src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author huffman Wed Mar 26 11:05:25 2014 -0700 (2014-03-26) changeset 56290 801a72ad52d3 parent 56189 c4daa97ac57a child 56371 fb9ae0727548 permissions -rw-r--r--
tuned proofs
     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_on_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 definition (in euclidean_space) eucl_less (infix "<e" 50)

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

   830

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

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

   833

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

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

   836   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)"

   837     "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)"

   838   by (auto simp: box_eucl_less eucl_less_def cbox_def)

   839

   840 lemma mem_box_real[simp]:

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

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

   843   by (auto simp: mem_box)

   844

   845 lemma box_real[simp]:

   846   fixes a b:: real

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

   848   by auto

   849

   850 lemma rational_boxes:

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

   852   assumes "e > 0"

   853   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"

   854 proof -

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

   856   then have e: "e' > 0"

   857     using assms by (auto intro!: divide_pos_pos simp: DIM_positive)

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

   859   proof

   860     fix i

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

   862     show "?th i" by auto

   863   qed

   864   from choice[OF this] obtain a where

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

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

   867   proof

   868     fix i

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

   870     show "?th i" by auto

   871   qed

   872   from choice[OF this] obtain b where

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

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

   875   show ?thesis

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

   877     fix y :: 'a

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

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

   880       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   882     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   883       fix i :: "'a"

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

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

   886         using * i by (auto simp: box_def)

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

   888         using a by auto

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

   890         using b by auto

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

   892         by auto

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

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

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

   896         by (rule power_strict_mono) auto

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

   898         by (simp add: power_divide)

   899     qed auto

   900     also have "\<dots> = e"

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

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

   903       by (auto simp: ball_def)

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

   905 qed

   906

   907 lemma open_UNION_box:

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

   909   assumes "open M"

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

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

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

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

   914 proof -

   915   {

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

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

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

   919     moreover obtain a b where ab:

   920       "x \<in> box a b"

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

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

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

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

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

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

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

   928   }

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

   930 qed

   931

   932 lemma box_eq_empty:

   933   fixes a :: "'a::euclidean_space"

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

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

   936 proof -

   937   {

   938     fix i x

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

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

   941       unfolding mem_box by (auto simp: box_def)

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

   943     then have False using as by auto

   944   }

   945   moreover

   946   {

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

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

   949     {

   950       fix i :: 'a

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

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

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

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

   955         by (auto simp: inner_add_left)

   956     }

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

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

   959   }

   960   ultimately show ?th1 by blast

   961

   962   {

   963     fix i x

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

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

   966       unfolding mem_box by auto

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

   968     then have False using as by auto

   969   }

   970   moreover

   971   {

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

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

   974     {

   975       fix i :: 'a

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

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

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

   979       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"

   980         by (auto simp: inner_add_left)

   981     }

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

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

   984   }

   985   ultimately show ?th2 by blast

   986 qed

   987

   988 lemma box_ne_empty:

   989   fixes a :: "'a::euclidean_space"

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

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

   992   unfolding box_eq_empty[of a b] by fastforce+

   993

   994 lemma

   995   fixes a :: "'a::euclidean_space"

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

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

   998   unfolding set_eq_iff mem_box eq_iff [symmetric]

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

  1000      (metis all_not_in_conv nonempty_Basis)

  1001

  1002 lemma subset_box_imp:

  1003   fixes a :: "'a::euclidean_space"

  1004   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"

  1005     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"

  1006     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"

  1007      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"

  1008   unfolding subset_eq[unfolded Ball_def] unfolding mem_box

  1009    by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  1010

  1011 lemma box_subset_cbox:

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

  1013   shows "box a b \<subseteq> cbox a b"

  1014   unfolding subset_eq [unfolded Ball_def] mem_box

  1015   by (fast intro: less_imp_le)

  1016

  1017 lemma subset_box:

  1018   fixes a :: "'a::euclidean_space"

  1019   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)

  1020     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)

  1021     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)

  1022     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)

  1023 proof -

  1024   show ?th1

  1025     unfolding subset_eq and Ball_def and mem_box

  1026     by (auto intro: order_trans)

  1027   show ?th2

  1028     unfolding subset_eq and Ball_def and mem_box

  1029     by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)

  1030   {

  1031     assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  1032     then have "box c d \<noteq> {}"

  1033       unfolding box_eq_empty by auto

  1034     fix i :: 'a

  1035     assume i: "i \<in> Basis"

  1036     (** TODO combine the following two parts as done in the HOL_light version. **)

  1037     {

  1038       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"

  1039       assume as2: "a\<bullet>i > c\<bullet>i"

  1040       {

  1041         fix j :: 'a

  1042         assume j: "j \<in> Basis"

  1043         then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  1044           apply (cases "j = i")

  1045           using as(2)[THEN bspec[where x=j]] i

  1046           apply (auto simp add: as2)

  1047           done

  1048       }

  1049       then have "?x\<in>box c d"

  1050         using i unfolding mem_box by auto

  1051       moreover

  1052       have "?x \<notin> cbox a b"

  1053         unfolding mem_box

  1054         apply auto

  1055         apply (rule_tac x=i in bexI)

  1056         using as(2)[THEN bspec[where x=i]] and as2 i

  1057         apply auto

  1058         done

  1059       ultimately have False using as by auto

  1060     }

  1061     then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto

  1062     moreover

  1063     {

  1064       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"

  1065       assume as2: "b\<bullet>i < d\<bullet>i"

  1066       {

  1067         fix j :: 'a

  1068         assume "j\<in>Basis"

  1069         then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  1070           apply (cases "j = i")

  1071           using as(2)[THEN bspec[where x=j]]

  1072           apply (auto simp add: as2)

  1073           done

  1074       }

  1075       then have "?x\<in>box c d"

  1076         unfolding mem_box by auto

  1077       moreover

  1078       have "?x\<notin>cbox a b"

  1079         unfolding mem_box

  1080         apply auto

  1081         apply (rule_tac x=i in bexI)

  1082         using as(2)[THEN bspec[where x=i]] and as2 using i

  1083         apply auto

  1084         done

  1085       ultimately have False using as by auto

  1086     }

  1087     then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto

  1088     ultimately

  1089     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  1090   } note part1 = this

  1091   show ?th3

  1092     unfolding subset_eq and Ball_def and mem_box

  1093     apply (rule, rule, rule, rule)

  1094     apply (rule part1)

  1095     unfolding subset_eq and Ball_def and mem_box

  1096     prefer 4

  1097     apply auto

  1098     apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+

  1099     done

  1100   {

  1101     assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  1102     fix i :: 'a

  1103     assume i:"i\<in>Basis"

  1104     from as(1) have "box c d \<subseteq> cbox a b"

  1105       using box_subset_cbox[of a b] by auto

  1106     then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"

  1107       using part1 and as(2) using i by auto

  1108   } note * = this

  1109   show ?th4

  1110     unfolding subset_eq and Ball_def and mem_box

  1111     apply (rule, rule, rule, rule)

  1112     apply (rule *)

  1113     unfolding subset_eq and Ball_def and mem_box

  1114     prefer 4

  1115     apply auto

  1116     apply (erule_tac x=xa in allE, simp)+

  1117     done

  1118 qed

  1119

  1120 lemma inter_interval:

  1121   fixes a :: "'a::euclidean_space"

  1122   shows "cbox a b \<inter> cbox c d =

  1123     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)"

  1124   unfolding set_eq_iff and Int_iff and mem_box

  1125   by auto

  1126

  1127 lemma disjoint_interval:

  1128   fixes a::"'a::euclidean_space"

  1129   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)

  1130     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)

  1131     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)

  1132     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)

  1133 proof -

  1134   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"

  1135   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  1136       (\<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)"

  1137     by blast

  1138   note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)

  1139   show ?th1 unfolding * by (intro **) auto

  1140   show ?th2 unfolding * by (intro **) auto

  1141   show ?th3 unfolding * by (intro **) auto

  1142   show ?th4 unfolding * by (intro **) auto

  1143 qed

  1144

  1145 text {* Intervals in general, including infinite and mixtures of open and closed. *}

  1146

  1147 definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>

  1148   (\<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)"

  1149

  1150 lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)

  1151   and is_interval_box: "is_interval (box a b)" (is ?th2)

  1152   unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff

  1153   by (meson order_trans le_less_trans less_le_trans less_trans)+

  1154

  1155 lemma is_interval_empty:

  1156  "is_interval {}"

  1157   unfolding is_interval_def

  1158   by simp

  1159

  1160 lemma is_interval_univ:

  1161  "is_interval UNIV"

  1162   unfolding is_interval_def

  1163   by simp

  1164

  1165 lemma mem_is_intervalI:

  1166   assumes "is_interval s"

  1167   assumes "a \<in> s" "b \<in> s"

  1168   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"

  1169   shows "x \<in> s"

  1170   by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

  1171

  1172 lemma interval_subst:

  1173   fixes S::"'a::euclidean_space set"

  1174   assumes "is_interval S"

  1175   assumes "x \<in> S" "y j \<in> S"

  1176   assumes "j \<in> Basis"

  1177   shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"

  1178   by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

  1179

  1180 lemma mem_box_componentwiseI:

  1181   fixes S::"'a::euclidean_space set"

  1182   assumes "is_interval S"

  1183   assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i)  S)"

  1184   shows "x \<in> S"

  1185 proof -

  1186   from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"

  1187     by auto

  1188   with finite_Basis obtain s and bs::"'a list" where

  1189     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

  1190     bs: "set bs = Basis" "distinct bs"

  1191     by (metis finite_distinct_list)

  1192   from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast

  1193   def y \<equiv> "rec_list

  1194     (s j)

  1195     (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"

  1196   have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"

  1197     using bs by (auto simp add: s(1)[symmetric] euclidean_representation)

  1198   also have [symmetric]: "y bs = \<dots>"

  1199     using bs(2) bs(1)[THEN equalityD1]

  1200     by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])

  1201   also have "y bs \<in> S"

  1202     using bs(1)[THEN equalityD1]

  1203     apply (induct bs)

  1204     apply (auto simp: y_def j)

  1205     apply (rule interval_subst[OF assms(1)])

  1206     apply (auto simp: s)

  1207     done

  1208   finally show ?thesis .

  1209 qed

  1210

  1211

  1212 subsection{* Connectedness *}

  1213

  1214 lemma connected_local:

  1215  "connected S \<longleftrightarrow>

  1216   \<not> (\<exists>e1 e2.

  1217       openin (subtopology euclidean S) e1 \<and>

  1218       openin (subtopology euclidean S) e2 \<and>

  1219       S \<subseteq> e1 \<union> e2 \<and>

  1220       e1 \<inter> e2 = {} \<and>

  1221       e1 \<noteq> {} \<and>

  1222       e2 \<noteq> {})"

  1223   unfolding connected_def openin_open

  1224   by blast

  1225

  1226 lemma exists_diff:

  1227   fixes P :: "'a set \<Rightarrow> bool"

  1228   shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1229 proof -

  1230   {

  1231     assume "?lhs"

  1232     then have ?rhs by blast

  1233   }

  1234   moreover

  1235   {

  1236     fix S

  1237     assume H: "P S"

  1238     have "S = - (- S)" by auto

  1239     with H have "P (- (- S))" by metis

  1240   }

  1241   ultimately show ?thesis by metis

  1242 qed

  1243

  1244 lemma connected_clopen: "connected S \<longleftrightarrow>

  1245   (\<forall>T. openin (subtopology euclidean S) T \<and>

  1246      closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")

  1247 proof -

  1248   have "\<not> connected S \<longleftrightarrow>

  1249     (\<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> {})"

  1250     unfolding connected_def openin_open closedin_closed

  1251     by (metis double_complement)

  1252   then have th0: "connected S \<longleftrightarrow>

  1253     \<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> {})"

  1254     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")

  1255     apply (simp add: closed_def)

  1256     apply metis

  1257     done

  1258   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'))"

  1259     (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")

  1260     unfolding connected_def openin_open closedin_closed by auto

  1261   {

  1262     fix e2

  1263     {

  1264       fix e1

  1265       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)"

  1266         by auto

  1267     }

  1268     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1269       by metis

  1270   }

  1271   then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"

  1272     by blast

  1273   then show ?thesis

  1274     unfolding th0 th1 by simp

  1275 qed

  1276

  1277

  1278 subsection{* Limit points *}

  1279

  1280 definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)

  1281   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))"

  1282

  1283 lemma islimptI:

  1284   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1285   shows "x islimpt S"

  1286   using assms unfolding islimpt_def by auto

  1287

  1288 lemma islimptE:

  1289   assumes "x islimpt S" and "x \<in> T" and "open T"

  1290   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"

  1291   using assms unfolding islimpt_def by auto

  1292

  1293 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

  1294   unfolding islimpt_def eventually_at_topological by auto

  1295

  1296 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"

  1297   unfolding islimpt_def by fast

  1298

  1299 lemma islimpt_approachable:

  1300   fixes x :: "'a::metric_space"

  1301   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  1302   unfolding islimpt_iff_eventually eventually_at by fast

  1303

  1304 lemma islimpt_approachable_le:

  1305   fixes x :: "'a::metric_space"

  1306   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"

  1307   unfolding islimpt_approachable

  1308   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

  1309     THEN arg_cong [where f=Not]]

  1310   by (simp add: Bex_def conj_commute conj_left_commute)

  1311

  1312 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  1313   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

  1314

  1315 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

  1316   unfolding islimpt_def by blast

  1317

  1318 text {* A perfect space has no isolated points. *}

  1319

  1320 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  1321   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

  1322

  1323 lemma perfect_choose_dist:

  1324   fixes x :: "'a::{perfect_space, metric_space}"

  1325   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

  1326   using islimpt_UNIV [of x]

  1327   by (simp add: islimpt_approachable)

  1328

  1329 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  1330   unfolding closed_def

  1331   apply (subst open_subopen)

  1332   apply (simp add: islimpt_def subset_eq)

  1333   apply (metis ComplE ComplI)

  1334   done

  1335

  1336 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  1337   unfolding islimpt_def by auto

  1338

  1339 lemma finite_set_avoid:

  1340   fixes a :: "'a::metric_space"

  1341   assumes fS: "finite S"

  1342   shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"

  1343 proof (induct rule: finite_induct[OF fS])

  1344   case 1

  1345   then show ?case by (auto intro: zero_less_one)

  1346 next

  1347   case (2 x F)

  1348   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"

  1349     by blast

  1350   show ?case

  1351   proof (cases "x = a")

  1352     case True

  1353     then show ?thesis using d by auto

  1354   next

  1355     case False

  1356     let ?d = "min d (dist a x)"

  1357     have dp: "?d > 0"

  1358       using False d(1) using dist_nz by auto

  1359     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"

  1360       by auto

  1361     with dp False show ?thesis

  1362       by (auto intro!: exI[where x="?d"])

  1363   qed

  1364 qed

  1365

  1366 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  1367   by (simp add: islimpt_iff_eventually eventually_conj_iff)

  1368

  1369 lemma discrete_imp_closed:

  1370   fixes S :: "'a::metric_space set"

  1371   assumes e: "0 < e"

  1372     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  1373   shows "closed S"

  1374 proof -

  1375   {

  1376     fix x

  1377     assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

  1378     from e have e2: "e/2 > 0" by arith

  1379     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"

  1380       by blast

  1381     let ?m = "min (e/2) (dist x y) "

  1382     from e2 y(2) have mp: "?m > 0"

  1383       by (simp add: dist_nz[symmetric])

  1384     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"

  1385       by blast

  1386     have th: "dist z y < e" using z y

  1387       by (intro dist_triangle_lt [where z=x], simp)

  1388     from d[rule_format, OF y(1) z(1) th] y z

  1389     have False by (auto simp add: dist_commute)}

  1390   then show ?thesis

  1391     by (metis islimpt_approachable closed_limpt [where 'a='a])

  1392 qed

  1393

  1394

  1395 subsection {* Interior of a Set *}

  1396

  1397 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

  1398

  1399 lemma interiorI [intro?]:

  1400   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  1401   shows "x \<in> interior S"

  1402   using assms unfolding interior_def by fast

  1403

  1404 lemma interiorE [elim?]:

  1405   assumes "x \<in> interior S"

  1406   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  1407   using assms unfolding interior_def by fast

  1408

  1409 lemma open_interior [simp, intro]: "open (interior S)"

  1410   by (simp add: interior_def open_Union)

  1411

  1412 lemma interior_subset: "interior S \<subseteq> S"

  1413   by (auto simp add: interior_def)

  1414

  1415 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  1416   by (auto simp add: interior_def)

  1417

  1418 lemma interior_open: "open S \<Longrightarrow> interior S = S"

  1419   by (intro equalityI interior_subset interior_maximal subset_refl)

  1420

  1421 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  1422   by (metis open_interior interior_open)

  1423

  1424 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  1425   by (metis interior_maximal interior_subset subset_trans)

  1426

  1427 lemma interior_empty [simp]: "interior {} = {}"

  1428   using open_empty by (rule interior_open)

  1429

  1430 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  1431   using open_UNIV by (rule interior_open)

  1432

  1433 lemma interior_interior [simp]: "interior (interior S) = interior S"

  1434   using open_interior by (rule interior_open)

  1435

  1436 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  1437   by (auto simp add: interior_def)

  1438

  1439 lemma interior_unique:

  1440   assumes "T \<subseteq> S" and "open T"

  1441   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

  1442   shows "interior S = T"

  1443   by (intro equalityI assms interior_subset open_interior interior_maximal)

  1444

  1445 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

  1446   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1447     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1448

  1449 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1450   using open_contains_ball_eq [where S="interior S"]

  1451   by (simp add: open_subset_interior)

  1452

  1453 lemma interior_limit_point [intro]:

  1454   fixes x :: "'a::perfect_space"

  1455   assumes x: "x \<in> interior S"

  1456   shows "x islimpt S"

  1457   using x islimpt_UNIV [of x]

  1458   unfolding interior_def islimpt_def

  1459   apply (clarsimp, rename_tac T T')

  1460   apply (drule_tac x="T \<inter> T'" in spec)

  1461   apply (auto simp add: open_Int)

  1462   done

  1463

  1464 lemma interior_closed_Un_empty_interior:

  1465   assumes cS: "closed S"

  1466     and iT: "interior T = {}"

  1467   shows "interior (S \<union> T) = interior S"

  1468 proof

  1469   show "interior S \<subseteq> interior (S \<union> T)"

  1470     by (rule interior_mono) (rule Un_upper1)

  1471   show "interior (S \<union> T) \<subseteq> interior S"

  1472   proof

  1473     fix x

  1474     assume "x \<in> interior (S \<union> T)"

  1475     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1476     show "x \<in> interior S"

  1477     proof (rule ccontr)

  1478       assume "x \<notin> interior S"

  1479       with x \<in> R open R obtain y where "y \<in> R - S"

  1480         unfolding interior_def by fast

  1481       from open R closed S have "open (R - S)"

  1482         by (rule open_Diff)

  1483       from R \<subseteq> S \<union> T have "R - S \<subseteq> T"

  1484         by fast

  1485       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {} show False

  1486         unfolding interior_def by fast

  1487     qed

  1488   qed

  1489 qed

  1490

  1491 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1492 proof (rule interior_unique)

  1493   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1494     by (intro Sigma_mono interior_subset)

  1495   show "open (interior A \<times> interior B)"

  1496     by (intro open_Times open_interior)

  1497   fix T

  1498   assume "T \<subseteq> A \<times> B" and "open T"

  1499   then show "T \<subseteq> interior A \<times> interior B"

  1500   proof safe

  1501     fix x y

  1502     assume "(x, y) \<in> T"

  1503     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1504       using open T unfolding open_prod_def by fast

  1505     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1506       using T \<subseteq> A \<times> B by auto

  1507     then show "x \<in> interior A" and "y \<in> interior B"

  1508       by (auto intro: interiorI)

  1509   qed

  1510 qed

  1511

  1512

  1513 subsection {* Closure of a Set *}

  1514

  1515 definition "closure S = S \<union> {x | x. x islimpt S}"

  1516

  1517 lemma interior_closure: "interior S = - (closure (- S))"

  1518   unfolding interior_def closure_def islimpt_def by auto

  1519

  1520 lemma closure_interior: "closure S = - interior (- S)"

  1521   unfolding interior_closure by simp

  1522

  1523 lemma closed_closure[simp, intro]: "closed (closure S)"

  1524   unfolding closure_interior by (simp add: closed_Compl)

  1525

  1526 lemma closure_subset: "S \<subseteq> closure S"

  1527   unfolding closure_def by simp

  1528

  1529 lemma closure_hull: "closure S = closed hull S"

  1530   unfolding hull_def closure_interior interior_def by auto

  1531

  1532 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1533   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1534

  1535 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1536   unfolding closure_eq .

  1537

  1538 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1539   unfolding closure_hull by (rule hull_hull)

  1540

  1541 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1542   unfolding closure_hull by (rule hull_mono)

  1543

  1544 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1545   unfolding closure_hull by (rule hull_minimal)

  1546

  1547 lemma closure_unique:

  1548   assumes "S \<subseteq> T"

  1549     and "closed T"

  1550     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1551   shows "closure S = T"

  1552   using assms unfolding closure_hull by (rule hull_unique)

  1553

  1554 lemma closure_empty [simp]: "closure {} = {}"

  1555   using closed_empty by (rule closure_closed)

  1556

  1557 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1558   using closed_UNIV by (rule closure_closed)

  1559

  1560 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1561   unfolding closure_interior by simp

  1562

  1563 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1564   using closure_empty closure_subset[of S]

  1565   by blast

  1566

  1567 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1568   using closure_eq[of S] closure_subset[of S]

  1569   by simp

  1570

  1571 lemma open_inter_closure_eq_empty:

  1572   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1573   using open_subset_interior[of S "- T"]

  1574   using interior_subset[of "- T"]

  1575   unfolding closure_interior

  1576   by auto

  1577

  1578 lemma open_inter_closure_subset:

  1579   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1580 proof

  1581   fix x

  1582   assume as: "open S" "x \<in> S \<inter> closure T"

  1583   {

  1584     assume *: "x islimpt T"

  1585     have "x islimpt (S \<inter> T)"

  1586     proof (rule islimptI)

  1587       fix A

  1588       assume "x \<in> A" "open A"

  1589       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1590         by (simp_all add: open_Int)

  1591       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1592         by (rule islimptE)

  1593       then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1594         by simp_all

  1595       then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1596     qed

  1597   }

  1598   then show "x \<in> closure (S \<inter> T)" using as

  1599     unfolding closure_def

  1600     by blast

  1601 qed

  1602

  1603 lemma closure_complement: "closure (- S) = - interior S"

  1604   unfolding closure_interior by simp

  1605

  1606 lemma interior_complement: "interior (- S) = - closure S"

  1607   unfolding closure_interior by simp

  1608

  1609 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1610 proof (rule closure_unique)

  1611   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1612     by (intro Sigma_mono closure_subset)

  1613   show "closed (closure A \<times> closure B)"

  1614     by (intro closed_Times closed_closure)

  1615   fix T

  1616   assume "A \<times> B \<subseteq> T" and "closed T"

  1617   then show "closure A \<times> closure B \<subseteq> T"

  1618     apply (simp add: closed_def open_prod_def, clarify)

  1619     apply (rule ccontr)

  1620     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1621     apply (simp add: closure_interior interior_def)

  1622     apply (drule_tac x=C in spec)

  1623     apply (drule_tac x=D in spec)

  1624     apply auto

  1625     done

  1626 qed

  1627

  1628 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1629   unfolding closure_def using islimpt_punctured by blast

  1630

  1631

  1632 subsection {* Frontier (aka boundary) *}

  1633

  1634 definition "frontier S = closure S - interior S"

  1635

  1636 lemma frontier_closed: "closed (frontier S)"

  1637   by (simp add: frontier_def closed_Diff)

  1638

  1639 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1640   by (auto simp add: frontier_def interior_closure)

  1641

  1642 lemma frontier_straddle:

  1643   fixes a :: "'a::metric_space"

  1644   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))"

  1645   unfolding frontier_def closure_interior

  1646   by (auto simp add: mem_interior subset_eq ball_def)

  1647

  1648 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1649   by (metis frontier_def closure_closed Diff_subset)

  1650

  1651 lemma frontier_empty[simp]: "frontier {} = {}"

  1652   by (simp add: frontier_def)

  1653

  1654 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1655 proof-

  1656   {

  1657     assume "frontier S \<subseteq> S"

  1658     then have "closure S \<subseteq> S"

  1659       using interior_subset unfolding frontier_def by auto

  1660     then have "closed S"

  1661       using closure_subset_eq by auto

  1662   }

  1663   then show ?thesis using frontier_subset_closed[of S] ..

  1664 qed

  1665

  1666 lemma frontier_complement: "frontier(- S) = frontier S"

  1667   by (auto simp add: frontier_def closure_complement interior_complement)

  1668

  1669 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1670   using frontier_complement frontier_subset_eq[of "- S"]

  1671   unfolding open_closed by auto

  1672

  1673 subsection {* Filters and the eventually true'' quantifier *}

  1674

  1675 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1676     (infixr "indirection" 70)

  1677   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1678

  1679 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1680

  1681 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1682 proof

  1683   assume "trivial_limit (at a within S)"

  1684   then show "\<not> a islimpt S"

  1685     unfolding trivial_limit_def

  1686     unfolding eventually_at_topological

  1687     unfolding islimpt_def

  1688     apply (clarsimp simp add: set_eq_iff)

  1689     apply (rename_tac T, rule_tac x=T in exI)

  1690     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1691     done

  1692 next

  1693   assume "\<not> a islimpt S"

  1694   then show "trivial_limit (at a within S)"

  1695     unfolding trivial_limit_def eventually_at_topological islimpt_def

  1696     by metis

  1697 qed

  1698

  1699 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1700   using trivial_limit_within [of a UNIV] by simp

  1701

  1702 lemma trivial_limit_at:

  1703   fixes a :: "'a::perfect_space"

  1704   shows "\<not> trivial_limit (at a)"

  1705   by (rule at_neq_bot)

  1706

  1707 lemma trivial_limit_at_infinity:

  1708   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1709   unfolding trivial_limit_def eventually_at_infinity

  1710   apply clarsimp

  1711   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1712    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1713   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1714   apply (drule_tac x=UNIV in spec, simp)

  1715   done

  1716

  1717 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"

  1718   using islimpt_in_closure

  1719   by (metis trivial_limit_within)

  1720

  1721 text {* Some property holds "sufficiently close" to the limit point. *}

  1722

  1723 lemma eventually_at2:

  1724   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1725   unfolding eventually_at dist_nz by auto

  1726

  1727 lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"

  1728   unfolding trivial_limit_def

  1729   by (auto elim: eventually_rev_mp)

  1730

  1731 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1732   by simp

  1733

  1734 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1735   by (simp add: filter_eq_iff)

  1736

  1737 text{* Combining theorems for "eventually" *}

  1738

  1739 lemma eventually_rev_mono:

  1740   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1741   using eventually_mono [of P Q] by fast

  1742

  1743 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"

  1744   by (simp add: eventually_False)

  1745

  1746

  1747 subsection {* Limits *}

  1748

  1749 lemma Lim:

  1750   "(f ---> l) net \<longleftrightarrow>

  1751         trivial_limit net \<or>

  1752         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1753   unfolding tendsto_iff trivial_limit_eq by auto

  1754

  1755 text{* Show that they yield usual definitions in the various cases. *}

  1756

  1757 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1758     (\<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)"

  1759   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1760

  1761 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1762     (\<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)"

  1763   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1764

  1765 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1766     (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"

  1767   by (auto simp add: tendsto_iff eventually_at2)

  1768

  1769 lemma Lim_at_infinity:

  1770   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"

  1771   by (auto simp add: tendsto_iff eventually_at_infinity)

  1772

  1773 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1774   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1775

  1776 text{* The expected monotonicity property. *}

  1777

  1778 lemma Lim_Un:

  1779   assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1780   shows "(f ---> l) (at x within (S \<union> T))"

  1781   using assms unfolding at_within_union by (rule filterlim_sup)

  1782

  1783 lemma Lim_Un_univ:

  1784   "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>

  1785     S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"

  1786   by (metis Lim_Un)

  1787

  1788 text{* Interrelations between restricted and unrestricted limits. *}

  1789

  1790 lemma Lim_at_within: (* FIXME: rename *)

  1791   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1792   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1793

  1794 lemma eventually_within_interior:

  1795   assumes "x \<in> interior S"

  1796   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"

  1797   (is "?lhs = ?rhs")

  1798 proof

  1799   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1800   {

  1801     assume "?lhs"

  1802     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"

  1803       unfolding eventually_at_topological

  1804       by auto

  1805     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"

  1806       by auto

  1807     then show "?rhs"

  1808       unfolding eventually_at_topological by auto

  1809   next

  1810     assume "?rhs"

  1811     then show "?lhs"

  1812       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1813   }

  1814 qed

  1815

  1816 lemma at_within_interior:

  1817   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1818   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1819

  1820 lemma Lim_within_LIMSEQ:

  1821   fixes a :: "'a::first_countable_topology"

  1822   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1823   shows "(X ---> L) (at a within T)"

  1824   using assms unfolding tendsto_def [where l=L]

  1825   by (simp add: sequentially_imp_eventually_within)

  1826

  1827 lemma Lim_right_bound:

  1828   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1829     'b::{linorder_topology, conditionally_complete_linorder}"

  1830   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"

  1831     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1832   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1833 proof (cases "{x<..} \<inter> I = {}")

  1834   case True

  1835   then show ?thesis by simp

  1836 next

  1837   case False

  1838   show ?thesis

  1839   proof (rule order_tendstoI)

  1840     fix a

  1841     assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1842     {

  1843       fix y

  1844       assume "y \<in> {x<..} \<inter> I"

  1845       with False bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1846         by (auto intro!: cInf_lower bdd_belowI2 simp del: Inf_image_eq)

  1847       with a have "a < f y"

  1848         by (blast intro: less_le_trans)

  1849     }

  1850     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1851       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1852   next

  1853     fix a

  1854     assume "Inf (f  ({x<..} \<inter> I)) < a"

  1855     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"

  1856       by auto

  1857     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1858       unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

  1859     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1860       unfolding eventually_at_filter by eventually_elim simp

  1861   qed

  1862 qed

  1863

  1864 text{* Another limit point characterization. *}

  1865

  1866 lemma islimpt_sequential:

  1867   fixes x :: "'a::first_countable_topology"

  1868   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1869     (is "?lhs = ?rhs")

  1870 proof

  1871   assume ?lhs

  1872   from countable_basis_at_decseq[of x] obtain A where A:

  1873       "\<And>i. open (A i)"

  1874       "\<And>i. x \<in> A i"

  1875       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  1876     by blast

  1877   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1878   {

  1879     fix n

  1880     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1881       unfolding islimpt_def using A(1,2)[of n] by auto

  1882     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1883       unfolding f_def by (rule someI_ex)

  1884     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto

  1885   }

  1886   then have "\<forall>n. f n \<in> S - {x}" by auto

  1887   moreover have "(\<lambda>n. f n) ----> x"

  1888   proof (rule topological_tendstoI)

  1889     fix S

  1890     assume "open S" "x \<in> S"

  1891     from A(3)[OF this] \<And>n. f n \<in> A n

  1892     show "eventually (\<lambda>x. f x \<in> S) sequentially"

  1893       by (auto elim!: eventually_elim1)

  1894   qed

  1895   ultimately show ?rhs by fast

  1896 next

  1897   assume ?rhs

  1898   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"

  1899     by auto

  1900   show ?lhs

  1901     unfolding islimpt_def

  1902   proof safe

  1903     fix T

  1904     assume "open T" "x \<in> T"

  1905     from lim[THEN topological_tendstoD, OF this] f

  1906     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1907       unfolding eventually_sequentially by auto

  1908   qed

  1909 qed

  1910

  1911 lemma Lim_null:

  1912   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1913   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1914   by (simp add: Lim dist_norm)

  1915

  1916 lemma Lim_null_comparison:

  1917   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1918   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1919   shows "(f ---> 0) net"

  1920   using assms(2)

  1921 proof (rule metric_tendsto_imp_tendsto)

  1922   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1923     using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

  1924 qed

  1925

  1926 lemma Lim_transform_bound:

  1927   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1928     and g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1929   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"

  1930     and "(g ---> 0) net"

  1931   shows "(f ---> 0) net"

  1932   using assms(1) tendsto_norm_zero [OF assms(2)]

  1933   by (rule Lim_null_comparison)

  1934

  1935 text{* Deducing things about the limit from the elements. *}

  1936

  1937 lemma Lim_in_closed_set:

  1938   assumes "closed S"

  1939     and "eventually (\<lambda>x. f(x) \<in> S) net"

  1940     and "\<not> trivial_limit net" "(f ---> l) net"

  1941   shows "l \<in> S"

  1942 proof (rule ccontr)

  1943   assume "l \<notin> S"

  1944   with closed S have "open (- S)" "l \<in> - S"

  1945     by (simp_all add: open_Compl)

  1946   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1947     by (rule topological_tendstoD)

  1948   with assms(2) have "eventually (\<lambda>x. False) net"

  1949     by (rule eventually_elim2) simp

  1950   with assms(3) show "False"

  1951     by (simp add: eventually_False)

  1952 qed

  1953

  1954 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1955

  1956 lemma Lim_dist_ubound:

  1957   assumes "\<not>(trivial_limit net)"

  1958     and "(f ---> l) net"

  1959     and "eventually (\<lambda>x. dist a (f x) \<le> e) net"

  1960   shows "dist a l \<le> e"

  1961   using assms by (fast intro: tendsto_le tendsto_intros)

  1962

  1963 lemma Lim_norm_ubound:

  1964   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1965   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"

  1966   shows "norm(l) \<le> e"

  1967   using assms by (fast intro: tendsto_le tendsto_intros)

  1968

  1969 lemma Lim_norm_lbound:

  1970   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1971   assumes "\<not> trivial_limit net"

  1972     and "(f ---> l) net"

  1973     and "eventually (\<lambda>x. e \<le> norm (f x)) net"

  1974   shows "e \<le> norm l"

  1975   using assms by (fast intro: tendsto_le tendsto_intros)

  1976

  1977 text{* Limit under bilinear function *}

  1978

  1979 lemma Lim_bilinear:

  1980   assumes "(f ---> l) net"

  1981     and "(g ---> m) net"

  1982     and "bounded_bilinear h"

  1983   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1984   using bounded_bilinear h (f ---> l) net (g ---> m) net

  1985   by (rule bounded_bilinear.tendsto)

  1986

  1987 text{* These are special for limits out of the same vector space. *}

  1988

  1989 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1990   unfolding id_def by (rule tendsto_ident_at)

  1991

  1992 lemma Lim_at_id: "(id ---> a) (at a)"

  1993   unfolding id_def by (rule tendsto_ident_at)

  1994

  1995 lemma Lim_at_zero:

  1996   fixes a :: "'a::real_normed_vector"

  1997     and l :: "'b::topological_space"

  1998   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"

  1999   using LIM_offset_zero LIM_offset_zero_cancel ..

  2000

  2001 text{* It's also sometimes useful to extract the limit point from the filter. *}

  2002

  2003 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  2004   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  2005

  2006 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  2007   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  2008

  2009 lemma netlimit_at:

  2010   fixes a :: "'a::{perfect_space,t2_space}"

  2011   shows "netlimit (at a) = a"

  2012   using netlimit_within [of a UNIV] by simp

  2013

  2014 lemma lim_within_interior:

  2015   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  2016   by (metis at_within_interior)

  2017

  2018 lemma netlimit_within_interior:

  2019   fixes x :: "'a::{t2_space,perfect_space}"

  2020   assumes "x \<in> interior S"

  2021   shows "netlimit (at x within S) = x"

  2022   using assms by (metis at_within_interior netlimit_at)

  2023

  2024 text{* Transformation of limit. *}

  2025

  2026 lemma Lim_transform:

  2027   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  2028   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  2029   shows "(g ---> l) net"

  2030   using tendsto_diff [OF assms(2) assms(1)] by simp

  2031

  2032 lemma Lim_transform_eventually:

  2033   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  2034   apply (rule topological_tendstoI)

  2035   apply (drule (2) topological_tendstoD)

  2036   apply (erule (1) eventually_elim2, simp)

  2037   done

  2038

  2039 lemma Lim_transform_within:

  2040   assumes "0 < d"

  2041     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2042     and "(f ---> l) (at x within S)"

  2043   shows "(g ---> l) (at x within S)"

  2044 proof (rule Lim_transform_eventually)

  2045   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  2046     using assms(1,2) by (auto simp: dist_nz eventually_at)

  2047   show "(f ---> l) (at x within S)" by fact

  2048 qed

  2049

  2050 lemma Lim_transform_at:

  2051   assumes "0 < d"

  2052     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  2053     and "(f ---> l) (at x)"

  2054   shows "(g ---> l) (at x)"

  2055   using _ assms(3)

  2056 proof (rule Lim_transform_eventually)

  2057   show "eventually (\<lambda>x. f x = g x) (at x)"

  2058     unfolding eventually_at2

  2059     using assms(1,2) by auto

  2060 qed

  2061

  2062 text{* Common case assuming being away from some crucial point like 0. *}

  2063

  2064 lemma Lim_transform_away_within:

  2065   fixes a b :: "'a::t1_space"

  2066   assumes "a \<noteq> b"

  2067     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2068     and "(f ---> l) (at a within S)"

  2069   shows "(g ---> l) (at a within S)"

  2070 proof (rule Lim_transform_eventually)

  2071   show "(f ---> l) (at a within S)" by fact

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

  2073     unfolding eventually_at_topological

  2074     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  2075 qed

  2076

  2077 lemma Lim_transform_away_at:

  2078   fixes a b :: "'a::t1_space"

  2079   assumes ab: "a\<noteq>b"

  2080     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  2081     and fl: "(f ---> l) (at a)"

  2082   shows "(g ---> l) (at a)"

  2083   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  2084

  2085 text{* Alternatively, within an open set. *}

  2086

  2087 lemma Lim_transform_within_open:

  2088   assumes "open S" and "a \<in> S"

  2089     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  2090     and "(f ---> l) (at a)"

  2091   shows "(g ---> l) (at a)"

  2092 proof (rule Lim_transform_eventually)

  2093   show "eventually (\<lambda>x. f x = g x) (at a)"

  2094     unfolding eventually_at_topological

  2095     using assms(1,2,3) by auto

  2096   show "(f ---> l) (at a)" by fact

  2097 qed

  2098

  2099 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  2100

  2101 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  2102

  2103 lemma Lim_cong_within(*[cong add]*):

  2104   assumes "a = b"

  2105     and "x = y"

  2106     and "S = T"

  2107     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  2108   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  2109   unfolding tendsto_def eventually_at_topological

  2110   using assms by simp

  2111

  2112 lemma Lim_cong_at(*[cong add]*):

  2113   assumes "a = b" "x = y"

  2114     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  2115   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  2116   unfolding tendsto_def eventually_at_topological

  2117   using assms by simp

  2118

  2119 text{* Useful lemmas on closure and set of possible sequential limits.*}

  2120

  2121 lemma closure_sequential:

  2122   fixes l :: "'a::first_countable_topology"

  2123   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"

  2124   (is "?lhs = ?rhs")

  2125 proof

  2126   assume "?lhs"

  2127   moreover

  2128   {

  2129     assume "l \<in> S"

  2130     then have "?rhs" using tendsto_const[of l sequentially] by auto

  2131   }

  2132   moreover

  2133   {

  2134     assume "l islimpt S"

  2135     then have "?rhs" unfolding islimpt_sequential by auto

  2136   }

  2137   ultimately show "?rhs"

  2138     unfolding closure_def by auto

  2139 next

  2140   assume "?rhs"

  2141   then show "?lhs" unfolding closure_def islimpt_sequential by auto

  2142 qed

  2143

  2144 lemma closed_sequential_limits:

  2145   fixes S :: "'a::first_countable_topology set"

  2146   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  2147 by (metis closure_sequential closure_subset_eq subset_iff)

  2148

  2149 lemma closure_approachable:

  2150   fixes S :: "'a::metric_space set"

  2151   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  2152   apply (auto simp add: closure_def islimpt_approachable)

  2153   apply (metis dist_self)

  2154   done

  2155

  2156 lemma closed_approachable:

  2157   fixes S :: "'a::metric_space set"

  2158   shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  2159   by (metis closure_closed closure_approachable)

  2160

  2161 lemma closure_contains_Inf:

  2162   fixes S :: "real set"

  2163   assumes "S \<noteq> {}" "bdd_below S"

  2164   shows "Inf S \<in> closure S"

  2165 proof -

  2166   have *: "\<forall>x\<in>S. Inf S \<le> x"

  2167     using cInf_lower[of _ S] assms by metis

  2168   {

  2169     fix e :: real

  2170     assume "e > 0"

  2171     then have "Inf S < Inf S + e" by simp

  2172     with assms obtain x where "x \<in> S" "x < Inf S + e"

  2173       by (subst (asm) cInf_less_iff) auto

  2174     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  2175       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  2176   }

  2177   then show ?thesis unfolding closure_approachable by auto

  2178 qed

  2179

  2180 lemma closed_contains_Inf:

  2181   fixes S :: "real set"

  2182   shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"

  2183   by (metis closure_contains_Inf closure_closed assms)

  2184

  2185 lemma not_trivial_limit_within_ball:

  2186   "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  2187   (is "?lhs = ?rhs")

  2188 proof -

  2189   {

  2190     assume "?lhs"

  2191     {

  2192       fix e :: real

  2193       assume "e > 0"

  2194       then obtain y where "y \<in> S - {x}" and "dist y x < e"

  2195         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2196         by auto

  2197       then have "y \<in> S \<inter> ball x e - {x}"

  2198         unfolding ball_def by (simp add: dist_commute)

  2199       then have "S \<inter> ball x e - {x} \<noteq> {}" by blast

  2200     }

  2201     then have "?rhs" by auto

  2202   }

  2203   moreover

  2204   {

  2205     assume "?rhs"

  2206     {

  2207       fix e :: real

  2208       assume "e > 0"

  2209       then obtain y where "y \<in> S \<inter> ball x e - {x}"

  2210         using ?rhs by blast

  2211       then have "y \<in> S - {x}" and "dist y x < e"

  2212         unfolding ball_def by (simp_all add: dist_commute)

  2213       then have "\<exists>y \<in> S - {x}. dist y x < e"

  2214         by auto

  2215     }

  2216     then have "?lhs"

  2217       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  2218       by auto

  2219   }

  2220   ultimately show ?thesis by auto

  2221 qed

  2222

  2223

  2224 subsection {* Infimum Distance *}

  2225

  2226 definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"

  2227

  2228 lemma bdd_below_infdist[intro, simp]: "bdd_below (dist xA)"

  2229   by (auto intro!: zero_le_dist)

  2230

  2231 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"

  2232   by (simp add: infdist_def)

  2233

  2234 lemma infdist_nonneg: "0 \<le> infdist x A"

  2235   by (auto simp add: infdist_def intro: cINF_greatest)

  2236

  2237 lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"

  2238   by (auto intro: cINF_lower simp add: infdist_def)

  2239

  2240 lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"

  2241   by (auto intro!: cINF_lower2 simp add: infdist_def)

  2242

  2243 lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"

  2244   by (auto intro!: antisym infdist_nonneg infdist_le2)

  2245

  2246 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  2247 proof (cases "A = {}")

  2248   case True

  2249   then show ?thesis by (simp add: infdist_def)

  2250 next

  2251   case False

  2252   then obtain a where "a \<in> A" by auto

  2253   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  2254   proof (rule cInf_greatest)

  2255     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  2256       by simp

  2257     fix d

  2258     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  2259     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  2260       by auto

  2261     show "infdist x A \<le> d"

  2262       unfolding infdist_notempty[OF A \<noteq> {}]

  2263     proof (rule cINF_lower2)

  2264       show "a \<in> A" by fact

  2265       show "dist x a \<le> d"

  2266         unfolding d by (rule dist_triangle)

  2267     qed simp

  2268   qed

  2269   also have "\<dots> = dist x y + infdist y A"

  2270   proof (rule cInf_eq, safe)

  2271     fix a

  2272     assume "a \<in> A"

  2273     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  2274       by (auto intro: infdist_le)

  2275   next

  2276     fix i

  2277     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  2278     then have "i - dist x y \<le> infdist y A"

  2279       unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  2280       by (intro cINF_greatest) (auto simp: field_simps)

  2281     then show "i \<le> dist x y + infdist y A"

  2282       by simp

  2283   qed

  2284   finally show ?thesis by simp

  2285 qed

  2286

  2287 lemma in_closure_iff_infdist_zero:

  2288   assumes "A \<noteq> {}"

  2289   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2290 proof

  2291   assume "x \<in> closure A"

  2292   show "infdist x A = 0"

  2293   proof (rule ccontr)

  2294     assume "infdist x A \<noteq> 0"

  2295     with infdist_nonneg[of x A] have "infdist x A > 0"

  2296       by auto

  2297     then have "ball x (infdist x A) \<inter> closure A = {}"

  2298       apply auto

  2299       apply (metis x \<in> closure A closure_approachable dist_commute infdist_le not_less)

  2300       done

  2301     then have "x \<notin> closure A"

  2302       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  2303     then show False using x \<in> closure A by simp

  2304   qed

  2305 next

  2306   assume x: "infdist x A = 0"

  2307   then obtain a where "a \<in> A"

  2308     by atomize_elim (metis all_not_in_conv assms)

  2309   show "x \<in> closure A"

  2310     unfolding closure_approachable

  2311     apply safe

  2312   proof (rule ccontr)

  2313     fix e :: real

  2314     assume "e > 0"

  2315     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  2316     then have "infdist x A \<ge> e" using a \<in> A

  2317       unfolding infdist_def

  2318       by (force simp: dist_commute intro: cINF_greatest)

  2319     with x e > 0 show False by auto

  2320   qed

  2321 qed

  2322

  2323 lemma in_closed_iff_infdist_zero:

  2324   assumes "closed A" "A \<noteq> {}"

  2325   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  2326 proof -

  2327   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2328     by (rule in_closure_iff_infdist_zero) fact

  2329   with assms show ?thesis by simp

  2330 qed

  2331

  2332 lemma tendsto_infdist [tendsto_intros]:

  2333   assumes f: "(f ---> l) F"

  2334   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  2335 proof (rule tendstoI)

  2336   fix e ::real

  2337   assume "e > 0"

  2338   from tendstoD[OF f this]

  2339   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  2340   proof (eventually_elim)

  2341     fix x

  2342     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  2343     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  2344       by (simp add: dist_commute dist_real_def)

  2345     also assume "dist (f x) l < e"

  2346     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  2347   qed

  2348 qed

  2349

  2350 text{* Some other lemmas about sequences. *}

  2351

  2352 lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)

  2353   assumes "eventually (\<lambda>i. P i) sequentially"

  2354   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  2355   using assms by (rule eventually_sequentially_seg [THEN iffD2])

  2356

  2357 lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)

  2358   "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  2359   apply (erule filterlim_compose)

  2360   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)

  2361   apply arith

  2362   done

  2363

  2364 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  2365   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

  2366

  2367 subsection {* More properties of closed balls *}

  2368

  2369 lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *)

  2370   assumes "closed s" and "continuous_on UNIV f"

  2371   shows "closed (vimage f s)"

  2372   using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]

  2373   by simp

  2374

  2375 lemma closed_cball: "closed (cball x e)"

  2376 proof -

  2377   have "closed (dist x - {..e})"

  2378     by (intro closed_vimage closed_atMost continuous_on_intros)

  2379   also have "dist x - {..e} = cball x e"

  2380     by auto

  2381   finally show ?thesis .

  2382 qed

  2383

  2384 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  2385 proof -

  2386   {

  2387     fix x and e::real

  2388     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  2389     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  2390   }

  2391   moreover

  2392   {

  2393     fix x and e::real

  2394     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  2395     then have "\<exists>d>0. ball x d \<subseteq> S"

  2396       unfolding subset_eq

  2397       apply(rule_tac x="e/2" in exI)

  2398       apply auto

  2399       done

  2400   }

  2401   ultimately show ?thesis

  2402     unfolding open_contains_ball by auto

  2403 qed

  2404

  2405 lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  2406   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  2407

  2408 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  2409   apply (simp add: interior_def, safe)

  2410   apply (force simp add: open_contains_cball)

  2411   apply (rule_tac x="ball x e" in exI)

  2412   apply (simp add: subset_trans [OF ball_subset_cball])

  2413   done

  2414

  2415 lemma islimpt_ball:

  2416   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  2417   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"

  2418   (is "?lhs = ?rhs")

  2419 proof

  2420   assume "?lhs"

  2421   {

  2422     assume "e \<le> 0"

  2423     then have *:"ball x e = {}"

  2424       using ball_eq_empty[of x e] by auto

  2425     have False using ?lhs

  2426       unfolding * using islimpt_EMPTY[of y] by auto

  2427   }

  2428   then have "e > 0" by (metis not_less)

  2429   moreover

  2430   have "y \<in> cball x e"

  2431     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  2432       ball_subset_cball[of x e] ?lhs

  2433     unfolding closed_limpt by auto

  2434   ultimately show "?rhs" by auto

  2435 next

  2436   assume "?rhs"

  2437   then have "e > 0" by auto

  2438   {

  2439     fix d :: real

  2440     assume "d > 0"

  2441     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2442     proof (cases "d \<le> dist x y")

  2443       case True

  2444       then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2445       proof (cases "x = y")

  2446         case True

  2447         then have False

  2448           using d \<le> dist x y d>0 by auto

  2449         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2450           by auto

  2451       next

  2452         case False

  2453         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

  2454           norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2455           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

  2456           by auto

  2457         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  2458           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

  2459           unfolding scaleR_minus_left scaleR_one

  2460           by (auto simp add: norm_minus_commute)

  2461         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  2462           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  2463           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm]

  2464           by auto

  2465         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs

  2466           by (auto simp add: dist_norm)

  2467         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0

  2468           by auto

  2469         moreover

  2470         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2471           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff

  2472           by (auto simp add: dist_commute)

  2473         moreover

  2474         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

  2475           unfolding dist_norm

  2476           apply simp

  2477           unfolding norm_minus_cancel

  2478           using d > 0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2479           unfolding dist_norm

  2480           apply auto

  2481           done

  2482         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2483           apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

  2484           apply auto

  2485           done

  2486       qed

  2487     next

  2488       case False

  2489       then have "d > dist x y" by auto

  2490       show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

  2491       proof (cases "x = y")

  2492         case True

  2493         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2494           using perfect_choose_dist[of "min e d" y]

  2495           using d > 0 e>0 by auto

  2496         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2497           unfolding x = y

  2498           using z \<noteq> y **

  2499           apply (rule_tac x=z in bexI)

  2500           apply (auto simp add: dist_commute)

  2501           done

  2502       next

  2503         case False

  2504         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2505           using d>0 d > dist x y ?rhs

  2506           apply (rule_tac x=x in bexI)

  2507           apply auto

  2508           done

  2509       qed

  2510     qed

  2511   }

  2512   then show "?lhs"

  2513     unfolding mem_cball islimpt_approachable mem_ball by auto

  2514 qed

  2515

  2516 lemma closure_ball_lemma:

  2517   fixes x y :: "'a::real_normed_vector"

  2518   assumes "x \<noteq> y"

  2519   shows "y islimpt ball x (dist x y)"

  2520 proof (rule islimptI)

  2521   fix T

  2522   assume "y \<in> T" "open T"

  2523   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2524     unfolding open_dist by fast

  2525   (* choose point between x and y, within distance r of y. *)

  2526   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2527   def z \<equiv> "y + scaleR k (x - y)"

  2528   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2529     unfolding z_def by (simp add: algebra_simps)

  2530   have "dist z y < r"

  2531     unfolding z_def k_def using 0 < r

  2532     by (simp add: dist_norm min_def)

  2533   then have "z \<in> T"

  2534     using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2535   have "dist x z < dist x y"

  2536     unfolding z_def2 dist_norm

  2537     apply (simp add: norm_minus_commute)

  2538     apply (simp only: dist_norm [symmetric])

  2539     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2540     apply (rule mult_strict_right_mono)

  2541     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2542     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2543     done

  2544   then have "z \<in> ball x (dist x y)"

  2545     by simp

  2546   have "z \<noteq> y"

  2547     unfolding z_def k_def using x \<noteq> y 0 < r

  2548     by (simp add: min_def)

  2549   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2550     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2551     by fast

  2552 qed

  2553

  2554 lemma closure_ball:

  2555   fixes x :: "'a::real_normed_vector"

  2556   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2557   apply (rule equalityI)

  2558   apply (rule closure_minimal)

  2559   apply (rule ball_subset_cball)

  2560   apply (rule closed_cball)

  2561   apply (rule subsetI, rename_tac y)

  2562   apply (simp add: le_less [where 'a=real])

  2563   apply (erule disjE)

  2564   apply (rule subsetD [OF closure_subset], simp)

  2565   apply (simp add: closure_def)

  2566   apply clarify

  2567   apply (rule closure_ball_lemma)

  2568   apply (simp add: zero_less_dist_iff)

  2569   done

  2570

  2571 (* In a trivial vector space, this fails for e = 0. *)

  2572 lemma interior_cball:

  2573   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2574   shows "interior (cball x e) = ball x e"

  2575 proof (cases "e \<ge> 0")

  2576   case False note cs = this

  2577   from cs have "ball x e = {}"

  2578     using ball_empty[of e x] by auto

  2579   moreover

  2580   {

  2581     fix y

  2582     assume "y \<in> cball x e"

  2583     then have False

  2584       unfolding mem_cball using dist_nz[of x y] cs by auto

  2585   }

  2586   then have "cball x e = {}" by auto

  2587   then have "interior (cball x e) = {}"

  2588     using interior_empty by auto

  2589   ultimately show ?thesis by blast

  2590 next

  2591   case True note cs = this

  2592   have "ball x e \<subseteq> cball x e"

  2593     using ball_subset_cball by auto

  2594   moreover

  2595   {

  2596     fix S y

  2597     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2598     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

  2599       unfolding open_dist by blast

  2600     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2601       using perfect_choose_dist [of d] by auto

  2602     have "xa \<in> S"

  2603       using d[THEN spec[where x = xa]]

  2604       using xa by (auto simp add: dist_commute)

  2605     then have xa_cball: "xa \<in> cball x e"

  2606       using as(1) by auto

  2607     then have "y \<in> ball x e"

  2608     proof (cases "x = y")

  2609       case True

  2610       then have "e > 0"

  2611         using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2612         by (auto simp add: dist_commute)

  2613       then show "y \<in> ball x e"

  2614         using x = y  by simp

  2615     next

  2616       case False

  2617       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

  2618         unfolding dist_norm

  2619         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2620       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2621         using d as(1)[unfolded subset_eq] by blast

  2622       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2623       then have **:"d / (2 * norm (y - x)) > 0"

  2624         unfolding zero_less_norm_iff[symmetric]

  2625         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2626       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

  2627         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2628         by (auto simp add: dist_norm algebra_simps)

  2629       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2630         by (auto simp add: algebra_simps)

  2631       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2632         using ** by auto

  2633       also have "\<dots> = (dist y x) + d/2"

  2634         using ** by (auto simp add: distrib_right dist_norm)

  2635       finally have "e \<ge> dist x y +d/2"

  2636         using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2637       then show "y \<in> ball x e"

  2638         unfolding mem_ball using d>0 by auto

  2639     qed

  2640   }

  2641   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

  2642     by auto

  2643   ultimately show ?thesis

  2644     using interior_unique[of "ball x e" "cball x e"]

  2645     using open_ball[of x e]

  2646     by auto

  2647 qed

  2648

  2649 lemma frontier_ball:

  2650   fixes a :: "'a::real_normed_vector"

  2651   shows "0 < e \<Longrightarrow> frontier(ball a e) = {x. dist a x = e}"

  2652   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2653   apply (simp add: set_eq_iff)

  2654   apply arith

  2655   done

  2656

  2657 lemma frontier_cball:

  2658   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2659   shows "frontier (cball a e) = {x. dist a x = e}"

  2660   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2661   apply (simp add: set_eq_iff)

  2662   apply arith

  2663   done

  2664

  2665 lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"

  2666   apply (simp add: set_eq_iff not_le)

  2667   apply (metis zero_le_dist dist_self order_less_le_trans)

  2668   done

  2669

  2670 lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"

  2671   by (simp add: cball_eq_empty)

  2672

  2673 lemma cball_eq_sing:

  2674   fixes x :: "'a::{metric_space,perfect_space}"

  2675   shows "cball x e = {x} \<longleftrightarrow> e = 0"

  2676 proof (rule linorder_cases)

  2677   assume e: "0 < e"

  2678   obtain a where "a \<noteq> x" "dist a x < e"

  2679     using perfect_choose_dist [OF e] by auto

  2680   then have "a \<noteq> x" "dist x a \<le> e"

  2681     by (auto simp add: dist_commute)

  2682   with e show ?thesis by (auto simp add: set_eq_iff)

  2683 qed auto

  2684

  2685 lemma cball_sing:

  2686   fixes x :: "'a::metric_space"

  2687   shows "e = 0 \<Longrightarrow> cball x e = {x}"

  2688   by (auto simp add: set_eq_iff)

  2689

  2690

  2691 subsection {* Boundedness *}

  2692

  2693   (* FIXME: This has to be unified with BSEQ!! *)

  2694 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2695   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2696

  2697 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2698   unfolding bounded_def subset_eq by auto

  2699

  2700 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2701   unfolding bounded_def

  2702   by auto (metis add_commute add_le_cancel_right dist_commute dist_triangle_le)

  2703

  2704 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2705   unfolding bounded_any_center [where a=0]

  2706   by (simp add: dist_norm)

  2707

  2708 lemma bounded_realI:

  2709   assumes "\<forall>x\<in>s. abs (x::real) \<le> B"

  2710   shows "bounded s"

  2711   unfolding bounded_def dist_real_def

  2712   by (metis abs_minus_commute assms diff_0_right)

  2713

  2714 lemma bounded_empty [simp]: "bounded {}"

  2715   by (simp add: bounded_def)

  2716

  2717 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"

  2718   by (metis bounded_def subset_eq)

  2719

  2720 lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"

  2721   by (metis bounded_subset interior_subset)

  2722

  2723 lemma bounded_closure[intro]:

  2724   assumes "bounded S"

  2725   shows "bounded (closure S)"

  2726 proof -

  2727   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2728     unfolding bounded_def by auto

  2729   {

  2730     fix y

  2731     assume "y \<in> closure S"

  2732     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2733       unfolding closure_sequential by auto

  2734     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2735     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2736       by (rule eventually_mono, simp add: f(1))

  2737     have "dist x y \<le> a"

  2738       apply (rule Lim_dist_ubound [of sequentially f])

  2739       apply (rule trivial_limit_sequentially)

  2740       apply (rule f(2))

  2741       apply fact

  2742       done

  2743   }

  2744   then show ?thesis

  2745     unfolding bounded_def by auto

  2746 qed

  2747

  2748 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2749   apply (simp add: bounded_def)

  2750   apply (rule_tac x=x in exI)

  2751   apply (rule_tac x=e in exI)

  2752   apply auto

  2753   done

  2754

  2755 lemma bounded_ball[simp,intro]: "bounded (ball x e)"

  2756   by (metis ball_subset_cball bounded_cball bounded_subset)

  2757

  2758 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2759   apply (auto simp add: bounded_def)

  2760   by (metis Un_iff add_le_cancel_left dist_triangle le_max_iff_disj max.order_iff)

  2761

  2762 lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"

  2763   by (induct rule: finite_induct[of F]) auto

  2764

  2765 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2766   by (induct set: finite) auto

  2767

  2768 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2769 proof -

  2770   have "\<forall>y\<in>{x}. dist x y \<le> 0"

  2771     by simp

  2772   then have "bounded {x}"

  2773     unfolding bounded_def by fast

  2774   then show ?thesis

  2775     by (metis insert_is_Un bounded_Un)

  2776 qed

  2777

  2778 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2779   by (induct set: finite) simp_all

  2780

  2781 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"

  2782   apply (simp add: bounded_iff)

  2783   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")

  2784   apply metis

  2785   apply arith

  2786   done

  2787

  2788 lemma Bseq_eq_bounded:

  2789   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  2790   shows "Bseq f \<longleftrightarrow> bounded (range f)"

  2791   unfolding Bseq_def bounded_pos by auto

  2792

  2793 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2794   by (metis Int_lower1 Int_lower2 bounded_subset)

  2795

  2796 lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"

  2797   by (metis Diff_subset bounded_subset)

  2798

  2799 lemma not_bounded_UNIV[simp, intro]:

  2800   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2801 proof (auto simp add: bounded_pos not_le)

  2802   obtain x :: 'a where "x \<noteq> 0"

  2803     using perfect_choose_dist [OF zero_less_one] by fast

  2804   fix b :: real

  2805   assume b: "b >0"

  2806   have b1: "b +1 \<ge> 0"

  2807     using b by simp

  2808   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2809     by (simp add: norm_sgn)

  2810   then show "\<exists>x::'a. b < norm x" ..

  2811 qed

  2812

  2813 lemma bounded_linear_image:

  2814   assumes "bounded S"

  2815     and "bounded_linear f"

  2816   shows "bounded (f  S)"

  2817 proof -

  2818   from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2819     unfolding bounded_pos by auto

  2820   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"

  2821     using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2822   {

  2823     fix x

  2824     assume "x \<in> S"

  2825     then have "norm x \<le> b"

  2826       using b by auto

  2827     then have "norm (f x) \<le> B * b"

  2828       using B(2)

  2829       apply (erule_tac x=x in allE)

  2830       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2831       done

  2832   }

  2833   then show ?thesis

  2834     unfolding bounded_pos

  2835     apply (rule_tac x="b*B" in exI)

  2836     using b B mult_pos_pos [of b B]

  2837     apply (auto simp add: mult_commute)

  2838     done

  2839 qed

  2840

  2841 lemma bounded_scaling:

  2842   fixes S :: "'a::real_normed_vector set"

  2843   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2844   apply (rule bounded_linear_image)

  2845   apply assumption

  2846   apply (rule bounded_linear_scaleR_right)

  2847   done

  2848

  2849 lemma bounded_translation:

  2850   fixes S :: "'a::real_normed_vector set"

  2851   assumes "bounded S"

  2852   shows "bounded ((\<lambda>x. a + x)  S)"

  2853 proof -

  2854   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2855     unfolding bounded_pos by auto

  2856   {

  2857     fix x

  2858     assume "x \<in> S"

  2859     then have "norm (a + x) \<le> b + norm a"

  2860       using norm_triangle_ineq[of a x] b by auto

  2861   }

  2862   then show ?thesis

  2863     unfolding bounded_pos

  2864     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2865     by (auto intro!: exI[of _ "b + norm a"])

  2866 qed

  2867

  2868

  2869 text{* Some theorems on sups and infs using the notion "bounded". *}

  2870

  2871 lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"

  2872   by (simp add: bounded_iff)

  2873

  2874 lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"

  2875   by (auto simp: bounded_def bdd_above_def dist_real_def)

  2876      (metis abs_le_D1 abs_minus_commute diff_le_eq)

  2877

  2878 lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"

  2879   by (auto simp: bounded_def bdd_below_def dist_real_def)

  2880      (metis abs_le_D1 add_commute diff_le_eq)

  2881

  2882 (* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)

  2883

  2884 lemma bounded_has_Sup:

  2885   fixes S :: "real set"

  2886   assumes "bounded S"

  2887     and "S \<noteq> {}"

  2888   shows "\<forall>x\<in>S. x \<le> Sup S"

  2889     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2890 proof

  2891   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2892     using assms by (metis cSup_least)

  2893 qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

  2894

  2895 lemma Sup_insert:

  2896   fixes S :: "real set"

  2897   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2898   by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)

  2899

  2900 lemma Sup_insert_finite:

  2901   fixes S :: "real set"

  2902   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2903   apply (rule Sup_insert)

  2904   apply (rule finite_imp_bounded)

  2905   apply simp

  2906   done

  2907

  2908 lemma bounded_has_Inf:

  2909   fixes S :: "real set"

  2910   assumes "bounded S"

  2911     and "S \<noteq> {}"

  2912   shows "\<forall>x\<in>S. x \<ge> Inf S"

  2913     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2914 proof

  2915   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2916     using assms by (metis cInf_greatest)

  2917 qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

  2918

  2919 lemma Inf_insert:

  2920   fixes S :: "real set"

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

  2922   by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)

  2923

  2924 lemma Inf_insert_finite:

  2925   fixes S :: "real set"

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

  2927   apply (rule Inf_insert)

  2928   apply (rule finite_imp_bounded)

  2929   apply simp

  2930   done

  2931

  2932 subsection {* Compactness *}

  2933

  2934 subsubsection {* Bolzano-Weierstrass property *}

  2935

  2936 lemma heine_borel_imp_bolzano_weierstrass:

  2937   assumes "compact s"

  2938     and "infinite t"

  2939     and "t \<subseteq> s"

  2940   shows "\<exists>x \<in> s. x islimpt t"

  2941 proof (rule ccontr)

  2942   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2943   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)"

  2944     unfolding islimpt_def

  2945     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]

  2946     by auto

  2947   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2948     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2949     using f by auto

  2950   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"

  2951     by auto

  2952   {

  2953     fix x y

  2954     assume "x \<in> t" "y \<in> t" "f x = f y"

  2955     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2956       using f[THEN bspec[where x=x]] and t \<subseteq> s by auto

  2957     then have "x = y"

  2958       using f x = f y and f[THEN bspec[where x=y]] and y \<in> t and t \<subseteq> s

  2959       by auto

  2960   }

  2961   then have "inj_on f t"

  2962     unfolding inj_on_def by simp

  2963   then have "infinite (f  t)"

  2964     using assms(2) using finite_imageD by auto

  2965   moreover

  2966   {

  2967     fix x

  2968     assume "x \<in> t" "f x \<notin> g"

  2969     from g(3) assms(3) x \<in> t obtain h where "h \<in> g" and "x \<in> h"

  2970       by auto

  2971     then obtain y where "y \<in> s" "h = f y"

  2972       using g'[THEN bspec[where x=h]] by auto

  2973     then have "y = x"

  2974       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y]

  2975       by auto

  2976     then have False

  2977       using f x \<notin> g h \<in> g unfolding h = f y

  2978       by auto

  2979   }

  2980   then have "f  t \<subseteq> g" by auto

  2981   ultimately show False

  2982     using g(2) using finite_subset by auto

  2983 qed

  2984

  2985 lemma acc_point_range_imp_convergent_subsequence:

  2986   fixes l :: "'a :: first_countable_topology"

  2987   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2988   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2989 proof -

  2990   from countable_basis_at_decseq[of l]

  2991   obtain A where A:

  2992       "\<And>i. open (A i)"

  2993       "\<And>i. l \<in> A i"

  2994       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  2995     by blast

  2996   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2997   {

  2998     fix n i

  2999     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  3000       using l A by auto

  3001     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  3002       unfolding ex_in_conv by (intro notI) simp

  3003     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  3004       by auto

  3005     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  3006       by (auto simp: not_le)

  3007     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  3008       unfolding s_def by (auto intro: someI2_ex)

  3009   }

  3010   note s = this

  3011   def r \<equiv> "rec_nat (s 0 0) s"

  3012   have "subseq r"

  3013     by (auto simp: r_def s subseq_Suc_iff)

  3014   moreover

  3015   have "(\<lambda>n. f (r n)) ----> l"

  3016   proof (rule topological_tendstoI)

  3017     fix S

  3018     assume "open S" "l \<in> S"

  3019     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3020       by auto

  3021     moreover

  3022     {

  3023       fix i

  3024       assume "Suc 0 \<le> i"

  3025       then have "f (r i) \<in> A i"

  3026         by (cases i) (simp_all add: r_def s)

  3027     }

  3028     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  3029       by (auto simp: eventually_sequentially)

  3030     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  3031       by eventually_elim auto

  3032   qed

  3033   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3034     by (auto simp: convergent_def comp_def)

  3035 qed

  3036

  3037 lemma sequence_infinite_lemma:

  3038   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  3039   assumes "\<forall>n. f n \<noteq> l"

  3040     and "(f ---> l) sequentially"

  3041   shows "infinite (range f)"

  3042 proof

  3043   assume "finite (range f)"

  3044   then have "closed (range f)"

  3045     by (rule finite_imp_closed)

  3046   then have "open (- range f)"

  3047     by (rule open_Compl)

  3048   from assms(1) have "l \<in> - range f"

  3049     by auto

  3050   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  3051     using open (- range f) l \<in> - range f

  3052     by (rule topological_tendstoD)

  3053   then show False

  3054     unfolding eventually_sequentially

  3055     by auto

  3056 qed

  3057

  3058 lemma closure_insert:

  3059   fixes x :: "'a::t1_space"

  3060   shows "closure (insert x s) = insert x (closure s)"

  3061   apply (rule closure_unique)

  3062   apply (rule insert_mono [OF closure_subset])

  3063   apply (rule closed_insert [OF closed_closure])

  3064   apply (simp add: closure_minimal)

  3065   done

  3066

  3067 lemma islimpt_insert:

  3068   fixes x :: "'a::t1_space"

  3069   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  3070 proof

  3071   assume *: "x islimpt (insert a s)"

  3072   show "x islimpt s"

  3073   proof (rule islimptI)

  3074     fix t

  3075     assume t: "x \<in> t" "open t"

  3076     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  3077     proof (cases "x = a")

  3078       case True

  3079       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  3080         using * t by (rule islimptE)

  3081       with x = a show ?thesis by auto

  3082     next

  3083       case False

  3084       with t have t': "x \<in> t - {a}" "open (t - {a})"

  3085         by (simp_all add: open_Diff)

  3086       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  3087         using * t' by (rule islimptE)

  3088       then show ?thesis by auto

  3089     qed

  3090   qed

  3091 next

  3092   assume "x islimpt s"

  3093   then show "x islimpt (insert a s)"

  3094     by (rule islimpt_subset) auto

  3095 qed

  3096

  3097 lemma islimpt_finite:

  3098   fixes x :: "'a::t1_space"

  3099   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  3100   by (induct set: finite) (simp_all add: islimpt_insert)

  3101

  3102 lemma islimpt_union_finite:

  3103   fixes x :: "'a::t1_space"

  3104   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  3105   by (simp add: islimpt_Un islimpt_finite)

  3106

  3107 lemma islimpt_eq_acc_point:

  3108   fixes l :: "'a :: t1_space"

  3109   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  3110 proof (safe intro!: islimptI)

  3111   fix U

  3112   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  3113   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  3114     by (auto intro: finite_imp_closed)

  3115   then show False

  3116     by (rule islimptE) auto

  3117 next

  3118   fix T

  3119   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  3120   then have "infinite (T \<inter> S - {l})"

  3121     by auto

  3122   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  3123     unfolding ex_in_conv by (intro notI) simp

  3124   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  3125     by auto

  3126 qed

  3127

  3128 lemma islimpt_range_imp_convergent_subsequence:

  3129   fixes l :: "'a :: {t1_space, first_countable_topology}"

  3130   assumes l: "l islimpt (range f)"

  3131   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3132   using l unfolding islimpt_eq_acc_point

  3133   by (rule acc_point_range_imp_convergent_subsequence)

  3134

  3135 lemma sequence_unique_limpt:

  3136   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  3137   assumes "(f ---> l) sequentially"

  3138     and "l' islimpt (range f)"

  3139   shows "l' = l"

  3140 proof (rule ccontr)

  3141   assume "l' \<noteq> l"

  3142   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  3143     using hausdorff [OF l' \<noteq> l] by auto

  3144   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  3145     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  3146   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  3147     unfolding eventually_sequentially by auto

  3148

  3149   have "UNIV = {..<N} \<union> {N..}"

  3150     by auto

  3151   then have "l' islimpt (f  ({..<N} \<union> {N..}))"

  3152     using assms(2) by simp

  3153   then have "l' islimpt (f  {..<N} \<union> f  {N..})"

  3154     by (simp add: image_Un)

  3155   then have "l' islimpt (f  {N..})"

  3156     by (simp add: islimpt_union_finite)

  3157   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  3158     using l' \<in> s open s by (rule islimptE)

  3159   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"

  3160     by auto

  3161   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t"

  3162     by simp

  3163   with s \<inter> t = {} show False

  3164     by simp

  3165 qed

  3166

  3167 lemma bolzano_weierstrass_imp_closed:

  3168   fixes s :: "'a::{first_countable_topology,t2_space} set"

  3169   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  3170   shows "closed s"

  3171 proof -

  3172   {

  3173     fix x l

  3174     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  3175     then have "l \<in> s"

  3176     proof (cases "\<forall>n. x n \<noteq> l")

  3177       case False

  3178       then show "l\<in>s" using as(1) by auto

  3179     next

  3180       case True note cas = this

  3181       with as(2) have "infinite (range x)"

  3182         using sequence_infinite_lemma[of x l] by auto

  3183       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  3184         using assms[THEN spec[where x="range x"]] as(1) by auto

  3185       then show "l\<in>s" using sequence_unique_limpt[of x l l']

  3186         using as cas by auto

  3187     qed

  3188   }

  3189   then show ?thesis

  3190     unfolding closed_sequential_limits by fast

  3191 qed

  3192

  3193 lemma compact_imp_bounded:

  3194   assumes "compact U"

  3195   shows "bounded U"

  3196 proof -

  3197   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  3198     using assms by auto

  3199   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  3200     by (rule compactE_image)

  3201   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  3202     by (simp add: bounded_UN)

  3203   then show "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  3204     by (rule bounded_subset)

  3205 qed

  3206

  3207 text{* In particular, some common special cases. *}

  3208

  3209 lemma compact_union [intro]:

  3210   assumes "compact s"

  3211     and "compact t"

  3212   shows " compact (s \<union> t)"

  3213 proof (rule compactI)

  3214   fix f

  3215   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  3216   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  3217     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3218   moreover

  3219   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  3220     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])

  3221   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  3222     by (auto intro!: exI[of _ "s' \<union> t'"])

  3223 qed

  3224

  3225 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3226   by (induct set: finite) auto

  3227

  3228 lemma compact_UN [intro]:

  3229   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3230   unfolding SUP_def by (rule compact_Union) auto

  3231

  3232 lemma closed_inter_compact [intro]:

  3233   assumes "closed s"

  3234     and "compact t"

  3235   shows "compact (s \<inter> t)"

  3236   using compact_inter_closed [of t s] assms

  3237   by (simp add: Int_commute)

  3238

  3239 lemma compact_inter [intro]:

  3240   fixes s t :: "'a :: t2_space set"

  3241   assumes "compact s"

  3242     and "compact t"

  3243   shows "compact (s \<inter> t)"

  3244   using assms by (intro compact_inter_closed compact_imp_closed)

  3245

  3246 lemma compact_sing [simp]: "compact {a}"

  3247   unfolding compact_eq_heine_borel by auto

  3248

  3249 lemma compact_insert [simp]:

  3250   assumes "compact s"

  3251   shows "compact (insert x s)"

  3252 proof -

  3253   have "compact ({x} \<union> s)"

  3254     using compact_sing assms by (rule compact_union)

  3255   then show ?thesis by simp

  3256 qed

  3257

  3258 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  3259   by (induct set: finite) simp_all

  3260

  3261 lemma open_delete:

  3262   fixes s :: "'a::t1_space set"

  3263   shows "open s \<Longrightarrow> open (s - {x})"

  3264   by (simp add: open_Diff)

  3265

  3266 text{*Compactness expressed with filters*}

  3267

  3268 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3269

  3270 lemma eventually_filter_from_subbase:

  3271   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3272     (is "_ \<longleftrightarrow> ?R P")

  3273   unfolding filter_from_subbase_def

  3274 proof (rule eventually_Abs_filter is_filter.intro)+

  3275   show "?R (\<lambda>x. True)"

  3276     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  3277 next

  3278   fix P Q

  3279   assume "?R P" then guess X ..

  3280   moreover

  3281   assume "?R Q" then guess Y ..

  3282   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  3283     by (intro exI[of _ "X \<union> Y"]) auto

  3284 next

  3285   fix P Q

  3286   assume "?R P" then guess X ..

  3287   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  3288   ultimately show "?R Q"

  3289     by (intro exI[of _ X]) auto

  3290 qed

  3291

  3292 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  3293   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  3294

  3295 lemma filter_from_subbase_not_bot:

  3296   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  3297   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  3298

  3299 lemma closure_iff_nhds_not_empty:

  3300   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  3301 proof safe

  3302   assume x: "x \<in> closure X"

  3303   fix S A

  3304   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  3305   then have "x \<notin> closure (-S)"

  3306     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  3307   with x have "x \<in> closure X - closure (-S)"

  3308     by auto

  3309   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  3310     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  3311   finally have "X \<inter> S \<noteq> {}" by auto

  3312   then show False using X \<inter> A = {} S \<subseteq> A by auto

  3313 next

  3314   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  3315   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  3316   show "x \<in> closure X"

  3317     by (simp add: closure_subset open_Compl)

  3318 qed

  3319

  3320 lemma compact_filter:

  3321   "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))"

  3322 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  3323   fix F

  3324   assume "compact U"

  3325   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  3326   then have "U \<noteq> {}"

  3327     by (auto simp: eventually_False)

  3328

  3329   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  3330   then have "\<forall>z\<in>Z. closed z"

  3331     by auto

  3332   moreover

  3333   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  3334     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  3335   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  3336   proof (intro allI impI)

  3337     fix B assume "finite B" "B \<subseteq> Z"

  3338     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  3339       by (auto intro!: eventually_Ball_finite)

  3340     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  3341       by eventually_elim auto

  3342     with F show "U \<inter> \<Inter>B \<noteq> {}"

  3343       by (intro notI) (simp add: eventually_False)

  3344   qed

  3345   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  3346     using compact U unfolding compact_fip by blast

  3347   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"

  3348     by auto

  3349

  3350   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  3351     unfolding eventually_inf eventually_nhds

  3352   proof safe

  3353     fix P Q R S

  3354     assume "eventually R F" "open S" "x \<in> S"

  3355     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  3356     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  3357     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  3358     ultimately show False by (auto simp: set_eq_iff)

  3359   qed

  3360   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  3361     by (metis eventually_bot)

  3362 next

  3363   fix A

  3364   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  3365   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  3366   then have inj_P': "\<And>A. inj_on P' A"

  3367     by (auto intro!: inj_onI simp: fun_eq_iff)

  3368   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  3369   have "F \<noteq> bot"

  3370     unfolding F_def

  3371   proof (safe intro!: filter_from_subbase_not_bot)

  3372     fix X

  3373     assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  3374     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  3375       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD simp del: Inf_image_eq)

  3376     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}"

  3377       by auto

  3378     with B show False

  3379       by (auto simp: P'_def fun_eq_iff)

  3380   qed

  3381   moreover have "eventually (\<lambda>x. x \<in> U) F"

  3382     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  3383   moreover

  3384   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)"

  3385   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  3386     by auto

  3387

  3388   {

  3389     fix V

  3390     assume "V \<in> A"

  3391     then have V: "eventually (\<lambda>x. x \<in> V) F"

  3392       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  3393     have "x \<in> closure V"

  3394       unfolding closure_iff_nhds_not_empty

  3395     proof (intro impI allI)

  3396       fix S A

  3397       assume "open S" "x \<in> S" "S \<subseteq> A"

  3398       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"

  3399         by (auto simp: eventually_nhds)

  3400       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  3401         by (auto simp: eventually_inf)

  3402       with x show "V \<inter> A \<noteq> {}"

  3403         by (auto simp del: Int_iff simp add: trivial_limit_def)

  3404     qed

  3405     then have "x \<in> V"

  3406       using V \<in> A A(1) by simp

  3407   }

  3408   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  3409   with U \<inter> \<Inter>A = {} show False by auto

  3410 qed

  3411

  3412 definition "countably_compact U \<longleftrightarrow>

  3413     (\<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))"

  3414

  3415 lemma countably_compactE:

  3416   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  3417   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  3418   using assms unfolding countably_compact_def by metis

  3419

  3420 lemma countably_compactI:

  3421   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')"

  3422   shows "countably_compact s"

  3423   using assms unfolding countably_compact_def by metis

  3424

  3425 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  3426   by (auto simp: compact_eq_heine_borel countably_compact_def)

  3427

  3428 lemma countably_compact_imp_compact:

  3429   assumes "countably_compact U"

  3430     and ccover: "countable B" "\<forall>b\<in>B. open b"

  3431     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"

  3432   shows "compact U"

  3433   using countably_compact U

  3434   unfolding compact_eq_heine_borel countably_compact_def

  3435 proof safe

  3436   fix A

  3437   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3438   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)"

  3439

  3440   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  3441   ultimately have "countable C" "\<forall>a\<in>C. open a"

  3442     unfolding C_def using ccover by auto

  3443   moreover

  3444   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  3445   proof safe

  3446     fix x a

  3447     assume "x \<in> U" "x \<in> a" "a \<in> A"

  3448     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"

  3449       by blast

  3450     with a \<in> A show "x \<in> \<Union>C"

  3451       unfolding C_def by auto

  3452   qed

  3453   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  3454   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  3455     using * by metis

  3456   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  3457     by (auto simp: C_def)

  3458   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"

  3459     unfolding bchoice_iff Bex_def ..

  3460   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3461     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  3462 qed

  3463

  3464 lemma countably_compact_imp_compact_second_countable:

  3465   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3466 proof (rule countably_compact_imp_compact)

  3467   fix T and x :: 'a

  3468   assume "open T" "x \<in> T"

  3469   from topological_basisE[OF is_basis this] obtain b where

  3470     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .

  3471   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  3472     by blast

  3473 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  3474

  3475 lemma countably_compact_eq_compact:

  3476   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3477   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  3478

  3479 subsubsection{* Sequential compactness *}

  3480

  3481 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"

  3482   where "seq_compact S \<longleftrightarrow>

  3483     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"

  3484

  3485 lemma seq_compactI:

  3486   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3487   shows "seq_compact S"

  3488   unfolding seq_compact_def using assms by fast

  3489

  3490 lemma seq_compactE:

  3491   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3492   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3493   using assms unfolding seq_compact_def by fast

  3494

  3495 lemma closed_sequentially: (* TODO: move upwards *)

  3496   assumes "closed s" and "\<forall>n. f n \<in> s" and "f ----> l"

  3497   shows "l \<in> s"

  3498 proof (rule ccontr)

  3499   assume "l \<notin> s"

  3500   with closed s and f ----> l have "eventually (\<lambda>n. f n \<in> - s) sequentially"

  3501     by (fast intro: topological_tendstoD)

  3502   with \<forall>n. f n \<in> s show "False"

  3503     by simp

  3504 qed

  3505

  3506 lemma seq_compact_inter_closed:

  3507   assumes "seq_compact s" and "closed t"

  3508   shows "seq_compact (s \<inter> t)"

  3509 proof (rule seq_compactI)

  3510   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"

  3511   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  3512     by simp_all

  3513   from seq_compact s and \<forall>n. f n \<in> s

  3514   obtain l r where "l \<in> s" and r: "subseq r" and l: "(f \<circ> r) ----> l"

  3515     by (rule seq_compactE)

  3516   from \<forall>n. f n \<in> t have "\<forall>n. (f \<circ> r) n \<in> t"

  3517     by simp

  3518   from closed t and this and l have "l \<in> t"

  3519     by (rule closed_sequentially)

  3520   with l \<in> s and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3521     by fast

  3522 qed

  3523

  3524 lemma seq_compact_closed_subset:

  3525   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"

  3526   shows "seq_compact s"

  3527   using assms seq_compact_inter_closed [of t s] by (simp add: Int_absorb1)

  3528

  3529 lemma seq_compact_imp_countably_compact:

  3530   fixes U :: "'a :: first_countable_topology set"

  3531   assumes "seq_compact U"

  3532   shows "countably_compact U"

  3533 proof (safe intro!: countably_compactI)

  3534   fix A

  3535   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  3536   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  3537     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  3538   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3539   proof cases

  3540     assume "finite A"

  3541     with A show ?thesis by auto

  3542   next

  3543     assume "infinite A"

  3544     then have "A \<noteq> {}" by auto

  3545     show ?thesis

  3546     proof (rule ccontr)

  3547       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  3548       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"

  3549         by auto

  3550       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"

  3551         by metis

  3552       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  3553       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  3554         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  3555       then have "range X \<subseteq> U"

  3556         by auto

  3557       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"

  3558         by auto

  3559       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  3560       obtain n where "x \<in> from_nat_into A n" by auto

  3561       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  3562       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  3563         unfolding tendsto_def by (auto simp: comp_def)

  3564       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  3565         by (auto simp: eventually_sequentially)

  3566       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  3567         by auto

  3568       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  3569         by (auto intro!: exI[of _ "max n N"])

  3570       ultimately show False

  3571         by auto

  3572     qed

  3573   qed

  3574 qed

  3575

  3576 lemma compact_imp_seq_compact:

  3577   fixes U :: "'a :: first_countable_topology set"

  3578   assumes "compact U"

  3579   shows "seq_compact U"

  3580   unfolding seq_compact_def

  3581 proof safe

  3582   fix X :: "nat \<Rightarrow> 'a"

  3583   assume "\<forall>n. X n \<in> U"

  3584   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  3585     by (auto simp: eventually_filtermap)

  3586   moreover

  3587   have "filtermap X sequentially \<noteq> bot"

  3588     by (simp add: trivial_limit_def eventually_filtermap)

  3589   ultimately

  3590   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  3591     using compact U by (auto simp: compact_filter)

  3592

  3593   from countable_basis_at_decseq[of x]

  3594   obtain A where A:

  3595       "\<And>i. open (A i)"

  3596       "\<And>i. x \<in> A i"

  3597       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3598     by blast

  3599   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3600   {

  3601     fix n i

  3602     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3603     proof (rule ccontr)

  3604       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3605       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"

  3606         by auto

  3607       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3608         by (auto simp: eventually_filtermap eventually_sequentially)

  3609       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3610         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3611       ultimately have "eventually (\<lambda>x. False) ?F"

  3612         by (auto simp add: eventually_inf)

  3613       with x show False

  3614         by (simp add: eventually_False)

  3615     qed

  3616     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3617       unfolding s_def by (auto intro: someI2_ex)

  3618   }

  3619   note s = this

  3620   def r \<equiv> "rec_nat (s 0 0) s"

  3621   have "subseq r"

  3622     by (auto simp: r_def s subseq_Suc_iff)

  3623   moreover

  3624   have "(\<lambda>n. X (r n)) ----> x"

  3625   proof (rule topological_tendstoI)

  3626     fix S

  3627     assume "open S" "x \<in> S"

  3628     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3629       by auto

  3630     moreover

  3631     {

  3632       fix i

  3633       assume "Suc 0 \<le> i"

  3634       then have "X (r i) \<in> A i"

  3635         by (cases i) (simp_all add: r_def s)

  3636     }

  3637     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3638       by (auto simp: eventually_sequentially)

  3639     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3640       by eventually_elim auto

  3641   qed

  3642   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3643     using x \<in> U by (auto simp: convergent_def comp_def)

  3644 qed

  3645

  3646 lemma countably_compact_imp_acc_point:

  3647   assumes "countably_compact s"

  3648     and "countable t"

  3649     and "infinite t"

  3650     and "t \<subseteq> s"

  3651   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3652 proof (rule ccontr)

  3653   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3654   note countably_compact s

  3655   moreover have "\<forall>t\<in>C. open t"

  3656     by (auto simp: C_def)

  3657   moreover

  3658   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3659   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3660   have "s \<subseteq> \<Union>C"

  3661     using t \<subseteq> s

  3662     unfolding C_def Union_image_eq

  3663     apply (safe dest!: s)

  3664     apply (rule_tac a="U \<inter> t" in UN_I)

  3665     apply (auto intro!: interiorI simp add: finite_subset)

  3666     done

  3667   moreover

  3668   from countable t have "countable C"

  3669     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3670   ultimately

  3671   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"

  3672     by (rule countably_compactE)

  3673   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"

  3674     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3675     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3676   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3677     using interior_subset by blast

  3678   moreover have "finite (\<Union>E)"

  3679     using E by auto

  3680   ultimately show False using infinite t

  3681     by (auto simp: finite_subset)

  3682 qed

  3683

  3684 lemma countable_acc_point_imp_seq_compact:

  3685   fixes s :: "'a::first_countable_topology set"

  3686   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>

  3687     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3688   shows "seq_compact s"

  3689 proof -

  3690   {

  3691     fix f :: "nat \<Rightarrow> 'a"

  3692     assume f: "\<forall>n. f n \<in> s"

  3693     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3694     proof (cases "finite (range f)")

  3695       case True

  3696       obtain l where "infinite {n. f n = f l}"

  3697         using pigeonhole_infinite[OF _ True] by auto

  3698       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3699         using infinite_enumerate by blast

  3700       then have "subseq r \<and> (f \<circ> r) ----> f l"

  3701         by (simp add: fr tendsto_const o_def)

  3702       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3703         by auto

  3704     next

  3705       case False

  3706       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"

  3707         by auto

  3708       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3709       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3710         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3711       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3712     qed

  3713   }

  3714   then show ?thesis

  3715     unfolding seq_compact_def by auto

  3716 qed

  3717

  3718 lemma seq_compact_eq_countably_compact:

  3719   fixes U :: "'a :: first_countable_topology set"

  3720   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3721   using

  3722     countable_acc_point_imp_seq_compact

  3723     countably_compact_imp_acc_point

  3724     seq_compact_imp_countably_compact

  3725   by metis

  3726

  3727 lemma seq_compact_eq_acc_point:

  3728   fixes s :: "'a :: first_countable_topology set"

  3729   shows "seq_compact s \<longleftrightarrow>

  3730     (\<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)))"

  3731   using

  3732     countable_acc_point_imp_seq_compact[of s]

  3733     countably_compact_imp_acc_point[of s]

  3734     seq_compact_imp_countably_compact[of s]

  3735   by metis

  3736

  3737 lemma seq_compact_eq_compact:

  3738   fixes U :: "'a :: second_countable_topology set"

  3739   shows "seq_compact U \<longleftrightarrow> compact U"

  3740   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3741

  3742 lemma bolzano_weierstrass_imp_seq_compact:

  3743   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3744   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3745   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3746

  3747 subsubsection{* Total boundedness *}

  3748

  3749 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)"

  3750   unfolding Cauchy_def by metis

  3751

  3752 fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a"

  3753 where

  3754   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3755 declare helper_1.simps[simp del]

  3756

  3757 lemma seq_compact_imp_totally_bounded:

  3758   assumes "seq_compact s"

  3759   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3760 proof (rule, rule, rule ccontr)

  3761   fix e::real

  3762   assume "e > 0"

  3763   assume assm: "\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3764   def x \<equiv> "helper_1 s e"

  3765   {

  3766     fix n

  3767     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3768     proof (induct n rule: nat_less_induct)

  3769       fix n

  3770       def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3771       assume as: "\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3772       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3773         using assm

  3774         apply simp

  3775         apply (erule_tac x="x  {0 ..< n}" in allE)

  3776         using as

  3777         apply auto

  3778         done

  3779       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3780         unfolding subset_eq by auto

  3781       have "Q (x n)"

  3782         unfolding x_def and helper_1.simps[of s e n]

  3783         apply (rule someI2[where a=z])

  3784         unfolding x_def[symmetric] and Q_def

  3785         using z

  3786         apply auto

  3787         done

  3788       then show "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3789         unfolding Q_def by auto

  3790     qed

  3791   }

  3792   then have "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"

  3793     by blast+

  3794   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3795     using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3796   from this(3) have "Cauchy (x \<circ> r)"

  3797     using LIMSEQ_imp_Cauchy by auto

  3798   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"

  3799     unfolding cauchy_def using e>0 by auto

  3800   show False

  3801     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3802     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3803     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

  3804     by auto

  3805 qed

  3806

  3807 subsubsection{* Heine-Borel theorem *}

  3808

  3809 lemma seq_compact_imp_heine_borel:

  3810   fixes s :: "'a :: metric_space set"

  3811   assumes "seq_compact s"

  3812   shows "compact s"

  3813 proof -

  3814   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3815   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)"

  3816     unfolding choice_iff' ..

  3817   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3818   have "countably_compact s"

  3819     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3820   then show "compact s"

  3821   proof (rule countably_compact_imp_compact)

  3822     show "countable K"

  3823       unfolding K_def using f

  3824       by (auto intro: countable_finite countable_subset countable_rat

  3825                intro!: countable_image countable_SIGMA countable_UN)

  3826     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3827   next

  3828     fix T x

  3829     assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3830     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"

  3831       by auto

  3832     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"

  3833       by auto

  3834     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"

  3835       by auto

  3836     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3837       unfolding Union_image_eq by auto

  3838     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K"

  3839       by (auto simp: K_def)

  3840     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3841     proof (rule bexI[rotated], safe)

  3842       fix y

  3843       assume "y \<in> ball k r"

  3844       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3845         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3846       with ball x e \<subseteq> T show "y \<in> T"

  3847         by auto

  3848     next

  3849       show "x \<in> ball k r" by fact

  3850     qed

  3851   qed

  3852 qed

  3853

  3854 lemma compact_eq_seq_compact_metric:

  3855   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3856   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3857

  3858 lemma compact_def:

  3859   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3860    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"

  3861   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3862

  3863 subsubsection {* Complete the chain of compactness variants *}

  3864

  3865 lemma compact_eq_bolzano_weierstrass:

  3866   fixes s :: "'a::metric_space set"

  3867   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"

  3868   (is "?lhs = ?rhs")

  3869 proof

  3870   assume ?lhs

  3871   then show ?rhs

  3872     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3873 next

  3874   assume ?rhs

  3875   then show ?lhs

  3876     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3877 qed

  3878

  3879 lemma bolzano_weierstrass_imp_bounded:

  3880   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3881   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3882

  3883 subsection {* Metric spaces with the Heine-Borel property *}

  3884

  3885 text {*

  3886   A metric space (or topological vector space) is said to have the

  3887   Heine-Borel property if every closed and bounded subset is compact.

  3888 *}

  3889

  3890 class heine_borel = metric_space +

  3891   assumes bounded_imp_convergent_subsequence:

  3892     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3893

  3894 lemma bounded_closed_imp_seq_compact:

  3895   fixes s::"'a::heine_borel set"

  3896   assumes "bounded s"

  3897     and "closed s"

  3898   shows "seq_compact s"

  3899 proof (unfold seq_compact_def, clarify)

  3900   fix f :: "nat \<Rightarrow> 'a"

  3901   assume f: "\<forall>n. f n \<in> s"

  3902   with bounded s have "bounded (range f)"

  3903     by (auto intro: bounded_subset)

  3904   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3905     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3906   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"

  3907     by simp

  3908   have "l \<in> s" using closed s fr l

  3909     by (rule closed_sequentially)

  3910   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3911     using l \<in> s r l by blast

  3912 qed

  3913

  3914 lemma compact_eq_bounded_closed:

  3915   fixes s :: "'a::heine_borel set"

  3916   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"

  3917   (is "?lhs = ?rhs")

  3918 proof

  3919   assume ?lhs

  3920   then show ?rhs

  3921     using compact_imp_closed compact_imp_bounded

  3922     by blast

  3923 next

  3924   assume ?rhs

  3925   then show ?lhs

  3926     using bounded_closed_imp_seq_compact[of s]

  3927     unfolding compact_eq_seq_compact_metric

  3928     by auto

  3929 qed

  3930

  3931 (* TODO: is this lemma necessary? *)

  3932 lemma bounded_increasing_convergent:

  3933   fixes s :: "nat \<Rightarrow> real"

  3934   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3935   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3936   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3937

  3938 instance real :: heine_borel

  3939 proof

  3940   fix f :: "nat \<Rightarrow> real"

  3941   assume f: "bounded (range f)"

  3942   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3943     unfolding comp_def by (metis seq_monosub)

  3944   then have "Bseq (f \<circ> r)"

  3945     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3946   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3947     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3948 qed

  3949

  3950 lemma compact_lemma:

  3951   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3952   assumes "bounded (range f)"

  3953   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.

  3954     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3955 proof safe

  3956   fix d :: "'a set"

  3957   assume d: "d \<subseteq> Basis"

  3958   with finite_Basis have "finite d"

  3959     by (blast intro: finite_subset)

  3960   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3961     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3962   proof (induct d)

  3963     case empty

  3964     then show ?case

  3965       unfolding subseq_def by auto

  3966   next

  3967     case (insert k d)

  3968     have k[intro]: "k \<in> Basis"

  3969       using insert by auto

  3970     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)"

  3971       using bounded (range f)

  3972       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3973     obtain l1::"'a" and r1 where r1: "subseq r1"

  3974       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3975       using insert(3) using insert(4) by auto

  3976     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f"

  3977       by simp

  3978     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3979       by (metis (lifting) bounded_subset f' image_subsetI s')

  3980     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3981       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]

  3982       by (auto simp: o_def)

  3983     def r \<equiv> "r1 \<circ> r2"

  3984     have r:"subseq r"

  3985       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3986     moreover

  3987     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3988     {

  3989       fix e::real

  3990       assume "e > 0"

  3991       from lr1 e > 0 have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3992         by blast

  3993       from lr2 e > 0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3994         by (rule tendstoD)

  3995       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3996         by (rule eventually_subseq)

  3997       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3998         using N1' N2

  3999         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  4000     }

  4001     ultimately show ?case by auto

  4002   qed

  4003 qed

  4004

  4005 instance euclidean_space \<subseteq> heine_borel

  4006 proof

  4007   fix f :: "nat \<Rightarrow> 'a"

  4008   assume f: "bounded (range f)"

  4009   then obtain l::'a and r where r: "subseq r"

  4010     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  4011     using compact_lemma [OF f] by blast

  4012   {

  4013     fix e::real

  4014     assume "e > 0"

  4015     then have "e / real_of_nat DIM('a) > 0"

  4016       by (auto intro!: divide_pos_pos DIM_positive)

  4017     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"

  4018       by simp

  4019     moreover

  4020     {

  4021       fix n

  4022       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  4023       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  4024         apply (subst euclidean_dist_l2)

  4025         using zero_le_dist

  4026         apply (rule setL2_le_setsum)

  4027         done

  4028       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  4029         apply (rule setsum_strict_mono)

  4030         using n

  4031         apply auto

  4032         done

  4033       finally have "dist (f (r n)) l < e"

  4034         by auto

  4035     }

  4036     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  4037       by (rule eventually_elim1)

  4038   }

  4039   then have *: "((f \<circ> r) ---> l) sequentially"

  4040     unfolding o_def tendsto_iff by simp

  4041   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4042     by auto

  4043 qed

  4044

  4045 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  4046   unfolding bounded_def

  4047   by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)

  4048

  4049 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  4050   unfolding bounded_def

  4051   by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)

  4052

  4053 instance prod :: (heine_borel, heine_borel) heine_borel

  4054 proof

  4055   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  4056   assume f: "bounded (range f)"

  4057   then have "bounded (fst  range f)"

  4058     by (rule bounded_fst)

  4059   then have s1: "bounded (range (fst \<circ> f))"

  4060     by (simp add: image_comp)

  4061   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  4062     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  4063   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  4064     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  4065   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  4066     using bounded_imp_convergent_subsequence [OF s2]

  4067     unfolding o_def by fast

  4068   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  4069     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  4070   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  4071     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  4072   have r: "subseq (r1 \<circ> r2)"

  4073     using r1 r2 unfolding subseq_def by simp

  4074   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  4075     using l r by fast

  4076 qed

  4077

  4078 subsubsection {* Completeness *}

  4079

  4080 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  4081   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  4082

  4083 lemma completeI:

  4084   assumes "\<And>f. \<forall>n. f n \<in> s \<Longrightarrow> Cauchy f \<Longrightarrow> \<exists>l\<in>s. f ----> l"

  4085   shows "complete s"

  4086   using assms unfolding complete_def by fast

  4087

  4088 lemma completeE:

  4089   assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"

  4090   obtains l where "l \<in> s" and "f ----> l"

  4091   using assms unfolding complete_def by fast

  4092

  4093 lemma compact_imp_complete:

  4094   assumes "compact s"

  4095   shows "complete s"

  4096 proof -

  4097   {

  4098     fix f

  4099     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  4100     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  4101       using assms unfolding compact_def by blast

  4102

  4103     note lr' = seq_suble [OF lr(2)]

  4104     {

  4105       fix e :: real

  4106       assume "e > 0"

  4107       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"

  4108         unfolding cauchy_def

  4109         using e > 0

  4110         apply (erule_tac x="e/2" in allE)

  4111         apply auto

  4112         done

  4113       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  4114       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"

  4115         using e > 0 by auto

  4116       {

  4117         fix n :: nat

  4118         assume n: "n \<ge> max N M"

  4119         have "dist ((f \<circ> r) n) l < e/2"

  4120           using n M by auto

  4121         moreover have "r n \<ge> N"

  4122           using lr'[of n] n by auto

  4123         then have "dist (f n) ((f \<circ> r) n) < e / 2"

  4124           using N and n by auto

  4125         ultimately have "dist (f n) l < e"

  4126           using dist_triangle_half_r[of "f (r n)" "f n" e l]

  4127           by (auto simp add: dist_commute)

  4128       }

  4129       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  4130     }

  4131     then have "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  4132       unfolding LIMSEQ_def by auto

  4133   }

  4134   then show ?thesis unfolding complete_def by auto

  4135 qed

  4136

  4137 lemma nat_approx_posE:

  4138   fixes e::real

  4139   assumes "0 < e"

  4140   obtains n :: nat where "1 / (Suc n) < e"

  4141 proof atomize_elim

  4142   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  4143     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  4144   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  4145     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  4146   also have "\<dots> = e" by simp

  4147   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  4148 qed

  4149

  4150 lemma compact_eq_totally_bounded:

  4151   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  4152     (is "_ \<longleftrightarrow> ?rhs")

  4153 proof

  4154   assume assms: "?rhs"

  4155   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)"

  4156     by (auto simp: choice_iff')

  4157

  4158   show "compact s"

  4159   proof cases

  4160     assume "s = {}"

  4161     then show "compact s" by (simp add: compact_def)

  4162   next

  4163     assume "s \<noteq> {}"

  4164     show ?thesis

  4165       unfolding compact_def

  4166     proof safe

  4167       fix f :: "nat \<Rightarrow> 'a"

  4168       assume f: "\<forall>n. f n \<in> s"

  4169

  4170       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  4171       then have [simp]: "\<And>n. 0 < e n" by auto

  4172       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)"

  4173       {

  4174         fix n U

  4175         assume "infinite {n. f n \<in> U}"

  4176         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  4177           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  4178         then obtain a where

  4179           "a \<in> k (e n)"

  4180           "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..

  4181         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  4182           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  4183         from someI_ex[OF this]

  4184         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  4185           unfolding B_def by auto

  4186       }

  4187       note B = this

  4188

  4189       def F \<equiv> "rec_nat (B 0 UNIV) B"

  4190       {

  4191         fix n

  4192         have "infinite {i. f i \<in> F n}"

  4193           by (induct n) (auto simp: F_def B)

  4194       }

  4195       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  4196         using B by (simp add: F_def)

  4197       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  4198         using decseq_SucI[of F] by (auto simp: decseq_def)

  4199

  4200       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  4201       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  4202         fix k i

  4203         have "infinite ({n. f n \<in> F k} - {.. i})"

  4204           using infinite {n. f n \<in> F k} by auto

  4205         from infinite_imp_nonempty[OF this]

  4206         show "\<exists>x>i. f x \<in> F k"

  4207           by (simp add: set_eq_iff not_le conj_commute)

  4208       qed

  4209

  4210       def t \<equiv> "rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  4211       have "subseq t"

  4212         unfolding subseq_Suc_iff by (simp add: t_def sel)

  4213       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  4214         using f by auto

  4215       moreover

  4216       {

  4217         fix n

  4218         have "(f \<circ> t) n \<in> F n"

  4219           by (cases n) (simp_all add: t_def sel)

  4220       }

  4221       note t = this

  4222

  4223       have "Cauchy (f \<circ> t)"

  4224       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  4225         fix r :: real and N n m

  4226         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  4227         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  4228           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  4229         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  4230           by (auto simp: subset_eq)

  4231         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  4232         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  4233           by (simp add: dist_commute)

  4234       qed

  4235

  4236       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  4237         using assms unfolding complete_def by blast

  4238     qed

  4239   qed

  4240 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  4241

  4242 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  4243 proof -

  4244   {

  4245     assume ?rhs

  4246     {

  4247       fix e::real

  4248       assume "e>0"

  4249       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  4250         by (erule_tac x="e/2" in allE) auto

  4251       {

  4252         fix n m

  4253         assume nm:"N \<le> m \<and> N \<le> n"

  4254         then have "dist (s m) (s n) < e" using N

  4255           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  4256           by blast

  4257       }

  4258       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  4259         by blast

  4260     }

  4261     then have ?lhs

  4262       unfolding cauchy_def

  4263       by blast

  4264   }

  4265   then show ?thesis

  4266     unfolding cauchy_def

  4267     using dist_triangle_half_l

  4268     by blast

  4269 qed

  4270

  4271 lemma cauchy_imp_bounded:

  4272   assumes "Cauchy s"

  4273   shows "bounded (range s)"

  4274 proof -

  4275   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  4276     unfolding cauchy_def

  4277     apply (erule_tac x= 1 in allE)

  4278     apply auto

  4279     done

  4280   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  4281   moreover

  4282   have "bounded (s  {0..N})"

  4283     using finite_imp_bounded[of "s  {1..N}"] by auto

  4284   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  4285     unfolding bounded_any_center [where a="s N"] by auto

  4286   ultimately show "?thesis"

  4287     unfolding bounded_any_center [where a="s N"]

  4288     apply (rule_tac x="max a 1" in exI)

  4289     apply auto

  4290     apply (erule_tac x=y in allE)

  4291     apply (erule_tac x=y in ballE)

  4292     apply auto

  4293     done

  4294 qed

  4295

  4296 instance heine_borel < complete_space

  4297 proof

  4298   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4299   then have "bounded (range f)"

  4300     by (rule cauchy_imp_bounded)

  4301   then have "compact (closure (range f))"

  4302     unfolding compact_eq_bounded_closed by auto

  4303   then have "complete (closure (range f))"

  4304     by (rule compact_imp_complete)

  4305   moreover have "\<forall>n. f n \<in> closure (range f)"

  4306     using closure_subset [of "range f"] by auto

  4307   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  4308     using Cauchy f unfolding complete_def by auto

  4309   then show "convergent f"

  4310     unfolding convergent_def by auto

  4311 qed

  4312

  4313 instance euclidean_space \<subseteq> banach ..

  4314

  4315 lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"

  4316 proof (rule completeI)

  4317   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4318   then have "convergent f" by (rule Cauchy_convergent)

  4319   then show "\<exists>l\<in>UNIV. f ----> l" unfolding convergent_def by simp

  4320 qed

  4321

  4322 lemma complete_imp_closed:

  4323   assumes "complete s"

  4324   shows "closed s"

  4325 proof (unfold closed_sequential_limits, clarify)

  4326   fix f x assume "\<forall>n. f n \<in> s" and "f ----> x"

  4327   from f ----> x have "Cauchy f"

  4328     by (rule LIMSEQ_imp_Cauchy)

  4329   with complete s and \<forall>n. f n \<in> s obtain l where "l \<in> s" and "f ----> l"

  4330     by (rule completeE)

  4331   from f ----> x and f ----> l have "x = l"

  4332     by (rule LIMSEQ_unique)

  4333   with l \<in> s show "x \<in> s"

  4334     by simp

  4335 qed

  4336

  4337 lemma complete_inter_closed:

  4338   assumes "complete s" and "closed t"

  4339   shows "complete (s \<inter> t)"

  4340 proof (rule completeI)

  4341   fix f assume "\<forall>n. f n \<in> s \<inter> t" and "Cauchy f"

  4342   then have "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"

  4343     by simp_all

  4344   from complete s obtain l where "l \<in> s" and "f ----> l"

  4345     using \<forall>n. f n \<in> s and Cauchy f by (rule completeE)

  4346   from closed t and \<forall>n. f n \<in> t and f ----> l have "l \<in> t"

  4347     by (rule closed_sequentially)

  4348   with l \<in> s and f ----> l show "\<exists>l\<in>s \<inter> t. f ----> l"

  4349     by fast

  4350 qed

  4351

  4352 lemma complete_closed_subset:

  4353   assumes "closed s" and "s \<subseteq> t" and "complete t"

  4354   shows "complete s"

  4355   using assms complete_inter_closed [of t s] by (simp add: Int_absorb1)

  4356

  4357 lemma complete_eq_closed:

  4358   fixes s :: "('a::complete_space) set"

  4359   shows "complete s \<longleftrightarrow> closed s"

  4360 proof

  4361   assume "closed s" then show "complete s"

  4362     using subset_UNIV complete_UNIV by (rule complete_closed_subset)

  4363 next

  4364   assume "complete s" then show "closed s"

  4365     by (rule complete_imp_closed)

  4366 qed

  4367

  4368 lemma convergent_eq_cauchy:

  4369   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  4370   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  4371   unfolding Cauchy_convergent_iff convergent_def ..

  4372

  4373 lemma convergent_imp_bounded:

  4374   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  4375   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  4376   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  4377

  4378 lemma compact_cball[simp]:

  4379   fixes x :: "'a::heine_borel"

  4380   shows "compact (cball x e)"

  4381   using compact_eq_bounded_closed bounded_cball closed_cball

  4382   by blast

  4383

  4384 lemma compact_frontier_bounded[intro]:

  4385   fixes s :: "'a::heine_borel set"

  4386   shows "bounded s \<Longrightarrow> compact (frontier s)"

  4387   unfolding frontier_def

  4388   using compact_eq_bounded_closed

  4389   by blast

  4390

  4391 lemma compact_frontier[intro]:

  4392   fixes s :: "'a::heine_borel set"

  4393   shows "compact s \<Longrightarrow> compact (frontier s)"

  4394   using compact_eq_bounded_closed compact_frontier_bounded

  4395   by blast

  4396

  4397 lemma frontier_subset_compact:

  4398   fixes s :: "'a::heine_borel set"

  4399   shows "compact s \<Longrightarrow> frontier s \<subseteq> s"

  4400   using frontier_subset_closed compact_eq_bounded_closed

  4401   by blast

  4402

  4403 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  4404

  4405 lemma bounded_closed_nest:

  4406   fixes s :: "nat \<Rightarrow> ('a::heine_borel) set"

  4407   assumes "\<forall>n. closed (s n)"

  4408     and "\<forall>n. s n \<noteq> {}"

  4409     and "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4410     and "bounded (s 0)"

  4411   shows "\<exists>a. \<forall>n. a \<in> s n"

  4412 proof -

  4413   from assms(2) obtain x where x: "\<forall>n. x n \<in> s n"

  4414     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4415   from assms(4,1) have "seq_compact (s 0)"

  4416     by (simp add: bounded_closed_imp_seq_compact)

  4417   then obtain l r where lr: "l \<in> s 0" "subseq r" "(x \<circ> r) ----> l"

  4418     using x and assms(3) unfolding seq_compact_def by blast

  4419   have "\<forall>n. l \<in> s n"

  4420   proof

  4421     fix n :: nat

  4422     have "closed (s n)"

  4423       using assms(1) by simp

  4424     moreover have "\<forall>i. (x \<circ> r) i \<in> s i"

  4425       using x and assms(3) and lr(2) [THEN seq_suble] by auto

  4426     then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"

  4427       using assms(3) by (fast intro!: le_add2)

  4428     moreover have "(\<lambda>i. (x \<circ> r) (i + n)) ----> l"

  4429       using lr(3) by (rule LIMSEQ_ignore_initial_segment)

  4430     ultimately show "l \<in> s n"

  4431       by (rule closed_sequentially)

  4432   qed

  4433   then show ?thesis ..

  4434 qed

  4435

  4436 text {* Decreasing case does not even need compactness, just completeness. *}

  4437

  4438 lemma decreasing_closed_nest:

  4439   fixes s :: "nat \<Rightarrow> ('a::complete_space) set"

  4440   assumes

  4441     "\<forall>n. closed (s n)"

  4442     "\<forall>n. s n \<noteq> {}"

  4443     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4444     "\<forall>e>0. \<exists>n. \<forall>x\<in>s n. \<forall>y\<in>s n. dist x y < e"

  4445   shows "\<exists>a. \<forall>n. a \<in> s n"

  4446 proof -

  4447   have "\<forall>n. \<exists>x. x \<in> s n"

  4448     using assms(2) by auto

  4449   then have "\<exists>t. \<forall>n. t n \<in> s n"

  4450     using choice[of "\<lambda>n x. x \<in> s n"] by auto

  4451   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  4452   {

  4453     fix e :: real

  4454     assume "e > 0"

  4455     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"

  4456       using assms(4) by auto

  4457     {

  4458       fix m n :: nat

  4459       assume "N \<le> m \<and> N \<le> n"

  4460       then have "t m \<in> s N" "t n \<in> s N"

  4461         using assms(3) t unfolding  subset_eq t by blast+

  4462       then have "dist (t m) (t n) < e"

  4463         using N by auto

  4464     }

  4465     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  4466       by auto

  4467   }

  4468   then have "Cauchy t"

  4469     unfolding cauchy_def by auto

  4470   then obtain l where l:"(t ---> l) sequentially"

  4471     using complete_UNIV unfolding complete_def by auto

  4472   {

  4473     fix n :: nat

  4474     {

  4475       fix e :: real

  4476       assume "e > 0"

  4477       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  4478         using l[unfolded LIMSEQ_def] by auto

  4479       have "t (max n N) \<in> s n"

  4480         using assms(3)

  4481         unfolding subset_eq

  4482         apply (erule_tac x=n in allE)

  4483         apply (erule_tac x="max n N" in allE)

  4484         using t

  4485         apply auto

  4486         done

  4487       then have "\<exists>y\<in>s n. dist y l < e"

  4488         apply (rule_tac x="t (max n N)" in bexI)

  4489         using N

  4490         apply auto

  4491         done

  4492     }

  4493     then have "l \<in> s n"

  4494       using closed_approachable[of "s n" l] assms(1) by auto

  4495   }

  4496   then show ?thesis by auto

  4497 qed

  4498

  4499 text {* Strengthen it to the intersection actually being a singleton. *}

  4500

  4501 lemma decreasing_closed_nest_sing:

  4502   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  4503   assumes

  4504     "\<forall>n. closed(s n)"

  4505     "\<forall>n. s n \<noteq> {}"

  4506     "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"

  4507     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  4508   shows "\<exists>a. \<Inter>(range s) = {a}"

  4509 proof -

  4510   obtain a where a: "\<forall>n. a \<in> s n"

  4511     using decreasing_closed_nest[of s] using assms by auto

  4512   {

  4513     fix b

  4514     assume b: "b \<in> \<Inter>(range s)"

  4515     {

  4516       fix e :: real

  4517       assume "e > 0"

  4518       then have "dist a b < e"

  4519         using assms(4) and b and a by blast

  4520     }

  4521     then have "dist a b = 0"

  4522       by (metis dist_eq_0_iff dist_nz less_le)

  4523   }

  4524   with a have "\<Inter>(range s) = {a}"

  4525     unfolding image_def by auto

  4526   then show ?thesis ..

  4527 qed

  4528

  4529 text{* Cauchy-type criteria for uniform convergence. *}

  4530

  4531 lemma uniformly_convergent_eq_cauchy:

  4532   fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"

  4533   shows

  4534     "(\<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>

  4535       (\<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)"

  4536   (is "?lhs = ?rhs")

  4537 proof

  4538   assume ?lhs

  4539   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"

  4540     by auto

  4541   {

  4542     fix e :: real

  4543     assume "e > 0"

  4544     then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"

  4545       using l[THEN spec[where x="e/2"]] by auto

  4546     {

  4547       fix n m :: nat and x :: "'b"

  4548       assume "N \<le> m \<and> N \<le> n \<and> P x"

  4549       then have "dist (s m x) (s n x) < e"

  4550         using N[THEN spec[where x=m], THEN spec[where x=x]]

  4551         using N[THEN spec[where x=n], THEN spec[where x=x]]

  4552         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

  4553     }

  4554     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

  4555   }

  4556   then show ?rhs by auto

  4557 next

  4558   assume ?rhs

  4559   then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"

  4560     unfolding cauchy_def

  4561     apply auto

  4562     apply (erule_tac x=e in allE)

  4563     apply auto

  4564     done

  4565   then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"

  4566     unfolding convergent_eq_cauchy[symmetric]

  4567     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]

  4568     by auto

  4569   {

  4570     fix e :: real

  4571     assume "e > 0"

  4572     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"

  4573       using ?rhs[THEN spec[where x="e/2"]] by auto

  4574     {

  4575       fix x

  4576       assume "P x"

  4577       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  4578         using l[THEN spec[where x=x], unfolded LIMSEQ_def] and e > 0

  4579         by (auto elim!: allE[where x="e/2"])

  4580       fix n :: nat

  4581       assume "n \<ge> N"

  4582       then have "dist(s n x)(l x) < e"

  4583         using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  4584         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

  4585         by (auto simp add: dist_commute)

  4586     }

  4587     then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4588       by auto

  4589   }

  4590   then show ?lhs by auto

  4591 qed

  4592

  4593 lemma uniformly_cauchy_imp_uniformly_convergent:

  4594   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  4595   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"

  4596     and "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n \<longrightarrow> dist(s n x)(l x) < e)"

  4597   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4598 proof -

  4599   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"

  4600     using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto

  4601   moreover

  4602   {

  4603     fix x

  4604     assume "P x"

  4605     then have "l x = l' x"

  4606       using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  4607       using l and assms(2) unfolding LIMSEQ_def by blast

  4608   }

  4609   ultimately show ?thesis by auto

  4610 qed

  4611

  4612

  4613 subsection {* Continuity *}

  4614

  4615 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  4616

  4617 lemma continuous_within_eps_delta:

  4618   "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)"

  4619   unfolding continuous_within and Lim_within

  4620   apply auto

  4621   apply (metis dist_nz dist_self)

  4622   apply blast

  4623   done

  4624

  4625 lemma continuous_at_eps_delta:

  4626   "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  4627   using continuous_within_eps_delta [of x UNIV f] by simp

  4628

  4629 text{* Versions in terms of open balls. *}

  4630

  4631 lemma continuous_within_ball:

  4632   "continuous (at x within s) f \<longleftrightarrow>

  4633     (\<forall>e > 0. \<exists>d > 0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e)"

  4634   (is "?lhs = ?rhs")

  4635 proof

  4636   assume ?lhs

  4637   {

  4638     fix e :: real

  4639     assume "e > 0"

  4640     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"

  4641       using ?lhs[unfolded continuous_within Lim_within] by auto

  4642     {

  4643       fix y

  4644       assume "y \<in> f  (ball x d \<inter> s)"

  4645       then have "y \<in> ball (f x) e"

  4646         using d(2)

  4647         unfolding dist_nz[symmetric]

  4648         apply (auto simp add: dist_commute)

  4649         apply (erule_tac x=xa in ballE)

  4650         apply auto

  4651         using e > 0

  4652         apply auto

  4653         done

  4654     }

  4655     then have "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e"

  4656       using d > 0

  4657       unfolding subset_eq ball_def by (auto simp add: dist_commute)

  4658   }

  4659   then show ?rhs by auto

  4660 next

  4661   assume ?rhs

  4662   then show ?lhs

  4663     unfolding continuous_within Lim_within ball_def subset_eq

  4664     apply (auto simp add: dist_commute)

  4665     apply (erule_tac x=e in allE)

  4666     apply auto

  4667     done

  4668 qed

  4669

  4670 lemma continuous_at_ball:

  4671   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  4672 proof

  4673   assume ?lhs

  4674   then show ?rhs

  4675     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4676     apply auto

  4677     apply (erule_tac x=e in allE)

  4678     apply auto

  4679     apply (rule_tac x=d in exI)

  4680     apply auto

  4681     apply (erule_tac x=xa in allE)

  4682     apply (auto simp add: dist_commute dist_nz)

  4683     unfolding dist_nz[symmetric]

  4684     apply auto

  4685     done

  4686 next

  4687   assume ?rhs

  4688   then show ?lhs

  4689     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4690     apply auto

  4691     apply (erule_tac x=e in allE)

  4692     apply auto

  4693     apply (rule_tac x=d in exI)

  4694     apply auto

  4695     apply (erule_tac x="f xa" in allE)

  4696     apply (auto simp add: dist_commute dist_nz)

  4697     done

  4698 qed

  4699

  4700 text{* Define setwise continuity in terms of limits within the set. *}

  4701

  4702 lemma continuous_on_iff:

  4703   "continuous_on s f \<longleftrightarrow>

  4704     (\<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)"

  4705   unfolding continuous_on_def Lim_within

  4706   by (metis dist_pos_lt dist_self)

  4707

  4708 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  4709   where "uniformly_continuous_on s f \<longleftrightarrow>

  4710     (\<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)"

  4711

  4712 text{* Some simple consequential lemmas. *}

  4713

  4714 lemma uniformly_continuous_imp_continuous:

  4715   "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"

  4716   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  4717

  4718 lemma continuous_at_imp_continuous_within:

  4719   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"

  4720   unfolding continuous_within continuous_at using Lim_at_within by auto

  4721

  4722 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  4723   by simp

  4724

  4725 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4726

  4727 lemma continuous_within_subset:

  4728   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"

  4729   unfolding continuous_within by(metis tendsto_within_subset)

  4730

  4731 lemma continuous_on_interior:

  4732   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4733   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)

  4734

  4735 lemma continuous_on_eq:

  4736   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4737   unfolding continuous_on_def tendsto_def eventually_at_topological

  4738   by simp

  4739

  4740 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4741

  4742 lemma continuous_within_sequentially:

  4743   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4744   shows "continuous (at a within s) f \<longleftrightarrow>

  4745     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4746          \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"

  4747   (is "?lhs = ?rhs")

  4748 proof

  4749   assume ?lhs

  4750   {

  4751     fix x :: "nat \<Rightarrow> 'a"

  4752     assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4753     fix T :: "'b set"

  4754     assume "open T" and "f a \<in> T"

  4755     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"

  4756       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  4757     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4758       using x(2) d>0 by simp

  4759     then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4760     proof eventually_elim

  4761       case (elim n)

  4762       then show ?case

  4763         using d x(1) f a \<in> T unfolding dist_nz[symmetric] by auto

  4764     qed

  4765   }

  4766   then show ?rhs

  4767     unfolding tendsto_iff tendsto_def by simp

  4768 next

  4769   assume ?rhs

  4770   then show ?lhs

  4771     unfolding continuous_within tendsto_def [where l="f a"]

  4772     by (simp add: sequentially_imp_eventually_within)

  4773 qed

  4774

  4775 lemma continuous_at_sequentially:

  4776   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4777   shows "continuous (at a) f \<longleftrightarrow>

  4778     (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"

  4779   using continuous_within_sequentially[of a UNIV f] by simp

  4780

  4781 lemma continuous_on_sequentially:

  4782   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4783   shows "continuous_on s f \<longleftrightarrow>

  4784     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4785       --> ((f \<circ> x) ---> f a) sequentially)"

  4786   (is "?lhs = ?rhs")

  4787 proof

  4788   assume ?rhs

  4789   then show ?lhs

  4790     using continuous_within_sequentially[of _ s f]

  4791     unfolding continuous_on_eq_continuous_within

  4792     by auto

  4793 next

  4794   assume ?lhs

  4795   then show ?rhs

  4796     unfolding continuous_on_eq_continuous_within

  4797     using continuous_within_sequentially[of _ s f]

  4798     by auto

  4799 qed

  4800

  4801 lemma uniformly_continuous_on_sequentially:

  4802   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4803                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4804                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4805 proof

  4806   assume ?lhs

  4807   {

  4808     fix x y

  4809     assume x: "\<forall>n. x n \<in> s"

  4810       and y: "\<forall>n. y n \<in> s"

  4811       and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4812     {

  4813       fix e :: real

  4814       assume "e > 0"

  4815       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"

  4816         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4817       obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"

  4818         using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4819       {

  4820         fix n

  4821         assume "n\<ge>N"

  4822         then have "dist (f (x n)) (f (y n)) < e"

  4823           using N[THEN spec[where x=n]]

  4824           using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

  4825           using x and y

  4826           unfolding dist_commute

  4827           by simp

  4828       }

  4829       then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4830         by auto

  4831     }

  4832     then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"

  4833       unfolding LIMSEQ_def and dist_real_def by auto

  4834   }

  4835   then show ?rhs by auto

  4836 next

  4837   assume ?rhs

  4838   {

  4839     assume "\<not> ?lhs"

  4840     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"

  4841       unfolding uniformly_continuous_on_def by auto

  4842     then obtain fa where fa:

  4843       "\<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"

  4844       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"]

  4845       unfolding Bex_def

  4846       by (auto simp add: dist_commute)

  4847     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4848     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4849     have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"

  4850       and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"

  4851       and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4852       unfolding x_def and y_def using fa

  4853       by auto

  4854     {

  4855       fix e :: real

  4856       assume "e > 0"

  4857       then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"

  4858         unfolding real_arch_inv[of e] by auto

  4859       {

  4860         fix n :: nat

  4861         assume "n \<ge> N"

  4862         then have "inverse (real n + 1) < inverse (real N)"

  4863           using real_of_nat_ge_zero and N\<noteq>0 by auto

  4864         also have "\<dots> < e" using N by auto

  4865         finally have "inverse (real n + 1) < e" by auto

  4866         then have "dist (x n) (y n) < e"

  4867           using xy0[THEN spec[where x=n]] by auto

  4868       }

  4869       then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto

  4870     }

  4871     then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4872       using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn

  4873       unfolding LIMSEQ_def dist_real_def by auto

  4874     then have False using fxy and e>0 by auto

  4875   }

  4876   then show ?lhs

  4877     unfolding uniformly_continuous_on_def by blast

  4878 qed

  4879

  4880 text{* The usual transformation theorems. *}

  4881

  4882 lemma continuous_transform_within:

  4883   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4884   assumes "0 < d"

  4885     and "x \<in> s"

  4886     and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4887     and "continuous (at x within s) f"

  4888   shows "continuous (at x within s) g"

  4889   unfolding continuous_within

  4890 proof (rule Lim_transform_within)

  4891   show "0 < d" by fact

  4892   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4893     using assms(3) by auto

  4894   have "f x = g x"

  4895     using assms(1,2,3) by auto

  4896   then show "(f ---> g x) (at x within s)"

  4897     using assms(4) unfolding continuous_within by simp

  4898 qed

  4899

  4900 lemma continuous_transform_at:

  4901   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4902   assumes "0 < d"

  4903     and "\<forall>x'. dist x' x < d --> f x' = g x'"

  4904     and "continuous (at x) f"

  4905   shows "continuous (at x) g"

  4906   using continuous_transform_within [of d x UNIV f g] assms by simp

  4907

  4908

  4909 subsubsection {* Structural rules for pointwise continuity *}

  4910

  4911 lemmas continuous_within_id = continuous_ident

  4912

  4913 lemmas continuous_at_id = isCont_ident

  4914

  4915 lemma continuous_infdist[continuous_intros]:

  4916   assumes "continuous F f"

  4917   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4918   using assms unfolding continuous_def by (rule tendsto_infdist)

  4919

  4920 lemma continuous_infnorm[continuous_intros]:

  4921   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4922   unfolding continuous_def by (rule tendsto_infnorm)

  4923

  4924 lemma continuous_inner[continuous_intros]:

  4925   assumes "continuous F f"

  4926     and "continuous F g"

  4927   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4928   using assms unfolding continuous_def by (rule tendsto_inner)

  4929

  4930 lemmas continuous_at_inverse = isCont_inverse

  4931

  4932 subsubsection {* Structural rules for setwise continuity *}

  4933

  4934 lemma continuous_on_infnorm[continuous_on_intros]:

  4935   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4936   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4937

  4938 lemma continuous_on_inner[continuous_on_intros]:

  4939   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4940   assumes "continuous_on s f"

  4941     and "continuous_on s g"

  4942   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4943   using bounded_bilinear_inner assms

  4944   by (rule bounded_bilinear.continuous_on)

  4945

  4946 subsubsection {* Structural rules for uniform continuity *}

  4947

  4948 lemma uniformly_continuous_on_id[continuous_on_intros]:

  4949   "uniformly_continuous_on s (\<lambda>x. x)"

  4950   unfolding uniformly_continuous_on_def by auto

  4951

  4952 lemma uniformly_continuous_on_const[continuous_on_intros]:

  4953   "uniformly_continuous_on s (\<lambda>x. c)"

  4954   unfolding uniformly_continuous_on_def by simp

  4955

  4956 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  4957   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4958   assumes "uniformly_continuous_on s f"

  4959     and "uniformly_continuous_on s g"

  4960   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4961 proof -

  4962   {

  4963     fix a b c d :: 'b

  4964     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4965       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4966       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4967       by arith

  4968   } note le = this

  4969   {

  4970     fix x y

  4971     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4972     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4973     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4974       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4975         simp add: le)

  4976   }

  4977   then show ?thesis

  4978     using assms unfolding uniformly_continuous_on_sequentially

  4979     unfolding dist_real_def by simp

  4980 qed

  4981

  4982 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  4983   assumes "uniformly_continuous_on s f"

  4984   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4985   unfolding norm_conv_dist using assms

  4986   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4987

  4988 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  4989   assumes "uniformly_continuous_on s g"

  4990   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4991   using assms unfolding uniformly_continuous_on_sequentially

  4992   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4993   by (auto intro: tendsto_zero)

  4994

  4995 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  4996   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4997   assumes "uniformly_continuous_on s f"

  4998   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4999   using bounded_linear_scaleR_right assms

  5000   by (rule bounded_linear.uniformly_continuous_on)

  5001

  5002 lemma dist_minus:

  5003   fixes x y :: "'a::real_normed_vector"

  5004   shows "dist (- x) (- y) = dist x y"

  5005   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  5006

  5007 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  5008   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5009   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  5010   unfolding uniformly_continuous_on_def dist_minus .

  5011

  5012 lemma uniformly_continuous_on_add[continuous_on_intros]:

  5013   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5014   assumes "uniformly_continuous_on s f"

  5015     and "uniformly_continuous_on s g"

  5016   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  5017   using assms

  5018   unfolding uniformly_continuous_on_sequentially

  5019   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  5020   by (auto intro: tendsto_add_zero)

  5021

  5022 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  5023   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5024   assumes "uniformly_continuous_on s f"

  5025     and "uniformly_continuous_on s g"

  5026   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  5027   using assms uniformly_continuous_on_add [of s f "- g"]

  5028     by (simp add: fun_Compl_def uniformly_continuous_on_minus)

  5029

  5030 text{* Continuity of all kinds is preserved under composition. *}

  5031

  5032 lemmas continuous_at_compose = isCont_o

  5033

  5034 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  5035   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  5036   shows "uniformly_continuous_on s (g \<circ> f)"

  5037 proof -

  5038   {

  5039     fix e :: real

  5040     assume "e > 0"

  5041     then obtain d where "d > 0"

  5042       and d: "\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5043       using assms(2) unfolding uniformly_continuous_on_def by auto

  5044     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"

  5045       using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto

  5046     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"

  5047       using d>0 using d by auto

  5048   }

  5049   then show ?thesis

  5050     using assms unfolding uniformly_continuous_on_def by auto

  5051 qed

  5052

  5053 text{* Continuity in terms of open preimages. *}

  5054

  5055 lemma continuous_at_open:

  5056   "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)))"

  5057   unfolding continuous_within_topological [of x UNIV f]

  5058   unfolding imp_conjL

  5059   by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  5060

  5061 lemma continuous_imp_tendsto:

  5062   assumes "continuous (at x0) f"

  5063     and "x ----> x0"

  5064   shows "(f \<circ> x) ----> (f x0)"

  5065 proof (rule topological_tendstoI)

  5066   fix S

  5067   assume "open S" "f x0 \<in> S"

  5068   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  5069      using assms continuous_at_open by metis

  5070   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  5071     using assms T_def by (auto simp: tendsto_def)

  5072   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  5073     using T_def by (auto elim!: eventually_elim1)

  5074 qed

  5075

  5076 lemma continuous_on_open:

  5077   "continuous_on s f \<longleftrightarrow>

  5078     (\<forall>t. openin (subtopology euclidean (f  s)) t \<longrightarrow>

  5079       openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5080   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  5081   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5082

  5083 text {* Similarly in terms of closed sets. *}

  5084

  5085 lemma continuous_on_closed:

  5086   "continuous_on s f \<longleftrightarrow>

  5087     (\<forall>t. closedin (subtopology euclidean (f  s)) t \<longrightarrow>

  5088       closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  5089   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  5090   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  5091

  5092 text {* Half-global and completely global cases. *}

  5093

  5094 lemma continuous_open_in_preimage:

  5095   assumes "continuous_on s f"  "open t"

  5096   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5097 proof -

  5098   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5099     by auto

  5100   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5101     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  5102   then show ?thesis

  5103     using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]]

  5104     using * by auto

  5105 qed

  5106

  5107 lemma continuous_closed_in_preimage:

  5108   assumes "continuous_on s f" and "closed t"

  5109   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  5110 proof -

  5111   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  5112     by auto

  5113   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  5114     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute

  5115     by auto

  5116   then show ?thesis

  5117     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]]

  5118     using * by auto

  5119 qed

  5120

  5121 lemma continuous_open_preimage:

  5122   assumes "continuous_on s f"

  5123     and "open s"

  5124     and "open t"

  5125   shows "open {x \<in> s. f x \<in> t}"

  5126 proof-

  5127   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5128     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  5129   then show ?thesis

  5130     using open_Int[of s T, OF assms(2)] by auto

  5131 qed

  5132

  5133 lemma continuous_closed_preimage:

  5134   assumes "continuous_on s f"

  5135     and "closed s"

  5136     and "closed t"

  5137   shows "closed {x \<in> s. f x \<in> t}"

  5138 proof-

  5139   obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  5140     using continuous_closed_in_preimage[OF assms(1,3)]

  5141     unfolding closedin_closed by auto

  5142   then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

  5143 qed

  5144

  5145 lemma continuous_open_preimage_univ:

  5146   "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  5147   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  5148

  5149 lemma continuous_closed_preimage_univ:

  5150   "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s \<Longrightarrow> closed {x. f x \<in> s}"

  5151   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  5152

  5153 lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  5154   unfolding vimage_def by (rule continuous_open_preimage_univ)

  5155

  5156 lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  5157   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  5158

  5159 lemma interior_image_subset:

  5160   assumes "\<forall>x. continuous (at x) f"

  5161     and "inj f"

  5162   shows "interior (f  s) \<subseteq> f  (interior s)"

  5163 proof

  5164   fix x assume "x \<in> interior (f  s)"

  5165   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  5166   then have "x \<in> f  s" by auto

  5167   then obtain y where y: "y \<in> s" "x = f y" by auto

  5168   have "open (vimage f T)"

  5169     using assms(1) open T by (rule continuous_open_vimage)

  5170   moreover have "y \<in> vimage f T"

  5171     using x = f y x \<in> T by simp

  5172   moreover have "vimage f T \<subseteq> s"

  5173     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  5174   ultimately have "y \<in> interior s" ..

  5175   with x = f y show "x \<in> f  interior s" ..

  5176 qed

  5177

  5178 text {* Equality of continuous functions on closure and related results. *}

  5179

  5180 lemma continuous_closed_in_preimage_constant:

  5181   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5182   shows "continuous_on s f \<Longrightarrow> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  5183   using continuous_closed_in_preimage[of s f "{a}"] by auto

  5184

  5185 lemma continuous_closed_preimage_constant:

  5186   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5187   shows "continuous_on s f \<Longrightarrow> closed s \<Longrightarrow> closed {x \<in> s. f x = a}"

  5188   using continuous_closed_preimage[of s f "{a}"] by auto

  5189

  5190 lemma continuous_constant_on_closure:

  5191   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5192   assumes "continuous_on (closure s) f"

  5193     and "\<forall>x \<in> s. f x = a"

  5194   shows "\<forall>x \<in> (closure s). f x = a"

  5195     using continuous_closed_preimage_constant[of "closure s" f a]

  5196       assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset

  5197     unfolding subset_eq

  5198     by auto

  5199

  5200 lemma image_closure_subset:

  5201   assumes "continuous_on (closure s) f"

  5202     and "closed t"

  5203     and "(f  s) \<subseteq> t"

  5204   shows "f  (closure s) \<subseteq> t"

  5205 proof -

  5206   have "s \<subseteq> {x \<in> closure s. f x \<in> t}"

  5207     using assms(3) closure_subset by auto

  5208   moreover have "closed {x \<in> closure s. f x \<in> t}"

  5209     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  5210   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  5211     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  5212   then show ?thesis by auto

  5213 qed

  5214

  5215 lemma continuous_on_closure_norm_le:

  5216   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5217   assumes "continuous_on (closure s) f"

  5218     and "\<forall>y \<in> s. norm(f y) \<le> b"

  5219     and "x \<in> (closure s)"

  5220   shows "norm (f x) \<le> b"

  5221 proof -

  5222   have *: "f  s \<subseteq> cball 0 b"

  5223     using assms(2)[unfolded mem_cball_0[symmetric]] by auto

  5224   show ?thesis

  5225     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  5226     unfolding subset_eq

  5227     apply (erule_tac x="f x" in ballE)

  5228     apply (auto simp add: dist_norm)

  5229     done

  5230 qed

  5231

  5232 text {* Making a continuous function avoid some value in a neighbourhood. *}

  5233

  5234 lemma continuous_within_avoid:

  5235   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5236   assumes "continuous (at x within s) f"

  5237     and "f x \<noteq> a"

  5238   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  5239 proof -

  5240   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  5241     using t1_space [OF f x \<noteq> a] by fast

  5242   have "(f ---> f x) (at x within s)"

  5243     using assms(1) by (simp add: continuous_within)

  5244   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  5245     using open U and f x \<in> U

  5246     unfolding tendsto_def by fast

  5247   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  5248     using a \<notin> U by (fast elim: eventually_mono [rotated])

  5249   then show ?thesis

  5250     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  5251 qed

  5252

  5253 lemma continuous_at_avoid:

  5254   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5255   assumes "continuous (at x) f"

  5256     and "f x \<noteq> a"

  5257   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5258   using assms continuous_within_avoid[of x UNIV f a] by simp

  5259

  5260 lemma continuous_on_avoid:

  5261   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5262   assumes "continuous_on s f"

  5263     and "x \<in> s"

  5264     and "f x \<noteq> a"

  5265   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  5266   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  5267     OF assms(2)] continuous_within_avoid[of x s f a]

  5268   using assms(3)

  5269   by auto

  5270

  5271 lemma continuous_on_open_avoid:

  5272   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5273   assumes "continuous_on s f"

  5274     and "open s"

  5275     and "x \<in> s"

  5276     and "f x \<noteq> a"

  5277   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5278   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  5279   using continuous_at_avoid[of x f a] assms(4)

  5280   by auto

  5281

  5282 text {* Proving a function is constant by proving open-ness of level set. *}

  5283

  5284 lemma continuous_levelset_open_in_cases:

  5285   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5286   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5287         openin (subtopology euclidean s) {x \<in> s. f x = a}

  5288         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  5289   unfolding connected_clopen

  5290   using continuous_closed_in_preimage_constant by auto

  5291

  5292 lemma continuous_levelset_open_in:

  5293   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5294   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5295         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  5296         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"

  5297   using continuous_levelset_open_in_cases[of s f ]

  5298   by meson

  5299

  5300 lemma continuous_levelset_open:

  5301   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5302   assumes "connected s"

  5303     and "continuous_on s f"

  5304     and "open {x \<in> s. f x = a}"

  5305     and "\<exists>x \<in> s.  f x = a"

  5306   shows "\<forall>x \<in> s. f x = a"

  5307   using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

  5308   using assms (3,4)

  5309   by fast

  5310

  5311 text {* Some arithmetical combinations (more to prove). *}

  5312

  5313 lemma open_scaling[intro]:

  5314   fixes s :: "'a::real_normed_vector set"

  5315   assumes "c \<noteq> 0"

  5316     and "open s"

  5317   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  5318 proof -

  5319   {

  5320     fix x

  5321     assume "x \<in> s"

  5322     then obtain e where "e>0"

  5323       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  5324       by auto

  5325     have "e * abs c > 0"

  5326       using assms(1)[unfolded zero_less_abs_iff[symmetric]]

  5327       using mult_pos_pos[OF e>0]

  5328       by auto

  5329     moreover

  5330     {

  5331       fix y

  5332       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  5333       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  5334         unfolding dist_norm

  5335         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  5336           assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

  5337       then have "y \<in> op *\<^sub>R c  s"

  5338         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]

  5339         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  5340         using assms(1)

  5341         unfolding dist_norm scaleR_scaleR

  5342         by auto

  5343     }

  5344     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s"

  5345       apply (rule_tac x="e * abs c" in exI)

  5346       apply auto

  5347       done

  5348   }

  5349   then show ?thesis unfolding open_dist by auto

  5350 qed

  5351

  5352 lemma minus_image_eq_vimage:

  5353   fixes A :: "'a::ab_group_add set"

  5354   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  5355   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  5356

  5357 lemma open_negations:

  5358   fixes s :: "'a::real_normed_vector set"

  5359   shows "open s \<Longrightarrow> open ((\<lambda>x. - x)  s)"

  5360   using open_scaling [of "- 1" s] by simp

  5361

  5362 lemma open_translation:

  5363   fixes s :: "'a::real_normed_vector set"

  5364   assumes "open s"

  5365   shows "open((\<lambda>x. a + x)  s)"

  5366 proof -

  5367   {

  5368     fix x

  5369     have "continuous (at x) (\<lambda>x. x - a)"

  5370       by (intro continuous_diff continuous_at_id continuous_const)

  5371   }

  5372   moreover have "{x. x - a \<in> s} = op + a  s"

  5373     by force

  5374   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]

  5375     using assms by auto

  5376 qed

  5377

  5378 lemma open_affinity:

  5379   fixes s :: "'a::real_normed_vector set"

  5380   assumes "open s"  "c \<noteq> 0"

  5381   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5382 proof -

  5383   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  5384     unfolding o_def ..

  5385   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s"

  5386     by auto

  5387   then show ?thesis

  5388     using assms open_translation[of "op *\<^sub>R c  s" a]

  5389     unfolding *

  5390     by auto

  5391 qed

  5392

  5393 lemma interior_translation:

  5394   fixes s :: "'a::real_normed_vector set"

  5395   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  5396 proof (rule set_eqI, rule)

  5397   fix x

  5398   assume "x \<in> interior (op + a  s)"

  5399   then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a  s"

  5400     unfolding mem_interior by auto

  5401   then have "ball (x - a) e \<subseteq> s"

  5402     unfolding subset_eq Ball_def mem_ball dist_norm

  5403     apply auto

  5404     apply (erule_tac x="a + xa" in allE)

  5405     unfolding ab_group_add_class.diff_diff_eq[symmetric]

  5406     apply auto

  5407     done

  5408   then show "x \<in> op + a  interior s"

  5409     unfolding image_iff

  5410     apply (rule_tac x="x - a" in bexI)

  5411     unfolding mem_interior

  5412     using e > 0

  5413     apply auto

  5414     done

  5415 next

  5416   fix x

  5417   assume "x \<in> op + a  interior s"

  5418   then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"

  5419     unfolding image_iff Bex_def mem_interior by auto

  5420   {

  5421     fix z

  5422     have *: "a + y - z = y + a - z" by auto

  5423     assume "z \<in> ball x e"

  5424     then have "z - a \<in> s"

  5425       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  5426       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  5427       by auto

  5428     then have "z \<in> op + a  s"

  5429       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  5430   }

  5431   then have "ball x e \<subseteq> op + a  s"

  5432     unfolding subset_eq by auto

  5433   then show "x \<in> interior (op + a  s)"

  5434     unfolding mem_interior using e > 0 by auto

  5435 qed

  5436

  5437 text {* Topological properties of linear functions. *}

  5438

  5439 lemma linear_lim_0:

  5440   assumes "bounded_linear f"

  5441   shows "(f ---> 0) (at (0))"

  5442 proof -

  5443   interpret f: bounded_linear f by fact

  5444   have "(f ---> f 0) (at 0)"

  5445     using tendsto_ident_at by (rule f.tendsto)

  5446   then show ?thesis unfolding f.zero .

  5447 qed

  5448

  5449 lemma linear_continuous_at:

  5450   assumes "bounded_linear f"

  5451   shows "continuous (at a) f"

  5452   unfolding continuous_at using assms

  5453   apply (rule bounded_linear.tendsto)

  5454   apply (rule tendsto_ident_at)

  5455   done

  5456

  5457 lemma linear_continuous_within:

  5458   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"

  5459   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  5460

  5461 lemma linear_continuous_on:

  5462   "bounded_linear f \<Longrightarrow> continuous_on s f"

  5463   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  5464

  5465 text {* Also bilinear functions, in composition form. *}

  5466

  5467 lemma bilinear_continuous_at_compose:

  5468   "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5469     continuous (at x) (\<lambda>x. h (f x) (g x))"

  5470   unfolding continuous_at

  5471   using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

  5472   by auto

  5473

  5474 lemma bilinear_continuous_within_compose:

  5475   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5476     continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  5477   unfolding continuous_within

  5478   using Lim_bilinear[of f "f x"]

  5479   by auto

  5480

  5481 lemma bilinear_continuous_on_compose:

  5482   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5483     continuous_on s (\<lambda>x. h (f x) (g x))"

  5484   unfolding continuous_on_def

  5485   by (fast elim: bounded_bilinear.tendsto)

  5486

  5487 text {* Preservation of compactness and connectedness under continuous function. *}

  5488

  5489 lemma compact_eq_openin_cover:

  5490   "compact S \<longleftrightarrow>

  5491     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5492       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  5493 proof safe

  5494   fix C

  5495   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  5496   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}"

  5497     unfolding openin_open by force+

  5498   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"

  5499     by (rule compactE)

  5500   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)"

  5501     by auto

  5502   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5503 next

  5504   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5505         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  5506   show "compact S"

  5507   proof (rule compactI)

  5508     fix C

  5509     let ?C = "image (\<lambda>T. S \<inter> T) C"

  5510     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  5511     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  5512       unfolding openin_open by auto

  5513     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  5514       by metis

  5515     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  5516     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  5517     proof (intro conjI)

  5518       from D \<subseteq> ?C show "?D \<subseteq> C"

  5519         by (fast intro: inv_into_into)

  5520       from finite D show "finite ?D"

  5521         by (rule finite_imageI)

  5522       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  5523         apply (rule subset_trans)

  5524         apply clarsimp

  5525         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  5526         apply (erule rev_bexI, fast)

  5527         done

  5528     qed

  5529     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5530   qed

  5531 qed

  5532

  5533 lemma connected_continuous_image:

  5534   assumes "continuous_on s f"

  5535     and "connected s"

  5536   shows "connected(f  s)"

  5537 proof -

  5538   {

  5539     fix T

  5540     assume as:

  5541       "T \<noteq> {}"

  5542       "T \<noteq> f  s"

  5543       "openin (subtopology euclidean (f  s)) T"

  5544       "closedin (subtopology euclidean (f  s)) T"

  5545     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  5546       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  5547       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  5548       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  5549     then have False using as(1,2)

  5550       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

  5551   }

  5552   then show ?thesis

  5553     unfolding connected_clopen by auto

  5554 qed

  5555

  5556 text {* Continuity implies uniform continuity on a compact domain. *}

  5557

  5558 lemma compact_uniformly_continuous:

  5559   assumes f: "continuous_on s f"

  5560     and s: "compact s"

  5561   shows "uniformly_continuous_on s f"

  5562   unfolding uniformly_continuous_on_def

  5563 proof (cases, safe)

  5564   fix e :: real

  5565   assume "0 < e" "s \<noteq> {}"

  5566   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) } }"

  5567   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  5568   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  5569   proof safe

  5570     fix y

  5571     assume "y \<in> s"

  5572     from continuous_open_in_preimage[OF f open_ball]

  5573     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  5574       unfolding openin_subtopology open_openin by metis

  5575     then obtain d where "ball y d \<subseteq> T" "0 < d"

  5576       using 0 < e y \<in> s by (auto elim!: openE)

  5577     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  5578       by (intro UN_I[of "(y, d)"]) auto

  5579   qed auto

  5580   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  5581     by (rule compactE_image)

  5582   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  5583     by (subst Min_gr_iff) auto

  5584   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  5585   proof (rule, safe)

  5586     fix x x'

  5587     assume in_s: "x' \<in> s" "x \<in> s"

  5588     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  5589       by blast

  5590     moreover assume "dist x x' < Min (sndD) / 2"

  5591     ultimately have "dist y x' < d"

  5592       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  5593     with D x in_s show  "dist (f x) (f x') < e"

  5594       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  5595   qed (insert D, auto)

  5596 qed auto

  5597

  5598 text {* A uniformly convergent limit of continuous functions is continuous. *}

  5599

  5600 lemma continuous_uniform_limit:

  5601   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  5602   assumes "\<not> trivial_limit F"

  5603     and "eventually (\<lambda>n. continuous_on s (f n)) F"

  5604     and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  5605   shows "continuous_on s g"

  5606 proof -

  5607   {

  5608     fix x and e :: real

  5609     assume "x\<in>s" "e>0"

  5610     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  5611       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  5612     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  5613     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  5614       using assms(1) by blast

  5615     have "e / 3 > 0" using e>0 by auto

  5616     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"

  5617       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  5618     {

  5619       fix y

  5620       assume "y \<in> s" and "dist y x < d"

  5621       then have "dist (f n y) (f n x) < e / 3"

  5622         by (rule d [rule_format])

  5623       then have "dist (f n y) (g x) < 2 * e / 3"

  5624         using dist_triangle [of "f n y" "g x" "f n x"]

  5625         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  5626         by auto

  5627       then have "dist (g y) (g x) < e"

  5628         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  5629         using dist_triangle3 [of "g y" "g x" "f n y"]

  5630         by auto

  5631     }

  5632     then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5633       using d>0 by auto

  5634   }

  5635   then show ?thesis

  5636     unfolding continuous_on_iff by auto

  5637 qed

  5638

  5639

  5640 subsection {* Topological stuff lifted from and dropped to R *}

  5641

  5642 lemma open_real:

  5643   fixes s :: "real set"

  5644   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"

  5645   unfolding open_dist dist_norm by simp

  5646

  5647 lemma islimpt_approachable_real:

  5648   fixes s :: "real set"

  5649   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  5650   unfolding islimpt_approachable dist_norm by simp

  5651

  5652 lemma closed_real:

  5653   fixes s :: "real set"

  5654   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)"

  5655   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5656

  5657 lemma continuous_at_real_range:

  5658   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5659   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5660   unfolding continuous_at

  5661   unfolding Lim_at

  5662   unfolding dist_nz[symmetric]

  5663   unfolding dist_norm

  5664   apply auto

  5665   apply (erule_tac x=e in allE)

  5666   apply auto

  5667   apply (rule_tac x=d in exI)

  5668   apply auto

  5669   apply (erule_tac x=x' in allE)

  5670   apply auto

  5671   apply (erule_tac x=e in allE)

  5672   apply auto

  5673   done

  5674

  5675 lemma continuous_on_real_range:

  5676   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5677   shows "continuous_on s f \<longleftrightarrow>

  5678     (\<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))"

  5679   unfolding continuous_on_iff dist_norm by simp

  5680

  5681 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5682

  5683 lemma distance_attains_sup:

  5684   assumes "compact s" "s \<noteq> {}"

  5685   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  5686 proof (rule continuous_attains_sup [OF assms])

  5687   {

  5688     fix x

  5689     assume "x\<in>s"

  5690     have "(dist a ---> dist a x) (at x within s)"

  5691       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  5692   }

  5693   then show "continuous_on s (dist a)"

  5694     unfolding continuous_on ..

  5695 qed

  5696

  5697 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5698

  5699 lemma distance_attains_inf:

  5700   fixes a :: "'a::heine_borel"

  5701   assumes "closed s"

  5702     and "s \<noteq> {}"

  5703   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  5704 proof -

  5705   from assms(2) obtain b where "b \<in> s" by auto

  5706   let ?B = "s \<inter> cball a (dist b a)"

  5707   have "?B \<noteq> {}" using b \<in> s

  5708     by (auto simp add: dist_commute)

  5709   moreover have "continuous_on ?B (dist a)"

  5710     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  5711   moreover have "compact ?B"

  5712     by (intro closed_inter_compact closed s compact_cball)

  5713   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  5714     by (metis continuous_attains_inf)

  5715   then show ?thesis by fastforce

  5716 qed

  5717

  5718

  5719 subsection {* Pasted sets *}

  5720

  5721 lemma bounded_Times:

  5722   assumes "bounded s" "bounded t"

  5723   shows "bounded (s \<times> t)"

  5724 proof -

  5725   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5726     using assms [unfolded bounded_def] by auto

  5727   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  5728     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5729   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5730 qed

  5731

  5732 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5733   by (induct x) simp

  5734

  5735 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5736   unfolding seq_compact_def

  5737   apply clarify

  5738   apply (drule_tac x="fst \<circ> f" in spec)

  5739   apply (drule mp, simp add: mem_Times_iff)

  5740   apply (clarify, rename_tac l1 r1)

  5741   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5742   apply (drule mp, simp add: mem_Times_iff)

  5743   apply (clarify, rename_tac l2 r2)

  5744   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5745   apply (rule_tac x="r1 \<circ> r2" in exI)

  5746   apply (rule conjI, simp add: subseq_def)

  5747   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  5748   apply (drule (1) tendsto_Pair) back

  5749   apply (simp add: o_def)

  5750   done

  5751

  5752 lemma compact_Times:

  5753   assumes "compact s" "compact t"

  5754   shows "compact (s \<times> t)"

  5755 proof (rule compactI)

  5756   fix C

  5757   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  5758   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)"

  5759   proof

  5760     fix x

  5761     assume "x \<in> s"

  5762     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")

  5763     proof

  5764       fix y

  5765       assume "y \<in> t"

  5766       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  5767       then show "?P y" by (auto elim!: open_prod_elim)

  5768     qed

  5769     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  5770       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"

  5771       by metis

  5772     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  5773     from compactE_image[OF compact t this] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  5774       by auto

  5775     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  5776       by (fastforce simp: subset_eq)

  5777     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  5778       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  5779   qed

  5780   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  5781     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"

  5782     unfolding subset_eq UN_iff by metis

  5783   moreover

  5784   from compactE_image[OF compact s a]

  5785   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  5786     by auto

  5787   moreover

  5788   {

  5789     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  5790       by auto

  5791     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  5792       using d e \<subseteq> s by (intro UN_mono) auto

  5793     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  5794   }

  5795   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  5796     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  5797 qed

  5798

  5799 text{* Hence some useful properties follow quite easily. *}

  5800

  5801 lemma compact_scaling:

  5802   fixes s :: "'a::real_normed_vector set"

  5803   assumes "compact s"

  5804   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5805 proof -

  5806   let ?f = "\<lambda>x. scaleR c x"

  5807   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5808   show ?thesis

  5809     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5810     using linear_continuous_at[OF *] assms

  5811     by auto

  5812 qed

  5813

  5814 lemma compact_negations:

  5815   fixes s :: "'a::real_normed_vector set"

  5816   assumes "compact s"

  5817   shows "compact ((\<lambda>x. - x)  s)"

  5818   using compact_scaling [OF assms, of "- 1"] by auto

  5819

  5820 lemma compact_sums:

  5821   fixes s t :: "'a::real_normed_vector set"

  5822   assumes "compact s"

  5823     and "compact t"

  5824   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5825 proof -

  5826   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5827     apply auto

  5828     unfolding image_iff

  5829     apply (rule_tac x="(xa, y)" in bexI)

  5830     apply auto

  5831     done

  5832   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5833     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5834   then show ?thesis

  5835     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5836 qed

  5837

  5838 lemma compact_differences:

  5839   fixes s t :: "'a::real_normed_vector set"

  5840   assumes "compact s"

  5841     and "compact t"

  5842   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5843 proof-

  5844   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5845     apply auto

  5846     apply (rule_tac x= xa in exI)

  5847     apply auto

  5848     done

  5849   then show ?thesis

  5850     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5851 qed

  5852

  5853 lemma compact_translation:

  5854   fixes s :: "'a::real_normed_vector set"

  5855   assumes "compact s"

  5856   shows "compact ((\<lambda>x. a + x)  s)"

  5857 proof -

  5858   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  5859     by auto

  5860   then show ?thesis

  5861     using compact_sums[OF assms compact_sing[of a]] by auto

  5862 qed

  5863

  5864 lemma compact_affinity:

  5865   fixes s :: "'a::real_normed_vector set"

  5866   assumes "compact s"

  5867   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5868 proof -

  5869   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  5870     by auto

  5871   then show ?thesis

  5872     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5873 qed

  5874

  5875 text {* Hence we get the following. *}

  5876

  5877 lemma compact_sup_maxdistance:

  5878   fixes s :: "'a::metric_space set"

  5879   assumes "compact s"

  5880     and "s \<noteq> {}"

  5881   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5882 proof -

  5883   have "compact (s \<times> s)"

  5884     using compact s by (intro compact_Times)

  5885   moreover have "s \<times> s \<noteq> {}"

  5886     using s \<noteq> {} by auto

  5887   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  5888     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  5889   ultimately show ?thesis

  5890     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  5891 qed

  5892

  5893 text {* We can state this in terms of diameter of a set. *}

  5894

  5895 definition diameter :: "'a::metric_space set \<Rightarrow> real" where

  5896   "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"

  5897

  5898 lemma diameter_bounded_bound:

  5899   fixes s :: "'a :: metric_space set"

  5900   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  5901   shows "dist x y \<le> diameter s"

  5902 proof -

  5903   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  5904     unfolding bounded_def by auto

  5905   have "bdd_above (split dist  (s\<times>s))"

  5906   proof (intro bdd_aboveI, safe)

  5907     fix a b

  5908     assume "a \<in> s" "b \<in> s"

  5909     with z[of a] z[of b] dist_triangle[of a b z]

  5910     show "dist a b \<le> 2 * d"

  5911       by (simp add: dist_commute)

  5912   qed

  5913   moreover have "(x,y) \<in> s\<times>s" using s by auto

  5914   ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"

  5915     by (rule cSUP_upper2) simp

  5916   with x \<in> s show ?thesis

  5917     by (auto simp add: diameter_def)

  5918 qed

  5919

  5920 lemma diameter_lower_bounded:

  5921   fixes s :: "'a :: metric_space set"

  5922   assumes s: "bounded s"

  5923     and d: "0 < d" "d < diameter s"

  5924   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  5925 proof (rule ccontr)

  5926   assume contr: "\<not> ?thesis"

  5927   moreover have "s \<noteq> {}"

  5928     using d by (auto simp add: diameter_def)

  5929   ultimately have "diameter s \<le> d"

  5930     by (auto simp: not_less diameter_def intro!: cSUP_least)

  5931   with d < diameter s show False by auto

  5932 qed

  5933

  5934 lemma diameter_bounded:

  5935   assumes "bounded s"

  5936   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  5937     and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  5938   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  5939   by auto

  5940

  5941 lemma diameter_compact_attained:

  5942   assumes "compact s"

  5943     and "s \<noteq> {}"

  5944   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  5945 proof -

  5946   have b: "bounded s" using assms(1)

  5947     by (rule compact_imp_bounded)

  5948   then obtain x y where xys: "x\<in>s" "y\<in>s"

  5949     and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5950     using compact_sup_maxdistance[OF assms] by auto

  5951   then have "diameter s \<le> dist x y"

  5952     unfolding diameter_def

  5953     apply clarsimp

  5954     apply (rule cSUP_least)

  5955     apply fast+

  5956     done

  5957   then show ?thesis

  5958     by (metis b diameter_bounded_bound order_antisym xys)

  5959 qed

  5960

  5961 text {* Related results with closure as the conclusion. *}

  5962

  5963 lemma closed_scaling:

  5964   fixes s :: "'a::real_normed_vector set"

  5965   assumes "closed s"

  5966   shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5967 proof (cases "c = 0")

  5968   case True then show ?thesis

  5969     by (auto simp add: image_constant_conv)

  5970 next

  5971   case False

  5972   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) - s)"

  5973     by (simp add: continuous_closed_vimage)

  5974   also have "(\<lambda>x. inverse c *\<^sub>R x) - s = (\<lambda>x. c *\<^sub>R x)  s"

  5975     using c \<noteq> 0 by (auto elim: image_eqI [rotated])

  5976   finally show ?thesis .

  5977 qed

  5978

  5979 lemma closed_negations:

  5980   fixes s :: "'a::real_normed_vector set"

  5981   assumes "closed s"

  5982   shows "closed ((\<lambda>x. -x)  s)"

  5983   using closed_scaling[OF assms, of "- 1"] by simp

  5984

  5985 lemma compact_closed_sums:

  5986   fixes s :: "'a::real_normed_vector set"

  5987   assumes "compact s" and "closed t"

  5988   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5989 proof -

  5990   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  5991   {

  5992     fix x l

  5993     assume as: "\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  5994     from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> s"  "\<forall>n. snd (f n) \<in> t"

  5995       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  5996     obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  5997       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  5998     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  5999       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

  6000       unfolding o_def

  6001       by auto

  6002     then have "l - l' \<in> t"

  6003       using assms(2)[unfolded closed_sequential_limits,

  6004         THEN spec[where x="\<lambda> n. snd (f (r n))"],

  6005         THEN spec[where x="l - l'"]]

  6006       using f(3)

  6007       by auto

  6008     then have "l \<in> ?S"

  6009       using l' \<in> s

  6010       apply auto

  6011       apply (rule_tac x=l' in exI)

  6012       apply (rule_tac x="l - l'" in exI)

  6013       apply auto

  6014       done

  6015   }

  6016   then show ?thesis

  6017     unfolding closed_sequential_limits by fast

  6018 qed
`