src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author huffman Tue Sep 24 15:03:49 2013 -0700 (2013-09-24) changeset 53860 f2d683432580 parent 53859 e6cb01686f7b child 53861 5ba90fca32bc permissions -rw-r--r--
factor out new lemma
     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 \<circ> 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 \<longleftrightarrow>

    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 \<longleftrightarrow> (\<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" and "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   "\<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" and "open (a \<times> b)"

   287       unfolding mem_Times_iff

   288       by (auto intro: open_Times)

   289   next

   290     fix S

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

   292     from open_prod_elim[OF this] guess a' b' . note a'b' = this

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

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

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

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

   297   qed (simp add: A B)

   298 qed

   299

   300 class second_countable_topology = topological_space +

   301   assumes ex_countable_subbasis:

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

   303 begin

   304

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

   306 proof -

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

   308     by blast

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

   310

   311   show ?thesis

   312   proof (intro exI conjI)

   313     show "countable ?B"

   314       by (intro countable_image countable_Collect_finite_subset B)

   315     {

   316       fix S

   317       assume "open S"

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

   319         unfolding B

   320       proof induct

   321         case UNIV

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

   323       next

   324         case (Int a b)

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

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

   327           by blast

   328         show ?case

   329           unfolding x y Int_UN_distrib2

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

   331       next

   332         case (UN K)

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

   334         then guess k unfolding bchoice_iff ..

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

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

   337       next

   338         case (Basis S)

   339         then show ?case

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

   341       qed

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

   343         unfolding subset_image_iff by blast }

   344     then show "topological_basis ?B"

   345       unfolding topological_space_class.topological_basis_def

   346       by (safe intro!: topological_space_class.open_Inter)

   347          (simp_all add: B generate_topology.Basis subset_eq)

   348   qed

   349 qed

   350

   351 end

   352

   353 sublocale second_countable_topology <

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

   355   using someI_ex[OF ex_countable_basis]

   356   by unfold_locales safe

   357

   358 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   359 proof

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

   361     using ex_countable_basis by auto

   362   moreover

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

   364     using ex_countable_basis by auto

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

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

   367       topological_basis_imp_subbasis)

   368 qed

   369

   370 instance second_countable_topology \<subseteq> first_countable_topology

   371 proof

   372   fix x :: 'a

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

   374   then have B: "countable B" "topological_basis B"

   375     using countable_basis is_basis

   376     by (auto simp: countable_basis is_basis)

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

   378     (\<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))"

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

   380        (fastforce simp: topological_space_class.topological_basis_def)+

   381 qed

   382

   383

   384 subsection {* Polish spaces *}

   385

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

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

   388

   389 class polish_space = complete_space + second_countable_topology

   390

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

   392

   393 definition "istopology L \<longleftrightarrow>

   394   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))"

   395

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

   397   morphisms "openin" "topology"

   398   unfolding istopology_def by blast

   399

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

   401   using openin[of U] by blast

   402

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

   404   using topology_inverse[unfolded mem_Collect_eq] .

   405

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

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

   408

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

   410 proof

   411   assume "T1 = T2"

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

   413 next

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

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

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

   417   then show "T1 = T2" unfolding openin_inverse .

   418 qed

   419

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

   421

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

   423

   424 subsubsection {* Main properties of open sets *}

   425

   426 lemma openin_clauses:

   427   fixes U :: "'a topology"

   428   shows

   429     "openin U {}"

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

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

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

   433

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

   435   unfolding topspace_def by blast

   436

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

   438   by (simp add: openin_clauses)

   439

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

   441   using openin_clauses by simp

   442

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

   444   using openin_clauses by simp

   445

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

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

   448

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

   450   by (simp add: openin_Union topspace_def)

   451

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

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

   454 proof

   455   assume ?lhs

   456   then show ?rhs by auto

   457 next

   458   assume H: ?rhs

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

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

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

   462   finally show "openin U S" .

   463 qed

   464

   465

   466 subsubsection {* Closed sets *}

   467

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

   469

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

   471   by (metis closedin_def)

   472

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

   474   by (simp add: closedin_def)

   475

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

   477   by (simp add: closedin_def)

   478

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

   480   by (auto simp add: Diff_Un closedin_def)

   481

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

   483   by auto

   484

   485 lemma closedin_Inter[intro]:

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

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

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

   489   using Ke Kc unfolding closedin_def Diff_Inter by auto

   490

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

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

   493

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

   495   by blast

   496

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

   498   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   499   apply (metis openin_subset subset_eq)

   500   done

   501

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

   503   by (simp add: openin_closedin_eq)

   504

   505 lemma openin_diff[intro]:

   506   assumes oS: "openin U S"

   507     and cT: "closedin U T"

   508   shows "openin U (S - T)"

   509 proof -

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

   511     by (auto simp add: topspace_def openin_subset)

   512   then show ?thesis using oS cT

   513     by (auto simp add: closedin_def)

   514 qed

   515

   516 lemma closedin_diff[intro]:

   517   assumes oS: "closedin U S"

   518     and cT: "openin U T"

   519   shows "closedin U (S - T)"

   520 proof -

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

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

   523   then show ?thesis

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

   525 qed

   526

   527

   528 subsubsection {* Subspace topology *}

   529

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

   531

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

   533   (is "istopology ?L")

   534 proof -

   535   have "?L {}" by blast

   536   {

   537     fix A B

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

   539     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"

   540       by blast

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

   542       using Sa Sb by blast+

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

   544   }

   545   moreover

   546   {

   547     fix K

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

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

   550       apply (rule set_eqI)

   551       apply (simp add: Ball_def image_iff)

   552       apply metis

   553       done

   554     from K[unfolded th0 subset_image_iff]

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

   556       by blast

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

   558       using Sk by auto

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

   560       using Sk by (auto simp add: subset_eq)

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

   562   }

   563   ultimately show ?thesis

   564     unfolding subset_eq mem_Collect_eq istopology_def by blast

   565 qed

   566

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

   568   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   569   by auto

   570

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

   572   by (auto simp add: topspace_def openin_subtopology)

   573

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

   575   unfolding closedin_def topspace_subtopology

   576   apply (simp add: openin_subtopology)

   577   apply (rule iffI)

   578   apply clarify

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

   580   apply auto

   581   done

   582

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

   584   unfolding openin_subtopology

   585   apply (rule iffI, clarify)

   586   apply (frule openin_subset[of U])

   587   apply blast

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

   589   apply auto

   590   done

   591

   592 lemma subtopology_superset:

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

   594   shows "subtopology U V = U"

   595 proof -

   596   {

   597     fix S

   598     {

   599       fix T

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

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

   602         by blast

   603       have "openin U S"

   604         unfolding eq using T by blast

   605     }

   606     moreover

   607     {

   608       assume S: "openin U S"

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

   610         using openin_subset[OF S] UV by auto

   611     }

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

   613       by blast

   614   }

   615   then show ?thesis

   616     unfolding topology_eq openin_subtopology by blast

   617 qed

   618

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

   620   by (simp add: subtopology_superset)

   621

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

   623   by (simp add: subtopology_superset)

   624

   625

   626 subsubsection {* The standard Euclidean topology *}

   627

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

   629   where "euclidean = topology open"

   630

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

   632   unfolding euclidean_def

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

   634   apply (rule topology_inverse[symmetric])

   635   apply (auto simp add: istopology_def)

   636   done

   637

   638 lemma topspace_euclidean: "topspace euclidean = UNIV"

   639   apply (simp add: topspace_def)

   640   apply (rule set_eqI)

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

   642   done

   643

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

   645   by (simp add: topspace_euclidean topspace_subtopology)

   646

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

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

   649

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

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

   652

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

   654

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

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

   657

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

   659   by (auto simp add: openin_open)

   660

   661 lemma open_openin_trans[trans]:

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

   663   by (metis Int_absorb1  openin_open_Int)

   664

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

   666   by (auto simp add: openin_open)

   667

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

   669   by (simp add: closedin_subtopology closed_closedin Int_ac)

   670

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

   672   by (metis closedin_closed)

   673

   674 lemma closed_closedin_trans:

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

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

   677   apply auto

   678   apply (frule closedin_closed_Int[of T S])

   679   apply simp

   680   done

   681

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

   683   by (auto simp add: closedin_closed)

   684

   685 lemma openin_euclidean_subtopology_iff:

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

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

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

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

   690 proof

   691   assume ?lhs

   692   then show ?rhs

   693     unfolding openin_open open_dist by blast

   694 next

   695   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}"

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

   697     unfolding T_def

   698     apply clarsimp

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

   700     apply (clarsimp simp add: less_diff_eq)

   701     apply (erule rev_bexI)

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

   703     apply (erule le_less_trans [OF dist_triangle])

   704     done

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

   706     unfolding T_def

   707     apply auto

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

   709     apply auto

   710     done

   711   from 1 2 show ?lhs

   712     unfolding openin_open open_dist by fast

   713 qed

   714

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

   716

   717 lemma openin_trans[trans]:

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

   719     openin (subtopology euclidean U) S"

   720   unfolding open_openin openin_open by blast

   721

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

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

   724

   725 lemma closedin_trans[trans]:

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

   727     closedin (subtopology euclidean U) S"

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

   729

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

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

   732

   733

   734 subsection {* Open and closed balls *}

   735

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

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

   738

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

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

   741

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

   743   by (simp add: ball_def)

   744

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

   746   by (simp add: cball_def)

   747

   748 lemma mem_ball_0:

   749   fixes x :: "'a::real_normed_vector"

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

   751   by (simp add: dist_norm)

   752

   753 lemma mem_cball_0:

   754   fixes x :: "'a::real_normed_vector"

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

   756   by (simp add: dist_norm)

   757

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

   759   by simp

   760

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

   762   by simp

   763

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

   765   by (simp add: subset_eq)

   766

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

   768   by (simp add: subset_eq)

   769

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

   771   by (simp add: subset_eq)

   772

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

   774   by (simp add: set_eq_iff) arith

   775

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

   777   by (simp add: set_eq_iff)

   778

   779 lemma diff_less_iff:

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

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

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

   783   by arith+

   784

   785 lemma diff_le_iff:

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

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

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

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

   790   by arith+

   791

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

   793   unfolding open_dist ball_def mem_Collect_eq Ball_def

   794   unfolding dist_commute

   795   apply clarify

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

   797   using dist_triangle_alt[where z=x]

   798   apply (clarsimp simp add: diff_less_iff)

   799   apply atomize

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

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

   802   apply arith

   803   done

   804

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

   806   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   807

   808 lemma openE[elim?]:

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

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

   811   using assms unfolding open_contains_ball by auto

   812

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

   814   by (metis open_contains_ball subset_eq centre_in_ball)

   815

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

   817   unfolding mem_ball set_eq_iff

   818   apply (simp add: not_less)

   819   apply (metis zero_le_dist order_trans dist_self)

   820   done

   821

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

   823

   824 lemma euclidean_dist_l2:

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

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

   827   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   829

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

   831

   832 lemma rational_boxes:

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

   834   assumes "e > 0"

   835   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"

   836 proof -

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

   838   then have e: "e' > 0"

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

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

   841   proof

   842     fix i

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

   844     show "?th i" by auto

   845   qed

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

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

   848   proof

   849     fix i

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

   851     show "?th i" by auto

   852   qed

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

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

   855   show ?thesis

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

   857     fix y :: 'a

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

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

   860       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   862     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   863       fix i :: "'a"

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

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

   866         using * i by (auto simp: box_def)

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

   868         using a by auto

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

   870         using b by auto

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

   872         by auto

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

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

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

   876         by (rule power_strict_mono) auto

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

   878         by (simp add: power_divide)

   879     qed auto

   880     also have "\<dots> = e"

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

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

   883       by (auto simp: ball_def)

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

   885 qed

   886

   887 lemma open_UNION_box:

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

   889   assumes "open M"

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

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

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

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

   894 proof -

   895   {

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

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

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

   899     moreover obtain a b where ab:

   900       "x \<in> box a b"

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

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

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

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

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

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

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

   908   }

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

   910 qed

   911

   912

   913 subsection{* Connectedness *}

   914

   915 lemma connected_local:

   916  "connected S \<longleftrightarrow>

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

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

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

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

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

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

   923       e2 \<noteq> {})"

   924   unfolding connected_def openin_open

   925   apply safe

   926   apply blast+

   927   done

   928

   929 lemma exists_diff:

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

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

   932 proof -

   933   {

   934     assume "?lhs"

   935     then have ?rhs by blast

   936   }

   937   moreover

   938   {

   939     fix S

   940     assume H: "P S"

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

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

   943   }

   944   ultimately show ?thesis by metis

   945 qed

   946

   947 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   950 proof -

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

   952     (\<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> {})"

   953     unfolding connected_def openin_open closedin_closed

   954     apply (subst exists_diff)

   955     apply blast

   956     done

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

   958     \<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> {})"

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

   960     apply (simp add: closed_def)

   961     apply metis

   962     done

   963   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'))"

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

   965     unfolding connected_def openin_open closedin_closed by auto

   966   {

   967     fix e2

   968     {

   969       fix e1

   970       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)"

   971         by auto

   972     }

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

   974       by metis

   975   }

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

   977     by blast

   978   then show ?thesis

   979     unfolding th0 th1 by simp

   980 qed

   981

   982

   983 subsection{* Limit points *}

   984

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

   986   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))"

   987

   988 lemma islimptI:

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

   990   shows "x islimpt S"

   991   using assms unfolding islimpt_def by auto

   992

   993 lemma islimptE:

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

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

   996   using assms unfolding islimpt_def by auto

   997

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

   999   unfolding islimpt_def eventually_at_topological by auto

  1000

  1001 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"

  1002   unfolding islimpt_def by fast

  1003

  1004 lemma islimpt_approachable:

  1005   fixes x :: "'a::metric_space"

  1006   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"

  1007   unfolding islimpt_iff_eventually eventually_at by fast

  1008

  1009 lemma islimpt_approachable_le:

  1010   fixes x :: "'a::metric_space"

  1011   shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"

  1012   unfolding islimpt_approachable

  1013   using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",

  1014     THEN arg_cong [where f=Not]]

  1015   by (simp add: Bex_def conj_commute conj_left_commute)

  1016

  1017 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"

  1018   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

  1019

  1020 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"

  1021   unfolding islimpt_def by blast

  1022

  1023 text {* A perfect space has no isolated points. *}

  1024

  1025 lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"

  1026   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

  1027

  1028 lemma perfect_choose_dist:

  1029   fixes x :: "'a::{perfect_space, metric_space}"

  1030   shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"

  1031   using islimpt_UNIV [of x]

  1032   by (simp add: islimpt_approachable)

  1033

  1034 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"

  1035   unfolding closed_def

  1036   apply (subst open_subopen)

  1037   apply (simp add: islimpt_def subset_eq)

  1038   apply (metis ComplE ComplI)

  1039   done

  1040

  1041 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"

  1042   unfolding islimpt_def by auto

  1043

  1044 lemma finite_set_avoid:

  1045   fixes a :: "'a::metric_space"

  1046   assumes fS: "finite S"

  1047   shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"

  1048 proof (induct rule: finite_induct[OF fS])

  1049   case 1

  1050   then show ?case by (auto intro: zero_less_one)

  1051 next

  1052   case (2 x F)

  1053   from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"

  1054     by blast

  1055   show ?case

  1056   proof (cases "x = a")

  1057     case True

  1058     then show ?thesis using d by auto

  1059   next

  1060     case False

  1061     let ?d = "min d (dist a x)"

  1062     have dp: "?d > 0"

  1063       using False d(1) using dist_nz by auto

  1064     from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"

  1065       by auto

  1066     with dp False show ?thesis

  1067       by (auto intro!: exI[where x="?d"])

  1068   qed

  1069 qed

  1070

  1071 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

  1072   by (simp add: islimpt_iff_eventually eventually_conj_iff)

  1073

  1074 lemma discrete_imp_closed:

  1075   fixes S :: "'a::metric_space set"

  1076   assumes e: "0 < e"

  1077     and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

  1078   shows "closed S"

  1079 proof -

  1080   {

  1081     fix x

  1082     assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"

  1083     from e have e2: "e/2 > 0" by arith

  1084     from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"

  1085       by blast

  1086     let ?m = "min (e/2) (dist x y) "

  1087     from e2 y(2) have mp: "?m > 0"

  1088       by (simp add: dist_nz[symmetric])

  1089     from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"

  1090       by blast

  1091     have th: "dist z y < e" using z y

  1092       by (intro dist_triangle_lt [where z=x], simp)

  1093     from d[rule_format, OF y(1) z(1) th] y z

  1094     have False by (auto simp add: dist_commute)}

  1095   then show ?thesis

  1096     by (metis islimpt_approachable closed_limpt [where 'a='a])

  1097 qed

  1098

  1099

  1100 subsection {* Interior of a Set *}

  1101

  1102 definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

  1103

  1104 lemma interiorI [intro?]:

  1105   assumes "open T" and "x \<in> T" and "T \<subseteq> S"

  1106   shows "x \<in> interior S"

  1107   using assms unfolding interior_def by fast

  1108

  1109 lemma interiorE [elim?]:

  1110   assumes "x \<in> interior S"

  1111   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"

  1112   using assms unfolding interior_def by fast

  1113

  1114 lemma open_interior [simp, intro]: "open (interior S)"

  1115   by (simp add: interior_def open_Union)

  1116

  1117 lemma interior_subset: "interior S \<subseteq> S"

  1118   by (auto simp add: interior_def)

  1119

  1120 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

  1121   by (auto simp add: interior_def)

  1122

  1123 lemma interior_open: "open S \<Longrightarrow> interior S = S"

  1124   by (intro equalityI interior_subset interior_maximal subset_refl)

  1125

  1126 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"

  1127   by (metis open_interior interior_open)

  1128

  1129 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"

  1130   by (metis interior_maximal interior_subset subset_trans)

  1131

  1132 lemma interior_empty [simp]: "interior {} = {}"

  1133   using open_empty by (rule interior_open)

  1134

  1135 lemma interior_UNIV [simp]: "interior UNIV = UNIV"

  1136   using open_UNIV by (rule interior_open)

  1137

  1138 lemma interior_interior [simp]: "interior (interior S) = interior S"

  1139   using open_interior by (rule interior_open)

  1140

  1141 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

  1142   by (auto simp add: interior_def)

  1143

  1144 lemma interior_unique:

  1145   assumes "T \<subseteq> S" and "open T"

  1146   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"

  1147   shows "interior S = T"

  1148   by (intro equalityI assms interior_subset open_interior interior_maximal)

  1149

  1150 lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"

  1151   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1152     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1153

  1154 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1155   using open_contains_ball_eq [where S="interior S"]

  1156   by (simp add: open_subset_interior)

  1157

  1158 lemma interior_limit_point [intro]:

  1159   fixes x :: "'a::perfect_space"

  1160   assumes x: "x \<in> interior S"

  1161   shows "x islimpt S"

  1162   using x islimpt_UNIV [of x]

  1163   unfolding interior_def islimpt_def

  1164   apply (clarsimp, rename_tac T T')

  1165   apply (drule_tac x="T \<inter> T'" in spec)

  1166   apply (auto simp add: open_Int)

  1167   done

  1168

  1169 lemma interior_closed_Un_empty_interior:

  1170   assumes cS: "closed S"

  1171     and iT: "interior T = {}"

  1172   shows "interior (S \<union> T) = interior S"

  1173 proof

  1174   show "interior S \<subseteq> interior (S \<union> T)"

  1175     by (rule interior_mono) (rule Un_upper1)

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

  1177   proof

  1178     fix x

  1179     assume "x \<in> interior (S \<union> T)"

  1180     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1181     show "x \<in> interior S"

  1182     proof (rule ccontr)

  1183       assume "x \<notin> interior S"

  1184       with x \<in> R open R obtain y where "y \<in> R - S"

  1185         unfolding interior_def by fast

  1186       from open R closed S have "open (R - S)"

  1187         by (rule open_Diff)

  1188       from R \<subseteq> S \<union> T have "R - S \<subseteq> T"

  1189         by fast

  1190       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {} show False

  1191         unfolding interior_def by fast

  1192     qed

  1193   qed

  1194 qed

  1195

  1196 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1197 proof (rule interior_unique)

  1198   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1199     by (intro Sigma_mono interior_subset)

  1200   show "open (interior A \<times> interior B)"

  1201     by (intro open_Times open_interior)

  1202   fix T

  1203   assume "T \<subseteq> A \<times> B" and "open T"

  1204   then show "T \<subseteq> interior A \<times> interior B"

  1205   proof safe

  1206     fix x y

  1207     assume "(x, y) \<in> T"

  1208     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1209       using open T unfolding open_prod_def by fast

  1210     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1211       using T \<subseteq> A \<times> B by auto

  1212     then show "x \<in> interior A" and "y \<in> interior B"

  1213       by (auto intro: interiorI)

  1214   qed

  1215 qed

  1216

  1217

  1218 subsection {* Closure of a Set *}

  1219

  1220 definition "closure S = S \<union> {x | x. x islimpt S}"

  1221

  1222 lemma interior_closure: "interior S = - (closure (- S))"

  1223   unfolding interior_def closure_def islimpt_def by auto

  1224

  1225 lemma closure_interior: "closure S = - interior (- S)"

  1226   unfolding interior_closure by simp

  1227

  1228 lemma closed_closure[simp, intro]: "closed (closure S)"

  1229   unfolding closure_interior by (simp add: closed_Compl)

  1230

  1231 lemma closure_subset: "S \<subseteq> closure S"

  1232   unfolding closure_def by simp

  1233

  1234 lemma closure_hull: "closure S = closed hull S"

  1235   unfolding hull_def closure_interior interior_def by auto

  1236

  1237 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1238   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1239

  1240 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1241   unfolding closure_eq .

  1242

  1243 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1244   unfolding closure_hull by (rule hull_hull)

  1245

  1246 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1247   unfolding closure_hull by (rule hull_mono)

  1248

  1249 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1250   unfolding closure_hull by (rule hull_minimal)

  1251

  1252 lemma closure_unique:

  1253   assumes "S \<subseteq> T"

  1254     and "closed T"

  1255     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1256   shows "closure S = T"

  1257   using assms unfolding closure_hull by (rule hull_unique)

  1258

  1259 lemma closure_empty [simp]: "closure {} = {}"

  1260   using closed_empty by (rule closure_closed)

  1261

  1262 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1263   using closed_UNIV by (rule closure_closed)

  1264

  1265 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1266   unfolding closure_interior by simp

  1267

  1268 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1269   using closure_empty closure_subset[of S]

  1270   by blast

  1271

  1272 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1273   using closure_eq[of S] closure_subset[of S]

  1274   by simp

  1275

  1276 lemma open_inter_closure_eq_empty:

  1277   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1278   using open_subset_interior[of S "- T"]

  1279   using interior_subset[of "- T"]

  1280   unfolding closure_interior

  1281   by auto

  1282

  1283 lemma open_inter_closure_subset:

  1284   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1285 proof

  1286   fix x

  1287   assume as: "open S" "x \<in> S \<inter> closure T"

  1288   {

  1289     assume *: "x islimpt T"

  1290     have "x islimpt (S \<inter> T)"

  1291     proof (rule islimptI)

  1292       fix A

  1293       assume "x \<in> A" "open A"

  1294       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1295         by (simp_all add: open_Int)

  1296       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1297         by (rule islimptE)

  1298       then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1299         by simp_all

  1300       then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1301     qed

  1302   }

  1303   then show "x \<in> closure (S \<inter> T)" using as

  1304     unfolding closure_def

  1305     by blast

  1306 qed

  1307

  1308 lemma closure_complement: "closure (- S) = - interior S"

  1309   unfolding closure_interior by simp

  1310

  1311 lemma interior_complement: "interior (- S) = - closure S"

  1312   unfolding closure_interior by simp

  1313

  1314 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1315 proof (rule closure_unique)

  1316   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1317     by (intro Sigma_mono closure_subset)

  1318   show "closed (closure A \<times> closure B)"

  1319     by (intro closed_Times closed_closure)

  1320   fix T

  1321   assume "A \<times> B \<subseteq> T" and "closed T"

  1322   then show "closure A \<times> closure B \<subseteq> T"

  1323     apply (simp add: closed_def open_prod_def, clarify)

  1324     apply (rule ccontr)

  1325     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1326     apply (simp add: closure_interior interior_def)

  1327     apply (drule_tac x=C in spec)

  1328     apply (drule_tac x=D in spec)

  1329     apply auto

  1330     done

  1331 qed

  1332

  1333 lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"

  1334   unfolding closure_def using islimpt_punctured by blast

  1335

  1336

  1337 subsection {* Frontier (aka boundary) *}

  1338

  1339 definition "frontier S = closure S - interior S"

  1340

  1341 lemma frontier_closed: "closed (frontier S)"

  1342   by (simp add: frontier_def closed_Diff)

  1343

  1344 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1345   by (auto simp add: frontier_def interior_closure)

  1346

  1347 lemma frontier_straddle:

  1348   fixes a :: "'a::metric_space"

  1349   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))"

  1350   unfolding frontier_def closure_interior

  1351   by (auto simp add: mem_interior subset_eq ball_def)

  1352

  1353 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1354   by (metis frontier_def closure_closed Diff_subset)

  1355

  1356 lemma frontier_empty[simp]: "frontier {} = {}"

  1357   by (simp add: frontier_def)

  1358

  1359 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1360 proof-

  1361   {

  1362     assume "frontier S \<subseteq> S"

  1363     then have "closure S \<subseteq> S"

  1364       using interior_subset unfolding frontier_def by auto

  1365     then have "closed S"

  1366       using closure_subset_eq by auto

  1367   }

  1368   then show ?thesis using frontier_subset_closed[of S] ..

  1369 qed

  1370

  1371 lemma frontier_complement: "frontier(- S) = frontier S"

  1372   by (auto simp add: frontier_def closure_complement interior_complement)

  1373

  1374 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1375   using frontier_complement frontier_subset_eq[of "- S"]

  1376   unfolding open_closed by auto

  1377

  1378 subsection {* Filters and the eventually true'' quantifier *}

  1379

  1380 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1381     (infixr "indirection" 70)

  1382   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1383

  1384 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1385

  1386 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1387 proof

  1388   assume "trivial_limit (at a within S)"

  1389   then show "\<not> a islimpt S"

  1390     unfolding trivial_limit_def

  1391     unfolding eventually_at_topological

  1392     unfolding islimpt_def

  1393     apply (clarsimp simp add: set_eq_iff)

  1394     apply (rename_tac T, rule_tac x=T in exI)

  1395     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1396     done

  1397 next

  1398   assume "\<not> a islimpt S"

  1399   then show "trivial_limit (at a within S)"

  1400     unfolding trivial_limit_def

  1401     unfolding eventually_at_topological

  1402     unfolding islimpt_def

  1403     apply clarsimp

  1404     apply (rule_tac x=T in exI)

  1405     apply auto

  1406     done

  1407 qed

  1408

  1409 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1410   using trivial_limit_within [of a UNIV] by simp

  1411

  1412 lemma trivial_limit_at:

  1413   fixes a :: "'a::perfect_space"

  1414   shows "\<not> trivial_limit (at a)"

  1415   by (rule at_neq_bot)

  1416

  1417 lemma trivial_limit_at_infinity:

  1418   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1419   unfolding trivial_limit_def eventually_at_infinity

  1420   apply clarsimp

  1421   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1422    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1423   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1424   apply (drule_tac x=UNIV in spec, simp)

  1425   done

  1426

  1427 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"

  1428   using islimpt_in_closure

  1429   by (metis trivial_limit_within)

  1430

  1431 text {* Some property holds "sufficiently close" to the limit point. *}

  1432

  1433 lemma eventually_at2:

  1434   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1435   unfolding eventually_at dist_nz by auto

  1436

  1437 lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"

  1438   unfolding trivial_limit_def

  1439   by (auto elim: eventually_rev_mp)

  1440

  1441 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1442   by simp

  1443

  1444 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1445   by (simp add: filter_eq_iff)

  1446

  1447 text{* Combining theorems for "eventually" *}

  1448

  1449 lemma eventually_rev_mono:

  1450   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1451   using eventually_mono [of P Q] by fast

  1452

  1453 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"

  1454   by (simp add: eventually_False)

  1455

  1456

  1457 subsection {* Limits *}

  1458

  1459 lemma Lim:

  1460   "(f ---> l) net \<longleftrightarrow>

  1461         trivial_limit net \<or>

  1462         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1463   unfolding tendsto_iff trivial_limit_eq by auto

  1464

  1465 text{* Show that they yield usual definitions in the various cases. *}

  1466

  1467 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1468     (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"

  1469   by (auto simp add: tendsto_iff eventually_at_le dist_nz)

  1470

  1471 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1472     (\<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)"

  1473   by (auto simp add: tendsto_iff eventually_at dist_nz)

  1474

  1475 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1476     (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"

  1477   by (auto simp add: tendsto_iff eventually_at2)

  1478

  1479 lemma Lim_at_infinity:

  1480   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"

  1481   by (auto simp add: tendsto_iff eventually_at_infinity)

  1482

  1483 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1484   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1485

  1486 text{* The expected monotonicity property. *}

  1487

  1488 lemma Lim_Un:

  1489   assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"

  1490   shows "(f ---> l) (at x within (S \<union> T))"

  1491   using assms unfolding at_within_union by (rule filterlim_sup)

  1492

  1493 lemma Lim_Un_univ:

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

  1495     S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"

  1496   by (metis Lim_Un)

  1497

  1498 text{* Interrelations between restricted and unrestricted limits. *}

  1499

  1500 lemma Lim_at_within: (* FIXME: rename *)

  1501   "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"

  1502   by (metis order_refl filterlim_mono subset_UNIV at_le)

  1503

  1504 lemma eventually_within_interior:

  1505   assumes "x \<in> interior S"

  1506   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"

  1507   (is "?lhs = ?rhs")

  1508 proof

  1509   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1510   {

  1511     assume "?lhs"

  1512     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1513       unfolding eventually_at_topological

  1514       by auto

  1515     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"

  1516       by auto

  1517     then show "?rhs"

  1518       unfolding eventually_at_topological by auto

  1519   next

  1520     assume "?rhs"

  1521     then show "?lhs"

  1522       by (auto elim: eventually_elim1 simp: eventually_at_filter)

  1523   }

  1524 qed

  1525

  1526 lemma at_within_interior:

  1527   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1528   unfolding filter_eq_iff by (intro allI eventually_within_interior)

  1529

  1530 lemma Lim_within_LIMSEQ:

  1531   fixes a :: "'a::metric_space"

  1532   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1533   shows "(X ---> L) (at a within T)"

  1534   using assms unfolding tendsto_def [where l=L]

  1535   by (simp add: sequentially_imp_eventually_within)

  1536

  1537 lemma Lim_right_bound:

  1538   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>

  1539     'b::{linorder_topology, conditionally_complete_linorder}"

  1540   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"

  1541     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1542   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1543 proof (cases "{x<..} \<inter> I = {}")

  1544   case True

  1545   then show ?thesis by simp

  1546 next

  1547   case False

  1548   show ?thesis

  1549   proof (rule order_tendstoI)

  1550     fix a

  1551     assume a: "a < Inf (f  ({x<..} \<inter> I))"

  1552     {

  1553       fix y

  1554       assume "y \<in> {x<..} \<inter> I"

  1555       with False bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"

  1556         by (auto intro: cInf_lower)

  1557       with a have "a < f y"

  1558         by (blast intro: less_le_trans)

  1559     }

  1560     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"

  1561       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)

  1562   next

  1563     fix a

  1564     assume "Inf (f  ({x<..} \<inter> I)) < a"

  1565     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"

  1566       by auto

  1567     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"

  1568       unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)

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

  1570       unfolding eventually_at_filter by eventually_elim simp

  1571   qed

  1572 qed

  1573

  1574 text{* Another limit point characterization. *}

  1575

  1576 lemma islimpt_sequential:

  1577   fixes x :: "'a::first_countable_topology"

  1578   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1579     (is "?lhs = ?rhs")

  1580 proof

  1581   assume ?lhs

  1582   from countable_basis_at_decseq[of x] guess A . note A = this

  1583   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1584   {

  1585     fix n

  1586     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1587       unfolding islimpt_def using A(1,2)[of n] by auto

  1588     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1589       unfolding f_def by (rule someI_ex)

  1590     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto

  1591   }

  1592   then have "\<forall>n. f n \<in> S - {x}" by auto

  1593   moreover have "(\<lambda>n. f n) ----> x"

  1594   proof (rule topological_tendstoI)

  1595     fix S

  1596     assume "open S" "x \<in> S"

  1597     from A(3)[OF this] \<And>n. f n \<in> A n

  1598     show "eventually (\<lambda>x. f x \<in> S) sequentially"

  1599       by (auto elim!: eventually_elim1)

  1600   qed

  1601   ultimately show ?rhs by fast

  1602 next

  1603   assume ?rhs

  1604   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"

  1605     by auto

  1606   show ?lhs

  1607     unfolding islimpt_def

  1608   proof safe

  1609     fix T

  1610     assume "open T" "x \<in> T"

  1611     from lim[THEN topological_tendstoD, OF this] f

  1612     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1613       unfolding eventually_sequentially by auto

  1614   qed

  1615 qed

  1616

  1617 lemma Lim_inv: (* TODO: delete *)

  1618   fixes f :: "'a \<Rightarrow> real"

  1619     and A :: "'a filter"

  1620   assumes "(f ---> l) A"

  1621     and "l \<noteq> 0"

  1622   shows "((inverse \<circ> f) ---> inverse l) A"

  1623   unfolding o_def using assms by (rule tendsto_inverse)

  1624

  1625 lemma Lim_null:

  1626   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1627   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1628   by (simp add: Lim dist_norm)

  1629

  1630 lemma Lim_null_comparison:

  1631   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1632   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1633   shows "(f ---> 0) net"

  1634   using assms(2)

  1635 proof (rule metric_tendsto_imp_tendsto)

  1636   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1637     using assms(1) by (rule eventually_elim1) (simp add: dist_norm)

  1638 qed

  1639

  1640 lemma Lim_transform_bound:

  1641   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1642     and g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1643   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"

  1644     and "(g ---> 0) net"

  1645   shows "(f ---> 0) net"

  1646   using assms(1) tendsto_norm_zero [OF assms(2)]

  1647   by (rule Lim_null_comparison)

  1648

  1649 text{* Deducing things about the limit from the elements. *}

  1650

  1651 lemma Lim_in_closed_set:

  1652   assumes "closed S"

  1653     and "eventually (\<lambda>x. f(x) \<in> S) net"

  1654     and "\<not> trivial_limit net" "(f ---> l) net"

  1655   shows "l \<in> S"

  1656 proof (rule ccontr)

  1657   assume "l \<notin> S"

  1658   with closed S have "open (- S)" "l \<in> - S"

  1659     by (simp_all add: open_Compl)

  1660   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1661     by (rule topological_tendstoD)

  1662   with assms(2) have "eventually (\<lambda>x. False) net"

  1663     by (rule eventually_elim2) simp

  1664   with assms(3) show "False"

  1665     by (simp add: eventually_False)

  1666 qed

  1667

  1668 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1669

  1670 lemma Lim_dist_ubound:

  1671   assumes "\<not>(trivial_limit net)"

  1672     and "(f ---> l) net"

  1673     and "eventually (\<lambda>x. dist a (f x) \<le> e) net"

  1674   shows "dist a l \<le> e"

  1675 proof -

  1676   have "dist a l \<in> {..e}"

  1677   proof (rule Lim_in_closed_set)

  1678     show "closed {..e}"

  1679       by simp

  1680     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net"

  1681       by (simp add: assms)

  1682     show "\<not> trivial_limit net"

  1683       by fact

  1684     show "((\<lambda>x. dist a (f x)) ---> dist a l) net"

  1685       by (intro tendsto_intros assms)

  1686   qed

  1687   then show ?thesis by simp

  1688 qed

  1689

  1690 lemma Lim_norm_ubound:

  1691   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1692   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"

  1693   shows "norm(l) \<le> e"

  1694 proof -

  1695   have "norm l \<in> {..e}"

  1696   proof (rule Lim_in_closed_set)

  1697     show "closed {..e}"

  1698       by simp

  1699     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net"

  1700       by (simp add: assms)

  1701     show "\<not> trivial_limit net"

  1702       by fact

  1703     show "((\<lambda>x. norm (f x)) ---> norm l) net"

  1704       by (intro tendsto_intros assms)

  1705   qed

  1706   then show ?thesis by simp

  1707 qed

  1708

  1709 lemma Lim_norm_lbound:

  1710   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1711   assumes "\<not> trivial_limit net"

  1712     and "(f ---> l) net"

  1713     and "eventually (\<lambda>x. e \<le> norm (f x)) net"

  1714   shows "e \<le> norm l"

  1715 proof -

  1716   have "norm l \<in> {e..}"

  1717   proof (rule Lim_in_closed_set)

  1718     show "closed {e..}"

  1719       by simp

  1720     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net"

  1721       by (simp add: assms)

  1722     show "\<not> trivial_limit net"

  1723       by fact

  1724     show "((\<lambda>x. norm (f x)) ---> norm l) net"

  1725       by (intro tendsto_intros assms)

  1726   qed

  1727   then show ?thesis by simp

  1728 qed

  1729

  1730 text{* Limit under bilinear function *}

  1731

  1732 lemma Lim_bilinear:

  1733   assumes "(f ---> l) net"

  1734     and "(g ---> m) net"

  1735     and "bounded_bilinear h"

  1736   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1737   using bounded_bilinear h (f ---> l) net (g ---> m) net

  1738   by (rule bounded_bilinear.tendsto)

  1739

  1740 text{* These are special for limits out of the same vector space. *}

  1741

  1742 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1743   unfolding id_def by (rule tendsto_ident_at)

  1744

  1745 lemma Lim_at_id: "(id ---> a) (at a)"

  1746   unfolding id_def by (rule tendsto_ident_at)

  1747

  1748 lemma Lim_at_zero:

  1749   fixes a :: "'a::real_normed_vector"

  1750     and l :: "'b::topological_space"

  1751   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"

  1752   using LIM_offset_zero LIM_offset_zero_cancel ..

  1753

  1754 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1755

  1756 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"

  1757   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"

  1758

  1759 lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"

  1760   by (rule tendsto_Lim) (auto intro: tendsto_intros)

  1761

  1762 lemma netlimit_at:

  1763   fixes a :: "'a::{perfect_space,t2_space}"

  1764   shows "netlimit (at a) = a"

  1765   using netlimit_within [of a UNIV] by simp

  1766

  1767 lemma lim_within_interior:

  1768   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1769   by (metis at_within_interior)

  1770

  1771 lemma netlimit_within_interior:

  1772   fixes x :: "'a::{t2_space,perfect_space}"

  1773   assumes "x \<in> interior S"

  1774   shows "netlimit (at x within S) = x"

  1775   using assms by (metis at_within_interior netlimit_at)

  1776

  1777 text{* Transformation of limit. *}

  1778

  1779 lemma Lim_transform:

  1780   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"

  1781   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1782   shows "(g ---> l) net"

  1783   using tendsto_diff [OF assms(2) assms(1)] by simp

  1784

  1785 lemma Lim_transform_eventually:

  1786   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1787   apply (rule topological_tendstoI)

  1788   apply (drule (2) topological_tendstoD)

  1789   apply (erule (1) eventually_elim2, simp)

  1790   done

  1791

  1792 lemma Lim_transform_within:

  1793   assumes "0 < d"

  1794     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1795     and "(f ---> l) (at x within S)"

  1796   shows "(g ---> l) (at x within S)"

  1797 proof (rule Lim_transform_eventually)

  1798   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1799     using assms(1,2) by (auto simp: dist_nz eventually_at)

  1800   show "(f ---> l) (at x within S)" by fact

  1801 qed

  1802

  1803 lemma Lim_transform_at:

  1804   assumes "0 < d"

  1805     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1806     and "(f ---> l) (at x)"

  1807   shows "(g ---> l) (at x)"

  1808   using _ assms(3)

  1809 proof (rule Lim_transform_eventually)

  1810   show "eventually (\<lambda>x. f x = g x) (at x)"

  1811     unfolding eventually_at2

  1812     using assms(1,2) by auto

  1813 qed

  1814

  1815 text{* Common case assuming being away from some crucial point like 0. *}

  1816

  1817 lemma Lim_transform_away_within:

  1818   fixes a b :: "'a::t1_space"

  1819   assumes "a \<noteq> b"

  1820     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1821     and "(f ---> l) (at a within S)"

  1822   shows "(g ---> l) (at a within S)"

  1823 proof (rule Lim_transform_eventually)

  1824   show "(f ---> l) (at a within S)" by fact

  1825   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1826     unfolding eventually_at_topological

  1827     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1828 qed

  1829

  1830 lemma Lim_transform_away_at:

  1831   fixes a b :: "'a::t1_space"

  1832   assumes ab: "a\<noteq>b"

  1833     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1834     and fl: "(f ---> l) (at a)"

  1835   shows "(g ---> l) (at a)"

  1836   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

  1837

  1838 text{* Alternatively, within an open set. *}

  1839

  1840 lemma Lim_transform_within_open:

  1841   assumes "open S" and "a \<in> S"

  1842     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1843     and "(f ---> l) (at a)"

  1844   shows "(g ---> l) (at a)"

  1845 proof (rule Lim_transform_eventually)

  1846   show "eventually (\<lambda>x. f x = g x) (at a)"

  1847     unfolding eventually_at_topological

  1848     using assms(1,2,3) by auto

  1849   show "(f ---> l) (at a)" by fact

  1850 qed

  1851

  1852 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1853

  1854 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1855

  1856 lemma Lim_cong_within(*[cong add]*):

  1857   assumes "a = b"

  1858     and "x = y"

  1859     and "S = T"

  1860     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1861   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1862   unfolding tendsto_def eventually_at_topological

  1863   using assms by simp

  1864

  1865 lemma Lim_cong_at(*[cong add]*):

  1866   assumes "a = b" "x = y"

  1867     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1868   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1869   unfolding tendsto_def eventually_at_topological

  1870   using assms by simp

  1871

  1872 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1873

  1874 lemma closure_sequential:

  1875   fixes l :: "'a::first_countable_topology"

  1876   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"

  1877   (is "?lhs = ?rhs")

  1878 proof

  1879   assume "?lhs"

  1880   moreover

  1881   {

  1882     assume "l \<in> S"

  1883     then have "?rhs" using tendsto_const[of l sequentially] by auto

  1884   }

  1885   moreover

  1886   {

  1887     assume "l islimpt S"

  1888     then have "?rhs" unfolding islimpt_sequential by auto

  1889   }

  1890   ultimately show "?rhs"

  1891     unfolding closure_def by auto

  1892 next

  1893   assume "?rhs"

  1894   then show "?lhs" unfolding closure_def islimpt_sequential by auto

  1895 qed

  1896

  1897 lemma closed_sequential_limits:

  1898   fixes S :: "'a::first_countable_topology set"

  1899   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1900   unfolding closed_limpt

  1901   using closure_sequential [where 'a='a] closure_closed [where 'a='a]

  1902     closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1903   by metis

  1904

  1905 lemma closure_approachable:

  1906   fixes S :: "'a::metric_space set"

  1907   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1908   apply (auto simp add: closure_def islimpt_approachable)

  1909   apply (metis dist_self)

  1910   done

  1911

  1912 lemma closed_approachable:

  1913   fixes S :: "'a::metric_space set"

  1914   shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1915   by (metis closure_closed closure_approachable)

  1916

  1917 lemma closure_contains_Inf:

  1918   fixes S :: "real set"

  1919   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1920   shows "Inf S \<in> closure S"

  1921 proof -

  1922   have *: "\<forall>x\<in>S. Inf S \<le> x"

  1923     using cInf_lower_EX[of _ S] assms by metis

  1924   {

  1925     fix e :: real

  1926     assume "e > 0"

  1927     then have "Inf S < Inf S + e" by simp

  1928     with assms obtain x where "x \<in> S" "x < Inf S + e"

  1929       by (subst (asm) cInf_less_iff[of _ B]) auto

  1930     with * have "\<exists>x\<in>S. dist x (Inf S) < e"

  1931       by (intro bexI[of _ x]) (auto simp add: dist_real_def)

  1932   }

  1933   then show ?thesis unfolding closure_approachable by auto

  1934 qed

  1935

  1936 lemma closed_contains_Inf:

  1937   fixes S :: "real set"

  1938   assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"

  1939     and "closed S"

  1940   shows "Inf S \<in> S"

  1941   by (metis closure_contains_Inf closure_closed assms)

  1942

  1943

  1944 lemma not_trivial_limit_within_ball:

  1945   "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"

  1946   (is "?lhs = ?rhs")

  1947 proof -

  1948   {

  1949     assume "?lhs"

  1950     {

  1951       fix e :: real

  1952       assume "e > 0"

  1953       then obtain y where "y \<in> S - {x}" and "dist y x < e"

  1954         using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1955         by auto

  1956       then have "y \<in> S \<inter> ball x e - {x}"

  1957         unfolding ball_def by (simp add: dist_commute)

  1958       then have "S \<inter> ball x e - {x} \<noteq> {}" by blast

  1959     }

  1960     then have "?rhs" by auto

  1961   }

  1962   moreover

  1963   {

  1964     assume "?rhs"

  1965     {

  1966       fix e :: real

  1967       assume "e > 0"

  1968       then obtain y where "y \<in> S \<inter> ball x e - {x}"

  1969         using ?rhs by blast

  1970       then have "y \<in> S - {x}" and "dist y x < e"

  1971         unfolding ball_def by (simp_all add: dist_commute)

  1972       then have "\<exists>y \<in> S - {x}. dist y x < e"

  1973         by auto

  1974     }

  1975     then have "?lhs"

  1976       using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]

  1977       by auto

  1978   }

  1979   ultimately show ?thesis by auto

  1980 qed

  1981

  1982

  1983 subsection {* Infimum Distance *}

  1984

  1985 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1986

  1987 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1988   by (simp add: infdist_def)

  1989

  1990 lemma infdist_nonneg: "0 \<le> infdist x A"

  1991   by (auto simp add: infdist_def intro: cInf_greatest)

  1992

  1993 lemma infdist_le:

  1994   assumes "a \<in> A"

  1995     and "d = dist x a"

  1996   shows "infdist x A \<le> d"

  1997   using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

  1998

  1999 lemma infdist_zero[simp]:

  2000   assumes "a \<in> A"

  2001   shows "infdist a A = 0"

  2002 proof -

  2003   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0"

  2004     by auto

  2005   with infdist_nonneg[of a A] assms show "infdist a A = 0"

  2006     by auto

  2007 qed

  2008

  2009 lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"

  2010 proof (cases "A = {}")

  2011   case True

  2012   then show ?thesis by (simp add: infdist_def)

  2013 next

  2014   case False

  2015   then obtain a where "a \<in> A" by auto

  2016   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  2017   proof (rule cInf_greatest)

  2018     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"

  2019       by simp

  2020     fix d

  2021     assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  2022     then obtain a where d: "d = dist x y + dist y a" "a \<in> A"

  2023       by auto

  2024     show "infdist x A \<le> d"

  2025       unfolding infdist_notempty[OF A \<noteq> {}]

  2026     proof (rule cInf_lower2)

  2027       show "dist x a \<in> {dist x a |a. a \<in> A}"

  2028         using a \<in> A by auto

  2029       show "dist x a \<le> d"

  2030         unfolding d by (rule dist_triangle)

  2031       fix d

  2032       assume "d \<in> {dist x a |a. a \<in> A}"

  2033       then obtain a where "a \<in> A" "d = dist x a"

  2034         by auto

  2035       then show "infdist x A \<le> d"

  2036         by (rule infdist_le)

  2037     qed

  2038   qed

  2039   also have "\<dots> = dist x y + infdist y A"

  2040   proof (rule cInf_eq, safe)

  2041     fix a

  2042     assume "a \<in> A"

  2043     then show "dist x y + infdist y A \<le> dist x y + dist y a"

  2044       by (auto intro: infdist_le)

  2045   next

  2046     fix i

  2047     assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  2048     then have "i - dist x y \<le> infdist y A"

  2049       unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  2050       by (intro cInf_greatest) (auto simp: field_simps)

  2051     then show "i \<le> dist x y + infdist y A"

  2052       by simp

  2053   qed

  2054   finally show ?thesis by simp

  2055 qed

  2056

  2057 lemma in_closure_iff_infdist_zero:

  2058   assumes "A \<noteq> {}"

  2059   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2060 proof

  2061   assume "x \<in> closure A"

  2062   show "infdist x A = 0"

  2063   proof (rule ccontr)

  2064     assume "infdist x A \<noteq> 0"

  2065     with infdist_nonneg[of x A] have "infdist x A > 0"

  2066       by auto

  2067     then have "ball x (infdist x A) \<inter> closure A = {}"

  2068       apply auto

  2069       apply (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  2070         eucl_less_not_refl euclidean_trans(2) infdist_le)

  2071       done

  2072     then have "x \<notin> closure A"

  2073       by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  2074     then show False using x \<in> closure A by simp

  2075   qed

  2076 next

  2077   assume x: "infdist x A = 0"

  2078   then obtain a where "a \<in> A"

  2079     by atomize_elim (metis all_not_in_conv assms)

  2080   show "x \<in> closure A"

  2081     unfolding closure_approachable

  2082     apply safe

  2083   proof (rule ccontr)

  2084     fix e :: real

  2085     assume "e > 0"

  2086     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  2087     then have "infdist x A \<ge> e" using a \<in> A

  2088       unfolding infdist_def

  2089       by (force simp: dist_commute intro: cInf_greatest)

  2090     with x e > 0 show False by auto

  2091   qed

  2092 qed

  2093

  2094 lemma in_closed_iff_infdist_zero:

  2095   assumes "closed A" "A \<noteq> {}"

  2096   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  2097 proof -

  2098   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  2099     by (rule in_closure_iff_infdist_zero) fact

  2100   with assms show ?thesis by simp

  2101 qed

  2102

  2103 lemma tendsto_infdist [tendsto_intros]:

  2104   assumes f: "(f ---> l) F"

  2105   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  2106 proof (rule tendstoI)

  2107   fix e ::real

  2108   assume "e > 0"

  2109   from tendstoD[OF f this]

  2110   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  2111   proof (eventually_elim)

  2112     fix x

  2113     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  2114     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  2115       by (simp add: dist_commute dist_real_def)

  2116     also assume "dist (f x) l < e"

  2117     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  2118   qed

  2119 qed

  2120

  2121 text{* Some other lemmas about sequences. *}

  2122

  2123 lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)

  2124   assumes "eventually (\<lambda>i. P i) sequentially"

  2125   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  2126   using assms by (rule eventually_sequentially_seg [THEN iffD2])

  2127

  2128 lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)

  2129   "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  2130   apply (erule filterlim_compose)

  2131   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)

  2132   apply arith

  2133   done

  2134

  2135 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  2136   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

  2137

  2138 subsection {* More properties of closed balls *}

  2139

  2140 lemma closed_cball: "closed (cball x e)"

  2141   unfolding cball_def closed_def

  2142   unfolding Collect_neg_eq [symmetric] not_le

  2143   apply (clarsimp simp add: open_dist, rename_tac y)

  2144   apply (rule_tac x="dist x y - e" in exI, clarsimp)

  2145   apply (rename_tac x')

  2146   apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  2147   apply simp

  2148   done

  2149

  2150 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  2151 proof -

  2152   {

  2153     fix x and e::real

  2154     assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  2155     then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  2156   }

  2157   moreover

  2158   {

  2159     fix x and e::real

  2160     assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  2161     then have "\<exists>d>0. ball x d \<subseteq> S"

  2162       unfolding subset_eq

  2163       apply(rule_tac x="e/2" in exI)

  2164       apply auto

  2165       done

  2166   }

  2167   ultimately show ?thesis

  2168     unfolding open_contains_ball by auto

  2169 qed

  2170

  2171 lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  2172   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  2173

  2174 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  2175   apply (simp add: interior_def, safe)

  2176   apply (force simp add: open_contains_cball)

  2177   apply (rule_tac x="ball x e" in exI)

  2178   apply (simp add: subset_trans [OF ball_subset_cball])

  2179   done

  2180

  2181 lemma islimpt_ball:

  2182   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  2183   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"

  2184   (is "?lhs = ?rhs")

  2185 proof

  2186   assume "?lhs"

  2187   {

  2188     assume "e \<le> 0"

  2189     then have *:"ball x e = {}"

  2190       using ball_eq_empty[of x e] by auto

  2191     have False using ?lhs

  2192       unfolding * using islimpt_EMPTY[of y] by auto

  2193   }

  2194   then have "e > 0" by (metis not_less)

  2195   moreover

  2196   have "y \<in> cball x e"

  2197     using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]

  2198       ball_subset_cball[of x e] ?lhs

  2199     unfolding closed_limpt by auto

  2200   ultimately show "?rhs" by auto

  2201 next

  2202   assume "?rhs"

  2203   then have "e > 0" by auto

  2204   {

  2205     fix d :: real

  2206     assume "d > 0"

  2207     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2208     proof (cases "d \<le> dist x y")

  2209       case True

  2210       then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2211       proof (cases "x = y")

  2212         case True

  2213         then have False

  2214           using d \<le> dist x y d>0 by auto

  2215         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2216           by auto

  2217       next

  2218         case False

  2219         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =

  2220           norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2221           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]

  2222           by auto

  2223         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  2224           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]

  2225           unfolding scaleR_minus_left scaleR_one

  2226           by (auto simp add: norm_minus_commute)

  2227         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  2228           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  2229           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm]

  2230           by auto

  2231         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs

  2232           by (auto simp add: dist_norm)

  2233         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0

  2234           by auto

  2235         moreover

  2236         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  2237           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff

  2238           by (auto simp add: dist_commute)

  2239         moreover

  2240         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"

  2241           unfolding dist_norm

  2242           apply simp

  2243           unfolding norm_minus_cancel

  2244           using d > 0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  2245           unfolding dist_norm

  2246           apply auto

  2247           done

  2248         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2249           apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)

  2250           apply auto

  2251           done

  2252       qed

  2253     next

  2254       case False

  2255       then have "d > dist x y" by auto

  2256       show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"

  2257       proof (cases "x = y")

  2258         case True

  2259         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  2260           using perfect_choose_dist[of "min e d" y]

  2261           using d > 0 e>0 by auto

  2262         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2263           unfolding x = y

  2264           using z \<noteq> y **

  2265           apply (rule_tac x=z in bexI)

  2266           apply (auto simp add: dist_commute)

  2267           done

  2268       next

  2269         case False

  2270         then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  2271           using d>0 d > dist x y ?rhs

  2272           apply (rule_tac x=x in bexI)

  2273           apply auto

  2274           done

  2275       qed

  2276     qed

  2277   }

  2278   then show "?lhs"

  2279     unfolding mem_cball islimpt_approachable mem_ball by auto

  2280 qed

  2281

  2282 lemma closure_ball_lemma:

  2283   fixes x y :: "'a::real_normed_vector"

  2284   assumes "x \<noteq> y"

  2285   shows "y islimpt ball x (dist x y)"

  2286 proof (rule islimptI)

  2287   fix T

  2288   assume "y \<in> T" "open T"

  2289   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2290     unfolding open_dist by fast

  2291   (* choose point between x and y, within distance r of y. *)

  2292   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2293   def z \<equiv> "y + scaleR k (x - y)"

  2294   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2295     unfolding z_def by (simp add: algebra_simps)

  2296   have "dist z y < r"

  2297     unfolding z_def k_def using 0 < r

  2298     by (simp add: dist_norm min_def)

  2299   then have "z \<in> T"

  2300     using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2301   have "dist x z < dist x y"

  2302     unfolding z_def2 dist_norm

  2303     apply (simp add: norm_minus_commute)

  2304     apply (simp only: dist_norm [symmetric])

  2305     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2306     apply (rule mult_strict_right_mono)

  2307     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2308     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2309     done

  2310   then have "z \<in> ball x (dist x y)"

  2311     by simp

  2312   have "z \<noteq> y"

  2313     unfolding z_def k_def using x \<noteq> y 0 < r

  2314     by (simp add: min_def)

  2315   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2316     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2317     by fast

  2318 qed

  2319

  2320 lemma closure_ball:

  2321   fixes x :: "'a::real_normed_vector"

  2322   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2323   apply (rule equalityI)

  2324   apply (rule closure_minimal)

  2325   apply (rule ball_subset_cball)

  2326   apply (rule closed_cball)

  2327   apply (rule subsetI, rename_tac y)

  2328   apply (simp add: le_less [where 'a=real])

  2329   apply (erule disjE)

  2330   apply (rule subsetD [OF closure_subset], simp)

  2331   apply (simp add: closure_def)

  2332   apply clarify

  2333   apply (rule closure_ball_lemma)

  2334   apply (simp add: zero_less_dist_iff)

  2335   done

  2336

  2337 (* In a trivial vector space, this fails for e = 0. *)

  2338 lemma interior_cball:

  2339   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2340   shows "interior (cball x e) = ball x e"

  2341 proof (cases "e \<ge> 0")

  2342   case False note cs = this

  2343   from cs have "ball x e = {}"

  2344     using ball_empty[of e x] by auto

  2345   moreover

  2346   {

  2347     fix y

  2348     assume "y \<in> cball x e"

  2349     then have False

  2350       unfolding mem_cball using dist_nz[of x y] cs by auto

  2351   }

  2352   then have "cball x e = {}" by auto

  2353   then have "interior (cball x e) = {}"

  2354     using interior_empty by auto

  2355   ultimately show ?thesis by blast

  2356 next

  2357   case True note cs = this

  2358   have "ball x e \<subseteq> cball x e"

  2359     using ball_subset_cball by auto

  2360   moreover

  2361   {

  2362     fix S y

  2363     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2364     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"

  2365       unfolding open_dist by blast

  2366     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2367       using perfect_choose_dist [of d] by auto

  2368     have "xa \<in> S"

  2369       using d[THEN spec[where x = xa]]

  2370       using xa by (auto simp add: dist_commute)

  2371     then have xa_cball: "xa \<in> cball x e"

  2372       using as(1) by auto

  2373     then have "y \<in> ball x e"

  2374     proof (cases "x = y")

  2375       case True

  2376       then have "e > 0"

  2377         using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]

  2378         by (auto simp add: dist_commute)

  2379       then show "y \<in> ball x e"

  2380         using x = y  by simp

  2381     next

  2382       case False

  2383       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"

  2384         unfolding dist_norm

  2385         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2386       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"

  2387         using d as(1)[unfolded subset_eq] by blast

  2388       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2389       then have **:"d / (2 * norm (y - x)) > 0"

  2390         unfolding zero_less_norm_iff[symmetric]

  2391         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2392       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =

  2393         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2394         by (auto simp add: dist_norm algebra_simps)

  2395       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2396         by (auto simp add: algebra_simps)

  2397       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2398         using ** by auto

  2399       also have "\<dots> = (dist y x) + d/2"

  2400         using ** by (auto simp add: distrib_right dist_norm)

  2401       finally have "e \<ge> dist x y +d/2"

  2402         using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2403       then show "y \<in> ball x e"

  2404         unfolding mem_ball using d>0 by auto

  2405     qed

  2406   }

  2407   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"

  2408     by auto

  2409   ultimately show ?thesis

  2410     using interior_unique[of "ball x e" "cball x e"]

  2411     using open_ball[of x e]

  2412     by auto

  2413 qed

  2414

  2415 lemma frontier_ball:

  2416   fixes a :: "'a::real_normed_vector"

  2417   shows "0 < e \<Longrightarrow> frontier(ball a e) = {x. dist a x = e}"

  2418   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2419   apply (simp add: set_eq_iff)

  2420   apply arith

  2421   done

  2422

  2423 lemma frontier_cball:

  2424   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2425   shows "frontier (cball a e) = {x. dist a x = e}"

  2426   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2427   apply (simp add: set_eq_iff)

  2428   apply arith

  2429   done

  2430

  2431 lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"

  2432   apply (simp add: set_eq_iff not_le)

  2433   apply (metis zero_le_dist dist_self order_less_le_trans)

  2434   done

  2435

  2436 lemma cball_empty: "e < 0 \<Longrightarrow> cball x e = {}"

  2437   by (simp add: cball_eq_empty)

  2438

  2439 lemma cball_eq_sing:

  2440   fixes x :: "'a::{metric_space,perfect_space}"

  2441   shows "cball x e = {x} \<longleftrightarrow> e = 0"

  2442 proof (rule linorder_cases)

  2443   assume e: "0 < e"

  2444   obtain a where "a \<noteq> x" "dist a x < e"

  2445     using perfect_choose_dist [OF e] by auto

  2446   then have "a \<noteq> x" "dist x a \<le> e"

  2447     by (auto simp add: dist_commute)

  2448   with e show ?thesis by (auto simp add: set_eq_iff)

  2449 qed auto

  2450

  2451 lemma cball_sing:

  2452   fixes x :: "'a::metric_space"

  2453   shows "e = 0 \<Longrightarrow> cball x e = {x}"

  2454   by (auto simp add: set_eq_iff)

  2455

  2456

  2457 subsection {* Boundedness *}

  2458

  2459   (* FIXME: This has to be unified with BSEQ!! *)

  2460 definition (in metric_space) bounded :: "'a set \<Rightarrow> bool"

  2461   where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2462

  2463 lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"

  2464   unfolding bounded_def subset_eq by auto

  2465

  2466 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2467   unfolding bounded_def

  2468   apply safe

  2469   apply (rule_tac x="dist a x + e" in exI)

  2470   apply clarify

  2471   apply (drule (1) bspec)

  2472   apply (erule order_trans [OF dist_triangle add_left_mono])

  2473   apply auto

  2474   done

  2475

  2476 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2477   unfolding bounded_any_center [where a=0]

  2478   by (simp add: dist_norm)

  2479

  2480 lemma bounded_realI:

  2481   assumes "\<forall>x\<in>s. abs (x::real) \<le> B"

  2482   shows "bounded s"

  2483   unfolding bounded_def dist_real_def

  2484   apply (rule_tac x=0 in exI)

  2485   using assms

  2486   apply auto

  2487   done

  2488

  2489 lemma bounded_empty [simp]: "bounded {}"

  2490   by (simp add: bounded_def)

  2491

  2492 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"

  2493   by (metis bounded_def subset_eq)

  2494

  2495 lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"

  2496   by (metis bounded_subset interior_subset)

  2497

  2498 lemma bounded_closure[intro]:

  2499   assumes "bounded S"

  2500   shows "bounded (closure S)"

  2501 proof -

  2502   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"

  2503     unfolding bounded_def by auto

  2504   {

  2505     fix y

  2506     assume "y \<in> closure S"

  2507     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2508       unfolding closure_sequential by auto

  2509     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2510     then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2511       by (rule eventually_mono, simp add: f(1))

  2512     have "dist x y \<le> a"

  2513       apply (rule Lim_dist_ubound [of sequentially f])

  2514       apply (rule trivial_limit_sequentially)

  2515       apply (rule f(2))

  2516       apply fact

  2517       done

  2518   }

  2519   then show ?thesis

  2520     unfolding bounded_def by auto

  2521 qed

  2522

  2523 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2524   apply (simp add: bounded_def)

  2525   apply (rule_tac x=x in exI)

  2526   apply (rule_tac x=e in exI)

  2527   apply auto

  2528   done

  2529

  2530 lemma bounded_ball[simp,intro]: "bounded (ball x e)"

  2531   by (metis ball_subset_cball bounded_cball bounded_subset)

  2532

  2533 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2534   apply (auto simp add: bounded_def)

  2535   apply (rename_tac x y r s)

  2536   apply (rule_tac x=x in exI)

  2537   apply (rule_tac x="max r (dist x y + s)" in exI)

  2538   apply (rule ballI)

  2539   apply safe

  2540   apply (drule (1) bspec)

  2541   apply simp

  2542   apply (drule (1) bspec)

  2543   apply (rule min_max.le_supI2)

  2544   apply (erule order_trans [OF dist_triangle add_left_mono])

  2545   done

  2546

  2547 lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"

  2548   by (induct rule: finite_induct[of F]) auto

  2549

  2550 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2551   by (induct set: finite) auto

  2552

  2553 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2554 proof -

  2555   have "\<forall>y\<in>{x}. dist x y \<le> 0"

  2556     by simp

  2557   then have "bounded {x}"

  2558     unfolding bounded_def by fast

  2559   then show ?thesis

  2560     by (metis insert_is_Un bounded_Un)

  2561 qed

  2562

  2563 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2564   by (induct set: finite) simp_all

  2565

  2566 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"

  2567   apply (simp add: bounded_iff)

  2568   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")

  2569   apply metis

  2570   apply arith

  2571   done

  2572

  2573 lemma Bseq_eq_bounded:

  2574   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"

  2575   shows "Bseq f \<longleftrightarrow> bounded (range f)"

  2576   unfolding Bseq_def bounded_pos by auto

  2577

  2578 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2579   by (metis Int_lower1 Int_lower2 bounded_subset)

  2580

  2581 lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"

  2582   by (metis Diff_subset bounded_subset)

  2583

  2584 lemma not_bounded_UNIV[simp, intro]:

  2585   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2586 proof (auto simp add: bounded_pos not_le)

  2587   obtain x :: 'a where "x \<noteq> 0"

  2588     using perfect_choose_dist [OF zero_less_one] by fast

  2589   fix b :: real

  2590   assume b: "b >0"

  2591   have b1: "b +1 \<ge> 0"

  2592     using b by simp

  2593   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2594     by (simp add: norm_sgn)

  2595   then show "\<exists>x::'a. b < norm x" ..

  2596 qed

  2597

  2598 lemma bounded_linear_image:

  2599   assumes "bounded S"

  2600     and "bounded_linear f"

  2601   shows "bounded (f  S)"

  2602 proof -

  2603   from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2604     unfolding bounded_pos by auto

  2605   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"

  2606     using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2607   {

  2608     fix x

  2609     assume "x \<in> S"

  2610     then have "norm x \<le> b"

  2611       using b by auto

  2612     then have "norm (f x) \<le> B * b"

  2613       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)

  2630   apply assumption

  2631   apply (rule bounded_linear_scaleR_right)

  2632   done

  2633

  2634 lemma bounded_translation:

  2635   fixes S :: "'a::real_normed_vector set"

  2636   assumes "bounded S"

  2637   shows "bounded ((\<lambda>x. a + x)  S)"

  2638 proof -

  2639   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"

  2640     unfolding bounded_pos by auto

  2641   {

  2642     fix x

  2643     assume "x \<in> S"

  2644     then have "norm (a + x) \<le> b + norm a"

  2645       using norm_triangle_ineq[of a x] b by auto

  2646   }

  2647   then show ?thesis

  2648     unfolding bounded_pos

  2649     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]

  2650     by (auto intro!: exI[of _ "b + norm a"])

  2651 qed

  2652

  2653

  2654 text{* Some theorems on sups and infs using the notion "bounded". *}

  2655

  2656 lemma bounded_real:

  2657   fixes S :: "real set"

  2658   shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x \<le> a)"

  2659   by (simp add: bounded_iff)

  2660

  2661 lemma bounded_has_Sup:

  2662   fixes S :: "real set"

  2663   assumes "bounded S"

  2664     and "S \<noteq> {}"

  2665   shows "\<forall>x\<in>S. x \<le> Sup S"

  2666     and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2667 proof

  2668   fix x

  2669   assume "x\<in>S"

  2670   then show "x \<le> Sup S"

  2671     by (metis cSup_upper abs_le_D1 assms(1) bounded_real)

  2672 next

  2673   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"

  2674     using assms by (metis cSup_least)

  2675 qed

  2676

  2677 lemma Sup_insert:

  2678   fixes S :: "real set"

  2679   shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2680   apply (subst cSup_insert_If)

  2681   apply (rule bounded_has_Sup(1)[of S, rule_format])

  2682   apply (auto simp: sup_max)

  2683   done

  2684

  2685 lemma Sup_insert_finite:

  2686   fixes S :: "real set"

  2687   shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"

  2688   apply (rule Sup_insert)

  2689   apply (rule finite_imp_bounded)

  2690   apply simp

  2691   done

  2692

  2693 lemma bounded_has_Inf:

  2694   fixes S :: "real set"

  2695   assumes "bounded S"

  2696     and "S \<noteq> {}"

  2697   shows "\<forall>x\<in>S. x \<ge> Inf S"

  2698     and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2699 proof

  2700   fix x

  2701   assume "x \<in> S"

  2702   from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"

  2703     unfolding bounded_real by auto

  2704   then show "x \<ge> Inf S" using x \<in> S

  2705     by (metis cInf_lower_EX abs_le_D2 minus_le_iff)

  2706 next

  2707   show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"

  2708     using assms by (metis cInf_greatest)

  2709 qed

  2710

  2711 lemma Inf_insert:

  2712   fixes S :: "real set"

  2713   shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2714   apply (subst cInf_insert_if)

  2715   apply (rule bounded_has_Inf(1)[of S, rule_format])

  2716   apply (auto simp: inf_min)

  2717   done

  2718

  2719 lemma Inf_insert_finite:

  2720   fixes S :: "real set"

  2721   shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"

  2722   apply (rule Inf_insert)

  2723   apply (rule finite_imp_bounded)

  2724   apply simp

  2725   done

  2726

  2727 subsection {* Compactness *}

  2728

  2729 subsubsection {* Bolzano-Weierstrass property *}

  2730

  2731 lemma heine_borel_imp_bolzano_weierstrass:

  2732   assumes "compact s"

  2733     and "infinite t"

  2734     and "t \<subseteq> s"

  2735   shows "\<exists>x \<in> s. x islimpt t"

  2736 proof (rule ccontr)

  2737   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2738   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)"

  2739     unfolding islimpt_def

  2740     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]

  2741     by auto

  2742   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2743     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]

  2744     using f by auto

  2745   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"

  2746     by auto

  2747   {

  2748     fix x y

  2749     assume "x \<in> t" "y \<in> t" "f x = f y"

  2750     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"

  2751       using f[THEN bspec[where x=x]] and t \<subseteq> s by auto

  2752     then have "x = y"

  2753       using f x = f y and f[THEN bspec[where x=y]] and y \<in> t and t \<subseteq> s

  2754       by auto

  2755   }

  2756   then have "inj_on f t"

  2757     unfolding inj_on_def by simp

  2758   then have "infinite (f  t)"

  2759     using assms(2) using finite_imageD by auto

  2760   moreover

  2761   {

  2762     fix x

  2763     assume "x \<in> t" "f x \<notin> g"

  2764     from g(3) assms(3) x \<in> t obtain h where "h \<in> g" and "x \<in> h"

  2765       by auto

  2766     then obtain y where "y \<in> s" "h = f y"

  2767       using g'[THEN bspec[where x=h]] by auto

  2768     then have "y = x"

  2769       using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y]

  2770       by auto

  2771     then have False

  2772       using f x \<notin> g h \<in> g unfolding h = f y

  2773       by auto

  2774   }

  2775   then have "f  t \<subseteq> g" by auto

  2776   ultimately show False

  2777     using g(2) using finite_subset by auto

  2778 qed

  2779

  2780 lemma acc_point_range_imp_convergent_subsequence:

  2781   fixes l :: "'a :: first_countable_topology"

  2782   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2783   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2784 proof -

  2785   from countable_basis_at_decseq[of l] guess A . note A = this

  2786

  2787   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2788   {

  2789     fix n i

  2790     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2791       using l A by auto

  2792     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2793       unfolding ex_in_conv by (intro notI) simp

  2794     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2795       by auto

  2796     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2797       by (auto simp: not_le)

  2798     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2799       unfolding s_def by (auto intro: someI2_ex)

  2800   }

  2801   note s = this

  2802   def r \<equiv> "nat_rec (s 0 0) s"

  2803   have "subseq r"

  2804     by (auto simp: r_def s subseq_Suc_iff)

  2805   moreover

  2806   have "(\<lambda>n. f (r n)) ----> l"

  2807   proof (rule topological_tendstoI)

  2808     fix S

  2809     assume "open S" "l \<in> S"

  2810     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  2811       by auto

  2812     moreover

  2813     {

  2814       fix i

  2815       assume "Suc 0 \<le> i"

  2816       then have "f (r i) \<in> A i"

  2817         by (cases i) (simp_all add: r_def s)

  2818     }

  2819     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"

  2820       by (auto simp: eventually_sequentially)

  2821     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2822       by eventually_elim auto

  2823   qed

  2824   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2825     by (auto simp: convergent_def comp_def)

  2826 qed

  2827

  2828 lemma sequence_infinite_lemma:

  2829   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2830   assumes "\<forall>n. f n \<noteq> l"

  2831     and "(f ---> l) sequentially"

  2832   shows "infinite (range f)"

  2833 proof

  2834   assume "finite (range f)"

  2835   then have "closed (range f)"

  2836     by (rule finite_imp_closed)

  2837   then have "open (- range f)"

  2838     by (rule open_Compl)

  2839   from assms(1) have "l \<in> - range f"

  2840     by auto

  2841   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2842     using open (- range f) l \<in> - range f

  2843     by (rule topological_tendstoD)

  2844   then show False

  2845     unfolding eventually_sequentially

  2846     by auto

  2847 qed

  2848

  2849 lemma closure_insert:

  2850   fixes x :: "'a::t1_space"

  2851   shows "closure (insert x s) = insert x (closure s)"

  2852   apply (rule closure_unique)

  2853   apply (rule insert_mono [OF closure_subset])

  2854   apply (rule closed_insert [OF closed_closure])

  2855   apply (simp add: closure_minimal)

  2856   done

  2857

  2858 lemma islimpt_insert:

  2859   fixes x :: "'a::t1_space"

  2860   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2861 proof

  2862   assume *: "x islimpt (insert a s)"

  2863   show "x islimpt s"

  2864   proof (rule islimptI)

  2865     fix t

  2866     assume t: "x \<in> t" "open t"

  2867     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2868     proof (cases "x = a")

  2869       case True

  2870       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2871         using * t by (rule islimptE)

  2872       with x = a show ?thesis by auto

  2873     next

  2874       case False

  2875       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2876         by (simp_all add: open_Diff)

  2877       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2878         using * t' by (rule islimptE)

  2879       then show ?thesis by auto

  2880     qed

  2881   qed

  2882 next

  2883   assume "x islimpt s"

  2884   then show "x islimpt (insert a s)"

  2885     by (rule islimpt_subset) auto

  2886 qed

  2887

  2888 lemma islimpt_finite:

  2889   fixes x :: "'a::t1_space"

  2890   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2891   by (induct set: finite) (simp_all add: islimpt_insert)

  2892

  2893 lemma islimpt_union_finite:

  2894   fixes x :: "'a::t1_space"

  2895   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2896   by (simp add: islimpt_Un islimpt_finite)

  2897

  2898 lemma islimpt_eq_acc_point:

  2899   fixes l :: "'a :: t1_space"

  2900   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2901 proof (safe intro!: islimptI)

  2902   fix U

  2903   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2904   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2905     by (auto intro: finite_imp_closed)

  2906   then show False

  2907     by (rule islimptE) auto

  2908 next

  2909   fix T

  2910   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2911   then have "infinite (T \<inter> S - {l})"

  2912     by auto

  2913   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2914     unfolding ex_in_conv by (intro notI) simp

  2915   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2916     by auto

  2917 qed

  2918

  2919 lemma islimpt_range_imp_convergent_subsequence:

  2920   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2921   assumes l: "l islimpt (range f)"

  2922   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2923   using l unfolding islimpt_eq_acc_point

  2924   by (rule acc_point_range_imp_convergent_subsequence)

  2925

  2926 lemma sequence_unique_limpt:

  2927   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2928   assumes "(f ---> l) sequentially"

  2929     and "l' islimpt (range f)"

  2930   shows "l' = l"

  2931 proof (rule ccontr)

  2932   assume "l' \<noteq> l"

  2933   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2934     using hausdorff [OF l' \<noteq> l] by auto

  2935   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2936     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2937   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2938     unfolding eventually_sequentially by auto

  2939

  2940   have "UNIV = {..<N} \<union> {N..}"

  2941     by auto

  2942   then have "l' islimpt (f  ({..<N} \<union> {N..}))"

  2943     using assms(2) by simp

  2944   then have "l' islimpt (f  {..<N} \<union> f  {N..})"

  2945     by (simp add: image_Un)

  2946   then have "l' islimpt (f  {N..})"

  2947     by (simp add: islimpt_union_finite)

  2948   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2949     using l' \<in> s open s by (rule islimptE)

  2950   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"

  2951     by auto

  2952   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t"

  2953     by simp

  2954   with s \<inter> t = {} show False

  2955     by simp

  2956 qed

  2957

  2958 lemma bolzano_weierstrass_imp_closed:

  2959   fixes s :: "'a::{first_countable_topology,t2_space} set"

  2960   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2961   shows "closed s"

  2962 proof -

  2963   {

  2964     fix x l

  2965     assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2966     then have "l \<in> s"

  2967     proof (cases "\<forall>n. x n \<noteq> l")

  2968       case False

  2969       then show "l\<in>s" using as(1) by auto

  2970     next

  2971       case True note cas = this

  2972       with as(2) have "infinite (range x)"

  2973         using sequence_infinite_lemma[of x l] by auto

  2974       then obtain l' where "l'\<in>s" "l' islimpt (range x)"

  2975         using assms[THEN spec[where x="range x"]] as(1) by auto

  2976       then show "l\<in>s" using sequence_unique_limpt[of x l l']

  2977         using as cas by auto

  2978     qed

  2979   }

  2980   then show ?thesis

  2981     unfolding closed_sequential_limits by fast

  2982 qed

  2983

  2984 lemma compact_imp_bounded:

  2985   assumes "compact U"

  2986   shows "bounded U"

  2987 proof -

  2988   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"

  2989     using assms by auto

  2990   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2991     by (rule compactE_image)

  2992   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2993     by (simp add: bounded_UN)

  2994   then show "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2995     by (rule bounded_subset)

  2996 qed

  2997

  2998 text{* In particular, some common special cases. *}

  2999

  3000 lemma compact_union [intro]:

  3001   assumes "compact s"

  3002     and "compact t"

  3003   shows " compact (s \<union> t)"

  3004 proof (rule compactI)

  3005   fix f

  3006   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  3007   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  3008     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  3009   moreover

  3010   from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  3011     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  3012   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  3013     by (auto intro!: exI[of _ "s' \<union> t'"])

  3014 qed

  3015

  3016 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  3017   by (induct set: finite) auto

  3018

  3019 lemma compact_UN [intro]:

  3020   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  3021   unfolding SUP_def by (rule compact_Union) auto

  3022

  3023 lemma closed_inter_compact [intro]:

  3024   assumes "closed s"

  3025     and "compact t"

  3026   shows "compact (s \<inter> t)"

  3027   using compact_inter_closed [of t s] assms

  3028   by (simp add: Int_commute)

  3029

  3030 lemma compact_inter [intro]:

  3031   fixes s t :: "'a :: t2_space set"

  3032   assumes "compact s"

  3033     and "compact t"

  3034   shows "compact (s \<inter> t)"

  3035   using assms by (intro compact_inter_closed compact_imp_closed)

  3036

  3037 lemma compact_sing [simp]: "compact {a}"

  3038   unfolding compact_eq_heine_borel by auto

  3039

  3040 lemma compact_insert [simp]:

  3041   assumes "compact s"

  3042   shows "compact (insert x s)"

  3043 proof -

  3044   have "compact ({x} \<union> s)"

  3045     using compact_sing assms by (rule compact_union)

  3046   then show ?thesis by simp

  3047 qed

  3048

  3049 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"

  3050   by (induct set: finite) simp_all

  3051

  3052 lemma open_delete:

  3053   fixes s :: "'a::t1_space set"

  3054   shows "open s \<Longrightarrow> open (s - {x})"

  3055   by (simp add: open_Diff)

  3056

  3057 text{* Finite intersection property *}

  3058

  3059 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  3060   by (auto simp: inj_on_def)

  3061

  3062 lemma compact_fip:

  3063   "compact U \<longleftrightarrow>

  3064     (\<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> {})"

  3065   (is "_ \<longleftrightarrow> ?R")

  3066 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  3067   fix A

  3068   assume "compact U"

  3069     and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  3070     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  3071   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>(uminusA)"

  3072     by auto

  3073   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  3074     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  3075   with fi[THEN spec, of B] show False

  3076     by (auto dest: finite_imageD intro: inj_setminus)

  3077 next

  3078   fix A

  3079   assume ?R

  3080   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3081   then have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  3082     by auto

  3083   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>(uminusB) = {}"

  3084     by (metis subset_image_iff)

  3085   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3086     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  3087 qed

  3088

  3089 lemma compact_imp_fip:

  3090   "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>

  3091     s \<inter> (\<Inter> f) \<noteq> {}"

  3092   unfolding compact_fip by auto

  3093

  3094 text{*Compactness expressed with filters*}

  3095

  3096 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3097

  3098 lemma eventually_filter_from_subbase:

  3099   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  3100     (is "_ \<longleftrightarrow> ?R P")

  3101   unfolding filter_from_subbase_def

  3102 proof (rule eventually_Abs_filter is_filter.intro)+

  3103   show "?R (\<lambda>x. True)"

  3104     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  3105 next

  3106   fix P Q assume "?R P" then guess X ..

  3107   moreover assume "?R Q" then guess Y ..

  3108   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  3109     by (intro exI[of _ "X \<union> Y"]) auto

  3110 next

  3111   fix P Q

  3112   assume "?R P" then guess X ..

  3113   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  3114   ultimately show "?R Q"

  3115     by (intro exI[of _ X]) auto

  3116 qed

  3117

  3118 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  3119   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  3120

  3121 lemma filter_from_subbase_not_bot:

  3122   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  3123   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  3124

  3125 lemma closure_iff_nhds_not_empty:

  3126   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  3127 proof safe

  3128   assume x: "x \<in> closure X"

  3129   fix S A

  3130   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  3131   then have "x \<notin> closure (-S)"

  3132     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  3133   with x have "x \<in> closure X - closure (-S)"

  3134     by auto

  3135   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  3136     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  3137   finally have "X \<inter> S \<noteq> {}" by auto

  3138   then show False using X \<inter> A = {} S \<subseteq> A by auto

  3139 next

  3140   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  3141   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  3142   show "x \<in> closure X"

  3143     by (simp add: closure_subset open_Compl)

  3144 qed

  3145

  3146 lemma compact_filter:

  3147   "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))"

  3148 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  3149   fix F

  3150   assume "compact U"

  3151   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  3152   then have "U \<noteq> {}"

  3153     by (auto simp: eventually_False)

  3154

  3155   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  3156   then have "\<forall>z\<in>Z. closed z"

  3157     by auto

  3158   moreover

  3159   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  3160     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  3161   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  3162   proof (intro allI impI)

  3163     fix B assume "finite B" "B \<subseteq> Z"

  3164     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  3165       by (auto intro!: eventually_Ball_finite)

  3166     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  3167       by eventually_elim auto

  3168     with F show "U \<inter> \<Inter>B \<noteq> {}"

  3169       by (intro notI) (simp add: eventually_False)

  3170   qed

  3171   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  3172     using compact U unfolding compact_fip by blast

  3173   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"

  3174     by auto

  3175

  3176   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  3177     unfolding eventually_inf eventually_nhds

  3178   proof safe

  3179     fix P Q R S

  3180     assume "eventually R F" "open S" "x \<in> S"

  3181     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  3182     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  3183     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  3184     ultimately show False by (auto simp: set_eq_iff)

  3185   qed

  3186   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  3187     by (metis eventually_bot)

  3188 next

  3189   fix A

  3190   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  3191   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  3192   then have inj_P': "\<And>A. inj_on P' A"

  3193     by (auto intro!: inj_onI simp: fun_eq_iff)

  3194   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  3195   have "F \<noteq> bot"

  3196     unfolding F_def

  3197   proof (safe intro!: filter_from_subbase_not_bot)

  3198     fix X

  3199     assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  3200     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  3201       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  3202     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}"

  3203       by auto

  3204     with B show False

  3205       by (auto simp: P'_def fun_eq_iff)

  3206   qed

  3207   moreover have "eventually (\<lambda>x. x \<in> U) F"

  3208     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  3209   moreover

  3210   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)"

  3211   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  3212     by auto

  3213

  3214   {

  3215     fix V

  3216     assume "V \<in> A"

  3217     then have V: "eventually (\<lambda>x. x \<in> V) F"

  3218       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  3219     have "x \<in> closure V"

  3220       unfolding closure_iff_nhds_not_empty

  3221     proof (intro impI allI)

  3222       fix S A

  3223       assume "open S" "x \<in> S" "S \<subseteq> A"

  3224       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"

  3225         by (auto simp: eventually_nhds)

  3226       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  3227         by (auto simp: eventually_inf)

  3228       with x show "V \<inter> A \<noteq> {}"

  3229         by (auto simp del: Int_iff simp add: trivial_limit_def)

  3230     qed

  3231     then have "x \<in> V"

  3232       using V \<in> A A(1) by simp

  3233   }

  3234   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  3235   with U \<inter> \<Inter>A = {} show False by auto

  3236 qed

  3237

  3238 definition "countably_compact U \<longleftrightarrow>

  3239     (\<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))"

  3240

  3241 lemma countably_compactE:

  3242   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  3243   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  3244   using assms unfolding countably_compact_def by metis

  3245

  3246 lemma countably_compactI:

  3247   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')"

  3248   shows "countably_compact s"

  3249   using assms unfolding countably_compact_def by metis

  3250

  3251 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  3252   by (auto simp: compact_eq_heine_borel countably_compact_def)

  3253

  3254 lemma countably_compact_imp_compact:

  3255   assumes "countably_compact U"

  3256     and ccover: "countable B" "\<forall>b\<in>B. open b"

  3257     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"

  3258   shows "compact U"

  3259   using countably_compact U

  3260   unfolding compact_eq_heine_borel countably_compact_def

  3261 proof safe

  3262   fix A

  3263   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  3264   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)"

  3265

  3266   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  3267   ultimately have "countable C" "\<forall>a\<in>C. open a"

  3268     unfolding C_def using ccover by auto

  3269   moreover

  3270   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  3271   proof safe

  3272     fix x a

  3273     assume "x \<in> U" "x \<in> a" "a \<in> A"

  3274     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"

  3275       by blast

  3276     with a \<in> A show "x \<in> \<Union>C"

  3277       unfolding C_def by auto

  3278   qed

  3279   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  3280   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  3281     using * by metis

  3282   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  3283     by (auto simp: C_def)

  3284   then guess f unfolding bchoice_iff Bex_def ..

  3285   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3286     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  3287 qed

  3288

  3289 lemma countably_compact_imp_compact_second_countable:

  3290   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3291 proof (rule countably_compact_imp_compact)

  3292   fix T and x :: 'a

  3293   assume "open T" "x \<in> T"

  3294   from topological_basisE[OF is_basis this] guess b .

  3295   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"

  3296     by auto

  3297 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  3298

  3299 lemma countably_compact_eq_compact:

  3300   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  3301   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  3302

  3303 subsubsection{* Sequential compactness *}

  3304

  3305 definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"

  3306   where "seq_compact S \<longleftrightarrow>

  3307     (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"

  3308

  3309 lemma seq_compact_imp_countably_compact:

  3310   fixes U :: "'a :: first_countable_topology set"

  3311   assumes "seq_compact U"

  3312   shows "countably_compact U"

  3313 proof (safe intro!: countably_compactI)

  3314   fix A

  3315   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  3316   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  3317     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  3318   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  3319   proof cases

  3320     assume "finite A"

  3321     with A show ?thesis by auto

  3322   next

  3323     assume "infinite A"

  3324     then have "A \<noteq> {}" by auto

  3325     show ?thesis

  3326     proof (rule ccontr)

  3327       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  3328       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"

  3329         by auto

  3330       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"

  3331         by metis

  3332       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  3333       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  3334         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  3335       then have "range X \<subseteq> U"

  3336         by auto

  3337       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x"

  3338         by auto

  3339       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  3340       obtain n where "x \<in> from_nat_into A n" by auto

  3341       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  3342       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  3343         unfolding tendsto_def by (auto simp: comp_def)

  3344       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  3345         by (auto simp: eventually_sequentially)

  3346       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  3347         by auto

  3348       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  3349         by (auto intro!: exI[of _ "max n N"])

  3350       ultimately show False

  3351         by auto

  3352     qed

  3353   qed

  3354 qed

  3355

  3356 lemma compact_imp_seq_compact:

  3357   fixes U :: "'a :: first_countable_topology set"

  3358   assumes "compact U"

  3359   shows "seq_compact U"

  3360   unfolding seq_compact_def

  3361 proof safe

  3362   fix X :: "nat \<Rightarrow> 'a"

  3363   assume "\<forall>n. X n \<in> U"

  3364   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  3365     by (auto simp: eventually_filtermap)

  3366   moreover

  3367   have "filtermap X sequentially \<noteq> bot"

  3368     by (simp add: trivial_limit_def eventually_filtermap)

  3369   ultimately

  3370   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  3371     using compact U by (auto simp: compact_filter)

  3372

  3373   from countable_basis_at_decseq[of x] guess A . note A = this

  3374   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  3375   {

  3376     fix n i

  3377     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  3378     proof (rule ccontr)

  3379       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  3380       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"

  3381         by auto

  3382       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  3383         by (auto simp: eventually_filtermap eventually_sequentially)

  3384       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  3385         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  3386       ultimately have "eventually (\<lambda>x. False) ?F"

  3387         by (auto simp add: eventually_inf)

  3388       with x show False

  3389         by (simp add: eventually_False)

  3390     qed

  3391     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  3392       unfolding s_def by (auto intro: someI2_ex)

  3393   }

  3394   note s = this

  3395   def r \<equiv> "nat_rec (s 0 0) s"

  3396   have "subseq r"

  3397     by (auto simp: r_def s subseq_Suc_iff)

  3398   moreover

  3399   have "(\<lambda>n. X (r n)) ----> x"

  3400   proof (rule topological_tendstoI)

  3401     fix S

  3402     assume "open S" "x \<in> S"

  3403     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"

  3404       by auto

  3405     moreover

  3406     {

  3407       fix i

  3408       assume "Suc 0 \<le> i"

  3409       then have "X (r i) \<in> A i"

  3410         by (cases i) (simp_all add: r_def s)

  3411     }

  3412     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"

  3413       by (auto simp: eventually_sequentially)

  3414     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  3415       by eventually_elim auto

  3416   qed

  3417   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  3418     using x \<in> U by (auto simp: convergent_def comp_def)

  3419 qed

  3420

  3421 lemma seq_compactI:

  3422   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3423   shows "seq_compact S"

  3424   unfolding seq_compact_def using assms by fast

  3425

  3426 lemma seq_compactE:

  3427   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  3428   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  3429   using assms unfolding seq_compact_def by fast

  3430

  3431 lemma countably_compact_imp_acc_point:

  3432   assumes "countably_compact s"

  3433     and "countable t"

  3434     and "infinite t"

  3435     and "t \<subseteq> s"

  3436   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3437 proof (rule ccontr)

  3438   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3439   note countably_compact s

  3440   moreover have "\<forall>t\<in>C. open t"

  3441     by (auto simp: C_def)

  3442   moreover

  3443   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3444   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3445   have "s \<subseteq> \<Union>C"

  3446     using t \<subseteq> s

  3447     unfolding C_def Union_image_eq

  3448     apply (safe dest!: s)

  3449     apply (rule_tac a="U \<inter> t" in UN_I)

  3450     apply (auto intro!: interiorI simp add: finite_subset)

  3451     done

  3452   moreover

  3453   from countable t have "countable C"

  3454     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3455   ultimately guess D by (rule countably_compactE)

  3456   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"

  3457     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3458     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3459   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3460     using interior_subset by blast

  3461   moreover have "finite (\<Union>E)"

  3462     using E by auto

  3463   ultimately show False using infinite t

  3464     by (auto simp: finite_subset)

  3465 qed

  3466

  3467 lemma countable_acc_point_imp_seq_compact:

  3468   fixes s :: "'a::first_countable_topology set"

  3469   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>

  3470     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3471   shows "seq_compact s"

  3472 proof -

  3473   {

  3474     fix f :: "nat \<Rightarrow> 'a"

  3475     assume f: "\<forall>n. f n \<in> s"

  3476     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3477     proof (cases "finite (range f)")

  3478       case True

  3479       obtain l where "infinite {n. f n = f l}"

  3480         using pigeonhole_infinite[OF _ True] by auto

  3481       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3482         using infinite_enumerate by blast

  3483       then have "subseq r \<and> (f \<circ> r) ----> f l"

  3484         by (simp add: fr tendsto_const o_def)

  3485       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3486         by auto

  3487     next

  3488       case False

  3489       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"

  3490         by auto

  3491       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3492       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3493         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3494       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3495     qed

  3496   }

  3497   then show ?thesis

  3498     unfolding seq_compact_def by auto

  3499 qed

  3500

  3501 lemma seq_compact_eq_countably_compact:

  3502   fixes U :: "'a :: first_countable_topology set"

  3503   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3504   using

  3505     countable_acc_point_imp_seq_compact

  3506     countably_compact_imp_acc_point

  3507     seq_compact_imp_countably_compact

  3508   by metis

  3509

  3510 lemma seq_compact_eq_acc_point:

  3511   fixes s :: "'a :: first_countable_topology set"

  3512   shows "seq_compact s \<longleftrightarrow>

  3513     (\<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)))"

  3514   using

  3515     countable_acc_point_imp_seq_compact[of s]

  3516     countably_compact_imp_acc_point[of s]

  3517     seq_compact_imp_countably_compact[of s]

  3518   by metis

  3519

  3520 lemma seq_compact_eq_compact:

  3521   fixes U :: "'a :: second_countable_topology set"

  3522   shows "seq_compact U \<longleftrightarrow> compact U"

  3523   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3524

  3525 lemma bolzano_weierstrass_imp_seq_compact:

  3526   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3527   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3528   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3529

  3530 subsubsection{* Total boundedness *}

  3531

  3532 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)"

  3533   unfolding Cauchy_def by metis

  3534

  3535 fun helper_1 :: "('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a"

  3536 where

  3537   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3538 declare helper_1.simps[simp del]

  3539

  3540 lemma seq_compact_imp_totally_bounded:

  3541   assumes "seq_compact s"

  3542   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3543 proof (rule, rule, rule ccontr)

  3544   fix e::real

  3545   assume "e > 0"

  3546   assume assm: "\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"

  3547   def x \<equiv> "helper_1 s e"

  3548   {

  3549     fix n

  3550     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3551     proof (induct n rule: nat_less_induct)

  3552       fix n

  3553       def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3554       assume as: "\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3555       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3556         using assm

  3557         apply simp

  3558         apply (erule_tac x="x  {0 ..< n}" in allE)

  3559         using as

  3560         apply auto

  3561         done

  3562       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)"

  3563         unfolding subset_eq by auto

  3564       have "Q (x n)"

  3565         unfolding x_def and helper_1.simps[of s e n]

  3566         apply (rule someI2[where a=z])

  3567         unfolding x_def[symmetric] and Q_def

  3568         using z

  3569         apply auto

  3570         done

  3571       then show "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3572         unfolding Q_def by auto

  3573     qed

  3574   }

  3575   then have "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)"

  3576     by blast+

  3577   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially"

  3578     using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3579   from this(3) have "Cauchy (x \<circ> r)"

  3580     using LIMSEQ_imp_Cauchy by auto

  3581   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"

  3582     unfolding cauchy_def using e>0 by auto

  3583   show False

  3584     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3585     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3586     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]

  3587     by auto

  3588 qed

  3589

  3590 subsubsection{* Heine-Borel theorem *}

  3591

  3592 lemma seq_compact_imp_heine_borel:

  3593   fixes s :: "'a :: metric_space set"

  3594   assumes "seq_compact s"

  3595   shows "compact s"

  3596 proof -

  3597   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3598   guess f unfolding choice_iff' .. note f = this

  3599   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3600   have "countably_compact s"

  3601     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3602   then show "compact s"

  3603   proof (rule countably_compact_imp_compact)

  3604     show "countable K"

  3605       unfolding K_def using f

  3606       by (auto intro: countable_finite countable_subset countable_rat

  3607                intro!: countable_image countable_SIGMA countable_UN)

  3608     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3609   next

  3610     fix T x

  3611     assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3612     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"

  3613       by auto

  3614     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"

  3615       by auto

  3616     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"

  3617       by auto

  3618     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3619       unfolding Union_image_eq by auto

  3620     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K"

  3621       by (auto simp: K_def)

  3622     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3623     proof (rule bexI[rotated], safe)

  3624       fix y

  3625       assume "y \<in> ball k r"

  3626       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3627         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3628       with ball x e \<subseteq> T show "y \<in> T"

  3629         by auto

  3630     next

  3631       show "x \<in> ball k r" by fact

  3632     qed

  3633   qed

  3634 qed

  3635

  3636 lemma compact_eq_seq_compact_metric:

  3637   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3638   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3639

  3640 lemma compact_def:

  3641   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3642    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"

  3643   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3644

  3645 subsubsection {* Complete the chain of compactness variants *}

  3646

  3647 lemma compact_eq_bolzano_weierstrass:

  3648   fixes s :: "'a::metric_space set"

  3649   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"

  3650   (is "?lhs = ?rhs")

  3651 proof

  3652   assume ?lhs

  3653   then show ?rhs

  3654     using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3655 next

  3656   assume ?rhs

  3657   then show ?lhs

  3658     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3659 qed

  3660

  3661 lemma bolzano_weierstrass_imp_bounded:

  3662   "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3663   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3664

  3665 text {*

  3666   A metric space (or topological vector space) is said to have the

  3667   Heine-Borel property if every closed and bounded subset is compact.

  3668 *}

  3669

  3670 class heine_borel = metric_space +

  3671   assumes bounded_imp_convergent_subsequence:

  3672     "bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3673

  3674 lemma bounded_closed_imp_seq_compact:

  3675   fixes s::"'a::heine_borel set"

  3676   assumes "bounded s"

  3677     and "closed s"

  3678   shows "seq_compact s"

  3679 proof (unfold seq_compact_def, clarify)

  3680   fix f :: "nat \<Rightarrow> 'a"

  3681   assume f: "\<forall>n. f n \<in> s"

  3682   with bounded s have "bounded (range f)"

  3683     by (auto intro: bounded_subset)

  3684   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3685     using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto

  3686   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"

  3687     by simp

  3688   have "l \<in> s" using closed s fr l

  3689     unfolding closed_sequential_limits by blast

  3690   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3691     using l \<in> s r l by blast

  3692 qed

  3693

  3694 lemma compact_eq_bounded_closed:

  3695   fixes s :: "'a::heine_borel set"

  3696   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"

  3697   (is "?lhs = ?rhs")

  3698 proof

  3699   assume ?lhs

  3700   then show ?rhs

  3701     using compact_imp_closed compact_imp_bounded

  3702     by blast

  3703 next

  3704   assume ?rhs

  3705   then show ?lhs

  3706     using bounded_closed_imp_seq_compact[of s]

  3707     unfolding compact_eq_seq_compact_metric

  3708     by auto

  3709 qed

  3710

  3711 (* TODO: is this lemma necessary? *)

  3712 lemma bounded_increasing_convergent:

  3713   fixes s :: "nat \<Rightarrow> real"

  3714   shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"

  3715   using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]

  3716   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

  3717

  3718 instance real :: heine_borel

  3719 proof

  3720   fix f :: "nat \<Rightarrow> real"

  3721   assume f: "bounded (range f)"

  3722   obtain r where r: "subseq r" "monoseq (f \<circ> r)"

  3723     unfolding comp_def by (metis seq_monosub)

  3724   then have "Bseq (f \<circ> r)"

  3725     unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)

  3726   with r show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"

  3727     using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)

  3728 qed

  3729

  3730 lemma compact_lemma:

  3731   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3732   assumes "bounded (range f)"

  3733   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r.

  3734     subseq r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3735 proof safe

  3736   fix d :: "'a set"

  3737   assume d: "d \<subseteq> Basis"

  3738   with finite_Basis have "finite d"

  3739     by (blast intro: finite_subset)

  3740   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3741     (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3742   proof (induct d)

  3743     case empty

  3744     then show ?case

  3745       unfolding subseq_def by auto

  3746   next

  3747     case (insert k d)

  3748     have k[intro]: "k \<in> Basis"

  3749       using insert by auto

  3750     have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)"

  3751       using bounded (range f)

  3752       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3753     obtain l1::"'a" and r1 where r1: "subseq r1"

  3754       and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3755       using insert(3) using insert(4) by auto

  3756     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f"

  3757       by simp

  3758     have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"

  3759       by (metis (lifting) bounded_subset f' image_subsetI s')

  3760     then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3761       using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"]

  3762       by (auto simp: o_def)

  3763     def r \<equiv> "r1 \<circ> r2"

  3764     have r:"subseq r"

  3765       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3766     moreover

  3767     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3768     {

  3769       fix e::real

  3770       assume "e > 0"

  3771       from lr1 e > 0 have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3772         by blast

  3773       from lr2 e > 0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially"

  3774         by (rule tendstoD)

  3775       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3776         by (rule eventually_subseq)

  3777       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3778         using N1' N2

  3779         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3780     }

  3781     ultimately show ?case by auto

  3782   qed

  3783 qed

  3784

  3785 instance euclidean_space \<subseteq> heine_borel

  3786 proof

  3787   fix f :: "nat \<Rightarrow> 'a"

  3788   assume f: "bounded (range f)"

  3789   then obtain l::'a and r where r: "subseq r"

  3790     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3791     using compact_lemma [OF f] by blast

  3792   {

  3793     fix e::real

  3794     assume "e > 0"

  3795     then have "e / real_of_nat DIM('a) > 0"

  3796       by (auto intro!: divide_pos_pos DIM_positive)

  3797     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"

  3798       by simp

  3799     moreover

  3800     {

  3801       fix n

  3802       assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3803       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3804         apply (subst euclidean_dist_l2)

  3805         using zero_le_dist

  3806         apply (rule setL2_le_setsum)

  3807         done

  3808       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3809         apply (rule setsum_strict_mono)

  3810         using n

  3811         apply auto

  3812         done

  3813       finally have "dist (f (r n)) l < e"

  3814         by auto

  3815     }

  3816     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3817       by (rule eventually_elim1)

  3818   }

  3819   then have *: "((f \<circ> r) ---> l) sequentially"

  3820     unfolding o_def tendsto_iff by simp

  3821   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3822     by auto

  3823 qed

  3824

  3825 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3826   unfolding bounded_def

  3827   apply clarify

  3828   apply (rule_tac x="a" in exI)

  3829   apply (rule_tac x="e" in exI)

  3830   apply clarsimp

  3831   apply (drule (1) bspec)

  3832   apply (simp add: dist_Pair_Pair)

  3833   apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3834   done

  3835

  3836 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3837   unfolding bounded_def

  3838   apply clarify

  3839   apply (rule_tac x="b" in exI)

  3840   apply (rule_tac x="e" in exI)

  3841   apply clarsimp

  3842   apply (drule (1) bspec)

  3843   apply (simp add: dist_Pair_Pair)

  3844   apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3845   done

  3846

  3847 instance prod :: (heine_borel, heine_borel) heine_borel

  3848 proof

  3849   fix f :: "nat \<Rightarrow> 'a \<times> 'b"

  3850   assume f: "bounded (range f)"

  3851   from f have s1: "bounded (range (fst \<circ> f))"

  3852     unfolding image_comp by (rule bounded_fst)

  3853   obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"

  3854     using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast

  3855   from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"

  3856     by (auto simp add: image_comp intro: bounded_snd bounded_subset)

  3857   obtain l2 r2 where r2: "subseq r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3858     using bounded_imp_convergent_subsequence [OF s2]

  3859     unfolding o_def by fast

  3860   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3861     using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .

  3862   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3863     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3864   have r: "subseq (r1 \<circ> r2)"

  3865     using r1 r2 unfolding subseq_def by simp

  3866   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3867     using l r by fast

  3868 qed

  3869

  3870 subsubsection{* Completeness *}

  3871

  3872 definition complete :: "'a::metric_space set \<Rightarrow> bool"

  3873   where "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

  3874

  3875 lemma compact_imp_complete:

  3876   assumes "compact s"

  3877   shows "complete s"

  3878 proof -

  3879   {

  3880     fix f

  3881     assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3882     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"

  3883       using assms unfolding compact_def by blast

  3884

  3885     note lr' = seq_suble [OF lr(2)]

  3886

  3887     {

  3888       fix e :: real

  3889       assume "e > 0"

  3890       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"

  3891         unfolding cauchy_def

  3892         using e > 0

  3893         apply (erule_tac x="e/2" in allE)

  3894         apply auto

  3895         done

  3896       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]]

  3897       obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"

  3898         using e > 0 by auto

  3899       {

  3900         fix n :: nat

  3901         assume n: "n \<ge> max N M"

  3902         have "dist ((f \<circ> r) n) l < e/2"

  3903           using n M by auto

  3904         moreover have "r n \<ge> N"

  3905           using lr'[of n] n by auto

  3906         then have "dist (f n) ((f \<circ> r) n) < e / 2"

  3907           using N and n by auto

  3908         ultimately have "dist (f n) l < e"

  3909           using dist_triangle_half_r[of "f (r n)" "f n" e l]

  3910           by (auto simp add: dist_commute)

  3911       }

  3912       then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast

  3913     }

  3914     then have "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s

  3915       unfolding LIMSEQ_def by auto

  3916   }

  3917   then show ?thesis unfolding complete_def by auto

  3918 qed

  3919

  3920 lemma nat_approx_posE:

  3921   fixes e::real

  3922   assumes "0 < e"

  3923   obtains n :: nat where "1 / (Suc n) < e"

  3924 proof atomize_elim

  3925   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3926     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3927   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3928     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3929   also have "\<dots> = e" by simp

  3930   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3931 qed

  3932

  3933 lemma compact_eq_totally_bounded:

  3934   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3935     (is "_ \<longleftrightarrow> ?rhs")

  3936 proof

  3937   assume assms: "?rhs"

  3938   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)"

  3939     by (auto simp: choice_iff')

  3940

  3941   show "compact s"

  3942   proof cases

  3943     assume "s = {}"

  3944     then show "compact s" by (simp add: compact_def)

  3945   next

  3946     assume "s \<noteq> {}"

  3947     show ?thesis

  3948       unfolding compact_def

  3949     proof safe

  3950       fix f :: "nat \<Rightarrow> 'a"

  3951       assume f: "\<forall>n. f n \<in> s"

  3952

  3953       def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"

  3954       then have [simp]: "\<And>n. 0 < e n" by auto

  3955       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)"

  3956       {

  3957         fix n U

  3958         assume "infinite {n. f n \<in> U}"

  3959         then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"

  3960           using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)

  3961         then guess a ..

  3962         then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"

  3963           by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)

  3964         from someI_ex[OF this]

  3965         have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"

  3966           unfolding B_def by auto

  3967       }

  3968       note B = this

  3969

  3970       def F \<equiv> "nat_rec (B 0 UNIV) B"

  3971       {

  3972         fix n

  3973         have "infinite {i. f i \<in> F n}"

  3974           by (induct n) (auto simp: F_def B)

  3975       }

  3976       then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"

  3977         using B by (simp add: F_def)

  3978       then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"

  3979         using decseq_SucI[of F] by (auto simp: decseq_def)

  3980

  3981       obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"

  3982       proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)

  3983         fix k i

  3984         have "infinite ({n. f n \<in> F k} - {.. i})"

  3985           using infinite {n. f n \<in> F k} by auto

  3986         from infinite_imp_nonempty[OF this]

  3987         show "\<exists>x>i. f x \<in> F k"

  3988           by (simp add: set_eq_iff not_le conj_commute)

  3989       qed

  3990

  3991       def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"

  3992       have "subseq t"

  3993         unfolding subseq_Suc_iff by (simp add: t_def sel)

  3994       moreover have "\<forall>i. (f \<circ> t) i \<in> s"

  3995         using f by auto

  3996       moreover

  3997       {

  3998         fix n

  3999         have "(f \<circ> t) n \<in> F n"

  4000           by (cases n) (simp_all add: t_def sel)

  4001       }

  4002       note t = this

  4003

  4004       have "Cauchy (f \<circ> t)"

  4005       proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)

  4006         fix r :: real and N n m

  4007         assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"

  4008         then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"

  4009           using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)

  4010         with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"

  4011           by (auto simp: subset_eq)

  4012         with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r

  4013         show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"

  4014           by (simp add: dist_commute)

  4015       qed

  4016

  4017       ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  4018         using assms unfolding complete_def by blast

  4019     qed

  4020   qed

  4021 qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

  4022

  4023 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  4024 proof -

  4025   {

  4026     assume ?rhs

  4027     {

  4028       fix e::real

  4029       assume "e>0"

  4030       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  4031         by (erule_tac x="e/2" in allE) auto

  4032       {

  4033         fix n m

  4034         assume nm:"N \<le> m \<and> N \<le> n"

  4035         then have "dist (s m) (s n) < e" using N

  4036           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  4037           by blast

  4038       }

  4039       then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  4040         by blast

  4041     }

  4042     then have ?lhs

  4043       unfolding cauchy_def

  4044       by blast

  4045   }

  4046   then show ?thesis

  4047     unfolding cauchy_def

  4048     using dist_triangle_half_l

  4049     by blast

  4050 qed

  4051

  4052 lemma cauchy_imp_bounded:

  4053   assumes "Cauchy s"

  4054   shows "bounded (range s)"

  4055 proof -

  4056   from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"

  4057     unfolding cauchy_def

  4058     apply (erule_tac x= 1 in allE)

  4059     apply auto

  4060     done

  4061   then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  4062   moreover

  4063   have "bounded (s  {0..N})"

  4064     using finite_imp_bounded[of "s  {1..N}"] by auto

  4065   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  4066     unfolding bounded_any_center [where a="s N"] by auto

  4067   ultimately show "?thesis"

  4068     unfolding bounded_any_center [where a="s N"]

  4069     apply (rule_tac x="max a 1" in exI)

  4070     apply auto

  4071     apply (erule_tac x=y in allE)

  4072     apply (erule_tac x=y in ballE)

  4073     apply auto

  4074     done

  4075 qed

  4076

  4077 instance heine_borel < complete_space

  4078 proof

  4079   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  4080   then have "bounded (range f)"

  4081     by (rule cauchy_imp_bounded)

  4082   then have "compact (closure (range f))"

  4083     unfolding compact_eq_bounded_closed by auto

  4084   then have "complete (closure (range f))"

  4085     by (rule compact_imp_complete)

  4086   moreover have "\<forall>n. f n \<in> closure (range f)"

  4087     using closure_subset [of "range f"] by auto

  4088   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  4089     using Cauchy f unfolding complete_def by auto

  4090   then show "convergent f"

  4091     unfolding convergent_def by auto

  4092 qed

  4093

  4094 instance euclidean_space \<subseteq> banach ..

  4095

  4096 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  4097 proof (simp add: complete_def, rule, rule)

  4098   fix f :: "nat \<Rightarrow> 'a"

  4099   assume "Cauchy f"

  4100   then have "convergent f" by (rule Cauchy_convergent)

  4101   then show "\<exists>l. f ----> l" unfolding convergent_def .

  4102 qed

  4103

  4104 lemma complete_imp_closed:

  4105   assumes "complete s"

  4106   shows "closed s"

  4107 proof -

  4108   {

  4109     fix x

  4110     assume "x islimpt s"

  4111     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  4112       unfolding islimpt_sequential by auto

  4113     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  4114       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  4115     then have "x \<in> s"

  4116       using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  4117   }

  4118   then show "closed s" unfolding closed_limpt by auto

  4119 qed

  4120

  4121 lemma complete_eq_closed:

  4122   fixes s :: "'a::complete_space set"

  4123   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  4124 proof

  4125   assume ?lhs

  4126   then show ?rhs by (rule complete_imp_closed)

  4127 next

  4128   assume ?rhs

  4129   {

  4130     fix f

  4131     assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  4132     then obtain l where "(f ---> l) sequentially"

  4133       using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  4134     then have "\<exists>l\<in>s. (f ---> l) sequentially"

  4135       using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]]

  4136       using as(1) by auto

  4137   }

  4138   then show ?lhs unfolding complete_def by auto

  4139 qed

  4140

  4141 lemma convergent_eq_cauchy:

  4142   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  4143   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  4144   unfolding Cauchy_convergent_iff convergent_def ..

  4145

  4146 lemma convergent_imp_bounded:

  4147   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  4148   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  4149   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  4150

  4151 lemma compact_cball[simp]:

  4152   fixes x :: "'a::heine_borel"

  4153   shows "compact(cball x e)"

  4154   using compact_eq_bounded_closed bounded_cball closed_cball

  4155   by blast

  4156

  4157 lemma compact_frontier_bounded[intro]:

  4158   fixes s :: "'a::heine_borel set"

  4159   shows "bounded s \<Longrightarrow> compact(frontier s)"

  4160   unfolding frontier_def

  4161   using compact_eq_bounded_closed

  4162   by blast

  4163

  4164 lemma compact_frontier[intro]:

  4165   fixes s :: "'a::heine_borel set"

  4166   shows "compact s \<Longrightarrow> compact (frontier s)"

  4167   using compact_eq_bounded_closed compact_frontier_bounded

  4168   by blast

  4169

  4170 lemma frontier_subset_compact:

  4171   fixes s :: "'a::heine_borel set"

  4172   shows "compact s \<Longrightarrow> frontier s \<subseteq> s"

  4173   using frontier_subset_closed compact_eq_bounded_closed

  4174   by blast

  4175

  4176 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  4177

  4178 lemma bounded_closed_nest:

  4179   assumes "\<forall>n. closed(s n)"

  4180     and "\<forall>n. (s n \<noteq> {})"

  4181     and "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"

  4182     and "bounded(s 0)"

  4183   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  4184 proof -

  4185   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n"

  4186     using choice[of "\<lambda>n x. x\<in> s n"] by auto

  4187   from assms(4,1) have *:"seq_compact (s 0)"

  4188     using bounded_closed_imp_seq_compact[of "s 0"] by auto

  4189

  4190   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  4191     unfolding seq_compact_def

  4192     apply (erule_tac x=x in allE)

  4193     using x using assms(3)

  4194     apply blast

  4195     done

  4196

  4197   {

  4198     fix n :: nat

  4199     {

  4200       fix e :: real

  4201       assume "e>0"

  4202       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e"

  4203         unfolding LIMSEQ_def by auto

  4204       then have "dist ((x \<circ> r) (max N n)) l < e" by auto

  4205       moreover

  4206       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"]

  4207         by auto

  4208       then have "(x \<circ> r) (max N n) \<in> s n"

  4209         using x

  4210         apply (erule_tac x=n in allE)

  4211         using x

  4212         apply (erule_tac x="r (max N n)" in allE)

  4213         using assms(3)

  4214         apply (erule_tac x=n in allE)

  4215         apply (erule_tac x="r (max N n)" in allE)

  4216         apply auto

  4217         done

  4218       ultimately have "\<exists>y\<in>s n. dist y l < e"

  4219         by auto

  4220     }

  4221     then have "l \<in> s n"

  4222       using closed_approachable[of "s n" l] assms(1) by blast

  4223   }

  4224   then show ?thesis by auto

  4225 qed

  4226

  4227 text {* Decreasing case does not even need compactness, just completeness. *}

  4228

  4229 lemma decreasing_closed_nest:

  4230   assumes

  4231     "\<forall>n. closed(s n)"

  4232     "\<forall>n. (s n \<noteq> {})"

  4233     "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  4234     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  4235   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  4236 proof-

  4237   have "\<forall>n. \<exists> x. x\<in>s n"

  4238     using assms(2) by auto

  4239   then have "\<exists>t. \<forall>n. t n \<in> s n"

  4240     using choice[of "\<lambda> n x. x \<in> s n"] by auto

  4241   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  4242   {

  4243     fix e :: real

  4244     assume "e > 0"

  4245     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"

  4246       using assms(4) by auto

  4247     {

  4248       fix m n :: nat

  4249       assume "N \<le> m \<and> N \<le> n"

  4250       then have "t m \<in> s N" "t n \<in> s N"

  4251         using assms(3) t unfolding  subset_eq t by blast+

  4252       then have "dist (t m) (t n) < e"

  4253         using N by auto

  4254     }

  4255     then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"

  4256       by auto

  4257   }

  4258   then have "Cauchy t"

  4259     unfolding cauchy_def by auto

  4260   then obtain l where l:"(t ---> l) sequentially"

  4261     using complete_univ unfolding complete_def by auto

  4262   {

  4263     fix n :: nat

  4264     {

  4265       fix e :: real

  4266       assume "e > 0"

  4267       then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"

  4268         using l[unfolded LIMSEQ_def] by auto

  4269       have "t (max n N) \<in> s n"

  4270         using assms(3)

  4271         unfolding subset_eq

  4272         apply (erule_tac x=n in allE)

  4273         apply (erule_tac x="max n N" in allE)

  4274         using t

  4275         apply auto

  4276         done

  4277       then have "\<exists>y\<in>s n. dist y l < e"

  4278         apply (rule_tac x="t (max n N)" in bexI)

  4279         using N

  4280         apply auto

  4281         done

  4282     }

  4283     then have "l \<in> s n"

  4284       using closed_approachable[of "s n" l] assms(1) by auto

  4285   }

  4286   then show ?thesis by auto

  4287 qed

  4288

  4289 text {* Strengthen it to the intersection actually being a singleton. *}

  4290

  4291 lemma decreasing_closed_nest_sing:

  4292   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  4293   assumes

  4294     "\<forall>n. closed(s n)"

  4295     "\<forall>n. s n \<noteq> {}"

  4296     "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  4297     "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  4298   shows "\<exists>a. \<Inter>(range s) = {a}"

  4299 proof -

  4300   obtain a where a: "\<forall>n. a \<in> s n"

  4301     using decreasing_closed_nest[of s] using assms by auto

  4302   {

  4303     fix b

  4304     assume b: "b \<in> \<Inter>(range s)"

  4305     {

  4306       fix e :: real

  4307       assume "e > 0"

  4308       then have "dist a b < e"

  4309         using assms(4) and b and a by blast

  4310     }

  4311     then have "dist a b = 0"

  4312       by (metis dist_eq_0_iff dist_nz less_le)

  4313   }

  4314   with a have "\<Inter>(range s) = {a}"

  4315     unfolding image_def by auto

  4316   then show ?thesis ..

  4317 qed

  4318

  4319 text{* Cauchy-type criteria for uniform convergence. *}

  4320

  4321 lemma uniformly_convergent_eq_cauchy:

  4322   fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space"

  4323   shows

  4324     "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e) \<longleftrightarrow>

  4325       (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  \<longrightarrow> dist (s m x) (s n x) < e)"

  4326   (is "?lhs = ?rhs")

  4327 proof

  4328   assume ?lhs

  4329   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"

  4330     by auto

  4331   {

  4332     fix e :: real

  4333     assume "e > 0"

  4334     then obtain N :: nat where N: "\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2"

  4335       using l[THEN spec[where x="e/2"]] by auto

  4336     {

  4337       fix n m :: nat and x :: "'b"

  4338       assume "N \<le> m \<and> N \<le> n \<and> P x"

  4339       then have "dist (s m x) (s n x) < e"

  4340         using N[THEN spec[where x=m], THEN spec[where x=x]]

  4341         using N[THEN spec[where x=n], THEN spec[where x=x]]

  4342         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto

  4343     }

  4344     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

  4345   }

  4346   then show ?rhs by auto

  4347 next

  4348   assume ?rhs

  4349   then have "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)"

  4350     unfolding cauchy_def

  4351     apply auto

  4352     apply (erule_tac x=e in allE)

  4353     apply auto

  4354     done

  4355   then obtain l where l: "\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially"

  4356     unfolding convergent_eq_cauchy[symmetric]

  4357     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"]

  4358     by auto

  4359   {

  4360     fix e :: real

  4361     assume "e > 0"

  4362     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"

  4363       using ?rhs[THEN spec[where x="e/2"]] by auto

  4364     {

  4365       fix x

  4366       assume "P x"

  4367       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  4368         using l[THEN spec[where x=x], unfolded LIMSEQ_def] and e > 0

  4369         by (auto elim!: allE[where x="e/2"])

  4370       fix n :: nat

  4371       assume "n \<ge> N"

  4372       then have "dist(s n x)(l x) < e"

  4373         using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  4374         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"]

  4375         by (auto simp add: dist_commute)

  4376     }

  4377     then have "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4378       by auto

  4379   }

  4380   then show ?lhs by auto

  4381 qed

  4382

  4383 lemma uniformly_cauchy_imp_uniformly_convergent:

  4384   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"

  4385   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"

  4386     and "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n \<longrightarrow> dist(s n x)(l x) < e)"

  4387   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e"

  4388 proof -

  4389   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"

  4390     using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto

  4391   moreover

  4392   {

  4393     fix x

  4394     assume "P x"

  4395     then have "l x = l' x"

  4396       using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  4397       using l and assms(2) unfolding LIMSEQ_def by blast

  4398   }

  4399   ultimately show ?thesis by auto

  4400 qed

  4401

  4402

  4403 subsection {* Continuity *}

  4404

  4405 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  4406

  4407 lemma continuous_within_eps_delta:

  4408   "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)"

  4409   unfolding continuous_within and Lim_within

  4410   apply auto

  4411   unfolding dist_nz[symmetric]

  4412   apply (auto del: allE elim!:allE)

  4413   apply(rule_tac x=d in exI)

  4414   apply auto

  4415   done

  4416

  4417 lemma continuous_at_eps_delta:

  4418   "continuous (at x) f \<longleftrightarrow> (\<forall>e > 0. \<exists>d > 0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

  4419   using continuous_within_eps_delta [of x UNIV f] by simp

  4420

  4421 text{* Versions in terms of open balls. *}

  4422

  4423 lemma continuous_within_ball:

  4424   "continuous (at x within s) f \<longleftrightarrow>

  4425     (\<forall>e > 0. \<exists>d > 0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e)"

  4426   (is "?lhs = ?rhs")

  4427 proof

  4428   assume ?lhs

  4429   {

  4430     fix e :: real

  4431     assume "e > 0"

  4432     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"

  4433       using ?lhs[unfolded continuous_within Lim_within] by auto

  4434     {

  4435       fix y

  4436       assume "y \<in> f  (ball x d \<inter> s)"

  4437       then have "y \<in> ball (f x) e"

  4438         using d(2)

  4439         unfolding dist_nz[symmetric]

  4440         apply (auto simp add: dist_commute)

  4441         apply (erule_tac x=xa in ballE)

  4442         apply auto

  4443         using e > 0

  4444         apply auto

  4445         done

  4446     }

  4447     then have "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e"

  4448       using d > 0

  4449       unfolding subset_eq ball_def by (auto simp add: dist_commute)

  4450   }

  4451   then show ?rhs by auto

  4452 next

  4453   assume ?rhs

  4454   then show ?lhs

  4455     unfolding continuous_within Lim_within ball_def subset_eq

  4456     apply (auto simp add: dist_commute)

  4457     apply (erule_tac x=e in allE)

  4458     apply auto

  4459     done

  4460 qed

  4461

  4462 lemma continuous_at_ball:

  4463   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  4464 proof

  4465   assume ?lhs

  4466   then show ?rhs

  4467     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4468     apply auto

  4469     apply (erule_tac x=e in allE)

  4470     apply auto

  4471     apply (rule_tac x=d in exI)

  4472     apply auto

  4473     apply (erule_tac x=xa in allE)

  4474     apply (auto simp add: dist_commute dist_nz)

  4475     unfolding dist_nz[symmetric]

  4476     apply auto

  4477     done

  4478 next

  4479   assume ?rhs

  4480   then show ?lhs

  4481     unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  4482     apply auto

  4483     apply (erule_tac x=e in allE)

  4484     apply auto

  4485     apply (rule_tac x=d in exI)

  4486     apply auto

  4487     apply (erule_tac x="f xa" in allE)

  4488     apply (auto simp add: dist_commute dist_nz)

  4489     done

  4490 qed

  4491

  4492 text{* Define setwise continuity in terms of limits within the set. *}

  4493

  4494 lemma continuous_on_iff:

  4495   "continuous_on s f \<longleftrightarrow>

  4496     (\<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)"

  4497   unfolding continuous_on_def Lim_within

  4498   apply (intro ball_cong [OF refl] all_cong ex_cong)

  4499   apply (rename_tac y, case_tac "y = x")

  4500   apply simp

  4501   apply (simp add: dist_nz)

  4502   done

  4503

  4504 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  4505   where "uniformly_continuous_on s f \<longleftrightarrow>

  4506     (\<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)"

  4507

  4508 text{* Some simple consequential lemmas. *}

  4509

  4510 lemma uniformly_continuous_imp_continuous:

  4511   "uniformly_continuous_on s f \<Longrightarrow> continuous_on s f"

  4512   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  4513

  4514 lemma continuous_at_imp_continuous_within:

  4515   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"

  4516   unfolding continuous_within continuous_at using Lim_at_within by auto

  4517

  4518 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  4519   by simp

  4520

  4521 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4522

  4523 lemma continuous_within_subset:

  4524   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"

  4525   unfolding continuous_within by(metis tendsto_within_subset)

  4526

  4527 lemma continuous_on_interior:

  4528   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4529   apply (erule interiorE)

  4530   apply (drule (1) continuous_on_subset)

  4531   apply (simp add: continuous_on_eq_continuous_at)

  4532   done

  4533

  4534 lemma continuous_on_eq:

  4535   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4536   unfolding continuous_on_def tendsto_def eventually_at_topological

  4537   by simp

  4538

  4539 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4540

  4541 lemma continuous_within_sequentially:

  4542   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4543   shows "continuous (at a within s) f \<longleftrightarrow>

  4544     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4545          \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"

  4546   (is "?lhs = ?rhs")

  4547 proof

  4548   assume ?lhs

  4549   {

  4550     fix x :: "nat \<Rightarrow> 'a"

  4551     assume x: "\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4552     fix T :: "'b set"

  4553     assume "open T" and "f a \<in> T"

  4554     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"

  4555       unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)

  4556     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4557       using x(2) d>0 by simp

  4558     then have "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4559     proof eventually_elim

  4560       case (elim n)

  4561       then show ?case

  4562         using d x(1) f a \<in> T unfolding dist_nz[symmetric] by auto

  4563     qed

  4564   }

  4565   then show ?rhs

  4566     unfolding tendsto_iff tendsto_def by simp

  4567 next

  4568   assume ?rhs

  4569   then show ?lhs

  4570     unfolding continuous_within tendsto_def [where l="f a"]

  4571     by (simp add: sequentially_imp_eventually_within)

  4572 qed

  4573

  4574 lemma continuous_at_sequentially:

  4575   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4576   shows "continuous (at a) f \<longleftrightarrow>

  4577     (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"

  4578   using continuous_within_sequentially[of a UNIV f] by simp

  4579

  4580 lemma continuous_on_sequentially:

  4581   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4582   shows "continuous_on s f \<longleftrightarrow>

  4583     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4584       --> ((f \<circ> x) ---> f a) sequentially)"

  4585   (is "?lhs = ?rhs")

  4586 proof

  4587   assume ?rhs

  4588   then show ?lhs

  4589     using continuous_within_sequentially[of _ s f]

  4590     unfolding continuous_on_eq_continuous_within

  4591     by auto

  4592 next

  4593   assume ?lhs

  4594   then show ?rhs

  4595     unfolding continuous_on_eq_continuous_within

  4596     using continuous_within_sequentially[of _ s f]

  4597     by auto

  4598 qed

  4599

  4600 lemma uniformly_continuous_on_sequentially:

  4601   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4602                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4603                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4604 proof

  4605   assume ?lhs

  4606   {

  4607     fix x y

  4608     assume x: "\<forall>n. x n \<in> s"

  4609       and y: "\<forall>n. y n \<in> s"

  4610       and xy: "((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4611     {

  4612       fix e :: real

  4613       assume "e > 0"

  4614       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"

  4615         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4616       obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"

  4617         using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4618       {

  4619         fix n

  4620         assume "n\<ge>N"

  4621         then have "dist (f (x n)) (f (y n)) < e"

  4622           using N[THEN spec[where x=n]]

  4623           using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]

  4624           using x and y

  4625           unfolding dist_commute

  4626           by simp

  4627       }

  4628       then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4629         by auto

  4630     }

  4631     then have "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially"

  4632       unfolding LIMSEQ_def and dist_real_def by auto

  4633   }

  4634   then show ?rhs by auto

  4635 next

  4636   assume ?rhs

  4637   {

  4638     assume "\<not> ?lhs"

  4639     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"

  4640       unfolding uniformly_continuous_on_def by auto

  4641     then obtain fa where fa:

  4642       "\<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"

  4643       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"]

  4644       unfolding Bex_def

  4645       by (auto simp add: dist_commute)

  4646     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4647     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4648     have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"

  4649       and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"

  4650       and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4651       unfolding x_def and y_def using fa

  4652       by auto

  4653     {

  4654       fix e :: real

  4655       assume "e > 0"

  4656       then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"

  4657         unfolding real_arch_inv[of e] by auto

  4658       {

  4659         fix n :: nat

  4660         assume "n \<ge> N"

  4661         then have "inverse (real n + 1) < inverse (real N)"

  4662           using real_of_nat_ge_zero and N\<noteq>0 by auto

  4663         also have "\<dots> < e" using N by auto

  4664         finally have "inverse (real n + 1) < e" by auto

  4665         then have "dist (x n) (y n) < e"

  4666           using xy0[THEN spec[where x=n]] by auto

  4667       }

  4668       then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto

  4669     }

  4670     then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"

  4671       using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn

  4672       unfolding LIMSEQ_def dist_real_def by auto

  4673     then have False using fxy and e>0 by auto

  4674   }

  4675   then show ?lhs

  4676     unfolding uniformly_continuous_on_def by blast

  4677 qed

  4678

  4679 text{* The usual transformation theorems. *}

  4680

  4681 lemma continuous_transform_within:

  4682   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4683   assumes "0 < d"

  4684     and "x \<in> s"

  4685     and "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4686     and "continuous (at x within s) f"

  4687   shows "continuous (at x within s) g"

  4688   unfolding continuous_within

  4689 proof (rule Lim_transform_within)

  4690   show "0 < d" by fact

  4691   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4692     using assms(3) by auto

  4693   have "f x = g x"

  4694     using assms(1,2,3) by auto

  4695   then show "(f ---> g x) (at x within s)"

  4696     using assms(4) unfolding continuous_within by simp

  4697 qed

  4698

  4699 lemma continuous_transform_at:

  4700   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4701   assumes "0 < d"

  4702     and "\<forall>x'. dist x' x < d --> f x' = g x'"

  4703     and "continuous (at x) f"

  4704   shows "continuous (at x) g"

  4705   using continuous_transform_within [of d x UNIV f g] assms by simp

  4706

  4707

  4708 subsubsection {* Structural rules for pointwise continuity *}

  4709

  4710 lemmas continuous_within_id = continuous_ident

  4711

  4712 lemmas continuous_at_id = isCont_ident

  4713

  4714 lemma continuous_infdist[continuous_intros]:

  4715   assumes "continuous F f"

  4716   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4717   using assms unfolding continuous_def by (rule tendsto_infdist)

  4718

  4719 lemma continuous_infnorm[continuous_intros]:

  4720   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4721   unfolding continuous_def by (rule tendsto_infnorm)

  4722

  4723 lemma continuous_inner[continuous_intros]:

  4724   assumes "continuous F f"

  4725     and "continuous F g"

  4726   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4727   using assms unfolding continuous_def by (rule tendsto_inner)

  4728

  4729 lemmas continuous_at_inverse = isCont_inverse

  4730

  4731 subsubsection {* Structural rules for setwise continuity *}

  4732

  4733 lemma continuous_on_infnorm[continuous_on_intros]:

  4734   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4735   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4736

  4737 lemma continuous_on_inner[continuous_on_intros]:

  4738   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4739   assumes "continuous_on s f"

  4740     and "continuous_on s g"

  4741   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4742   using bounded_bilinear_inner assms

  4743   by (rule bounded_bilinear.continuous_on)

  4744

  4745 subsubsection {* Structural rules for uniform continuity *}

  4746

  4747 lemma uniformly_continuous_on_id[continuous_on_intros]:

  4748   "uniformly_continuous_on s (\<lambda>x. x)"

  4749   unfolding uniformly_continuous_on_def by auto

  4750

  4751 lemma uniformly_continuous_on_const[continuous_on_intros]:

  4752   "uniformly_continuous_on s (\<lambda>x. c)"

  4753   unfolding uniformly_continuous_on_def by simp

  4754

  4755 lemma uniformly_continuous_on_dist[continuous_on_intros]:

  4756   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4757   assumes "uniformly_continuous_on s f"

  4758     and "uniformly_continuous_on s g"

  4759   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4760 proof -

  4761   {

  4762     fix a b c d :: 'b

  4763     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4764       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4765       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4766       by arith

  4767   } note le = this

  4768   {

  4769     fix x y

  4770     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4771     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4772     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4773       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4774         simp add: le)

  4775   }

  4776   then show ?thesis

  4777     using assms unfolding uniformly_continuous_on_sequentially

  4778     unfolding dist_real_def by simp

  4779 qed

  4780

  4781 lemma uniformly_continuous_on_norm[continuous_on_intros]:

  4782   assumes "uniformly_continuous_on s f"

  4783   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4784   unfolding norm_conv_dist using assms

  4785   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4786

  4787 lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:

  4788   assumes "uniformly_continuous_on s g"

  4789   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4790   using assms unfolding uniformly_continuous_on_sequentially

  4791   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4792   by (auto intro: tendsto_zero)

  4793

  4794 lemma uniformly_continuous_on_cmul[continuous_on_intros]:

  4795   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4796   assumes "uniformly_continuous_on s f"

  4797   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4798   using bounded_linear_scaleR_right assms

  4799   by (rule bounded_linear.uniformly_continuous_on)

  4800

  4801 lemma dist_minus:

  4802   fixes x y :: "'a::real_normed_vector"

  4803   shows "dist (- x) (- y) = dist x y"

  4804   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4805

  4806 lemma uniformly_continuous_on_minus[continuous_on_intros]:

  4807   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4808   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4809   unfolding uniformly_continuous_on_def dist_minus .

  4810

  4811 lemma uniformly_continuous_on_add[continuous_on_intros]:

  4812   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4813   assumes "uniformly_continuous_on s f"

  4814     and "uniformly_continuous_on s g"

  4815   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4816   using assms

  4817   unfolding uniformly_continuous_on_sequentially

  4818   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4819   by (auto intro: tendsto_add_zero)

  4820

  4821 lemma uniformly_continuous_on_diff[continuous_on_intros]:

  4822   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4823   assumes "uniformly_continuous_on s f"

  4824     and "uniformly_continuous_on s g"

  4825   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4826   unfolding ab_diff_minus using assms

  4827   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4828

  4829 text{* Continuity of all kinds is preserved under composition. *}

  4830

  4831 lemmas continuous_at_compose = isCont_o

  4832

  4833 lemma uniformly_continuous_on_compose[continuous_on_intros]:

  4834   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4835   shows "uniformly_continuous_on s (g \<circ> f)"

  4836 proof -

  4837   {

  4838     fix e :: real

  4839     assume "e > 0"

  4840     then obtain d where "d > 0"

  4841       and d: "\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4842       using assms(2) unfolding uniformly_continuous_on_def by auto

  4843     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d"

  4844       using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto

  4845     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"

  4846       using d>0 using d by auto

  4847   }

  4848   then show ?thesis

  4849     using assms unfolding uniformly_continuous_on_def by auto

  4850 qed

  4851

  4852 text{* Continuity in terms of open preimages. *}

  4853

  4854 lemma continuous_at_open:

  4855   "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)))"

  4856   unfolding continuous_within_topological [of x UNIV f]

  4857   unfolding imp_conjL

  4858   by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4859

  4860 lemma continuous_imp_tendsto:

  4861   assumes "continuous (at x0) f"

  4862     and "x ----> x0"

  4863   shows "(f \<circ> x) ----> (f x0)"

  4864 proof (rule topological_tendstoI)

  4865   fix S

  4866   assume "open S" "f x0 \<in> S"

  4867   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"

  4868      using assms continuous_at_open by metis

  4869   then have "eventually (\<lambda>n. x n \<in> T) sequentially"

  4870     using assms T_def by (auto simp: tendsto_def)

  4871   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"

  4872     using T_def by (auto elim!: eventually_elim1)

  4873 qed

  4874

  4875 lemma continuous_on_open:

  4876   "continuous_on s f \<longleftrightarrow>

  4877     (\<forall>t. openin (subtopology euclidean (f  s)) t \<longrightarrow>

  4878       openin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  4879   unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute

  4880   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4881

  4882 text {* Similarly in terms of closed sets. *}

  4883

  4884 lemma continuous_on_closed:

  4885   "continuous_on s f \<longleftrightarrow>

  4886     (\<forall>t. closedin (subtopology euclidean (f  s)) t \<longrightarrow>

  4887       closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})"

  4888   unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute

  4889   by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

  4890

  4891 text {* Half-global and completely global cases. *}

  4892

  4893 lemma continuous_open_in_preimage:

  4894   assumes "continuous_on s f"  "open t"

  4895   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4896 proof -

  4897   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  4898     by auto

  4899   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4900     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4901   then show ?thesis

  4902     using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]]

  4903     using * by auto

  4904 qed

  4905

  4906 lemma continuous_closed_in_preimage:

  4907   assumes "continuous_on s f" and "closed t"

  4908   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4909 proof -

  4910   have *: "\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)"

  4911     by auto

  4912   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4913     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute

  4914     by auto

  4915   then show ?thesis

  4916     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]]

  4917     using * by auto

  4918 qed

  4919

  4920 lemma continuous_open_preimage:

  4921   assumes "continuous_on s f"

  4922     and "open s"

  4923     and "open t"

  4924   shows "open {x \<in> s. f x \<in> t}"

  4925 proof-

  4926   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4927     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4928   then show ?thesis

  4929     using open_Int[of s T, OF assms(2)] by auto

  4930 qed

  4931

  4932 lemma continuous_closed_preimage:

  4933   assumes "continuous_on s f"

  4934     and "closed s"

  4935     and "closed t"

  4936   shows "closed {x \<in> s. f x \<in> t}"

  4937 proof-

  4938   obtain T where "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4939     using continuous_closed_in_preimage[OF assms(1,3)]

  4940     unfolding closedin_closed by auto

  4941   then show ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4942 qed

  4943

  4944 lemma continuous_open_preimage_univ:

  4945   "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4946   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4947

  4948 lemma continuous_closed_preimage_univ:

  4949   "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s \<Longrightarrow> closed {x. f x \<in> s}"

  4950   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4951

  4952 lemma continuous_open_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4953   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4954

  4955 lemma continuous_closed_vimage: "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4956   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4957

  4958 lemma interior_image_subset:

  4959   assumes "\<forall>x. continuous (at x) f"

  4960     and "inj f"

  4961   shows "interior (f  s) \<subseteq> f  (interior s)"

  4962 proof

  4963   fix x assume "x \<in> interior (f  s)"

  4964   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4965   then have "x \<in> f  s" by auto

  4966   then obtain y where y: "y \<in> s" "x = f y" by auto

  4967   have "open (vimage f T)"

  4968     using assms(1) open T by (rule continuous_open_vimage)

  4969   moreover have "y \<in> vimage f T"

  4970     using x = f y x \<in> T by simp

  4971   moreover have "vimage f T \<subseteq> s"

  4972     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4973   ultimately have "y \<in> interior s" ..

  4974   with x = f y show "x \<in> f  interior s" ..

  4975 qed

  4976

  4977 text {* Equality of continuous functions on closure and related results. *}

  4978

  4979 lemma continuous_closed_in_preimage_constant:

  4980   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4981   shows "continuous_on s f \<Longrightarrow> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4982   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4983

  4984 lemma continuous_closed_preimage_constant:

  4985   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4986   shows "continuous_on s f \<Longrightarrow> closed s \<Longrightarrow> closed {x \<in> s. f x = a}"

  4987   using continuous_closed_preimage[of s f "{a}"] by auto

  4988

  4989 lemma continuous_constant_on_closure:

  4990   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4991   assumes "continuous_on (closure s) f"

  4992     and "\<forall>x \<in> s. f x = a"

  4993   shows "\<forall>x \<in> (closure s). f x = a"

  4994     using continuous_closed_preimage_constant[of "closure s" f a]

  4995       assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset

  4996     unfolding subset_eq

  4997     by auto

  4998

  4999 lemma image_closure_subset:

  5000   assumes "continuous_on (closure s) f"

  5001     and "closed t"

  5002     and "(f  s) \<subseteq> t"

  5003   shows "f  (closure s) \<subseteq> t"

  5004 proof -

  5005   have "s \<subseteq> {x \<in> closure s. f x \<in> t}"

  5006     using assms(3) closure_subset by auto

  5007   moreover have "closed {x \<in> closure s. f x \<in> t}"

  5008     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  5009   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  5010     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  5011   then show ?thesis by auto

  5012 qed

  5013

  5014 lemma continuous_on_closure_norm_le:

  5015   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  5016   assumes "continuous_on (closure s) f"

  5017     and "\<forall>y \<in> s. norm(f y) \<le> b"

  5018     and "x \<in> (closure s)"

  5019   shows "norm (f x) \<le> b"

  5020 proof -

  5021   have *: "f  s \<subseteq> cball 0 b"

  5022     using assms(2)[unfolded mem_cball_0[symmetric]] by auto

  5023   show ?thesis

  5024     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  5025     unfolding subset_eq

  5026     apply (erule_tac x="f x" in ballE)

  5027     apply (auto simp add: dist_norm)

  5028     done

  5029 qed

  5030

  5031 text {* Making a continuous function avoid some value in a neighbourhood. *}

  5032

  5033 lemma continuous_within_avoid:

  5034   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5035   assumes "continuous (at x within s) f"

  5036     and "f x \<noteq> a"

  5037   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  5038 proof -

  5039   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  5040     using t1_space [OF f x \<noteq> a] by fast

  5041   have "(f ---> f x) (at x within s)"

  5042     using assms(1) by (simp add: continuous_within)

  5043   then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  5044     using open U and f x \<in> U

  5045     unfolding tendsto_def by fast

  5046   then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  5047     using a \<notin> U by (fast elim: eventually_mono [rotated])

  5048   then show ?thesis

  5049     using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)

  5050 qed

  5051

  5052 lemma continuous_at_avoid:

  5053   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5054   assumes "continuous (at x) f"

  5055     and "f x \<noteq> a"

  5056   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5057   using assms continuous_within_avoid[of x UNIV f a] by simp

  5058

  5059 lemma continuous_on_avoid:

  5060   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5061   assumes "continuous_on s f"

  5062     and "x \<in> s"

  5063     and "f x \<noteq> a"

  5064   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  5065   using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],

  5066     OF assms(2)] continuous_within_avoid[of x s f a]

  5067   using assms(3)

  5068   by auto

  5069

  5070 lemma continuous_on_open_avoid:

  5071   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  5072   assumes "continuous_on s f"

  5073     and "open s"

  5074     and "x \<in> s"

  5075     and "f x \<noteq> a"

  5076   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  5077   using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]

  5078   using continuous_at_avoid[of x f a] assms(4)

  5079   by auto

  5080

  5081 text {* Proving a function is constant by proving open-ness of level set. *}

  5082

  5083 lemma continuous_levelset_open_in_cases:

  5084   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5085   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5086         openin (subtopology euclidean s) {x \<in> s. f x = a}

  5087         \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  5088   unfolding connected_clopen

  5089   using continuous_closed_in_preimage_constant by auto

  5090

  5091 lemma continuous_levelset_open_in:

  5092   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5093   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  5094         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  5095         (\<exists>x \<in> s. f x = a)  \<Longrightarrow> (\<forall>x \<in> s. f x = a)"

  5096   using continuous_levelset_open_in_cases[of s f ]

  5097   by meson

  5098

  5099 lemma continuous_levelset_open:

  5100   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  5101   assumes "connected s"

  5102     and "continuous_on s f"

  5103     and "open {x \<in> s. f x = a}"

  5104     and "\<exists>x \<in> s.  f x = a"

  5105   shows "\<forall>x \<in> s. f x = a"

  5106   using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open]

  5107   using assms (3,4)

  5108   by fast

  5109

  5110 text {* Some arithmetical combinations (more to prove). *}

  5111

  5112 lemma open_scaling[intro]:

  5113   fixes s :: "'a::real_normed_vector set"

  5114   assumes "c \<noteq> 0"

  5115     and "open s"

  5116   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  5117 proof -

  5118   {

  5119     fix x

  5120     assume "x \<in> s"

  5121     then obtain e where "e>0"

  5122       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]

  5123       by auto

  5124     have "e * abs c > 0"

  5125       using assms(1)[unfolded zero_less_abs_iff[symmetric]]

  5126       using mult_pos_pos[OF e>0]

  5127       by auto

  5128     moreover

  5129     {

  5130       fix y

  5131       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  5132       then have "norm ((1 / c) *\<^sub>R y - x) < e"

  5133         unfolding dist_norm

  5134         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  5135           assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)

  5136       then have "y \<in> op *\<^sub>R c  s"

  5137         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]

  5138         using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]

  5139         using assms(1)

  5140         unfolding dist_norm scaleR_scaleR

  5141         by auto

  5142     }

  5143     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s"

  5144       apply (rule_tac x="e * abs c" in exI)

  5145       apply auto

  5146       done

  5147   }

  5148   then show ?thesis unfolding open_dist by auto

  5149 qed

  5150

  5151 lemma minus_image_eq_vimage:

  5152   fixes A :: "'a::ab_group_add set"

  5153   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  5154   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  5155

  5156 lemma open_negations:

  5157   fixes s :: "'a::real_normed_vector set"

  5158   shows "open s \<Longrightarrow> open ((\<lambda> x. -x)  s)"

  5159   unfolding scaleR_minus1_left [symmetric]

  5160   by (rule open_scaling, auto)

  5161

  5162 lemma open_translation:

  5163   fixes s :: "'a::real_normed_vector set"

  5164   assumes "open s"

  5165   shows "open((\<lambda>x. a + x)  s)"

  5166 proof -

  5167   {

  5168     fix x

  5169     have "continuous (at x) (\<lambda>x. x - a)"

  5170       by (intro continuous_diff continuous_at_id continuous_const)

  5171   }

  5172   moreover have "{x. x - a \<in> s} = op + a  s"

  5173     by force

  5174   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s]

  5175     using assms by auto

  5176 qed

  5177

  5178 lemma open_affinity:

  5179   fixes s :: "'a::real_normed_vector set"

  5180   assumes "open s"  "c \<noteq> 0"

  5181   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5182 proof -

  5183   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"

  5184     unfolding o_def ..

  5185   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s"

  5186     by auto

  5187   then show ?thesis

  5188     using assms open_translation[of "op *\<^sub>R c  s" a]

  5189     unfolding *

  5190     by auto

  5191 qed

  5192

  5193 lemma interior_translation:

  5194   fixes s :: "'a::real_normed_vector set"

  5195   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  5196 proof (rule set_eqI, rule)

  5197   fix x

  5198   assume "x \<in> interior (op + a  s)"

  5199   then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a  s"

  5200     unfolding mem_interior by auto

  5201   then have "ball (x - a) e \<subseteq> s"

  5202     unfolding subset_eq Ball_def mem_ball dist_norm

  5203     apply auto

  5204     apply (erule_tac x="a + xa" in allE)

  5205     unfolding ab_group_add_class.diff_diff_eq[symmetric]

  5206     apply auto

  5207     done

  5208   then show "x \<in> op + a  interior s"

  5209     unfolding image_iff

  5210     apply (rule_tac x="x - a" in bexI)

  5211     unfolding mem_interior

  5212     using e > 0

  5213     apply auto

  5214     done

  5215 next

  5216   fix x

  5217   assume "x \<in> op + a  interior s"

  5218   then obtain y e where "e > 0" and e: "ball y e \<subseteq> s" and y: "x = a + y"

  5219     unfolding image_iff Bex_def mem_interior by auto

  5220   {

  5221     fix z

  5222     have *: "a + y - z = y + a - z" by auto

  5223     assume "z \<in> ball x e"

  5224     then have "z - a \<in> s"

  5225       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]

  5226       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *

  5227       by auto

  5228     then have "z \<in> op + a  s"

  5229       unfolding image_iff by (auto intro!: bexI[where x="z - a"])

  5230   }

  5231   then have "ball x e \<subseteq> op + a  s"

  5232     unfolding subset_eq by auto

  5233   then show "x \<in> interior (op + a  s)"

  5234     unfolding mem_interior using e > 0 by auto

  5235 qed

  5236

  5237 text {* Topological properties of linear functions. *}

  5238

  5239 lemma linear_lim_0:

  5240   assumes "bounded_linear f"

  5241   shows "(f ---> 0) (at (0))"

  5242 proof -

  5243   interpret f: bounded_linear f by fact

  5244   have "(f ---> f 0) (at 0)"

  5245     using tendsto_ident_at by (rule f.tendsto)

  5246   then show ?thesis unfolding f.zero .

  5247 qed

  5248

  5249 lemma linear_continuous_at:

  5250   assumes "bounded_linear f"

  5251   shows "continuous (at a) f"

  5252   unfolding continuous_at using assms

  5253   apply (rule bounded_linear.tendsto)

  5254   apply (rule tendsto_ident_at)

  5255   done

  5256

  5257 lemma linear_continuous_within:

  5258   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"

  5259   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  5260

  5261 lemma linear_continuous_on:

  5262   "bounded_linear f \<Longrightarrow> continuous_on s f"

  5263   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  5264

  5265 text {* Also bilinear functions, in composition form. *}

  5266

  5267 lemma bilinear_continuous_at_compose:

  5268   "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5269     continuous (at x) (\<lambda>x. h (f x) (g x))"

  5270   unfolding continuous_at

  5271   using Lim_bilinear[of f "f x" "(at x)" g "g x" h]

  5272   by auto

  5273

  5274 lemma bilinear_continuous_within_compose:

  5275   "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5276     continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  5277   unfolding continuous_within

  5278   using Lim_bilinear[of f "f x"]

  5279   by auto

  5280

  5281 lemma bilinear_continuous_on_compose:

  5282   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>

  5283     continuous_on s (\<lambda>x. h (f x) (g x))"

  5284   unfolding continuous_on_def

  5285   by (fast elim: bounded_bilinear.tendsto)

  5286

  5287 text {* Preservation of compactness and connectedness under continuous function. *}

  5288

  5289 lemma compact_eq_openin_cover:

  5290   "compact S \<longleftrightarrow>

  5291     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5292       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  5293 proof safe

  5294   fix C

  5295   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  5296   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}"

  5297     unfolding openin_open by force+

  5298   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"

  5299     by (rule compactE)

  5300   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)"

  5301     by auto

  5302   then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5303 next

  5304   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  5305         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  5306   show "compact S"

  5307   proof (rule compactI)

  5308     fix C

  5309     let ?C = "image (\<lambda>T. S \<inter> T) C"

  5310     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  5311     then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  5312       unfolding openin_open by auto

  5313     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  5314       by metis

  5315     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  5316     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  5317     proof (intro conjI)

  5318       from D \<subseteq> ?C show "?D \<subseteq> C"

  5319         by (fast intro: inv_into_into)

  5320       from finite D show "finite ?D"

  5321         by (rule finite_imageI)

  5322       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  5323         apply (rule subset_trans)

  5324         apply clarsimp

  5325         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  5326         apply (erule rev_bexI, fast)

  5327         done

  5328     qed

  5329     then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  5330   qed

  5331 qed

  5332

  5333 lemma connected_continuous_image:

  5334   assumes "continuous_on s f"

  5335     and "connected s"

  5336   shows "connected(f  s)"

  5337 proof -

  5338   {

  5339     fix T

  5340     assume as:

  5341       "T \<noteq> {}"

  5342       "T \<noteq> f  s"

  5343       "openin (subtopology euclidean (f  s)) T"

  5344       "closedin (subtopology euclidean (f  s)) T"

  5345     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  5346       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  5347       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  5348       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  5349     then have False using as(1,2)

  5350       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto

  5351   }

  5352   then show ?thesis

  5353     unfolding connected_clopen by auto

  5354 qed

  5355

  5356 text {* Continuity implies uniform continuity on a compact domain. *}

  5357

  5358 lemma compact_uniformly_continuous:

  5359   assumes f: "continuous_on s f"

  5360     and s: "compact s"

  5361   shows "uniformly_continuous_on s f"

  5362   unfolding uniformly_continuous_on_def

  5363 proof (cases, safe)

  5364   fix e :: real

  5365   assume "0 < e" "s \<noteq> {}"

  5366   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) } }"

  5367   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  5368   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  5369   proof safe

  5370     fix y

  5371     assume "y \<in> s"

  5372     from continuous_open_in_preimage[OF f open_ball]

  5373     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  5374       unfolding openin_subtopology open_openin by metis

  5375     then obtain d where "ball y d \<subseteq> T" "0 < d"

  5376       using 0 < e y \<in> s by (auto elim!: openE)

  5377     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  5378       by (intro UN_I[of "(y, d)"]) auto

  5379   qed auto

  5380   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  5381     by (rule compactE_image)

  5382   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  5383     by (subst Min_gr_iff) auto

  5384   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  5385   proof (rule, safe)

  5386     fix x x'

  5387     assume in_s: "x' \<in> s" "x \<in> s"

  5388     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  5389       by blast

  5390     moreover assume "dist x x' < Min (sndD) / 2"

  5391     ultimately have "dist y x' < d"

  5392       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  5393     with D x in_s show  "dist (f x) (f x') < e"

  5394       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  5395   qed (insert D, auto)

  5396 qed auto

  5397

  5398 text {* A uniformly convergent limit of continuous functions is continuous. *}

  5399

  5400 lemma continuous_uniform_limit:

  5401   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  5402   assumes "\<not> trivial_limit F"

  5403     and "eventually (\<lambda>n. continuous_on s (f n)) F"

  5404     and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  5405   shows "continuous_on s g"

  5406 proof -

  5407   {

  5408     fix x and e :: real

  5409     assume "x\<in>s" "e>0"

  5410     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  5411       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  5412     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  5413     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  5414       using assms(1) by blast

  5415     have "e / 3 > 0" using e>0 by auto

  5416     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"

  5417       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  5418     {

  5419       fix y

  5420       assume "y \<in> s" and "dist y x < d"

  5421       then have "dist (f n y) (f n x) < e / 3"

  5422         by (rule d [rule_format])

  5423       then have "dist (f n y) (g x) < 2 * e / 3"

  5424         using dist_triangle [of "f n y" "g x" "f n x"]

  5425         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  5426         by auto

  5427       then have "dist (g y) (g x) < e"

  5428         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  5429         using dist_triangle3 [of "g y" "g x" "f n y"]

  5430         by auto

  5431     }

  5432     then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  5433       using d>0 by auto

  5434   }

  5435   then show ?thesis

  5436     unfolding continuous_on_iff by auto

  5437 qed

  5438

  5439

  5440 subsection {* Topological stuff lifted from and dropped to R *}

  5441

  5442 lemma open_real:

  5443   fixes s :: "real set"

  5444   shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)"

  5445   unfolding open_dist dist_norm by simp

  5446

  5447 lemma islimpt_approachable_real:

  5448   fixes s :: "real set"

  5449   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  5450   unfolding islimpt_approachable dist_norm by simp

  5451

  5452 lemma closed_real:

  5453   fixes s :: "real set"

  5454   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)"

  5455   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5456

  5457 lemma continuous_at_real_range:

  5458   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5459   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5460   unfolding continuous_at

  5461   unfolding Lim_at

  5462   unfolding dist_nz[symmetric]

  5463   unfolding dist_norm

  5464   apply auto

  5465   apply (erule_tac x=e in allE)

  5466   apply auto

  5467   apply (rule_tac x=d in exI)

  5468   apply auto

  5469   apply (erule_tac x=x' in allE)

  5470   apply auto

  5471   apply (erule_tac x=e in allE)

  5472   apply auto

  5473   done

  5474

  5475 lemma continuous_on_real_range:

  5476   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5477   shows "continuous_on s f \<longleftrightarrow>

  5478     (\<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))"

  5479   unfolding continuous_on_iff dist_norm by simp

  5480

  5481 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5482

  5483 lemma distance_attains_sup:

  5484   assumes "compact s" "s \<noteq> {}"

  5485   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"

  5486 proof (rule continuous_attains_sup [OF assms])

  5487   {

  5488     fix x

  5489     assume "x\<in>s"

  5490     have "(dist a ---> dist a x) (at x within s)"

  5491       by (intro tendsto_dist tendsto_const tendsto_ident_at)

  5492   }

  5493   then show "continuous_on s (dist a)"

  5494     unfolding continuous_on ..

  5495 qed

  5496

  5497 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5498

  5499 lemma distance_attains_inf:

  5500   fixes a :: "'a::heine_borel"

  5501   assumes "closed s"

  5502     and "s \<noteq> {}"

  5503   shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"

  5504 proof -

  5505   from assms(2) obtain b where "b \<in> s" by auto

  5506   let ?B = "s \<inter> cball a (dist b a)"

  5507   have "?B \<noteq> {}" using b \<in> s

  5508     by (auto simp add: dist_commute)

  5509   moreover have "continuous_on ?B (dist a)"

  5510     by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)

  5511   moreover have "compact ?B"

  5512     by (intro closed_inter_compact closed s compact_cball)

  5513   ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"

  5514     by (metis continuous_attains_inf)

  5515   then show ?thesis by fastforce

  5516 qed

  5517

  5518

  5519 subsection {* Pasted sets *}

  5520

  5521 lemma bounded_Times:

  5522   assumes "bounded s" "bounded t"

  5523   shows "bounded (s \<times> t)"

  5524 proof -

  5525   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5526     using assms [unfolded bounded_def] by auto

  5527   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"

  5528     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5529   then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5530 qed

  5531

  5532 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5533   by (induct x) simp

  5534

  5535 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5536   unfolding seq_compact_def

  5537   apply clarify

  5538   apply (drule_tac x="fst \<circ> f" in spec)

  5539   apply (drule mp, simp add: mem_Times_iff)

  5540   apply (clarify, rename_tac l1 r1)

  5541   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5542   apply (drule mp, simp add: mem_Times_iff)

  5543   apply (clarify, rename_tac l2 r2)

  5544   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5545   apply (rule_tac x="r1 \<circ> r2" in exI)

  5546   apply (rule conjI, simp add: subseq_def)

  5547   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)

  5548   apply (drule (1) tendsto_Pair) back

  5549   apply (simp add: o_def)

  5550   done

  5551

  5552 lemma compact_Times:

  5553   assumes "compact s" "compact t"

  5554   shows "compact (s \<times> t)"

  5555 proof (rule compactI)

  5556   fix C

  5557   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"

  5558   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)"

  5559   proof

  5560     fix x

  5561     assume "x \<in> s"

  5562     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")

  5563     proof

  5564       fix y

  5565       assume "y \<in> t"

  5566       with x \<in> s C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto

  5567       then show "?P y" by (auto elim!: open_prod_elim)

  5568     qed

  5569     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"

  5570       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"

  5571       by metis

  5572     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto

  5573     from compactE_image[OF compact t this] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"

  5574       by auto

  5575     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"

  5576       by (fastforce simp: subset_eq)

  5577     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"

  5578       using c by (intro exI[of _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)

  5579   qed

  5580   then obtain a d where a: "\<forall>x\<in>s. open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"

  5581     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"

  5582     unfolding subset_eq UN_iff by metis

  5583   moreover

  5584   from compactE_image[OF compact s a]

  5585   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"

  5586     by auto

  5587   moreover

  5588   {

  5589     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"

  5590       by auto

  5591     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"

  5592       using d e \<subseteq> s by (intro UN_mono) auto

  5593     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .

  5594   }

  5595   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"

  5596     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp add: subset_eq)

  5597 qed

  5598

  5599 text{* Hence some useful properties follow quite easily. *}

  5600

  5601 lemma compact_scaling:

  5602   fixes s :: "'a::real_normed_vector set"

  5603   assumes "compact s"

  5604   shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5605 proof -

  5606   let ?f = "\<lambda>x. scaleR c x"

  5607   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5608   show ?thesis

  5609     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5610     using linear_continuous_at[OF *] assms

  5611     by auto

  5612 qed

  5613

  5614 lemma compact_negations:

  5615   fixes s :: "'a::real_normed_vector set"

  5616   assumes "compact s"

  5617   shows "compact ((\<lambda>x. - x)  s)"

  5618   using compact_scaling [OF assms, of "- 1"] by auto

  5619

  5620 lemma compact_sums:

  5621   fixes s t :: "'a::real_normed_vector set"

  5622   assumes "compact s"

  5623     and "compact t"

  5624   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5625 proof -

  5626   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5627     apply auto

  5628     unfolding image_iff

  5629     apply (rule_tac x="(xa, y)" in bexI)

  5630     apply auto

  5631     done

  5632   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5633     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5634   then show ?thesis

  5635     unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5636 qed

  5637

  5638 lemma compact_differences:

  5639   fixes s t :: "'a::real_normed_vector set"

  5640   assumes "compact s"

  5641     and "compact t"

  5642   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5643 proof-

  5644   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5645     apply auto

  5646     apply (rule_tac x= xa in exI)

  5647     apply auto

  5648     apply (rule_tac x=xa in exI)

  5649     apply auto

  5650     done

  5651   then show ?thesis

  5652     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5653 qed

  5654

  5655 lemma compact_translation:

  5656   fixes s :: "'a::real_normed_vector set"

  5657   assumes "compact s"

  5658   shows "compact ((\<lambda>x. a + x)  s)"

  5659 proof -

  5660   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s"

  5661     by auto

  5662   then show ?thesis

  5663     using compact_sums[OF assms compact_sing[of a]] by auto

  5664 qed

  5665

  5666 lemma compact_affinity:

  5667   fixes s :: "'a::real_normed_vector set"

  5668   assumes "compact s"

  5669   shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5670 proof -

  5671   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s"

  5672     by auto

  5673   then show ?thesis

  5674     using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5675 qed

  5676

  5677 text {* Hence we get the following. *}

  5678

  5679 lemma compact_sup_maxdistance:

  5680   fixes s :: "'a::metric_space set"

  5681   assumes "compact s"

  5682     and "s \<noteq> {}"

  5683   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5684 proof -

  5685   have "compact (s \<times> s)"

  5686     using compact s by (intro compact_Times)

  5687   moreover have "s \<times> s \<noteq> {}"

  5688     using s \<noteq> {} by auto

  5689   moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"

  5690     by (intro continuous_at_imp_continuous_on ballI continuous_intros)

  5691   ultimately show ?thesis

  5692     using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto

  5693 qed

  5694

  5695 text {* We can state this in terms of diameter of a set. *}

  5696

  5697 definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

  5698

  5699 lemma diameter_bounded_bound:

  5700   fixes s :: "'a :: metric_space set"

  5701   assumes s: "bounded s" "x \<in> s" "y \<in> s"

  5702   shows "dist x y \<le> diameter s"

  5703 proof -

  5704   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  5705   from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"

  5706     unfolding bounded_def by auto

  5707   have "dist x y \<le> Sup ?D"

  5708   proof (rule cSup_upper, safe)

  5709     fix a b

  5710     assume "a \<in> s" "b \<in> s"

  5711     with z[of a] z[of b] dist_triangle[of a b z]

  5712     show "dist a b \<le> 2 * d"

  5713       by (simp add: dist_commute)

  5714   qed (insert s, auto)

  5715   with x \<in> s show ?thesis

  5716     by (auto simp add: diameter_def)

  5717 qed

  5718

  5719 lemma diameter_lower_bounded:

  5720   fixes s :: "'a :: metric_space set"

  5721   assumes s: "bounded s"

  5722     and d: "0 < d" "d < diameter s"

  5723   shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"

  5724 proof (rule ccontr)

  5725   let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"

  5726   assume contr: "\<not> ?thesis"

  5727   moreover

  5728   from d have "s \<noteq> {}"

  5729     by (auto simp: diameter_def)

  5730   then have "?D \<noteq> {}" by auto

  5731   ultimately have "Sup ?D \<le> d"

  5732     by (intro cSup_least) (auto simp: not_less)

  5733   with d < diameter s s \<noteq> {} show False

  5734     by (auto simp: diameter_def)

  5735 qed

  5736

  5737 lemma diameter_bounded:

  5738   assumes "bounded s"

  5739   shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"

  5740     and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"

  5741   using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms

  5742   by auto

  5743

  5744 lemma diameter_compact_attained:

  5745   assumes "compact s"

  5746     and "s \<noteq> {}"

  5747   shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"

  5748 proof -

  5749   have b: "bounded s" using assms(1)

  5750     by (rule compact_imp_bounded)

  5751   then obtain x y where xys: "x\<in>s" "y\<in>s"

  5752     and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"

  5753     using compact_sup_maxdistance[OF assms] by auto

  5754   then have "diameter s \<le> dist x y"

  5755     unfolding diameter_def

  5756     apply clarsimp

  5757     apply (rule cSup_least)

  5758     apply fast+

  5759     done

  5760   then show ?thesis

  5761     by (metis b diameter_bounded_bound order_antisym xys)

  5762 qed

  5763

  5764 text {* Related results with closure as the conclusion. *}

  5765

  5766 lemma closed_scaling:

  5767   fixes s :: "'a::real_normed_vector set"

  5768   assumes "closed s"

  5769   shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5770 proof (cases "c = 0")

  5771   case True then show ?thesis

  5772     by (auto simp add: image_constant_conv)

  5773 next

  5774   case False

  5775   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) - s)"

  5776     by (simp add: continuous_closed_vimage)

  5777   also have "(\<lambda>x. inverse c *\<^sub>R x) - s = (\<lambda>x. c *\<^sub>R x)  s"

  5778     using c \<noteq> 0 by (auto elim: image_eqI [rotated])

  5779   finally show ?thesis .

  5780 qed

  5781

  5782 lemma closed_negations:

  5783   fixes s :: "'a::real_normed_vector set"

  5784   assumes "closed s"

  5785   shows "closed ((\<lambda>x. -x)  s)"

  5786   using closed_scaling[OF assms, of "- 1"] by simp

  5787

  5788 lemma compact_closed_sums:

  5789   fixes s :: "'a::real_normed_vector set"

  5790   assumes "compact s" and "closed t"

  5791   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5792 proof -

  5793   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  5794   {

  5795     fix x l

  5796     assume as: "\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  5797     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"

  5798       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  5799     obtain l' r where "l'\<in>s" and r: "subseq r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  5800       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  5801     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  5802       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)

  5803       unfolding o_def

  5804       by auto

  5805     then have "l - l' \<in> t"

  5806       using assms(2)[unfolded closed_sequential_limits,

  5807         THEN spec[where x="\<lambda> n. snd (f (r n))"],

  5808         THEN spec[where x="l - l'"]]

  5809       using f(3)

  5810       by auto

  5811     then have "l \<in> ?S"

  5812       using l' \<in> s

  5813       apply auto

  5814       apply (rule_tac x=l' in exI)

  5815       apply (rule_tac x="l - l'" in exI)

  5816       apply auto

  5817       done

  5818   }

  5819   then show ?thesis

  5820     unfolding closed_sequential_limits by fast

  5821 qed

  5822

  5823 lemma closed_compact_sums:

  5824   fixes s t :: "'a::real_normed_vector set"

  5825   assumes "closed s"

  5826     and "compact t"

  5827   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5828 proof -

  5829   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}"

  5830     apply auto

  5831     apply (rule_tac x=y in exI)

  5832     apply auto

  5833     apply (rule_tac x=y in exI)

  5834     apply auto

  5835     done

  5836   then show ?thesis

  5837     using compact_closed_sums[OF assms(2,1)] by simp

  5838 qed

  5839

  5840 lemma compact_closed_differences:

  5841   fixes s t :: "'a::real_normed_vector set"

  5842   assumes "compact s"

  5843     and "closed t"

  5844   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5845 proof -

  5846   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  5847     apply auto

  5848     apply (rule_tac x=xa in exI)

  5849     apply auto

  5850     apply (rule_tac x=xa in exI)

  5851     apply auto

  5852     done

  5853   then show ?thesis

  5854     using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  5855 qed

  5856

  5857 lemma closed_compact_differences:

  5858   fixes s t :: "'a::real_normed_vector set"

  5859   assumes "closed s"

  5860     and "compact t"

  5861   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5862 proof -

  5863   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  5864     apply auto

  5865     apply (rule_tac x=xa in exI)

  5866     apply auto

  5867     apply (rule_tac x=xa in exI)

  5868     apply auto

  5869     done

  5870  then show ?thesis

  5871   using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  5872 qed

  5873

  5874 lemma closed_translation:

  5875   fixes a :: "'a::real_normed_vector"

  5876   assumes "closed s"

  5877   shows "closed ((\<lambda>x. a + x)  s)"

  5878 proof -

  5879   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  5880   then show ?thesis

  5881     using compact_closed_sums[OF compact_sing[of a] assms] by auto

  5882 qed

  5883

  5884 lemma translation_Compl:

  5885   fixes a :: "'a::ab_group_add"

  5886   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5887   apply (auto simp add: image_iff)

  5888   apply (rule_tac x="x - a" in bexI)

  5889   apply auto

  5890   done

  5891

  5892 lemma translation_UNIV:

  5893   fixes a :: "'a::ab_group_add"

  5894   shows "range (\<lambda>x. a + x) = UNIV"

  5895   apply (auto simp add: image_iff)

  5896   apply (rule_tac x="x - a" in exI)

  5897   apply auto

  5898   done

  5899

  5900 lemma translation_diff:

  5901   fixes a :: "'a::ab_group_add"

  5902   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5903   by auto

  5904

  5905 lemma closure_translation:

  5906   fixes a :: "'a::real_normed_vector"

  5907   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5908 proof -

  5909   have *: "op + a  (- s) = - op + a  s"

  5910     apply auto

  5911     unfolding image_iff

  5912     apply (rule_tac x="x - a" in bexI)

  5913     apply auto

  5914     done

  5915   show ?thesis

  5916     unfolding closure_interior translation_Compl

  5917     using interior_translation[of a "- s"]

  5918     unfolding *

  5919     by auto

  5920 qed

  5921

  5922 lemma frontier_translation:

  5923   fixes a :: "'a::real_normed_vector"

  5924   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5925   unfolding frontier_def translation_diff interior_translation closure_translation

  5926   by auto

  5927

  5928

  5929 subsection {* Separation between points and sets *}

  5930

  5931 lemma separate_point_closed:

  5932   fixes s :: "'a::heine_borel set"

  5933   assumes "closed s"

  5934     and "a \<notin> s"

  5935   shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"

  5936 proof (cases "s = {}")

  5937   case True

  5938   then show ?thesis by(auto intro!: exI[where x=1])

  5939 next

  5940   case False

  5941   from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"

  5942     using s \<noteq> {} distance_attains_inf [of s a] by blast

  5943   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s

  5944     by blast

  5945 qed

  5946

  5947 lemma separate_compact_closed:

  5948   fixes s t :: "'a::heine_borel set"

  5949   assumes "compact s"

  5950     and t: "closed t" "s \<inter> t = {}"

  5951   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5952 proof cases

  5953   assume "s \<noteq> {} \<and> t \<noteq> {}"

  5954   then have "s \<noteq> {}" "t \<noteq> {}" by auto

  5955   let ?inf = "\<lambda>x. infdist x t"

  5956   have "continuous_on s ?inf"

  5957     by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_at_id)

  5958   then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"

  5959     using continuous_attains_inf[OF compact s s \<noteq> {}] by auto

  5960   then have "0 < ?inf x"

  5961     using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)

  5962   moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"

  5963     using x by (auto intro: order_trans infdist_le)

  5964   ultimately show ?thesis by auto

  5965 qed (auto intro!: exI[of _ 1])

  5966

  5967 lemma separate_closed_compact:

  5968   fixes s t :: "'a::heine_borel set"

  5969   assumes "closed s"

  5970     and "compact t"

  5971     and "s \<inter> t = {}"

  5972   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5973 proof -

  5974   have *: "t \<inter> s = {}"

  5975     using assms(3) by auto

  5976   show ?thesis

  5977     using separate_compact_closed[OF assms(2,1) *]

  5978     apply auto

  5979     apply (rule_tac x=d in exI)

  5980     apply auto

  5981     apply (erule_tac x=y in ballE)

  5982     apply (auto simp add: dist_commute)

  5983     done

  5984 qed

  5985

  5986

  5987 subsection {* Intervals *}

  5988

  5989 lemma interval:

  5990   fixes a :: "'a::ordered_euclidean_space"

  5991   shows "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}"

  5992     and "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5993   by (auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5994

  5995 lemma mem_interval:

  5996   fixes a :: "'a::ordered_euclidean_space"

  5997   shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5998     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)"

  5999   using interval[of a b]

  6000   by (auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  6001

  6002 lemma interval_eq_empty:

  6003   fixes a :: "'a::ordered_euclidean_space"

  6004   shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)

  6005     and "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  6006 proof -

  6007   {

  6008     fix i x

  6009     assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  6010     then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"

  6011       unfolding mem_interval by auto

  6012     then have "a\<bullet>i < b\<bullet>i" by auto

  6013     then have False using as by auto

  6014   }

  6015   moreover

  6016   {

  6017     assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  6018     let ?x = "(1/2) *\<^sub>R (a + b)"

  6019     {

  6020       fix i :: 'a

  6021       assume i: "i \<in> Basis"

  6022       have "a\<bullet>i < b\<bullet>i"

  6023         using as[THEN bspec[where x=i]] i by auto

  6024       then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  6025         by (auto simp: inner_add_left)

  6026     }

  6027     then have "{a <..< b} \<noteq> {}"

  6028       using mem_interval(1)[of "?x" a b] by auto

  6029   }

  6030   ultimately show ?th1 by blast

  6031

  6032   {

  6033     fix i x

  6034     assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  6035     then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  6036       unfolding mem_interval by auto

  6037     then have "a\<bullet>i \<le> b\<bullet>i" by auto

  6038     then have False using as by auto

  6039   }

  6040   moreover

  6041   {

  6042     assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  6043     let ?x = "(1/2) *\<^sub>R (a + b)"

  6044     {

  6045       fix i :: 'a

  6046       assume i:"i \<in> Basis"

  6047       have "a\<bullet>i \<le> b\<bullet>i"

  6048         using as[THEN bspec[where x=i]] i by auto

  6049       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"

  6050         by (auto simp: inner_add_left)

  6051     }

  6052     then have "{a .. b} \<noteq> {}"

  6053       using mem_interval(2)[of "?x" a b] by auto

  6054   }

  6055   ultimately show ?th2 by blast

  6056 qed

  6057

  6058 lemma interval_ne_empty:

  6059   fixes a :: "'a::ordered_euclidean_space"

  6060   shows "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"

  6061   and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  6062   unfolding interval_eq_empty[of a b] by fastforce+

  6063

  6064 lemma interval_sing:

  6065   fixes a :: "'a::ordered_euclidean_space"

  6066   shows "{a .. a} = {a}"

  6067     and "{a<..<a} = {}"

  6068   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  6069   by (auto intro: euclidean_eqI simp: ex_in_conv)

  6070

  6071 lemma subset_interval_imp:

  6072   fixes a :: "'a::ordered_euclidean_space"

  6073   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}"

  6074     and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"

  6075     and "(\<forall>i\<in>Basis. a\<bu`