src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author wenzelm Thu Aug 29 19:20:35 2013 +0200 (2013-08-29) changeset 53282 9d6e263fa921 parent 53255 addd7b9b2bff child 53291 f7fa953bd15b 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/Glbs"

    14   "~~/src/HOL/Library/FuncSet"

    15   Linear_Algebra

    16   Norm_Arith

    17 begin

    18

    19 lemma dist_0_norm:

    20   fixes x :: "'a::real_normed_vector"

    21   shows "dist 0 x = norm x"

    22 unfolding dist_norm by simp

    23

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

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

    26

    27 (* LEGACY *)

    28 lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"

    29   by (rule LIMSEQ_subseq_LIMSEQ)

    30

    31 lemmas real_isGlb_unique = isGlb_unique[where 'a=real]

    32

    33 lemma countable_PiE:

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

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

    36

    37 lemma Lim_within_open:

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

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

    40   by (fact tendsto_within_open)

    41

    42 lemma continuous_on_union:

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

    44   by (fact continuous_on_closed_Un)

    45

    46 lemma continuous_on_cases:

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

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

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

    50   by (rule continuous_on_If) auto

    51

    52

    53 subsection {* Topological Basis *}

    54

    55 context topological_space

    56 begin

    57

    58 definition "topological_basis B =

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

    60

    61 lemma topological_basis:

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

    63   unfolding topological_basis_def

    64   apply safe

    65      apply fastforce

    66     apply fastforce

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

    68    apply simp

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

    70   apply auto

    71   done

    72

    73 lemma topological_basis_iff:

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

    75   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'))"

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

    77 proof safe

    78   fix O' and x::'a

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

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

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

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

    83 next

    84   assume H: ?rhs

    85   show "topological_basis B"

    86     using assms unfolding topological_basis_def

    87   proof safe

    88     fix O'::"'a set"

    89     assume "open O'"

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

    91       by (force intro: bchoice simp: Bex_def)

    92     then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"

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

    94   qed

    95 qed

    96

    97 lemma topological_basisI:

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

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

   100   shows "topological_basis B"

   101   using assms by (subst topological_basis_iff) auto

   102

   103 lemma topological_basisE:

   104   fixes O'

   105   assumes "topological_basis B"

   106     and "open O'"

   107     and "x \<in> O'"

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

   109 proof atomize_elim

   110   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"

   111     by (simp add: topological_basis_def)

   112   with topological_basis_iff assms

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

   114     using assms by (simp add: Bex_def)

   115 qed

   116

   117 lemma topological_basis_open:

   118   assumes "topological_basis B"

   119     and "X \<in> B"

   120   shows "open X"

   121   using assms by (simp add: topological_basis_def)

   122

   123 lemma topological_basis_imp_subbasis:

   124   assumes B: "topological_basis B"

   125   shows "open = generate_topology B"

   126 proof (intro ext iffI)

   127   fix S :: "'a set"

   128   assume "open S"

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

   130     unfolding topological_basis_def by blast

   131   then show "generate_topology B S"

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

   133 next

   134   fix S :: "'a set"

   135   assume "generate_topology B S"

   136   then show "open S"

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

   138 qed

   139

   140 lemma basis_dense:

   141   fixes B::"'a set set"

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

   143   assumes "topological_basis B"

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

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

   146 proof (intro allI impI)

   147   fix X::"'a set"

   148   assume "open X" "X \<noteq> {}"

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

   150   guess B' . note B' = this

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

   152     by (auto intro!: choosefrom_basis)

   153 qed

   154

   155 end

   156

   157 lemma topological_basis_prod:

   158   assumes A: "topological_basis A"

   159     and B: "topological_basis B"

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

   161   unfolding topological_basis_def

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

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

   164   assume "open S"

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

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

   167     fix x y

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

   169     from open_prod_elim[OF open S this]

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

   171       by (metis mem_Sigma_iff)

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

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

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

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

   176   qed auto

   177 qed (metis A B topological_basis_open open_Times)

   178

   179

   180 subsection {* Countable Basis *}

   181

   182 locale countable_basis =

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

   184   assumes is_basis: "topological_basis B"

   185     and countable_basis: "countable B"

   186 begin

   187

   188 lemma open_countable_basis_ex:

   189   assumes "open X"

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

   191   using assms countable_basis is_basis

   192   unfolding topological_basis_def by blast

   193

   194 lemma open_countable_basisE:

   195   assumes "open X"

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

   197   using assms open_countable_basis_ex

   198   by (atomize_elim) simp

   199

   200 lemma countable_dense_exists:

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

   202 proof -

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

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

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

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

   207 qed

   208

   209 lemma countable_dense_setE:

   210   obtains D :: "'a set"

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

   212   using countable_dense_exists by blast

   213

   214 end

   215

   216 lemma (in first_countable_topology) first_countable_basisE:

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

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

   219   using first_countable_basis[of x]

   220   apply atomize_elim

   221   apply (elim exE)

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

   223   apply auto

   224   done

   225

   226 lemma (in first_countable_topology) first_countable_basis_Int_stableE:

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

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

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

   230 proof atomize_elim

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

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

   233   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>

   234         (\<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)"

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

   236     show "countable A"

   237       unfolding A_def by (intro countable_image countable_Collect_finite)

   238     fix a

   239     assume "a \<in> A"

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

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

   242   next

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

   244     fix a b

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

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

   247       by (auto simp: A_def)

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

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

   250   next

   251     fix S

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

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

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

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

   256   qed

   257 qed

   258

   259 lemma (in topological_space) first_countableI:

   260   assumes "countable A"

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

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

   263   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))"

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

   265   fix i

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

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

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

   269 next

   270   fix S

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

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

   273     using subset_range_from_nat_into[OF countable A] by auto

   274 qed

   275

   276 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

   277 proof

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

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

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

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

   282     (\<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))"

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

   284     fix a b

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

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

   287       unfolding mem_Times_iff by (auto intro: open_Times)

   288   next

   289     fix S

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

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

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

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

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

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

   296   qed (simp add: A B)

   297 qed

   298

   299 class second_countable_topology = topological_space +

   300   assumes ex_countable_subbasis:

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

   302 begin

   303

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

   305 proof -

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

   307     by blast

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

   309

   310   show ?thesis

   311   proof (intro exI conjI)

   312     show "countable ?B"

   313       by (intro countable_image countable_Collect_finite_subset B)

   314     {

   315       fix S

   316       assume "open S"

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

   318         unfolding B

   319       proof induct

   320         case UNIV

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

   322       next

   323         case (Int a b)

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

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

   326           by blast

   327         show ?case

   328           unfolding x y Int_UN_distrib2

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

   330       next

   331         case (UN K)

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

   333         then guess k unfolding bchoice_iff ..

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

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

   336       next

   337         case (Basis S)

   338         then show ?case

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

   340       qed

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

   342         unfolding subset_image_iff by blast }

   343     then show "topological_basis ?B"

   344       unfolding topological_space_class.topological_basis_def

   345       by (safe intro!: topological_space_class.open_Inter)

   346          (simp_all add: B generate_topology.Basis subset_eq)

   347   qed

   348 qed

   349

   350 end

   351

   352 sublocale second_countable_topology <

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

   354   using someI_ex[OF ex_countable_basis]

   355   by unfold_locales safe

   356

   357 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   358 proof

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

   360     using ex_countable_basis by auto

   361   moreover

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

   363     using ex_countable_basis by auto

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

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

   366       topological_basis_imp_subbasis)

   367 qed

   368

   369 instance second_countable_topology \<subseteq> first_countable_topology

   370 proof

   371   fix x :: 'a

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

   373   then have B: "countable B" "topological_basis B"

   374     using countable_basis is_basis

   375     by (auto simp: countable_basis is_basis)

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

   377     (\<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))"

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

   379        (fastforce simp: topological_space_class.topological_basis_def)+

   380 qed

   381

   382

   383 subsection {* Polish spaces *}

   384

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

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

   387

   388 class polish_space = complete_space + second_countable_topology

   389

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

   391

   392 definition "istopology L \<longleftrightarrow>

   393   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))"

   394

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

   396   morphisms "openin" "topology"

   397   unfolding istopology_def by blast

   398

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

   400   using openin[of U] by blast

   401

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

   403   using topology_inverse[unfolded mem_Collect_eq] .

   404

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

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

   407

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

   409 proof

   410   assume "T1 = T2"

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

   412 next

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

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

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

   416   then show "T1 = T2" unfolding openin_inverse .

   417 qed

   418

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

   420

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

   422

   423 subsubsection {* Main properties of open sets *}

   424

   425 lemma openin_clauses:

   426   fixes U :: "'a topology"

   427   shows

   428     "openin U {}"

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

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

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

   432

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

   434   unfolding topspace_def by blast

   435

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

   437   by (simp add: openin_clauses)

   438

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

   440   using openin_clauses by simp

   441

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

   443   using openin_clauses by simp

   444

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

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

   447

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

   449   by (simp add: openin_Union topspace_def)

   450

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

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

   453 proof

   454   assume ?lhs

   455   then show ?rhs by auto

   456 next

   457   assume H: ?rhs

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

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

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

   461   finally show "openin U S" .

   462 qed

   463

   464

   465 subsubsection {* Closed sets *}

   466

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

   468

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

   470   by (metis closedin_def)

   471

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

   473   by (simp add: closedin_def)

   474

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

   476   by (simp add: closedin_def)

   477

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

   479   by (auto simp add: Diff_Un closedin_def)

   480

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

   482   by auto

   483

   484 lemma closedin_Inter[intro]:

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

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

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

   488   using Ke Kc unfolding closedin_def Diff_Inter by auto

   489

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

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

   492

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

   494   by blast

   495

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

   497   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   498   apply (metis openin_subset subset_eq)

   499   done

   500

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

   502   by (simp add: openin_closedin_eq)

   503

   504 lemma openin_diff[intro]:

   505   assumes oS: "openin U S"

   506     and cT: "closedin U T"

   507   shows "openin U (S - T)"

   508 proof -

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

   510     by (auto simp add: topspace_def openin_subset)

   511   then show ?thesis using oS cT

   512     by (auto simp add: closedin_def)

   513 qed

   514

   515 lemma closedin_diff[intro]:

   516   assumes oS: "closedin U S"

   517     and cT: "openin U T"

   518   shows "closedin U (S - T)"

   519 proof -

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

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

   522   then show ?thesis

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

   524 qed

   525

   526

   527 subsubsection {* Subspace topology *}

   528

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

   530

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

   532   (is "istopology ?L")

   533 proof -

   534   have "?L {}" by blast

   535   {

   536     fix A B

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

   538     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"

   539       by blast

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

   541       using Sa Sb by blast+

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

   543   }

   544   moreover

   545   {

   546     fix K

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

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

   549       apply (rule set_eqI)

   550       apply (simp add: Ball_def image_iff)

   551       apply metis

   552       done

   553     from K[unfolded th0 subset_image_iff]

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

   555       by blast

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

   557       using Sk by auto

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

   559       using Sk by (auto simp add: subset_eq)

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

   561   }

   562   ultimately show ?thesis

   563     unfolding subset_eq mem_Collect_eq istopology_def by blast

   564 qed

   565

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

   567   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   568   by auto

   569

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

   571   by (auto simp add: topspace_def openin_subtopology)

   572

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

   574   unfolding closedin_def topspace_subtopology

   575   apply (simp add: openin_subtopology)

   576   apply (rule iffI)

   577   apply clarify

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

   579   apply auto

   580   done

   581

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

   583   unfolding openin_subtopology

   584   apply (rule iffI, clarify)

   585   apply (frule openin_subset[of U])

   586   apply blast

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

   588   apply auto

   589   done

   590

   591 lemma subtopology_superset:

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

   593   shows "subtopology U V = U"

   594 proof -

   595   {

   596     fix S

   597     {

   598       fix T

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

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

   601         by blast

   602       have "openin U S"

   603         unfolding eq using T by blast

   604     }

   605     moreover

   606     {

   607       assume S: "openin U S"

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

   609         using openin_subset[OF S] UV by auto

   610     }

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

   612       by blast

   613   }

   614   then show ?thesis

   615     unfolding topology_eq openin_subtopology by blast

   616 qed

   617

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

   619   by (simp add: subtopology_superset)

   620

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

   622   by (simp add: subtopology_superset)

   623

   624

   625 subsubsection {* The standard Euclidean topology *}

   626

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

   628   where "euclidean = topology open"

   629

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

   631   unfolding euclidean_def

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

   633   apply (rule topology_inverse[symmetric])

   634   apply (auto simp add: istopology_def)

   635   done

   636

   637 lemma topspace_euclidean: "topspace euclidean = UNIV"

   638   apply (simp add: topspace_def)

   639   apply (rule set_eqI)

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

   641   done

   642

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

   644   by (simp add: topspace_euclidean topspace_subtopology)

   645

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

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

   648

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

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

   651

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

   653

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

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

   656

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

   658   by (auto simp add: openin_open)

   659

   660 lemma open_openin_trans[trans]:

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

   662   by (metis Int_absorb1  openin_open_Int)

   663

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

   665   by (auto simp add: openin_open)

   666

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

   668   by (simp add: closedin_subtopology closed_closedin Int_ac)

   669

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

   671   by (metis closedin_closed)

   672

   673 lemma closed_closedin_trans:

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

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

   676   apply auto

   677   apply (frule closedin_closed_Int[of T S])

   678   apply simp

   679   done

   680

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

   682   by (auto simp add: closedin_closed)

   683

   684 lemma openin_euclidean_subtopology_iff:

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

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

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

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

   689 proof

   690   assume ?lhs

   691   then show ?rhs

   692     unfolding openin_open open_dist by blast

   693 next

   694   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}"

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

   696     unfolding T_def

   697     apply clarsimp

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

   699     apply (clarsimp simp add: less_diff_eq)

   700     apply (erule rev_bexI)

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

   702     apply (erule le_less_trans [OF dist_triangle])

   703     done

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

   705     unfolding T_def

   706     apply auto

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

   708     apply auto

   709     done

   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   unfolding open_dist ball_def mem_Collect_eq Ball_def

   793   unfolding dist_commute

   794   apply clarify

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

   796   using dist_triangle_alt[where z=x]

   797   apply (clarsimp simp add: diff_less_iff)

   798   apply atomize

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

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

   801   apply arith

   802   done

   803

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

   805   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   806

   807 lemma openE[elim?]:

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

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

   810   using assms unfolding open_contains_ball by auto

   811

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

   813   by (metis open_contains_ball subset_eq centre_in_ball)

   814

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

   816   unfolding mem_ball set_eq_iff

   817   apply (simp add: not_less)

   818   apply (metis zero_le_dist order_trans dist_self)

   819   done

   820

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

   822

   823 lemma euclidean_dist_l2:

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

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

   826   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   828

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

   830

   831 lemma rational_boxes:

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

   833   assumes "0 < e"

   834   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"

   835 proof -

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

   837   then have e: "0 < e'"

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

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

   840   proof

   841     fix i

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

   843     show "?th i" by auto

   844   qed

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

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

   847   proof

   848     fix i

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

   850     show "?th i" by auto

   851   qed

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

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

   854   show ?thesis

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

   856     fix y :: 'a

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

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

   859       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   861     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   862       fix i :: "'a"

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

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

   865         using * i by (auto simp: box_def)

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

   867         using a by auto

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

   869         using b by auto

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

   871         by auto

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

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

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

   875         by (rule power_strict_mono) auto

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

   877         by (simp add: power_divide)

   878     qed auto

   879     also have "\<dots> = e"

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

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

   882       by (auto simp: ball_def)

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

   884 qed

   885

   886 lemma open_UNION_box:

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

   888   assumes "open M"

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

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

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

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

   893 proof -

   894   {

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

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

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

   898     moreover obtain a b where ab:

   899       "x \<in> box a b"

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

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

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

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

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

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

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

   907   }

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

   909 qed

   910

   911

   912 subsection{* Connectedness *}

   913

   914 lemma connected_local:

   915  "connected S \<longleftrightarrow>

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

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

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

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

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

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

   922       e2 \<noteq> {})"

   923   unfolding connected_def openin_open

   924   apply safe

   925   apply blast+

   926   done

   927

   928 lemma exists_diff:

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

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

   931 proof -

   932   {

   933     assume "?lhs"

   934     then have ?rhs by blast

   935   }

   936   moreover

   937   {

   938     fix S

   939     assume H: "P S"

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

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

   942   }

   943   ultimately show ?thesis by metis

   944 qed

   945

   946 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   949 proof -

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

   951     (\<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> {})"

   952     unfolding connected_def openin_open closedin_closed

   953     apply (subst exists_diff)

   954     apply blast

   955     done

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

   957     \<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> {})"

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

   959     apply (simp add: closed_def)

   960     apply metis

   961     done

   962   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'))"

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

   964     unfolding connected_def openin_open closedin_closed by auto

   965   {

   966     fix e2

   967     {

   968       fix e1

   969       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)"

   970         by auto

   971     }

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

   973       by metis

   974   }

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

   976     by blast

   977   then show ?thesis

   978     unfolding th0 th1 by simp

   979 qed

   980

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

   982   by simp

   983

   984

   985 subsection{* Limit points *}

   986

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

   988   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))"

   989

   990 lemma islimptI:

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

   992   shows "x islimpt S"

   993   using assms unfolding islimpt_def by auto

   994

   995 lemma islimptE:

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

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

   998   using assms unfolding islimpt_def by auto

   999

  1000 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"

  1001   unfolding islimpt_def eventually_at_topological by auto

  1002

  1003 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"

  1004   unfolding islimpt_def by fast

  1005

  1006 lemma islimpt_approachable:

  1007   fixes x :: "'a::metric_space"

  1008   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  1009   unfolding islimpt_iff_eventually eventually_at by fast

  1010

  1011 lemma islimpt_approachable_le:

  1012   fixes x :: "'a::metric_space"

  1013   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"

  1014   unfolding islimpt_approachable

  1015   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

  1016     THEN arg_cong [where f=Not]]

  1017   by (simp add: Bex_def conj_commute conj_left_commute)

  1018

  1019 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  1020   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

  1021

  1022 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

  1023   unfolding islimpt_def by blast

  1024

  1025 text {* A perfect space has no isolated points. *}

  1026

  1027 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  1028   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

  1029

  1030 lemma perfect_choose_dist:

  1031   fixes x :: "'a::{perfect_space, metric_space}"

  1032   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

  1033   using islimpt_UNIV [of x]

  1034   by (simp add: islimpt_approachable)

  1035

  1036 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  1037   unfolding closed_def

  1038   apply (subst open_subopen)

  1039   apply (simp add: islimpt_def subset_eq)

  1040   apply (metis ComplE ComplI)

  1041   done

  1042

  1043 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  1044   unfolding islimpt_def by auto

  1045

  1046 lemma finite_set_avoid:

  1047   fixes a :: "'a::metric_space"

  1048   assumes fS: "finite S"

  1049   shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"

  1050 proof (induct rule: finite_induct[OF fS])

  1051   case 1

  1052   then show ?case by (auto intro: zero_less_one)

  1053 next

  1054   case (2 x F)

  1055   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"

  1056     by blast

  1057   show ?case

  1058   proof (cases "x = a")

  1059     case True

  1060     then show ?thesis using d by auto

  1061   next

  1062     case False

  1063     let ?d = "min d (dist a x)"

  1064     have dp: "?d > 0"

  1065       using False d(1) using dist_nz by auto

  1066     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"

  1067       by auto

  1068     with dp False show ?thesis

  1069       by (auto intro!: exI[where x="?d"])

  1070   qed

  1071 qed

  1072

  1073 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  1074   by (simp add: islimpt_iff_eventually eventually_conj_iff)

  1075

  1076 lemma discrete_imp_closed:

  1077   fixes S :: "'a::metric_space set"

  1078   assumes e: "0 < e"

  1079     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  1080   shows "closed S"

  1081 proof -

  1082   {

  1083     fix x

  1084     assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

  1085     from e have e2: "e/2 > 0" by arith

  1086     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"

  1087       by blast

  1088     let ?m = "min (e/2) (dist x y) "

  1089     from e2 y(2) have mp: "?m > 0"

  1090       by (simp add: dist_nz[THEN sym])

  1091     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"

  1092       by blast

  1093     have th: "dist z y < e" using z y

  1094       by (intro dist_triangle_lt [where z=x], simp)

  1095     from d[rule_format, OF y(1) z(1) th] y z

  1096     have False by (auto simp add: dist_commute)}

  1097   then show ?thesis

  1098     by (metis islimpt_approachable closed_limpt [where 'a='a])

  1099 qed

  1100

  1101

  1102 subsection {* Interior of a Set *}

  1103

  1104 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

  1105

  1106 lemma interiorI [intro?]:

  1107   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  1108   shows "x \<in> interior S"

  1109   using assms unfolding interior_def by fast

  1110

  1111 lemma interiorE [elim?]:

  1112   assumes "x \<in> interior S"

  1113   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  1114   using assms unfolding interior_def by fast

  1115

  1116 lemma open_interior [simp, intro]: "open (interior S)"

  1117   by (simp add: interior_def open_Union)

  1118

  1119 lemma interior_subset: "interior S \<subseteq> S"

  1120   by (auto simp add: interior_def)

  1121

  1122 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  1123   by (auto simp add: interior_def)

  1124

  1125 lemma interior_open: "open S \<Longrightarrow> interior S = S"

  1126   by (intro equalityI interior_subset interior_maximal subset_refl)

  1127

  1128 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  1129   by (metis open_interior interior_open)

  1130

  1131 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  1132   by (metis interior_maximal interior_subset subset_trans)

  1133

  1134 lemma interior_empty [simp]: "interior {} = {}"

  1135   using open_empty by (rule interior_open)

  1136

  1137 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  1138   using open_UNIV by (rule interior_open)

  1139

  1140 lemma interior_interior [simp]: "interior (interior S) = interior S"

  1141   using open_interior by (rule interior_open)

  1142

  1143 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  1144   by (auto simp add: interior_def)

  1145

  1146 lemma interior_unique:

  1147   assumes "T \<subseteq> S" and "open T"

  1148   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

  1149   shows "interior S = T"

  1150   by (intro equalityI assms interior_subset open_interior interior_maximal)

  1151

  1152 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

  1153   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1154     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1155

  1156 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1157   using open_contains_ball_eq [where S="interior S"]

  1158   by (simp add: open_subset_interior)

  1159

  1160 lemma interior_limit_point [intro]:

  1161   fixes x :: "'a::perfect_space"

  1162   assumes x: "x \<in> interior S"

  1163   shows "x islimpt S"

  1164   using x islimpt_UNIV [of x]

  1165   unfolding interior_def islimpt_def

  1166   apply (clarsimp, rename_tac T T')

  1167   apply (drule_tac x="T \<inter> T'" in spec)

  1168   apply (auto simp add: open_Int)

  1169   done

  1170

  1171 lemma interior_closed_Un_empty_interior:

  1172   assumes cS: "closed S"

  1173     and iT: "interior T = {}"

  1174   shows "interior (S \<union> T) = interior S"

  1175 proof

  1176   show "interior S \<subseteq> interior (S \<union> T)"

  1177     by (rule interior_mono) (rule Un_upper1)

  1178   show "interior (S \<union> T) \<subseteq> interior S"

  1179   proof

  1180     fix x

  1181     assume "x \<in> interior (S \<union> T)"

  1182     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1183     show "x \<in> interior S"

  1184     proof (rule ccontr)

  1185       assume "x \<notin> interior S"

  1186       with x \<in> R open R obtain y where "y \<in> R - S"

  1187         unfolding interior_def by fast

  1188       from open R closed S have "open (R - S)"

  1189         by (rule open_Diff)

  1190       from R \<subseteq> S \<union> T have "R - S \<subseteq> T"

  1191         by fast

  1192       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {} show False

  1193         unfolding interior_def by fast

  1194     qed

  1195   qed

  1196 qed

  1197

  1198 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1199 proof (rule interior_unique)

  1200   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1201     by (intro Sigma_mono interior_subset)

  1202   show "open (interior A \<times> interior B)"

  1203     by (intro open_Times open_interior)

  1204   fix T

  1205   assume "T \<subseteq> A \<times> B" and "open T"

  1206   then show "T \<subseteq> interior A \<times> interior B"

  1207   proof safe

  1208     fix x y

  1209     assume "(x, y) \<in> T"

  1210     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1211       using open T unfolding open_prod_def by fast

  1212     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1213       using T \<subseteq> A \<times> B by auto

  1214     then show "x \<in> interior A" and "y \<in> interior B"

  1215       by (auto intro: interiorI)

  1216   qed

  1217 qed

  1218

  1219

  1220 subsection {* Closure of a Set *}

  1221

  1222 definition "closure S = S \<union> {x | x. x islimpt S}"

  1223

  1224 lemma interior_closure: "interior S = - (closure (- S))"

  1225   unfolding interior_def closure_def islimpt_def by auto

  1226

  1227 lemma closure_interior: "closure S = - interior (- S)"

  1228   unfolding interior_closure by simp

  1229

  1230 lemma closed_closure[simp, intro]: "closed (closure S)"

  1231   unfolding closure_interior by (simp add: closed_Compl)

  1232

  1233 lemma closure_subset: "S \<subseteq> closure S"

  1234   unfolding closure_def by simp

  1235

  1236 lemma closure_hull: "closure S = closed hull S"

  1237   unfolding hull_def closure_interior interior_def by auto

  1238

  1239 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1240   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1241

  1242 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1243   unfolding closure_eq .

  1244

  1245 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1246   unfolding closure_hull by (rule hull_hull)

  1247

  1248 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1249   unfolding closure_hull by (rule hull_mono)

  1250

  1251 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1252   unfolding closure_hull by (rule hull_minimal)

  1253

  1254 lemma closure_unique:

  1255   assumes "S \<subseteq> T"

  1256     and "closed T"

  1257     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1258   shows "closure S = T"

  1259   using assms unfolding closure_hull by (rule hull_unique)

  1260

  1261 lemma closure_empty [simp]: "closure {} = {}"

  1262   using closed_empty by (rule closure_closed)

  1263

  1264 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1265   using closed_UNIV by (rule closure_closed)

  1266

  1267 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1268   unfolding closure_interior by simp

  1269

  1270 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1271   using closure_empty closure_subset[of S]

  1272   by blast

  1273

  1274 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1275   using closure_eq[of S] closure_subset[of S]

  1276   by simp

  1277

  1278 lemma open_inter_closure_eq_empty:

  1279   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1280   using open_subset_interior[of S "- T"]

  1281   using interior_subset[of "- T"]

  1282   unfolding closure_interior

  1283   by auto

  1284

  1285 lemma open_inter_closure_subset:

  1286   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1287 proof

  1288   fix x

  1289   assume as: "open S" "x \<in> S \<inter> closure T"

  1290   {

  1291     assume *: "x islimpt T"

  1292     have "x islimpt (S \<inter> T)"

  1293     proof (rule islimptI)

  1294       fix A

  1295       assume "x \<in> A" "open A"

  1296       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1297         by (simp_all add: open_Int)

  1298       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1299         by (rule islimptE)

  1300       then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1301         by simp_all

  1302       then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1303     qed

  1304   }

  1305   then show "x \<in> closure (S \<inter> T)" using as

  1306     unfolding closure_def

  1307     by blast

  1308 qed

  1309

  1310 lemma closure_complement: "closure (- S) = - interior S"

  1311   unfolding closure_interior by simp

  1312

  1313 lemma interior_complement: "interior (- S) = - closure S"

  1314   unfolding closure_interior by simp

  1315

  1316 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1317 proof (rule closure_unique)

  1318   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1319     by (intro Sigma_mono closure_subset)

  1320   show "closed (closure A \<times> closure B)"

  1321     by (intro closed_Times closed_closure)

  1322   fix T

  1323   assume "A \<times> B \<subseteq> T" and "closed T"

  1324   then show "closure A \<times> closure B \<subseteq> T"

  1325     apply (simp add: closed_def open_prod_def, clarify)

  1326     apply (rule ccontr)

  1327     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1328     apply (simp add: closure_interior interior_def)

  1329     apply (drule_tac x=C in spec)

  1330     apply (drule_tac x=D in spec)

  1331     apply auto

  1332     done

  1333 qed

  1334

  1335 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1336   unfolding closure_def using islimpt_punctured by blast

  1337

  1338

  1339 subsection {* Frontier (aka boundary) *}

  1340

  1341 definition "frontier S = closure S - interior S"

  1342

  1343 lemma frontier_closed: "closed (frontier S)"

  1344   by (simp add: frontier_def closed_Diff)

  1345

  1346 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1347   by (auto simp add: frontier_def interior_closure)

  1348

  1349 lemma frontier_straddle:

  1350   fixes a :: "'a::metric_space"

  1351   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))"

  1352   unfolding frontier_def closure_interior

  1353   by (auto simp add: mem_interior subset_eq ball_def)

  1354

  1355 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1356   by (metis frontier_def closure_closed Diff_subset)

  1357

  1358 lemma frontier_empty[simp]: "frontier {} = {}"

  1359   by (simp add: frontier_def)

  1360

  1361 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1362 proof-

  1363   {

  1364     assume "frontier S \<subseteq> S"

  1365     then have "closure S \<subseteq> S"

  1366       using interior_subset unfolding frontier_def by auto

  1367     then have "closed S"

  1368       using closure_subset_eq by auto

  1369   }

  1370   then show ?thesis using frontier_subset_closed[of S] ..

  1371 qed

  1372

  1373 lemma frontier_complement: "frontier(- S) = frontier S"

  1374   by (auto simp add: frontier_def closure_complement interior_complement)

  1375

  1376 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1377   using frontier_complement frontier_subset_eq[of "- S"]

  1378   unfolding open_closed by auto

  1379

  1380 subsection {* Filters and the eventually true'' quantifier *}

  1381

  1382 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1383     (infixr "indirection" 70)

  1384   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1385

  1386 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1387

  1388 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1389 proof

  1390   assume "trivial_limit (at a within S)"

  1391   then show "\<not> a islimpt S"

  1392     unfolding trivial_limit_def

  1393     unfolding eventually_at_topological

  1394     unfolding islimpt_def

  1395     apply (clarsimp simp add: set_eq_iff)

  1396     apply (rename_tac T, rule_tac x=T in exI)

  1397     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1398     done

  1399 next

  1400   assume "\<not> a islimpt S"

  1401   then show "trivial_limit (at a within S)"

  1402     unfolding trivial_limit_def

  1403     unfolding eventually_at_topological

  1404     unfolding islimpt_def

  1405     apply clarsimp

  1406     apply (rule_tac x=T in exI)

  1407     apply auto

  1408     done

  1409 qed

  1410

  1411 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1412   using trivial_limit_within [of a UNIV] by simp

  1413

  1414 lemma trivial_limit_at:

  1415   fixes a :: "'a::perfect_space"

  1416   shows "\<not> trivial_limit (at a)"

  1417   by (rule at_neq_bot)

  1418

  1419 lemma trivial_limit_at_infinity:

  1420   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1421   unfolding trivial_limit_def eventually_at_infinity

  1422   apply clarsimp

  1423   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1424    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1425   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1426   apply (drule_tac x=UNIV in spec, simp)

  1427   done

  1428

  1429 lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"

  1430   using islimpt_in_closure by (metis trivial_limit_within)

  1431

  1432 text {* Some property holds "sufficiently close" to the limit point. *}

  1433

  1434 lemma eventually_at2:

  1435   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1436   unfolding eventually_at dist_nz by auto

  1437

  1438 lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"

  1439   unfolding trivial_limit_def

  1440   by (auto elim: eventually_rev_mp)

  1441

  1442 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1443   by simp

  1444

  1445 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1446   by (simp add: filter_eq_iff)

  1447

  1448 text{* Combining theorems for "eventually" *}

  1449

  1450 lemma eventually_rev_mono:

  1451   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1452   using eventually_mono [of P Q] by fast

  1453

  1454 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"

  1455   by (simp add: eventually_False)

  1456

  1457

  1458 subsection {* Limits *}

  1459

  1460 lemma Lim:

  1461   "(f ---> l) net \<longleftrightarrow>

  1462         trivial_limit net \<or>

  1463         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1464   unfolding tendsto_iff trivial_limit_eq by auto

  1465

  1466 text{* Show that they yield usual definitions in the various cases. *}

  1467

  1468 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1469            (\<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)"

  1470   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1471

  1472 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1473         (\<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)"

  1474   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1475

  1476 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1477         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1478   by (auto simp add: tendsto_iff eventually_at2)

  1479

  1480 lemma Lim_at_infinity:

  1481   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1482   by (auto simp add: tendsto_iff eventually_at_infinity)

  1483

  1484 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1485   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1486

  1487 text{* The expected monotonicity property. *}

  1488

  1489 lemma Lim_within_empty: "(f ---> l) (at x within {})"

  1490   unfolding tendsto_def eventually_at_filter by simp

  1491

  1492 lemma Lim_Un:

  1493   assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1494   shows "(f ---> l) (at x within (S \<union> T))"

  1495   using assms unfolding tendsto_def eventually_at_filter

  1496   apply clarify

  1497   apply (drule spec, drule (1) mp, drule (1) mp)

  1498   apply (drule spec, drule (1) mp, drule (1) mp)

  1499   apply (auto elim: eventually_elim2)

  1500   done

  1501

  1502 lemma Lim_Un_univ:

  1503   "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>

  1504     S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"

  1505   by (metis Lim_Un)

  1506

  1507 text{* Interrelations between restricted and unrestricted limits. *}

  1508

  1509 lemma Lim_at_within: (* FIXME: rename *)

  1510   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1511   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1512

  1513 lemma eventually_within_interior:

  1514   assumes "x \<in> interior S"

  1515   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"

  1516   (is "?lhs = ?rhs")

  1517 proof

  1518   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1519   {

  1520     assume "?lhs"

  1521     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1522       unfolding eventually_at_topological

  1523       by auto

  1524     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1525       by auto

  1526     then show "?rhs"

  1527       unfolding eventually_at_topological by auto

  1528   next

  1529     assume "?rhs"

  1530     then show "?lhs"

  1531       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1532   }

  1533 qed

  1534

  1535 lemma at_within_interior:

  1536   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1537   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1538

  1539 lemma Lim_within_LIMSEQ:

  1540   fixes a :: "'a::metric_space"

  1541   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1542   shows "(X ---> L) (at a within T)"

  1543   using assms unfolding tendsto_def [where l=L]

  1544   by (simp add: sequentially_imp_eventually_within)

  1545

  1546 lemma Lim_right_bound:

  1547   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1548     'b::{linorder_topology, conditionally_complete_linorder}"

  1549   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"

  1550     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1551   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1552 proof cases

  1553   assume "{x<..} \<inter> I = {}"

  1554   then show ?thesis by (simp add: Lim_within_empty)

  1555 next

  1556   assume e: "{x<..} \<inter> I \<noteq> {}"

  1557   show ?thesis

  1558   proof (rule order_tendstoI)

  1559     fix a

  1560     assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1561     {

  1562       fix y

  1563       assume "y \<in> {x<..} \<inter> I"

  1564       with e bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1565         by (auto intro: cInf_lower)

  1566       with a have "a < f y"

  1567         by (blast intro: less_le_trans)

  1568     }

  1569     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1570       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1571   next

  1572     fix a

  1573     assume "Inf (f  ({x<..} \<inter> I)) < a"

  1574     from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a"

  1575       by auto

  1576     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1577       unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

  1578     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"

  1579       unfolding eventually_at_filter by eventually_elim simp

  1580   qed

  1581 qed

  1582

  1583 text{* Another limit point characterization. *}

  1584

  1585 lemma islimpt_sequential:

  1586   fixes x :: "'a::first_countable_topology"

  1587   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1588     (is "?lhs = ?rhs")

  1589 proof

  1590   assume ?lhs

  1591   from countable_basis_at_decseq[of x] guess A . note A = this

  1592   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1593   {

  1594     fix n

  1595     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1596       unfolding islimpt_def using A(1,2)[of n] by auto

  1597     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1598       unfolding f_def by (rule someI_ex)

  1599     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto

  1600   }

  1601   then have "\<forall>n. f n \<in> S - {x}" by auto

  1602   moreover have "(\<lambda>n. f n) ----> x"

  1603   proof (rule topological_tendstoI)

  1604     fix S

  1605     assume "open S" "x \<in> S"

  1606     from A(3)[OF this] \<And>n. f n \<in> A n

  1607     show "eventually (\<lambda>x. f x \<in> S) sequentially"

  1608       by (auto elim!: eventually_elim1)

  1609   qed

  1610   ultimately show ?rhs by fast

  1611 next

  1612   assume ?rhs

  1613   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"

  1614     by auto

  1615   show ?lhs

  1616     unfolding islimpt_def

  1617   proof safe

  1618     fix T

  1619     assume "open T" "x \<in> T"

  1620     from lim[THEN topological_tendstoD, OF this] f

  1621     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1622       unfolding eventually_sequentially by auto

  1623   qed

  1624 qed

  1625

  1626 lemma Lim_inv: (* TODO: delete *)

  1627   fixes f :: "'a \<Rightarrow> real"

  1628     and A :: "'a filter"

  1629   assumes "(f ---> l) A"

  1630     and "l \<noteq> 0"

  1631   shows "((inverse o f) ---> inverse l) A"

  1632   unfolding o_def using assms by (rule tendsto_inverse)

  1633

  1634 lemma Lim_null:

  1635   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1636   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1637   by (simp add: Lim dist_norm)

  1638

  1639 lemma Lim_null_comparison:

  1640   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1641   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1642   shows "(f ---> 0) net"

  1643   using assms(2)

  1644 proof (rule metric_tendsto_imp_tendsto)

  1645   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1646     using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

  1647 qed

  1648

  1649 lemma Lim_transform_bound:

  1650   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1651     and g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1652   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"

  1653     and "(g ---> 0) net"

  1654   shows "(f ---> 0) net"

  1655   using assms(1) tendsto_norm_zero [OF assms(2)]

  1656   by (rule Lim_null_comparison)

  1657

  1658 text{* Deducing things about the limit from the elements. *}

  1659

  1660 lemma Lim_in_closed_set:

  1661   assumes "closed S"

  1662     and "eventually (\<lambda>x. f(x) \<in> S) net"

  1663     and "\<not>(trivial_limit net)" "(f ---> l) net"

  1664   shows "l \<in> S"

  1665 proof (rule ccontr)

  1666   assume "l \<notin> S"

  1667   with closed S have "open (- S)" "l \<in> - S"

  1668     by (simp_all add: open_Compl)

  1669   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1670     by (rule topological_tendstoD)

  1671   with assms(2) have "eventually (\<lambda>x. False) net"

  1672     by (rule eventually_elim2) simp

  1673   with assms(3) show "False"

  1674     by (simp add: eventually_False)

  1675 qed

  1676

  1677 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1678

  1679 lemma Lim_dist_ubound:

  1680   assumes "\<not>(trivial_limit net)"

  1681     and "(f ---> l) net"

  1682     and "eventually (\<lambda>x. dist a (f x) <= e) net"

  1683   shows "dist a l <= e"

  1684 proof -

  1685   have "dist a l \<in> {..e}"

  1686   proof (rule Lim_in_closed_set)

  1687     show "closed {..e}"

  1688       by simp

  1689     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net"

  1690       by (simp add: assms)

  1691     show "\<not> trivial_limit net"

  1692       by fact

  1693     show "((\<lambda>x. dist a (f x)) ---> dist a l) net"

  1694       by (intro tendsto_intros assms)

  1695   qed

  1696   then show ?thesis by simp

  1697 qed

  1698

  1699 lemma Lim_norm_ubound:

  1700   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1701   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"

  1702   shows "norm(l) \<le> e"

  1703 proof -

  1704   have "norm l \<in> {..e}"

  1705   proof (rule Lim_in_closed_set)

  1706     show "closed {..e}"

  1707       by simp

  1708     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net"

  1709       by (simp add: assms)

  1710     show "\<not> trivial_limit net"

  1711       by fact

  1712     show "((\<lambda>x. norm (f x)) ---> norm l) net"

  1713       by (intro tendsto_intros assms)

  1714   qed

  1715   then show ?thesis by simp

  1716 qed

  1717

  1718 lemma Lim_norm_lbound:

  1719   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1720   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1721   shows "e \<le> norm l"

  1722 proof -

  1723   have "norm l \<in> {e..}"

  1724   proof (rule Lim_in_closed_set)

  1725     show "closed {e..}"

  1726       by simp

  1727     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net"

  1728       by (simp add: assms)

  1729     show "\<not> trivial_limit net"

  1730       by fact

  1731     show "((\<lambda>x. norm (f x)) ---> norm l) net"

  1732       by (intro tendsto_intros assms)

  1733   qed

  1734   then show ?thesis by simp

  1735 qed

  1736

  1737 text{* Limit under bilinear function *}

  1738

  1739 lemma Lim_bilinear:

  1740   assumes "(f ---> l) net"

  1741     and "(g ---> m) net"

  1742     and "bounded_bilinear h"

  1743   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1744   using bounded_bilinear h (f ---> l) net (g ---> m) net

  1745   by (rule bounded_bilinear.tendsto)

  1746

  1747 text{* These are special for limits out of the same vector space. *}

  1748

  1749 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1750   unfolding id_def by (rule tendsto_ident_at)

  1751

  1752 lemma Lim_at_id: "(id ---> a) (at a)"

  1753   unfolding id_def by (rule tendsto_ident_at)

  1754

  1755 lemma Lim_at_zero:

  1756   fixes a :: "'a::real_normed_vector"

  1757   fixes l :: "'b::topological_space"

  1758   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"

  1759   using LIM_offset_zero LIM_offset_zero_cancel ..

  1760

  1761 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1762

  1763 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  1764   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  1765

  1766 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  1767   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  1768

  1769 lemma netlimit_at:

  1770   fixes a :: "'a::{perfect_space,t2_space}"

  1771   shows "netlimit (at a) = a"

  1772   using netlimit_within [of a UNIV] by simp

  1773

  1774 lemma lim_within_interior:

  1775   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1776   by (metis at_within_interior)

  1777

  1778 lemma netlimit_within_interior:

  1779   fixes x :: "'a::{t2_space,perfect_space}"

  1780   assumes "x \<in> interior S"

  1781   shows "netlimit (at x within S) = x"

  1782   using assms by (metis at_within_interior netlimit_at)

  1783

  1784 text{* Transformation of limit. *}

  1785

  1786 lemma Lim_transform:

  1787   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1788   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1789   shows "(g ---> l) net"

  1790   using tendsto_diff [OF assms(2) assms(1)] by simp

  1791

  1792 lemma Lim_transform_eventually:

  1793   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1794   apply (rule topological_tendstoI)

  1795   apply (drule (2) topological_tendstoD)

  1796   apply (erule (1) eventually_elim2, simp)

  1797   done

  1798

  1799 lemma Lim_transform_within:

  1800   assumes "0 < d"

  1801     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1802     and "(f ---> l) (at x within S)"

  1803   shows "(g ---> l) (at x within S)"

  1804 proof (rule Lim_transform_eventually)

  1805   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1806     using assms(1,2) by (auto simp: dist_nz eventually_at)

  1807   show "(f ---> l) (at x within S)" by fact

  1808 qed

  1809

  1810 lemma Lim_transform_at:

  1811   assumes "0 < d"

  1812     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1813     and "(f ---> l) (at x)"

  1814   shows "(g ---> l) (at x)"

  1815   using _ assms(3)

  1816 proof (rule Lim_transform_eventually)

  1817   show "eventually (\<lambda>x. f x = g x) (at x)"

  1818     unfolding eventually_at2

  1819     using assms(1,2) by auto

  1820 qed

  1821

  1822 text{* Common case assuming being away from some crucial point like 0. *}

  1823

  1824 lemma Lim_transform_away_within:

  1825   fixes a b :: "'a::t1_space"

  1826   assumes "a \<noteq> b"

  1827     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1828     and "(f ---> l) (at a within S)"

  1829   shows "(g ---> l) (at a within S)"

  1830 proof (rule Lim_transform_eventually)

  1831   show "(f ---> l) (at a within S)" by fact

  1832   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1833     unfolding eventually_at_topological

  1834     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1835 qed

  1836

  1837 lemma Lim_transform_away_at:

  1838   fixes a b :: "'a::t1_space"

  1839   assumes ab: "a\<noteq>b"

  1840     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1841     and fl: "(f ---> l) (at a)"

  1842   shows "(g ---> l) (at a)"

  1843   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  1844

  1845 text{* Alternatively, within an open set. *}

  1846

  1847 lemma Lim_transform_within_open:

  1848   assumes "open S" and "a \<in> S"

  1849     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1850     and "(f ---> l) (at a)"

  1851   shows "(g ---> l) (at a)"

  1852 proof (rule Lim_transform_eventually)

  1853   show "eventually (\<lambda>x. f x = g x) (at a)"

  1854     unfolding eventually_at_topological

  1855     using assms(1,2,3) by auto

  1856   show "(f ---> l) (at a)" by fact

  1857 qed

  1858

  1859 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1860

  1861 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1862

  1863 lemma Lim_cong_within(*[cong add]*):

  1864   assumes "a = b"

  1865     and "x = y"

  1866     and "S = T"

  1867     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1868   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1869   unfolding tendsto_def eventually_at_topological

  1870   using assms by simp

  1871

  1872 lemma Lim_cong_at(*[cong add]*):

  1873   assumes "a = b" "x = y"

  1874     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1875   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1876   unfolding tendsto_def eventually_at_topological

  1877   using assms by simp

  1878

  1879 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1880

  1881 lemma closure_sequential:

  1882   fixes l :: "'a::first_countable_topology"

  1883   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1884 proof

  1885   assume "?lhs"

  1886   moreover

  1887   {

  1888     assume "l \<in> S"

  1889     then have "?rhs" using tendsto_const[of l sequentially] by auto

  1890   }

  1891   moreover

  1892   {

  1893     assume "l islimpt S"

  1894     then have "?rhs" unfolding islimpt_sequential by auto

  1895   }

  1896   ultimately show "?rhs"

  1897     unfolding closure_def by auto

  1898 next

  1899   assume "?rhs"

  1900   then show "?lhs" unfolding closure_def islimpt_sequential by auto

  1901 qed

  1902

  1903 lemma closed_sequential_limits:

  1904   fixes S :: "'a::first_countable_topology set"

  1905   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1906   unfolding closed_limpt

  1907   using closure_sequential [where 'a='a] closure_closed [where 'a='a]

  1908     closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1909   by metis

  1910

  1911 lemma closure_approachable:

  1912   fixes S :: "'a::metric_space set"

  1913   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1914   apply (auto simp add: closure_def islimpt_approachable)

  1915   apply (metis dist_self)

  1916   done

  1917

  1918 lemma closed_approachable:

  1919   fixes S :: "'a::metric_space set"

  1920   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1921   by (metis closure_closed closure_approachable)

  1922

  1923 lemma closure_contains_Inf:

  1924   fixes S :: "real set"

  1925   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1926   shows "Inf S \<in> closure S"

  1927 proof -

  1928   have *: "\<forall>x\<in>S. Inf S \<le> x"

  1929     using cInf_lower_EX[of _ S] assms by metis

  1930   {

  1931     fix e :: real

  1932     assume "e > 0"

  1933     then have "Inf S < Inf S + e" by simp

  1934     with assms obtain x where "x \<in> S" "x < Inf S + e"

  1935       by (subst (asm) cInf_less_iff[of _ B]) auto

  1936     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  1937       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  1938   }

  1939   then show ?thesis unfolding closure_approachable by auto

  1940 qed

  1941

  1942 lemma closed_contains_Inf:

  1943   fixes S :: "real set"

  1944   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1945     and "closed S"

  1946   shows "Inf S \<in> S"

  1947   by (metis closure_contains_Inf closure_closed assms)

  1948

  1949

  1950 lemma not_trivial_limit_within_ball:

  1951   "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  1952   (is "?lhs = ?rhs")

  1953 proof -

  1954   {

  1955     assume "?lhs"

  1956     {

  1957       fix e :: real

  1958       assume "e > 0"

  1959       then obtain y where "y:(S-{x}) & dist y x < e"

  1960         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1961         by auto

  1962       then have "y : (S Int ball x e - {x})"

  1963         unfolding ball_def by (simp add: dist_commute)

  1964       then have "S Int ball x e - {x} ~= {}" by blast

  1965     }

  1966     then have "?rhs" by auto

  1967   }

  1968   moreover

  1969   {

  1970     assume "?rhs"

  1971     {

  1972       fix e :: real

  1973       assume "e > 0"

  1974       then obtain y where "y : (S Int ball x e - {x})"

  1975         using ?rhs by blast

  1976       then have "y:(S-{x}) & dist y x < e"

  1977         unfolding ball_def by (simp add: dist_commute)

  1978       then have "EX y:(S-{x}). dist y x < e"

  1979         by auto

  1980     }

  1981     then have "?lhs"

  1982       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1983       by auto

  1984   }

  1985   ultimately show ?thesis by auto

  1986 qed

  1987

  1988

  1989 subsection {* Infimum Distance *}

  1990

  1991 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1992

  1993 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1994   by (simp add: infdist_def)

  1995

  1996 lemma infdist_nonneg: "0 \<le> infdist x A"

  1997   by (auto simp add: infdist_def intro: cInf_greatest)

  1998

  1999 lemma infdist_le:

  2000   assumes "a \<in> A"

  2001     and "d = dist x a"

  2002   shows "infdist x A \<le> d"

  2003   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

  2004

  2005 lemma infdist_zero[simp]:

  2006   assumes "a \<in> A"

  2007   shows "infdist a A = 0"

  2008 proof -

  2009   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  2010   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  2011 qed

  2012

  2013 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  2014 proof cases

  2015   assume "A = {}"

  2016   then show ?thesis by (simp add: infdist_def)

  2017 next

  2018   assume "A \<noteq> {}"

  2019   then obtain a where "a \<in> A" by auto

  2020   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  2021   proof (rule cInf_greatest)

  2022     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  2023       by simp

  2024     fix d

  2025     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  2026     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  2027       by auto

  2028     show "infdist x A \<le> d"

  2029       unfolding infdist_notempty[OF A \<noteq> {}]

  2030     proof (rule cInf_lower2)

  2031       show "dist x a \<in> {dist x a |a. a \<in> A}"

  2032         using a \<in> A by auto

  2033       show "dist x a \<le> d"

  2034         unfolding d by (rule dist_triangle)

  2035       fix d

  2036       assume "d \<in> {dist x a |a. a \<in> A}"

  2037       then obtain a where "a \<in> A" "d = dist x a"

  2038         by auto

  2039       then show "infdist x A \<le> d"

  2040         by (rule infdist_le)

  2041     qed

  2042   qed

  2043   also have "\<dots> = dist x y + infdist y A"

  2044   proof (rule cInf_eq, safe)

  2045     fix a

  2046     assume "a \<in> A"

  2047     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  2048       by (auto intro: infdist_le)

  2049   next

  2050     fix i

  2051     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  2052     then have "i - dist x y \<le> infdist y A"

  2053       unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  2054       by (intro cInf_greatest) (auto simp: field_simps)

  2055     then show "i \<le> dist x y + infdist y A"

  2056       by simp

  2057   qed

  2058   finally show ?thesis by simp

  2059 qed

  2060

  2061 lemma in_closure_iff_infdist_zero:

  2062   assumes "A \<noteq> {}"

  2063   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2064 proof

  2065   assume "x \<in> closure A"

  2066   show "infdist x A = 0"

  2067   proof (rule ccontr)

  2068     assume "infdist x A \<noteq> 0"

  2069     with infdist_nonneg[of x A] have "infdist x A > 0"

  2070       by auto

  2071     then have "ball x (infdist x A) \<inter> closure A = {}"

  2072       apply auto

  2073       apply (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  2074         eucl_less_not_refl euclidean_trans(2) infdist_le)

  2075       done

  2076     then have "x \<notin> closure A"

  2077       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  2078     then show False using x \<in> closure A by simp

  2079   qed

  2080 next

  2081   assume x: "infdist x A = 0"

  2082   then obtain a where "a \<in> A"

  2083     by atomize_elim (metis all_not_in_conv assms)

  2084   show "x \<in> closure A"

  2085     unfolding closure_approachable

  2086     apply safe

  2087   proof (rule ccontr)

  2088     fix e :: real

  2089     assume "e > 0"

  2090     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  2091     then have "infdist x A \<ge> e" using a \<in> A

  2092       unfolding infdist_def

  2093       by (force simp: dist_commute intro: cInf_greatest)

  2094     with x e > 0 show False by auto

  2095   qed

  2096 qed

  2097

  2098 lemma in_closed_iff_infdist_zero:

  2099   assumes "closed A" "A \<noteq> {}"

  2100   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  2101 proof -

  2102   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2103     by (rule in_closure_iff_infdist_zero) fact

  2104   with assms show ?thesis by simp

  2105 qed

  2106

  2107 lemma tendsto_infdist [tendsto_intros]:

  2108   assumes f: "(f ---> l) F"

  2109   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  2110 proof (rule tendstoI)

  2111   fix e ::real

  2112   assume "e > 0"

  2113   from tendstoD[OF f this]

  2114   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  2115   proof (eventually_elim)

  2116     fix x

  2117     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  2118     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  2119       by (simp add: dist_commute dist_real_def)

  2120     also assume "dist (f x) l < e"

  2121     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  2122   qed

  2123 qed

  2124

  2125 text{* Some other lemmas about sequences. *}

  2126

  2127 lemma sequentially_offset:

  2128   assumes "eventually (\<lambda>i. P i) sequentially"

  2129   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  2130   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  2131

  2132 lemma seq_offset:

  2133   assumes "(f ---> l) sequentially"

  2134   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  2135   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  2136

  2137 lemma seq_offset_neg:

  2138   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  2139   apply (rule topological_tendstoI)

  2140   apply (drule (2) topological_tendstoD)

  2141   apply (simp only: eventually_sequentially)

  2142   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  2143   apply metis

  2144   apply arith

  2145   done

  2146

  2147 lemma seq_offset_rev:

  2148   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  2149   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  2150

  2151 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  2152   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  2153

  2154 subsection {* More properties of closed balls *}

  2155

  2156 lemma closed_cball: "closed (cball x e)"

  2157   unfolding cball_def closed_def

  2158   unfolding Collect_neg_eq [symmetric] not_le

  2159   apply (clarsimp simp add: open_dist, rename_tac y)

  2160   apply (rule_tac x="dist x y - e" in exI, clarsimp)

  2161   apply (rename_tac x')

  2162   apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  2163   apply simp

  2164   done

  2165

  2166 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  2167 proof -

  2168   {

  2169     fix x and e::real

  2170     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  2171     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  2172   }

  2173   moreover

  2174   {

  2175     fix x and e::real

  2176     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  2177     then have "\<exists>d>0. ball x d \<subseteq> S"

  2178       unfolding subset_eq

  2179       apply(rule_tac x="e/2" in exI)

  2180       apply auto

  2181       done

  2182   }

  2183   ultimately show ?thesis

  2184     unfolding open_contains_ball by auto

  2185 qed

  2186

  2187 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  2188   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  2189

  2190 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  2191   apply (simp add: interior_def, safe)

  2192   apply (force simp add: open_contains_cball)

  2193   apply (rule_tac x="ball x e" in exI)

  2194   apply (simp add: subset_trans [OF ball_subset_cball])

  2195   done

  2196

  2197 lemma islimpt_ball:

  2198   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  2199   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  2200 proof

  2201   assume "?lhs"

  2202   {

  2203     assume "e \<le> 0"

  2204     then have *:"ball x e = {}"

  2205       using ball_eq_empty[of x e] by auto

  2206     have False using ?lhs

  2207       unfolding * using islimpt_EMPTY[of y] by auto

  2208   }

  2209   then have "e > 0" by (metis not_less)

  2210   moreover

  2211   have "y \<in> cball x e"

  2212     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  2213       ball_subset_cball[of x e] ?lhs

  2214     unfolding closed_limpt by auto

  2215   ultimately show "?rhs" by auto

  2216 next

  2217   assume "?rhs"

  2218   then have "e>0" by auto

  2219   {

  2220     fix d :: real

  2221     assume "d > 0"

  2222     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2223     proof (cases "d \<le> dist x y")

  2224       case True

  2225       then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2226       proof (cases "x = y")

  2227         case True

  2228         then have False

  2229           using d \<le> dist x y d>0 by auto

  2230         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2231           by auto

  2232       next

  2233         case False

  2234         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

  2235           norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2236           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym]

  2237           by auto

  2238         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  2239           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  2240           unfolding scaleR_minus_left scaleR_one

  2241           by (auto simp add: norm_minus_commute)

  2242         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  2243           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  2244           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm]

  2245           by auto

  2246         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs

  2247           by (auto simp add: dist_norm)

  2248         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0

  2249           by auto

  2250         moreover

  2251         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2252           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff

  2253           by (auto simp add: dist_commute)

  2254         moreover

  2255         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

  2256           unfolding dist_norm

  2257           apply simp

  2258           unfolding norm_minus_cancel

  2259           using d > 0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2260           unfolding dist_norm

  2261           apply auto

  2262           done

  2263         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2264           apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

  2265           apply auto

  2266           done

  2267       qed

  2268     next

  2269       case False

  2270       then have "d > dist x y" by auto

  2271       show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

  2272       proof (cases "x = y")

  2273         case True

  2274         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2275           using perfect_choose_dist[of "min e d" y]

  2276           using d > 0 e>0 by auto

  2277         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2278           unfolding x = y

  2279           using z \<noteq> y **

  2280           apply (rule_tac x=z in bexI)

  2281           apply (auto simp add: dist_commute)

  2282           done

  2283       next

  2284         case False

  2285         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2286           using d>0 d > dist x y ?rhs

  2287           apply (rule_tac x=x in bexI)

  2288           apply auto

  2289           done

  2290       qed

  2291     qed

  2292   }

  2293   then show "?lhs"

  2294     unfolding mem_cball islimpt_approachable mem_ball by auto

  2295 qed

  2296

  2297 lemma closure_ball_lemma:

  2298   fixes x y :: "'a::real_normed_vector"

  2299   assumes "x \<noteq> y"

  2300   shows "y islimpt ball x (dist x y)"

  2301 proof (rule islimptI)

  2302   fix T

  2303   assume "y \<in> T" "open T"

  2304   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2305     unfolding open_dist by fast

  2306   (* choose point between x and y, within distance r of y. *)

  2307   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2308   def z \<equiv> "y + scaleR k (x - y)"

  2309   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2310     unfolding z_def by (simp add: algebra_simps)

  2311   have "dist z y < r"

  2312     unfolding z_def k_def using 0 < r

  2313     by (simp add: dist_norm min_def)

  2314   then have "z \<in> T"

  2315     using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2316   have "dist x z < dist x y"

  2317     unfolding z_def2 dist_norm

  2318     apply (simp add: norm_minus_commute)

  2319     apply (simp only: dist_norm [symmetric])

  2320     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2321     apply (rule mult_strict_right_mono)

  2322     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2323     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2324     done

  2325   then have "z \<in> ball x (dist x y)"

  2326     by simp

  2327   have "z \<noteq> y"

  2328     unfolding z_def k_def using x \<noteq> y 0 < r

  2329     by (simp add: min_def)

  2330   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2331     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2332     by fast

  2333 qed

  2334

  2335 lemma closure_ball:

  2336   fixes x :: "'a::real_normed_vector"

  2337   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2338   apply (rule equalityI)

  2339   apply (rule closure_minimal)

  2340   apply (rule ball_subset_cball)

  2341   apply (rule closed_cball)

  2342   apply (rule subsetI, rename_tac y)

  2343   apply (simp add: le_less [where 'a=real])

  2344   apply (erule disjE)

  2345   apply (rule subsetD [OF closure_subset], simp)

  2346   apply (simp add: closure_def)

  2347   apply clarify

  2348   apply (rule closure_ball_lemma)

  2349   apply (simp add: zero_less_dist_iff)

  2350   done

  2351

  2352 (* In a trivial vector space, this fails for e = 0. *)

  2353 lemma interior_cball:

  2354   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2355   shows "interior (cball x e) = ball x e"

  2356 proof (cases "e\<ge>0")

  2357   case False note cs = this

  2358   from cs have "ball x e = {}"

  2359     using ball_empty[of e x] by auto

  2360   moreover

  2361   {

  2362     fix y

  2363     assume "y \<in> cball x e"

  2364     then have False

  2365       unfolding mem_cball using dist_nz[of x y] cs by auto

  2366   }

  2367   then have "cball x e = {}" by auto

  2368   then have "interior (cball x e) = {}"

  2369     using interior_empty by auto

  2370   ultimately show ?thesis by blast

  2371 next

  2372   case True note cs = this

  2373   have "ball x e \<subseteq> cball x e"

  2374     using ball_subset_cball by auto

  2375   moreover

  2376   {

  2377     fix S y

  2378     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2379     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

  2380       unfolding open_dist by blast

  2381     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2382       using perfect_choose_dist [of d] by auto

  2383     have "xa \<in> S"

  2384       using d[THEN spec[where x = xa]]

  2385       using xa by (auto simp add: dist_commute)

  2386     then have xa_cball: "xa \<in> cball x e"

  2387       using as(1) by auto

  2388     then have "y \<in> ball x e"

  2389     proof (cases "x = y")

  2390       case True

  2391       then have "e > 0"

  2392         using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2393         by (auto simp add: dist_commute)

  2394       then show "y \<in> ball x e"

  2395         using x = y  by simp

  2396     next

  2397       case False

  2398       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

  2399         unfolding dist_norm

  2400         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2401       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2402         using d as(1)[unfolded subset_eq] by blast

  2403       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2404       then have **:"d / (2 * norm (y - x)) > 0"

  2405         unfolding zero_less_norm_iff[THEN sym]

  2406         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2407       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

  2408         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2409         by (auto simp add: dist_norm algebra_simps)

  2410       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2411         by (auto simp add: algebra_simps)

  2412       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2413         using ** by auto

  2414       also have "\<dots> = (dist y x) + d/2"

  2415         using ** by (auto simp add: distrib_right dist_norm)

  2416       finally have "e \<ge> dist x y +d/2"

  2417         using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2418       then show "y \<in> ball x e"

  2419         unfolding mem_ball using d>0 by auto

  2420     qed

  2421   }

  2422   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

  2423     by auto

  2424   ultimately show ?thesis

  2425     using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  2426 qed

  2427

  2428 lemma frontier_ball:

  2429   fixes a :: "'a::real_normed_vector"

  2430   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  2431   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2432   apply (simp add: set_eq_iff)

  2433   apply arith

  2434   done

  2435

  2436 lemma frontier_cball:

  2437   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2438   shows "frontier(cball a e) = {x. dist a x = e}"

  2439   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2440   apply (simp add: set_eq_iff)

  2441   apply arith

  2442   done

  2443

  2444 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  2445   apply (simp add: set_eq_iff not_le)

  2446   apply (metis zero_le_dist dist_self order_less_le_trans)

  2447   done

  2448

  2449 lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"

  2450   by (simp add: cball_eq_empty)

  2451

  2452 lemma cball_eq_sing:

  2453   fixes x :: "'a::{metric_space,perfect_space}"

  2454   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2455 proof (rule linorder_cases)

  2456   assume e: "0 < e"

  2457   obtain a where "a \<noteq> x" "dist a x < e"

  2458     using perfect_choose_dist [OF e] by auto

  2459   then have "a \<noteq> x" "dist x a \<le> e"

  2460     by (auto simp add: dist_commute)

  2461   with e show ?thesis by (auto simp add: set_eq_iff)

  2462 qed auto

  2463

  2464 lemma cball_sing:

  2465   fixes x :: "'a::metric_space"

  2466   shows "e = 0 ==> cball x e = {x}"

  2467   by (auto simp add: set_eq_iff)

  2468

  2469

  2470 subsection {* Boundedness *}

  2471

  2472   (* FIXME: This has to be unified with BSEQ!! *)

  2473 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2474   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2475

  2476 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2477   unfolding bounded_def subset_eq by auto

  2478

  2479 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2480   unfolding bounded_def

  2481   apply safe

  2482   apply (rule_tac x="dist a x + e" in exI, clarify)

  2483   apply (drule (1) bspec)

  2484   apply (erule order_trans [OF dist_triangle add_left_mono])

  2485   apply auto

  2486   done

  2487

  2488 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2489   unfolding bounded_any_center [where a=0]

  2490   by (simp add: dist_norm)

  2491

  2492 lemma bounded_realI:

  2493   assumes "\<forall>x\<in>s. abs (x::real) \<le> B"

  2494   shows "bounded s"

  2495   unfolding bounded_def dist_real_def

  2496   apply (rule_tac x=0 in exI)

  2497   using assms

  2498   apply auto

  2499   done

  2500

  2501 lemma bounded_empty [simp]: "bounded {}"

  2502   by (simp add: bounded_def)

  2503

  2504 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2505   by (metis bounded_def subset_eq)

  2506

  2507 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2508   by (metis bounded_subset interior_subset)

  2509

  2510 lemma bounded_closure[intro]:

  2511   assumes "bounded S"

  2512   shows "bounded (closure S)"

  2513 proof -

  2514   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2515     unfolding bounded_def by auto

  2516   {

  2517     fix y

  2518     assume "y \<in> closure S"

  2519     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2520       unfolding closure_sequential by auto

  2521     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2522     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2523       by (rule eventually_mono, simp add: f(1))

  2524     have "dist x y \<le> a"

  2525       apply (rule Lim_dist_ubound [of sequentially f])

  2526       apply (rule trivial_limit_sequentially)

  2527       apply (rule f(2))

  2528       apply fact

  2529       done

  2530   }

  2531   then show ?thesis

  2532     unfolding bounded_def by auto

  2533 qed

  2534

  2535 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2536   apply (simp add: bounded_def)

  2537   apply (rule_tac x=x in exI)

  2538   apply (rule_tac x=e in exI)

  2539   apply auto

  2540   done

  2541

  2542 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2543   by (metis ball_subset_cball bounded_cball bounded_subset)

  2544

  2545 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2546   apply (auto simp add: bounded_def)

  2547   apply (rename_tac x y r s)

  2548   apply (rule_tac x=x in exI)

  2549   apply (rule_tac x="max r (dist x y + s)" in exI)

  2550   apply (rule ballI, rename_tac z, safe)

  2551   apply (drule (1) bspec, simp)

  2552   apply (drule (1) bspec)

  2553   apply (rule min_max.le_supI2)

  2554   apply (erule order_trans [OF dist_triangle add_left_mono])

  2555   done

  2556

  2557 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2558   by (induct rule: finite_induct[of F]) auto

  2559

  2560 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2561   by (induct set: finite) auto

  2562

  2563 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2564 proof -

  2565   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp

  2566   then have "bounded {x}" unfolding bounded_def by fast

  2567   then show ?thesis by (metis insert_is_Un bounded_Un)

  2568 qed

  2569

  2570 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2571   by (induct set: finite) simp_all

  2572

  2573 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2574   apply (simp add: bounded_iff)

  2575   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2576   apply metis

  2577   apply arith

  2578   done

  2579

  2580 lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"

  2581   unfolding Bseq_def bounded_pos by auto

  2582

  2583 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2584   by (metis Int_lower1 Int_lower2 bounded_subset)

  2585

  2586 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2587   by (metis Diff_subset bounded_subset)

  2588

  2589 lemma not_bounded_UNIV[simp, intro]:

  2590   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2591 proof(auto simp add: bounded_pos not_le)

  2592   obtain x :: 'a where "x \<noteq> 0"

  2593     using perfect_choose_dist [OF zero_less_one] by fast

  2594   fix b::real  assume b: "b >0"

  2595   have b1: "b +1 \<ge> 0" using b by simp

  2596   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2597     by (simp add: norm_sgn)

  2598   then show "\<exists>x::'a. b < norm x" ..

  2599 qed

  2600

  2601 lemma bounded_linear_image:

  2602   assumes "bounded S" "bounded_linear f"

  2603   shows "bounded(f  S)"

  2604 proof -

  2605   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"

  2606     unfolding bounded_pos by auto

  2607   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"

  2608     using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2609   {

  2610     fix x

  2611     assume "x\<in>S"

  2612     then have "norm x \<le> b" using b by auto

  2613     then have "norm (f x) \<le> B * b" using B(2)

  2614       apply (erule_tac x=x in allE)

  2615       apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2616       done

  2617   }

  2618   then show ?thesis

  2619     unfolding bounded_pos

  2620     apply (rule_tac x="b*B" in exI)

  2621     using b B mult_pos_pos [of b B]

  2622     apply (auto simp add: mult_commute)

  2623     done

  2624 qed

  2625

  2626 lemma bounded_scaling:

  2627   fixes S :: "'a::real_normed_vector set"

  2628   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2629   apply (rule bounded_linear_image, assumption)

  2630   apply (rule bounded_linear_scaleR_right)

  2631   done

  2632

  2633 lemma bounded_translation:

  2634   fixes S :: "'a::real_normed_vector set"

  2635   assumes "bounded S"

  2636   shows "bounded ((\<lambda>x. a + x)  S)"

  2637 proof -

  2638   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"

  2639     unfolding bounded_pos by auto

  2640   {

  2641     fix x

  2642     assume "x\<in>S"

  2643     then have "norm (a + x) \<le> b + norm a"

  2644       using norm_triangle_ineq[of a x] b by auto

  2645   }

  2646   then show ?thesis

  2647     unfolding bounded_pos

  2648     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2649     by (auto intro!: exI[of _ "b + norm a"])

  2650 qed

  2651

  2652

  2653 text{* Some theorems on sups and infs using the notion "bounded". *}

  2654

  2655 lemma bounded_real:

  2656   fixes S :: "real set"

  2657   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2658   by (simp add: bounded_iff)

  2659

  2660 lemma bounded_has_Sup:

  2661   fixes S :: "real set"

  2662   assumes "bounded S" "S \<noteq> {}"

  2663   shows "\<forall>x\<in>S. x \<le> Sup S"

  2664     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2665 proof

  2666   fix x

  2667   assume "x\<in>S"

  2668   then show "x \<le> Sup S"

  2669     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

  2670 next

  2671   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2672     using assms by (metis cSup_least)

  2673 qed

  2674

  2675 lemma Sup_insert:

  2676   fixes S :: "real set"

  2677   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2678   apply (subst cSup_insert_If)

  2679   apply (rule bounded_has_Sup(1)[of S, rule_format])

  2680   apply (auto simp: sup_max)

  2681   done

  2682

  2683 lemma Sup_insert_finite:

  2684   fixes S :: "real set"

  2685   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2686   apply (rule Sup_insert)

  2687   apply (rule finite_imp_bounded)

  2688   apply simp

  2689   done

  2690

  2691 lemma bounded_has_Inf:

  2692   fixes S :: "real set"

  2693   assumes "bounded S"  "S \<noteq> {}"

  2694   shows "\<forall>x\<in>S. x \<ge> Inf S"

  2695     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2696 proof

  2697   fix x

  2698   assume "x\<in>S"

  2699   from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"

  2700     unfolding bounded_real by auto

  2701   then show "x \<ge> Inf S" using x\<in>S

  2702     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

  2703 next

  2704   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b"

  2705     using assms by (metis cInf_greatest)

  2706 qed

  2707

  2708 lemma Inf_insert:

  2709   fixes S :: "real set"

  2710   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2711   apply (subst cInf_insert_if)

  2712   apply (rule bounded_has_Inf(1)[of S, rule_format])

  2713   apply (auto simp: inf_min)

  2714   done

  2715

  2716 lemma Inf_insert_finite:

  2717   fixes S :: "real set"

  2718   shows "finite S \<Longrightarrow> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2719   apply (rule Inf_insert)

  2720   apply (rule finite_imp_bounded)

  2721   apply simp

  2722   done

  2723

  2724 subsection {* Compactness *}

  2725

  2726 subsubsection {* Bolzano-Weierstrass property *}

  2727

  2728 lemma heine_borel_imp_bolzano_weierstrass:

  2729   assumes "compact s" "infinite t"  "t \<subseteq> s"

  2730   shows "\<exists>x \<in> s. x islimpt t"

  2731 proof(rule ccontr)

  2732   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2733   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)"

  2734     unfolding islimpt_def

  2735     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]

  2736     by auto

  2737   obtain g where g: "g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2738     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2739     using f by auto

  2740   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2741   {

  2742     fix x y

  2743     assume "x\<in>t" "y\<in>t" "f x = f y"

  2744     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2745       using f[THEN bspec[where x=x]] and t\<subseteq>s by auto

  2746     then have "x = y"

  2747       using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto

  2748   }

  2749   then have "inj_on f t"

  2750     unfolding inj_on_def by simp

  2751   then have "infinite (f  t)"

  2752     using assms(2) using finite_imageD by auto

  2753   moreover

  2754   {

  2755     fix x

  2756     assume "x\<in>t" "f x \<notin> g"

  2757     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2758     then obtain y where "y\<in>s" "h = f y"

  2759       using g'[THEN bspec[where x=h]] by auto

  2760     then have "y = x"

  2761       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2762     then have False

  2763       using f x \<notin> g h\<in>g unfolding h = f y by auto

  2764   }

  2765   then have "f  t \<subseteq> g" by auto

  2766   ultimately show False

  2767     using g(2) using finite_subset by auto

  2768 qed

  2769

  2770 lemma acc_point_range_imp_convergent_subsequence:

  2771   fixes l :: "'a :: first_countable_topology"

  2772   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2773   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2774 proof -

  2775   from countable_basis_at_decseq[of l] guess A . note A = this

  2776

  2777   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2778   {

  2779     fix n i

  2780     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2781       using l A by auto

  2782     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2783       unfolding ex_in_conv by (intro notI) simp

  2784     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2785       by auto

  2786     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2787       by (auto simp: not_le)

  2788     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2789       unfolding s_def by (auto intro: someI2_ex)

  2790   }

  2791   note s = this

  2792   def r \<equiv> "nat_rec (s 0 0) s"

  2793   have "subseq r"

  2794     by (auto simp: r_def s subseq_Suc_iff)

  2795   moreover

  2796   have "(\<lambda>n. f (r n)) ----> l"

  2797   proof (rule topological_tendstoI)

  2798     fix S

  2799     assume "open S" "l \<in> S"

  2800     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2801     moreover

  2802     {

  2803       fix i

  2804       assume "Suc 0 \<le> i"

  2805       then have "f (r i) \<in> A i"

  2806         by (cases i) (simp_all add: r_def s)

  2807     }

  2808     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  2809       by (auto simp: eventually_sequentially)

  2810     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2811       by eventually_elim auto

  2812   qed

  2813   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2814     by (auto simp: convergent_def comp_def)

  2815 qed

  2816

  2817 lemma sequence_infinite_lemma:

  2818   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2819   assumes "\<forall>n. f n \<noteq> l"

  2820     and "(f ---> l) sequentially"

  2821   shows "infinite (range f)"

  2822 proof

  2823   assume "finite (range f)"

  2824   then have "closed (range f)" by (rule finite_imp_closed)

  2825   then have "open (- range f)" by (rule open_Compl)

  2826   from assms(1) have "l \<in> - range f" by auto

  2827   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2828     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2829   then show False unfolding eventually_sequentially by auto

  2830 qed

  2831

  2832 lemma closure_insert:

  2833   fixes x :: "'a::t1_space"

  2834   shows "closure (insert x s) = insert x (closure s)"

  2835   apply (rule closure_unique)

  2836   apply (rule insert_mono [OF closure_subset])

  2837   apply (rule closed_insert [OF closed_closure])

  2838   apply (simp add: closure_minimal)

  2839   done

  2840

  2841 lemma islimpt_insert:

  2842   fixes x :: "'a::t1_space"

  2843   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2844 proof

  2845   assume *: "x islimpt (insert a s)"

  2846   show "x islimpt s"

  2847   proof (rule islimptI)

  2848     fix t

  2849     assume t: "x \<in> t" "open t"

  2850     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2851     proof (cases "x = a")

  2852       case True

  2853       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2854         using * t by (rule islimptE)

  2855       with x = a show ?thesis by auto

  2856     next

  2857       case False

  2858       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2859         by (simp_all add: open_Diff)

  2860       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2861         using * t' by (rule islimptE)

  2862       then show ?thesis by auto

  2863     qed

  2864   qed

  2865 next

  2866   assume "x islimpt s"

  2867   then show "x islimpt (insert a s)"

  2868     by (rule islimpt_subset) auto

  2869 qed

  2870

  2871 lemma islimpt_finite:

  2872   fixes x :: "'a::t1_space"

  2873   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2874   by (induct set: finite) (simp_all add: islimpt_insert)

  2875

  2876 lemma islimpt_union_finite:

  2877   fixes x :: "'a::t1_space"

  2878   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2879   by (simp add: islimpt_Un islimpt_finite)

  2880

  2881 lemma islimpt_eq_acc_point:

  2882   fixes l :: "'a :: t1_space"

  2883   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2884 proof (safe intro!: islimptI)

  2885   fix U

  2886   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2887   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2888     by (auto intro: finite_imp_closed)

  2889   then show False

  2890     by (rule islimptE) auto

  2891 next

  2892   fix T

  2893   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2894   then have "infinite (T \<inter> S - {l})"

  2895     by auto

  2896   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2897     unfolding ex_in_conv by (intro notI) simp

  2898   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2899     by auto

  2900 qed

  2901

  2902 lemma islimpt_range_imp_convergent_subsequence:

  2903   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2904   assumes l: "l islimpt (range f)"

  2905   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2906   using l unfolding islimpt_eq_acc_point

  2907   by (rule acc_point_range_imp_convergent_subsequence)

  2908

  2909 lemma sequence_unique_limpt:

  2910   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2911   assumes "(f ---> l) sequentially"

  2912     and "l' islimpt (range f)"

  2913   shows "l' = l"

  2914 proof (rule ccontr)

  2915   assume "l' \<noteq> l"

  2916   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2917     using hausdorff [OF l' \<noteq> l] by auto

  2918   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2919     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2920   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2921     unfolding eventually_sequentially by auto

  2922

  2923   have "UNIV = {..<N} \<union> {N..}"

  2924     by auto

  2925   then have "l' islimpt (f  ({..<N} \<union> {N..}))"

  2926     using assms(2) by simp

  2927   then have "l' islimpt (f  {..<N} \<union> f  {N..})"

  2928     by (simp add: image_Un)

  2929   then have "l' islimpt (f  {N..})"

  2930     by (simp add: islimpt_union_finite)

  2931   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2932     using l' \<in> s open s by (rule islimptE)

  2933   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"

  2934     by auto

  2935   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t"

  2936     by simp

  2937   with s \<inter> t = {} show False

  2938     by simp

  2939 qed

  2940

  2941 lemma bolzano_weierstrass_imp_closed:

  2942   fixes s :: "'a::{first_countable_topology, t2_space} set"

  2943   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2944   shows "closed s"

  2945 proof -

  2946   {

  2947     fix x l

  2948     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2949     then have "l \<in> s"

  2950     proof (cases "\<forall>n. x n \<noteq> l")

  2951       case False

  2952       then show "l\<in>s" using as(1) by auto

  2953     next

  2954       case True note cas = this

  2955       with as(2) have "infinite (range x)"

  2956         using sequence_infinite_lemma[of x l] by auto

  2957       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  2958         using assms[THEN spec[where x="range x"]] as(1) by auto

  2959       then show "l\<in>s" using sequence_unique_limpt[of x l l']

  2960         using as cas by auto

  2961     qed

  2962   }

  2963   then show ?thesis

  2964     unfolding closed_sequential_limits by fast

  2965 qed

  2966

  2967 lemma compact_imp_bounded:

  2968   assumes "compact U"

  2969   shows "bounded U"

  2970 proof -

  2971   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  2972     using assms by auto

  2973   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2974     by (rule compactE_image)

  2975   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2976     by (simp add: bounded_UN)

  2977   then show "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2978     by (rule bounded_subset)

  2979 qed

  2980

  2981 text{* In particular, some common special cases. *}

  2982

  2983 lemma compact_union [intro]:

  2984   assumes "compact s" "compact t"

  2985   shows " compact (s \<union> t)"

  2986 proof (rule compactI)

  2987   fix f

  2988   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  2989   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  2990     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2991   moreover

  2992   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  2993     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2994   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  2995     by (auto intro!: exI[of _ "s' \<union> t'"])

  2996 qed

  2997

  2998 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  2999   by (induct set: finite) auto

  3000

  3001 lemma compact_UN [intro]:

  3002   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3003   unfolding SUP_def by (rule compact_Union) auto

  3004

  3005 lemma closed_inter_compact [intro]:

  3006   assumes "closed s"

  3007     and "compact t"

  3008   shows "compact (s \<inter> t)"

  3009   using compact_inter_closed [of t s] assms

  3010   by (simp add: Int_commute)

  3011

  3012 lemma compact_inter [intro]:

  3013   fixes s t :: "'a :: t2_space set"

  3014   assumes "compact s"

  3015     and "compact t"

  3016   shows "compact (s \<inter> t)"

  3017   using assms by (intro compact_inter_closed compact_imp_closed)

  3018

  3019 lemma compact_sing [simp]: "compact {a}"

  3020   unfolding compact_eq_heine_borel by auto

  3021

  3022 lemma compact_insert [simp]:

  3023   assumes "compact s"

  3024   shows "compact (insert x s)"

  3025 proof -

  3026   have "compact ({x} \<union> s)"

  3027     using compact_sing assms by (rule compact_union)

  3028   then show ?thesis by simp

  3029 qed

  3030

  3031 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  3032   by (induct set: finite) simp_all

  3033

  3034 lemma open_delete:

  3035   fixes s :: "'a::t1_space set"

  3036   shows "open s \<Longrightarrow> open (s - {x})"

  3037   by (simp add: open_Diff)

  3038

  3039 text{* Finite intersection property *}

  3040

  3041 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  3042   by (auto simp: inj_on_def)

  3043

  3044 lemma compact_fip:

  3045   "compact U \<longleftrightarrow>

  3046     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"

  3047   (is "_ \<longleftrightarrow> ?R")

  3048 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  3049   fix A

  3050   assume "compact U"

  3051     and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  3052     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  3053   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>(uminusA)"

  3054     by auto

  3055   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  3056     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  3057   with fi[THEN spec, of B] show False

  3058     by (auto dest: finite_imageD intro: inj_setminus)

  3059 next

  3060   fix A

  3061   assume ?R

  3062   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3063   then have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  3064     by auto

  3065   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>(uminusB) = {}"

  3066     by (metis subset_image_iff)

  3067   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3068     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  3069 qed

  3070

  3071 lemma compact_imp_fip:

  3072   "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>

  3073     s \<inter> (\<Inter> f) \<noteq> {}"

  3074   unfolding compact_fip by auto

  3075

  3076 text{*Compactness expressed with filters*}

  3077

  3078 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3079

  3080 lemma eventually_filter_from_subbase:

  3081   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3082     (is "_ \<longleftrightarrow> ?R P")

  3083   unfolding filter_from_subbase_def

  3084 proof (rule eventually_Abs_filter is_filter.intro)+

  3085   show "?R (\<lambda>x. True)"

  3086     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  3087 next

  3088   fix P Q assume "?R P" then guess X ..

  3089   moreover assume "?R Q" then guess Y ..

  3090   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  3091     by (intro exI[of _ "X \<union> Y"]) auto

  3092 next

  3093   fix P Q

  3094   assume "?R P" then guess X ..

  3095   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  3096   ultimately show "?R Q"

  3097     by (intro exI[of _ X]) auto

  3098 qed

  3099

  3100 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  3101   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  3102

  3103 lemma filter_from_subbase_not_bot:

  3104   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  3105   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  3106

  3107 lemma closure_iff_nhds_not_empty:

  3108   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  3109 proof safe

  3110   assume x: "x \<in> closure X"

  3111   fix S A

  3112   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  3113   then have "x \<notin> closure (-S)"

  3114     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  3115   with x have "x \<in> closure X - closure (-S)"

  3116     by auto

  3117   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  3118     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  3119   finally have "X \<inter> S \<noteq> {}" by auto

  3120   then show False using X \<inter> A = {} S \<subseteq> A by auto

  3121 next

  3122   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  3123   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  3124   show "x \<in> closure X"

  3125     by (simp add: closure_subset open_Compl)

  3126 qed

  3127

  3128 lemma compact_filter:

  3129   "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))"

  3130 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  3131   fix F

  3132   assume "compact U"

  3133   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  3134   then have "U \<noteq> {}"

  3135     by (auto simp: eventually_False)

  3136

  3137   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  3138   then have "\<forall>z\<in>Z. closed z"

  3139     by auto

  3140   moreover

  3141   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  3142     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  3143   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  3144   proof (intro allI impI)

  3145     fix B assume "finite B" "B \<subseteq> Z"

  3146     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  3147       by (auto intro!: eventually_Ball_finite)

  3148     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  3149       by eventually_elim auto

  3150     with F show "U \<inter> \<Inter>B \<noteq> {}"

  3151       by (intro notI) (simp add: eventually_False)

  3152   qed

  3153   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  3154     using compact U unfolding compact_fip by blast

  3155   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"

  3156     by auto

  3157

  3158   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  3159     unfolding eventually_inf eventually_nhds

  3160   proof safe

  3161     fix P Q R S

  3162     assume "eventually R F" "open S" "x \<in> S"

  3163     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  3164     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  3165     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  3166     ultimately show False by (auto simp: set_eq_iff)

  3167   qed

  3168   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  3169     by (metis eventually_bot)

  3170 next

  3171   fix A

  3172   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  3173   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  3174   then have inj_P': "\<And>A. inj_on P' A"

  3175     by (auto intro!: inj_onI simp: fun_eq_iff)

  3176   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  3177   have "F \<noteq> bot"

  3178     unfolding F_def

  3179   proof (safe intro!: filter_from_subbase_not_bot)

  3180     fix X

  3181     assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  3182     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  3183       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  3184     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}"

  3185       by auto

  3186     with B show False

  3187       by (auto simp: P'_def fun_eq_iff)

  3188   qed

  3189   moreover have "eventually (\<lambda>x. x \<in> U) F"

  3190     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  3191   moreover

  3192   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)"

  3193   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  3194     by auto

  3195

  3196   {

  3197     fix V

  3198     assume "V \<in> A"

  3199     then have V: "eventually (\<lambda>x. x \<in> V) F"

  3200       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  3201     have "x \<in> closure V"

  3202       unfolding closure_iff_nhds_not_empty

  3203     proof (intro impI allI)

  3204       fix S A

  3205       assume "open S" "x \<in> S" "S \<subseteq> A"

  3206       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"

  3207         by (auto simp: eventually_nhds)

  3208       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  3209         by (auto simp: eventually_inf)

  3210       with x show "V \<inter> A \<noteq> {}"

  3211         by (auto simp del: Int_iff simp add: trivial_limit_def)

  3212     qed

  3213     then have "x \<in> V"

  3214       using V \<in> A A(1) by simp

  3215   }

  3216   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  3217   with U \<inter> \<Inter>A = {} show False by auto

  3218 qed

  3219

  3220 definition "countably_compact U \<longleftrightarrow>

  3221     (\<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))"

  3222

  3223 lemma countably_compactE:

  3224   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  3225   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  3226   using assms unfolding countably_compact_def by metis

  3227

  3228 lemma countably_compactI:

  3229   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')"

  3230   shows "countably_compact s"

  3231   using assms unfolding countably_compact_def by metis

  3232

  3233 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  3234   by (auto simp: compact_eq_heine_borel countably_compact_def)

  3235

  3236 lemma countably_compact_imp_compact:

  3237   assumes "countably_compact U"

  3238     and ccover: "countable B" "\<forall>b\<in>B. open b"

  3239     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"

  3240   shows "compact U"

  3241   using countably_compact U

  3242   unfolding compact_eq_heine_borel countably_compact_def

  3243 proof safe

  3244   fix A

  3245   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3246   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)"

  3247

  3248   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  3249   ultimately have "countable C" "\<forall>a\<in>C. open a"

  3250     unfolding C_def using ccover by auto

  3251   moreover

  3252   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  3253   proof safe

  3254     fix x a

  3255     assume "x \<in> U" "x \<in> a" "a \<in> A"

  3256     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"

  3257       by blast

  3258     with a \<in> A show "x \<in> \<Union>C"

  3259       unfolding C_def by auto

  3260   qed

  3261   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  3262   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  3263     using * by metis

  3264   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  3265     by (auto simp: C_def)

  3266   then guess f unfolding bchoice_iff Bex_def ..

  3267   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3268     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  3269 qed

  3270

  3271 lemma countably_compact_imp_compact_second_countable:

  3272   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3273 proof (rule countably_compact_imp_compact)

  3274   fix T and x :: 'a

  3275   assume "open T" "x \<in> T"

  3276   from topological_basisE[OF is_basis this] guess b .

  3277   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  3278     by auto

  3279 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  3280

  3281 lemma countably_compact_eq_compact:

  3282   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3283   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  3284

  3285 subsubsection{* Sequential compactness *}

  3286

  3287 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"

  3288   where "seq_compact S \<longleftrightarrow>

  3289     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  3290

  3291 lemma seq_compact_imp_countably_compact:

  3292   fixes U :: "'a :: first_countable_topology set"

  3293   assumes "seq_compact U"

  3294   shows "countably_compact U"

  3295 proof (safe intro!: countably_compactI)

  3296   fix A

  3297   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  3298   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  3299     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  3300   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3301   proof cases

  3302     assume "finite A"

  3303     with A show ?thesis by auto

  3304   next

  3305     assume "infinite A"

  3306     then have "A \<noteq> {}" by auto

  3307     show ?thesis

  3308     proof (rule ccontr)

  3309       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  3310       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"

  3311         by auto

  3312       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"

  3313         by metis

  3314       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  3315       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  3316         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  3317       then have "range X \<subseteq> U"

  3318         by auto

  3319       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"

  3320         by auto

  3321       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  3322       obtain n where "x \<in> from_nat_into A n" by auto

  3323       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  3324       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  3325         unfolding tendsto_def by (auto simp: comp_def)

  3326       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  3327         by (auto simp: eventually_sequentially)

  3328       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  3329         by auto

  3330       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  3331         by (auto intro!: exI[of _ "max n N"])

  3332       ultimately show False

  3333         by auto

  3334     qed

  3335   qed

  3336 qed

  3337

  3338 lemma compact_imp_seq_compact:

  3339   fixes U :: "'a :: first_countable_topology set"

  3340   assumes "compact U"

  3341   shows "seq_compact U"

  3342   unfolding seq_compact_def

  3343 proof safe

  3344   fix X :: "nat \<Rightarrow> 'a"

  3345   assume "\<forall>n. X n \<in> U"

  3346   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  3347     by (auto simp: eventually_filtermap)

  3348   moreover

  3349   have "filtermap X sequentially \<noteq> bot"

  3350     by (simp add: trivial_limit_def eventually_filtermap)

  3351   ultimately

  3352   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  3353     using compact U by (auto simp: compact_filter)

  3354

  3355   from countable_basis_at_decseq[of x] guess A . note A = this

  3356   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3357   {

  3358     fix n i

  3359     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3360     proof (rule ccontr)

  3361       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3362       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"

  3363         by auto

  3364       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3365         by (auto simp: eventually_filtermap eventually_sequentially)

  3366       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3367         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3368       ultimately have "eventually (\<lambda>x. False) ?F"

  3369         by (auto simp add: eventually_inf)

  3370       with x show False

  3371         by (simp add: eventually_False)

  3372     qed

  3373     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3374       unfolding s_def by (auto intro: someI2_ex)

  3375   }

  3376   note s = this

  3377   def r \<equiv> "nat_rec (s 0 0) s"

  3378   have "subseq r"

  3379     by (auto simp: r_def s subseq_Suc_iff)

  3380   moreover

  3381   have "(\<lambda>n. X (r n)) ----> x"

  3382   proof (rule topological_tendstoI)

  3383     fix S

  3384     assume "open S" "x \<in> S"

  3385     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3386       by auto

  3387     moreover

  3388     {

  3389       fix i

  3390       assume "Suc 0 \<le> i"

  3391       then have "X (r i) \<in> A i"

  3392         by (cases i) (simp_all add: r_def s)

  3393     }

  3394     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3395       by (auto simp: eventually_sequentially)

  3396     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3397       by eventually_elim auto

  3398   qed

  3399   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3400     using x \<in> U by (auto simp: convergent_def comp_def)

  3401 qed

  3402

  3403 lemma seq_compactI:

  3404   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  3405   shows "seq_compact S"

  3406   unfolding seq_compact_def using assms by fast

  3407

  3408 lemma seq_compactE:

  3409   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3410   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3411   using assms unfolding seq_compact_def by fast

  3412

  3413 lemma countably_compact_imp_acc_point:

  3414   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"

  3415   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3416 proof (rule ccontr)

  3417   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3418   note countably_compact s

  3419   moreover have "\<forall>t\<in>C. open t"

  3420     by (auto simp: C_def)

  3421   moreover

  3422   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3423   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3424   have "s \<subseteq> \<Union>C"

  3425     using t \<subseteq> s

  3426     unfolding C_def Union_image_eq

  3427     apply (safe dest!: s)

  3428     apply (rule_tac a="U \<inter> t" in UN_I)

  3429     apply (auto intro!: interiorI simp add: finite_subset)

  3430     done

  3431   moreover

  3432   from countable t have "countable C"

  3433     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3434   ultimately guess D by (rule countably_compactE)

  3435   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"

  3436     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3437     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3438   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3439     using interior_subset by blast

  3440   moreover have "finite (\<Union>E)"

  3441     using E by auto

  3442   ultimately show False using infinite t

  3443     by (auto simp: finite_subset)

  3444 qed

  3445

  3446 lemma countable_acc_point_imp_seq_compact:

  3447   fixes s :: "'a::first_countable_topology set"

  3448   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3449   shows "seq_compact s"

  3450 proof -

  3451   {

  3452     fix f :: "nat \<Rightarrow> 'a"

  3453     assume f: "\<forall>n. f n \<in> s"

  3454     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3455     proof (cases "finite (range f)")

  3456       case True

  3457       obtain l where "infinite {n. f n = f l}"

  3458         using pigeonhole_infinite[OF _ True] by auto

  3459       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3460         using infinite_enumerate by blast

  3461       then have "subseq r \<and> (f \<circ> r) ----> f l"

  3462         by (simp add: fr tendsto_const o_def)

  3463       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3464         by auto

  3465     next

  3466       case False

  3467       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"

  3468         by auto

  3469       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3470       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3471         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3472       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3473     qed

  3474   }

  3475   then show ?thesis

  3476     unfolding seq_compact_def by auto

  3477 qed

  3478

  3479 lemma seq_compact_eq_countably_compact:

  3480   fixes U :: "'a :: first_countable_topology set"

  3481   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3482   using

  3483     countable_acc_point_imp_seq_compact

  3484     countably_compact_imp_acc_point

  3485     seq_compact_imp_countably_compact

  3486   by metis

  3487

  3488 lemma seq_compact_eq_acc_point:

  3489   fixes s :: "'a :: first_countable_topology set"

  3490   shows "seq_compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"

  3491   using

  3492     countable_acc_point_imp_seq_compact[of s]

  3493     countably_compact_imp_acc_point[of s]

  3494     seq_compact_imp_countably_compact[of s]

  3495   by metis

  3496

  3497 lemma seq_compact_eq_compact:

  3498   fixes U :: "'a :: second_countable_topology set"

  3499   shows "seq_compact U \<longleftrightarrow> compact U"

  3500   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3501

  3502 lemma bolzano_weierstrass_imp_seq_compact:

  3503   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3504   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3505   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3506

  3507 subsubsection{* Total boundedness *}

  3508

  3509 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)"

  3510   unfolding Cauchy_def by metis

  3511

  3512 fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a"

  3513 where

  3514   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3515 declare helper_1.simps[simp del]

  3516

  3517 lemma seq_compact_imp_totally_bounded:

  3518   assumes "seq_compact s"

  3519   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3520 proof (rule, rule, rule ccontr)

  3521   fix e::real

  3522   assume "e > 0"

  3523   assume assm: "\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3524   def x \<equiv> "helper_1 s e"

  3525   {

  3526     fix n

  3527     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3528     proof (induct n rule: nat_less_induct)

  3529       fix n

  3530       def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3531       assume as: "\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3532       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3533         using assm

  3534         apply simp

  3535         apply (erule_tac x="x  {0 ..< n}" in allE)

  3536         using as

  3537         apply auto

  3538         done

  3539       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3540         unfolding subset_eq by auto

  3541       have "Q (x n)"

  3542         unfolding x_def and helper_1.simps[of s e n]

  3543         apply (rule someI2[where a=z])

  3544         unfolding x_def[symmetric] and Q_def

  3545         using z

  3546         apply auto

  3547         done

  3548       then show "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3549         unfolding Q_def by auto

  3550     qed

  3551   }

  3552   then have "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"

  3553     by blast+

  3554   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3555     using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3556   from this(3) have "Cauchy (x \<circ> r)"

  3557     using LIMSEQ_imp_Cauchy by auto

  3558   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"

  3559     unfolding cauchy_def using e>0 by auto

  3560   show False

  3561     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3562     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3563     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

  3564     by auto

  3565 qed

  3566

  3567 subsubsection{* Heine-Borel theorem *}

  3568

  3569 lemma seq_compact_imp_heine_borel:

  3570   fixes s :: "'a :: metric_space set"

  3571   assumes "seq_compact s"

  3572   shows "compact s"

  3573 proof -

  3574   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3575   guess f unfolding choice_iff' .. note f = this

  3576   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3577   have "countably_compact s"

  3578     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3579   then show "compact s"

  3580   proof (rule countably_compact_imp_compact)

  3581     show "countable K"

  3582       unfolding K_def using f

  3583       by (auto intro: countable_finite countable_subset countable_rat

  3584                intro!: countable_image countable_SIGMA countable_UN)

  3585     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3586   next

  3587     fix T x

  3588     assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3589     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"

  3590       by auto

  3591     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"

  3592       by auto

  3593     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"

  3594       by auto

  3595     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3596       unfolding Union_image_eq by auto

  3597     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K"

  3598       by (auto simp: K_def)

  3599     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3600     proof (rule bexI[rotated], safe)

  3601       fix y

  3602       assume "y \<in> ball k r"

  3603       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3604         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3605       with ball x e \<subseteq> T show "y \<in> T"

  3606         by auto

  3607     next

  3608       show "x \<in> ball k r" by fact

  3609     qed

  3610   qed

  3611 qed

  3612

  3613 lemma compact_eq_seq_compact_metric:

  3614   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3615   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3616

  3617 lemma compact_def:

  3618   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3619    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"

  3620   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3621

  3622 subsubsection {* Complete the chain of compactness variants *}

  3623

  3624 lemma compact_eq_bolzano_weierstrass:

  3625   fixes s :: "'a::metric_space set"

  3626   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"

  3627   (is "?lhs = ?rhs")

  3628 proof

  3629   assume ?lhs

  3630   then show ?rhs

  3631     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3632 next

  3633   assume ?rhs

  3634   then show ?lhs

  3635     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3636 qed

  3637

  3638 lemma bolzano_weierstrass_imp_bounded:

  3639   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3640   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3641

  3642 text {*

  3643   A metric space (or topological vector space) is said to have the

  3644   Heine-Borel property if every closed and bounded subset is compact.

  3645 *}

  3646

  3647 class heine_borel = metric_space +

  3648   assumes bounded_imp_convergent_subsequence:

  3649     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3650

  3651 lemma bounded_closed_imp_seq_compact:

  3652   fixes s::"'a::heine_borel set"

  3653   assumes "bounded s"

  3654     and "closed s"

  3655   shows "seq_compact s"

  3656 proof (unfold seq_compact_def, clarify)

  3657   fix f :: "nat \<Rightarrow> 'a"

  3658   assume f: "\<forall>n. f n \<in> s"

  3659   with bounded s have "bounded (range f)"

  3660     by (auto intro: bounded_subset)

  3661   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3662     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3663   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"

  3664     by simp

  3665   have "l \<in> s" using closed s fr l

  3666     unfolding closed_sequential_limits by blast

  3667   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3668     using l \<in> s r l by blast

  3669 qed

  3670

  3671 lemma compact_eq_bounded_closed:

  3672   fixes s :: "'a::heine_borel set"

  3673   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3674 proof

  3675   assume ?lhs

  3676   then show ?rhs

  3677     using compact_imp_closed compact_imp_bounded

  3678     by blast

  3679 next

  3680   assume ?rhs

  3681   then show ?lhs

  3682     using bounded_closed_imp_seq_compact[of s]

  3683     unfolding compact_eq_seq_compact_metric

  3684     by auto

  3685 qed

  3686

  3687 (* TODO: is this lemma necessary? *)

  3688 lemma bounded_increasing_convergent:

  3689   fixes s :: "nat \<Rightarrow> real"

  3690   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3691   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3692   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3693

  3694 instance real :: heine_borel

  3695 proof

  3696   fix f :: "nat \<Rightarrow> real"

  3697   assume f: "bounded (range f)"

  3698   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3699     unfolding comp_def by (metis seq_monosub)

  3700   moreover

  3701   then have "Bseq (f \<circ> r)"

  3702     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3703   ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3704     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3705 qed

  3706

  3707 lemma compact_lemma:

  3708   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3709   assumes "bounded (range f)"

  3710   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  3711         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3712 proof safe

  3713   fix d :: "'a set"

  3714   assume d: "d \<subseteq> Basis"

  3715   with finite_Basis have "finite d"

  3716     by (blast intro: finite_subset)

  3717   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3718     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3719   proof (induct d)

  3720     case empty

  3721     then show ?case

  3722       unfolding subseq_def by auto

  3723   next

  3724     case (insert k d)

  3725     have k[intro]: "k \<in> Basis"

  3726       using insert by auto

  3727     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)"

  3728       using bounded (range f)

  3729       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3730     obtain l1::"'a" and r1 where r1: "subseq r1"

  3731       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3732       using insert(3) using insert(4) by auto

  3733     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f"

  3734       by simp

  3735     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3736       by (metis (lifting) bounded_subset f' image_subsetI s')

  3737     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3738       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]

  3739       by (auto simp: o_def)

  3740     def r \<equiv> "r1 \<circ> r2"

  3741     have r:"subseq r"

  3742       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3743     moreover

  3744     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3745     {

  3746       fix e::real

  3747       assume "e > 0"

  3748       from lr1 e > 0 have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3749         by blast

  3750       from lr2 e > 0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3751         by (rule tendstoD)

  3752       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3753         by (rule eventually_subseq)

  3754       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3755         using N1' N2

  3756         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3757     }

  3758     ultimately show ?case by auto

  3759   qed

  3760 qed

  3761

  3762 instance euclidean_space \<subseteq> heine_borel

  3763 proof

  3764   fix f :: "nat \<Rightarrow> 'a"

  3765   assume f: "bounded (range f)"

  3766   then obtain l::'a and r where r: "subseq r"

  3767     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3768     using compact_lemma [OF f] by blast

  3769   {

  3770     fix e::real

  3771     assume "e > 0"

  3772     then have "e / real_of_nat DIM('a) > 0"

  3773       by (auto intro!: divide_pos_pos DIM_positive)

  3774     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"

  3775       by simp

  3776     moreover

  3777     {

  3778       fix n

  3779       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3780       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3781         apply (subst euclidean_dist_l2)

  3782         using zero_le_dist

  3783         apply (rule setL2_le_setsum)

  3784         done

  3785       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3786         apply (rule setsum_strict_mono)

  3787         using n

  3788         apply auto

  3789         done

  3790       finally have "dist (f (r n)) l < e"

  3791         by auto

  3792     }

  3793     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3794       by (rule eventually_elim1)

  3795   }

  3796   then have *: "((f \<circ> r) ---> l) sequentially"

  3797     unfolding o_def tendsto_iff by simp

  3798   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3799     by auto

  3800 qed

  3801

  3802 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3803   unfolding bounded_def

  3804   apply clarify

  3805   apply (rule_tac x="a" in exI)

  3806   apply (rule_tac x="e" in exI)

  3807   apply clarsimp

  3808   apply (drule (1) bspec)

  3809   apply (simp add: dist_Pair_Pair)

  3810   apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3811   done

  3812

  3813 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3814   unfolding bounded_def

  3815   apply clarify

  3816   apply (rule_tac x="b" in exI)

  3817   apply (rule_tac x="e" in exI)

  3818   apply clarsimp

  3819   apply (drule (1) bspec)

  3820   apply (simp add: dist_Pair_Pair)

  3821   apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3822   done

  3823

  3824 instance prod :: (heine_borel, heine_borel) heine_borel

  3825 proof

  3826   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  3827   assume f: "bounded (range f)"

  3828   from f have s1: "bounded (range (fst \<circ> f))"

  3829     unfolding image_comp by (rule bounded_fst)

  3830   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  3831     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  3832   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  3833     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  3834   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3835     using bounded_imp_convergent_subsequence [OF s2]

  3836     unfolding o_def by fast

  3837   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3838     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  3839   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3840     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3841   have r: "subseq (r1 \<circ> r2)"

  3842     using r1 r2 unfolding subseq_def by simp

  3843   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3844     using l r by fast

  3845 qed

  3846

  3847 subsubsection{* Completeness *}

  3848

  3849 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  3850   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  3851

  3852 lemma compact_imp_complete:

  3853   assumes "compact s"

  3854   shows "complete s"

  3855 proof -

  3856   {

  3857     fix f

  3858     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3859     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  3860       using assms unfolding compact_def by blast

  3861

  3862     note lr' = seq_suble [OF lr(2)]

  3863

  3864     {

  3865       fix e :: real

  3866       assume "e > 0"

  3867       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"

  3868         unfolding cauchy_def

  3869         using e > 0

  3870         apply (erule_tac x="e/2" in allE)

  3871         apply auto

  3872         done

  3873       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  3874       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"

  3875         using e > 0 by auto

  3876       {

  3877         fix n :: nat

  3878         assume n: "n \<ge> max N M"

  3879         have "dist ((f \<circ> r) n) l < e/2"

  3880           using n M by auto

  3881         moreover have "r n \<ge> N"

  3882           using lr'[of n] n by auto

  3883         then have "dist (f n) ((f \<circ> r) n) < e / 2"

  3884           using N and n by auto

  3885         ultimately have "dist (f n) l < e"

  3886           using dist_triangle_half_r[of "f (r n)" "f n" e l]

  3887           by (auto simp add: dist_commute)

  3888       }

  3889       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  3890     }

  3891     then have "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  3892       unfolding LIMSEQ_def by auto

  3893   }

  3894   then show ?thesis unfolding complete_def by auto

  3895 qed

  3896

  3897 lemma nat_approx_posE:

  3898   fixes e::real

  3899   assumes "0 < e"

  3900   obtains n :: nat where "1 / (Suc n) < e"

  3901 proof atomize_elim

  3902   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3903     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3904   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3905     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3906   also have "\<dots> = e" by simp

  3907   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3908 qed

  3909

  3910 lemma compact_eq_totally_bounded:

  3911   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3912     (is "_ \<longleftrightarrow> ?rhs")

  3913 proof

  3914   assume assms: "?rhs"

  3915   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)"

  3916     by (auto simp: choice_iff')

  3917

  3918   show "compact s"

  3919   proof cases

  3920     assume "s = {}"

  3921     then show "compact s" by (simp add: compact_def)

  3922   next

  3923     assume "s \<noteq> {}"

  3924     show ?thesis

  3925       unfolding compact_def

  3926     proof safe

  3927       fix f :: "nat \<Rightarrow> 'a"

  3928       assume f: "\<forall>n. f n \<in> s"

  3929

  3930       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  3931       then have [simp]: "\<And>n. 0 < e n" by auto

  3932       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)"

  3933       {

  3934         fix n U

  3935         assume "infinite {n. f n \<in> U}"

  3936         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  3937           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  3938         then guess a ..

  3939         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3940           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  3941         from someI_ex[OF this]

  3942         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  3943           unfolding B_def by auto

  3944       }

  3945       note B = this

  3946

  3947       def F \<equiv> "nat_rec (B 0 UNIV) B"

  3948       {

  3949         fix n

  3950         have "infinite {i. f i \<in> F n}"

  3951           by (induct n) (auto simp: F_def B)

  3952       }

  3953       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  3954         using B by (simp add: F_def)

  3955       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  3956         using decseq_SucI[of F] by (auto simp: decseq_def)

  3957

  3958       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  3959       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  3960         fix k i

  3961         have "infinite ({n. f n \<in> F k} - {.. i})"

  3962           using infinite {n. f n \<in> F k} by auto

  3963         from infinite_imp_nonempty[OF this]

  3964         show "\<exists>x>i. f x \<in> F k"

  3965           by (simp add: set_eq_iff not_le conj_commute)

  3966       qed

  3967

  3968       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  3969       have "subseq t"

  3970         unfolding subseq_Suc_iff by (simp add: t_def sel)

  3971       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  3972         using f by auto

  3973       moreover

  3974       {

  3975         fix n

  3976         have "(f \<circ> t) n \<in> F n"

  3977           by (cases n) (simp_all add: t_def sel)

  3978       }

  3979       note t = this

  3980

  3981       have "Cauchy (f \<circ> t)"

  3982       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  3983         fix r :: real and N n m

  3984         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  3985         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  3986           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  3987         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  3988           by (auto simp: subset_eq)

  3989         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  3990         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  3991           by (simp add: dist_commute)

  3992       qed

  3993

  3994       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3995         using assms unfolding complete_def by blast

  3996     qed

  3997   qed

  3998 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  3999

  4000 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  4001 proof -

  4002   {

  4003     assume ?rhs

  4004     {

  4005       fix e::real

  4006       assume "e>0"

  4007       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  4008         by (erule_tac x="e/2" in allE) auto

  4009       {

  4010         fix n m

  4011         assume nm:"N \<le> m \<and> N \<le> n"

  4012         then have "dist (s m) (s n) < e" using N

  4013           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  4014           by blast

  4015       }

  4016       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  4017         by blast

  4018     }

  4019     then have ?lhs

  4020       unfolding cauchy_def

  4021       by blast

  4022   }

  4023   then show ?thesis

  4024     unfolding cauchy_def

  4025     using dist_triangle_half_l

  4026     by blast

  4027 qed

  4028

  4029 lemma cauchy_imp_bounded:

  4030   assumes "Cauchy s"

  4031   shows "bounded (range s)"

  4032 proof -

  4033   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  4034     unfolding cauchy_def

  4035     apply (erule_tac x= 1 in allE)

  4036     apply auto

  4037     done

  4038   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  4039   moreover

  4040   have "bounded (s  {0..N})"

  4041     using finite_imp_bounded[of "s  {1..N}"] by auto

  4042   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  4043     unfolding bounded_any_center [where a="s N"] by auto

  4044   ultimately show "?thesis"

  4045     unfolding bounded_any_center [where a="s N"]

  4046     apply (rule_tac x="max a 1" in exI)

  4047     apply auto

  4048     apply (erule_tac x=y in allE)

  4049     apply (erule_tac x=y in ballE)

  4050     apply auto

  4051     done

  4052 qed

  4053

  4054 instance heine_borel < complete_space

  4055 proof

  4056   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4057   then have "bounded (range f)"

  4058     by (rule cauchy_imp_bounded)

  4059   then have "compact (closure (range f))"

  4060     unfolding compact_eq_bounded_closed by auto

  4061   then have "complete (closure (range f))"

  4062     by (rule compact_imp_complete)

  4063   moreover have "\<forall>n. f n \<in> closure (range f)"

  4064     using closure_subset [of "range f"] by auto

  4065   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  4066     using Cauchy f unfolding complete_def by auto

  4067   then show "convergent f"

  4068     unfolding convergent_def by auto

  4069 qed

  4070

  4071 instance euclidean_space \<subseteq> banach ..

  4072

  4073 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  4074 proof (simp add: complete_def, rule, rule)

  4075   fix f :: "nat \<Rightarrow> 'a"

  4076   assume "Cauchy f"

  4077   then have "convergent f" by (rule Cauchy_convergent)

  4078   then show "\<exists>l. f ----> l" unfolding convergent_def .

  4079 qed

  4080

  4081 lemma complete_imp_closed:

  4082   assumes "complete s"

  4083   shows "closed s"

  4084 proof -

  4085   {

  4086     fix x

  4087     assume "x islimpt s"

  4088     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  4089       unfolding islimpt_sequential by auto

  4090     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  4091       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  4092     then have "x \<in> s"

  4093       using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  4094   }

  4095   then show "closed s" unfolding closed_limpt by auto

  4096 qed

  4097

  4098 lemma complete_eq_closed:

  4099   fixes s :: "'a::complete_space set"

  4100   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  4101 proof

  4102   assume ?lhs

  4103   then show ?rhs by (rule complete_imp_closed)

  4104 next

  4105   assume ?rhs

  4106   {

  4107     fix f

  4108     assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  4109     then obtain l where "(f ---> l) sequentially"

  4110       using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  4111     then have "\<exists>l\<in>s. (f ---> l) sequentially"

  4112       using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]

  4113       using as(1) by auto

  4114   }

  4115   then show ?lhs unfolding complete_def by auto

  4116 qed

  4117

  4118 lemma convergent_eq_cauchy:

  4119   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  4120   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  4121   unfolding Cauchy_convergent_iff convergent_def ..

  4122

  4123 lemma convergent_imp_bounded:

  4124   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  4125   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  4126   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  4127

  4128 lemma compact_cball[simp]:

  4129   fixes x :: "'a::heine_borel"

  4130   shows "compact(cball x e)"

  4131   using compact_eq_bounded_closed bounded_cball closed_cball

  4132   by blast

  4133

  4134 lemma compact_frontier_bounded[intro]:

  4135   fixes s :: "'a::heine_borel set"

  4136   shows "bounded s ==> compact(frontier s)"

  4137   unfolding frontier_def

  4138   using compact_eq_bounded_closed

  4139   by blast

  4140

  4141 lemma compact_frontier[intro]:

  4142   fixes s :: "'a::heine_borel set"

  4143   shows "compact s ==> compact (frontier s)"

  4144   using compact_eq_bounded_closed compact_frontier_bounded

  4145   by blast

  4146

  4147 lemma frontier_subset_compact:

  4148   fixes s :: "'a::heine_borel set"

  4149   shows "compact s ==> frontier s \<subseteq> s"

  4150   using frontier_subset_closed compact_eq_bounded_closed

  4151   by blast

  4152

  4153 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  4154

  4155 lemma bounded_closed_nest:

  4156   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  4157     "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  4158   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  4159 proof -

  4160   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n"

  4161     using choice[of "\<lambda>n x. x\<in> s n"] by auto

  4162   from assms(4,1) have *:"seq_compact (s 0)"

  4163     using bounded_closed_imp_seq_compact[of "s 0"] by auto

  4164

  4165   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  4166     unfolding seq_compact_def

  4167     apply (erule_tac x=x in allE)

  4168     using x using assms(3)

  4169     apply blast

  4170     done

  4171

  4172   {

  4173     fix n :: nat

  4174     {

  4175       fix e :: real

  4176       assume "e>0"

  4177       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e"

  4178         unfolding LIMSEQ_def by auto

  4179       then have "dist ((x \<circ> r) (max N n)) l < e" by auto

  4180       moreover

  4181       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"]

  4182         by auto

  4183       then have "(x \<circ> r) (max N n) \<in> s n"

  4184         using x

  4185         apply (erule_tac x=n in allE)

  4186         using x

  4187         apply (erule_tac x="r (max N n)" in allE)

  4188         using assms(3)

  4189         apply (erule_tac x=n in allE)

  4190         apply (erule_tac x="r (max N n)" in allE)

  4191         apply auto

  4192         done

  4193       ultimately have "\<exists>y\<in>s n. dist y l < e"

  4194         by auto

  4195     }

  4196     then have "l \<in> s n"

  4197       using closed_approachable[of "s n" l] assms(1) by blast

  4198   }

  4199   then show ?thesis by auto

  4200 qed

  4201

  4202 text {* Decreasing case does not even need compactness, just completeness. *}

  4203

  4204 lemma decreasing_closed_nest:

  4205   assumes

  4206     "\<forall>n. closed(s n)"

  4207     "\<forall>n. (s n \<noteq> {})"

  4208     "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  4209     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  4210   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  4211 proof-

  4212   have "\<forall>n. \<exists> x. x\<in>s n"

  4213     using assms(2) by auto

  4214   then have "\<exists>t. \<forall>n. t n \<in> s n"

  4215     using choice[of "\<lambda> n x. x \<in> s n"] by auto

  4216   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  4217   {

  4218     fix e :: real

  4219     assume "e > 0"

  4220     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"

  4221       using assms(4) by auto

  4222     {

  4223       fix m n :: nat

  4224       assume "N \<le> m \<and> N \<le> n"

  4225       then have "t m \<in> s N" "t n \<in> s N"

  4226         using assms(3) t unfolding  subset_eq t by blast+

  4227       then have "dist (t m) (t n) < e"

  4228         using N by auto

  4229     }

  4230     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  4231       by auto

  4232   }

  4233   then have "Cauchy t"

  4234     unfolding cauchy_def by auto

  4235   then obtain l where l:"(t ---> l) sequentially"

  4236     using complete_univ unfolding complete_def by auto

  4237   {

  4238     fix n :: nat

  4239     {

  4240       fix e :: real

  4241       assume "e > 0"

  4242       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  4243         using l[unfolded LIMSEQ_def] by auto

  4244       have "t (max n N) \<in> s n"

  4245         using assms(3)

  4246         unfolding subset_eq

  4247         apply (erule_tac x=n in allE)

  4248         apply (erule_tac x="max n N" in allE)

  4249         using t

  4250         apply auto

  4251         done

  4252       then have "\<exists>y\<in>s n. dist y l < e"

  4253         apply (rule_tac x="t (max n N)" in bexI)

  4254         using N

  4255         apply auto

  4256         done

  4257     }

  4258     then have "l \<in> s n"

  4259       using closed_approachable[of "s n" l] assms(1) by auto

  4260   }

  4261   then show ?thesis by auto

  4262 qed

  4263

  4264 text {* Strengthen it to the intersection actually being a singleton. *}

  4265

  4266 lemma decreasing_closed_nest_sing:

  4267   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  4268   assumes

  4269     "\<forall>n. closed(s n)"

  4270     "\<forall>n. s n \<noteq> {}"

  4271     "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  4272     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  4273   shows "\<exists>a. \<Inter>(range s) = {a}"

  4274 proof -

  4275   obtain a where a: "\<forall>n. a \<in> s n"

  4276     using decreasing_closed_nest[of s] using assms by auto

  4277   {

  4278     fix b

  4279     assume b: "b \<in> \<Inter>(range s)"

  4280     {

  4281       fix e :: real

  4282       assume "e > 0"

  4283       then have "dist a b < e"

  4284         using assms(4) and b and a by blast

  4285     }

  4286     then have "dist a b = 0"

  4287       by (metis dist_eq_0_iff dist_nz less_le)

  4288   }

  4289   with a have "\<Inter>(range s) = {a}"

  4290     unfolding image_def by auto

  4291   then show ?thesis ..

  4292 qed

  4293

  4294 text{* Cauchy-type criteria for uniform convergence. *}

  4295

  4296 lemma uniformly_convergent_eq_cauchy:

  4297   fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"

  4298   shows

  4299     "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  4300       (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)"

  4301   (is "?lhs = ?rhs")

  4302 proof

  4303   assume ?lhs

  4304   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"

  4305     by auto

  4306   {

  4307     fix e :: real

  4308     assume "e > 0"

  4309     then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"

  4310       using l[THEN spec[where x="e/2"]] by auto

  4311     {

  4312       fix n m :: nat and x :: "'b"

  4313       assume "N \<le> m \<and> N \<le> n \<and> P x"

  4314       then have "dist (s m x) (s n x) < e"

  4315         using N[THEN spec[where x=m], THEN spec[where x=x]]

  4316         using N[THEN spec[where x=n], THEN spec[where x=x]]

  4317         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

  4318     }

  4319     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

  4320   }

  4321   then show ?rhs by auto

  4322 next

  4323   assume ?rhs

  4324   then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"

  4325     unfolding cauchy_def

  4326     apply auto

  4327     apply (erule_tac x=e in allE)

  4328     apply auto

  4329     done

  4330   then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"

  4331     unfolding convergent_eq_cauchy[THEN sym]

  4332     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]

  4333     by auto

  4334   {

  4335     fix e :: real

  4336     assume "e > 0"

  4337     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"

  4338       using ?rhs[THEN spec[where x="e/2"]] by auto

  4339     {

  4340       fix x

  4341       assume "P x"

  4342       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  4343         using l[THEN spec[where x=x], unfolded LIMSEQ_def] and e > 0

  4344         by (auto elim!: allE[where x="e/2"])

  4345       fix n :: nat

  4346       assume "n \<ge> N"

  4347       then have "dist(s n x)(l x) < e"

  4348         using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  4349         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

  4350         by (auto simp add: dist_commute)

  4351     }

  4352     then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4353       by auto

  4354   }

  4355   then show ?lhs by auto

  4356 qed

  4357

  4358 lemma uniformly_cauchy_imp_uniformly_convergent:

  4359   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  4360   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"

  4361           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  4362   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  4363 proof -

  4364   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"

  4365     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  4366   moreover

  4367   {

  4368     fix x

  4369     assume "P x"

  4370     then have "l x = l' x"

  4371       using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  4372       using l and assms(2) unfolding LIMSEQ_def by blast

  4373   }

  4374   ultimately show ?thesis by auto

  4375 qed

  4376

  4377

  4378 subsection {* Continuity *}

  4379

  4380 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  4381

  4382 lemma continuous_within_eps_delta:

  4383   "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)"

  4384   unfolding continuous_within and Lim_within

  4385   apply auto

  4386   unfolding dist_nz[THEN sym]

  4387   apply (auto del: allE elim!:allE)

  4388   apply(rule_tac x=d in exI)

  4389   apply auto

  4390   done

  4391

  4392 lemma continuous_at_eps_delta:

  4393   "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  4394   using continuous_within_eps_delta [of x UNIV f] by simp

  4395

  4396 text{* Versions in terms of open balls. *}

  4397

  4398 lemma continuous_within_ball:

  4399   "continuous (at x within s) f \<longleftrightarrow>

  4400     (\<forall>e > 0. \<exists>d > 0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e)"

  4401   (is "?lhs = ?rhs")

  4402 proof

  4403   assume ?lhs

  4404   {

  4405     fix e :: real

  4406     assume "e > 0"

  4407     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"

  4408       using ?lhs[unfolded continuous_within Lim_within] by auto

  4409     {

  4410       fix y

  4411       assume "y \<in> f  (ball x d \<inter> s)"

  4412       then have "y \<in> ball (f x) e"

  4413         using d(2)

  4414         unfolding dist_nz[THEN sym]

  4415         apply (auto simp add: dist_commute)

  4416         apply (erule_tac x=xa in ballE)

  4417         apply auto

  4418         using e > 0

  4419         apply auto

  4420         done

  4421     }

  4422     then have "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e"

  4423       using d > 0

  4424       unfolding subset_eq ball_def by (auto simp add: dist_commute)

  4425   }

  4426   then show ?rhs by auto

  4427 next

  4428   assume ?rhs

  4429   then show ?lhs

  4430     unfolding continuous_within Lim_within ball_def subset_eq

  4431     apply (auto simp add: dist_commute)

  4432     apply (erule_tac x=e in allE)

  4433     apply auto

  4434     done

  4435 qed

  4436

  4437 lemma continuous_at_ball:

  4438   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  4439 proof

  4440   assume ?lhs

  4441   then show ?rhs

  4442     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4443     apply auto

  4444     apply (erule_tac x=e in allE)

  4445     apply auto

  4446     apply (rule_tac x=d in exI)

  4447     apply auto

  4448     apply (erule_tac x=xa in allE)

  4449     apply (auto simp add: dist_commute dist_nz)

  4450     unfolding dist_nz[THEN sym]

  4451     apply auto

  4452     done

  4453 next

  4454   assume ?rhs

  4455   then show ?lhs

  4456     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4457     apply auto

  4458     apply (erule_tac x=e in allE)

  4459     apply auto

  4460     apply (rule_tac x=d in exI)

  4461     apply auto

  4462     apply (erule_tac x="f xa" in allE)

  4463     apply (auto simp add: dist_commute dist_nz)

  4464     done

  4465 qed

  4466

  4467 text{* Define setwise continuity in terms of limits within the set. *}

  4468

  4469 lemma continuous_on_iff:

  4470   "continuous_on s f \<longleftrightarrow>

  4471     (\<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)"

  4472   unfolding continuous_on_def Lim_within

  4473   apply (intro ball_cong [OF refl] all_cong ex_cong)

  4474   apply (rename_tac y, case_tac "y = x")

  4475   apply simp

  4476   apply (simp add: dist_nz)

  4477   done

  4478

  4479 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  4480   where "uniformly_continuous_on s f \<longleftrightarrow>

  4481     (\<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)"

  4482

  4483 text{* Some simple consequential lemmas. *}

  4484

  4485 lemma uniformly_continuous_imp_continuous:

  4486   "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"

  4487   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  4488

  4489 lemma continuous_at_imp_continuous_within:

  4490   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"

  4491   unfolding continuous_within continuous_at using Lim_at_within by auto

  4492

  4493 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  4494   by simp

  4495

  4496 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4497

  4498 lemma continuous_within_subset:

  4499   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"

  4500   unfolding continuous_within by(metis tendsto_within_subset)

  4501

  4502 lemma continuous_on_interior:

  4503   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4504   apply (erule interiorE)

  4505   apply (drule (1) continuous_on_subset)

  4506   apply (simp add: continuous_on_eq_continuous_at)

  4507   done

  4508

  4509 lemma continuous_on_eq:

  4510   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4511   unfolding continuous_on_def tendsto_def eventually_at_topological

  4512   by simp

  4513

  4514 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4515

  4516 lemma continuous_within_sequentially:

  4517   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4518   shows "continuous (at a within s) f \<longleftrightarrow>

  4519     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4520          \<longrightarrow> ((f o x) ---> f a) sequentially)"

  4521   (is "?lhs = ?rhs")

  4522 proof

  4523   assume ?lhs

  4524   {

  4525     fix x :: "nat \<Rightarrow> 'a"

  4526     assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4527     fix T :: "'b set"

  4528     assume "open T" and "f a \<in> T"

  4529     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"

  4530       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  4531     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4532       using x(2) d>0 by simp

  4533     then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4534     proof eventually_elim

  4535       case (elim n)

  4536       then show ?case

  4537         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  4538     qed

  4539   }

  4540   then show ?rhs

  4541     unfolding tendsto_iff tendsto_def by simp

  4542 next

  4543   assume ?rhs

  4544   then show ?lhs

  4545     unfolding continuous_within tendsto_def [where l="f a"]

  4546     by (simp add: sequentially_imp_eventually_within)

  4547 qed

  4548

  4549 lemma continuous_at_sequentially:

  4550   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4551   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  4552                   --> ((f o x) ---> f a) sequentially)"

  4553   using continuous_within_sequentially[of a UNIV f] by simp

  4554

  4555 lemma continuous_on_sequentially:

  4556   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4557   shows "continuous_on s f \<longleftrightarrow>

  4558     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4559                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  4560 proof

  4561   assume ?rhs

  4562   then show ?lhs

  4563     using continuous_within_sequentially[of _ s f]

  4564     unfolding continuous_on_eq_continuous_within

  4565     by auto

  4566 next

  4567   assume ?lhs

  4568   then show ?rhs

  4569     unfolding continuous_on_eq_continuous_within

  4570     using continuous_within_sequentially[of _ s f]

  4571     by auto

  4572 qed

  4573

  4574 lemma uniformly_continuous_on_sequentially:

  4575   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4576                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4577                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4578 proof

  4579   assume ?lhs

  4580   {

  4581     fix x y

  4582     assume x: "\<forall>n. x n \<in> s"

  4583       and y: "\<forall>n. y n \<in> s"

  4584       and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4585     {

  4586       fix e :: real

  4587       assume "e > 0"

  4588       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"

  4589         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4590       obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"

  4591         using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4592       {

  4593         fix n

  4594         assume "n\<ge>N"

  4595         then have "dist (f (x n)) (f (y n)) < e"

  4596           using N[THEN spec[where x=n]]

  4597           using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

  4598           using x and y

  4599           unfolding dist_commute

  4600           by simp

  4601       }

  4602       then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4603         by auto

  4604     }

  4605     then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"

  4606       unfolding LIMSEQ_def and dist_real_def by auto

  4607   }

  4608   then show ?rhs by auto

  4609 next

  4610   assume ?rhs

  4611   {

  4612     assume "\<not> ?lhs"

  4613     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"

  4614       unfolding uniformly_continuous_on_def by auto

  4615     then obtain fa where fa:

  4616       "\<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"

  4617       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"]

  4618       unfolding Bex_def

  4619       by (auto simp add: dist_commute)

  4620     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4621     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4622     have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"

  4623       and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"

  4624       and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4625       unfolding x_def and y_def using fa

  4626       by auto

  4627     {

  4628       fix e :: real

  4629       assume "e > 0"

  4630       then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"

  4631         unfolding real_arch_inv[of e] by auto

  4632       {

  4633         fix n :: nat

  4634         assume "n \<ge> N"

  4635         then have "inverse (real n + 1) < inverse (real N)"

  4636           using real_of_nat_ge_zero and N\<noteq>0 by auto

  4637         also have "\<dots> < e" using N by auto

  4638         finally have "inverse (real n + 1) < e" by auto

  4639         then have "dist (x n) (y n) < e"

  4640           using xy0[THEN spec[where x=n]] by auto

  4641       }

  4642       then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto

  4643     }

  4644     then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4645       using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn

  4646       unfolding LIMSEQ_def dist_real_def by auto

  4647     then have False using fxy and e>0 by auto

  4648   }

  4649   then show ?lhs

  4650     unfolding uniformly_continuous_on_def by blast

  4651 qed

  4652

  4653 text{* The usual transformation theorems. *}

  4654

  4655 lemma continuous_transform_within:

  4656   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4657   assumes "0 < d"

  4658     and "x \<in> s"

  4659     and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4660     and "continuous (at x within s) f"

  4661   shows "continuous (at x within s) g"

  4662   unfolding continuous_within

  4663 proof (rule Lim_transform_within)

  4664   show "0 < d" by fact

  4665   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4666     using assms(3) by auto

  4667   have "f x = g x"

  4668     using assms(1,2,3) by auto

  4669   then show "(f ---> g x) (at x within s)"

  4670     using assms(4) unfolding continuous_within by simp

  4671 qed

  4672

  4673 lemma continuous_transform_at:

  4674   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4675   assumes "0 < d"

  4676     and "\<forall>x'. dist x' x < d --> f x' = g x'"

  4677     and "continuous (at x) f"

  4678   shows "continuous (at x) g"

  4679   using continuous_transform_within [of d x UNIV f g] assms by simp

  4680

  4681

  4682 subsubsection {* Structural rules for pointwise continuity *}

  4683

  4684 lemmas continuous_within_id = continuous_ident

  4685

  4686 lemmas continuous_at_id = isCont_ident

  4687

  4688 lemma continuous_infdist[continuous_intros]:

  4689   assumes "continuous F f"

  4690   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4691   using assms unfolding continuous_def by (rule tendsto_infdist)

  4692

  4693 lemma continuous_infnorm[continuous_intros]:

  4694   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4695   unfolding continuous_def by (rule tendsto_infnorm)

  4696

  4697 lemma continuous_inner[continuous_intros]:

  4698   assumes "continuous F f"

  4699     and "continuous F g"

  4700   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4701   using assms unfolding continuous_def by (rule tendsto_inner)

  4702

  4703 lemmas continuous_at_inverse = isCont_inverse

  4704

  4705 subsubsection {* Structural rules for setwise continuity *}

  4706

  4707 lemma continuous_on_infnorm[continuous_on_intros]:

  4708   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4709   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4710

  4711 lemma continuous_on_inner[continuous_on_intros]:

  4712   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4713   assumes "continuous_on s f"

  4714     and "continuous_on s g"

  4715   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4716   using bounded_bilinear_inner assms

  4717   by (rule bounded_bilinear.continuous_on)

  4718

  4719 subsubsection {* Structural rules for uniform continuity *}

  4720

  4721 lemma uniformly_continuous_on_id[continuous_on_intros]:

  4722   "uniformly_continuous_on s (\<lambda>x. x)"

  4723   unfolding uniformly_continuous_on_def by auto

  4724

  4725 lemma uniformly_continuous_on_const[continuous_on_intros]:

  4726   "uniformly_continuous_on s (\<lambda>x. c)"

  4727   unfolding uniformly_continuous_on_def by simp

  4728

  4729 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  4730   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4731   assumes "uniformly_continuous_on s f"

  4732     and "uniformly_continuous_on s g"

  4733   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4734 proof -

  4735   {

  4736     fix a b c d :: 'b

  4737     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4738       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4739       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4740       by arith

  4741   } note le = this

  4742   {

  4743     fix x y

  4744     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4745     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4746     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4747       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4748         simp add: le)

  4749   }

  4750   then show ?thesis

  4751     using assms unfolding uniformly_continuous_on_sequentially

  4752     unfolding dist_real_def by simp

  4753 qed

  4754

  4755 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  4756   assumes "uniformly_continuous_on s f"

  4757   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4758   unfolding norm_conv_dist using assms

  4759   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4760

  4761 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  4762   assumes "uniformly_continuous_on s g"

  4763   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4764   using assms unfolding uniformly_continuous_on_sequentially

  4765   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4766   by (auto intro: tendsto_zero)

  4767

  4768 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  4769   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4770   assumes "uniformly_continuous_on s f"

  4771   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4772   using bounded_linear_scaleR_right assms

  4773   by (rule bounded_linear.uniformly_continuous_on)

  4774

  4775 lemma dist_minus:

  4776   fixes x y :: "'a::real_normed_vector"

  4777   shows "dist (- x) (- y) = dist x y"

  4778   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4779

  4780 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  4781   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4782   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4783   unfolding uniformly_continuous_on_def dist_minus .

  4784

  4785 lemma uniformly_continuous_on_add[continuous_on_intros]:

  4786   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4787   assumes "uniformly_continuous_on s f"

  4788     and "uniformly_continuous_on s g"

  4789   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4790   using assms

  4791   unfolding uniformly_continuous_on_sequentially

  4792   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4793   by (auto intro: tendsto_add_zero)

  4794

  4795 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  4796   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4797   assumes "uniformly_continuous_on s f"

  4798     and "uniformly_continuous_on s g"

  4799   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4800   unfolding ab_diff_minus using assms

  4801   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4802

  4803 text{* Continuity of all kinds is preserved under composition. *}

  4804

  4805 lemmas continuous_at_compose = isCont_o

  4806

  4807 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  4808   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4809   shows "uniformly_continuous_on s (g o f)"

  4810 proof-

  4811   {

  4812     fix e :: real

  4813     assume "e > 0"

  4814     then obtain d where "d > 0"

  4815       and d: "\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4816       using assms(2) unfolding uniformly_continuous_on_def by auto

  4817     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"

  4818       using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto

  4819     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"

  4820       using d>0 using d by auto

  4821   }

  4822   then show ?thesis

  4823     using assms unfolding uniformly_continuous_on_def by auto

  4824 qed

  4825

  4826 text{* Continuity in terms of open preimages. *}

  4827

  4828 lemma continuous_at_open:

  4829   "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)))"

  4830   unfolding continuous_within_topological [of x UNIV f]

  4831   unfolding imp_conjL

  4832   by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4833

  4834 lemma continuous_imp_tendsto:

  4835   assumes "continuous (at x0) f"

  4836     and "x ----> x0"

  4837   shows "(f \<circ> x) ----> (f x0)"

  4838 proof (rule topological_tendstoI)

  4839   fix S

  4840   assume "open S" "f x0 \<in> S"

  4841   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  4842      using assms continuous_at_open by metis

  4843   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  4844     using assms T_def by (auto simp: tendsto_def)

  4845   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  4846     using T_def by (auto elim!: eventually_elim1)

  4847 qed

  4848

  4849 lemma continuous_on_open:

  4850   "continuous_on s f \<longleftrightarrow>

  4851     (\<forall>t. openin (subtopology euclidean (f  s)) t \<longrightarrow>

  4852       openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  4853   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  4854   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4855

  4856 text {* Similarly in terms of closed sets. *}

  4857

  4858 lemma continuous_on_closed:

  4859   "continuous_on s f \<longleftrightarrow>

  4860     (\<forall>t. closedin (subtopology euclidean (f  s)) t \<longrightarrow>

  4861       closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  4862   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  4863   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4864

  4865 text {* Half-global and completely global cases. *}

  4866

  4867 lemma continuous_open_in_preimage:

  4868   assumes "continuous_on s f"  "open t"

  4869   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4870 proof -

  4871   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  4872     by auto

  4873   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4874     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4875   then show ?thesis

  4876     using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]]

  4877     using * by auto

  4878 qed

  4879

  4880 lemma continuous_closed_in_preimage:

  4881   assumes "continuous_on s f"  "closed t"

  4882   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4883 proof -

  4884   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  4885     by auto

  4886   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4887     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute

  4888     by auto

  4889   then show ?thesis

  4890     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]]

  4891     using * by auto

  4892 qed

  4893

  4894 lemma continuous_open_preimage:

  4895   assumes "continuous_on s f" "open s" "open t"

  4896   shows "open {x \<in> s. f x \<in> t}"

  4897 proof-

  4898   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4899     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4900   then show ?thesis

  4901     using open_Int[of s T, OF assms(2)] by auto

  4902 qed

  4903

  4904 lemma continuous_closed_preimage:

  4905   assumes "continuous_on s f" "closed s" "closed t"

  4906   shows "closed {x \<in> s. f x \<in> t}"

  4907 proof-

  4908   obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4909     using continuous_closed_in_preimage[OF assms(1,3)]

  4910     unfolding closedin_closed by auto

  4911   then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4912 qed

  4913

  4914 lemma continuous_open_preimage_univ:

  4915   "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4916   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4917

  4918 lemma continuous_closed_preimage_univ:

  4919   "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4920   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4921

  4922 lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4923   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4924

  4925 lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4926   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4927

  4928 lemma interior_image_subset:

  4929   assumes "\<forall>x. continuous (at x) f" "inj f"

  4930   shows "interior (f  s) \<subseteq> f  (interior s)"

  4931 proof

  4932   fix x assume "x \<in> interior (f  s)"

  4933   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4934   then have "x \<in> f  s" by auto

  4935   then obtain y where y: "y \<in> s" "x = f y" by auto

  4936   have "open (vimage f T)"

  4937     using assms(1) open T by (rule continuous_open_vimage)

  4938   moreover have "y \<in> vimage f T"

  4939     using x = f y x \<in> T by simp

  4940   moreover have "vimage f T \<subseteq> s"

  4941     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4942   ultimately have "y \<in> interior s" ..

  4943   with x = f y show "x \<in> f  interior s" ..

  4944 qed

  4945

  4946 text {* Equality of continuous functions on closure and related results. *}

  4947

  4948 lemma continuous_closed_in_preimage_constant:

  4949   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4950   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4951   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4952

  4953 lemma continuous_closed_preimage_constant:

  4954   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4955   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4956   using continuous_closed_preimage[of s f "{a}"] by auto

  4957

  4958 lemma continuous_constant_on_closure:

  4959   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4960   assumes "continuous_on (closure s) f"

  4961     and "\<forall>x \<in> s. f x = a"

  4962   shows "\<forall>x \<in> (closure s). f x = a"

  4963     using continuous_closed_preimage_constant[of "closure s" f a]

  4964       assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset

  4965     unfolding subset_eq

  4966     by auto

  4967

  4968 lemma image_closure_subset:

  4969   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4970   shows "f  (closure s) \<subseteq> t"

  4971 proof -

  4972   have "s \<subseteq> {x \<in> closure s. f x \<in> t}"

  4973     using assms(3) closure_subset by auto

  4974   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4975     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4976   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4977     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4978   then show ?thesis by auto

  4979 qed

  4980

  4981 lemma continuous_on_closure_norm_le:

  4982   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4983   assumes "continuous_on (closure s) f"

  4984     and "\<forall>y \<in> s. norm(f y) \<le> b"

  4985     and "x \<in> (closure s)"

  4986   shows "norm(f x) \<le> b"

  4987 proof -

  4988   have *: "f  s \<subseteq> cball 0 b"

  4989     using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4990   show ?thesis

  4991     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4992     unfolding subset_eq

  4993     apply (erule_tac x="f x" in ballE)

  4994     apply (auto simp add: dist_norm)

  4995     done

  4996 qed

  4997

  4998 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4999

  5000 lemma continuous_within_avoid:

  5001   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5002   assumes "continuous (at x within s) f"

  5003     and "f x \<noteq> a"

  5004   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  5005 proof-

  5006   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  5007     using t1_space [OF f x \<noteq> a] by fast

  5008   have "(f ---> f x) (at x within s)"

  5009     using assms(1) by (simp add: continuous_within)

  5010   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  5011     using open U and f x \<in> U

  5012     unfolding tendsto_def by fast

  5013   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  5014     using a \<notin> U by (fast elim: eventually_mono [rotated])

  5015   then show ?thesis

  5016     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  5017 qed

  5018

  5019 lemma continuous_at_avoid:

  5020   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5021   assumes "continuous (at x) f"

  5022     and "f x \<noteq> a"

  5023   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5024   using assms continuous_within_avoid[of x UNIV f a] by simp

  5025

  5026 lemma continuous_on_avoid:

  5027   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5028   assumes "continuous_on s f"

  5029     and "x \<in> s"

  5030     and "f x \<noteq> a"

  5031   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  5032   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  5033     OF assms(2)] continuous_within_avoid[of x s f a]

  5034   using assms(3)

  5035   by auto

  5036

  5037 lemma continuous_on_open_avoid:

  5038   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5039   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  5040   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5041   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  5042   using continuous_at_avoid[of x f a] assms(4)

  5043   by auto

  5044

  5045 text {* Proving a function is constant by proving open-ness of level set. *}

  5046

  5047 lemma continuous_levelset_open_in_cases:

  5048   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5049   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5050         openin (subtopology euclidean s) {x \<in> s. f x = a}

  5051         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  5052   unfolding connected_clopen

  5053   using continuous_closed_in_preimage_constant by auto

  5054

  5055 lemma continuous_levelset_open_in:

  5056   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5057   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5058         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  5059         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  5060   using continuous_levelset_open_in_cases[of s f ]

  5061   by meson

  5062

  5063 lemma continuous_levelset_open:

  5064   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5065   assumes "connected s"

  5066     and "continuous_on s f"

  5067     and "open {x \<in> s. f x = a}"

  5068     and "\<exists>x \<in> s.  f x = a"

  5069   shows "\<forall>x \<in> s. f x = a"

  5070   using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

  5071   using assms (3,4)

  5072   by fast

  5073

  5074 text {* Some arithmetical combinations (more to prove). *}

  5075

  5076 lemma open_scaling[intro]:

  5077   fixes s :: "'a::real_normed_vector set"

  5078   assumes "c \<noteq> 0"  "open s"

  5079   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  5080 proof -

  5081   {

  5082     fix x

  5083     assume "x \<in> s"

  5084     then obtain e where "e>0"

  5085       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  5086       by auto

  5087     have "e * abs c > 0"

  5088       using assms(1)[unfolded zero_less_abs_iff[THEN sym]]

  5089       using mult_pos_pos[OF e>0]

  5090       by auto

  5091     moreover

  5092     {

  5093       fix y

  5094       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  5095       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  5096         unfolding dist_norm

  5097         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  5098           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  5099       then have "y \<in> op *\<^sub>R c  s"

  5100         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]

  5101         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  5102         using assms(1)

  5103         unfolding dist_norm scaleR_scaleR

  5104         by auto

  5105     }

  5106     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s"

  5107       apply (rule_tac x="e * abs c" in exI)

  5108       apply auto

  5109       done

  5110   }

  5111   then show ?thesis unfolding open_dist by auto

  5112 qed

  5113

  5114 lemma minus_image_eq_vimage:

  5115   fixes A :: "'a::ab_group_add set"

  5116   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  5117   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  5118

  5119 lemma open_negations:

  5120   fixes s :: "'a::real_normed_vector set"

  5121   shows "open s ==> open ((\<lambda> x. -x)  s)"

  5122   unfolding scaleR_minus1_left [symmetric]

  5123   by (rule open_scaling, auto)

  5124

  5125 lemma open_translation:

  5126   fixes s :: "'a::real_normed_vector set"

  5127   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  5128 proof -

  5129   {

  5130     fix x

  5131     have "continuous (at x) (\<lambda>x. x - a)"

  5132       by (intro continuous_diff continuous_at_id continuous_const)

  5133   }

  5134   moreover have "{x. x - a \<in> s} = op + a  s"

  5135     by force

  5136   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]

  5137     using assms by auto

  5138 qed

  5139

  5140 lemma open_affinity:

  5141   fixes s :: "'a::real_normed_vector set"

  5142   assumes "open s"  "c \<noteq> 0"

  5143   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5144 proof -

  5145   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  5146     unfolding o_def ..

  5147   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s"

  5148     by auto

  5149   then show ?thesis

  5150     using assms open_translation[of "op *\<^sub>R c  s" a]

  5151     unfolding *

  5152     by auto

  5153 qed

  5154

  5155 lemma interior_translation:

  5156   fixes s :: "'a::real_normed_vector set"

  5157   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  5158 proof (rule set_eqI, rule)

  5159   fix x

  5160   assume "x \<in> interior (op + a  s)"

  5161   then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a  s"

  5162     unfolding mem_interior by auto

  5163   then have "ball (x - a) e \<subseteq> s"

  5164     unfolding subset_eq Ball_def mem_ball dist_norm

  5165     apply auto

  5166     apply (erule_tac x="a + xa" in allE)

  5167     unfolding ab_group_add_class.diff_diff_eq[THEN sym]

  5168     apply auto

  5169     done

  5170   then show "x \<in> op + a  interior s"

  5171     unfolding image_iff

  5172     apply (rule_tac x="x - a" in bexI)

  5173     unfolding mem_interior

  5174     using e > 0

  5175     apply auto

  5176     done

  5177 next

  5178   fix x

  5179   assume "x \<in> op + a  interior s"

  5180   then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"

  5181     unfolding image_iff Bex_def mem_interior by auto

  5182   {

  5183     fix z

  5184     have *: "a + y - z = y + a - z" by auto

  5185     assume "z \<in> ball x e"

  5186     then have "z - a \<in> s"

  5187       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  5188       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  5189       by auto

  5190     then have "z \<in> op + a  s"

  5191       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  5192   }

  5193   then have "ball x e \<subseteq> op + a  s"

  5194     unfolding subset_eq by auto

  5195   then show "x \<in> interior (op + a  s)"

  5196     unfolding mem_interior using e > 0 by auto

  5197 qed

  5198

  5199 text {* Topological properties of linear functions. *}

  5200

  5201 lemma linear_lim_0:

  5202   assumes "bounded_linear f"

  5203   shows "(f ---> 0) (at (0))"

  5204 proof -

  5205   interpret f: bounded_linear f by fact

  5206   have "(f ---> f 0) (at 0)"

  5207     using tendsto_ident_at by (rule f.tendsto)

  5208   then show ?thesis unfolding f.zero .

  5209 qed

  5210

  5211 lemma linear_continuous_at:

  5212   assumes "bounded_linear f"

  5213   shows "continuous (at a) f"

  5214   unfolding continuous_at using assms

  5215   apply (rule bounded_linear.tendsto)

  5216   apply (rule tendsto_ident_at)

  5217   done

  5218

  5219 lemma linear_continuous_within:

  5220   "bounded_linear f ==> continuous (at x within s) f"

  5221   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  5222

  5223 lemma linear_continuous_on:

  5224   "bounded_linear f ==> continuous_on s f"

  5225   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  5226

  5227 text {* Also bilinear functions, in composition form. *}

  5228

  5229 lemma bilinear_continuous_at_compose:

  5230   "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5231     continuous (at x) (\<lambda>x. h (f x) (g x))"

  5232   unfolding continuous_at

  5233   using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

  5234   by auto

  5235

  5236 lemma bilinear_continuous_within_compose:

  5237   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5238     continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  5239   unfolding continuous_within

  5240   using Lim_bilinear[of f "f x"]

  5241   by auto

  5242

  5243 lemma bilinear_continuous_on_compose:

  5244   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5245     continuous_on s (\<lambda>x. h (f x) (g x))"

  5246   unfolding continuous_on_def

  5247   by (fast elim: bounded_bilinear.tendsto)

  5248

  5249 text {* Preservation of compactness and connectedness under continuous function. *}

  5250

  5251 lemma compact_eq_openin_cover:

  5252   "compact S \<longleftrightarrow>

  5253     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5254       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  5255 proof safe

  5256   fix C

  5257   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  5258   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}"

  5259     unfolding openin_open by force+

  5260   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"

  5261     by (rule compactE)

  5262   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)"

  5263     by auto

  5264   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5265 next

  5266   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5267         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  5268   show "compact S"

  5269   proof (rule compactI)

  5270     fix C

  5271     let ?C = "image (\<lambda>T. S \<inter> T) C"

  5272     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  5273     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  5274       unfolding openin_open by auto

  5275     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  5276       by metis

  5277     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  5278     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  5279     proof (intro conjI)

  5280       from D \<subseteq> ?C show "?D \<subseteq> C"

  5281         by (fast intro: inv_into_into)

  5282       from finite D show "finite ?D"

  5283         by (rule finite_imageI)

  5284       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  5285         apply (rule subset_trans)

  5286         apply clarsimp

  5287         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  5288         apply (erule rev_bexI, fast)

  5289         done

  5290     qed

  5291     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5292   qed

  5293 qed

  5294

  5295 lemma connected_continuous_image:

  5296   assumes "continuous_on s f"  "connected s"

  5297   shows "connected(f  s)"

  5298 proof -

  5299   {

  5300     fix T

  5301     assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  5302     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  5303       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  5304       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  5305       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  5306     then have False using as(1,2)

  5307       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

  5308   }

  5309   then show ?thesis

  5310     unfolding connected_clopen by auto

  5311 qed

  5312

  5313 text {* Continuity implies uniform continuity on a compact domain. *}

  5314

  5315 lemma compact_uniformly_continuous:

  5316   assumes f: "continuous_on s f" and s: "compact s"

  5317   shows "uniformly_continuous_on s f"

  5318   unfolding uniformly_continuous_on_def

  5319 proof (cases, safe)

  5320   fix e :: real

  5321   assume "0 < e" "s \<noteq> {}"

  5322   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) } }"

  5323   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  5324   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  5325   proof safe

  5326     fix y

  5327     assume "y \<in> s"

  5328     from continuous_open_in_preimage[OF f open_ball]

  5329     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  5330       unfolding openin_subtopology open_openin by metis

  5331     then obtain d where "ball y d \<subseteq> T" "0 < d"

  5332       using 0 < e y \<in> s by (auto elim!: openE)

  5333     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  5334       by (intro UN_I[of "(y, d)"]) auto

  5335   qed auto

  5336   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  5337     by (rule compactE_image)

  5338   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  5339     by (subst Min_gr_iff) auto

  5340   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  5341   proof (rule, safe)

  5342     fix x x'

  5343     assume in_s: "x' \<in> s" "x \<in> s"

  5344     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  5345       by blast

  5346     moreover assume "dist x x' < Min (sndD) / 2"

  5347     ultimately have "dist y x' < d"

  5348       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  5349     with D x in_s show  "dist (f x) (f x') < e"

  5350       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  5351   qed (insert D, auto)

  5352 qed auto

  5353

  5354 text {* A uniformly convergent limit of continuous functions is continuous. *}

  5355

  5356 lemma continuous_uniform_limit:

  5357   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  5358   assumes "\<not> trivial_limit F"

  5359     and "eventually (\<lambda>n. continuous_on s (f n)) F"

  5360     and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  5361   shows "continuous_on s g"

  5362 proof -

  5363   {

  5364     fix x and e :: real

  5365     assume "x\<in>s" "e>0"

  5366     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  5367       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  5368     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  5369     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  5370       using assms(1) by blast

  5371     have "e / 3 > 0" using e>0 by auto

  5372     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"

  5373       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  5374     {

  5375       fix y

  5376       assume "y \<in> s" and "dist y x < d"

  5377       then have "dist (f n y) (f n x) < e / 3"

  5378         by (rule d [rule_format])

  5379       then have "dist (f n y) (g x) < 2 * e / 3"

  5380         using dist_triangle [of "f n y" "g x" "f n x"]

  5381         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  5382         by auto

  5383       then have "dist (g y) (g x) < e"

  5384         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  5385         using dist_triangle3 [of "g y" "g x" "f n y"]

  5386         by auto

  5387     }

  5388     then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5389       using d>0 by auto

  5390   }

  5391   then show ?thesis

  5392     unfolding continuous_on_iff by auto

  5393 qed

  5394

  5395

  5396 subsection {* Topological stuff lifted from and dropped to R *}

  5397

  5398 lemma open_real:

  5399   fixes s :: "real set"

  5400   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"

  5401   unfolding open_dist dist_norm by simp

  5402

  5403 lemma islimpt_approachable_real:

  5404   fixes s :: "real set"

  5405   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  5406   unfolding islimpt_approachable dist_norm by simp

  5407

  5408 lemma closed_real:

  5409   fixes s :: "real set"

  5410   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)"

  5411   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5412

  5413 lemma continuous_at_real_range:

  5414   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5415   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5416   unfolding continuous_at

  5417   unfolding Lim_at

  5418   unfolding dist_nz[THEN sym]

  5419   unfolding dist_norm

  5420   apply auto

  5421   apply (erule_tac x=e in allE)

  5422   apply auto

  5423   apply (rule_tac x=d in exI)

  5424   apply auto

  5425   apply (erule_tac x=x' in allE)

  5426   apply auto

  5427   apply (erule_tac x=e in allE)

  5428   apply auto

  5429   done

  5430

  5431 lemma continuous_on_real_range:

  5432   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5433   shows "continuous_on s f \<longleftrightarrow>

  5434     (\<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))"

  5435   unfolding continuous_on_iff dist_norm by simp

  5436

  5437 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5438

  5439 lemma distance_attains_sup:

  5440   assumes "compact s" "s \<noteq> {}"

  5441   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  5442 proof (rule continuous_attains_sup [OF assms])

  5443   {

  5444     fix x

  5445     assume "x\<in>s"

  5446     have "(dist a ---> dist a x) (at x within s)"

  5447       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  5448   }

  5449   then show "continuous_on s (dist a)"

  5450     unfolding continuous_on ..

  5451 qed

  5452

  5453 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5454

  5455 lemma distance_attains_inf:

  5456   fixes a :: "'a::heine_borel"

  5457   assumes "closed s"  "s \<noteq> {}"

  5458   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  5459 proof -

  5460   from assms(2) obtain b where "b \<in> s" by auto

  5461   let ?B = "s \<inter> cball a (dist b a)"

  5462   have "?B \<noteq> {}" using b \<in> s

  5463     by (auto simp add: dist_commute)

  5464   moreover have "continuous_on ?B (dist a)"

  5465     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  5466   moreover have "compact ?B"

  5467     by (intro closed_inter_compact closed s compact_cball)

  5468   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  5469     by (metis continuous_attains_inf)

  5470   then show ?thesis by fastforce

  5471 qed

  5472

  5473

  5474 subsection {* Pasted sets *}

  5475

  5476 lemma bounded_Times:

  5477   assumes "bounded s" "bounded t"

  5478   shows "bounded (s \<times> t)"

  5479 proof -

  5480   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5481     using assms [unfolded bounded_def] by auto

  5482   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  5483     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5484   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5485 qed

  5486

  5487 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5488   by (induct x) simp

  5489

  5490 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5491   unfolding seq_compact_def

  5492   apply clarify

  5493   apply (drule_tac x="fst \<circ> f" in spec)

  5494   apply (drule mp, simp add: mem_Times_iff)

  5495   apply (clarify, rename_tac l1 r1)

  5496   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5497   apply (drule mp, simp add: mem_Times_iff)

  5498   apply (clarify, rename_tac l2 r2)

  5499   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5500   apply (rule_tac x="r1 \<circ> r2" in exI)

  5501   apply (rule conjI, simp add: subseq_def)

  5502   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  5503   apply (drule (1) tendsto_Pair) back

  5504   apply (simp add: o_def)

  5505   done

  5506

  5507 lemma compact_Times:

  5508   assumes "compact s" "compact t"

  5509   shows "compact (s \<times> t)"

  5510 proof (rule compactI)

  5511   fix C

  5512   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  5513   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)"

  5514   proof

  5515     fix x

  5516     assume "x \<in> s"

  5517     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")

  5518     proof

  5519       fix y

  5520       assume "y \<in> t"

  5521       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  5522       then show "?P y" by (auto elim!: open_prod_elim)

  5523     qed

  5524     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  5525       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"

  5526       by metis

  5527     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  5528     from compactE_image[OF compact t this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  5529       by auto

  5530     moreover with c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  5531       by (fastforce simp: subset_eq)

  5532     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  5533       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  5534   qed

  5535   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  5536     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"

  5537     unfolding subset_eq UN_iff by metis

  5538   moreover

  5539   from compactE_image[OF compact s a]

  5540   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  5541     by auto

  5542   moreover

  5543   {

  5544     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  5545       by auto

  5546     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  5547       using d e \<subseteq> s by (intro UN_mono) auto

  5548     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  5549   }

  5550   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  5551     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  5552 qed

  5553

  5554 text{* Hence some useful properties follow quite easily. *}

  5555

  5556 lemma compact_scaling:

  5557   fixes s :: "'a::real_normed_vector set"

  5558   assumes "compact s"

  5559   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5560 proof -

  5561   let ?f = "\<lambda>x. scaleR c x"

  5562   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5563   show ?thesis

  5564     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5565     using linear_continuous_at[OF *] assms

  5566     by auto

  5567 qed

  5568

  5569 lemma compact_negations:

  5570   fixes s :: "'a::real_normed_vector set"

  5571   assumes "compact s"

  5572   shows "compact ((\<lambda>x. -x)  s)"

  5573   using compact_scaling [OF assms, of "- 1"] by auto

  5574

  5575 lemma compact_sums:

  5576   fixes s t :: "'a::real_normed_vector set"

  5577   assumes "compact s" and "compact t"

  5578   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5579 proof -

  5580   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5581     apply auto

  5582     unfolding image_iff

  5583     apply (rule_tac x="(xa, y)" in bexI)

  5584     apply auto

  5585     done

  5586   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5587     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5588   then show ?thesis

  5589     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5590 qed

  5591

  5592 lemma compact_differences:

  5593   fixes s t :: "'a::real_normed_vector set"

  5594   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5595 proof-

  5596   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5597     apply auto

  5598     apply (rule_tac x= xa in exI)

  5599     apply auto

  5600     apply (rule_tac x=xa in exI)

  5601     apply auto

  5602     done

  5603   then show ?thesis

  5604     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5605 qed

  5606

  5607 lemma compact_translation:

  5608   fixes s :: "'a::real_normed_vector set"

  5609   assumes "compact s"

  5610   shows "compact ((\<lambda>x. a + x)  s)"

  5611 proof -

  5612   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  5613     by auto

  5614   then show ?thesis

  5615     using compact_sums[OF assms compact_sing[of a]] by auto

  5616 qed

  5617

  5618 lemma compact_affinity:

  5619   fixes s :: "'a::real_normed_vector set"

  5620   assumes "compact s"

  5621   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5622 proof -

  5623   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  5624     by auto

  5625   then show ?thesis

  5626     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5627 qed

  5628

  5629 text {* Hence we get the following. *}

  5630

  5631 lemma compact_sup_maxdistance:

  5632   fixes s :: "'a::metric_space set"

  5633   assumes "compact s"  "s \<noteq> {}"

  5634   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5635 proof -

  5636   have "compact (s \<times> s)"

  5637     using compact s by (intro compact_Times)

  5638   moreover have "s \<times> s \<noteq> {}"

  5639     using s \<noteq> {} by auto

  5640   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  5641     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  5642   ultimately show ?thesis

  5643     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  5644 qed

  5645

  5646 text {* We can state this in terms of diameter of a set. *}

  5647

  5648 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

  5649

  5650 lemma diameter_bounded_bound:

  5651   fixes s :: "'a :: metric_space set"

  5652   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  5653   shows "dist x y \<le> diameter s"

  5654 proof -

  5655   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  5656   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  5657     unfolding bounded_def by auto

  5658   have "dist x y \<le> Sup ?D"

  5659   proof (rule cSup_upper, safe)

  5660     fix a b

  5661     assume "a \<in> s" "b \<in> s"

  5662     with z[of a] z[of b] dist_triangle[of a b z]

  5663     show "dist a b \<le> 2 * d"

  5664       by (simp add: dist_commute)

  5665   qed (insert s, auto)

  5666   with x \<in> s show ?thesis

  5667     by (auto simp add: diameter_def)

  5668 qed

  5669

  5670 lemma diameter_lower_bounded:

  5671   fixes s :: "'a :: metric_space set"

  5672   assumes s: "bounded s"

  5673     and d: "0 < d" "d < diameter s"

  5674   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  5675 proof (rule ccontr)

  5676   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  5677   assume contr: "\<not> ?thesis"

  5678   moreover

  5679   from d have "s \<noteq> {}"

  5680     by (auto simp: diameter_def)

  5681   then have "?D \<noteq> {}" by auto

  5682   ultimately have "Sup ?D \<le> d"

  5683     by (intro cSup_least) (auto simp: not_less)

  5684   with d < diameter s s \<noteq> {} show False

  5685     by (auto simp: diameter_def)

  5686 qed

  5687

  5688 lemma diameter_bounded:

  5689   assumes "bounded s"

  5690   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  5691         "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  5692   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  5693   by auto

  5694

  5695 lemma diameter_compact_attained:

  5696   assumes "compact s"  "s \<noteq> {}"

  5697   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  5698 proof -

  5699   have b: "bounded s" using assms(1)

  5700     by (rule compact_imp_bounded)

  5701   then obtain x y where xys: "x\<in>s" "y\<in>s" and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5702     using compact_sup_maxdistance[OF assms] by auto

  5703   then have "diameter s \<le> dist x y"

  5704     unfolding diameter_def

  5705     apply clarsimp

  5706     apply (rule cSup_least)

  5707     apply fast+

  5708     done

  5709   then show ?thesis

  5710     by (metis b diameter_bounded_bound order_antisym xys)

  5711 qed

  5712

  5713 text {* Related results with closure as the conclusion. *}

  5714

  5715 lemma closed_scaling:

  5716   fixes s :: "'a::real_normed_vector set"

  5717   assumes "closed s"

  5718   shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5719 proof (cases "s = {}")

  5720   case True

  5721   then show ?thesis by auto

  5722 next

  5723   case False

  5724   show ?thesis

  5725   proof (cases "c = 0")

  5726     have *: "(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  5727     case True

  5728     then show ?thesis

  5729       apply auto

  5730       unfolding *

  5731       apply auto

  5732       done

  5733   next

  5734     case False

  5735     {

  5736       fix x l

  5737       assume as: "\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  5738       {

  5739         fix n :: nat

  5740         have "scaleR (1 / c) (x n) \<in> s"

  5741           using as(1)[THEN spec[where x=n]]

  5742           using c\<noteq>0

  5743           by auto

  5744       }

  5745       moreover

  5746       {

  5747         fix e :: real

  5748         assume "e > 0"

  5749         then have "0 < e *\<bar>c\<bar>"

  5750           using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  5751         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  5752           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  5753         then have "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  5754           unfolding dist_norm

  5755           unfolding scaleR_right_diff_distrib[THEN sym]

  5756           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto

  5757       }

  5758       then have "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially"

  5759         unfolding LIMSEQ_def by auto

  5760       ultimately have "l \<in> scaleR c  s"

  5761         using assms[unfolded closed_sequential_limits,

  5762           THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"],

  5763           THEN spec[where x="scaleR (1/c) l"]]

  5764         unfolding image_iff using c\<noteq>0

  5765           apply (rule_tac x="scaleR (1 / c) l" in bexI)

  5766           apply auto

  5767           done

  5768     }

  5769     then show ?thesis

  5770       unfolding closed_sequential_limits by fast

  5771   qed

  5772 qed

  5773

  5774 lemma closed_negations:

  5775   fixes s :: "'a::real_normed_vector set"

  5776   assumes "closed s"

  5777   shows "closed ((\<lambda>x. -x)  s)"

  5778   using closed_scaling[OF assms, of "- 1"] by simp

  5779

  5780 lemma compact_closed_sums:

  5781   fixes s :: "'a::real_normed_vector set"

  5782   assumes "compact s" and "closed t"

  5783   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5784 proof -

  5785   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  5786   {

  5787     fix x l

  5788     assume as: "\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  5789     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"

  5790       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  5791     obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  5792       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  5793     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  5794       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

  5795       unfolding o_def

  5796       by auto

  5797     then have "l - l' \<in> t"

  5798       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  5799       using f(3)

  5800       by auto

  5801     then have "l \<in> ?S"

  5802       using l' \<in> s

  5803       apply auto

  5804       apply (rule_tac x=l' in exI)

  5805       apply (rule_tac x="l - l'" in exI)

  5806       apply auto

  5807       done

  5808   }

  5809   then show ?thesis

  5810     unfolding closed_sequential_limits by fast

  5811 qed

  5812

  5813 lemma closed_compact_sums:

  5814   fixes s t :: "'a::real_normed_vector set"

  5815   assumes "closed s"  "compact t"

  5816   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5817 proof -

  5818   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}"

  5819     apply auto

  5820     apply (rule_tac x=y in exI)

  5821     apply auto

  5822     apply (rule_tac x=y in exI)

  5823     apply auto

  5824     done

  5825   then show ?thesis

  5826     using compact_closed_sums[OF assms(2,1)] by simp

  5827 qed

  5828

  5829 lemma compact_closed_differences:

  5830   fixes s t :: "'a::real_normed_vector set"

  5831   assumes "compact s" and "closed t"

  5832   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5833 proof -

  5834   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  5835     apply auto

  5836     apply (rule_tac x=xa in exI)

  5837     apply auto

  5838     apply (rule_tac x=xa in exI)

  5839     apply auto

  5840     done

  5841   then show ?thesis

  5842     using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  5843 qed

  5844

  5845 lemma closed_compact_differences:

  5846   fixes s t :: "'a::real_normed_vector set"

  5847   assumes "closed s" "compact t"

  5848   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5849 proof -

  5850   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  5851     apply auto

  5852     apply (rule_tac x=xa in exI)

  5853     apply auto

  5854     apply (rule_tac x=xa in exI)

  5855     apply auto

  5856     done

  5857  then show ?thesis

  5858   using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  5859 qed

  5860

  5861 lemma closed_translation:

  5862   fixes a :: "'a::real_normed_vector"

  5863   assumes "closed s"

  5864   shows "closed ((\<lambda>x. a + x)  s)"

  5865 proof -

  5866   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  5867   then show ?thesis

  5868     using compact_closed_sums[OF compact_sing[of a] assms] by auto

  5869 qed

  5870

  5871 lemma translation_Compl:

  5872   fixes a :: "'a::ab_group_add"

  5873   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5874   apply (auto simp add: image_iff)

  5875   apply (rule_tac x="x - a" in bexI)

  5876   apply auto

  5877   done

  5878

  5879 lemma translation_UNIV:

  5880   fixes a :: "'a::ab_group_add"

  5881   shows "range (\<lambda>x. a + x) = UNIV"

  5882   apply (auto simp add: image_iff)

  5883   apply (rule_tac x="x - a" in exI)

  5884   apply auto

  5885   done

  5886

  5887 lemma translation_diff:

  5888   fixes a :: "'a::ab_group_add"

  5889   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5890   by auto

  5891

  5892 lemma closure_translation:

  5893   fixes a :: "'a::real_normed_vector"

  5894   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5895 proof -

  5896   have *: "op + a  (- s) = - op + a  s"

  5897     apply auto

  5898     unfolding image_iff

  5899     apply (rule_tac x="x - a" in bexI)

  5900     apply auto

  5901     done

  5902   show ?thesis

  5903     unfolding closure_interior translation_Compl

  5904     using interior_translation[of a "- s"]

  5905     unfolding *

  5906     by auto

  5907 qed

  5908

  5909 lemma frontier_translation:

  5910   fixes a :: "'a::real_normed_vector"

  5911   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5912   unfolding frontier_def translation_diff interior_translation closure_translation

  5913   by auto

  5914

  5915

  5916 subsection {* Separation between points and sets *}

  5917

  5918 lemma separate_point_closed:

  5919   fixes s :: "'a::heine_borel set"

  5920   assumes "closed s" and "a \<notin> s"

  5921   shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"

  5922 proof (cases "s = {}")

  5923   case True

  5924   then show ?thesis by(auto intro!: exI[where x=1])

  5925 next

  5926   case False

  5927   from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"

  5928     using s \<noteq> {} distance_attains_inf [of s a] by blast

  5929   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s

  5930     by blast

  5931 qed

  5932

  5933 lemma separate_compact_closed:

  5934   fixes s t :: "'a::heine_borel set"

  5935   assumes "compact s"

  5936     and t: "closed t" "s \<inter> t = {}"

  5937   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5938 proof cases

  5939   assume "s \<noteq> {} \<and> t \<noteq> {}"

  5940   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  5941   let ?inf = "\<lambda>x. infdist x t"

  5942   have "continuous_on s ?inf"

  5943     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

  5944   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  5945     using continuous_attains_inf[OF compact s s \<noteq> {}] by auto

  5946   then have "0 < ?inf x"

  5947     using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  5948   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  5949     using x by (auto intro: order_trans infdist_le)

  5950   ultimately show ?thesis by auto

  5951 qed (auto intro!: exI[of _ 1])

  5952

  5953 lemma separate_closed_compact:

  5954   fixes s t :: "'a::heine_borel set"

  5955   assumes "closed s"

  5956     and "compact t"

  5957     and "s \<inter> t = {}"

  5958   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5959 proof -

  5960   have *: "t \<inter> s = {}"

  5961     using assms(3) by auto

  5962   show ?thesis

  5963     using separate_compact_closed[OF assms(2,1) *]

  5964     apply auto

  5965     apply (rule_tac x=d in exI)

  5966     apply auto

  5967     apply (erule_tac x=y in ballE)

  5968     apply (auto simp add: dist_commute)

  5969     done

  5970 qed

  5971

  5972

  5973 subsection {* Intervals *}

  5974

  5975 lemma interval:

  5976   fixes a :: "'a::ordered_euclidean_space"

  5977   shows "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}"

  5978     and "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5979   by (auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5980

  5981 lemma mem_interval:

  5982   fixes a :: "'a::ordered_euclidean_space"

  5983   shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5984     and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  5985   using interval[of a b]

  5986   by (auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5987

  5988 lemma interval_eq_empty:

  5989   fixes a :: "'a::ordered_euclidean_space"

  5990   shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)

  5991     and "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  5992 proof -

  5993   {

  5994     fix i x

  5995     assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  5996     then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"

  5997       unfolding mem_interval by auto

  5998     then have "a\<bullet>i < b\<bullet>i" by auto

  5999     then have False using as by auto

  6000   }

  6001   moreover

  6002   {

  6003     assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  6004     let ?x = "(1/2) *\<^sub>R (a + b)"

  6005     {

  6006       fix i :: 'a

  6007       assume i: "i \<in> Basis"

  6008       have "a\<bullet>i < b\<bullet>i"

  6009         using as[THEN bspec[where x=i]] i by auto

  6010       then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  6011         by (auto simp: inner_add_left)

  6012     }

  6013     then have "{a <..< b} \<noteq> {}"

  6014       using mem_interval(1)[of "?x" a b] by auto

  6015   }

  6016   ultimately show ?th1 by blast

  6017

  6018   {

  6019     fix i x

  6020     assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  6021     then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  6022       unfolding mem_interval by auto

  6023     then have "a\<bullet>i \<le> b\<bullet>i" by auto

  6024     then have False using as by auto

  6025   }

  6026   moreover

  6027   {

  6028     assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  6029     let ?x = "(1/2) *\<^sub>R (a + b)"

  6030     {

  6031       fix i :: 'a

  6032       assume i:"i \<in> Basis"

  6033       have "a\<bullet>i \<le> b\<bullet>i"

  6034         using as[THEN bspec[where x=i]] i by auto

  6035       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"

  6036         by (auto simp: inner_add_left)

  6037     }

  6038     then have "{a .. b} \<noteq> {}"

  6039       using mem_interval(2)[of "?x" a b] by auto

  6040   }

  6041   ultimately show ?th2 by blast

  6042 qed

  6043

  6044 lemma interval_ne_empty:

  6045   fixes a :: "'a::ordered_euclidean_space"

  6046   shows "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"

  6047   and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  6048   unfolding interval_eq_empty[of a b] by fastforce+

  6049

  6050 lemma interval_sing:

  6051   fixes a :: "'a::ordered_euclidean_space"

  6052   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  6053   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  6054   by (auto intro: euclidean_eqI simp: ex_in_conv)

  6055

  6056 lemma subset_interval_imp:

  6057   fixes a :: "'a::ordered_euclidean_space"

  6058   shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"

  6059     and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"

  6060     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"

  6061     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"

  6062   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
`