = {x} \<longleftrightarrow> e = 0"

  2702 proof (rule linorder_cases)

  2703   assume e: "0 < e"

  2704   obtain a where "a \<noteq> x" "dist a x < e"

  2705     using perfect_choose_dist [OF e] by auto

  2706   then have "a \<noteq> x" "dist x a \<le> e"

  2707     by (auto simp add: dist_commute)

  2708   with e show ?thesis by (auto simp add: set_eq_iff)

  2709 qed auto

  2710

  2711 lemma cball_sing:

  2712   fixes x :: "'a::metric_space"

  2713   shows "e = 0 \<Longrightarrow> cball x e = {x}"

  2714   by (auto simp add: set_eq_iff)

  2715

  2716

  2717 subsection {* Boundedness *}

  2718

  2719   (* FIXME: This has to be unified with BSEQ!! *)

  2720 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2721   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2722

  2723 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2724   unfolding bounded_def subset_eq by auto

  2725

  2726 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2727   unfolding bounded_def

  2728   by auto (metis add_commute add_le_cancel_right dist_commute dist_triangle_le)

  2729

  2730 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2731   unfolding bounded_any_center [where a=0]

  2732   by (simp add: dist_norm)

  2733

  2734 lemma bounded_realI:

  2735   assumes "\<forall>x\<in>s. abs (x::real) \<le> B"

  2736   shows "bounded s"

  2737   unfolding bounded_def dist_real_def

  2738   by (metis abs_minus_commute assms diff_0_right)

  2739

  2740 lemma bounded_empty [simp]: "bounded {}"

  2741   by (simp add: bounded_def)

  2742

  2743 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"

  2744   by (metis bounded_def subset_eq)

  2745

  2746 lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"

  2747   by (metis bounded_subset interior_subset)

  2748

  2749 lemma bounded_closure[intro]:

  2750   assumes "bounded S"

  2751   shows "bounded (closure S)"

  2752 proof -

  2753   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2754     unfolding bounded_def by auto

  2755   {

  2756     fix y

  2757     assume "y \<in> closure S"

  2758     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2759       unfolding closure_sequential by auto

  2760     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2761     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2762       by (rule eventually_mono, simp add: f(1))

  2763     have "dist x y \<le> a"

  2764       apply (rule Lim_dist_ubound [of sequentially f])

  2765       apply (rule trivial_limit_sequentially)

  2766       apply (rule f(2))

  2767       apply fact

  2768       done

  2769   }

  2770   then show ?thesis

  2771     unfolding bounded_def by auto

  2772 qed

  2773

  2774 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2775   apply (simp add: bounded_def)

  2776   apply (rule_tac x=x in exI)

  2777   apply (rule_tac x=e in exI)

  2778   apply auto

  2779   done

  2780

  2781 lemma bounded_ball[simp,intro]: "bounded (ball x e)"

  2782   by (metis ball_subset_cball bounded_cball bounded_subset)

  2783

  2784 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2785   apply (auto simp add: bounded_def)

  2786   by (metis Un_iff add_le_cancel_left dist_triangle le_max_iff_disj max.order_iff)

  2787

  2788 lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"

  2789   by (induct rule: finite_induct[of F]) auto

  2790

  2791 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2792   by (induct set: finite) auto

  2793

  2794 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2795 proof -

  2796   have "\<forall>y\<in>{x}. dist x y \<le> 0"

  2797     by simp

  2798   then have "bounded {x}"

  2799     unfolding bounded_def by fast

  2800   then show ?thesis

  2801     by (metis insert_is_Un bounded_Un)

  2802 qed

  2803

  2804 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2805   by (induct set: finite) simp_all

  2806

  2807 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"

  2808   apply (simp add: bounded_iff)

  2809   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")

  2810   apply metis

  2811   apply arith

  2812   done

  2813

  2814 lemma Bseq_eq_bounded:

  2815   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  2816   shows "Bseq f \<longleftrightarrow> bounded (range f)"

  2817   unfolding Bseq_def bounded_pos by auto

  2818

  2819 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2820   by (metis Int_lower1 Int_lower2 bounded_subset)

  2821

  2822 lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"

  2823   by (metis Diff_subset bounded_subset)

  2824

  2825 lemma not_bounded_UNIV[simp, intro]:

  2826   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2827 proof (auto simp add: bounded_pos not_le)

  2828   obtain x :: 'a where "x \<noteq> 0"

  2829     using perfect_choose_dist [OF zero_less_one] by fast

  2830   fix b :: real

  2831   assume b: "b >0"

  2832   have b1: "b +1 \<ge> 0"

  2833     using b by simp

  2834   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2835     by (simp add: norm_sgn)

  2836   then show "\<exists>x::'a. b < norm x" ..

  2837 qed

  2838

  2839 lemma bounded_linear_image:

  2840   assumes "bounded S"

  2841     and "bounded_linear f"

  2842   shows "bounded (f  S)"

  2843 proof -

  2844   from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2845     unfolding bounded_pos by auto

  2846   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"

  2847     using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2848   {

  2849     fix x

  2850     assume "x \<in> S"

  2851     then have "norm x \<le> b"

  2852       using b by auto

  2853     then have "norm (f x) \<le> B * b"

  2854       using B(2)

  2855       apply (erule_tac x=x in allE)

  2856       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2857       done

  2858   }

  2859   then show ?thesis

  2860     unfolding bounded_pos

  2861     apply (rule_tac x="b*B" in exI)

  2862     using b B by (auto simp add: mult_commute)

  2863 qed

  2864

  2865 lemma bounded_scaling:

  2866   fixes S :: "'a::real_normed_vector set"

  2867   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2868   apply (rule bounded_linear_image)

  2869   apply assumption

  2870   apply (rule bounded_linear_scaleR_right)

  2871   done

  2872

  2873 lemma bounded_translation:

  2874   fixes S :: "'a::real_normed_vector set"

  2875   assumes "bounded S"

  2876   shows "bounded ((\<lambda>x. a + x)  S)"

  2877 proof -

  2878   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2879     unfolding bounded_pos by auto

  2880   {

  2881     fix x

  2882     assume "x \<in> S"

  2883     then have "norm (a + x) \<le> b + norm a"

  2884       using norm_triangle_ineq[of a x] b by auto

  2885   }

  2886   then show ?thesis

  2887     unfolding bounded_pos

  2888     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2889     by (auto intro!: exI[of _ "b + norm a"])

  2890 qed

  2891

  2892

  2893 text{* Some theorems on sups and infs using the notion "bounded". *}

  2894

  2895 lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"

  2896   by (simp add: bounded_iff)

  2897

  2898 lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"

  2899   by (auto simp: bounded_def bdd_above_def dist_real_def)

  2900      (metis abs_le_D1 abs_minus_commute diff_le_eq)

  2901

  2902 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"

  2903   by (auto simp: bounded_def bdd_below_def dist_real_def)

  2904      (metis abs_le_D1 add_commute diff_le_eq)

  2905

  2906 (* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)

  2907

  2908 lemma bounded_has_Sup:

  2909   fixes S :: "real set"

  2910   assumes "bounded S"

  2911     and "S \<noteq> {}"

  2912   shows "\<forall>x\<in>S. x \<le> Sup S"

  2913     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2914 proof

  2915   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2916     using assms by (metis cSup_least)

  2917 qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

  2918

  2919 lemma Sup_insert:

  2920   fixes S :: "real set"

  2921   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2922   by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)

  2923

  2924 lemma Sup_insert_finite:

  2925   fixes S :: "real set"

  2926   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2927   apply (rule Sup_insert)

  2928   apply (rule finite_imp_bounded)

  2929   apply simp

  2930   done

  2931

  2932 lemma bounded_has_Inf:

  2933   fixes S :: "real set"

  2934   assumes "bounded S"

  2935     and "S \<noteq> {}"

  2936   shows "\<forall>x\<in>S. x \<ge> Inf S"

  2937     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2938 proof

  2939   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2940     using assms by (metis cInf_greatest)

  2941 qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

  2942

  2943 lemma Inf_insert:

  2944   fixes S :: "real set"

  2945   shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2946   by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)

  2947

  2948 lemma Inf_insert_finite:

  2949   fixes S :: "real set"

  2950   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2951   apply (rule Inf_insert)

  2952   apply (rule finite_imp_bounded)

  2953   apply simp

  2954   done

  2955

  2956 subsection {* Compactness *}

  2957

  2958 subsubsection {* Bolzano-Weierstrass property *}

  2959

  2960 lemma heine_borel_imp_bolzano_weierstrass:

  2961   assumes "compact s"

  2962     and "infinite t"

  2963     and "t \<subseteq> s"

  2964   shows "\<exists>x \<in> s. x islimpt t"

  2965 proof (rule ccontr)

  2966   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2967   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)"

  2968     unfolding islimpt_def

  2969     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]

  2970     by auto

  2971   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2972     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2973     using f by auto

  2974   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"

  2975     by auto

  2976   {

  2977     fix x y

  2978     assume "x \<in> t" "y \<in> t" "f x = f y"

  2979     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2980       using f[THEN bspec[where x=x]] and t \<subseteq> s by auto

  2981     then have "x = y"

  2982       using f x = f y and f[THEN bspec[where x=y]] and y \<in> t and t \<subseteq> s

  2983       by auto

  2984   }

  2985   then have "inj_on f t"

  2986     unfolding inj_on_def by simp

  2987   then have "infinite (f  t)"

  2988     using assms(2) using finite_imageD by auto

  2989   moreover

  2990   {

  2991     fix x

  2992     assume "x \<in> t" "f x \<notin> g"

  2993     from g(3) assms(3) x \<in> t obtain h where "h \<in> g" and "x \<in> h"

  2994       by auto

  2995     then obtain y where "y \<in> s" "h = f y"

  2996       using g'[THEN bspec[where x=h]] by auto

  2997     then have "y = x"

  2998       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y]

  2999       by auto

  3000     then have False

  3001       using f x \<notin> g h \<in> g unfolding h = f y

  3002       by auto

  3003   }

  3004   then have "f  t \<subseteq> g" by auto

  3005   ultimately show False

  3006     using g(2) using finite_subset by auto

  3007 qed

  3008

  3009 lemma acc_point_range_imp_convergent_subsequence:

  3010   fixes l :: "'a :: first_countable_topology"

  3011   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  3012   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3013 proof -

  3014   from countable_basis_at_decseq[of l]

  3015   obtain A where A:

  3016       "\<And>i. open (A i)"

  3017       "\<And>i. l \<in> A i"

  3018       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3019     by blast

  3020   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  3021   {

  3022     fix n i

  3023     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  3024       using l A by auto

  3025     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  3026       unfolding ex_in_conv by (intro notI) simp

  3027     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  3028       by auto

  3029     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  3030       by (auto simp: not_le)

  3031     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  3032       unfolding s_def by (auto intro: someI2_ex)

  3033   }

  3034   note s = this

  3035   def r \<equiv> "rec_nat (s 0 0) s"

  3036   have "subseq r"

  3037     by (auto simp: r_def s subseq_Suc_iff)

  3038   moreover

  3039   have "(\<lambda>n. f (r n)) ----> l"

  3040   proof (rule topological_tendstoI)

  3041     fix S

  3042     assume "open S" "l \<in> S"

  3043     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3044       by auto

  3045     moreover

  3046     {

  3047       fix i

  3048       assume "Suc 0 \<le> i"

  3049       then have "f (r i) \<in> A i"

  3050         by (cases i) (simp_all add: r_def s)

  3051     }

  3052     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  3053       by (auto simp: eventually_sequentially)

  3054     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  3055       by eventually_elim auto

  3056   qed

  3057   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3058     by (auto simp: convergent_def comp_def)

  3059 qed

  3060

  3061 lemma sequence_infinite_lemma:

  3062   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  3063   assumes "\<forall>n. f n \<noteq> l"

  3064     and "(f ---> l) sequentially"

  3065   shows "infinite (range f)"

  3066 proof

  3067   assume "finite (range f)"

  3068   then have "closed (range f)"

  3069     by (rule finite_imp_closed)

  3070   then have "open (- range f)"

  3071     by (rule open_Compl)

  3072   from assms(1) have "l \<in> - range f"

  3073     by auto

  3074   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  3075     using open (- range f) l \<in> - range f

  3076     by (rule topological_tendstoD)

  3077   then show False

  3078     unfolding eventually_sequentially

  3079     by auto

  3080 qed

  3081

  3082 lemma closure_insert:

  3083   fixes x :: "'a::t1_space"

  3084   shows "closure (insert x s) = insert x (closure s)"

  3085   apply (rule closure_unique)

  3086   apply (rule insert_mono [OF closure_subset])

  3087   apply (rule closed_insert [OF closed_closure])

  3088   apply (simp add: closure_minimal)

  3089   done

  3090

  3091 lemma islimpt_insert:

  3092   fixes x :: "'a::t1_space"

  3093   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  3094 proof

  3095   assume *: "x islimpt (insert a s)"

  3096   show "x islimpt s"

  3097   proof (rule islimptI)

  3098     fix t

  3099     assume t: "x \<in> t" "open t"

  3100     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  3101     proof (cases "x = a")

  3102       case True

  3103       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  3104         using * t by (rule islimptE)

  3105       with x = a show ?thesis by auto

  3106     next

  3107       case False

  3108       with t have t': "x \<in> t - {a}" "open (t - {a})"

  3109         by (simp_all add: open_Diff)

  3110       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  3111         using * t' by (rule islimptE)

  3112       then show ?thesis by auto

  3113     qed

  3114   qed

  3115 next

  3116   assume "x islimpt s"

  3117   then show "x islimpt (insert a s)"

  3118     by (rule islimpt_subset) auto

  3119 qed

  3120

  3121 lemma islimpt_finite:

  3122   fixes x :: "'a::t1_space"

  3123   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  3124   by (induct set: finite) (simp_all add: islimpt_insert)

  3125

  3126 lemma islimpt_union_finite:

  3127   fixes x :: "'a::t1_space"

  3128   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3129   by (simp add: islimpt_Un islimpt_finite)

  3130

  3131 lemma islimpt_eq_acc_point:

  3132   fixes l :: "'a :: t1_space"

  3133   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  3134 proof (safe intro!: islimptI)

  3135   fix U

  3136   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  3137   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  3138     by (auto intro: finite_imp_closed)

  3139   then show False

  3140     by (rule islimptE) auto

  3141 next

  3142   fix T

  3143   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  3144   then have "infinite (T \<inter> S - {l})"

  3145     by auto

  3146   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  3147     unfolding ex_in_conv by (intro notI) simp

  3148   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  3149     by auto

  3150 qed

  3151

  3152 lemma islimpt_range_imp_convergent_subsequence:

  3153   fixes l :: "'a :: {t1_space, first_countable_topology}"

  3154   assumes l: "l islimpt (range f)"

  3155   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3156   using l unfolding islimpt_eq_acc_point

  3157   by (rule acc_point_range_imp_convergent_subsequence)

  3158

  3159 lemma sequence_unique_limpt:

  3160   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3161   assumes "(f ---> l) sequentially"

  3162     and "l' islimpt (range f)"

  3163   shows "l' = l"

  3164 proof (rule ccontr)

  3165   assume "l' \<noteq> l"

  3166   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3167     using hausdorff [OF l' \<noteq> l] by auto

  3168   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3169     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3170   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3171     unfolding eventually_sequentially by auto

  3172

  3173   have "UNIV = {..<N} \<union> {N..}"

  3174     by auto

  3175   then have "l' islimpt (f  ({..<N} \<union> {N..}))"

  3176     using assms(2) by simp

  3177   then have "l' islimpt (f  {..<N} \<union> f  {N..})"

  3178     by (simp add: image_Un)

  3179   then have "l' islimpt (f  {N..})"

  3180     by (simp add: islimpt_union_finite)

  3181   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3182     using l' \<in> s open s by (rule islimptE)

  3183   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"

  3184     by auto

  3185   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t"

  3186     by simp

  3187   with s \<inter> t = {} show False

  3188     by simp

  3189 qed

  3190

  3191 lemma bolzano_weierstrass_imp_closed:

  3192   fixes s :: "'a::{first_countable_topology,t2_space} set"

  3193   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3194   shows "closed s"

  3195 proof -

  3196   {

  3197     fix x l

  3198     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3199     then have "l \<in> s"

  3200     proof (cases "\<forall>n. x n \<noteq> l")

  3201       case False

  3202       then show "l\<in>s" using as(1) by auto

  3203     next

  3204       case True note cas = this

  3205       with as(2) have "infinite (range x)"

  3206         using sequence_infinite_lemma[of x l] by auto

  3207       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  3208         using assms[THEN spec[where x="range x"]] as(1) by auto

  3209       then show "l\<in>s" using sequence_unique_limpt[of x l l']

  3210         using as cas by auto

  3211     qed

  3212   }

  3213   then show ?thesis

  3214     unfolding closed_sequential_limits by fast

  3215 qed

  3216

  3217 lemma compact_imp_bounded:

  3218   assumes "compact U"

  3219   shows "bounded U"

  3220 proof -

  3221   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  3222     using assms by auto

  3223   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  3224     by (rule compactE_image)

  3225   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  3226     by (simp add: bounded_UN)

  3227   then show "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  3228     by (rule bounded_subset)

  3229 qed

  3230

  3231 text{* In particular, some common special cases. *}

  3232

  3233 lemma compact_union [intro]:

  3234   assumes "compact s"

  3235     and "compact t"

  3236   shows " compact (s \<union> t)"

  3237 proof (rule compactI)

  3238   fix f

  3239   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  3240   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  3241     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3242   moreover

  3243   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  3244     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3245   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  3246     by (auto intro!: exI[of _ "s' \<union> t'"])

  3247 qed

  3248

  3249 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3250   by (induct set: finite) auto

  3251

  3252 lemma compact_UN [intro]:

  3253   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3254   unfolding SUP_def by (rule compact_Union) auto

  3255

  3256 lemma closed_inter_compact [intro]:

  3257   assumes "closed s"

  3258     and "compact t"

  3259   shows "compact (s \<inter> t)"

  3260   using compact_inter_closed [of t s] assms

  3261   by (simp add: Int_commute)

  3262

  3263 lemma compact_inter [intro]:

  3264   fixes s t :: "'a :: t2_space set"

  3265   assumes "compact s"

  3266     and "compact t"

  3267   shows "compact (s \<inter> t)"

  3268   using assms by (intro compact_inter_closed compact_imp_closed)

  3269

  3270 lemma compact_sing [simp]: "compact {a}"

  3271   unfolding compact_eq_heine_borel by auto

  3272

  3273 lemma compact_insert [simp]:

  3274   assumes "compact s"

  3275   shows "compact (insert x s)"

  3276 proof -

  3277   have "compact ({x} \<union> s)"

  3278     using compact_sing assms by (rule compact_union)

  3279   then show ?thesis by simp

  3280 qed

  3281

  3282 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  3283   by (induct set: finite) simp_all

  3284

  3285 lemma open_delete:

  3286   fixes s :: "'a::t1_space set"

  3287   shows "open s \<Longrightarrow> open (s - {x})"

  3288   by (simp add: open_Diff)

  3289

  3290 text{*Compactness expressed with filters*}

  3291

  3292 lemma closure_iff_nhds_not_empty:

  3293   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  3294 proof safe

  3295   assume x: "x \<in> closure X"

  3296   fix S A

  3297   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  3298   then have "x \<notin> closure (-S)"

  3299     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  3300   with x have "x \<in> closure X - closure (-S)"

  3301     by auto

  3302   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  3303     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  3304   finally have "X \<inter> S \<noteq> {}" by auto

  3305   then show False using X \<inter> A = {} S \<subseteq> A by auto

  3306 next

  3307   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  3308   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  3309   show "x \<in> closure X"

  3310     by (simp add: closure_subset open_Compl)

  3311 qed

  3312

  3313 lemma compact_filter:

  3314   "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))"

  3315 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  3316   fix F

  3317   assume "compact U"

  3318   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  3319   then have "U \<noteq> {}"

  3320     by (auto simp: eventually_False)

  3321

  3322   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  3323   then have "\<forall>z\<in>Z. closed z"

  3324     by auto

  3325   moreover

  3326   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  3327     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  3328   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  3329   proof (intro allI impI)

  3330     fix B assume "finite B" "B \<subseteq> Z"

  3331     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  3332       by (auto intro!: eventually_Ball_finite)

  3333     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  3334       by eventually_elim auto

  3335     with F show "U \<inter> \<Inter>B \<noteq> {}"

  3336       by (intro notI) (simp add: eventually_False)

  3337   qed

  3338   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  3339     using compact U unfolding compact_fip by blast

  3340   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"

  3341     by auto

  3342

  3343   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  3344     unfolding eventually_inf eventually_nhds

  3345   proof safe

  3346     fix P Q R S

  3347     assume "eventually R F" "open S" "x \<in> S"

  3348     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  3349     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  3350     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  3351     ultimately show False by (auto simp: set_eq_iff)

  3352   qed

  3353   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  3354     by (metis eventually_bot)

  3355 next

  3356   fix A

  3357   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  3358   def F \<equiv> "INF a:insert U A. principal a"

  3359   have "F \<noteq> bot"

  3360     unfolding F_def

  3361   proof (rule INF_filter_not_bot)

  3362     fix X assume "X \<subseteq> insert U A" "finite X"

  3363     moreover with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"

  3364       by auto

  3365     ultimately show "(INF a:X. principal a) \<noteq> bot"

  3366       by (auto simp add: INF_principal_finite principal_eq_bot_iff)

  3367   qed

  3368   moreover

  3369   have "F \<le> principal U"

  3370     unfolding F_def by auto

  3371   then have "eventually (\<lambda>x. x \<in> U) F"

  3372     by (auto simp: le_filter_def eventually_principal)

  3373   moreover

  3374   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)"

  3375   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  3376     by auto

  3377

  3378   { fix V assume "V \<in> A"

  3379     then have "F \<le> principal V"

  3380       unfolding F_def by (intro INF_lower2[of V]) auto

  3381     then have V: "eventually (\<lambda>x. x \<in> V) F"

  3382       by (auto simp: le_filter_def eventually_principal)

  3383     have "x \<in> closure V"

  3384       unfolding closure_iff_nhds_not_empty

  3385     proof (intro impI allI)

  3386       fix S A

  3387       assume "open S" "x \<in> S" "S \<subseteq> A"

  3388       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"

  3389         by (auto simp: eventually_nhds)

  3390       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  3391         by (auto simp: eventually_inf)

  3392       with x show "V \<inter> A \<noteq> {}"

  3393         by (auto simp del: Int_iff simp add: trivial_limit_def)

  3394     qed

  3395     then have "x \<in> V"

  3396       using V \<in> A A(1) by simp

  3397   }

  3398   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  3399   with U \<inter> \<Inter>A = {} show False by auto

  3400 qed

  3401

  3402 definition "countably_compact U \<longleftrightarrow>

  3403     (\<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))"

  3404

  3405 lemma countably_compactE:

  3406   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  3407   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  3408   using assms unfolding countably_compact_def by metis

  3409

  3410 lemma countably_compactI:

  3411   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')"

  3412   shows "countably_compact s"

  3413   using assms unfolding countably_compact_def by metis

  3414

  3415 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  3416   by (auto simp: compact_eq_heine_borel countably_compact_def)

  3417

  3418 lemma countably_compact_imp_compact:

  3419   assumes "countably_compact U"

  3420     and ccover: "countable B" "\<forall>b\<in>B. open b"

  3421     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"

  3422   shows "compact U"

  3423   using countably_compact U

  3424   unfolding compact_eq_heine_borel countably_compact_def

  3425 proof safe

  3426   fix A

  3427   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3428   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)"

  3429

  3430   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  3431   ultimately have "countable C" "\<forall>a\<in>C. open a"

  3432     unfolding C_def using ccover by auto

  3433   moreover

  3434   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  3435   proof safe

  3436     fix x a

  3437     assume "x \<in> U" "x \<in> a" "a \<in> A"

  3438     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"

  3439       by blast

  3440     with a \<in> A show "x \<in> \<Union>C"

  3441       unfolding C_def by auto

  3442   qed

  3443   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  3444   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  3445     using * by metis

  3446   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  3447     by (auto simp: C_def)

  3448   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"

  3449     unfolding bchoice_iff Bex_def ..

  3450   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3451     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  3452 qed

  3453

  3454 lemma countably_compact_imp_compact_second_countable:

  3455   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3456 proof (rule countably_compact_imp_compact)

  3457   fix T and x :: 'a

  3458   assume "open T" "x \<in> T"

  3459   from topological_basisE[OF is_basis this] obtain b where

  3460     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .

  3461   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  3462     by blast

  3463 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  3464

  3465 lemma countably_compact_eq_compact:

  3466   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3467   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  3468

  3469 subsubsection{* Sequential compactness *}

  3470

  3471 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"

  3472   where "seq_compact S \<longleftrightarrow>

  3473     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"

  3474

  3475 lemma seq_compactI:

  3476   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3477   shows "seq_compact S"

  3478   unfolding seq_compact_def using assms by fast

  3479

  3480 lemma seq_compactE:

  3481   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3482   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3483   using assms unfolding seq_compact_def by fast

  3484

  3485 lemma closed_sequentially: (* TODO: move upwards *)

  3486   assumes "closed s" and "\<forall>n. f n \<in> s" and "f ----> l"

  3487   shows "l \<in> s"

  3488 proof (rule ccontr)

  3489   assume "l \<notin> s"

  3490   with closed s and f ----> l have "eventually (\<lambda>n. f n \<in> - s) sequentially"

  3491     by (fast intro: topological_tendstoD)

  3492   with \<forall>n. f n \<in> s show "False"

  3493     by simp

  3494 qed

  3495

  3496 lemma seq_compact_inter_closed:

  3497   assumes "seq_compact s" and "closed t"

  3498   shows "seq_compact (s \<inter> t)"

  3499 proof (rule seq_compactI)

  3500   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"

  3501   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  3502     by simp_all

  3503   from seq_compact s and \<forall>n. f n \<in> s

  3504   obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) ----> l"

  3505     by (rule seq_compactE)

  3506   from \<forall>n. f n \<in> t have "\<forall>n. (f \<circ> r) n \<in> t"

  3507     by simp

  3508   from closed t and this and l have "l \<in> t"

  3509     by (rule closed_sequentially)

  3510   with l \<in> s and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3511     by fast

  3512 qed

  3513

  3514 lemma seq_compact_closed_subset:

  3515   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"

  3516   shows "seq_compact s"

  3517   using assms seq_compact_inter_closed [of t s] by (simp add: Int_absorb1)

  3518

  3519 lemma seq_compact_imp_countably_compact:

  3520   fixes U :: "'a :: first_countable_topology set"

  3521   assumes "seq_compact U"

  3522   shows "countably_compact U"

  3523 proof (safe intro!: countably_compactI)

  3524   fix A

  3525   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  3526   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  3527     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  3528   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3529   proof cases

  3530     assume "finite A"

  3531     with A show ?thesis by auto

  3532   next

  3533     assume "infinite A"

  3534     then have "A \<noteq> {}" by auto

  3535     show ?thesis

  3536     proof (rule ccontr)

  3537       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  3538       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"

  3539         by auto

  3540       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"

  3541         by metis

  3542       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  3543       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  3544         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  3545       then have "range X \<subseteq> U"

  3546         by auto

  3547       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"

  3548         by auto

  3549       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  3550       obtain n where "x \<in> from_nat_into A n" by auto

  3551       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  3552       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  3553         unfolding tendsto_def by (auto simp: comp_def)

  3554       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  3555         by (auto simp: eventually_sequentially)

  3556       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  3557         by auto

  3558       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  3559         by (auto intro!: exI[of _ "max n N"])

  3560       ultimately show False

  3561         by auto

  3562     qed

  3563   qed

  3564 qed

  3565

  3566 lemma compact_imp_seq_compact:

  3567   fixes U :: "'a :: first_countable_topology set"

  3568   assumes "compact U"

  3569   shows "seq_compact U"

  3570   unfolding seq_compact_def

  3571 proof safe

  3572   fix X :: "nat \<Rightarrow> 'a"

  3573   assume "\<forall>n. X n \<in> U"

  3574   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  3575     by (auto simp: eventually_filtermap)

  3576   moreover

  3577   have "filtermap X sequentially \<noteq> bot"

  3578     by (simp add: trivial_limit_def eventually_filtermap)

  3579   ultimately

  3580   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  3581     using compact U by (auto simp: compact_filter)

  3582

  3583   from countable_basis_at_decseq[of x]

  3584   obtain A where A:

  3585       "\<And>i. open (A i)"

  3586       "\<And>i. x \<in> A i"

  3587       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3588     by blast

  3589   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3590   {

  3591     fix n i

  3592     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3593     proof (rule ccontr)

  3594       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3595       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"

  3596         by auto

  3597       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3598         by (auto simp: eventually_filtermap eventually_sequentially)

  3599       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3600         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3601       ultimately have "eventually (\<lambda>x. False) ?F"

  3602         by (auto simp add: eventually_inf)

  3603       with x show False

  3604         by (simp add: eventually_False)

  3605     qed

  3606     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3607       unfolding s_def by (auto intro: someI2_ex)

  3608   }

  3609   note s = this

  3610   def r \<equiv> "rec_nat (s 0 0) s"

  3611   have "subseq r"

  3612     by (auto simp: r_def s subseq_Suc_iff)

  3613   moreover

  3614   have "(\<lambda>n. X (r n)) ----> x"

  3615   proof (rule topological_tendstoI)

  3616     fix S

  3617     assume "open S" "x \<in> S"

  3618     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3619       by auto

  3620     moreover

  3621     {

  3622       fix i

  3623       assume "Suc 0 \<le> i"

  3624       then have "X (r i) \<in> A i"

  3625         by (cases i) (simp_all add: r_def s)

  3626     }

  3627     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3628       by (auto simp: eventually_sequentially)

  3629     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3630       by eventually_elim auto

  3631   qed

  3632   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3633     using x \<in> U by (auto simp: convergent_def comp_def)

  3634 qed

  3635

  3636 lemma countably_compact_imp_acc_point:

  3637   assumes "countably_compact s"

  3638     and "countable t"

  3639     and "infinite t"

  3640     and "t \<subseteq> s"

  3641   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3642 proof (rule ccontr)

  3643   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3644   note countably_compact s

  3645   moreover have "\<forall>t\<in>C. open t"

  3646     by (auto simp: C_def)

  3647   moreover

  3648   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3649   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3650   have "s \<subseteq> \<Union>C"

  3651     using t \<subseteq> s

  3652     unfolding C_def Union_image_eq

  3653     apply (safe dest!: s)

  3654     apply (rule_tac a="U \<inter> t" in UN_I)

  3655     apply (auto intro!: interiorI simp add: finite_subset)

  3656     done

  3657   moreover

  3658   from countable t have "countable C"

  3659     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3660   ultimately

  3661   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"

  3662     by (rule countably_compactE)

  3663   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"

  3664     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3665     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3666   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3667     using interior_subset by blast

  3668   moreover have "finite (\<Union>E)"

  3669     using E by auto

  3670   ultimately show False using infinite t

  3671     by (auto simp: finite_subset)

  3672 qed

  3673

  3674 lemma countable_acc_point_imp_seq_compact:

  3675   fixes s :: "'a::first_countable_topology set"

  3676   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>

  3677     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3678   shows "seq_compact s"

  3679 proof -

  3680   {

  3681     fix f :: "nat \<Rightarrow> 'a"

  3682     assume f: "\<forall>n. f n \<in> s"

  3683     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3684     proof (cases "finite (range f)")

  3685       case True

  3686       obtain l where "infinite {n. f n = f l}"

  3687         using pigeonhole_infinite[OF _ True] by auto

  3688       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3689         using infinite_enumerate by blast

  3690       then have "subseq r \<and> (f \<circ> r) ----> f l"

  3691         by (simp add: fr tendsto_const o_def)

  3692       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3693         by auto

  3694     next

  3695       case False

  3696       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"

  3697         by auto

  3698       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3699       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3700         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3701       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3702     qed

  3703   }

  3704   then show ?thesis

  3705     unfolding seq_compact_def by auto

  3706 qed

  3707

  3708 lemma seq_compact_eq_countably_compact:

  3709   fixes U :: "'a :: first_countable_topology set"

  3710   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3711   using

  3712     countable_acc_point_imp_seq_compact

  3713     countably_compact_imp_acc_point

  3714     seq_compact_imp_countably_compact

  3715   by metis

  3716

  3717 lemma seq_compact_eq_acc_point:

  3718   fixes s :: "'a :: first_countable_topology set"

  3719   shows "seq_compact s \<longleftrightarrow>

  3720     (\<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)))"

  3721   using

  3722     countable_acc_point_imp_seq_compact[of s]

  3723     countably_compact_imp_acc_point[of s]

  3724     seq_compact_imp_countably_compact[of s]

  3725   by metis

  3726

  3727 lemma seq_compact_eq_compact:

  3728   fixes U :: "'a :: second_countable_topology set"

  3729   shows "seq_compact U \<longleftrightarrow> compact U"

  3730   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3731

  3732 lemma bolzano_weierstrass_imp_seq_compact:

  3733   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3734   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3735   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3736

  3737 subsubsection{* Total boundedness *}

  3738

  3739 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)"

  3740   unfolding Cauchy_def by metis

  3741

  3742 fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a"

  3743 where

  3744   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3745 declare helper_1.simps[simp del]

  3746

  3747 lemma seq_compact_imp_totally_bounded:

  3748   assumes "seq_compact s"

  3749   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3750 proof (rule, rule, rule ccontr)

  3751   fix e::real

  3752   assume "e > 0"

  3753   assume assm: "\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3754   def x \<equiv> "helper_1 s e"

  3755   {

  3756     fix n

  3757     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3758     proof (induct n rule: nat_less_induct)

  3759       fix n

  3760       def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3761       assume as: "\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3762       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3763         using assm

  3764         apply simp

  3765         apply (erule_tac x="x  {0 ..< n}" in allE)

  3766         using as

  3767         apply auto

  3768         done

  3769       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3770         unfolding subset_eq by auto

  3771       have "Q (x n)"

  3772         unfolding x_def and helper_1.simps[of s e n]

  3773         apply (rule someI2[where a=z])

  3774         unfolding x_def[symmetric] and Q_def

  3775         using z

  3776         apply auto

  3777         done

  3778       then show "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3779         unfolding Q_def by auto

  3780     qed

  3781   }

  3782   then have "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"

  3783     by blast+

  3784   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3785     using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3786   from this(3) have "Cauchy (x \<circ> r)"

  3787     using LIMSEQ_imp_Cauchy by auto

  3788   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"

  3789     unfolding cauchy_def using e>0 by auto

  3790   show False

  3791     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3792     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3793     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

  3794     by auto

  3795 qed

  3796

  3797 subsubsection{* Heine-Borel theorem *}

  3798

  3799 lemma seq_compact_imp_heine_borel:

  3800   fixes s :: "'a :: metric_space set"

  3801   assumes "seq_compact s"

  3802   shows "compact s"

  3803 proof -

  3804   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3805   obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  f e)"

  3806     unfolding choice_iff' ..

  3807   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3808   have "countably_compact s"

  3809     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3810   then show "compact s"

  3811   proof (rule countably_compact_imp_compact)

  3812     show "countable K"

  3813       unfolding K_def using f

  3814       by (auto intro: countable_finite countable_subset countable_rat

  3815                intro!: countable_image countable_SIGMA countable_UN)

  3816     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3817   next

  3818     fix T x

  3819     assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3820     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"

  3821       by auto

  3822     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"

  3823       by auto

  3824     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"

  3825       by auto

  3826     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3827       unfolding Union_image_eq by auto

  3828     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K"

  3829       by (auto simp: K_def)

  3830     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3831     proof (rule bexI[rotated], safe)

  3832       fix y

  3833       assume "y \<in> ball k r"

  3834       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3835         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3836       with ball x e \<subseteq> T show "y \<in> T"

  3837         by auto

  3838     next

  3839       show "x \<in> ball k r" by fact

  3840     qed

  3841   qed

  3842 qed

  3843

  3844 lemma compact_eq_seq_compact_metric:

  3845   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3846   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3847

  3848 lemma compact_def:

  3849   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3850    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"

  3851   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3852

  3853 subsubsection {* Complete the chain of compactness variants *}

  3854

  3855 lemma compact_eq_bolzano_weierstrass:

  3856   fixes s :: "'a::metric_space set"

  3857   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"

  3858   (is "?lhs = ?rhs")

  3859 proof

  3860   assume ?lhs

  3861   then show ?rhs

  3862     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3863 next

  3864   assume ?rhs

  3865   then show ?lhs

  3866     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3867 qed

  3868

  3869 lemma bolzano_weierstrass_imp_bounded:

  3870   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3871   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3872

  3873 subsection {* Metric spaces with the Heine-Borel property *}

  3874

  3875 text {*

  3876   A metric space (or topological vector space) is said to have the

  3877   Heine-Borel property if every closed and bounded subset is compact.

  3878 *}

  3879

  3880 class heine_borel = metric_space +

  3881   assumes bounded_imp_convergent_subsequence:

  3882     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3883

  3884 lemma bounded_closed_imp_seq_compact:

  3885   fixes s::"'a::heine_borel set"

  3886   assumes "bounded s"

  3887     and "closed s"

  3888   shows "seq_compact s"

  3889 proof (unfold seq_compact_def, clarify)

  3890   fix f :: "nat \<Rightarrow> 'a"

  3891   assume f: "\<forall>n. f n \<in> s"

  3892   with bounded s have "bounded (range f)"

  3893     by (auto intro: bounded_subset)

  3894   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3895     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3896   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"

  3897     by simp

  3898   have "l \<in> s" using closed s fr l

  3899     by (rule closed_sequentially)

  3900   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3901     using l \<in> s r l by blast

  3902 qed

  3903

  3904 lemma compact_eq_bounded_closed:

  3905   fixes s :: "'a::heine_borel set"

  3906   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"

  3907   (is "?lhs = ?rhs")

  3908 proof

  3909   assume ?lhs

  3910   then show ?rhs

  3911     using compact_imp_closed compact_imp_bounded

  3912     by blast

  3913 next

  3914   assume ?rhs

  3915   then show ?lhs

  3916     using bounded_closed_imp_seq_compact[of s]

  3917     unfolding compact_eq_seq_compact_metric

  3918     by auto

  3919 qed

  3920

  3921 (* TODO: is this lemma necessary? *)

  3922 lemma bounded_increasing_convergent:

  3923   fixes s :: "nat \<Rightarrow> real"

  3924   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3925   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3926   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3927

  3928 instance real :: heine_borel

  3929 proof

  3930   fix f :: "nat \<Rightarrow> real"

  3931   assume f: "bounded (range f)"

  3932   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3933     unfolding comp_def by (metis seq_monosub)

  3934   then have "Bseq (f \<circ> r)"

  3935     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3936   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3937     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3938 qed

  3939

  3940 lemma compact_lemma:

  3941   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3942   assumes "bounded (range f)"

  3943   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.

  3944     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3945 proof safe

  3946   fix d :: "'a set"

  3947   assume d: "d \<subseteq> Basis"

  3948   with finite_Basis have "finite d"

  3949     by (blast intro: finite_subset)

  3950   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3951     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3952   proof (induct d)

  3953     case empty

  3954     then show ?case

  3955       unfolding subseq_def by auto

  3956   next

  3957     case (insert k d)

  3958     have k[intro]: "k \<in> Basis"

  3959       using insert by auto

  3960     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)"

  3961       using bounded (range f)

  3962       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3963     obtain l1::"'a" and r1 where r1: "subseq r1"

  3964       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3965       using insert(3) using insert(4) by auto

  3966     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f"

  3967       by simp

  3968     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3969       by (metis (lifting) bounded_subset f' image_subsetI s')

  3970     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3971       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]

  3972       by (auto simp: o_def)

  3973     def r \<equiv> "r1 \<circ> r2"

  3974     have r:"subseq r"

  3975       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3976     moreover

  3977     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3978     {

  3979       fix e::real

  3980       assume "e > 0"

  3981       from lr1 e > 0 have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3982         by blast

  3983       from lr2 e > 0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3984         by (rule tendstoD)

  3985       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3986         by (rule eventually_subseq)

  3987       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3988         using N1' N2

  3989         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3990     }

  3991     ultimately show ?case by auto

  3992   qed

  3993 qed

  3994

  3995 instance euclidean_space \<subseteq> heine_borel

  3996 proof

  3997   fix f :: "nat \<Rightarrow> 'a"

  3998   assume f: "bounded (range f)"

  3999   then obtain l::'a and r where r: "subseq r"

  4000     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  4001     using compact_lemma [OF f] by blast

  4002   {

  4003     fix e::real

  4004     assume "e > 0"

  4005     hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)

  4006     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"

  4007       by simp

  4008     moreover

  4009     {

  4010       fix n

  4011       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  4012       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  4013         apply (subst euclidean_dist_l2)

  4014         using zero_le_dist

  4015         apply (rule setL2_le_setsum)

  4016         done

  4017       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  4018         apply (rule setsum_strict_mono)

  4019         using n

  4020         apply auto

  4021         done

  4022       finally have "dist (f (r n)) l < e"

  4023         by auto

  4024     }

  4025     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  4026       by (rule eventually_elim1)

  4027   }

  4028   then have *: "((f \<circ> r) ---> l) sequentially"

  4029     unfolding o_def tendsto_iff by simp

  4030   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4031     by auto

  4032 qed

  4033

  4034 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  4035   unfolding bounded_def

  4036   by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)

  4037

  4038 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  4039   unfolding bounded_def

  4040   by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)

  4041

  4042 instance prod :: (heine_borel, heine_borel) heine_borel

  4043 proof

  4044   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  4045   assume f: "bounded (range f)"

  4046   then have "bounded (fst  range f)"

  4047     by (rule bounded_fst)

  4048   then have s1: "bounded (range (fst \<circ> f))"

  4049     by (simp add: image_comp)

  4050   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  4051     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  4052   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  4053     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  4054   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  4055     using bounded_imp_convergent_subsequence [OF s2]

  4056     unfolding o_def by fast

  4057   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  4058     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  4059   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  4060     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  4061   have r: "subseq (r1 \<circ> r2)"

  4062     using r1 r2 unfolding subseq_def by simp

  4063   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4064     using l r by fast

  4065 qed

  4066

  4067 subsubsection {* Completeness *}

  4068

  4069 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  4070   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  4071

  4072 lemma completeI:

  4073   assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f ----> l"

  4074   shows "complete s"

  4075   using assms unfolding complete_def by fast

  4076

  4077 lemma completeE:

  4078   assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"

  4079   obtains l where "l \<in> s" and "f ----> l"

  4080   using assms unfolding complete_def by fast

  4081

  4082 lemma compact_imp_complete:

  4083   assumes "compact s"

  4084   shows "complete s"

  4085 proof -

  4086   {

  4087     fix f

  4088     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  4089     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  4090       using assms unfolding compact_def by blast

  4091

  4092     note lr' = seq_suble [OF lr(2)]

  4093     {

  4094       fix e :: real

  4095       assume "e > 0"

  4096       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"

  4097         unfolding cauchy_def

  4098         using e > 0

  4099         apply (erule_tac x="e/2" in allE)

  4100         apply auto

  4101         done

  4102       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  4103       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"

  4104         using e > 0 by auto

  4105       {

  4106         fix n :: nat

  4107         assume n: "n \<ge> max N M"

  4108         have "dist ((f \<circ> r) n) l < e/2"

  4109           using n M by auto

  4110         moreover have "r n \<ge> N"

  4111           using lr'[of n] n by auto

  4112         then have "dist (f n) ((f \<circ> r) n) < e / 2"

  4113           using N and n by auto

  4114         ultimately have "dist (f n) l < e"

  4115           using dist_triangle_half_r[of "f (r n)" "f n" e l]

  4116           by (auto simp add: dist_commute)

  4117       }

  4118       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  4119     }

  4120     then have "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  4121       unfolding LIMSEQ_def by auto

  4122   }

  4123   then show ?thesis unfolding complete_def by auto

  4124 qed

  4125

  4126 lemma nat_approx_posE:

  4127   fixes e::real

  4128   assumes "0 < e"

  4129   obtains n :: nat where "1 / (Suc n) < e"

  4130 proof atomize_elim

  4131   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  4132     by (rule divide_strict_left_mono) (auto simp: 0 < e)

  4133   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  4134     by (rule divide_left_mono) (auto simp: 0 < e)

  4135   also have "\<dots> = e" by simp

  4136   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  4137 qed

  4138

  4139 lemma compact_eq_totally_bounded:

  4140   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  4141     (is "_ \<longleftrightarrow> ?rhs")

  4142 proof

  4143   assume assms: "?rhs"

  4144   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)"

  4145     by (auto simp: choice_iff')

  4146

  4147   show "compact s"

  4148   proof cases

  4149     assume "s = {}"

  4150     then show "compact s" by (simp add: compact_def)

  4151   next

  4152     assume "s \<noteq> {}"

  4153     show ?thesis

  4154       unfolding compact_def

  4155     proof safe

  4156       fix f :: "nat \<Rightarrow> 'a"

  4157       assume f: "\<forall>n. f n \<in> s"

  4158

  4159       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  4160       then have [simp]: "\<And>n. 0 < e n" by auto

  4161       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)"

  4162       {

  4163         fix n U

  4164         assume "infinite {n. f n \<in> U}"

  4165         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  4166           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  4167         then obtain a where

  4168           "a \<in> k (e n)"

  4169           "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..

  4170         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  4171           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  4172         from someI_ex[OF this]

  4173         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  4174           unfolding B_def by auto

  4175       }

  4176       note B = this

  4177

  4178       def F \<equiv> "rec_nat (B 0 UNIV) B"

  4179       {

  4180         fix n

  4181         have "infinite {i. f i \<in> F n}"

  4182           by (induct n) (auto simp: F_def B)

  4183       }

  4184       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  4185         using B by (simp add: F_def)

  4186       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  4187         using decseq_SucI[of F] by (auto simp: decseq_def)

  4188

  4189       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  4190       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  4191         fix k i

  4192         have "infinite ({n. f n \<in> F k} - {.. i})"

  4193           using infinite {n. f n \<in> F k} by auto

  4194         from infinite_imp_nonempty[OF this]

  4195         show "\<exists>x>i. f x \<in> F k"

  4196           by (simp add: set_eq_iff not_le conj_commute)

  4197       qed

  4198

  4199       def t \<equiv> "rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  4200       have "subseq t"

  4201         unfolding subseq_Suc_iff by (simp add: t_def sel)

  4202       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  4203         using f by auto

  4204       moreover

  4205       {

  4206         fix n

  4207         have "(f \<circ> t) n \<in> F n"

  4208           by (cases n) (simp_all add: t_def sel)

  4209       }

  4210       note t = this

  4211

  4212       have "Cauchy (f \<circ> t)"

  4213       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  4214         fix r :: real and N n m

  4215         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  4216         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  4217           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  4218         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  4219           by (auto simp: subset_eq)

  4220         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  4221         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  4222           by (simp add: dist_commute)

  4223       qed

  4224

  4225       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  4226         using assms unfolding complete_def by blast

  4227     qed

  4228   qed

  4229 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  4230

  4231 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  4232 proof -

  4233   {

  4234     assume ?rhs

  4235     {

  4236       fix e::real

  4237       assume "e>0"

  4238       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  4239         by (erule_tac x="e/2" in allE) auto

  4240       {

  4241         fix n m

  4242         assume nm:"N \<le> m \<and> N \<le> n"

  4243         then have "dist (s m) (s n) < e" using N

  4244           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  4245           by blast

  4246       }

  4247       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  4248         by blast

  4249     }

  4250     then have ?lhs

  4251       unfolding cauchy_def

  4252       by blast

  4253   }

  4254   then show ?thesis

  4255     unfolding cauchy_def

  4256     using dist_triangle_half_l

  4257     by blast

  4258 qed

  4259

  4260 lemma cauchy_imp_bounded:

  4261   assumes "Cauchy s"

  4262   shows "bounded (range s)"

  4263 proof -

  4264   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  4265     unfolding cauchy_def

  4266     apply (erule_tac x= 1 in allE)

  4267     apply auto

  4268     done

  4269   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  4270   moreover

  4271   have "bounded (s  {0..N})"

  4272     using finite_imp_bounded[of "s  {1..N}"] by auto

  4273   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  4274     unfolding bounded_any_center [where a="s N"] by auto

  4275   ultimately show "?thesis"

  4276     unfolding bounded_any_center [where a="s N"]

  4277     apply (rule_tac x="max a 1" in exI)

  4278     apply auto

  4279     apply (erule_tac x=y in allE)

  4280     apply (erule_tac x=y in ballE)

  4281     apply auto

  4282     done

  4283 qed

  4284

  4285 instance heine_borel < complete_space

  4286 proof

  4287   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4288   then have "bounded (range f)"

  4289     by (rule cauchy_imp_bounded)

  4290   then have "compact (closure (range f))"

  4291     unfolding compact_eq_bounded_closed by auto

  4292   then have "complete (closure (range f))"

  4293     by (rule compact_imp_complete)

  4294   moreover have "\<forall>n. f n \<in> closure (range f)"

  4295     using closure_subset [of "range f"] by auto

  4296   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  4297     using Cauchy f unfolding complete_def by auto

  4298   then show "convergent f"

  4299     unfolding convergent_def by auto

  4300 qed

  4301

  4302 instance euclidean_space \<subseteq> banach ..

  4303

  4304 lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"

  4305 proof (rule completeI)

  4306   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4307   then have "convergent f" by (rule Cauchy_convergent)

  4308   then show "\<exists>l\<in>UNIV. f ----> l" unfolding convergent_def by simp

  4309 qed

  4310

  4311 lemma complete_imp_closed:

  4312   assumes "complete s"

  4313   shows "closed s"

  4314 proof (unfold closed_sequential_limits, clarify)

  4315   fix f x assume "\<forall>n. f n \<in> s" and "f ----> x"

  4316   from f ----> x have "Cauchy f"

  4317     by (rule LIMSEQ_imp_Cauchy)

  4318   with complete s and \<forall>n. f n \<in> s obtain l where "l \<in> s" and "f ----> l"

  4319     by (rule completeE)

  4320   from f ----> x and f ----> l have "x = l"

  4321     by (rule LIMSEQ_unique)

  4322   with l \<in> s show "x \<in> s"

  4323     by simp

  4324 qed

  4325

  4326 lemma complete_inter_closed:

  4327   assumes "complete s" and "closed t"

  4328   shows "complete (s \<inter> t)"

  4329 proof (rule completeI)

  4330   fix f assume "\<forall>n. f n \<in> s \<inter> t" and "Cauchy f"

  4331   then have "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  4332     by simp_all

  4333   from complete s obtain l where "l \<in> s" and "f ----> l"

  4334     using \<forall>n. f n \<in> s and Cauchy f by (rule completeE)

  4335   from closed t and \<forall>n. f n \<in> t and f ----> l have "l \<in> t"

  4336     by (rule closed_sequentially)

  4337   with l \<in> s and f ----> l show "\<exists>l\<in>s \<inter> t. f ----> l"

  4338     by fast

  4339 qed

  4340

  4341 lemma complete_closed_subset:

  4342   assumes "closed s" and "s \<subseteq> t" and "complete t"

  4343   shows "complete s"

  4344   using assms complete_inter_closed [of t s] by (simp add: Int_absorb1)

  4345

  4346 lemma complete_eq_closed:

  4347   fixes s :: "('a::complete_space) set"

  4348   shows "complete s \<longleftrightarrow> closed s"

  4349 proof

  4350   assume "closed s" then show "complete s"

  4351     using subset_UNIV complete_UNIV by (rule complete_closed_subset)

  4352 next

  4353   assume "complete s" then show "closed s"

  4354     by (rule complete_imp_closed)

  4355 qed

  4356

  4357 lemma convergent_eq_cauchy:

  4358   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  4359   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  4360   unfolding Cauchy_convergent_iff convergent_def ..

  4361

  4362 lemma convergent_imp_bounded:

  4363   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  4364   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  4365   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  4366

  4367 lemma compact_cball[simp]:

  4368   fixes x :: "'a::heine_borel"

  4369   shows "compact (cball x e)"

  4370   using compact_eq_bounded_closed bounded_cball closed_cball

  4371   by blast

  4372

  4373 lemma compact_frontier_bounded[intro]:

  4374   fixes s :: "'a::heine_borel set"

  4375   shows "bounded s \<Longrightarrow> compact (frontier s)"

  4376   unfolding frontier_def

  4377   using compact_eq_bounded_closed

  4378   by blast

  4379

  4380 lemma compact_frontier[intro]:

  4381   fixes s :: "'a::heine_borel set"

  4382   shows "compact s \<Longrightarrow> compact (frontier s)"

  4383   using compact_eq_bounded_closed compact_frontier_bounded

  4384   by blast

  4385

  4386 lemma frontier_subset_compact:

  4387   fixes s :: "'a::heine_borel set"

  4388   shows "compact s \<Longrightarrow> frontier s \<subseteq> s"

  4389   using frontier_subset_closed compact_eq_bounded_closed

  4390   by blast

  4391

  4392 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  4393

  4394 lemma bounded_closed_nest:

  4395   fixes s :: "nat \<Rightarrow> ('a::heine_borel) set"

  4396   assumes "\<forall>n. closed (s n)"

  4397     and "\<forall>n. s n \<noteq> {}"

  4398     and "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4399     and "bounded (s 0)"

  4400   shows "\<exists>a. \<forall>n. a \<in> s n"

  4401 proof -

  4402   from assms(2) obtain x where x: "\<forall>n. x n \<in> s n"

  4403     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4404   from assms(4,1) have "seq_compact (s 0)"

  4405     by (simp add: bounded_closed_imp_seq_compact)

  4406   then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) ----> l"

  4407     using x and assms(3) unfolding seq_compact_def by blast

  4408   have "\<forall>n. l \<in> s n"

  4409   proof

  4410     fix n :: nat

  4411     have "closed (s n)"

  4412       using assms(1) by simp

  4413     moreover have "\<forall>i. (x \<circ> r) i \<in> s i"

  4414       using x and assms(3) and lr(2) [THEN seq_suble] by auto

  4415     then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"

  4416       using assms(3) by (fast intro!: le_add2)

  4417     moreover have "(\<lambda>i. (x \<circ> r) (i + n)) ----> l"

  4418       using lr(3) by (rule LIMSEQ_ignore_initial_segment)

  4419     ultimately show "l \<in> s n"

  4420       by (rule closed_sequentially)

  4421   qed

  4422   then show ?thesis ..

  4423 qed

  4424

  4425 text {* Decreasing case does not even need compactness, just completeness. *}

  4426

  4427 lemma decreasing_closed_nest:

  4428   fixes s :: "nat \<Rightarrow> ('a::complete_space) set"

  4429   assumes

  4430     "\<forall>n. closed (s n)"

  4431     "\<forall>n. s n \<noteq> {}"

  4432     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4433     "\<forall>e>0. \<exists>n. \<forall>x\<in>s n. \<forall>y\<in>s n. dist x y < e"

  4434   shows "\<exists>a. \<forall>n. a \<in> s n"

  4435 proof -

  4436   have "\<forall>n. \<exists>x. x \<in> s n"

  4437     using assms(2) by auto

  4438   then have "\<exists>t. \<forall>n. t n \<in> s n"

  4439     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4440   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  4441   {

  4442     fix e :: real

  4443     assume "e > 0"

  4444     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"

  4445       using assms(4) by auto

  4446     {

  4447       fix m n :: nat

  4448       assume "N \<le> m \<and> N \<le> n"

  4449       then have "t m \<in> s N" "t n \<in> s N"

  4450         using assms(3) t unfolding  subset_eq t by blast+

  4451       then have "dist (t m) (t n) < e"

  4452         using N by auto

  4453     }

  4454     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  4455       by auto

  4456   }

  4457   then have "Cauchy t"

  4458     unfolding cauchy_def by auto

  4459   then obtain l where l:"(t ---> l) sequentially"

  4460     using complete_UNIV unfolding complete_def by auto

  4461   {

  4462     fix n :: nat

  4463     {

  4464       fix e :: real

  4465       assume "e > 0"

  4466       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  4467         using l[unfolded LIMSEQ_def] by auto

  4468       have "t (max n N) \<in> s n"

  4469         using assms(3)

  4470         unfolding subset_eq

  4471         apply (erule_tac x=n in allE)

  4472         apply (erule_tac x="max n N" in allE)

  4473         using t

  4474         apply auto

  4475         done

  4476       then have "\<exists>y\<in>s n. dist y l < e"

  4477         apply (rule_tac x="t (max n N)" in bexI)

  4478         using N

  4479         apply auto

  4480         done

  4481     }

  4482     then have "l \<in> s n"

  4483       using closed_approachable[of "s n" l] assms(1) by auto

  4484   }

  4485   then show ?thesis by auto

  4486 qed

  4487

  4488 text {* Strengthen it to the intersection actually being a singleton. *}

  4489

  4490 lemma decreasing_closed_nest_sing:

  4491   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  4492   assumes

  4493     "\<forall>n. closed(s n)"

  4494     "\<forall>n. s n \<noteq> {}"

  4495     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4496     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  4497   shows "\<exists>a. \<Inter>(range s) = {a}"

  4498 proof -

  4499   obtain a where a: "\<forall>n. a \<in> s n"

  4500     using decreasing_closed_nest[of s] using assms by auto

  4501   {

  4502     fix b

  4503     assume b: "b \<in> \<Inter>(range s)"

  4504     {

  4505       fix e :: real

  4506       assume "e > 0"

  4507       then have "dist a b < e"

  4508         using assms(4) and b and a by blast

  4509     }

  4510     then have "dist a b = 0"

  4511       by (metis dist_eq_0_iff dist_nz less_le)

  4512   }

  4513   with a have "\<Inter>(range s) = {a}"

  4514     unfolding image_def by auto

  4515   then show ?thesis ..

  4516 qed

  4517

  4518 text{* Cauchy-type criteria for uniform convergence. *}

  4519

  4520 lemma uniformly_convergent_eq_cauchy:

  4521   fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"

  4522   shows

  4523     "(\<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>

  4524       (\<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)"

  4525   (is "?lhs = ?rhs")

  4526 proof

  4527   assume ?lhs

  4528   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"

  4529     by auto

  4530   {

  4531     fix e :: real

  4532     assume "e > 0"

  4533     then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"

  4534       using l[THEN spec[where x="e/2"]] by auto

  4535     {

  4536       fix n m :: nat and x :: "'b"

  4537       assume "N \<le> m \<and> N \<le> n \<and> P x"

  4538       then have "dist (s m x) (s n x) < e"

  4539         using N[THEN spec[where x=m], THEN spec[where x=x]]

  4540         using N[THEN spec[where x=n], THEN spec[where x=x]]

  4541         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

  4542     }

  4543     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

  4544   }

  4545   then show ?rhs by auto

  4546 next

  4547   assume ?rhs

  4548   then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"

  4549     unfolding cauchy_def

  4550     apply auto

  4551     apply (erule_tac x=e in allE)

  4552     apply auto

  4553     done

  4554   then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"

  4555     unfolding convergent_eq_cauchy[symmetric]

  4556     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]

  4557     by auto

  4558   {

  4559     fix e :: real

  4560     assume "e > 0"

  4561     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"

  4562       using ?rhs[THEN spec[where x="e/2"]] by auto

  4563     {

  4564       fix x

  4565       assume "P x"

  4566       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  4567         using l[THEN spec[where x=x], unfolded LIMSEQ_def] and e > 0

  4568         by (auto elim!: allE[where x="e/2"])

  4569       fix n :: nat

  4570       assume "n \<ge> N"

  4571       then have "dist(s n x)(l x) < e"

  4572         using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  4573         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

  4574         by (auto simp add: dist_commute)

  4575     }

  4576     then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4577       by auto

  4578   }

  4579   then show ?lhs by auto

  4580 qed

  4581

  4582 lemma uniformly_cauchy_imp_uniformly_convergent:

  4583   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  4584   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"

  4585     and "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n \<longrightarrow> dist(s n x)(l x) < e)"

  4586   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4587 proof -

  4588   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"

  4589     using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto

  4590   moreover

  4591   {

  4592     fix x

  4593     assume "P x"

  4594     then have "l x = l' x"

  4595       using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  4596       using l and assms(2) unfolding LIMSEQ_def by blast

  4597   }

  4598   ultimately show ?thesis by auto

  4599 qed

  4600

  4601

  4602 subsection {* Continuity *}

  4603

  4604 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  4605

  4606 lemma continuous_within_eps_delta:

  4607   "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)"

  4608   unfolding continuous_within and Lim_within

  4609   apply auto

  4610   apply (metis dist_nz dist_self)

  4611   apply blast

  4612   done

  4613

  4614 lemma continuous_at_eps_delta:

  4615   "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  4616   using continuous_within_eps_delta [of x UNIV f] by simp

  4617

  4618 lemma continuous_at_right_real_increasing:

  4619   assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"

  4620   shows "(continuous (at_right (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f (a + delta) - f a < e)"

  4621   apply (auto simp add: continuous_within_eps_delta dist_real_def greaterThan_def)

  4622   apply (drule_tac x = e in spec, auto)

  4623   apply (drule_tac x = "a + d / 2" in spec)

  4624   apply (subst (asm) abs_of_nonneg)

  4625   apply (auto intro: nondecF simp add: field_simps)

  4626   apply (rule_tac x = "d / 2" in exI)

  4627   apply (auto simp add: field_simps)

  4628   apply (drule_tac x = e in spec, auto)

  4629   apply (rule_tac x = delta in exI, auto)

  4630   apply (subst abs_of_nonneg)

  4631   apply (auto intro: nondecF simp add: field_simps)

  4632   apply (rule le_less_trans)

  4633   prefer 2 apply assumption

  4634 by (rule nondecF, auto)

  4635

  4636 lemma continuous_at_left_real_increasing:

  4637   assumes nondecF: "\<And> x y. x \<le> y \<Longrightarrow> f x \<le> ((f y) :: real)"

  4638   shows "(continuous (at_left (a :: real)) f) = (\<forall>e > 0. \<exists>delta > 0. f a - f (a - delta) < e)"

  4639   apply (auto simp add: continuous_within_eps_delta dist_real_def lessThan_def)

  4640   apply (drule_tac x = e in spec, auto)

  4641   apply (drule_tac x = "a - d / 2" in spec)

  4642   apply (subst (asm) abs_of_nonpos)

  4643   apply (auto intro: nondecF simp add: field_simps)

  4644   apply (rule_tac x = "d / 2" in exI)

  4645   apply (auto simp add: field_simps)

  4646   apply (drule_tac x = e in spec, auto)

  4647   apply (rule_tac x = delta in exI, auto)

  4648   apply (subst abs_of_nonpos)

  4649   apply (auto intro: nondecF simp add: field_simps)

  4650   apply (rule less_le_trans)

  4651   apply assumption

  4652   apply auto

  4653 by (rule nondecF, auto)

  4654

  4655 text{* Versions in terms of open balls. *}

  4656

  4657 lemma continuous_within_ball:

  4658   "continuous (at x within s) f \<longleftrightarrow>

  4659     (\<forall>e > 0. \<exists>d > 0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e)"

  4660   (is "?lhs = ?rhs")

  4661 proof

  4662   assume ?lhs

  4663   {

  4664     fix e :: real

  4665     assume "e > 0"

  4666     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"

  4667       using ?lhs[unfolded continuous_within Lim_within] by auto

  4668     {

  4669       fix y

  4670       assume "y \<in> f  (ball x d \<inter> s)"

  4671       then have "y \<in> ball (f x) e"

  4672         using d(2)

  4673         unfolding dist_nz[symmetric]

  4674         apply (auto simp add: dist_commute)

  4675         apply (erule_tac x=xa in ballE)

  4676         apply auto

  4677         using e > 0

  4678         apply auto

  4679         done

  4680     }

  4681     then have "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e"

  4682       using d > 0

  4683       unfolding subset_eq ball_def by (auto simp add: dist_commute)

  4684   }

  4685   then show ?rhs by auto

  4686 next

  4687   assume ?rhs

  4688   then show ?lhs

  4689     unfolding continuous_within Lim_within ball_def subset_eq

  4690     apply (auto simp add: dist_commute)

  4691     apply (erule_tac x=e in allE)

  4692     apply auto

  4693     done

  4694 qed

  4695

  4696 lemma continuous_at_ball:

  4697   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  4698 proof

  4699   assume ?lhs

  4700   then show ?rhs

  4701     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4702     apply auto

  4703     apply (erule_tac x=e in allE)

  4704     apply auto

  4705     apply (rule_tac x=d in exI)

  4706     apply auto

  4707     apply (erule_tac x=xa in allE)

  4708     apply (auto simp add: dist_commute dist_nz)

  4709     unfolding dist_nz[symmetric]

  4710     apply auto

  4711     done

  4712 next

  4713   assume ?rhs

  4714   then show ?lhs

  4715     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4716     apply auto

  4717     apply (erule_tac x=e in allE)

  4718     apply auto

  4719     apply (rule_tac x=d in exI)

  4720     apply auto

  4721     apply (erule_tac x="f xa" in allE)

  4722     apply (auto simp add: dist_commute dist_nz)

  4723     done

  4724 qed

  4725

  4726 text{* Define setwise continuity in terms of limits within the set. *}

  4727

  4728 lemma continuous_on_iff:

  4729   "continuous_on s f \<longleftrightarrow>

  4730     (\<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)"

  4731   unfolding continuous_on_def Lim_within

  4732   by (metis dist_pos_lt dist_self)

  4733

  4734 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  4735   where "uniformly_continuous_on s f \<longleftrightarrow>

  4736     (\<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)"

  4737

  4738 text{* Some simple consequential lemmas. *}

  4739

  4740 lemma uniformly_continuous_imp_continuous:

  4741   "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"

  4742   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  4743

  4744 lemma continuous_at_imp_continuous_within:

  4745   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"

  4746   unfolding continuous_within continuous_at using Lim_at_within by auto

  4747

  4748 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  4749   by simp

  4750

  4751 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4752

  4753 lemma continuous_within_subset:

  4754   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"

  4755   unfolding continuous_within by(metis tendsto_within_subset)

  4756

  4757 lemma continuous_on_interior:

  4758   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4759   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)

  4760

  4761 lemma continuous_on_eq:

  4762   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4763   unfolding continuous_on_def tendsto_def eventually_at_topological

  4764   by simp

  4765

  4766 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4767

  4768 lemma continuous_within_sequentially:

  4769   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4770   shows "continuous (at a within s) f \<longleftrightarrow>

  4771     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4772          \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"

  4773   (is "?lhs = ?rhs")

  4774 proof

  4775   assume ?lhs

  4776   {

  4777     fix x :: "nat \<Rightarrow> 'a"

  4778     assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4779     fix T :: "'b set"

  4780     assume "open T" and "f a \<in> T"

  4781     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"

  4782       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  4783     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4784       using x(2) d>0 by simp

  4785     then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4786     proof eventually_elim

  4787       case (elim n)

  4788       then show ?case

  4789         using d x(1) f a \<in> T unfolding dist_nz[symmetric] by auto

  4790     qed

  4791   }

  4792   then show ?rhs

  4793     unfolding tendsto_iff tendsto_def by simp

  4794 next

  4795   assume ?rhs

  4796   then show ?lhs

  4797     unfolding continuous_within tendsto_def [where l="f a"]

  4798     by (simp add: sequentially_imp_eventually_within)

  4799 qed

  4800

  4801 lemma continuous_at_sequentially:

  4802   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4803   shows "continuous (at a) f \<longleftrightarrow>

  4804     (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"

  4805   using continuous_within_sequentially[of a UNIV f] by simp

  4806

  4807 lemma continuous_on_sequentially:

  4808   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4809   shows "continuous_on s f \<longleftrightarrow>

  4810     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4811       --> ((f \<circ> x) ---> f a) sequentially)"

  4812   (is "?lhs = ?rhs")

  4813 proof

  4814   assume ?rhs

  4815   then show ?lhs

  4816     using continuous_within_sequentially[of _ s f]

  4817     unfolding continuous_on_eq_continuous_within

  4818     by auto

  4819 next

  4820   assume ?lhs

  4821   then show ?rhs

  4822     unfolding continuous_on_eq_continuous_within

  4823     using continuous_within_sequentially[of _ s f]

  4824     by auto

  4825 qed

  4826

  4827 lemma uniformly_continuous_on_sequentially:

  4828   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4829                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4830                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4831 proof

  4832   assume ?lhs

  4833   {

  4834     fix x y

  4835     assume x: "\<forall>n. x n \<in> s"

  4836       and y: "\<forall>n. y n \<in> s"

  4837       and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4838     {

  4839       fix e :: real

  4840       assume "e > 0"

  4841       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"

  4842         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4843       obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"

  4844         using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4845       {

  4846         fix n

  4847         assume "n\<ge>N"

  4848         then have "dist (f (x n)) (f (y n)) < e"

  4849           using N[THEN spec[where x=n]]

  4850           using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

  4851           using x and y

  4852           unfolding dist_commute

  4853           by simp

  4854       }

  4855       then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4856         by auto

  4857     }

  4858     then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"

  4859       unfolding LIMSEQ_def and dist_real_def by auto

  4860   }

  4861   then show ?rhs by auto

  4862 next

  4863   assume ?rhs

  4864   {

  4865     assume "\<not> ?lhs"

  4866     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"

  4867       unfolding uniformly_continuous_on_def by auto

  4868     then obtain fa where fa:

  4869       "\<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"

  4870       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"]

  4871       unfolding Bex_def

  4872       by (auto simp add: dist_commute)

  4873     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4874     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4875     have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"

  4876       and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"

  4877       and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4878       unfolding x_def and y_def using fa

  4879       by auto

  4880     {

  4881       fix e :: real

  4882       assume "e > 0"

  4883       then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"

  4884         unfolding real_arch_inv[of e] by auto

  4885       {

  4886         fix n :: nat

  4887         assume "n \<ge> N"

  4888         then have "inverse (real n + 1) < inverse (real N)"

  4889           using real_of_nat_ge_zero and N\<noteq>0 by auto

  4890         also have "\<dots> < e" using N by auto

  4891         finally have "inverse (real n + 1) < e" by auto

  4892         then have "dist (x n) (y n) < e"

  4893           using xy0[THEN spec[where x=n]] by auto

  4894       }

  4895       then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto

  4896     }

  4897     then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4898       using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn

  4899       unfolding LIMSEQ_def dist_real_def by auto

  4900     then have False using fxy and e>0 by auto

  4901   }

  4902   then show ?lhs

  4903     unfolding uniformly_continuous_on_def by blast

  4904 qed

  4905

  4906 text{* The usual transformation theorems. *}

  4907

  4908 lemma continuous_transform_within:

  4909   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4910   assumes "0 < d"

  4911     and "x \<in> s"

  4912     and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4913     and "continuous (at x within s) f"

  4914   shows "continuous (at x within s) g"

  4915   unfolding continuous_within

  4916 proof (rule Lim_transform_within)

  4917   show "0 < d" by fact

  4918   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4919     using assms(3) by auto

  4920   have "f x = g x"

  4921     using assms(1,2,3) by auto

  4922   then show "(f ---> g x) (at x within s)"

  4923     using assms(4) unfolding continuous_within by simp

  4924 qed

  4925

  4926 lemma continuous_transform_at:

  4927   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4928   assumes "0 < d"

  4929     and "\<forall>x'. dist x' x < d --> f x' = g x'"

  4930     and "continuous (at x) f"

  4931   shows "continuous (at x) g"

  4932   using continuous_transform_within [of d x UNIV f g] assms by simp

  4933

  4934

  4935 subsubsection {* Structural rules for pointwise continuity *}

  4936

  4937 lemmas continuous_within_id = continuous_ident

  4938

  4939 lemmas continuous_at_id = isCont_ident

  4940

  4941 lemma continuous_infdist[continuous_intros]:

  4942   assumes "continuous F f"

  4943   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4944   using assms unfolding continuous_def by (rule tendsto_infdist)

  4945

  4946 lemma continuous_infnorm[continuous_intros]:

  4947   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4948   unfolding continuous_def by (rule tendsto_infnorm)

  4949

  4950 lemma continuous_inner[continuous_intros]:

  4951   assumes "continuous F f"

  4952     and "continuous F g"

  4953   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4954   using assms unfolding continuous_def by (rule tendsto_inner)

  4955

  4956 lemmas continuous_at_inverse = isCont_inverse

  4957

  4958 subsubsection {* Structural rules for setwise continuity *}

  4959

  4960 lemma continuous_on_infnorm[continuous_intros]:

  4961   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4962   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4963

  4964 lemma continuous_on_inner[continuous_intros]:

  4965   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4966   assumes "continuous_on s f"

  4967     and "continuous_on s g"

  4968   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4969   using bounded_bilinear_inner assms

  4970   by (rule bounded_bilinear.continuous_on)

  4971

  4972 subsubsection {* Structural rules for uniform continuity *}

  4973

  4974 lemma uniformly_continuous_on_id[continuous_intros]:

  4975   "uniformly_continuous_on s (\<lambda>x. x)"

  4976   unfolding uniformly_continuous_on_def by auto

  4977

  4978 lemma uniformly_continuous_on_const[continuous_intros]:

  4979   "uniformly_continuous_on s (\<lambda>x. c)"

  4980   unfolding uniformly_continuous_on_def by simp

  4981

  4982 lemma uniformly_continuous_on_dist[continuous_intros]:

  4983   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4984   assumes "uniformly_continuous_on s f"

  4985     and "uniformly_continuous_on s g"

  4986   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4987 proof -

  4988   {

  4989     fix a b c d :: 'b

  4990     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4991       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4992       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4993       by arith

  4994   } note le = this

  4995   {

  4996     fix x y

  4997     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4998     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4999     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  5000       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  5001         simp add: le)

  5002   }

  5003   then show ?thesis

  5004     using assms unfolding uniformly_continuous_on_sequentially

  5005     unfolding dist_real_def by simp

  5006 qed

  5007

  5008 lemma uniformly_continuous_on_norm[continuous_intros]:

  5009   assumes "uniformly_continuous_on s f"

  5010   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  5011   unfolding norm_conv_dist using assms

  5012   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  5013

  5014 lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:

  5015   assumes "uniformly_continuous_on s g"

  5016   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  5017   using assms unfolding uniformly_continuous_on_sequentially

  5018   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  5019   by (auto intro: tendsto_zero)

  5020

  5021 lemma uniformly_continuous_on_cmul[continuous_intros]:

  5022   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5023   assumes "uniformly_continuous_on s f"

  5024   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  5025   using bounded_linear_scaleR_right assms

  5026   by (rule bounded_linear.uniformly_continuous_on)

  5027

  5028 lemma dist_minus:

  5029   fixes x y :: "'a::real_normed_vector"

  5030   shows "dist (- x) (- y) = dist x y"

  5031   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  5032

  5033 lemma uniformly_continuous_on_minus[continuous_intros]:

  5034   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5035   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  5036   unfolding uniformly_continuous_on_def dist_minus .

  5037

  5038 lemma uniformly_continuous_on_add[continuous_intros]:

  5039   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5040   assumes "uniformly_continuous_on s f"

  5041     and "uniformly_continuous_on s g"

  5042   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  5043   using assms

  5044   unfolding uniformly_continuous_on_sequentially

  5045   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  5046   by (auto intro: tendsto_add_zero)

  5047

  5048 lemma uniformly_continuous_on_diff[continuous_intros]:

  5049   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5050   assumes "uniformly_continuous_on s f"

  5051     and "uniformly_continuous_on s g"

  5052   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  5053   using assms uniformly_continuous_on_add [of s f "- g"]

  5054     by (simp add: fun_Compl_def uniformly_continuous_on_minus)

  5055

  5056 text{* Continuity of all kinds is preserved under composition. *}

  5057

  5058 lemmas continuous_at_compose = isCont_o

  5059

  5060 lemma uniformly_continuous_on_compose[continuous_intros]:

  5061   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  5062   shows "uniformly_continuous_on s (g \<circ> f)"

  5063 proof -

  5064   {

  5065     fix e :: real

  5066     assume "e > 0"

  5067     then obtain d where "d > 0"

  5068       and d: "\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5069       using assms(2) unfolding uniformly_continuous_on_def by auto

  5070     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"

  5071       using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto

  5072     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"

  5073       using d>0 using d by auto

  5074   }

  5075   then show ?thesis

  5076     using assms unfolding uniformly_continuous_on_def by auto

  5077 qed

  5078

  5079 text{* Continuity in terms of open preimages. *}

  5080

  5081 lemma continuous_at_open:

  5082   "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)))"

  5083   unfolding continuous_within_topological [of x UNIV f]

  5084   unfolding imp_conjL

  5085   by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  5086

  5087 lemma continuous_imp_tendsto:

  5088   assumes "continuous (at x0) f"

  5089     and "x ----> x0"

  5090   shows "(f \<circ> x) ----> (f x0)"

  5091 proof (rule topological_tendstoI)

  5092   fix S

  5093   assume "open S" "f x0 \<in> S"

  5094   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  5095      using assms continuous_at_open by metis

  5096   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  5097     using assms T_def by (auto simp: tendsto_def)

  5098   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  5099     using T_def by (auto elim!: eventually_elim1)

  5100 qed

  5101

  5102 lemma continuous_on_open:

  5103   "continuous_on s f \<longleftrightarrow>

  5104     (\<forall>t. openin (subtopology euclidean (f  s)) t \<longrightarrow>

  5105       openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5106   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  5107   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5108

  5109 text {* Similarly in terms of closed sets. *}

  5110

  5111 lemma continuous_on_closed:

  5112   "continuous_on s f \<longleftrightarrow>

  5113     (\<forall>t. closedin (subtopology euclidean (f  s)) t \<longrightarrow>

  5114       closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5115   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  5116   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5117

  5118 text {* Half-global and completely global cases. *}

  5119

  5120 lemma continuous_open_in_preimage:

  5121   assumes "continuous_on s f"  "open t"

  5122   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5123 proof -

  5124   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5125     by auto

  5126   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5127     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  5128   then show ?thesis

  5129     using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]]

  5130     using * by auto

  5131 qed

  5132

  5133 lemma continuous_closed_in_preimage:

  5134   assumes "continuous_on s f" and "closed t"

  5135   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5136 proof -

  5137   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5138     by auto

  5139   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5140     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute

  5141     by auto

  5142   then show ?thesis

  5143     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]]

  5144     using * by auto

  5145 qed

  5146

  5147 lemma continuous_open_preimage:

  5148   assumes "continuous_on s f"

  5149     and "open s"

  5150     and "open t"

  5151   shows "open {x \<in> s. f x \<in> t}"

  5152 proof-

  5153   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5154     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  5155   then show ?thesis

  5156     using open_Int[of s T, OF assms(2)] by auto

  5157 qed

  5158

  5159 lemma continuous_closed_preimage:

  5160   assumes "continuous_on s f"

  5161     and "closed s"

  5162     and "closed t"

  5163   shows "closed {x \<in> s. f x \<in> t}"

  5164 proof-

  5165   obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5166     using continuous_closed_in_preimage[OF assms(1,3)]

  5167     unfolding closedin_closed by auto

  5168   then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

  5169 qed

  5170

  5171 lemma continuous_open_preimage_univ:

  5172   "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  5173   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  5174

  5175 lemma continuous_closed_preimage_univ:

  5176   "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s \<Longrightarrow> closed {x. f x \<in> s}"

  5177   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  5178

  5179 lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  5180   unfolding vimage_def by (rule continuous_open_preimage_univ)

  5181

  5182 lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  5183   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  5184

  5185 lemma interior_image_subset:

  5186   assumes "\<forall>x. continuous (at x) f"

  5187     and "inj f"

  5188   shows "interior (f  s) \<subseteq> f  (interior s)"

  5189 proof

  5190   fix x assume "x \<in> interior (f  s)"

  5191   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  5192   then have "x \<in> f  s" by auto

  5193   then obtain y where y: "y \<in> s" "x = f y" by auto

  5194   have "open (vimage f T)"

  5195     using assms(1) open T by (rule continuous_open_vimage)

  5196   moreover have "y \<in> vimage f T"

  5197     using x = f y x \<in> T by simp

  5198   moreover have "vimage f T \<subseteq> s"

  5199     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  5200   ultimately have "y \<in> interior s" ..

  5201   with x = f y show "x \<in> f  interior s" ..

  5202 qed

  5203

  5204 text {* Equality of continuous functions on closure and related results. *}

  5205

  5206 lemma continuous_closed_in_preimage_constant:

  5207   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5208   shows "continuous_on s f \<Longrightarrow> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  5209   using continuous_closed_in_preimage[of s f "{a}"] by auto

  5210

  5211 lemma continuous_closed_preimage_constant:

  5212   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5213   shows "continuous_on s f \<Longrightarrow> closed s \<Longrightarrow> closed {x \<in> s. f x = a}"

  5214   using continuous_closed_preimage[of s f "{a}"] by auto

  5215

  5216 lemma continuous_constant_on_closure:

  5217   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5218   assumes "continuous_on (closure s) f"

  5219     and "\<forall>x \<in> s. f x = a"

  5220   shows "\<forall>x \<in> (closure s). f x = a"

  5221     using continuous_closed_preimage_constant[of "closure s" f a]

  5222       assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset

  5223     unfolding subset_eq

  5224     by auto

  5225

  5226 lemma image_closure_subset:

  5227   assumes "continuous_on (closure s) f"

  5228     and "closed t"

  5229     and "(f  s) \<subseteq> t"

  5230   shows "f  (closure s) \<subseteq> t"

  5231 proof -

  5232   have "s \<subseteq> {x \<in> closure s. f x \<in> t}"

  5233     using assms(3) closure_subset by auto

  5234   moreover have "closed {x \<in> closure s. f x \<in> t}"

  5235     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  5236   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  5237     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  5238   then show ?thesis by auto

  5239 qed

  5240

  5241 lemma continuous_on_closure_norm_le:

  5242   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5243   assumes "continuous_on (closure s) f"

  5244     and "\<forall>y \<in> s. norm(f y) \<le> b"

  5245     and "x \<in> (closure s)"

  5246   shows "norm (f x) \<le> b"

  5247 proof -

  5248   have *: "f  s \<subseteq> cball 0 b"

  5249     using assms(2)[unfolded mem_cball_0[symmetric]] by auto

  5250   show ?thesis

  5251     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  5252     unfolding subset_eq

  5253     apply (erule_tac x="f x" in ballE)

  5254     apply (auto simp add: dist_norm)

  5255     done

  5256 qed

  5257

  5258 text {* Making a continuous function avoid some value in a neighbourhood. *}

  5259

  5260 lemma continuous_within_avoid:

  5261   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5262   assumes "continuous (at x within s) f"

  5263     and "f x \<noteq> a"

  5264   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  5265 proof -

  5266   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  5267     using t1_space [OF f x \<noteq> a] by fast

  5268   have "(f ---> f x) (at x within s)"

  5269     using assms(1) by (simp add: continuous_within)

  5270   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  5271     using open U and f x \<in> U

  5272     unfolding tendsto_def by fast

  5273   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  5274     using a \<notin> U by (fast elim: eventually_mono [rotated])

  5275   then show ?thesis

  5276     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  5277 qed

  5278

  5279 lemma continuous_at_avoid:

  5280   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5281   assumes "continuous (at x) f"

  5282     and "f x \<noteq> a"

  5283   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5284   using assms continuous_within_avoid[of x UNIV f a] by simp

  5285

  5286 lemma continuous_on_avoid:

  5287   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5288   assumes "continuous_on s f"

  5289     and "x \<in> s"

  5290     and "f x \<noteq> a"

  5291   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  5292   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  5293     OF assms(2)] continuous_within_avoid[of x s f a]

  5294   using assms(3)

  5295   by auto

  5296

  5297 lemma continuous_on_open_avoid:

  5298   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5299   assumes "continuous_on s f"

  5300     and "open s"

  5301     and "x \<in> s"

  5302     and "f x \<noteq> a"

  5303   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5304   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  5305   using continuous_at_avoid[of x f a] assms(4)

  5306   by auto

  5307

  5308 text {* Proving a function is constant by proving open-ness of level set. *}

  5309

  5310 lemma continuous_levelset_open_in_cases:

  5311   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5312   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5313         openin (subtopology euclidean s) {x \<in> s. f x = a}

  5314         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  5315   unfolding connected_clopen

  5316   using continuous_closed_in_preimage_constant by auto

  5317

  5318 lemma continuous_levelset_open_in:

  5319   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5320   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5321         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  5322         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"

  5323   using continuous_levelset_open_in_cases[of s f ]

  5324   by meson

  5325

  5326 lemma continuous_levelset_open:

  5327   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5328   assumes "connected s"

  5329     and "continuous_on s f"

  5330     and "open {x \<in> s. f x = a}"

  5331     and "\<exists>x \<in> s.  f x = a"

  5332   shows "\<forall>x \<in> s. f x = a"

  5333   using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

  5334   using assms (3,4)

  5335   by fast

  5336

  5337 text {* Some arithmetical combinations (more to prove). *}

  5338

  5339 lemma open_scaling[intro]:

  5340   fixes s :: "'a::real_normed_vector set"

  5341   assumes "c \<noteq> 0"

  5342     and "open s"

  5343   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  5344 proof -

  5345   {

  5346     fix x

  5347     assume "x \<in> s"

  5348     then obtain e where "e>0"

  5349       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  5350       by auto

  5351     have "e * abs c > 0"

  5352       using assms(1)[unfolded zero_less_abs_iff[symmetric]] e>0 by auto

  5353     moreover

  5354     {

  5355       fix y

  5356       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  5357       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  5358         unfolding dist_norm

  5359         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  5360           assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

  5361       then have "y \<in> op *\<^sub>R c  s"

  5362         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]

  5363         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  5364         using assms(1)

  5365         unfolding dist_norm scaleR_scaleR

  5366         by auto

  5367     }

  5368     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s"

  5369       apply (rule_tac x="e * abs c" in exI)

  5370       apply auto

  5371       done

  5372   }

  5373   then show ?thesis unfolding open_dist by auto

  5374 qed

  5375

  5376 lemma minus_image_eq_vimage:

  5377   fixes A :: "'a::ab_group_add set"

  5378   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  5379   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  5380

  5381 lemma open_negations:

  5382   fixes s :: "'a::real_normed_vector set"

  5383   shows "open s \<Longrightarrow> open ((\<lambda>x. - x)  s)"

  5384   using open_scaling [of "- 1" s] by simp

  5385

  5386 lemma open_translation:

  5387   fixes s :: "'a::real_normed_vector set"

  5388   assumes "open s"

  5389   shows "open((\<lambda>x. a + x)  s)"

  5390 proof -

  5391   {

  5392     fix x

  5393     have "continuous (at x) (\<lambda>x. x - a)"

  5394       by (intro continuous_diff continuous_at_id continuous_const)

  5395   }

  5396   moreover have "{x. x - a \<in> s} = op + a  s"

  5397     by force

  5398   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]

  5399     using assms by auto

  5400 qed

  5401

  5402 lemma open_affinity:

  5403   fixes s :: "'a::real_normed_vector set"

  5404   assumes "open s"  "c \<noteq> 0"

  5405   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5406 proof -

  5407   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  5408     unfolding o_def ..

  5409   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s"

  5410     by auto

  5411   then show ?thesis

  5412     using assms open_translation[of "op *\<^sub>R c  s" a]

  5413     unfolding *

  5414     by auto

  5415 qed

  5416

  5417 lemma interior_translation:

  5418   fixes s :: "'a::real_normed_vector set"

  5419   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  5420 proof (rule set_eqI, rule)

  5421   fix x

  5422   assume "x \<in> interior (op + a  s)"

  5423   then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a  s"

  5424     unfolding mem_interior by auto

  5425   then have "ball (x - a) e \<subseteq> s"

  5426     unfolding subset_eq Ball_def mem_ball dist_norm

  5427     apply auto

  5428     apply (erule_tac x="a + xa" in allE)

  5429     unfolding ab_group_add_class.diff_diff_eq[symmetric]

  5430     apply auto

  5431     done

  5432   then show "x \<in> op + a  interior s"

  5433     unfolding image_iff

  5434     apply (rule_tac x="x - a" in bexI)

  5435     unfolding mem_interior

  5436     using e > 0

  5437     apply auto

  5438     done

  5439 next

  5440   fix x

  5441   assume "x \<in> op + a  interior s"

  5442   then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"

  5443     unfolding image_iff Bex_def mem_interior by auto

  5444   {

  5445     fix z

  5446     have *: "a + y - z = y + a - z" by auto

  5447     assume "z \<in> ball x e"

  5448     then have "z - a \<in> s"

  5449       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  5450       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  5451       by auto

  5452     then have "z \<in> op + a  s"

  5453       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  5454   }

  5455   then have "ball x e \<subseteq> op + a  s"

  5456     unfolding subset_eq by auto

  5457   then show "x \<in> interior (op + a  s)"

  5458     unfolding mem_interior using e > 0 by auto

  5459 qed

  5460

  5461 text {* Topological properties of linear functions. *}

  5462

  5463 lemma linear_lim_0:

  5464   assumes "bounded_linear f"

  5465   shows "(f ---> 0) (at (0))"

  5466 proof -

  5467   interpret f: bounded_linear f by fact

  5468   have "(f ---> f 0) (at 0)"

  5469     using tendsto_ident_at by (rule f.tendsto)

  5470   then show ?thesis unfolding f.zero .

  5471 qed

  5472

  5473 lemma linear_continuous_at:

  5474   assumes "bounded_linear f"

  5475   shows "continuous (at a) f"

  5476   unfolding continuous_at using assms

  5477   apply (rule bounded_linear.tendsto)

  5478   apply (rule tendsto_ident_at)

  5479   done

  5480

  5481 lemma linear_continuous_within:

  5482   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"

  5483   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  5484

  5485 lemma linear_continuous_on:

  5486   "bounded_linear f \<Longrightarrow> continuous_on s f"

  5487   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  5488

  5489 text {* Also bilinear functions, in composition form. *}

  5490

  5491 lemma bilinear_continuous_at_compose:

  5492   "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5493     continuous (at x) (\<lambda>x. h (f x) (g x))"

  5494   unfolding continuous_at

  5495   using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

  5496   by auto

  5497

  5498 lemma bilinear_continuous_within_compose:

  5499   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5500     continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  5501   unfolding continuous_within

  5502   using Lim_bilinear[of f "f x"]

  5503   by auto

  5504

  5505 lemma bilinear_continuous_on_compose:

  5506   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5507     continuous_on s (\<lambda>x. h (f x) (g x))"

  5508   unfolding continuous_on_def

  5509   by (fast elim: bounded_bilinear.tendsto)

  5510

  5511 text {* Preservation of compactness and connectedness under continuous function. *}

  5512

  5513 lemma compact_eq_openin_cover:

  5514   "compact S \<longleftrightarrow>

  5515     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5516       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  5517 proof safe

  5518   fix C

  5519   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  5520   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}"

  5521     unfolding openin_open by force+

  5522   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"

  5523     by (rule compactE)

  5524   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)"

  5525     by auto

  5526   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5527 next

  5528   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5529         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  5530   show "compact S"

  5531   proof (rule compactI)

  5532     fix C

  5533     let ?C = "image (\<lambda>T. S \<inter> T) C"

  5534     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  5535     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  5536       unfolding openin_open by auto

  5537     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  5538       by metis

  5539     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  5540     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  5541     proof (intro conjI)

  5542       from D \<subseteq> ?C show "?D \<subseteq> C"

  5543         by (fast intro: inv_into_into)

  5544       from finite D show "finite ?D"

  5545         by (rule finite_imageI)

  5546       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  5547         apply (rule subset_trans)

  5548         apply clarsimp

  5549         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  5550         apply (erule rev_bexI, fast)

  5551         done

  5552     qed

  5553     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5554   qed

  5555 qed

  5556

  5557 lemma connected_continuous_image:

  5558   assumes "continuous_on s f"

  5559     and "connected s"

  5560   shows "connected(f  s)"

  5561 proof -

  5562   {

  5563     fix T

  5564     assume as:

  5565       "T \<noteq> {}"

  5566       "T \<noteq> f  s"

  5567       "openin (subtopology euclidean (f  s)) T"

  5568       "closedin (subtopology euclidean (f  s)) T"

  5569     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  5570       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  5571       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  5572       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  5573     then have False using as(1,2)

  5574       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

  5575   }

  5576   then show ?thesis

  5577     unfolding connected_clopen by auto

  5578 qed

  5579

  5580 text {* Continuity implies uniform continuity on a compact domain. *}

  5581

  5582 lemma compact_uniformly_continuous:

  5583   assumes f: "continuous_on s f"

  5584     and s: "compact s"

  5585   shows "uniformly_continuous_on s f"

  5586   unfolding uniformly_continuous_on_def

  5587 proof (cases, safe)

  5588   fix e :: real

  5589   assume "0 < e" "s \<noteq> {}"

  5590   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) } }"

  5591   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  5592   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  5593   proof safe

  5594     fix y

  5595     assume "y \<in> s"

  5596     from continuous_open_in_preimage[OF f open_ball]

  5597     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  5598       unfolding openin_subtopology open_openin by metis

  5599     then obtain d where "ball y d \<subseteq> T" "0 < d"

  5600       using 0 < e y \<in> s by (auto elim!: openE)

  5601     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  5602       by (intro UN_I[of "(y, d)"]) auto

  5603   qed auto

  5604   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  5605     by (rule compactE_image)

  5606   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  5607     by (subst Min_gr_iff) auto

  5608   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  5609   proof (rule, safe)

  5610     fix x x'

  5611     assume in_s: "x' \<in> s" "x \<in> s"

  5612     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  5613       by blast

  5614     moreover assume "dist x x' < Min (sndD) / 2"

  5615     ultimately have "dist y x' < d"

  5616       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  5617     with D x in_s show  "dist (f x) (f x') < e"

  5618       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  5619   qed (insert D, auto)

  5620 qed auto

  5621

  5622 text {* A uniformly convergent limit of continuous functions is continuous. *}

  5623

  5624 lemma continuous_uniform_limit:

  5625   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  5626   assumes "\<not> trivial_limit F"

  5627     and "eventually (\<lambda>n. continuous_on s (f n)) F"

  5628     and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  5629   shows "continuous_on s g"

  5630 proof -

  5631   {

  5632     fix x and e :: real

  5633     assume "x\<in>s" "e>0"

  5634     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  5635       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  5636     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  5637     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  5638       using assms(1) by blast

  5639     have "e / 3 > 0" using e>0 by auto

  5640     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"

  5641       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  5642     {

  5643       fix y

  5644       assume "y \<in> s" and "dist y x < d"

  5645       then have "dist (f n y) (f n x) < e / 3"

  5646         by (rule d [rule_format])

  5647       then have "dist (f n y) (g x) < 2 * e / 3"

  5648         using dist_triangle [of "f n y" "g x" "f n x"]

  5649         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  5650         by auto

  5651       then have "dist (g y) (g x) < e"

  5652         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  5653         using dist_triangle3 [of "g y" "g x" "f n y"]

  5654         by auto

  5655     }

  5656     then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5657       using d>0 by auto

  5658   }

  5659   then show ?thesis

  5660     unfolding continuous_on_iff by auto

  5661 qed

  5662

  5663

  5664 subsection {* Topological stuff lifted from and dropped to R *}

  5665

  5666 lemma open_real:

  5667   fixes s :: "real set"

  5668   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"

  5669   unfolding open_dist dist_norm by simp

  5670

  5671 lemma islimpt_approachable_real:

  5672   fixes s :: "real set"

  5673   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  5674   unfolding islimpt_approachable dist_norm by simp

  5675

  5676 lemma closed_real:

  5677   fixes s :: "real set"

  5678   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)"

  5679   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5680

  5681 lemma continuous_at_real_range:

  5682   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5683   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5684   unfolding continuous_at

  5685   unfolding Lim_at

  5686   unfolding dist_nz[symmetric]

  5687   unfolding dist_norm

  5688   apply auto

  5689   apply (erule_tac x=e in allE)

  5690   apply auto

  5691   apply (rule_tac x=d in exI)

  5692   apply auto

  5693   apply (erule_tac x=x' in allE)

  5694   apply auto

  5695   apply (erule_tac x=e in allE)

  5696   apply auto

  5697   done

  5698

  5699 lemma continuous_on_real_range:

  5700   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5701   shows "continuous_on s f \<longleftrightarrow>

  5702     (\<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))"

  5703   unfolding continuous_on_iff dist_norm by simp

  5704

  5705 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5706

  5707 lemma distance_attains_sup:

  5708   assumes "compact s" "s \<noteq> {}"

  5709   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  5710 proof (rule continuous_attains_sup [OF assms])

  5711   {

  5712     fix x

  5713     assume "x\<in>s"

  5714     have "(dist a ---> dist a x) (at x within s)"

  5715       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  5716   }

  5717   then show "continuous_on s (dist a)"

  5718     unfolding continuous_on ..

  5719 qed

  5720

  5721 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5722

  5723 lemma distance_attains_inf:

  5724   fixes a :: "'a::heine_borel"

  5725   assumes "closed s"

  5726     and "s \<noteq> {}"

  5727   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  5728 proof -

  5729   from assms(2) obtain b where "b \<in> s" by auto

  5730   let ?B = "s \<inter> cball a (dist b a)"

  5731   have "?B \<noteq> {}" using b \<in> s

  5732     by (auto simp add: dist_commute)

  5733   moreover have "continuous_on ?B (dist a)"

  5734     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  5735   moreover have "compact ?B"

  5736     by (intro closed_inter_compact closed s compact_cball)

  5737   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  5738     by (metis continuous_attains_inf)

  5739   then show ?thesis by fastforce

  5740 qed

  5741

  5742

  5743 subsection {* Pasted sets *}

  5744

  5745 lemma bounded_Times:

  5746   assumes "bounded s" "bounded t"

  5747   shows "bounded (s \<times> t)"

  5748 proof -

  5749   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5750     using assms [unfolded bounded_def] by auto

  5751   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  5752     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5753   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5754 qed

  5755

  5756 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5757   by (induct x) simp

  5758

  5759 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5760   unfolding seq_compact_def

  5761   apply clarify

  5762   apply (drule_tac x="fst \<circ> f" in spec)

  5763   apply (drule mp, simp add: mem_Times_iff)

  5764   apply (clarify, rename_tac l1 r1)

  5765   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5766   apply (drule mp, simp add: mem_Times_iff)

  5767   apply (clarify, rename_tac l2 r2)

  5768   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5769   apply (rule_tac x="r1 \<circ> r2" in exI)

  5770   apply (rule conjI, simp add: subseq_def)

  5771   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  5772   apply (drule (1) tendsto_Pair) back

  5773   apply (simp add: o_def)

  5774   done

  5775

  5776 lemma compact_Times:

  5777   assumes "compact s" "compact t"

  5778   shows "compact (s \<times> t)"

  5779 proof (rule compactI)

  5780   fix C

  5781   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  5782   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)"

  5783   proof

  5784     fix x

  5785     assume "x \<in> s"

  5786     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")

  5787     proof

  5788       fix y

  5789       assume "y \<in> t"

  5790       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  5791       then show "?P y" by (auto elim!: open_prod_elim)

  5792     qed

  5793     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  5794       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"

  5795       by metis

  5796     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  5797     from compactE_image[OF compact t this] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  5798       by auto

  5799     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  5800       by (fastforce simp: subset_eq)

  5801     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  5802       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  5803   qed

  5804   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  5805     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"

  5806     unfolding subset_eq UN_iff by metis

  5807   moreover

  5808   from compactE_image[OF compact s a]

  5809   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  5810     by auto

  5811   moreover

  5812   {

  5813     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  5814       by auto

  5815     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  5816       using d e \<subseteq> s by (intro UN_mono) auto

  5817     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  5818   }

  5819   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  5820     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  5821 qed

  5822

  5823 text{* Hence some useful properties follow quite easily. *}

  5824

  5825 lemma compact_scaling:

  5826   fixes s :: "'a::real_normed_vector set"

  5827   assumes "compact s"

  5828   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5829 proof -

  5830   let ?f = "\<lambda>x. scaleR c x"

  5831   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5832   show ?thesis

  5833     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5834     using linear_continuous_at[OF *] assms

  5835     by auto

  5836 qed

  5837

  5838 lemma compact_negations:

  5839   fixes s :: "'a::real_normed_vector set"

  5840   assumes "compact s"

  5841   shows "compact ((\<lambda>x. - x)  s)"

  5842   using compact_scaling [OF assms, of "- 1"] by auto

  5843

  5844 lemma compact_sums:

  5845   fixes s t :: "'a::real_normed_vector set"

  5846   assumes "compact s"

  5847     and "compact t"

  5848   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5849 proof -

  5850   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5851     apply auto

  5852     unfolding image_iff

  5853     apply (rule_tac x="(xa, y)" in bexI)

  5854     apply auto

  5855     done

  5856   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5857     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5858   then show ?thesis

  5859     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5860 qed

  5861

  5862 lemma compact_differences:

  5863   fixes s t :: "'a::real_normed_vector set"

  5864   assumes "compact s"

  5865     and "compact t"

  5866   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5867 proof-

  5868   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5869     apply auto

  5870     apply (rule_tac x= xa in exI)

  5871     apply auto

  5872     done

  5873   then show ?thesis

  5874     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5875 qed

  5876

  5877 lemma compact_translation:

  5878   fixes s :: "'a::real_normed_vector set"

  5879   assumes "compact s"

  5880   shows "compact ((\<lambda>x. a + x)  s)"

  5881 proof -

  5882   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  5883     by auto

  5884   then show ?thesis

  5885     using compact_sums[OF assms compact_sing[of a]] by auto

  5886 qed

  5887

  5888 lemma compact_affinity:

  5889   fixes s :: "'a::real_normed_vector set"

  5890   assumes "compact s"

  5891   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5892 proof -

  5893   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  5894     by auto

  5895   then show ?thesis

  5896     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5897 qed

  5898

  5899 text {* Hence we get the following. *}

  5900

  5901 lemma compact_sup_maxdistance:

  5902   fixes s :: "'a::metric_space set"

  5903   assumes "compact s"

  5904     and "s \<noteq> {}"

  5905   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5906 proof -

  5907   have "compact (s \<times> s)"

  5908     using compact s by (intro compact_Times)

  5909   moreover have "s \<times> s \<noteq> {}"

  5910     using s \<noteq> {} by auto

  5911   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  5912     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  5913   ultimately show ?thesis

  5914     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  5915 qed

  5916

  5917 text {* We can state this in terms of diameter of a set. *}

  5918

  5919 definition diameter :: "'a::metric_space set \<Rightarrow> real" where

  5920   "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"

  5921

  5922 lemma diameter_bounded_bound:

  5923   fixes s :: "'a :: metric_space set"

  5924   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  5925   shows "dist x y \<le> diameter s"

  5926 proof -

  5927   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  5928     unfolding bounded_def by auto

  5929   have "bdd_above (split dist  (s\<times>s))"

  5930   proof (intro bdd_aboveI, safe)

  5931     fix a b

  5932     assume "a \<in> s" "b \<in> s"

  5933     with z[of a] z[of b] dist_triangle[of a b z]

  5934     show "dist a b \<le> 2 * d"

  5935       by (simp add: dist_commute)

  5936   qed

  5937   moreover have "(x,y) \<in> s\<times>s" using s by auto

  5938   ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"

  5939     by (rule cSUP_upper2) simp

  5940   with x \<in> s show ?thesis

  5941     by (auto simp add: diameter_def)

  5942 qed

  5943

  5944 lemma diameter_lower_bounded:

  5945   fixes s :: "'a :: metric_space set"

  5946   assumes s: "bounded s"

  5947     and d: "0 < d" "d < diameter s"

  5948   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  5949 proof (rule ccontr)

  5950   assume contr: "\<not> ?thesis"

  5951   moreover have "s \<noteq> {}"

  5952     using d by (auto simp add: diameter_def)

  5953   ultimately have "diameter s \<le> d"

  5954     by (auto simp: not_less diameter_def intro!: cSUP_least)

  5955   with d < diameter s show False by auto

  5956 qed

  5957

  5958 lemma diameter_bounded:

  5959   assumes "bounded s"

  5960   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  5961     and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  5962   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  5963   by auto

  5964

  5965 lemma diameter_compact_attained:

  5966   assumes "compact s"

  5967     and "s \<noteq> {}"

  5968   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  5969 proof -

  5970   have b: "bounded s" using assms(1)

  5971     by (rule compact_imp_bounded)

  5972   then obtain x y where xys: "x\<in>s" "y\<in>s"

  5973     and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5974     using compact_sup_maxdistance[OF assms] by auto

  5975   then have "diameter s \<le> dist x y"

  5976     unfolding diameter_def

  5977     apply clarsimp

  5978     apply (rule cSUP_least)

  5979     apply fast+

  5980     done

  5981   then show ?thesis

  5982     by (metis b diameter_bounded_bound order_antisym xys)

  5983 qed

  5984

  5985 text {* Related results with closure as the conclusion. *}

  5986

  5987 lemma closed_scaling:

  5988   fixes s :: "'a::real_normed_vector set"

  5989   assumes "closed s"

  5990   shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5991 proof (cases "c = 0")

  5992   case True then show ?thesis

  5993     by (auto simp add: image_constant_conv)

  5994 next

  5995   case False

  5996   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) - s)"

  5997     by (simp add: continuous_closed_vimage)

  5998   also have "(\<lambda>x. inverse c *\<^sub>R x) - s = (\<lambda>x. c *\<^sub>R x)  s"

  5999     using c \<noteq> 0 by (auto elim: image_eqI [rotated])

  6000   finally show ?thesis .

  6001 qed

  6002

  6003 lemma closed_negations:

  6004   fixes s :: "'a::real_normed_vector set"

  6005   assumes "closed s"

  6006   shows "closed ((\<lambda>x. -x)  s)"

  6007   using closed_scaling[OF assms, of "- 1"] by simp

  6008

  6009 lemma compact_closed_sums:

  6010   fixes s :: "'a::real_normed_vector set"

  6011   assumes "compact s" and "closed t"

  6012   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  6013 proof -

  6014   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  6015   {

  6016     fix x l
`