src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author huffman Thu Jan 17 15:28:53 2013 -0800 (2013-01-17) changeset 50970 3e5b67f85bf9 parent 50955 ada575c605e1 child 50971 5e3d3d690975 permissions -rw-r--r--
generalized theorem edelstein_fix to class metric_space
     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/Diagonal_Subsequence"

    13   "~~/src/HOL/Library/Countable_Set"

    14   "~~/src/HOL/Library/Glbs"

    15   "~~/src/HOL/Library/FuncSet"

    16   Linear_Algebra

    17   Norm_Arith

    18 begin

    19

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

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

    22

    23 (* TODO: Move this to RComplete.thy -- would need to include Glb into RComplete *)

    24 lemma real_isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::real)"

    25   apply (frule isGlb_isLb)

    26   apply (frule_tac x = y in isGlb_isLb)

    27   apply (blast intro!: order_antisym dest!: isGlb_le_isLb)

    28   done

    29

    30 lemma countable_PiE:

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

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

    33

    34 subsection {* Topological Basis *}

    35

    36 context topological_space

    37 begin

    38

    39 definition "topological_basis B =

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

    41

    42 lemma topological_basis_iff:

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

    44   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'))"

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

    46 proof safe

    47   fix O' and x::'a

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

    49   hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)

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

    51   thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto

    52 next

    53   assume H: ?rhs

    54   show "topological_basis B" using assms unfolding topological_basis_def

    55   proof safe

    56     fix O'::"'a set" assume "open O'"

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

    58       by (force intro: bchoice simp: Bex_def)

    59     thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"

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

    61   qed

    62 qed

    63

    64 lemma topological_basisI:

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

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

    67   shows "topological_basis B"

    68   using assms by (subst topological_basis_iff) auto

    69

    70 lemma topological_basisE:

    71   fixes O'

    72   assumes "topological_basis B"

    73   assumes "open O'"

    74   assumes "x \<in> O'"

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

    76 proof atomize_elim

    77   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)

    78   with topological_basis_iff assms

    79   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)

    80 qed

    81

    82 lemma topological_basis_open:

    83   assumes "topological_basis B"

    84   assumes "X \<in> B"

    85   shows "open X"

    86   using assms

    87   by (simp add: topological_basis_def)

    88

    89 lemma basis_dense:

    90   fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"

    91   assumes "topological_basis B"

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

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

    94 proof (intro allI impI)

    95   fix X::"'a set" assume "open X" "X \<noteq> {}"

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

    97   guess B' . note B' = this

    98   thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)

    99 qed

   100

   101 end

   102

   103 lemma topological_basis_prod:

   104   assumes A: "topological_basis A" and B: "topological_basis B"

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

   106   unfolding topological_basis_def

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

   108   fix S :: "('a \<times> 'b) set" assume "open S"

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

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

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

   112     from open_prod_elim[OF open S this]

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

   114       by (metis mem_Sigma_iff)

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

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

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

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

   119   qed auto

   120 qed (metis A B topological_basis_open open_Times)

   121

   122 subsection {* Countable Basis *}

   123

   124 locale countable_basis =

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

   126   assumes is_basis: "topological_basis B"

   127   assumes countable_basis: "countable B"

   128 begin

   129

   130 lemma open_countable_basis_ex:

   131   assumes "open X"

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

   133   using assms countable_basis is_basis unfolding topological_basis_def by blast

   134

   135 lemma open_countable_basisE:

   136   assumes "open X"

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

   138   using assms open_countable_basis_ex by (atomize_elim) simp

   139

   140 lemma countable_dense_exists:

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

   142 proof -

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

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

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

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

   147 qed

   148

   149 lemma countable_dense_setE:

   150   obtains D :: "'a set"

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

   152   using countable_dense_exists by blast

   153

   154 text {* Construction of an increasing sequence approximating open sets,

   155   therefore basis which is closed under union. *}

   156

   157 definition union_closed_basis::"'a set set" where

   158   "union_closed_basis = (\<lambda>l. \<Union>set l)  lists B"

   159

   160 lemma basis_union_closed_basis: "topological_basis union_closed_basis"

   161 proof (rule topological_basisI)

   162   fix O' and x::'a assume "open O'" "x \<in> O'"

   163   from topological_basisE[OF is_basis this] guess B' . note B' = this

   164   thus "\<exists>B'\<in>union_closed_basis. x \<in> B' \<and> B' \<subseteq> O'" unfolding union_closed_basis_def

   165     by (auto intro!: bexI[where x="[B']"])

   166 next

   167   fix B' assume "B' \<in> union_closed_basis"

   168   thus "open B'"

   169     using topological_basis_open[OF is_basis]

   170     by (auto simp: union_closed_basis_def)

   171 qed

   172

   173 lemma countable_union_closed_basis: "countable union_closed_basis"

   174   unfolding union_closed_basis_def using countable_basis by simp

   175

   176 lemmas open_union_closed_basis = topological_basis_open[OF basis_union_closed_basis]

   177

   178 lemma union_closed_basis_ex:

   179  assumes X: "X \<in> union_closed_basis"

   180  shows "\<exists>B'. finite B' \<and> X = \<Union>B' \<and> B' \<subseteq> B"

   181 proof -

   182   from X obtain l where "\<And>x. x\<in>set l \<Longrightarrow> x\<in>B" "X = \<Union>set l" by (auto simp: union_closed_basis_def)

   183   thus ?thesis by auto

   184 qed

   185

   186 lemma union_closed_basisE:

   187   assumes "X \<in> union_closed_basis"

   188   obtains B' where "finite B'" "X = \<Union>B'" "B' \<subseteq> B" using union_closed_basis_ex[OF assms] by blast

   189

   190 lemma union_closed_basisI:

   191   assumes "finite B'" "X = \<Union>B'" "B' \<subseteq> B"

   192   shows "X \<in> union_closed_basis"

   193 proof -

   194   from finite_list[OF finite B'] guess l ..

   195   thus ?thesis using assms unfolding union_closed_basis_def by (auto intro!: image_eqI[where x=l])

   196 qed

   197

   198 lemma empty_basisI[intro]: "{} \<in> union_closed_basis"

   199   by (rule union_closed_basisI[of "{}"]) auto

   200

   201 lemma union_basisI[intro]:

   202   assumes "X \<in> union_closed_basis" "Y \<in> union_closed_basis"

   203   shows "X \<union> Y \<in> union_closed_basis"

   204   using assms by (auto intro: union_closed_basisI elim!:union_closed_basisE)

   205

   206 lemma open_imp_Union_of_incseq:

   207   assumes "open X"

   208   shows "\<exists>S. incseq S \<and> (\<Union>j. S j) = X \<and> range S \<subseteq> union_closed_basis"

   209 proof -

   210   from open_countable_basis_ex[OF open X]

   211   obtain B' where B': "B'\<subseteq>B" "X = \<Union>B'" by auto

   212   from this(1) countable_basis have "countable B'" by (rule countable_subset)

   213   show ?thesis

   214   proof cases

   215     assume "B' \<noteq> {}"

   216     def S \<equiv> "\<lambda>n. \<Union>i\<in>{0..n}. from_nat_into B' i"

   217     have S:"\<And>n. S n = \<Union>{from_nat_into B' i|i. i\<in>{0..n}}" unfolding S_def by force

   218     have "incseq S" by (force simp: S_def incseq_Suc_iff)

   219     moreover

   220     have "(\<Union>j. S j) = X" unfolding B'

   221     proof safe

   222       fix x X assume "X \<in> B'" "x \<in> X"

   223       then obtain n where "X = from_nat_into B' n"

   224         by (metis countable B' from_nat_into_surj)

   225       also have "\<dots> \<subseteq> S n" by (auto simp: S_def)

   226       finally show "x \<in> (\<Union>j. S j)" using x \<in> X by auto

   227     next

   228       fix x n

   229       assume "x \<in> S n"

   230       also have "\<dots> = (\<Union>i\<in>{0..n}. from_nat_into B' i)"

   231         by (simp add: S_def)

   232       also have "\<dots> \<subseteq> (\<Union>i. from_nat_into B' i)" by auto

   233       also have "\<dots> \<subseteq> \<Union>B'" using B' \<noteq> {} by (auto intro: from_nat_into)

   234       finally show "x \<in> \<Union>B'" .

   235     qed

   236     moreover have "range S \<subseteq> union_closed_basis" using B'

   237       by (auto intro!: union_closed_basisI[OF _ S] simp: from_nat_into B' \<noteq> {})

   238     ultimately show ?thesis by auto

   239   qed (auto simp: B')

   240 qed

   241

   242 lemma open_incseqE:

   243   assumes "open X"

   244   obtains S where "incseq S" "(\<Union>j. S j) = X" "range S \<subseteq> union_closed_basis"

   245   using open_imp_Union_of_incseq assms by atomize_elim

   246

   247 end

   248

   249 class first_countable_topology = topological_space +

   250   assumes first_countable_basis:

   251     "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

   252

   253 lemma (in first_countable_topology) countable_basis_at_decseq:

   254   obtains A :: "nat \<Rightarrow> 'a set" where

   255     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"

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

   257 proof atomize_elim

   258   from first_countable_basis[of x] obtain A

   259     where "countable A"

   260     and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"

   261     and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"  by auto

   262   then have "A \<noteq> {}" by auto

   263   with countable A have r: "A = range (from_nat_into A)" by auto

   264   def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"

   265   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>

   266       (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"

   267   proof (safe intro!: exI[of _ F])

   268     fix i

   269     show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)

   270     show "x \<in> F i" using nhds(2) r by (auto simp: F_def)

   271   next

   272     fix S assume "open S" "x \<in> S"

   273     from incl[OF this] obtain i where "F i \<subseteq> S"

   274       by (subst (asm) r) (auto simp: F_def)

   275     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"

   276       by (auto simp: F_def)

   277     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"

   278       by (auto simp: eventually_sequentially)

   279   qed

   280 qed

   281

   282 lemma (in first_countable_topology) first_countable_basisE:

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

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

   285   using first_countable_basis[of x]

   286   by atomize_elim auto

   287

   288 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

   289 proof

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

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

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

   293   show "\<exists>A::('a\<times>'b) set set. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

   294   proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b)  (A \<times> B)"], safe)

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

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

   297       unfolding mem_Times_iff by (auto intro: open_Times)

   298   next

   299     fix S assume "open S" "x \<in> S"

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

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

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

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

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

   305   qed (simp add: A B)

   306 qed

   307

   308 instance metric_space \<subseteq> first_countable_topology

   309 proof

   310   fix x :: 'a

   311   show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"

   312   proof (intro exI, safe)

   313     fix S assume "open S" "x \<in> S"

   314     then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"

   315       by (auto simp: open_dist dist_commute subset_eq)

   316     moreover from reals_Archimedean[OF 0 < r] guess n ..

   317     moreover

   318     then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"

   319       by (auto simp: inverse_eq_divide)

   320     ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"

   321       by auto

   322   qed (auto intro: open_ball)

   323 qed

   324

   325 class second_countable_topology = topological_space +

   326   assumes ex_countable_basis:

   327     "\<exists>B::'a::topological_space set set. countable B \<and> topological_basis B"

   328

   329 sublocale second_countable_topology < countable_basis "SOME B. countable B \<and> topological_basis B"

   330   using someI_ex[OF ex_countable_basis] by unfold_locales safe

   331

   332 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

   333 proof

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

   335     using ex_countable_basis by auto

   336   moreover

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

   338     using ex_countable_basis by auto

   339   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> topological_basis B"

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

   341 qed

   342

   343 instance second_countable_topology \<subseteq> first_countable_topology

   344 proof

   345   fix x :: 'a

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

   347   then have B: "countable B" "topological_basis B"

   348     using countable_basis is_basis

   349     by (auto simp: countable_basis is_basis)

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

   351     by (intro exI[of _ "{b\<in>B. x \<in> b}"])

   352        (fastforce simp: topological_space_class.topological_basis_def)

   353 qed

   354

   355 subsection {* Polish spaces *}

   356

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

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

   359

   360 class polish_space = complete_space + second_countable_topology

   361

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

   363

   364 definition "istopology L \<longleftrightarrow> 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))"

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

   366   morphisms "openin" "topology"

   367   unfolding istopology_def by blast

   368

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

   370   using openin[of U] by blast

   371

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

   373   using topology_inverse[unfolded mem_Collect_eq] .

   374

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

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

   377

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

   379 proof-

   380   { assume "T1=T2"

   381     hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }

   382   moreover

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

   384     hence "openin T1 = openin T2" by (simp add: fun_eq_iff)

   385     hence "topology (openin T1) = topology (openin T2)" by simp

   386     hence "T1 = T2" unfolding openin_inverse .

   387   }

   388   ultimately show ?thesis by blast

   389 qed

   390

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

   392

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

   394

   395 subsubsection {* Main properties of open sets *}

   396

   397 lemma openin_clauses:

   398   fixes U :: "'a topology"

   399   shows "openin U {}"

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

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

   402   using openin[of U] unfolding istopology_def mem_Collect_eq

   403   by fast+

   404

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

   406   unfolding topspace_def by blast

   407 lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)

   408

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

   410   using openin_clauses by simp

   411

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

   413   using openin_clauses by simp

   414

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

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

   417

   418 lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)

   419

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

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

   422 proof

   423   assume ?lhs

   424   then show ?rhs by auto

   425 next

   426   assume H: ?rhs

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

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

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

   430   finally show "openin U S" .

   431 qed

   432

   433

   434 subsubsection {* Closed sets *}

   435

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

   437

   438 lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)

   439 lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)

   440 lemma closedin_topspace[intro,simp]:

   441   "closedin U (topspace U)" by (simp add: closedin_def)

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

   443   by (auto simp add: Diff_Un closedin_def)

   444

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

   446 lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"

   447   shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto

   448

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

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

   451

   452 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast

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

   454   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   455   apply (metis openin_subset subset_eq)

   456   done

   457

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

   459   by (simp add: openin_closedin_eq)

   460

   461 lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"

   462 proof-

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

   464     by (auto simp add: topspace_def openin_subset)

   465   then show ?thesis using oS cT by (auto simp add: closedin_def)

   466 qed

   467

   468 lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"

   469 proof-

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

   471     by (auto simp add: topspace_def )

   472   then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)

   473 qed

   474

   475 subsubsection {* Subspace topology *}

   476

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

   478

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

   480   (is "istopology ?L")

   481 proof-

   482   have "?L {}" by blast

   483   {fix A B assume A: "?L A" and B: "?L B"

   484     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" by blast

   485     have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+

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

   487   moreover

   488   {fix K assume K: "K \<subseteq> Collect ?L"

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

   490       apply (rule set_eqI)

   491       apply (simp add: Ball_def image_iff)

   492       by metis

   493     from K[unfolded th0 subset_image_iff]

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

   495     have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto

   496     moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)

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

   498   ultimately show ?thesis

   499     unfolding subset_eq mem_Collect_eq istopology_def by blast

   500 qed

   501

   502 lemma openin_subtopology:

   503   "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"

   504   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   505   by auto

   506

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

   508   by (auto simp add: topspace_def openin_subtopology)

   509

   510 lemma closedin_subtopology:

   511   "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"

   512   unfolding closedin_def topspace_subtopology

   513   apply (simp add: openin_subtopology)

   514   apply (rule iffI)

   515   apply clarify

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

   517   by auto

   518

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

   520   unfolding openin_subtopology

   521   apply (rule iffI, clarify)

   522   apply (frule openin_subset[of U])  apply blast

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

   524   apply auto

   525   done

   526

   527 lemma subtopology_superset:

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

   529   shows "subtopology U V = U"

   530 proof-

   531   {fix S

   532     {fix T assume T: "openin U T" "S = T \<inter> V"

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

   534       have "openin U S" unfolding eq using T by blast}

   535     moreover

   536     {assume S: "openin U S"

   537       hence "\<exists>T. openin U T \<and> S = T \<inter> V"

   538         using openin_subset[OF S] UV by auto}

   539     ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}

   540   then show ?thesis unfolding topology_eq openin_subtopology by blast

   541 qed

   542

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

   544   by (simp add: subtopology_superset)

   545

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

   547   by (simp add: subtopology_superset)

   548

   549 subsubsection {* The standard Euclidean topology *}

   550

   551 definition

   552   euclidean :: "'a::topological_space topology" where

   553   "euclidean = topology open"

   554

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

   556   unfolding euclidean_def

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

   558   apply (rule topology_inverse[symmetric])

   559   apply (auto simp add: istopology_def)

   560   done

   561

   562 lemma topspace_euclidean: "topspace euclidean = UNIV"

   563   apply (simp add: topspace_def)

   564   apply (rule set_eqI)

   565   by (auto simp add: open_openin[symmetric])

   566

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

   568   by (simp add: topspace_euclidean topspace_subtopology)

   569

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

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

   572

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

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

   575

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

   577

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

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

   580

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

   582   by (auto simp add: openin_open)

   583

   584 lemma open_openin_trans[trans]:

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

   586   by (metis Int_absorb1  openin_open_Int)

   587

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

   589   by (auto simp add: openin_open)

   590

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

   592   by (simp add: closedin_subtopology closed_closedin Int_ac)

   593

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

   595   by (metis closedin_closed)

   596

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

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

   599   apply auto

   600   apply (frule closedin_closed_Int[of T S])

   601   by simp

   602

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

   604   by (auto simp add: closedin_closed)

   605

   606 lemma openin_euclidean_subtopology_iff:

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

   608   shows "openin (subtopology euclidean U) S

   609   \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")

   610 proof

   611   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast

   612 next

   613   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}"

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

   615     unfolding T_def

   616     apply clarsimp

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

   618     apply (clarsimp simp add: less_diff_eq)

   619     apply (erule rev_bexI)

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

   621     apply (erule le_less_trans [OF dist_triangle])

   622     done

   623   assume ?rhs hence 2: "S = U \<inter> T"

   624     unfolding T_def

   625     apply auto

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

   627     apply auto

   628     done

   629   from 1 2 show ?lhs

   630     unfolding openin_open open_dist by fast

   631 qed

   632

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

   634

   635 lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T

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

   637   unfolding open_openin openin_open by blast

   638

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

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

   641

   642 lemma closedin_trans[trans]:

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

   644            closedin (subtopology euclidean U) T

   645            ==> closedin (subtopology euclidean U) S"

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

   647

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

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

   650

   651

   652 subsection {* Open and closed balls *}

   653

   654 definition

   655   ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

   656   "ball x e = {y. dist x y < e}"

   657

   658 definition

   659   cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where

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

   661

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

   663   by (simp add: ball_def)

   664

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

   666   by (simp add: cball_def)

   667

   668 lemma mem_ball_0:

   669   fixes x :: "'a::real_normed_vector"

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

   671   by (simp add: dist_norm)

   672

   673 lemma mem_cball_0:

   674   fixes x :: "'a::real_normed_vector"

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

   676   by (simp add: dist_norm)

   677

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

   679   by simp

   680

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

   682   by simp

   683

   684 lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)

   685 lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)

   686 lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)

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

   688   by (simp add: set_eq_iff) arith

   689

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

   691   by (simp add: set_eq_iff)

   692

   693 lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"

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

   695   "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+

   696 lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"

   697   "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+

   698

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

   700   unfolding open_dist ball_def mem_Collect_eq Ball_def

   701   unfolding dist_commute

   702   apply clarify

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

   704   using dist_triangle_alt[where z=x]

   705   apply (clarsimp simp add: diff_less_iff)

   706   apply atomize

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

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

   709   by arith

   710

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

   712   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

   713

   714 lemma openE[elim?]:

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

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

   717   using assms unfolding open_contains_ball by auto

   718

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

   720   by (metis open_contains_ball subset_eq centre_in_ball)

   721

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

   723   unfolding mem_ball set_eq_iff

   724   apply (simp add: not_less)

   725   by (metis zero_le_dist order_trans dist_self)

   726

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

   728

   729 lemma euclidean_dist_l2:

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

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

   732   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

   734

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

   736

   737 lemma rational_boxes:

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

   739   assumes "0 < e"

   740   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"

   741 proof -

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

   743   then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)

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

   745   proof

   746     fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto

   747   qed

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

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

   750   proof

   751     fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto

   752   qed

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

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

   755   show ?thesis

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

   757     fix y :: 'a assume *: "y \<in> box ?a ?b"

   758     have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"

   759       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

   761     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

   762       fix i :: "'a" assume i: "i \<in> Basis"

   763       have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)

   764       moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto

   765       moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto

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

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

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

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

   770         by (rule power_strict_mono) auto

   771       then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"

   772         by (simp add: power_divide)

   773     qed auto

   774     also have "\<dots> = e" using 0 < e by (simp add: real_eq_of_nat)

   775     finally show "y \<in> ball x e" by (auto simp: ball_def)

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

   777 qed

   778

   779 lemma open_UNION_box:

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

   781   assumes "open M"

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

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

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

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

   786 proof safe

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

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

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

   790   moreover then obtain a b where ab: "x \<in> box a b"

   791     "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"

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

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

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

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

   796 qed (auto simp: I_def)

   797

   798 subsection{* Connectedness *}

   799

   800 definition "connected S \<longleftrightarrow>

   801   ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})

   802   \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"

   803

   804 lemma connected_local:

   805  "connected S \<longleftrightarrow> ~(\<exists>e1 e2.

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

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

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

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

   810                  ~(e1 = {}) \<and>

   811                  ~(e2 = {}))"

   812 unfolding connected_def openin_open by (safe, blast+)

   813

   814 lemma exists_diff:

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

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

   817 proof-

   818   {assume "?lhs" hence ?rhs by blast }

   819   moreover

   820   {fix S assume H: "P S"

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

   822     with H have "P (- (- S))" by metis }

   823   ultimately show ?thesis by metis

   824 qed

   825

   826 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

   829 proof-

   830   have " \<not> connected S \<longleftrightarrow> (\<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> {})"

   831     unfolding connected_def openin_open closedin_closed

   832     apply (subst exists_diff) by blast

   833   hence th0: "connected S \<longleftrightarrow> \<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> {})"

   834     (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis

   835

   836   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'))"

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

   838     unfolding connected_def openin_open closedin_closed by auto

   839   {fix e2

   840     {fix e1 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)"

   841         by auto}

   842     then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}

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

   844   then show ?thesis unfolding th0 th1 by simp

   845 qed

   846

   847 lemma connected_empty[simp, intro]: "connected {}"

   848   by (simp add: connected_def)

   849

   850

   851 subsection{* Limit points *}

   852

   853 definition (in topological_space)

   854   islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where

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

   856

   857 lemma islimptI:

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

   859   shows "x islimpt S"

   860   using assms unfolding islimpt_def by auto

   861

   862 lemma islimptE:

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

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

   865   using assms unfolding islimpt_def by auto

   866

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

   868   unfolding islimpt_def eventually_at_topological by auto

   869

   870 lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"

   871   unfolding islimpt_def by fast

   872

   873 lemma islimpt_approachable:

   874   fixes x :: "'a::metric_space"

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

   876   unfolding islimpt_iff_eventually eventually_at by fast

   877

   878 lemma islimpt_approachable_le:

   879   fixes x :: "'a::metric_space"

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

   881   unfolding islimpt_approachable

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

   883     THEN arg_cong [where f=Not]]

   884   by (simp add: Bex_def conj_commute conj_left_commute)

   885

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

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

   888

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

   890

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

   892   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

   893

   894 lemma perfect_choose_dist:

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

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

   897 using islimpt_UNIV [of x]

   898 by (simp add: islimpt_approachable)

   899

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

   901   unfolding closed_def

   902   apply (subst open_subopen)

   903   apply (simp add: islimpt_def subset_eq)

   904   by (metis ComplE ComplI)

   905

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

   907   unfolding islimpt_def by auto

   908

   909 lemma finite_set_avoid:

   910   fixes a :: "'a::metric_space"

   911   assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"

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

   913   case 1 thus ?case by (auto intro: zero_less_one)

   914 next

   915   case (2 x F)

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

   917   {assume "x = a" hence ?case using d by auto  }

   918   moreover

   919   {assume xa: "x\<noteq>a"

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

   921     have dp: "?d > 0" using xa d(1) using dist_nz by auto

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

   923     with dp xa have ?case by(auto intro!: exI[where x="?d"]) }

   924   ultimately show ?case by blast

   925 qed

   926

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

   928   by (simp add: islimpt_iff_eventually eventually_conj_iff)

   929

   930 lemma discrete_imp_closed:

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

   932   assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"

   933   shows "closed S"

   934 proof-

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

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

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

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

   939     from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])

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

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

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

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

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

   945   then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])

   946 qed

   947

   948

   949 subsection {* Interior of a Set *}

   950

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

   952

   953 lemma interiorI [intro?]:

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

   955   shows "x \<in> interior S"

   956   using assms unfolding interior_def by fast

   957

   958 lemma interiorE [elim?]:

   959   assumes "x \<in> interior S"

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

   961   using assms unfolding interior_def by fast

   962

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

   964   by (simp add: interior_def open_Union)

   965

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

   967   by (auto simp add: interior_def)

   968

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

   970   by (auto simp add: interior_def)

   971

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

   973   by (intro equalityI interior_subset interior_maximal subset_refl)

   974

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

   976   by (metis open_interior interior_open)

   977

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

   979   by (metis interior_maximal interior_subset subset_trans)

   980

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

   982   using open_empty by (rule interior_open)

   983

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

   985   using open_UNIV by (rule interior_open)

   986

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

   988   using open_interior by (rule interior_open)

   989

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

   991   by (auto simp add: interior_def)

   992

   993 lemma interior_unique:

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

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

   996   shows "interior S = T"

   997   by (intro equalityI assms interior_subset open_interior interior_maximal)

   998

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

  1000   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

  1001     Int_lower2 interior_maximal interior_subset open_Int open_interior)

  1002

  1003 lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"

  1004   using open_contains_ball_eq [where S="interior S"]

  1005   by (simp add: open_subset_interior)

  1006

  1007 lemma interior_limit_point [intro]:

  1008   fixes x :: "'a::perfect_space"

  1009   assumes x: "x \<in> interior S" shows "x islimpt S"

  1010   using x islimpt_UNIV [of x]

  1011   unfolding interior_def islimpt_def

  1012   apply (clarsimp, rename_tac T T')

  1013   apply (drule_tac x="T \<inter> T'" in spec)

  1014   apply (auto simp add: open_Int)

  1015   done

  1016

  1017 lemma interior_closed_Un_empty_interior:

  1018   assumes cS: "closed S" and iT: "interior T = {}"

  1019   shows "interior (S \<union> T) = interior S"

  1020 proof

  1021   show "interior S \<subseteq> interior (S \<union> T)"

  1022     by (rule interior_mono, rule Un_upper1)

  1023 next

  1024   show "interior (S \<union> T) \<subseteq> interior S"

  1025   proof

  1026     fix x assume "x \<in> interior (S \<union> T)"

  1027     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..

  1028     show "x \<in> interior S"

  1029     proof (rule ccontr)

  1030       assume "x \<notin> interior S"

  1031       with x \<in> R open R obtain y where "y \<in> R - S"

  1032         unfolding interior_def by fast

  1033       from open R closed S have "open (R - S)" by (rule open_Diff)

  1034       from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast

  1035       from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}

  1036       show "False" unfolding interior_def by fast

  1037     qed

  1038   qed

  1039 qed

  1040

  1041 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"

  1042 proof (rule interior_unique)

  1043   show "interior A \<times> interior B \<subseteq> A \<times> B"

  1044     by (intro Sigma_mono interior_subset)

  1045   show "open (interior A \<times> interior B)"

  1046     by (intro open_Times open_interior)

  1047   fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"

  1048   proof (safe)

  1049     fix x y assume "(x, y) \<in> T"

  1050     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"

  1051       using open T unfolding open_prod_def by fast

  1052     hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"

  1053       using T \<subseteq> A \<times> B by auto

  1054     thus "x \<in> interior A" and "y \<in> interior B"

  1055       by (auto intro: interiorI)

  1056   qed

  1057 qed

  1058

  1059

  1060 subsection {* Closure of a Set *}

  1061

  1062 definition "closure S = S \<union> {x | x. x islimpt S}"

  1063

  1064 lemma interior_closure: "interior S = - (closure (- S))"

  1065   unfolding interior_def closure_def islimpt_def by auto

  1066

  1067 lemma closure_interior: "closure S = - interior (- S)"

  1068   unfolding interior_closure by simp

  1069

  1070 lemma closed_closure[simp, intro]: "closed (closure S)"

  1071   unfolding closure_interior by (simp add: closed_Compl)

  1072

  1073 lemma closure_subset: "S \<subseteq> closure S"

  1074   unfolding closure_def by simp

  1075

  1076 lemma closure_hull: "closure S = closed hull S"

  1077   unfolding hull_def closure_interior interior_def by auto

  1078

  1079 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"

  1080   unfolding closure_hull using closed_Inter by (rule hull_eq)

  1081

  1082 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"

  1083   unfolding closure_eq .

  1084

  1085 lemma closure_closure [simp]: "closure (closure S) = closure S"

  1086   unfolding closure_hull by (rule hull_hull)

  1087

  1088 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"

  1089   unfolding closure_hull by (rule hull_mono)

  1090

  1091 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"

  1092   unfolding closure_hull by (rule hull_minimal)

  1093

  1094 lemma closure_unique:

  1095   assumes "S \<subseteq> T" and "closed T"

  1096   assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"

  1097   shows "closure S = T"

  1098   using assms unfolding closure_hull by (rule hull_unique)

  1099

  1100 lemma closure_empty [simp]: "closure {} = {}"

  1101   using closed_empty by (rule closure_closed)

  1102

  1103 lemma closure_UNIV [simp]: "closure UNIV = UNIV"

  1104   using closed_UNIV by (rule closure_closed)

  1105

  1106 lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"

  1107   unfolding closure_interior by simp

  1108

  1109 lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"

  1110   using closure_empty closure_subset[of S]

  1111   by blast

  1112

  1113 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"

  1114   using closure_eq[of S] closure_subset[of S]

  1115   by simp

  1116

  1117 lemma open_inter_closure_eq_empty:

  1118   "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"

  1119   using open_subset_interior[of S "- T"]

  1120   using interior_subset[of "- T"]

  1121   unfolding closure_interior

  1122   by auto

  1123

  1124 lemma open_inter_closure_subset:

  1125   "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"

  1126 proof

  1127   fix x

  1128   assume as: "open S" "x \<in> S \<inter> closure T"

  1129   { assume *:"x islimpt T"

  1130     have "x islimpt (S \<inter> T)"

  1131     proof (rule islimptI)

  1132       fix A

  1133       assume "x \<in> A" "open A"

  1134       with as have "x \<in> A \<inter> S" "open (A \<inter> S)"

  1135         by (simp_all add: open_Int)

  1136       with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"

  1137         by (rule islimptE)

  1138       hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"

  1139         by simp_all

  1140       thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..

  1141     qed

  1142   }

  1143   then show "x \<in> closure (S \<inter> T)" using as

  1144     unfolding closure_def

  1145     by blast

  1146 qed

  1147

  1148 lemma closure_complement: "closure (- S) = - interior S"

  1149   unfolding closure_interior by simp

  1150

  1151 lemma interior_complement: "interior (- S) = - closure S"

  1152   unfolding closure_interior by simp

  1153

  1154 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"

  1155 proof (rule closure_unique)

  1156   show "A \<times> B \<subseteq> closure A \<times> closure B"

  1157     by (intro Sigma_mono closure_subset)

  1158   show "closed (closure A \<times> closure B)"

  1159     by (intro closed_Times closed_closure)

  1160   fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"

  1161     apply (simp add: closed_def open_prod_def, clarify)

  1162     apply (rule ccontr)

  1163     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)

  1164     apply (simp add: closure_interior interior_def)

  1165     apply (drule_tac x=C in spec)

  1166     apply (drule_tac x=D in spec)

  1167     apply auto

  1168     done

  1169 qed

  1170

  1171

  1172 subsection {* Frontier (aka boundary) *}

  1173

  1174 definition "frontier S = closure S - interior S"

  1175

  1176 lemma frontier_closed: "closed(frontier S)"

  1177   by (simp add: frontier_def closed_Diff)

  1178

  1179 lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"

  1180   by (auto simp add: frontier_def interior_closure)

  1181

  1182 lemma frontier_straddle:

  1183   fixes a :: "'a::metric_space"

  1184   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))"

  1185   unfolding frontier_def closure_interior

  1186   by (auto simp add: mem_interior subset_eq ball_def)

  1187

  1188 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"

  1189   by (metis frontier_def closure_closed Diff_subset)

  1190

  1191 lemma frontier_empty[simp]: "frontier {} = {}"

  1192   by (simp add: frontier_def)

  1193

  1194 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"

  1195 proof-

  1196   { assume "frontier S \<subseteq> S"

  1197     hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto

  1198     hence "closed S" using closure_subset_eq by auto

  1199   }

  1200   thus ?thesis using frontier_subset_closed[of S] ..

  1201 qed

  1202

  1203 lemma frontier_complement: "frontier(- S) = frontier S"

  1204   by (auto simp add: frontier_def closure_complement interior_complement)

  1205

  1206 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"

  1207   using frontier_complement frontier_subset_eq[of "- S"]

  1208   unfolding open_closed by auto

  1209

  1210 subsection {* Filters and the eventually true'' quantifier *}

  1211

  1212 definition

  1213   indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"

  1214     (infixr "indirection" 70) where

  1215   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

  1216

  1217 text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

  1218

  1219 lemma trivial_limit_within:

  1220   shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"

  1221 proof

  1222   assume "trivial_limit (at a within S)"

  1223   thus "\<not> a islimpt S"

  1224     unfolding trivial_limit_def

  1225     unfolding eventually_within eventually_at_topological

  1226     unfolding islimpt_def

  1227     apply (clarsimp simp add: set_eq_iff)

  1228     apply (rename_tac T, rule_tac x=T in exI)

  1229     apply (clarsimp, drule_tac x=y in bspec, simp_all)

  1230     done

  1231 next

  1232   assume "\<not> a islimpt S"

  1233   thus "trivial_limit (at a within S)"

  1234     unfolding trivial_limit_def

  1235     unfolding eventually_within eventually_at_topological

  1236     unfolding islimpt_def

  1237     apply clarsimp

  1238     apply (rule_tac x=T in exI)

  1239     apply auto

  1240     done

  1241 qed

  1242

  1243 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"

  1244   using trivial_limit_within [of a UNIV] by simp

  1245

  1246 lemma trivial_limit_at:

  1247   fixes a :: "'a::perfect_space"

  1248   shows "\<not> trivial_limit (at a)"

  1249   by (rule at_neq_bot)

  1250

  1251 lemma trivial_limit_at_infinity:

  1252   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"

  1253   unfolding trivial_limit_def eventually_at_infinity

  1254   apply clarsimp

  1255   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)

  1256    apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)

  1257   apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])

  1258   apply (drule_tac x=UNIV in spec, simp)

  1259   done

  1260

  1261 text {* Some property holds "sufficiently close" to the limit point. *}

  1262

  1263 lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)

  1264   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1265 unfolding eventually_at dist_nz by auto

  1266

  1267 lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *)

  1268   "eventually P (at a within S) \<longleftrightarrow>

  1269         (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

  1270   by (rule eventually_within_less)

  1271

  1272 lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"

  1273   unfolding trivial_limit_def

  1274   by (auto elim: eventually_rev_mp)

  1275

  1276 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"

  1277   by simp

  1278

  1279 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

  1280   by (simp add: filter_eq_iff)

  1281

  1282 text{* Combining theorems for "eventually" *}

  1283

  1284 lemma eventually_rev_mono:

  1285   "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"

  1286 using eventually_mono [of P Q] by fast

  1287

  1288 lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

  1289   by (simp add: eventually_False)

  1290

  1291

  1292 subsection {* Limits *}

  1293

  1294 text{* Notation Lim to avoid collition with lim defined in analysis *}

  1295

  1296 definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"

  1297   where "Lim A f = (THE l. (f ---> l) A)"

  1298

  1299 lemma Lim:

  1300  "(f ---> l) net \<longleftrightarrow>

  1301         trivial_limit net \<or>

  1302         (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"

  1303   unfolding tendsto_iff trivial_limit_eq by auto

  1304

  1305 text{* Show that they yield usual definitions in the various cases. *}

  1306

  1307 lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>

  1308            (\<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)"

  1309   by (auto simp add: tendsto_iff eventually_within_le)

  1310

  1311 lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>

  1312         (\<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)"

  1313   by (auto simp add: tendsto_iff eventually_within)

  1314

  1315 lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>

  1316         (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"

  1317   by (auto simp add: tendsto_iff eventually_at)

  1318

  1319 lemma Lim_at_infinity:

  1320   "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"

  1321   by (auto simp add: tendsto_iff eventually_at_infinity)

  1322

  1323 lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"

  1324   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

  1325

  1326 text{* The expected monotonicity property. *}

  1327

  1328 lemma Lim_within_empty: "(f ---> l) (net within {})"

  1329   unfolding tendsto_def Limits.eventually_within by simp

  1330

  1331 lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"

  1332   unfolding tendsto_def Limits.eventually_within

  1333   by (auto elim!: eventually_elim1)

  1334

  1335 lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"

  1336   shows "(f ---> l) (net within (S \<union> T))"

  1337   using assms unfolding tendsto_def Limits.eventually_within

  1338   apply clarify

  1339   apply (drule spec, drule (1) mp, drule (1) mp)

  1340   apply (drule spec, drule (1) mp, drule (1) mp)

  1341   apply (auto elim: eventually_elim2)

  1342   done

  1343

  1344 lemma Lim_Un_univ:

  1345  "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV

  1346         ==> (f ---> l) net"

  1347   by (metis Lim_Un within_UNIV)

  1348

  1349 text{* Interrelations between restricted and unrestricted limits. *}

  1350

  1351 lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"

  1352   (* FIXME: rename *)

  1353   unfolding tendsto_def Limits.eventually_within

  1354   apply (clarify, drule spec, drule (1) mp, drule (1) mp)

  1355   by (auto elim!: eventually_elim1)

  1356

  1357 lemma eventually_within_interior:

  1358   assumes "x \<in> interior S"

  1359   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")

  1360 proof-

  1361   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..

  1362   { assume "?lhs"

  1363     then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"

  1364       unfolding Limits.eventually_within Limits.eventually_at_topological

  1365       by auto

  1366     with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"

  1367       by auto

  1368     then have "?rhs"

  1369       unfolding Limits.eventually_at_topological by auto

  1370   } moreover

  1371   { assume "?rhs" hence "?lhs"

  1372       unfolding Limits.eventually_within

  1373       by (auto elim: eventually_elim1)

  1374   } ultimately

  1375   show "?thesis" ..

  1376 qed

  1377

  1378 lemma at_within_interior:

  1379   "x \<in> interior S \<Longrightarrow> at x within S = at x"

  1380   by (simp add: filter_eq_iff eventually_within_interior)

  1381

  1382 lemma at_within_open:

  1383   "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"

  1384   by (simp only: at_within_interior interior_open)

  1385

  1386 lemma Lim_within_open:

  1387   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"

  1388   assumes"a \<in> S" "open S"

  1389   shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"

  1390   using assms by (simp only: at_within_open)

  1391

  1392 lemma Lim_within_LIMSEQ:

  1393   fixes a :: "'a::metric_space"

  1394   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"

  1395   shows "(X ---> L) (at a within T)"

  1396   using assms unfolding tendsto_def [where l=L]

  1397   by (simp add: sequentially_imp_eventually_within)

  1398

  1399 lemma Lim_right_bound:

  1400   fixes f :: "real \<Rightarrow> real"

  1401   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"

  1402   assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"

  1403   shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"

  1404 proof cases

  1405   assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)

  1406 next

  1407   assume [simp]: "{x<..} \<inter> I \<noteq> {}"

  1408   show ?thesis

  1409   proof (rule Lim_within_LIMSEQ, safe)

  1410     fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"

  1411

  1412     show "(\<lambda>n. f (S n)) ----> Inf (f  ({x<..} \<inter> I))"

  1413     proof (rule LIMSEQ_I, rule ccontr)

  1414       fix r :: real assume "0 < r"

  1415       with Inf_close[of "f  ({x<..} \<inter> I)" r]

  1416       obtain y where y: "x < y" "y \<in> I" "f y < Inf (f  ({x <..} \<inter> I)) + r" by auto

  1417       from x < y have "0 < y - x" by auto

  1418       from S(2)[THEN LIMSEQ_D, OF this]

  1419       obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto

  1420

  1421       assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f  ({x<..} \<inter> I))) < r)"

  1422       moreover have "\<And>n. Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1423         using S bnd by (intro Inf_lower[where z=K]) auto

  1424       ultimately obtain n where n: "N \<le> n" "r + Inf (f  ({x<..} \<inter> I)) \<le> f (S n)"

  1425         by (auto simp: not_less field_simps)

  1426       with N[OF n(1)] mono[OF _ y \<in> I, of "S n"] S(1)[THEN spec, of n] y

  1427       show False by auto

  1428     qed

  1429   qed

  1430 qed

  1431

  1432 text{* Another limit point characterization. *}

  1433

  1434 lemma islimpt_sequential:

  1435   fixes x :: "'a::first_countable_topology"

  1436   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"

  1437     (is "?lhs = ?rhs")

  1438 proof

  1439   assume ?lhs

  1440   from countable_basis_at_decseq[of x] guess A . note A = this

  1441   def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1442   { fix n

  1443     from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"

  1444       unfolding islimpt_def using A(1,2)[of n] by auto

  1445     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"

  1446       unfolding f_def by (rule someI_ex)

  1447     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }

  1448   then have "\<forall>n. f n \<in> S - {x}" by auto

  1449   moreover have "(\<lambda>n. f n) ----> x"

  1450   proof (rule topological_tendstoI)

  1451     fix S assume "open S" "x \<in> S"

  1452     from A(3)[OF this] \<And>n. f n \<in> A n

  1453     show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)

  1454   qed

  1455   ultimately show ?rhs by fast

  1456 next

  1457   assume ?rhs

  1458   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto

  1459   show ?lhs

  1460     unfolding islimpt_def

  1461   proof safe

  1462     fix T assume "open T" "x \<in> T"

  1463     from lim[THEN topological_tendstoD, OF this] f

  1464     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"

  1465       unfolding eventually_sequentially by auto

  1466   qed

  1467 qed

  1468

  1469 lemma Lim_inv: (* TODO: delete *)

  1470   fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"

  1471   assumes "(f ---> l) A" and "l \<noteq> 0"

  1472   shows "((inverse o f) ---> inverse l) A"

  1473   unfolding o_def using assms by (rule tendsto_inverse)

  1474

  1475 lemma Lim_null:

  1476   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1477   shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

  1478   by (simp add: Lim dist_norm)

  1479

  1480 lemma Lim_null_comparison:

  1481   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1482   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"

  1483   shows "(f ---> 0) net"

  1484 proof (rule metric_tendsto_imp_tendsto)

  1485   show "(g ---> 0) net" by fact

  1486   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"

  1487     using assms(1) by (rule eventually_elim1, simp add: dist_norm)

  1488 qed

  1489

  1490 lemma Lim_transform_bound:

  1491   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1492   fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"

  1493   assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"

  1494   shows "(f ---> 0) net"

  1495   using assms(1) tendsto_norm_zero [OF assms(2)]

  1496   by (rule Lim_null_comparison)

  1497

  1498 text{* Deducing things about the limit from the elements. *}

  1499

  1500 lemma Lim_in_closed_set:

  1501   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"

  1502   shows "l \<in> S"

  1503 proof (rule ccontr)

  1504   assume "l \<notin> S"

  1505   with closed S have "open (- S)" "l \<in> - S"

  1506     by (simp_all add: open_Compl)

  1507   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"

  1508     by (rule topological_tendstoD)

  1509   with assms(2) have "eventually (\<lambda>x. False) net"

  1510     by (rule eventually_elim2) simp

  1511   with assms(3) show "False"

  1512     by (simp add: eventually_False)

  1513 qed

  1514

  1515 text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

  1516

  1517 lemma Lim_dist_ubound:

  1518   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"

  1519   shows "dist a l <= e"

  1520 proof-

  1521   have "dist a l \<in> {..e}"

  1522   proof (rule Lim_in_closed_set)

  1523     show "closed {..e}" by simp

  1524     show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)

  1525     show "\<not> trivial_limit net" by fact

  1526     show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)

  1527   qed

  1528   thus ?thesis by simp

  1529 qed

  1530

  1531 lemma Lim_norm_ubound:

  1532   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1533   assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"

  1534   shows "norm(l) <= e"

  1535 proof-

  1536   have "norm l \<in> {..e}"

  1537   proof (rule Lim_in_closed_set)

  1538     show "closed {..e}" by simp

  1539     show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)

  1540     show "\<not> trivial_limit net" by fact

  1541     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1542   qed

  1543   thus ?thesis by simp

  1544 qed

  1545

  1546 lemma Lim_norm_lbound:

  1547   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"

  1548   assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"

  1549   shows "e \<le> norm l"

  1550 proof-

  1551   have "norm l \<in> {e..}"

  1552   proof (rule Lim_in_closed_set)

  1553     show "closed {e..}" by simp

  1554     show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)

  1555     show "\<not> trivial_limit net" by fact

  1556     show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)

  1557   qed

  1558   thus ?thesis by simp

  1559 qed

  1560

  1561 text{* Uniqueness of the limit, when nontrivial. *}

  1562

  1563 lemma tendsto_Lim:

  1564   fixes f :: "'a \<Rightarrow> 'b::t2_space"

  1565   shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"

  1566   unfolding Lim_def using tendsto_unique[of net f] by auto

  1567

  1568 text{* Limit under bilinear function *}

  1569

  1570 lemma Lim_bilinear:

  1571   assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"

  1572   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"

  1573 using bounded_bilinear h (f ---> l) net (g ---> m) net

  1574 by (rule bounded_bilinear.tendsto)

  1575

  1576 text{* These are special for limits out of the same vector space. *}

  1577

  1578 lemma Lim_within_id: "(id ---> a) (at a within s)"

  1579   unfolding id_def by (rule tendsto_ident_at_within)

  1580

  1581 lemma Lim_at_id: "(id ---> a) (at a)"

  1582   unfolding id_def by (rule tendsto_ident_at)

  1583

  1584 lemma Lim_at_zero:

  1585   fixes a :: "'a::real_normed_vector"

  1586   fixes l :: "'b::topological_space"

  1587   shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")

  1588   using LIM_offset_zero LIM_offset_zero_cancel ..

  1589

  1590 text{* It's also sometimes useful to extract the limit point from the filter. *}

  1591

  1592 definition

  1593   netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where

  1594   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"

  1595

  1596 lemma netlimit_within:

  1597   assumes "\<not> trivial_limit (at a within S)"

  1598   shows "netlimit (at a within S) = a"

  1599 unfolding netlimit_def

  1600 apply (rule some_equality)

  1601 apply (rule Lim_at_within)

  1602 apply (rule tendsto_ident_at)

  1603 apply (erule tendsto_unique [OF assms])

  1604 apply (rule Lim_at_within)

  1605 apply (rule tendsto_ident_at)

  1606 done

  1607

  1608 lemma netlimit_at:

  1609   fixes a :: "'a::{perfect_space,t2_space}"

  1610   shows "netlimit (at a) = a"

  1611   using netlimit_within [of a UNIV] by simp

  1612

  1613 lemma lim_within_interior:

  1614   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"

  1615   by (simp add: at_within_interior)

  1616

  1617 lemma netlimit_within_interior:

  1618   fixes x :: "'a::{t2_space,perfect_space}"

  1619   assumes "x \<in> interior S"

  1620   shows "netlimit (at x within S) = x"

  1621 using assms by (simp add: at_within_interior netlimit_at)

  1622

  1623 text{* Transformation of limit. *}

  1624

  1625 lemma Lim_transform:

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

  1627   assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"

  1628   shows "(g ---> l) net"

  1629   using tendsto_diff [OF assms(2) assms(1)] by simp

  1630

  1631 lemma Lim_transform_eventually:

  1632   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

  1633   apply (rule topological_tendstoI)

  1634   apply (drule (2) topological_tendstoD)

  1635   apply (erule (1) eventually_elim2, simp)

  1636   done

  1637

  1638 lemma Lim_transform_within:

  1639   assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1640   and "(f ---> l) (at x within S)"

  1641   shows "(g ---> l) (at x within S)"

  1642 proof (rule Lim_transform_eventually)

  1643   show "eventually (\<lambda>x. f x = g x) (at x within S)"

  1644     unfolding eventually_within

  1645     using assms(1,2) by auto

  1646   show "(f ---> l) (at x within S)" by fact

  1647 qed

  1648

  1649 lemma Lim_transform_at:

  1650   assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  1651   and "(f ---> l) (at x)"

  1652   shows "(g ---> l) (at x)"

  1653 proof (rule Lim_transform_eventually)

  1654   show "eventually (\<lambda>x. f x = g x) (at x)"

  1655     unfolding eventually_at

  1656     using assms(1,2) by auto

  1657   show "(f ---> l) (at x)" by fact

  1658 qed

  1659

  1660 text{* Common case assuming being away from some crucial point like 0. *}

  1661

  1662 lemma Lim_transform_away_within:

  1663   fixes a b :: "'a::t1_space"

  1664   assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1665   and "(f ---> l) (at a within S)"

  1666   shows "(g ---> l) (at a within S)"

  1667 proof (rule Lim_transform_eventually)

  1668   show "(f ---> l) (at a within S)" by fact

  1669   show "eventually (\<lambda>x. f x = g x) (at a within S)"

  1670     unfolding Limits.eventually_within eventually_at_topological

  1671     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

  1672 qed

  1673

  1674 lemma Lim_transform_away_at:

  1675   fixes a b :: "'a::t1_space"

  1676   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

  1677   and fl: "(f ---> l) (at a)"

  1678   shows "(g ---> l) (at a)"

  1679   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl

  1680   by simp

  1681

  1682 text{* Alternatively, within an open set. *}

  1683

  1684 lemma Lim_transform_within_open:

  1685   assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

  1686   and "(f ---> l) (at a)"

  1687   shows "(g ---> l) (at a)"

  1688 proof (rule Lim_transform_eventually)

  1689   show "eventually (\<lambda>x. f x = g x) (at a)"

  1690     unfolding eventually_at_topological

  1691     using assms(1,2,3) by auto

  1692   show "(f ---> l) (at a)" by fact

  1693 qed

  1694

  1695 text{* A congruence rule allowing us to transform limits assuming not at point. *}

  1696

  1697 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

  1698

  1699 lemma Lim_cong_within(*[cong add]*):

  1700   assumes "a = b" "x = y" "S = T"

  1701   assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

  1702   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

  1703   unfolding tendsto_def Limits.eventually_within eventually_at_topological

  1704   using assms by simp

  1705

  1706 lemma Lim_cong_at(*[cong add]*):

  1707   assumes "a = b" "x = y"

  1708   assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

  1709   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

  1710   unfolding tendsto_def eventually_at_topological

  1711   using assms by simp

  1712

  1713 text{* Useful lemmas on closure and set of possible sequential limits.*}

  1714

  1715 lemma closure_sequential:

  1716   fixes l :: "'a::first_countable_topology"

  1717   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")

  1718 proof

  1719   assume "?lhs" moreover

  1720   { assume "l \<in> S"

  1721     hence "?rhs" using tendsto_const[of l sequentially] by auto

  1722   } moreover

  1723   { assume "l islimpt S"

  1724     hence "?rhs" unfolding islimpt_sequential by auto

  1725   } ultimately

  1726   show "?rhs" unfolding closure_def by auto

  1727 next

  1728   assume "?rhs"

  1729   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

  1730 qed

  1731

  1732 lemma closed_sequential_limits:

  1733   fixes S :: "'a::first_countable_topology set"

  1734   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"

  1735   unfolding closed_limpt

  1736   using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]

  1737   by metis

  1738

  1739 lemma closure_approachable:

  1740   fixes S :: "'a::metric_space set"

  1741   shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"

  1742   apply (auto simp add: closure_def islimpt_approachable)

  1743   by (metis dist_self)

  1744

  1745 lemma closed_approachable:

  1746   fixes S :: "'a::metric_space set"

  1747   shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"

  1748   by (metis closure_closed closure_approachable)

  1749

  1750 subsection {* Infimum Distance *}

  1751

  1752 definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

  1753

  1754 lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

  1755   by (simp add: infdist_def)

  1756

  1757 lemma infdist_nonneg:

  1758   shows "0 \<le> infdist x A"

  1759   using assms by (auto simp add: infdist_def)

  1760

  1761 lemma infdist_le:

  1762   assumes "a \<in> A"

  1763   assumes "d = dist x a"

  1764   shows "infdist x A \<le> d"

  1765   using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)

  1766

  1767 lemma infdist_zero[simp]:

  1768   assumes "a \<in> A" shows "infdist a A = 0"

  1769 proof -

  1770   from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto

  1771   with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto

  1772 qed

  1773

  1774 lemma infdist_triangle:

  1775   shows "infdist x A \<le> infdist y A + dist x y"

  1776 proof cases

  1777   assume "A = {}" thus ?thesis by (simp add: infdist_def)

  1778 next

  1779   assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto

  1780   have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"

  1781   proof

  1782     from A \<noteq> {} show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp

  1783     fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"

  1784     then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto

  1785     show "infdist x A \<le> d"

  1786       unfolding infdist_notempty[OF A \<noteq> {}]

  1787     proof (rule Inf_lower2)

  1788       show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto

  1789       show "dist x a \<le> d" unfolding d by (rule dist_triangle)

  1790       fix d assume "d \<in> {dist x a |a. a \<in> A}"

  1791       then obtain a where "a \<in> A" "d = dist x a" by auto

  1792       thus "infdist x A \<le> d" by (rule infdist_le)

  1793     qed

  1794   qed

  1795   also have "\<dots> = dist x y + infdist y A"

  1796   proof (rule Inf_eq, safe)

  1797     fix a assume "a \<in> A"

  1798     thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)

  1799   next

  1800     fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"

  1801     hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A

  1802       by (intro Inf_greatest) (auto simp: field_simps)

  1803     thus "i \<le> dist x y + infdist y A" by simp

  1804   qed

  1805   finally show ?thesis by simp

  1806 qed

  1807

  1808 lemma

  1809   in_closure_iff_infdist_zero:

  1810   assumes "A \<noteq> {}"

  1811   shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1812 proof

  1813   assume "x \<in> closure A"

  1814   show "infdist x A = 0"

  1815   proof (rule ccontr)

  1816     assume "infdist x A \<noteq> 0"

  1817     with infdist_nonneg[of x A] have "infdist x A > 0" by auto

  1818     hence "ball x (infdist x A) \<inter> closure A = {}" apply auto

  1819       by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute

  1820         eucl_less_not_refl euclidean_trans(2) infdist_le)

  1821     hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)

  1822     thus False using x \<in> closure A by simp

  1823   qed

  1824 next

  1825   assume x: "infdist x A = 0"

  1826   then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)

  1827   show "x \<in> closure A" unfolding closure_approachable

  1828   proof (safe, rule ccontr)

  1829     fix e::real assume "0 < e"

  1830     assume "\<not> (\<exists>y\<in>A. dist y x < e)"

  1831     hence "infdist x A \<ge> e" using a \<in> A

  1832       unfolding infdist_def

  1833       by (force simp: dist_commute)

  1834     with x 0 < e show False by auto

  1835   qed

  1836 qed

  1837

  1838 lemma

  1839   in_closed_iff_infdist_zero:

  1840   assumes "closed A" "A \<noteq> {}"

  1841   shows "x \<in> A \<longleftrightarrow> infdist x A = 0"

  1842 proof -

  1843   have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"

  1844     by (rule in_closure_iff_infdist_zero) fact

  1845   with assms show ?thesis by simp

  1846 qed

  1847

  1848 lemma tendsto_infdist [tendsto_intros]:

  1849   assumes f: "(f ---> l) F"

  1850   shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"

  1851 proof (rule tendstoI)

  1852   fix e ::real assume "0 < e"

  1853   from tendstoD[OF f this]

  1854   show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"

  1855   proof (eventually_elim)

  1856     fix x

  1857     from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]

  1858     have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"

  1859       by (simp add: dist_commute dist_real_def)

  1860     also assume "dist (f x) l < e"

  1861     finally show "dist (infdist (f x) A) (infdist l A) < e" .

  1862   qed

  1863 qed

  1864

  1865 text{* Some other lemmas about sequences. *}

  1866

  1867 lemma sequentially_offset:

  1868   assumes "eventually (\<lambda>i. P i) sequentially"

  1869   shows "eventually (\<lambda>i. P (i + k)) sequentially"

  1870   using assms unfolding eventually_sequentially by (metis trans_le_add1)

  1871

  1872 lemma seq_offset:

  1873   assumes "(f ---> l) sequentially"

  1874   shows "((\<lambda>i. f (i + k)) ---> l) sequentially"

  1875   using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

  1876

  1877 lemma seq_offset_neg:

  1878   "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"

  1879   apply (rule topological_tendstoI)

  1880   apply (drule (2) topological_tendstoD)

  1881   apply (simp only: eventually_sequentially)

  1882   apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")

  1883   apply metis

  1884   by arith

  1885

  1886 lemma seq_offset_rev:

  1887   "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"

  1888   by (rule LIMSEQ_offset) (* FIXME: redundant *)

  1889

  1890 lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"

  1891   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

  1892

  1893 subsection {* More properties of closed balls *}

  1894

  1895 lemma closed_cball: "closed (cball x e)"

  1896 unfolding cball_def closed_def

  1897 unfolding Collect_neg_eq [symmetric] not_le

  1898 apply (clarsimp simp add: open_dist, rename_tac y)

  1899 apply (rule_tac x="dist x y - e" in exI, clarsimp)

  1900 apply (rename_tac x')

  1901 apply (cut_tac x=x and y=x' and z=y in dist_triangle)

  1902 apply simp

  1903 done

  1904

  1905 lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"

  1906 proof-

  1907   { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"

  1908     hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)

  1909   } moreover

  1910   { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"

  1911     hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto

  1912   } ultimately

  1913   show ?thesis unfolding open_contains_ball by auto

  1914 qed

  1915

  1916 lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"

  1917   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

  1918

  1919 lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"

  1920   apply (simp add: interior_def, safe)

  1921   apply (force simp add: open_contains_cball)

  1922   apply (rule_tac x="ball x e" in exI)

  1923   apply (simp add: subset_trans [OF ball_subset_cball])

  1924   done

  1925

  1926 lemma islimpt_ball:

  1927   fixes x y :: "'a::{real_normed_vector,perfect_space}"

  1928   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")

  1929 proof

  1930   assume "?lhs"

  1931   { assume "e \<le> 0"

  1932     hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto

  1933     have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto

  1934   }

  1935   hence "e > 0" by (metis not_less)

  1936   moreover

  1937   have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] ?lhs unfolding closed_limpt by auto

  1938   ultimately show "?rhs" by auto

  1939 next

  1940   assume "?rhs" hence "e>0"  by auto

  1941   { fix d::real assume "d>0"

  1942     have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1943     proof(cases "d \<le> dist x y")

  1944       case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1945       proof(cases "x=y")

  1946         case True hence False using d \<le> dist x y d>0 by auto

  1947         thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto

  1948       next

  1949         case False

  1950

  1951         have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))

  1952               = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  1953           unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto

  1954         also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"

  1955           using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]

  1956           unfolding scaleR_minus_left scaleR_one

  1957           by (auto simp add: norm_minus_commute)

  1958         also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"

  1959           unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]

  1960           unfolding distrib_right using x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto

  1961         also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)

  1962         finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

  1963

  1964         moreover

  1965

  1966         have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"

  1967           using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

  1968         moreover

  1969         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

  1970           using d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]

  1971           unfolding dist_norm by auto

  1972         ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto

  1973       qed

  1974     next

  1975       case False hence "d > dist x y" by auto

  1976       show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1977       proof(cases "x=y")

  1978         case True

  1979         obtain z where **: "z \<noteq> y" "dist z y < min e d"

  1980           using perfect_choose_dist[of "min e d" y]

  1981           using d > 0 e>0 by auto

  1982         show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1983           unfolding x = y

  1984           using z \<noteq> y **

  1985           by (rule_tac x=z in bexI, auto simp add: dist_commute)

  1986       next

  1987         case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"

  1988           using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)

  1989       qed

  1990     qed  }

  1991   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

  1992 qed

  1993

  1994 lemma closure_ball_lemma:

  1995   fixes x y :: "'a::real_normed_vector"

  1996   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

  1997 proof (rule islimptI)

  1998   fix T assume "y \<in> T" "open T"

  1999   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"

  2000     unfolding open_dist by fast

  2001   (* choose point between x and y, within distance r of y. *)

  2002   def k \<equiv> "min 1 (r / (2 * dist x y))"

  2003   def z \<equiv> "y + scaleR k (x - y)"

  2004   have z_def2: "z = x + scaleR (1 - k) (y - x)"

  2005     unfolding z_def by (simp add: algebra_simps)

  2006   have "dist z y < r"

  2007     unfolding z_def k_def using 0 < r

  2008     by (simp add: dist_norm min_def)

  2009   hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp

  2010   have "dist x z < dist x y"

  2011     unfolding z_def2 dist_norm

  2012     apply (simp add: norm_minus_commute)

  2013     apply (simp only: dist_norm [symmetric])

  2014     apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)

  2015     apply (rule mult_strict_right_mono)

  2016     apply (simp add: k_def divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)

  2017     apply (simp add: zero_less_dist_iff x \<noteq> y)

  2018     done

  2019   hence "z \<in> ball x (dist x y)" by simp

  2020   have "z \<noteq> y"

  2021     unfolding z_def k_def using x \<noteq> y 0 < r

  2022     by (simp add: min_def)

  2023   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"

  2024     using z \<in> ball x (dist x y) z \<in> T z \<noteq> y

  2025     by fast

  2026 qed

  2027

  2028 lemma closure_ball:

  2029   fixes x :: "'a::real_normed_vector"

  2030   shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"

  2031 apply (rule equalityI)

  2032 apply (rule closure_minimal)

  2033 apply (rule ball_subset_cball)

  2034 apply (rule closed_cball)

  2035 apply (rule subsetI, rename_tac y)

  2036 apply (simp add: le_less [where 'a=real])

  2037 apply (erule disjE)

  2038 apply (rule subsetD [OF closure_subset], simp)

  2039 apply (simp add: closure_def)

  2040 apply clarify

  2041 apply (rule closure_ball_lemma)

  2042 apply (simp add: zero_less_dist_iff)

  2043 done

  2044

  2045 (* In a trivial vector space, this fails for e = 0. *)

  2046 lemma interior_cball:

  2047   fixes x :: "'a::{real_normed_vector, perfect_space}"

  2048   shows "interior (cball x e) = ball x e"

  2049 proof(cases "e\<ge>0")

  2050   case False note cs = this

  2051   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

  2052   { fix y assume "y \<in> cball x e"

  2053     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

  2054   hence "cball x e = {}" by auto

  2055   hence "interior (cball x e) = {}" using interior_empty by auto

  2056   ultimately show ?thesis by blast

  2057 next

  2058   case True note cs = this

  2059   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

  2060   { fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"

  2061     then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast

  2062

  2063     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"

  2064       using perfect_choose_dist [of d] by auto

  2065     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

  2066     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

  2067

  2068     hence "y \<in> ball x e" proof(cases "x = y")

  2069       case True

  2070       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

  2071       thus "y \<in> ball x e" using x = y  by simp

  2072     next

  2073       case False

  2074       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm

  2075         using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto

  2076       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

  2077       have "y - x \<noteq> 0" using x \<noteq> y by auto

  2078       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

  2079         using d>0 divide_pos_pos[of d "2*norm (y - x)"] by auto

  2080

  2081       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"

  2082         by (auto simp add: dist_norm algebra_simps)

  2083       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"

  2084         by (auto simp add: algebra_simps)

  2085       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"

  2086         using ** by auto

  2087       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

  2088       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

  2089       thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto

  2090     qed  }

  2091   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

  2092   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

  2093 qed

  2094

  2095 lemma frontier_ball:

  2096   fixes a :: "'a::real_normed_vector"

  2097   shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"

  2098   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

  2099   apply (simp add: set_eq_iff)

  2100   by arith

  2101

  2102 lemma frontier_cball:

  2103   fixes a :: "'a::{real_normed_vector, perfect_space}"

  2104   shows "frontier(cball a e) = {x. dist a x = e}"

  2105   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

  2106   apply (simp add: set_eq_iff)

  2107   by arith

  2108

  2109 lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"

  2110   apply (simp add: set_eq_iff not_le)

  2111   by (metis zero_le_dist dist_self order_less_le_trans)

  2112 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

  2113

  2114 lemma cball_eq_sing:

  2115   fixes x :: "'a::{metric_space,perfect_space}"

  2116   shows "(cball x e = {x}) \<longleftrightarrow> e = 0"

  2117 proof (rule linorder_cases)

  2118   assume e: "0 < e"

  2119   obtain a where "a \<noteq> x" "dist a x < e"

  2120     using perfect_choose_dist [OF e] by auto

  2121   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

  2122   with e show ?thesis by (auto simp add: set_eq_iff)

  2123 qed auto

  2124

  2125 lemma cball_sing:

  2126   fixes x :: "'a::metric_space"

  2127   shows "e = 0 ==> cball x e = {x}"

  2128   by (auto simp add: set_eq_iff)

  2129

  2130

  2131 subsection {* Boundedness *}

  2132

  2133   (* FIXME: This has to be unified with BSEQ!! *)

  2134 definition (in metric_space)

  2135   bounded :: "'a set \<Rightarrow> bool" where

  2136   "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

  2137

  2138 lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"

  2139 unfolding bounded_def

  2140 apply safe

  2141 apply (rule_tac x="dist a x + e" in exI, clarify)

  2142 apply (drule (1) bspec)

  2143 apply (erule order_trans [OF dist_triangle add_left_mono])

  2144 apply auto

  2145 done

  2146

  2147 lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"

  2148 unfolding bounded_any_center [where a=0]

  2149 by (simp add: dist_norm)

  2150

  2151 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

  2152   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

  2153   using assms by auto

  2154

  2155 lemma bounded_empty [simp]: "bounded {}"

  2156   by (simp add: bounded_def)

  2157

  2158 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"

  2159   by (metis bounded_def subset_eq)

  2160

  2161 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

  2162   by (metis bounded_subset interior_subset)

  2163

  2164 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

  2165 proof-

  2166   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

  2167   { fix y assume "y \<in> closure S"

  2168     then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"

  2169       unfolding closure_sequential by auto

  2170     have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp

  2171     hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"

  2172       by (rule eventually_mono, simp add: f(1))

  2173     have "dist x y \<le> a"

  2174       apply (rule Lim_dist_ubound [of sequentially f])

  2175       apply (rule trivial_limit_sequentially)

  2176       apply (rule f(2))

  2177       apply fact

  2178       done

  2179   }

  2180   thus ?thesis unfolding bounded_def by auto

  2181 qed

  2182

  2183 lemma bounded_cball[simp,intro]: "bounded (cball x e)"

  2184   apply (simp add: bounded_def)

  2185   apply (rule_tac x=x in exI)

  2186   apply (rule_tac x=e in exI)

  2187   apply auto

  2188   done

  2189

  2190 lemma bounded_ball[simp,intro]: "bounded(ball x e)"

  2191   by (metis ball_subset_cball bounded_cball bounded_subset)

  2192

  2193 lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"

  2194   apply (auto simp add: bounded_def)

  2195   apply (rename_tac x y r s)

  2196   apply (rule_tac x=x in exI)

  2197   apply (rule_tac x="max r (dist x y + s)" in exI)

  2198   apply (rule ballI, rename_tac z, safe)

  2199   apply (drule (1) bspec, simp)

  2200   apply (drule (1) bspec)

  2201   apply (rule min_max.le_supI2)

  2202   apply (erule order_trans [OF dist_triangle add_left_mono])

  2203   done

  2204

  2205 lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"

  2206   by (induct rule: finite_induct[of F], auto)

  2207

  2208 lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"

  2209   by (induct set: finite, auto)

  2210

  2211 lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"

  2212 proof -

  2213   have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp

  2214   hence "bounded {x}" unfolding bounded_def by fast

  2215   thus ?thesis by (metis insert_is_Un bounded_Un)

  2216 qed

  2217

  2218 lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"

  2219   by (induct set: finite, simp_all)

  2220

  2221 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"

  2222   apply (simp add: bounded_iff)

  2223   apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")

  2224   by metis arith

  2225

  2226 lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"

  2227   by (metis Int_lower1 Int_lower2 bounded_subset)

  2228

  2229 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

  2230 apply (metis Diff_subset bounded_subset)

  2231 done

  2232

  2233 lemma not_bounded_UNIV[simp, intro]:

  2234   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"

  2235 proof(auto simp add: bounded_pos not_le)

  2236   obtain x :: 'a where "x \<noteq> 0"

  2237     using perfect_choose_dist [OF zero_less_one] by fast

  2238   fix b::real  assume b: "b >0"

  2239   have b1: "b +1 \<ge> 0" using b by simp

  2240   with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"

  2241     by (simp add: norm_sgn)

  2242   then show "\<exists>x::'a. b < norm x" ..

  2243 qed

  2244

  2245 lemma bounded_linear_image:

  2246   assumes "bounded S" "bounded_linear f"

  2247   shows "bounded(f  S)"

  2248 proof-

  2249   from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2250   from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)

  2251   { fix x assume "x\<in>S"

  2252     hence "norm x \<le> b" using b by auto

  2253     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

  2254       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

  2255   }

  2256   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

  2257     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

  2258 qed

  2259

  2260 lemma bounded_scaling:

  2261   fixes S :: "'a::real_normed_vector set"

  2262   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"

  2263   apply (rule bounded_linear_image, assumption)

  2264   apply (rule bounded_linear_scaleR_right)

  2265   done

  2266

  2267 lemma bounded_translation:

  2268   fixes S :: "'a::real_normed_vector set"

  2269   assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"

  2270 proof-

  2271   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

  2272   { fix x assume "x\<in>S"

  2273     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

  2274   }

  2275   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

  2276     by (auto intro!: exI[of _ "b + norm a"])

  2277 qed

  2278

  2279

  2280 text{* Some theorems on sups and infs using the notion "bounded". *}

  2281

  2282 lemma bounded_real:

  2283   fixes S :: "real set"

  2284   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

  2285   by (simp add: bounded_iff)

  2286

  2287 lemma bounded_has_Sup:

  2288   fixes S :: "real set"

  2289   assumes "bounded S" "S \<noteq> {}"

  2290   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

  2291 proof

  2292   fix x assume "x\<in>S"

  2293   thus "x \<le> Sup S"

  2294     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)

  2295 next

  2296   show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms

  2297     by (metis SupInf.Sup_least)

  2298 qed

  2299

  2300 lemma Sup_insert:

  2301   fixes S :: "real set"

  2302   shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2303 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal)

  2304

  2305 lemma Sup_insert_finite:

  2306   fixes S :: "real set"

  2307   shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"

  2308   apply (rule Sup_insert)

  2309   apply (rule finite_imp_bounded)

  2310   by simp

  2311

  2312 lemma bounded_has_Inf:

  2313   fixes S :: "real set"

  2314   assumes "bounded S"  "S \<noteq> {}"

  2315   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

  2316 proof

  2317   fix x assume "x\<in>S"

  2318   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

  2319   thus "x \<ge> Inf S" using x\<in>S

  2320     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)

  2321 next

  2322   show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms

  2323     by (metis SupInf.Inf_greatest)

  2324 qed

  2325

  2326 lemma Inf_insert:

  2327   fixes S :: "real set"

  2328   shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2329 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)

  2330

  2331 lemma Inf_insert_finite:

  2332   fixes S :: "real set"

  2333   shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"

  2334   by (rule Inf_insert, rule finite_imp_bounded, simp)

  2335

  2336 subsection {* Compactness *}

  2337

  2338 subsubsection{* Open-cover compactness *}

  2339

  2340 definition compact :: "'a::topological_space set \<Rightarrow> bool" where

  2341   compact_eq_heine_borel: -- "This name is used for backwards compatibility"

  2342     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  2343

  2344 lemma compactI:

  2345   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"

  2346   shows "compact s"

  2347   unfolding compact_eq_heine_borel using assms by metis

  2348

  2349 lemma compactE:

  2350   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"

  2351   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  2352   using assms unfolding compact_eq_heine_borel by metis

  2353

  2354 lemma compactE_image:

  2355   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"

  2356   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"

  2357   using assms unfolding ball_simps[symmetric] SUP_def

  2358   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])

  2359

  2360 subsubsection {* Bolzano-Weierstrass property *}

  2361

  2362 lemma heine_borel_imp_bolzano_weierstrass:

  2363   assumes "compact s" "infinite t"  "t \<subseteq> s"

  2364   shows "\<exists>x \<in> s. x islimpt t"

  2365 proof(rule ccontr)

  2366   assume "\<not> (\<exists>x \<in> s. x islimpt t)"

  2367   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)" unfolding islimpt_def

  2368     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto

  2369   obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"

  2370     using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto

  2371   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto

  2372   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

  2373     hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and t\<subseteq>s by auto

  2374     hence "x = y" using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto  }

  2375   hence "inj_on f t" unfolding inj_on_def by simp

  2376   hence "infinite (f  t)" using assms(2) using finite_imageD by auto

  2377   moreover

  2378   { fix x assume "x\<in>t" "f x \<notin> g"

  2379     from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto

  2380     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

  2381     hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto

  2382     hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }

  2383   hence "f  t \<subseteq> g" by auto

  2384   ultimately show False using g(2) using finite_subset by auto

  2385 qed

  2386

  2387 lemma acc_point_range_imp_convergent_subsequence:

  2388   fixes l :: "'a :: first_countable_topology"

  2389   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"

  2390   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2391 proof -

  2392   from countable_basis_at_decseq[of l] guess A . note A = this

  2393

  2394   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"

  2395   { fix n i

  2396     have "infinite (A (Suc n) \<inter> range f - f{.. i})"

  2397       using l A by auto

  2398     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"

  2399       unfolding ex_in_conv by (intro notI) simp

  2400     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

  2401       by auto

  2402     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

  2403       by (auto simp: not_le)

  2404     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

  2405       unfolding s_def by (auto intro: someI2_ex) }

  2406   note s = this

  2407   def r \<equiv> "nat_rec (s 0 0) s"

  2408   have "subseq r"

  2409     by (auto simp: r_def s subseq_Suc_iff)

  2410   moreover

  2411   have "(\<lambda>n. f (r n)) ----> l"

  2412   proof (rule topological_tendstoI)

  2413     fix S assume "open S" "l \<in> S"

  2414     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2415     moreover

  2416     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"

  2417         by (cases i) (simp_all add: r_def s) }

  2418     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2419     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

  2420       by eventually_elim auto

  2421   qed

  2422   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2423     by (auto simp: convergent_def comp_def)

  2424 qed

  2425

  2426 lemma sequence_infinite_lemma:

  2427   fixes f :: "nat \<Rightarrow> 'a::t1_space"

  2428   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

  2429   shows "infinite (range f)"

  2430 proof

  2431   assume "finite (range f)"

  2432   hence "closed (range f)" by (rule finite_imp_closed)

  2433   hence "open (- range f)" by (rule open_Compl)

  2434   from assms(1) have "l \<in> - range f" by auto

  2435   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

  2436     using open (- range f) l \<in> - range f by (rule topological_tendstoD)

  2437   thus False unfolding eventually_sequentially by auto

  2438 qed

  2439

  2440 lemma closure_insert:

  2441   fixes x :: "'a::t1_space"

  2442   shows "closure (insert x s) = insert x (closure s)"

  2443 apply (rule closure_unique)

  2444 apply (rule insert_mono [OF closure_subset])

  2445 apply (rule closed_insert [OF closed_closure])

  2446 apply (simp add: closure_minimal)

  2447 done

  2448

  2449 lemma islimpt_insert:

  2450   fixes x :: "'a::t1_space"

  2451   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

  2452 proof

  2453   assume *: "x islimpt (insert a s)"

  2454   show "x islimpt s"

  2455   proof (rule islimptI)

  2456     fix t assume t: "x \<in> t" "open t"

  2457     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

  2458     proof (cases "x = a")

  2459       case True

  2460       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

  2461         using * t by (rule islimptE)

  2462       with x = a show ?thesis by auto

  2463     next

  2464       case False

  2465       with t have t': "x \<in> t - {a}" "open (t - {a})"

  2466         by (simp_all add: open_Diff)

  2467       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

  2468         using * t' by (rule islimptE)

  2469       thus ?thesis by auto

  2470     qed

  2471   qed

  2472 next

  2473   assume "x islimpt s" thus "x islimpt (insert a s)"

  2474     by (rule islimpt_subset) auto

  2475 qed

  2476

  2477 lemma islimpt_finite:

  2478   fixes x :: "'a::t1_space"

  2479   shows "finite s \<Longrightarrow> \<not> x islimpt s"

  2480 by (induct set: finite, simp_all add: islimpt_insert)

  2481

  2482 lemma islimpt_union_finite:

  2483   fixes x :: "'a::t1_space"

  2484   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

  2485 by (simp add: islimpt_Un islimpt_finite)

  2486

  2487 lemma islimpt_eq_acc_point:

  2488   fixes l :: "'a :: t1_space"

  2489   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

  2490 proof (safe intro!: islimptI)

  2491   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

  2492   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

  2493     by (auto intro: finite_imp_closed)

  2494   then show False

  2495     by (rule islimptE) auto

  2496 next

  2497   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

  2498   then have "infinite (T \<inter> S - {l})" by auto

  2499   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

  2500     unfolding ex_in_conv by (intro notI) simp

  2501   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

  2502     by auto

  2503 qed

  2504

  2505 lemma islimpt_range_imp_convergent_subsequence:

  2506   fixes l :: "'a :: {t1_space, first_countable_topology}"

  2507   assumes l: "l islimpt (range f)"

  2508   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  2509   using l unfolding islimpt_eq_acc_point

  2510   by (rule acc_point_range_imp_convergent_subsequence)

  2511

  2512 lemma sequence_unique_limpt:

  2513   fixes f :: "nat \<Rightarrow> 'a::t2_space"

  2514   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

  2515   shows "l' = l"

  2516 proof (rule ccontr)

  2517   assume "l' \<noteq> l"

  2518   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

  2519     using hausdorff [OF l' \<noteq> l] by auto

  2520   have "eventually (\<lambda>n. f n \<in> t) sequentially"

  2521     using assms(1) open t l \<in> t by (rule topological_tendstoD)

  2522   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

  2523     unfolding eventually_sequentially by auto

  2524

  2525   have "UNIV = {..<N} \<union> {N..}" by auto

  2526   hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp

  2527   hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)

  2528   hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)

  2529   then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"

  2530     using l' \<in> s open s by (rule islimptE)

  2531   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

  2532   with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp

  2533   with s \<inter> t = {} show False by simp

  2534 qed

  2535

  2536 lemma bolzano_weierstrass_imp_closed:

  2537   fixes s :: "'a::{first_countable_topology, t2_space} set"

  2538   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

  2539   shows "closed s"

  2540 proof-

  2541   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

  2542     hence "l \<in> s"

  2543     proof(cases "\<forall>n. x n \<noteq> l")

  2544       case False thus "l\<in>s" using as(1) by auto

  2545     next

  2546       case True note cas = this

  2547       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

  2548       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

  2549       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

  2550     qed  }

  2551   thus ?thesis unfolding closed_sequential_limits by fast

  2552 qed

  2553

  2554 lemma compact_imp_closed:

  2555   fixes s :: "'a::t2_space set"

  2556   assumes "compact s" shows "closed s"

  2557 unfolding closed_def

  2558 proof (rule openI)

  2559   fix y assume "y \<in> - s"

  2560   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"

  2561   note compact s

  2562   moreover have "\<forall>u\<in>?C. open u" by simp

  2563   moreover have "s \<subseteq> \<Union>?C"

  2564   proof

  2565     fix x assume "x \<in> s"

  2566     with y \<in> - s have "x \<noteq> y" by clarsimp

  2567     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"

  2568       by (rule hausdorff)

  2569     with x \<in> s show "x \<in> \<Union>?C"

  2570       unfolding eventually_nhds by auto

  2571   qed

  2572   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"

  2573     by (rule compactE)

  2574   from D \<subseteq> ?C have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto

  2575   with finite D have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"

  2576     by (simp add: eventually_Ball_finite)

  2577   with s \<subseteq> \<Union>D have "eventually (\<lambda>y. y \<notin> s) (nhds y)"

  2578     by (auto elim!: eventually_mono [rotated])

  2579   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"

  2580     by (simp add: eventually_nhds subset_eq)

  2581 qed

  2582

  2583 lemma compact_imp_bounded:

  2584   assumes "compact U" shows "bounded U"

  2585 proof -

  2586   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto

  2587   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

  2588     by (elim compactE_image)

  2589   from finite D have "bounded (\<Union>x\<in>D. ball x 1)"

  2590     by (simp add: bounded_UN)

  2591   thus "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)

  2592     by (rule bounded_subset)

  2593 qed

  2594

  2595 text{* In particular, some common special cases. *}

  2596

  2597 lemma compact_empty[simp]:

  2598  "compact {}"

  2599   unfolding compact_eq_heine_borel

  2600   by auto

  2601

  2602 lemma compact_union [intro]:

  2603   assumes "compact s" "compact t" shows " compact (s \<union> t)"

  2604 proof (rule compactI)

  2605   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

  2606   from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

  2607     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2608   moreover from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

  2609     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

  2610   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

  2611     by (auto intro!: exI[of _ "s' \<union> t'"])

  2612 qed

  2613

  2614 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

  2615   by (induct set: finite) auto

  2616

  2617 lemma compact_UN [intro]:

  2618   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

  2619   unfolding SUP_def by (rule compact_Union) auto

  2620

  2621 lemma compact_inter_closed [intro]:

  2622   assumes "compact s" and "closed t"

  2623   shows "compact (s \<inter> t)"

  2624 proof (rule compactI)

  2625   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"

  2626   from C closed t have "\<forall>c\<in>C \<union> {-t}. open c" by auto

  2627   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto

  2628   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"

  2629     using compact s unfolding compact_eq_heine_borel by auto

  2630   then guess D ..

  2631   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"

  2632     by (intro exI[of _ "D - {-t}"]) auto

  2633 qed

  2634

  2635 lemma closed_inter_compact [intro]:

  2636   assumes "closed s" and "compact t"

  2637   shows "compact (s \<inter> t)"

  2638   using compact_inter_closed [of t s] assms

  2639   by (simp add: Int_commute)

  2640

  2641 lemma compact_inter [intro]:

  2642   fixes s t :: "'a :: t2_space set"

  2643   assumes "compact s" and "compact t"

  2644   shows "compact (s \<inter> t)"

  2645   using assms by (intro compact_inter_closed compact_imp_closed)

  2646

  2647 lemma compact_sing [simp]: "compact {a}"

  2648   unfolding compact_eq_heine_borel by auto

  2649

  2650 lemma compact_insert [simp]:

  2651   assumes "compact s" shows "compact (insert x s)"

  2652 proof -

  2653   have "compact ({x} \<union> s)"

  2654     using compact_sing assms by (rule compact_union)

  2655   thus ?thesis by simp

  2656 qed

  2657

  2658 lemma finite_imp_compact:

  2659   shows "finite s \<Longrightarrow> compact s"

  2660   by (induct set: finite) simp_all

  2661

  2662 lemma open_delete:

  2663   fixes s :: "'a::t1_space set"

  2664   shows "open s \<Longrightarrow> open (s - {x})"

  2665   by (simp add: open_Diff)

  2666

  2667 text{* Finite intersection property *}

  2668

  2669 lemma inj_setminus: "inj_on uminus (A::'a set set)"

  2670   by (auto simp: inj_on_def)

  2671

  2672 lemma compact_fip:

  2673   "compact U \<longleftrightarrow>

  2674     (\<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> {})"

  2675   (is "_ \<longleftrightarrow> ?R")

  2676 proof (safe intro!: compact_eq_heine_borel[THEN iffD2])

  2677   fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"

  2678     and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"

  2679   from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>uminusA"

  2680     by auto

  2681   with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"

  2682     unfolding compact_eq_heine_borel by (metis subset_image_iff)

  2683   with fi[THEN spec, of B] show False

  2684     by (auto dest: finite_imageD intro: inj_setminus)

  2685 next

  2686   fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2687   from cover have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"

  2688     by auto

  2689   with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>uminusB = {}"

  2690     by (metis subset_image_iff)

  2691   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2692     by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)

  2693 qed

  2694

  2695 lemma compact_imp_fip:

  2696   "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>

  2697     s \<inter> (\<Inter> f) \<noteq> {}"

  2698   unfolding compact_fip by auto

  2699

  2700 text{*Compactness expressed with filters*}

  2701

  2702 definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2703

  2704 lemma eventually_filter_from_subbase:

  2705   "eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

  2706     (is "_ \<longleftrightarrow> ?R P")

  2707   unfolding filter_from_subbase_def

  2708 proof (rule eventually_Abs_filter is_filter.intro)+

  2709   show "?R (\<lambda>x. True)"

  2710     by (rule exI[of _ "{}"]) (simp add: le_fun_def)

  2711 next

  2712   fix P Q assume "?R P" then guess X ..

  2713   moreover assume "?R Q" then guess Y ..

  2714   ultimately show "?R (\<lambda>x. P x \<and> Q x)"

  2715     by (intro exI[of _ "X \<union> Y"]) auto

  2716 next

  2717   fix P Q

  2718   assume "?R P" then guess X ..

  2719   moreover assume "\<forall>x. P x \<longrightarrow> Q x"

  2720   ultimately show "?R Q"

  2721     by (intro exI[of _ X]) auto

  2722 qed

  2723

  2724 lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"

  2725   by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

  2726

  2727 lemma filter_from_subbase_not_bot:

  2728   "\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"

  2729   unfolding trivial_limit_def eventually_filter_from_subbase by auto

  2730

  2731 lemma closure_iff_nhds_not_empty:

  2732   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"

  2733 proof safe

  2734   assume x: "x \<in> closure X"

  2735   fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"

  2736   then have "x \<notin> closure (-S)"

  2737     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)

  2738   with x have "x \<in> closure X - closure (-S)"

  2739     by auto

  2740   also have "\<dots> \<subseteq> closure (X \<inter> S)"

  2741     using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)

  2742   finally have "X \<inter> S \<noteq> {}" by auto

  2743   then show False using X \<inter> A = {} S \<subseteq> A by auto

  2744 next

  2745   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"

  2746   from this[THEN spec, of "- X", THEN spec, of "- closure X"]

  2747   show "x \<in> closure X"

  2748     by (simp add: closure_subset open_Compl)

  2749 qed

  2750

  2751 lemma compact_filter:

  2752   "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))"

  2753 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)

  2754   fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"

  2755   from F have "U \<noteq> {}"

  2756     by (auto simp: eventually_False)

  2757

  2758   def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"

  2759   then have "\<forall>z\<in>Z. closed z"

  2760     by auto

  2761   moreover

  2762   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"

  2763     unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])

  2764   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"

  2765   proof (intro allI impI)

  2766     fix B assume "finite B" "B \<subseteq> Z"

  2767     with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"

  2768       by (auto intro!: eventually_Ball_finite)

  2769     with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"

  2770       by eventually_elim auto

  2771     with F show "U \<inter> \<Inter>B \<noteq> {}"

  2772       by (intro notI) (simp add: eventually_False)

  2773   qed

  2774   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"

  2775     using compact U unfolding compact_fip by blast

  2776   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto

  2777

  2778   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"

  2779     unfolding eventually_inf eventually_nhds

  2780   proof safe

  2781     fix P Q R S

  2782     assume "eventually R F" "open S" "x \<in> S"

  2783     with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]

  2784     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)

  2785     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"

  2786     ultimately show False by (auto simp: set_eq_iff)

  2787   qed

  2788   with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"

  2789     by (metis eventually_bot)

  2790 next

  2791   fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

  2792

  2793   def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"

  2794   then have inj_P': "\<And>A. inj_on P' A"

  2795     by (auto intro!: inj_onI simp: fun_eq_iff)

  2796   def F \<equiv> "filter_from_subbase (P'  insert U A)"

  2797   have "F \<noteq> bot"

  2798     unfolding F_def

  2799   proof (safe intro!: filter_from_subbase_not_bot)

  2800     fix X assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"

  2801     then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"

  2802       unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)

  2803     with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto

  2804     with B show False by (auto simp: P'_def fun_eq_iff)

  2805   qed

  2806   moreover have "eventually (\<lambda>x. x \<in> U) F"

  2807     unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)

  2808   moreover 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)"

  2809   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"

  2810     by auto

  2811

  2812   { fix V assume "V \<in> A"

  2813     then have V: "eventually (\<lambda>x. x \<in> V) F"

  2814       by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)

  2815     have "x \<in> closure V"

  2816       unfolding closure_iff_nhds_not_empty

  2817     proof (intro impI allI)

  2818       fix S A assume "open S" "x \<in> S" "S \<subseteq> A"

  2819       then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)

  2820       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"

  2821         by (auto simp: eventually_inf)

  2822       with x show "V \<inter> A \<noteq> {}"

  2823         by (auto simp del: Int_iff simp add: trivial_limit_def)

  2824     qed

  2825     then have "x \<in> V"

  2826       using V \<in> A A(1) by simp }

  2827   with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto

  2828   with U \<inter> \<Inter>A = {} show False by auto

  2829 qed

  2830

  2831 definition "countably_compact U \<longleftrightarrow>

  2832     (\<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))"

  2833

  2834 lemma countably_compactE:

  2835   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"

  2836   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

  2837   using assms unfolding countably_compact_def by metis

  2838

  2839 lemma countably_compactI:

  2840   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')"

  2841   shows "countably_compact s"

  2842   using assms unfolding countably_compact_def by metis

  2843

  2844 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"

  2845   by (auto simp: compact_eq_heine_borel countably_compact_def)

  2846

  2847 lemma countably_compact_imp_compact:

  2848   assumes "countably_compact U"

  2849   assumes ccover: "countable B" "\<forall>b\<in>B. open b"

  2850   assumes 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"

  2851   shows "compact U"

  2852   using countably_compact U unfolding compact_eq_heine_borel countably_compact_def

  2853 proof safe

  2854   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"

  2855   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)"

  2856

  2857   moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"

  2858   ultimately have "countable C" "\<forall>a\<in>C. open a"

  2859     unfolding C_def using ccover by auto

  2860   moreover

  2861   have "\<Union>A \<inter> U \<subseteq> \<Union>C"

  2862   proof safe

  2863     fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"

  2864     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast

  2865     with a \<in> A show "x \<in> \<Union>C" unfolding C_def

  2866       by auto

  2867   qed

  2868   then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto

  2869   ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"

  2870     using * by metis

  2871   moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"

  2872     by (auto simp: C_def)

  2873   then guess f unfolding bchoice_iff Bex_def ..

  2874   ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2875     unfolding C_def by (intro exI[of _ "fT"]) fastforce

  2876 qed

  2877

  2878 lemma countably_compact_imp_compact_second_countable:

  2879   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2880 proof (rule countably_compact_imp_compact)

  2881   fix T and x :: 'a assume "open T" "x \<in> T"

  2882   from topological_basisE[OF is_basis this] guess b .

  2883   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto

  2884 qed (insert countable_basis topological_basis_open[OF is_basis], auto)

  2885

  2886 lemma countably_compact_eq_compact:

  2887   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"

  2888   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

  2889

  2890 subsubsection{* Sequential compactness *}

  2891

  2892 definition

  2893   seq_compact :: "'a::topological_space set \<Rightarrow> bool" where

  2894   "seq_compact S \<longleftrightarrow>

  2895    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

  2896        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

  2897

  2898 lemma seq_compact_imp_countably_compact:

  2899   fixes U :: "'a :: first_countable_topology set"

  2900   assumes "seq_compact U"

  2901   shows "countably_compact U"

  2902 proof (safe intro!: countably_compactI)

  2903   fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"

  2904   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"

  2905     using seq_compact U by (fastforce simp: seq_compact_def subset_eq)

  2906   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"

  2907   proof cases

  2908     assume "finite A" with A show ?thesis by auto

  2909   next

  2910     assume "infinite A"

  2911     then have "A \<noteq> {}" by auto

  2912     show ?thesis

  2913     proof (rule ccontr)

  2914       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

  2915       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto

  2916       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis

  2917       def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"

  2918       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"

  2919         using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)

  2920       then have "range X \<subseteq> U" by auto

  2921       with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto

  2922       from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]

  2923       obtain n where "x \<in> from_nat_into A n" by auto

  2924       with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]

  2925       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"

  2926         unfolding tendsto_def by (auto simp: comp_def)

  2927       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"

  2928         by (auto simp: eventually_sequentially)

  2929       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"

  2930         by auto

  2931       moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"

  2932         by (auto intro!: exI[of _ "max n N"])

  2933       ultimately show False

  2934         by auto

  2935     qed

  2936   qed

  2937 qed

  2938

  2939 lemma compact_imp_seq_compact:

  2940   fixes U :: "'a :: first_countable_topology set"

  2941   assumes "compact U" shows "seq_compact U"

  2942   unfolding seq_compact_def

  2943 proof safe

  2944   fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"

  2945   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"

  2946     by (auto simp: eventually_filtermap)

  2947   moreover have "filtermap X sequentially \<noteq> bot"

  2948     by (simp add: trivial_limit_def eventually_filtermap)

  2949   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")

  2950     using compact U by (auto simp: compact_filter)

  2951

  2952   from countable_basis_at_decseq[of x] guess A . note A = this

  2953   def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"

  2954   { fix n i

  2955     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"

  2956     proof (rule ccontr)

  2957       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"

  2958       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto

  2959       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"

  2960         by (auto simp: eventually_filtermap eventually_sequentially)

  2961       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"

  2962         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)

  2963       ultimately have "eventually (\<lambda>x. False) ?F"

  2964         by (auto simp add: eventually_inf)

  2965       with x show False

  2966         by (simp add: eventually_False)

  2967     qed

  2968     then have "i < s n i" "X (s n i) \<in> A (Suc n)"

  2969       unfolding s_def by (auto intro: someI2_ex) }

  2970   note s = this

  2971   def r \<equiv> "nat_rec (s 0 0) s"

  2972   have "subseq r"

  2973     by (auto simp: r_def s subseq_Suc_iff)

  2974   moreover

  2975   have "(\<lambda>n. X (r n)) ----> x"

  2976   proof (rule topological_tendstoI)

  2977     fix S assume "open S" "x \<in> S"

  2978     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

  2979     moreover

  2980     { fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"

  2981         by (cases i) (simp_all add: r_def s) }

  2982     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

  2983     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"

  2984       by eventually_elim auto

  2985   qed

  2986   ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"

  2987     using x \<in> U by (auto simp: convergent_def comp_def)

  2988 qed

  2989

  2990 lemma seq_compactI:

  2991   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

  2992   shows "seq_compact S"

  2993   unfolding seq_compact_def using assms by fast

  2994

  2995 lemma seq_compactE:

  2996   assumes "seq_compact S" "\<forall>n. f n \<in> S"

  2997   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

  2998   using assms unfolding seq_compact_def by fast

  2999

  3000 lemma countably_compact_imp_acc_point:

  3001   assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"

  3002   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"

  3003 proof (rule ccontr)

  3004   def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"

  3005   note countably_compact s

  3006   moreover have "\<forall>t\<in>C. open t"

  3007     by (auto simp: C_def)

  3008   moreover

  3009   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"

  3010   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis

  3011   have "s \<subseteq> \<Union>C"

  3012     using t \<subseteq> s

  3013     unfolding C_def Union_image_eq

  3014     apply (safe dest!: s)

  3015     apply (rule_tac a="U \<inter> t" in UN_I)

  3016     apply (auto intro!: interiorI simp add: finite_subset)

  3017     done

  3018   moreover

  3019   from countable t have "countable C"

  3020     unfolding C_def by (auto intro: countable_Collect_finite_subset)

  3021   ultimately guess D by (rule countably_compactE)

  3022   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and

  3023     s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"

  3024     by (metis (lifting) Union_image_eq finite_subset_image C_def)

  3025   from s t \<subseteq> s have "t \<subseteq> \<Union>E"

  3026     using interior_subset by blast

  3027   moreover have "finite (\<Union>E)"

  3028     using E by auto

  3029   ultimately show False using infinite t by (auto simp: finite_subset)

  3030 qed

  3031

  3032 lemma countable_acc_point_imp_seq_compact:

  3033   fixes s :: "'a::first_countable_topology set"

  3034   assumes "\<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))"

  3035   shows "seq_compact s"

  3036 proof -

  3037   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3038     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3039     proof (cases "finite (range f)")

  3040       case True

  3041       obtain l where "infinite {n. f n = f l}"

  3042         using pigeonhole_infinite[OF _ True] by auto

  3043       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"

  3044         using infinite_enumerate by blast

  3045       hence "subseq r \<and> (f \<circ> r) ----> f l"

  3046         by (simp add: fr tendsto_const o_def)

  3047       with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"

  3048         by auto

  3049     next

  3050       case False

  3051       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto

  3052       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..

  3053       from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3054         using acc_point_range_imp_convergent_subsequence[of l f] by auto

  3055       with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

  3056     qed

  3057   }

  3058   thus ?thesis unfolding seq_compact_def by auto

  3059 qed

  3060

  3061 lemma seq_compact_eq_countably_compact:

  3062   fixes U :: "'a :: first_countable_topology set"

  3063   shows "seq_compact U \<longleftrightarrow> countably_compact U"

  3064   using

  3065     countable_acc_point_imp_seq_compact

  3066     countably_compact_imp_acc_point

  3067     seq_compact_imp_countably_compact

  3068   by metis

  3069

  3070 lemma seq_compact_eq_acc_point:

  3071   fixes s :: "'a :: first_countable_topology set"

  3072   shows "seq_compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"

  3073   using

  3074     countable_acc_point_imp_seq_compact[of s]

  3075     countably_compact_imp_acc_point[of s]

  3076     seq_compact_imp_countably_compact[of s]

  3077   by metis

  3078

  3079 lemma seq_compact_eq_compact:

  3080   fixes U :: "'a :: second_countable_topology set"

  3081   shows "seq_compact U \<longleftrightarrow> compact U"

  3082   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

  3083

  3084 lemma bolzano_weierstrass_imp_seq_compact:

  3085   fixes s :: "'a::{t1_space, first_countable_topology} set"

  3086   shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"

  3087   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

  3088

  3089 subsubsection{* Total boundedness *}

  3090

  3091 lemma cauchy_def:

  3092   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

  3093 unfolding Cauchy_def by blast

  3094

  3095 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

  3096   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

  3097 declare helper_1.simps[simp del]

  3098

  3099 lemma seq_compact_imp_totally_bounded:

  3100   assumes "seq_compact s"

  3101   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"

  3102 proof(rule, rule, rule ccontr)

  3103   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k)"

  3104   def x \<equiv> "helper_1 s e"

  3105   { fix n

  3106     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

  3107     proof(induct_tac rule:nat_less_induct)

  3108       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

  3109       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

  3110       have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x  {0 ..< n}" in allE) using as by auto

  3111       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto

  3112       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

  3113         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

  3114       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

  3115     qed }

  3116   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

  3117   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto

  3118   from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto

  3119   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" unfolding cauchy_def using e>0 by auto

  3120   show False

  3121     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

  3122     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

  3123     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

  3124 qed

  3125

  3126 subsubsection{* Heine-Borel theorem *}

  3127

  3128 lemma seq_compact_imp_heine_borel:

  3129   fixes s :: "'a :: metric_space set"

  3130   assumes "seq_compact s" shows "compact s"

  3131 proof -

  3132   from seq_compact_imp_totally_bounded[OF seq_compact s]

  3133   guess f unfolding choice_iff' .. note f = this

  3134   def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"

  3135   have "countably_compact s"

  3136     using seq_compact s by (rule seq_compact_imp_countably_compact)

  3137   then show "compact s"

  3138   proof (rule countably_compact_imp_compact)

  3139     show "countable K"

  3140       unfolding K_def using f

  3141       by (auto intro: countable_finite countable_subset countable_rat

  3142                intro!: countable_image countable_SIGMA countable_UN)

  3143     show "\<forall>b\<in>K. open b" by (auto simp: K_def)

  3144   next

  3145     fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"

  3146     from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto

  3147     then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto

  3148     from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto

  3149     from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"

  3150       unfolding Union_image_eq by auto

  3151     from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K" by (auto simp: K_def)

  3152     then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"

  3153     proof (rule bexI[rotated], safe)

  3154       fix y assume "y \<in> ball k r"

  3155       with r < e / 2 x \<in> ball k r have "dist x y < e"

  3156         by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)

  3157       with ball x e \<subseteq> T show "y \<in> T" by auto

  3158     qed (rule x \<in> ball k r)

  3159   qed

  3160 qed

  3161

  3162 lemma compact_eq_seq_compact_metric:

  3163   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"

  3164   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

  3165

  3166 lemma compact_def:

  3167   "compact (S :: 'a::metric_space set) \<longleftrightarrow>

  3168    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"

  3169   unfolding compact_eq_seq_compact_metric seq_compact_def by auto

  3170

  3171 subsubsection {* Complete the chain of compactness variants *}

  3172

  3173 lemma compact_eq_bolzano_weierstrass:

  3174   fixes s :: "'a::metric_space set"

  3175   shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")

  3176 proof

  3177   assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto

  3178 next

  3179   assume ?rhs thus ?lhs

  3180     unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)

  3181 qed

  3182

  3183 lemma bolzano_weierstrass_imp_bounded:

  3184   "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"

  3185   using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

  3186

  3187 text {*

  3188   A metric space (or topological vector space) is said to have the

  3189   Heine-Borel property if every closed and bounded subset is compact.

  3190 *}

  3191

  3192 class heine_borel = metric_space +

  3193   assumes bounded_imp_convergent_subsequence:

  3194     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s

  3195       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3196

  3197 lemma bounded_closed_imp_seq_compact:

  3198   fixes s::"'a::heine_borel set"

  3199   assumes "bounded s" and "closed s" shows "seq_compact s"

  3200 proof (unfold seq_compact_def, clarify)

  3201   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

  3202   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3203     using bounded_imp_convergent_subsequence [OF bounded s \<forall>n. f n \<in> s] by auto

  3204   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

  3205   have "l \<in> s" using closed s fr l

  3206     unfolding closed_sequential_limits by blast

  3207   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3208     using l \<in> s r l by blast

  3209 qed

  3210

  3211 lemma compact_eq_bounded_closed:

  3212   fixes s :: "'a::heine_borel set"

  3213   shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")

  3214 proof

  3215   assume ?lhs thus ?rhs

  3216     using compact_imp_closed compact_imp_bounded by blast

  3217 next

  3218   assume ?rhs thus ?lhs

  3219     using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto

  3220 qed

  3221

  3222 lemma lim_subseq:

  3223   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"

  3224 unfolding tendsto_def eventually_sequentially o_def

  3225 by (metis seq_suble le_trans)

  3226

  3227 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"

  3228   unfolding Ex1_def

  3229   apply (rule_tac x="nat_rec e f" in exI)

  3230   apply (rule conjI)+

  3231 apply (rule def_nat_rec_0, simp)

  3232 apply (rule allI, rule def_nat_rec_Suc, simp)

  3233 apply (rule allI, rule impI, rule ext)

  3234 apply (erule conjE)

  3235 apply (induct_tac x)

  3236 apply simp

  3237 apply (erule_tac x="n" in allE)

  3238 apply (simp)

  3239 done

  3240

  3241 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"

  3242   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"

  3243   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"

  3244 proof-

  3245   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto

  3246   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto

  3247   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"

  3248     { fix n::nat

  3249       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto

  3250       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto

  3251       with n have "s N \<le> t - e" using e>0 by auto

  3252       hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using n\<le>N by auto  }

  3253     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto

  3254     hence False using isLub_le_isUb[OF t, of "t - e"] and e>0 by auto  }

  3255   thus ?thesis by blast

  3256 qed

  3257

  3258 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"

  3259   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"

  3260   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"

  3261   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]

  3262   unfolding monoseq_def incseq_def

  3263   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]

  3264   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto

  3265

  3266 (* TODO: merge this lemma with the ones above *)

  3267 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"

  3268   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"

  3269   shows "\<exists>l. (s ---> l) sequentially"

  3270 proof-

  3271   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto

  3272   { fix m::nat

  3273     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"

  3274       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)

  3275       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }

  3276   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto

  3277   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]

  3278     unfolding monoseq_def by auto

  3279   thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="l" in exI)

  3280     unfolding dist_norm  by auto

  3281 qed

  3282

  3283 lemma compact_real_lemma:

  3284   assumes "\<forall>n::nat. abs(s n) \<le> b"

  3285   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"

  3286 proof-

  3287   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"

  3288     using seq_monosub[of s] by auto

  3289   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms

  3290     unfolding tendsto_iff dist_norm eventually_sequentially by auto

  3291 qed

  3292

  3293 instance real :: heine_borel

  3294 proof

  3295   fix s :: "real set" and f :: "nat \<Rightarrow> real"

  3296   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  3297   then obtain b where b: "\<forall>n. abs (f n) \<le> b"

  3298     unfolding bounded_iff by auto

  3299   obtain l :: real and r :: "nat \<Rightarrow> nat" where

  3300     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

  3301     using compact_real_lemma [OF b] by auto

  3302   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3303     by auto

  3304 qed

  3305

  3306 lemma compact_lemma:

  3307   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

  3308   assumes "bounded s" and "\<forall>n. f n \<in> s"

  3309   shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>

  3310         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3311 proof safe

  3312   fix d :: "'a set" assume d: "d \<subseteq> Basis"

  3313   with finite_Basis have "finite d" by (blast intro: finite_subset)

  3314   from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>

  3315       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"

  3316   proof(induct d) case empty thus ?case unfolding subseq_def by auto

  3317   next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto

  3318     have s': "bounded ((\<lambda>x. x \<bullet> k)  s)" using bounded s

  3319       by (auto intro!: bounded_linear_image bounded_linear_inner_left)

  3320     obtain l1::"'a" and r1 where r1:"subseq r1" and

  3321       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3322       using insert(3) using insert(4) by auto

  3323     have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  s" using \<forall>n. f n \<in> s by simp

  3324     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"

  3325       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

  3326     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

  3327       using r1 and r2 unfolding r_def o_def subseq_def by auto

  3328     moreover

  3329     def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"

  3330     { fix e::real assume "e>0"

  3331       from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast

  3332       from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)

  3333       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"

  3334         by (rule eventually_subseq)

  3335       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3336         using N1' N2

  3337         by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)

  3338     }

  3339     ultimately show ?case by auto

  3340   qed

  3341 qed

  3342

  3343 instance euclidean_space \<subseteq> heine_borel

  3344 proof

  3345   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"

  3346   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  3347   then obtain l::'a and r where r: "subseq r"

  3348     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"

  3349     using compact_lemma [OF s f] by blast

  3350   { fix e::real assume "e>0"

  3351     hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)

  3352     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"

  3353       by simp

  3354     moreover

  3355     { fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"

  3356       have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"

  3357         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

  3358       also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"

  3359         apply(rule setsum_strict_mono) using n by auto

  3360       finally have "dist (f (r n)) l < e"

  3361         by auto

  3362     }

  3363     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

  3364       by (rule eventually_elim1)

  3365   }

  3366   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

  3367   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

  3368 qed

  3369

  3370 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"

  3371 unfolding bounded_def

  3372 apply clarify

  3373 apply (rule_tac x="a" in exI)

  3374 apply (rule_tac x="e" in exI)

  3375 apply clarsimp

  3376 apply (drule (1) bspec)

  3377 apply (simp add: dist_Pair_Pair)

  3378 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

  3379 done

  3380

  3381 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"

  3382 unfolding bounded_def

  3383 apply clarify

  3384 apply (rule_tac x="b" in exI)

  3385 apply (rule_tac x="e" in exI)

  3386 apply clarsimp

  3387 apply (drule (1) bspec)

  3388 apply (simp add: dist_Pair_Pair)

  3389 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

  3390 done

  3391

  3392 instance prod :: (heine_borel, heine_borel) heine_borel

  3393 proof

  3394   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"

  3395   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

  3396   from s have s1: "bounded (fst  s)" by (rule bounded_fst)

  3397   from f have f1: "\<forall>n. fst (f n) \<in> fst  s" by simp

  3398   obtain l1 r1 where r1: "subseq r1"

  3399     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"

  3400     using bounded_imp_convergent_subsequence [OF s1 f1]

  3401     unfolding o_def by fast

  3402   from s have s2: "bounded (snd  s)" by (rule bounded_snd)

  3403   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd  s" by simp

  3404   obtain l2 r2 where r2: "subseq r2"

  3405     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

  3406     using bounded_imp_convergent_subsequence [OF s2 f2]

  3407     unfolding o_def by fast

  3408   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

  3409     using lim_subseq [OF r2 l1] unfolding o_def .

  3410   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

  3411     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

  3412   have r: "subseq (r1 \<circ> r2)"

  3413     using r1 r2 unfolding subseq_def by simp

  3414   show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

  3415     using l r by fast

  3416 qed

  3417

  3418 subsubsection{* Completeness *}

  3419

  3420 definition

  3421   complete :: "'a::metric_space set \<Rightarrow> bool" where

  3422   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f

  3423                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"

  3424

  3425 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

  3426 proof-

  3427   { assume ?rhs

  3428     { fix e::real

  3429       assume "e>0"

  3430       with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

  3431         by (erule_tac x="e/2" in allE) auto

  3432       { fix n m

  3433         assume nm:"N \<le> m \<and> N \<le> n"

  3434         hence "dist (s m) (s n) < e" using N

  3435           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

  3436           by blast

  3437       }

  3438       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

  3439         by blast

  3440     }

  3441     hence ?lhs

  3442       unfolding cauchy_def

  3443       by blast

  3444   }

  3445   thus ?thesis

  3446     unfolding cauchy_def

  3447     using dist_triangle_half_l

  3448     by blast

  3449 qed

  3450

  3451 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

  3452 proof-

  3453   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto

  3454   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

  3455   moreover

  3456   have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto

  3457   then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"

  3458     unfolding bounded_any_center [where a="s N"] by auto

  3459   ultimately show "?thesis"

  3460     unfolding bounded_any_center [where a="s N"]

  3461     apply(rule_tac x="max a 1" in exI) apply auto

  3462     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

  3463 qed

  3464

  3465 lemma seq_compact_imp_complete: assumes "seq_compact s" shows "complete s"

  3466 proof-

  3467   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

  3468     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding seq_compact_def by blast

  3469

  3470     note lr' = seq_suble [OF lr(2)]

  3471

  3472     { fix e::real assume "e>0"

  3473       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" unfolding cauchy_def using e>0 apply (erule_tac x="e/2" in allE) by auto

  3474       from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using e>0 by auto

  3475       { fix n::nat assume n:"n \<ge> max N M"

  3476         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

  3477         moreover have "r n \<ge> N" using lr'[of n] n by auto

  3478         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

  3479         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }

  3480       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

  3481     hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }

  3482   thus ?thesis unfolding complete_def by auto

  3483 qed

  3484

  3485 instance heine_borel < complete_space

  3486 proof

  3487   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3488   hence "bounded (range f)"

  3489     by (rule cauchy_imp_bounded)

  3490   hence "seq_compact (closure (range f))"

  3491     using bounded_closed_imp_seq_compact [of "closure (range f)"] by auto

  3492   hence "complete (closure (range f))"

  3493     by (rule seq_compact_imp_complete)

  3494   moreover have "\<forall>n. f n \<in> closure (range f)"

  3495     using closure_subset [of "range f"] by auto

  3496   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

  3497     using Cauchy f unfolding complete_def by auto

  3498   then show "convergent f"

  3499     unfolding convergent_def by auto

  3500 qed

  3501

  3502 instance euclidean_space \<subseteq> banach ..

  3503

  3504 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

  3505 proof(simp add: complete_def, rule, rule)

  3506   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

  3507   hence "convergent f" by (rule Cauchy_convergent)

  3508   thus "\<exists>l. f ----> l" unfolding convergent_def .

  3509 qed

  3510

  3511 lemma complete_imp_closed: assumes "complete s" shows "closed s"

  3512 proof -

  3513   { fix x assume "x islimpt s"

  3514     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

  3515       unfolding islimpt_sequential by auto

  3516     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

  3517       using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto

  3518     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

  3519   }

  3520   thus "closed s" unfolding closed_limpt by auto

  3521 qed

  3522

  3523 lemma complete_eq_closed:

  3524   fixes s :: "'a::complete_space set"

  3525   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

  3526 proof

  3527   assume ?lhs thus ?rhs by (rule complete_imp_closed)

  3528 next

  3529   assume ?rhs

  3530   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

  3531     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

  3532     hence "\<exists>l\<in>s. (f ---> l) sequentially" using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }

  3533   thus ?lhs unfolding complete_def by auto

  3534 qed

  3535

  3536 lemma convergent_eq_cauchy:

  3537   fixes s :: "nat \<Rightarrow> 'a::complete_space"

  3538   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"

  3539   unfolding Cauchy_convergent_iff convergent_def ..

  3540

  3541 lemma convergent_imp_bounded:

  3542   fixes s :: "nat \<Rightarrow> 'a::metric_space"

  3543   shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"

  3544   by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

  3545

  3546 lemma nat_approx_posE:

  3547   fixes e::real

  3548   assumes "0 < e"

  3549   obtains n::nat where "1 / (Suc n) < e"

  3550 proof atomize_elim

  3551   have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"

  3552     by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)

  3553   also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"

  3554     by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)

  3555   also have "\<dots> = e" by simp

  3556   finally show  "\<exists>n. 1 / real (Suc n) < e" ..

  3557 qed

  3558

  3559 lemma compact_eq_totally_bounded:

  3560   "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"

  3561 proof (safe intro!: seq_compact_imp_complete[unfolded  compact_eq_seq_compact_metric[symmetric]])

  3562   fix e::real

  3563   def f \<equiv> "(\<lambda>x::'a. ball x e)  UNIV"

  3564   assume "0 < e" "compact s"

  3565   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f \<longrightarrow> (\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')"

  3566     by (simp add: compact_eq_heine_borel)

  3567   moreover

  3568   have d0: "\<And>x::'a. dist x x < e" using 0 < e by simp

  3569   hence "(\<forall>t\<in>f. open t) \<and> s \<subseteq> \<Union>f" by (auto simp: f_def intro!: d0)

  3570   ultimately have "(\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f')" ..

  3571   then guess K .. note K = this

  3572   have "\<forall>K'\<in>K. \<exists>k. K' = ball k e" using K by (auto simp: f_def)

  3573   then obtain k where "\<And>K'. K' \<in> K \<Longrightarrow> K' = ball (k K') e" unfolding bchoice_iff by blast

  3574   thus "\<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k" using K

  3575     by (intro exI[where x="k  K"]) (auto simp: f_def)

  3576 next

  3577   assume assms: "complete s" "\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e)  k"

  3578   show "compact s"

  3579   proof cases

  3580     assume "s = {}" thus "compact s" by (simp add: compact_def)

  3581   next

  3582     assume "s \<noteq> {}"

  3583     show ?thesis

  3584       unfolding compact_def

  3585     proof safe

  3586       fix f::"nat \<Rightarrow> _" assume "\<forall>n. f n \<in> s" hence f: "\<And>n. f n \<in> s" by simp

  3587       from assms have "\<forall>e. \<exists>k. e>0 \<longrightarrow> finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))" by simp

  3588       then obtain K where

  3589         K: "\<And>e. e > 0 \<Longrightarrow> finite (K e) \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3590         unfolding choice_iff by blast

  3591       {

  3592         fix e::real and f' have f': "\<And>n::nat. (f o f') n \<in> s" using f by auto

  3593         assume "e > 0"

  3594         from K[OF this] have K: "finite (K e)" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3595           by simp_all

  3596         have "\<exists>k\<in>(K e). \<exists>r. subseq r \<and> (\<forall>i. (f o f' o r) i \<in> ball k e)"

  3597         proof (rule ccontr)

  3598           from K have "finite (K e)" "K e \<noteq> {}" "s \<subseteq> (\<Union>((\<lambda>x. ball x e)  (K e)))"

  3599             using s \<noteq> {}

  3600             by auto

  3601           moreover

  3602           assume "\<not> (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f \<circ> f' o r) i \<in> ball k e))"

  3603           hence "\<And>r k. k \<in> K e \<Longrightarrow> subseq r \<Longrightarrow> (\<exists>i. (f o f' o r) i \<notin> ball k e)" by simp

  3604           ultimately

  3605           show False using f'

  3606           proof (induct arbitrary: s f f' rule: finite_ne_induct)

  3607             case (singleton x)

  3608             have "\<exists>i. (f \<circ> f' o id) i \<notin> ball x e" by (rule singleton) (auto simp: subseq_def)

  3609             thus ?case using singleton by (auto simp: ball_def)

  3610           next

  3611             case (insert x A)

  3612             show ?case

  3613             proof cases

  3614               have inf_ms: "infinite ((f o f') - s)" using insert by (simp add: vimage_def)

  3615               have "infinite ((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A)))"

  3616                 using insert by (intro infinite_super[OF _ inf_ms]) auto

  3617               also have "((f o f') - \<Union>((\<lambda>x. ball x e)  (insert x A))) =

  3618                 {m. (f o f') m \<in> ball x e} \<union> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by auto

  3619               finally have "infinite \<dots>" .

  3620               moreover assume "finite {m. (f o f') m \<in> ball x e}"

  3621               ultimately have inf: "infinite {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}" by blast

  3622               hence "A \<noteq> {}" by auto then obtain k where "k \<in> A" by auto

  3623               def r \<equiv> "enumerate {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3624               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3625                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3626               hence "subseq r" by (simp add: subseq_def)

  3627               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> \<Union>((\<lambda>x. ball x e)  A)}"

  3628                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3629               show False

  3630               proof (rule insert)

  3631                 show "\<Union>(\<lambda>x. ball x e)  A \<subseteq> \<Union>(\<lambda>x. ball x e)  A" by simp

  3632                 fix k s assume "k \<in> A" "subseq s"

  3633                 thus "\<exists>i. (f o f' o r o s) i \<notin> ball k e" using subseq r

  3634                   by (subst (2) o_assoc[symmetric]) (intro insert(6) subseq_o, simp_all)

  3635               next

  3636                 fix n show "(f \<circ> f' o r) n \<in> \<Union>(\<lambda>x. ball x e)  A" using r_in_set by auto

  3637               qed

  3638             next

  3639               assume inf: "infinite {m. (f o f') m \<in> ball x e}"

  3640               def r \<equiv> "enumerate {m. (f o f') m \<in> ball x e}"

  3641               have r_mono: "\<And>n m. n < m \<Longrightarrow> r n < r m"

  3642                 using enumerate_mono[OF _ inf] by (simp add: r_def)

  3643               hence "subseq r" by (simp add: subseq_def)

  3644               from insert(6)[OF insertI1 this] obtain i where "(f o f') (r i) \<notin> ball x e" by auto

  3645               moreover

  3646               have r_in_set: "\<And>n. r n \<in> {m. (f o f') m \<in> ball x e}"

  3647                 using enumerate_in_set[OF inf] by (simp add: r_def)

  3648               hence "(f o f') (r i) \<in> ball x e" by simp

  3649               ultimately show False by simp

  3650             qed

  3651           qed

  3652         qed

  3653       }

  3654       hence ex: "\<forall>f'. \<forall>e > 0. (\<exists>k\<in>K e. \<exists>r. subseq r \<and> (\<forall>i. (f o f' \<circ> r) i \<in> ball k e))" by simp

  3655       let ?e = "\<lambda>n. 1 / real (Suc n)"

  3656       let ?P = "\<lambda>n s. \<exists>k\<in>K (?e n). (\<forall>i. (f o s) i \<in> ball k (?e n))"

  3657       interpret subseqs ?P using ex by unfold_locales force

  3658       from complete s have limI: "\<And>f. (\<And>n. f n \<in> s) \<Longrightarrow> Cauchy f \<Longrightarrow> (\<exists>l\<in>s. f ----> l)"

  3659         by (simp add: complete_def)

  3660       have "\<exists>l\<in>s. (f o diagseq) ----> l"

  3661       proof (intro limI metric_CauchyI)

  3662         fix e::real assume "0 < e" hence "0 < e / 2" by auto

  3663         from nat_approx_posE[OF this] guess n . note n = this

  3664         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) n) < e"

  3665         proof (rule exI[where x="Suc n"], safe)

  3666           fix m mm assume "Suc n \<le> m" "Suc n \<le> mm"

  3667           let ?e = "1 / real (Suc n)"

  3668           from reducer_reduces[of n] obtain k where

  3669             "k\<in>K ?e"  "\<And>i. (f o seqseq (Suc n)) i \<in> ball k ?e"

  3670             unfolding seqseq_reducer by auto

  3671           moreover

  3672           note diagseq_sub[OF Suc n \<le> m] diagseq_sub[OF Suc n \<le> mm]

  3673           ultimately have "{(f o diagseq) m, (f o diagseq) mm} \<subseteq> ball k ?e" by auto

  3674           also have "\<dots> \<subseteq> ball k (e / 2)" using n by (intro subset_ball) simp

  3675           finally

  3676           have "dist k ((f \<circ> diagseq) m) + dist k ((f \<circ> diagseq) mm) < e / 2 + e /2"

  3677             by (intro add_strict_mono) auto

  3678           hence "dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k < e"

  3679             by (simp add: dist_commute)

  3680           moreover have "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) \<le>

  3681             dist ((f \<circ> diagseq) m) k + dist ((f \<circ> diagseq) mm) k"

  3682             by (rule dist_triangle2)

  3683           ultimately show "dist ((f \<circ> diagseq) m) ((f \<circ> diagseq) mm) < e"

  3684             by simp

  3685         qed

  3686       next

  3687         fix n show "(f o diagseq) n \<in> s" using f by simp

  3688       qed

  3689       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l" using subseq_diagseq by auto

  3690     qed

  3691   qed

  3692 qed

  3693

  3694 lemma compact_cball[simp]:

  3695   fixes x :: "'a::heine_borel"

  3696   shows "compact(cball x e)"

  3697   using compact_eq_bounded_closed bounded_cball closed_cball

  3698   by blast

  3699

  3700 lemma compact_frontier_bounded[intro]:

  3701   fixes s :: "'a::heine_borel set"

  3702   shows "bounded s ==> compact(frontier s)"

  3703   unfolding frontier_def

  3704   using compact_eq_bounded_closed

  3705   by blast

  3706

  3707 lemma compact_frontier[intro]:

  3708   fixes s :: "'a::heine_borel set"

  3709   shows "compact s ==> compact (frontier s)"

  3710   using compact_eq_bounded_closed compact_frontier_bounded

  3711   by blast

  3712

  3713 lemma frontier_subset_compact:

  3714   fixes s :: "'a::heine_borel set"

  3715   shows "compact s ==> frontier s \<subseteq> s"

  3716   using frontier_subset_closed compact_eq_bounded_closed

  3717   by blast

  3718

  3719 subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

  3720

  3721 lemma bounded_closed_nest:

  3722   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"

  3723   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"

  3724   shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"

  3725 proof-

  3726   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto

  3727   from assms(4,1) have *:"seq_compact (s 0)" using bounded_closed_imp_seq_compact[of "s 0"] by auto

  3728

  3729   then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"

  3730     unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

  3731

  3732   { fix n::nat

  3733     { fix e::real assume "e>0"

  3734       with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto

  3735       hence "dist ((x \<circ> r) (max N n)) l < e" by auto

  3736       moreover

  3737       have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto

  3738       hence "(x \<circ> r) (max N n) \<in> s n"

  3739         using x apply(erule_tac x=n in allE)

  3740         using x apply(erule_tac x="r (max N n)" in allE)

  3741         using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto

  3742       ultimately have "\<exists>y\<in>s n. dist y l < e" by auto

  3743     }

  3744     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast

  3745   }

  3746   thus ?thesis by auto

  3747 qed

  3748

  3749 text {* Decreasing case does not even need compactness, just completeness. *}

  3750

  3751 lemma decreasing_closed_nest:

  3752   assumes "\<forall>n. closed(s n)"

  3753           "\<forall>n. (s n \<noteq> {})"

  3754           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3755           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"

  3756   shows "\<exists>a::'a::complete_space. \<forall>n::nat. a \<in> s n"

  3757 proof-

  3758   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto

  3759   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto

  3760   then obtain t where t: "\<forall>n. t n \<in> s n" by auto

  3761   { fix e::real assume "e>0"

  3762     then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e" using assms(4) by auto

  3763     { fix m n ::nat assume "N \<le> m \<and> N \<le> n"

  3764       hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+

  3765       hence "dist (t m) (t n) < e" using N by auto

  3766     }

  3767     hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto

  3768   }

  3769   hence  "Cauchy t" unfolding cauchy_def by auto

  3770   then obtain l where l:"(t ---> l) sequentially" using complete_univ unfolding complete_def by auto

  3771   { fix n::nat

  3772     { fix e::real assume "e>0"

  3773       then obtain N::nat where N:"\<forall>n\<ge>N. dist (t n) l < e" using l[unfolded LIMSEQ_def] by auto

  3774       have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto

  3775       hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto

  3776     }

  3777     hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto

  3778   }

  3779   then show ?thesis by auto

  3780 qed

  3781

  3782 text {* Strengthen it to the intersection actually being a singleton. *}

  3783

  3784 lemma decreasing_closed_nest_sing:

  3785   fixes s :: "nat \<Rightarrow> 'a::complete_space set"

  3786   assumes "\<forall>n. closed(s n)"

  3787           "\<forall>n. s n \<noteq> {}"

  3788           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"

  3789           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"

  3790   shows "\<exists>a. \<Inter>(range s) = {a}"

  3791 proof-

  3792   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto

  3793   { fix b assume b:"b \<in> \<Inter>(range s)"

  3794     { fix e::real assume "e>0"

  3795       hence "dist a b < e" using assms(4 )using b using a by blast

  3796     }

  3797     hence "dist a b = 0" by (metis dist_eq_0_iff dist_nz less_le)

  3798   }

  3799   with a have "\<Inter>(range s) = {a}" unfolding image_def by auto

  3800   thus ?thesis ..

  3801 qed

  3802

  3803 text{* Cauchy-type criteria for uniform convergence. *}

  3804

  3805 lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::heine_borel" shows

  3806  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>

  3807   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")

  3808 proof(rule)

  3809   assume ?lhs

  3810   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" by auto

  3811   { fix e::real assume "e>0"

  3812     then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto

  3813     { fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"

  3814       hence "dist (s m x) (s n x) < e"

  3815         using N[THEN spec[where x=m], THEN spec[where x=x]]

  3816         using N[THEN spec[where x=n], THEN spec[where x=x]]

  3817         using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }

  3818     hence "\<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  }

  3819   thus ?rhs by auto

  3820 next

  3821   assume ?rhs

  3822   hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto

  3823   then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]

  3824     using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto

  3825   { fix e::real assume "e>0"

  3826     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"

  3827       using ?rhs[THEN spec[where x="e/2"]] by auto

  3828     { fix x assume "P x"

  3829       then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"

  3830         using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])

  3831       fix n::nat assume "n\<ge>N"

  3832       hence "dist(s n x)(l x) < e"  using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]

  3833         using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }

  3834     hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }

  3835   thus ?lhs by auto

  3836 qed

  3837

  3838 lemma uniformly_cauchy_imp_uniformly_convergent:

  3839   fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::heine_borel"

  3840   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"

  3841           "\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"

  3842   shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"

  3843 proof-

  3844   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"

  3845     using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto

  3846   moreover

  3847   { fix x assume "P x"

  3848     hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]

  3849       using l and assms(2) unfolding LIMSEQ_def by blast  }

  3850   ultimately show ?thesis by auto

  3851 qed

  3852

  3853

  3854 subsection {* Continuity *}

  3855

  3856 text {* Define continuity over a net to take in restrictions of the set. *}

  3857

  3858 definition

  3859   continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3860   where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"

  3861

  3862 lemma continuous_trivial_limit:

  3863  "trivial_limit net ==> continuous net f"

  3864   unfolding continuous_def tendsto_def trivial_limit_eq by auto

  3865

  3866 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f(x)) (at x within s)"

  3867   unfolding continuous_def

  3868   unfolding tendsto_def

  3869   using netlimit_within[of x s]

  3870   by (cases "trivial_limit (at x within s)") (auto simp add: trivial_limit_eventually)

  3871

  3872 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> (f ---> f(x)) (at x)"

  3873   using continuous_within [of x UNIV f] by simp

  3874

  3875 lemma continuous_at_within:

  3876   assumes "continuous (at x) f"  shows "continuous (at x within s) f"

  3877   using assms unfolding continuous_at continuous_within

  3878   by (rule Lim_at_within)

  3879

  3880 text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

  3881

  3882 lemma continuous_within_eps_delta:

  3883   "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)"

  3884   unfolding continuous_within and Lim_within

  3885   apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

  3886

  3887 lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.

  3888                            \<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"

  3889   using continuous_within_eps_delta [of x UNIV f] by simp

  3890

  3891 text{* Versions in terms of open balls. *}

  3892

  3893 lemma continuous_within_ball:

  3894  "continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  3895                             f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3896 proof

  3897   assume ?lhs

  3898   { fix e::real assume "e>0"

  3899     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"

  3900       using ?lhs[unfolded continuous_within Lim_within] by auto

  3901     { fix y assume "y\<in>f  (ball x d \<inter> s)"

  3902       hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]

  3903         apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto

  3904     }

  3905     hence "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e" using d>0 unfolding subset_eq ball_def by (auto simp add: dist_commute)  }

  3906   thus ?rhs by auto

  3907 next

  3908   assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq

  3909     apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto

  3910 qed

  3911

  3912 lemma continuous_at_ball:

  3913   "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")

  3914 proof

  3915   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3916     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)

  3917     unfolding dist_nz[THEN sym] by auto

  3918 next

  3919   assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball

  3920     apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)

  3921 qed

  3922

  3923 text{* Define setwise continuity in terms of limits within the set. *}

  3924

  3925 definition

  3926   continuous_on ::

  3927     "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"

  3928 where

  3929   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"

  3930

  3931 lemma continuous_on_topological:

  3932   "continuous_on s f \<longleftrightarrow>

  3933     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  3934       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  3935 unfolding continuous_on_def tendsto_def

  3936 unfolding Limits.eventually_within eventually_at_topological

  3937 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  3938

  3939 lemma continuous_on_iff:

  3940   "continuous_on s f \<longleftrightarrow>

  3941     (\<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)"

  3942 unfolding continuous_on_def Lim_within

  3943 apply (intro ball_cong [OF refl] all_cong ex_cong)

  3944 apply (rename_tac y, case_tac "y = x", simp)

  3945 apply (simp add: dist_nz)

  3946 done

  3947

  3948 definition

  3949   uniformly_continuous_on ::

  3950     "'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"

  3951 where

  3952   "uniformly_continuous_on s f \<longleftrightarrow>

  3953     (\<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)"

  3954

  3955 text{* Some simple consequential lemmas. *}

  3956

  3957 lemma uniformly_continuous_imp_continuous:

  3958  " uniformly_continuous_on s f ==> continuous_on s f"

  3959   unfolding uniformly_continuous_on_def continuous_on_iff by blast

  3960

  3961 lemma continuous_at_imp_continuous_within:

  3962  "continuous (at x) f ==> continuous (at x within s) f"

  3963   unfolding continuous_within continuous_at using Lim_at_within by auto

  3964

  3965 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"

  3966 unfolding tendsto_def by (simp add: trivial_limit_eq)

  3967

  3968 lemma continuous_at_imp_continuous_on:

  3969   assumes "\<forall>x\<in>s. continuous (at x) f"

  3970   shows "continuous_on s f"

  3971 unfolding continuous_on_def

  3972 proof

  3973   fix x assume "x \<in> s"

  3974   with assms have *: "(f ---> f (netlimit (at x))) (at x)"

  3975     unfolding continuous_def by simp

  3976   have "(f ---> f x) (at x)"

  3977   proof (cases "trivial_limit (at x)")

  3978     case True thus ?thesis

  3979       by (rule Lim_trivial_limit)

  3980   next

  3981     case False

  3982     hence 1: "netlimit (at x) = x"

  3983       using netlimit_within [of x UNIV] by simp

  3984     with * show ?thesis by simp

  3985   qed

  3986   thus "(f ---> f x) (at x within s)"

  3987     by (rule Lim_at_within)

  3988 qed

  3989

  3990 lemma continuous_on_eq_continuous_within:

  3991   "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x within s) f)"

  3992 unfolding continuous_on_def continuous_def

  3993 apply (rule ball_cong [OF refl])

  3994 apply (case_tac "trivial_limit (at x within s)")

  3995 apply (simp add: Lim_trivial_limit)

  3996 apply (simp add: netlimit_within)

  3997 done

  3998

  3999 lemmas continuous_on = continuous_on_def -- "legacy theorem name"

  4000

  4001 lemma continuous_on_eq_continuous_at:

  4002   shows "open s ==> (continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. continuous (at x) f))"

  4003   by (auto simp add: continuous_on continuous_at Lim_within_open)

  4004

  4005 lemma continuous_within_subset:

  4006  "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s

  4007              ==> continuous (at x within t) f"

  4008   unfolding continuous_within by(metis Lim_within_subset)

  4009

  4010 lemma continuous_on_subset:

  4011   shows "continuous_on s f \<Longrightarrow> t \<subseteq> s ==> continuous_on t f"

  4012   unfolding continuous_on by (metis subset_eq Lim_within_subset)

  4013

  4014 lemma continuous_on_interior:

  4015   shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"

  4016   by (erule interiorE, drule (1) continuous_on_subset,

  4017     simp add: continuous_on_eq_continuous_at)

  4018

  4019 lemma continuous_on_eq:

  4020   "(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"

  4021   unfolding continuous_on_def tendsto_def Limits.eventually_within

  4022   by simp

  4023

  4024 text {* Characterization of various kinds of continuity in terms of sequences. *}

  4025

  4026 lemma continuous_within_sequentially:

  4027   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4028   shows "continuous (at a within s) f \<longleftrightarrow>

  4029                 (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially

  4030                      --> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")

  4031 proof

  4032   assume ?lhs

  4033   { fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"

  4034     fix T::"'b set" assume "open T" and "f a \<in> T"

  4035     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"

  4036       unfolding continuous_within tendsto_def eventually_within by auto

  4037     have "eventually (\<lambda>n. dist (x n) a < d) sequentially"

  4038       using x(2) d>0 by simp

  4039     hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"

  4040     proof eventually_elim

  4041       case (elim n) thus ?case

  4042         using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto

  4043     qed

  4044   }

  4045   thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp

  4046 next

  4047   assume ?rhs thus ?lhs

  4048     unfolding continuous_within tendsto_def [where l="f a"]

  4049     by (simp add: sequentially_imp_eventually_within)

  4050 qed

  4051

  4052 lemma continuous_at_sequentially:

  4053   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4054   shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially

  4055                   --> ((f o x) ---> f a) sequentially)"

  4056   using continuous_within_sequentially[of a UNIV f] by simp

  4057

  4058 lemma continuous_on_sequentially:

  4059   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4060   shows "continuous_on s f \<longleftrightarrow>

  4061     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially

  4062                     --> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")

  4063 proof

  4064   assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto

  4065 next

  4066   assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto

  4067 qed

  4068

  4069 lemma uniformly_continuous_on_sequentially:

  4070   "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>

  4071                     ((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially

  4072                     \<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")

  4073 proof

  4074   assume ?lhs

  4075   { fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"

  4076     { fix e::real assume "e>0"

  4077       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"

  4078         using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto

  4079       obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto

  4080       { fix n assume "n\<ge>N"

  4081         hence "dist (f (x n)) (f (y n)) < e"

  4082           using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y

  4083           unfolding dist_commute by simp  }

  4084       hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }

  4085     hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }

  4086   thus ?rhs by auto

  4087 next

  4088   assume ?rhs

  4089   { assume "\<not> ?lhs"

  4090     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" unfolding uniformly_continuous_on_def by auto

  4091     then obtain fa where fa:"\<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"

  4092       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"] unfolding Bex_def

  4093       by (auto simp add: dist_commute)

  4094     def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"

  4095     def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"

  4096     have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"

  4097       unfolding x_def and y_def using fa by auto

  4098     { fix e::real assume "e>0"

  4099       then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto

  4100       { fix n::nat assume "n\<ge>N"

  4101         hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto

  4102         also have "\<dots> < e" using N by auto

  4103         finally have "inverse (real n + 1) < e" by auto

  4104         hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }

  4105       hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }

  4106     hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto

  4107     hence False using fxy and e>0 by auto  }

  4108   thus ?lhs unfolding uniformly_continuous_on_def by blast

  4109 qed

  4110

  4111 text{* The usual transformation theorems. *}

  4112

  4113 lemma continuous_transform_within:

  4114   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4115   assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"

  4116           "continuous (at x within s) f"

  4117   shows "continuous (at x within s) g"

  4118 unfolding continuous_within

  4119 proof (rule Lim_transform_within)

  4120   show "0 < d" by fact

  4121   show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

  4122     using assms(3) by auto

  4123   have "f x = g x"

  4124     using assms(1,2,3) by auto

  4125   thus "(f ---> g x) (at x within s)"

  4126     using assms(4) unfolding continuous_within by simp

  4127 qed

  4128

  4129 lemma continuous_transform_at:

  4130   fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"

  4131   assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"

  4132           "continuous (at x) f"

  4133   shows "continuous (at x) g"

  4134   using continuous_transform_within [of d x UNIV f g] assms by simp

  4135

  4136 subsubsection {* Structural rules for pointwise continuity *}

  4137

  4138 lemma continuous_within_id: "continuous (at a within s) (\<lambda>x. x)"

  4139   unfolding continuous_within by (rule tendsto_ident_at_within)

  4140

  4141 lemma continuous_at_id: "continuous (at a) (\<lambda>x. x)"

  4142   unfolding continuous_at by (rule tendsto_ident_at)

  4143

  4144 lemma continuous_const: "continuous F (\<lambda>x. c)"

  4145   unfolding continuous_def by (rule tendsto_const)

  4146

  4147 lemma continuous_dist:

  4148   assumes "continuous F f" and "continuous F g"

  4149   shows "continuous F (\<lambda>x. dist (f x) (g x))"

  4150   using assms unfolding continuous_def by (rule tendsto_dist)

  4151

  4152 lemma continuous_infdist:

  4153   assumes "continuous F f"

  4154   shows "continuous F (\<lambda>x. infdist (f x) A)"

  4155   using assms unfolding continuous_def by (rule tendsto_infdist)

  4156

  4157 lemma continuous_norm:

  4158   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"

  4159   unfolding continuous_def by (rule tendsto_norm)

  4160

  4161 lemma continuous_infnorm:

  4162   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"

  4163   unfolding continuous_def by (rule tendsto_infnorm)

  4164

  4165 lemma continuous_add:

  4166   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  4167   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"

  4168   unfolding continuous_def by (rule tendsto_add)

  4169

  4170 lemma continuous_minus:

  4171   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  4172   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"

  4173   unfolding continuous_def by (rule tendsto_minus)

  4174

  4175 lemma continuous_diff:

  4176   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  4177   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"

  4178   unfolding continuous_def by (rule tendsto_diff)

  4179

  4180 lemma continuous_scaleR:

  4181   fixes g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

  4182   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x *\<^sub>R g x)"

  4183   unfolding continuous_def by (rule tendsto_scaleR)

  4184

  4185 lemma continuous_mult:

  4186   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"

  4187   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x * g x)"

  4188   unfolding continuous_def by (rule tendsto_mult)

  4189

  4190 lemma continuous_inner:

  4191   assumes "continuous F f" and "continuous F g"

  4192   shows "continuous F (\<lambda>x. inner (f x) (g x))"

  4193   using assms unfolding continuous_def by (rule tendsto_inner)

  4194

  4195 lemma continuous_inverse:

  4196   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  4197   assumes "continuous F f" and "f (netlimit F) \<noteq> 0"

  4198   shows "continuous F (\<lambda>x. inverse (f x))"

  4199   using assms unfolding continuous_def by (rule tendsto_inverse)

  4200

  4201 lemma continuous_at_within_inverse:

  4202   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  4203   assumes "continuous (at a within s) f" and "f a \<noteq> 0"

  4204   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"

  4205   using assms unfolding continuous_within by (rule tendsto_inverse)

  4206

  4207 lemma continuous_at_inverse:

  4208   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

  4209   assumes "continuous (at a) f" and "f a \<noteq> 0"

  4210   shows "continuous (at a) (\<lambda>x. inverse (f x))"

  4211   using assms unfolding continuous_at by (rule tendsto_inverse)

  4212

  4213 lemmas continuous_intros = continuous_at_id continuous_within_id

  4214   continuous_const continuous_dist continuous_norm continuous_infnorm

  4215   continuous_add continuous_minus continuous_diff continuous_scaleR continuous_mult

  4216   continuous_inner continuous_at_inverse continuous_at_within_inverse

  4217

  4218 subsubsection {* Structural rules for setwise continuity *}

  4219

  4220 lemma continuous_on_id: "continuous_on s (\<lambda>x. x)"

  4221   unfolding continuous_on_def by (fast intro: tendsto_ident_at_within)

  4222

  4223 lemma continuous_on_const: "continuous_on s (\<lambda>x. c)"

  4224   unfolding continuous_on_def by (auto intro: tendsto_intros)

  4225

  4226 lemma continuous_on_norm:

  4227   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"

  4228   unfolding continuous_on_def by (fast intro: tendsto_norm)

  4229

  4230 lemma continuous_on_infnorm:

  4231   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"

  4232   unfolding continuous_on by (fast intro: tendsto_infnorm)

  4233

  4234 lemma continuous_on_minus:

  4235   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  4236   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"

  4237   unfolding continuous_on_def by (auto intro: tendsto_intros)

  4238

  4239 lemma continuous_on_add:

  4240   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  4241   shows "continuous_on s f \<Longrightarrow> continuous_on s g

  4242            \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"

  4243   unfolding continuous_on_def by (auto intro: tendsto_intros)

  4244

  4245 lemma continuous_on_diff:

  4246   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  4247   shows "continuous_on s f \<Longrightarrow> continuous_on s g

  4248            \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"

  4249   unfolding continuous_on_def by (auto intro: tendsto_intros)

  4250

  4251 lemma (in bounded_linear) continuous_on:

  4252   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"

  4253   unfolding continuous_on_def by (fast intro: tendsto)

  4254

  4255 lemma (in bounded_bilinear) continuous_on:

  4256   "\<lbrakk>continuous_on s f; continuous_on s g\<rbrakk> \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"

  4257   unfolding continuous_on_def by (fast intro: tendsto)

  4258

  4259 lemma continuous_on_scaleR:

  4260   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

  4261   assumes "continuous_on s f" and "continuous_on s g"

  4262   shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"

  4263   using bounded_bilinear_scaleR assms

  4264   by (rule bounded_bilinear.continuous_on)

  4265

  4266 lemma continuous_on_mult:

  4267   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"

  4268   assumes "continuous_on s f" and "continuous_on s g"

  4269   shows "continuous_on s (\<lambda>x. f x * g x)"

  4270   using bounded_bilinear_mult assms

  4271   by (rule bounded_bilinear.continuous_on)

  4272

  4273 lemma continuous_on_inner:

  4274   fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"

  4275   assumes "continuous_on s f" and "continuous_on s g"

  4276   shows "continuous_on s (\<lambda>x. inner (f x) (g x))"

  4277   using bounded_bilinear_inner assms

  4278   by (rule bounded_bilinear.continuous_on)

  4279

  4280 lemma continuous_on_inverse:

  4281   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"

  4282   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

  4283   shows "continuous_on s (\<lambda>x. inverse (f x))"

  4284   using assms unfolding continuous_on by (fast intro: tendsto_inverse)

  4285

  4286 subsubsection {* Structural rules for uniform continuity *}

  4287

  4288 lemma uniformly_continuous_on_id:

  4289   shows "uniformly_continuous_on s (\<lambda>x. x)"

  4290   unfolding uniformly_continuous_on_def by auto

  4291

  4292 lemma uniformly_continuous_on_const:

  4293   shows "uniformly_continuous_on s (\<lambda>x. c)"

  4294   unfolding uniformly_continuous_on_def by simp

  4295

  4296 lemma uniformly_continuous_on_dist:

  4297   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"

  4298   assumes "uniformly_continuous_on s f"

  4299   assumes "uniformly_continuous_on s g"

  4300   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"

  4301 proof -

  4302   { fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"

  4303       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]

  4304       using dist_triangle3 [of c d a] dist_triangle [of a d b]

  4305       by arith

  4306   } note le = this

  4307   { fix x y

  4308     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"

  4309     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"

  4310     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"

  4311       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],

  4312         simp add: le)

  4313   }

  4314   thus ?thesis using assms unfolding uniformly_continuous_on_sequentially

  4315     unfolding dist_real_def by simp

  4316 qed

  4317

  4318 lemma uniformly_continuous_on_norm:

  4319   assumes "uniformly_continuous_on s f"

  4320   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"

  4321   unfolding norm_conv_dist using assms

  4322   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

  4323

  4324 lemma (in bounded_linear) uniformly_continuous_on:

  4325   assumes "uniformly_continuous_on s g"

  4326   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"

  4327   using assms unfolding uniformly_continuous_on_sequentially

  4328   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]

  4329   by (auto intro: tendsto_zero)

  4330

  4331 lemma uniformly_continuous_on_cmul:

  4332   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4333   assumes "uniformly_continuous_on s f"

  4334   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"

  4335   using bounded_linear_scaleR_right assms

  4336   by (rule bounded_linear.uniformly_continuous_on)

  4337

  4338 lemma dist_minus:

  4339   fixes x y :: "'a::real_normed_vector"

  4340   shows "dist (- x) (- y) = dist x y"

  4341   unfolding dist_norm minus_diff_minus norm_minus_cancel ..

  4342

  4343 lemma uniformly_continuous_on_minus:

  4344   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4345   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"

  4346   unfolding uniformly_continuous_on_def dist_minus .

  4347

  4348 lemma uniformly_continuous_on_add:

  4349   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4350   assumes "uniformly_continuous_on s f"

  4351   assumes "uniformly_continuous_on s g"

  4352   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"

  4353   using assms unfolding uniformly_continuous_on_sequentially

  4354   unfolding dist_norm tendsto_norm_zero_iff add_diff_add

  4355   by (auto intro: tendsto_add_zero)

  4356

  4357 lemma uniformly_continuous_on_diff:

  4358   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4359   assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"

  4360   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"

  4361   unfolding ab_diff_minus using assms

  4362   by (intro uniformly_continuous_on_add uniformly_continuous_on_minus)

  4363

  4364 text{* Continuity of all kinds is preserved under composition. *}

  4365

  4366 lemma continuous_within_topological:

  4367   "continuous (at x within s) f \<longleftrightarrow>

  4368     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow>

  4369       (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"

  4370 unfolding continuous_within

  4371 unfolding tendsto_def Limits.eventually_within eventually_at_topological

  4372 by (intro ball_cong [OF refl] all_cong imp_cong ex_cong conj_cong refl) auto

  4373

  4374 lemma continuous_within_compose:

  4375   assumes "continuous (at x within s) f"

  4376   assumes "continuous (at (f x) within f  s) g"

  4377   shows "continuous (at x within s) (g o f)"

  4378 using assms unfolding continuous_within_topological by simp metis

  4379

  4380 lemma continuous_at_compose:

  4381   assumes "continuous (at x) f" and "continuous (at (f x)) g"

  4382   shows "continuous (at x) (g o f)"

  4383 proof-

  4384   have "continuous (at (f x) within range f) g" using assms(2)

  4385     using continuous_within_subset[of "f x" UNIV g "range f"] by simp

  4386   thus ?thesis using assms(1)

  4387     using continuous_within_compose[of x UNIV f g] by simp

  4388 qed

  4389

  4390 lemma continuous_on_compose:

  4391   "continuous_on s f \<Longrightarrow> continuous_on (f  s) g \<Longrightarrow> continuous_on s (g o f)"

  4392   unfolding continuous_on_topological by simp metis

  4393

  4394 lemma uniformly_continuous_on_compose:

  4395   assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"

  4396   shows "uniformly_continuous_on s (g o f)"

  4397 proof-

  4398   { fix e::real assume "e>0"

  4399     then obtain d where "d>0" and d:"\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto

  4400     obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using d>0 using assms(1) unfolding uniformly_continuous_on_def by auto

  4401     hence "\<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" using d>0 using d by auto  }

  4402   thus ?thesis using assms unfolding uniformly_continuous_on_def by auto

  4403 qed

  4404

  4405 lemmas continuous_on_intros = continuous_on_id continuous_on_const

  4406   continuous_on_compose continuous_on_norm continuous_on_infnorm

  4407   continuous_on_add continuous_on_minus continuous_on_diff

  4408   continuous_on_scaleR continuous_on_mult continuous_on_inverse

  4409   continuous_on_inner

  4410   uniformly_continuous_on_id uniformly_continuous_on_const

  4411   uniformly_continuous_on_dist uniformly_continuous_on_norm

  4412   uniformly_continuous_on_compose uniformly_continuous_on_add

  4413   uniformly_continuous_on_minus uniformly_continuous_on_diff

  4414   uniformly_continuous_on_cmul

  4415

  4416 text{* Continuity in terms of open preimages. *}

  4417

  4418 lemma continuous_at_open:

  4419   shows "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)))"

  4420 unfolding continuous_within_topological [of x UNIV f, unfolded within_UNIV]

  4421 unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

  4422

  4423 lemma continuous_on_open:

  4424   shows "continuous_on s f \<longleftrightarrow>

  4425         (\<forall>t. openin (subtopology euclidean (f  s)) t

  4426             --> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4427 proof (safe)

  4428   fix t :: "'b set"

  4429   assume 1: "continuous_on s f"

  4430   assume 2: "openin (subtopology euclidean (f  s)) t"

  4431   from 2 obtain B where B: "open B" and t: "t = f  s \<inter> B"

  4432     unfolding openin_open by auto

  4433   def U == "\<Union>{A. open A \<and> (\<forall>x\<in>s. x \<in> A \<longrightarrow> f x \<in> B)}"

  4434   have "open U" unfolding U_def by (simp add: open_Union)

  4435   moreover have "\<forall>x\<in>s. x \<in> U \<longleftrightarrow> f x \<in> t"

  4436   proof (intro ballI iffI)

  4437     fix x assume "x \<in> s" and "x \<in> U" thus "f x \<in> t"

  4438       unfolding U_def t by auto

  4439   next

  4440     fix x assume "x \<in> s" and "f x \<in> t"

  4441     hence "x \<in> s" and "f x \<in> B"

  4442       unfolding t by auto

  4443     with 1 B obtain A where "open A" "x \<in> A" "\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B"

  4444       unfolding t continuous_on_topological by metis

  4445     then show "x \<in> U"

  4446       unfolding U_def by auto

  4447   qed

  4448   ultimately have "open U \<and> {x \<in> s. f x \<in> t} = s \<inter> U" by auto

  4449   then show "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4450     unfolding openin_open by fast

  4451 next

  4452   assume "?rhs" show "continuous_on s f"

  4453   unfolding continuous_on_topological

  4454   proof (clarify)

  4455     fix x and B assume "x \<in> s" and "open B" and "f x \<in> B"

  4456     have "openin (subtopology euclidean (f  s)) (f  s \<inter> B)"

  4457       unfolding openin_open using open B by auto

  4458     then have "openin (subtopology euclidean s) {x \<in> s. f x \<in> f  s \<inter> B}"

  4459       using ?rhs by fast

  4460     then show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"

  4461       unfolding openin_open using x \<in> s and f x \<in> B by auto

  4462   qed

  4463 qed

  4464

  4465 text {* Similarly in terms of closed sets. *}

  4466

  4467 lemma continuous_on_closed:

  4468   shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f  s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")

  4469 proof

  4470   assume ?lhs

  4471   { fix t

  4472     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4473     have **:"f  s - (f  s - (f  s - t)) = f  s - t" by auto

  4474     assume as:"closedin (subtopology euclidean (f  s)) t"

  4475     hence "closedin (subtopology euclidean (f  s)) (f  s - (f  s - t))" unfolding closedin_def topspace_euclidean_subtopology unfolding ** by auto

  4476     hence "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?lhs[unfolded continuous_on_open, THEN spec[where x="(f  s) - t"]]

  4477       unfolding openin_closedin_eq topspace_euclidean_subtopology unfolding * by auto  }

  4478   thus ?rhs by auto

  4479 next

  4480   assume ?rhs

  4481   { fix t

  4482     have *:"s - {x \<in> s. f x \<in> f  s - t} = {x \<in> s. f x \<in> t}" by auto

  4483     assume as:"openin (subtopology euclidean (f  s)) t"

  4484     hence "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}" using ?rhs[THEN spec[where x="(f  s) - t"]]

  4485       unfolding openin_closedin_eq topspace_euclidean_subtopology *[THEN sym] closedin_subtopology by auto }

  4486   thus ?lhs unfolding continuous_on_open by auto

  4487 qed

  4488

  4489 text {* Half-global and completely global cases. *}

  4490

  4491 lemma continuous_open_in_preimage:

  4492   assumes "continuous_on s f"  "open t"

  4493   shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4494 proof-

  4495   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto

  4496   have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4497     using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto

  4498   thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4499 qed

  4500

  4501 lemma continuous_closed_in_preimage:

  4502   assumes "continuous_on s f"  "closed t"

  4503   shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"

  4504 proof-

  4505   have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto

  4506   have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"

  4507     using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto

  4508   thus ?thesis

  4509     using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto

  4510 qed

  4511

  4512 lemma continuous_open_preimage:

  4513   assumes "continuous_on s f" "open s" "open t"

  4514   shows "open {x \<in> s. f x \<in> t}"

  4515 proof-

  4516   obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4517     using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto

  4518   thus ?thesis using open_Int[of s T, OF assms(2)] by auto

  4519 qed

  4520

  4521 lemma continuous_closed_preimage:

  4522   assumes "continuous_on s f" "closed s" "closed t"

  4523   shows "closed {x \<in> s. f x \<in> t}"

  4524 proof-

  4525   obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"

  4526     using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto

  4527   thus ?thesis using closed_Int[of s T, OF assms(2)] by auto

  4528 qed

  4529

  4530 lemma continuous_open_preimage_univ:

  4531   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"

  4532   using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

  4533

  4534 lemma continuous_closed_preimage_univ:

  4535   shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"

  4536   using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

  4537

  4538 lemma continuous_open_vimage:

  4539   shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"

  4540   unfolding vimage_def by (rule continuous_open_preimage_univ)

  4541

  4542 lemma continuous_closed_vimage:

  4543   shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"

  4544   unfolding vimage_def by (rule continuous_closed_preimage_univ)

  4545

  4546 lemma interior_image_subset:

  4547   assumes "\<forall>x. continuous (at x) f" "inj f"

  4548   shows "interior (f  s) \<subseteq> f  (interior s)"

  4549 proof

  4550   fix x assume "x \<in> interior (f  s)"

  4551   then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..

  4552   hence "x \<in> f  s" by auto

  4553   then obtain y where y: "y \<in> s" "x = f y" by auto

  4554   have "open (vimage f T)"

  4555     using assms(1) open T by (rule continuous_open_vimage)

  4556   moreover have "y \<in> vimage f T"

  4557     using x = f y x \<in> T by simp

  4558   moreover have "vimage f T \<subseteq> s"

  4559     using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto

  4560   ultimately have "y \<in> interior s" ..

  4561   with x = f y show "x \<in> f  interior s" ..

  4562 qed

  4563

  4564 text {* Equality of continuous functions on closure and related results. *}

  4565

  4566 lemma continuous_closed_in_preimage_constant:

  4567   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4568   shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"

  4569   using continuous_closed_in_preimage[of s f "{a}"] by auto

  4570

  4571 lemma continuous_closed_preimage_constant:

  4572   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4573   shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"

  4574   using continuous_closed_preimage[of s f "{a}"] by auto

  4575

  4576 lemma continuous_constant_on_closure:

  4577   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4578   assumes "continuous_on (closure s) f"

  4579           "\<forall>x \<in> s. f x = a"

  4580   shows "\<forall>x \<in> (closure s). f x = a"

  4581     using continuous_closed_preimage_constant[of "closure s" f a]

  4582     assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

  4583

  4584 lemma image_closure_subset:

  4585   assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"

  4586   shows "f  (closure s) \<subseteq> t"

  4587 proof-

  4588   have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto

  4589   moreover have "closed {x \<in> closure s. f x \<in> t}"

  4590     using continuous_closed_preimage[OF assms(1)] and assms(2) by auto

  4591   ultimately have "closure s = {x \<in> closure s . f x \<in> t}"

  4592     using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto

  4593   thus ?thesis by auto

  4594 qed

  4595

  4596 lemma continuous_on_closure_norm_le:

  4597   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

  4598   assumes "continuous_on (closure s) f"  "\<forall>y \<in> s. norm(f y) \<le> b"  "x \<in> (closure s)"

  4599   shows "norm(f x) \<le> b"

  4600 proof-

  4601   have *:"f  s \<subseteq> cball 0 b" using assms(2)[unfolded mem_cball_0[THEN sym]] by auto

  4602   show ?thesis

  4603     using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)

  4604     unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)

  4605 qed

  4606

  4607 text {* Making a continuous function avoid some value in a neighbourhood. *}

  4608

  4609 lemma continuous_within_avoid:

  4610   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4611   assumes "continuous (at x within s) f" and "f x \<noteq> a"

  4612   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"

  4613 proof-

  4614   obtain U where "open U" and "f x \<in> U" and "a \<notin> U"

  4615     using t1_space [OF f x \<noteq> a] by fast

  4616   have "(f ---> f x) (at x within s)"

  4617     using assms(1) by (simp add: continuous_within)

  4618   hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"

  4619     using open U and f x \<in> U

  4620     unfolding tendsto_def by fast

  4621   hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"

  4622     using a \<notin> U by (fast elim: eventually_mono [rotated])

  4623   thus ?thesis

  4624     unfolding Limits.eventually_within Limits.eventually_at

  4625     by (rule ex_forward, cut_tac f x \<noteq> a, auto simp: dist_commute)

  4626 qed

  4627

  4628 lemma continuous_at_avoid:

  4629   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4630   assumes "continuous (at x) f" and "f x \<noteq> a"

  4631   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4632   using assms continuous_within_avoid[of x UNIV f a] by simp

  4633

  4634 lemma continuous_on_avoid:

  4635   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4636   assumes "continuous_on s f"  "x \<in> s"  "f x \<noteq> a"

  4637   shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"

  4638 using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x], OF assms(2)]  continuous_within_avoid[of x s f a]  assms(3) by auto

  4639

  4640 lemma continuous_on_open_avoid:

  4641   fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"

  4642   assumes "continuous_on s f"  "open s"  "x \<in> s"  "f x \<noteq> a"

  4643   shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"

  4644 using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]  continuous_at_avoid[of x f a]  assms(4) by auto

  4645

  4646 text {* Proving a function is constant by proving open-ness of level set. *}

  4647

  4648 lemma continuous_levelset_open_in_cases:

  4649   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4650   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4651         openin (subtopology euclidean s) {x \<in> s. f x = a}

  4652         ==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"

  4653 unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

  4654

  4655 lemma continuous_levelset_open_in:

  4656   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4657   shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>

  4658         openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>

  4659         (\<exists>x \<in> s. f x = a)  ==> (\<forall>x \<in> s. f x = a)"

  4660 using continuous_levelset_open_in_cases[of s f ]

  4661 by meson

  4662

  4663 lemma continuous_levelset_open:

  4664   fixes f :: "_ \<Rightarrow> 'b::t1_space"

  4665   assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"

  4666   shows "\<forall>x \<in> s. f x = a"

  4667 using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

  4668

  4669 text {* Some arithmetical combinations (more to prove). *}

  4670

  4671 lemma open_scaling[intro]:

  4672   fixes s :: "'a::real_normed_vector set"

  4673   assumes "c \<noteq> 0"  "open s"

  4674   shows "open((\<lambda>x. c *\<^sub>R x)  s)"

  4675 proof-

  4676   { fix x assume "x \<in> s"

  4677     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]] by auto

  4678     have "e * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto

  4679     moreover

  4680     { fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"

  4681       hence "norm ((1 / c) *\<^sub>R y - x) < e" unfolding dist_norm

  4682         using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)

  4683           assms(1)[unfolded zero_less_abs_iff[THEN sym]] by (simp del:zero_less_abs_iff)

  4684       hence "y \<in> op *\<^sub>R c  s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }

  4685     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s" apply(rule_tac x="e * abs c" in exI) by auto  }

  4686   thus ?thesis unfolding open_dist by auto

  4687 qed

  4688

  4689 lemma minus_image_eq_vimage:

  4690   fixes A :: "'a::ab_group_add set"

  4691   shows "(\<lambda>x. - x)  A = (\<lambda>x. - x) - A"

  4692   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])

  4693

  4694 lemma open_negations:

  4695   fixes s :: "'a::real_normed_vector set"

  4696   shows "open s ==> open ((\<lambda> x. -x)  s)"

  4697   unfolding scaleR_minus1_left [symmetric]

  4698   by (rule open_scaling, auto)

  4699

  4700 lemma open_translation:

  4701   fixes s :: "'a::real_normed_vector set"

  4702   assumes "open s"  shows "open((\<lambda>x. a + x)  s)"

  4703 proof-

  4704   { fix x have "continuous (at x) (\<lambda>x. x - a)"

  4705       by (intro continuous_diff continuous_at_id continuous_const) }

  4706   moreover have "{x. x - a \<in> s} = op + a  s" by force

  4707   ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto

  4708 qed

  4709

  4710 lemma open_affinity:

  4711   fixes s :: "'a::real_normed_vector set"

  4712   assumes "open s"  "c \<noteq> 0"

  4713   shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"

  4714 proof-

  4715   have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..

  4716   have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto

  4717   thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto

  4718 qed

  4719

  4720 lemma interior_translation:

  4721   fixes s :: "'a::real_normed_vector set"

  4722   shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"

  4723 proof (rule set_eqI, rule)

  4724   fix x assume "x \<in> interior (op + a  s)"

  4725   then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto

  4726   hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto

  4727   thus "x \<in> op + a  interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using e > 0 by auto

  4728 next

  4729   fix x assume "x \<in> op + a  interior s"

  4730   then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto

  4731   { fix z have *:"a + y - z = y + a - z" by auto

  4732     assume "z\<in>ball x e"

  4733     hence "z - a \<in> s" using e[unfolded subset_eq, THEN bspec[where x="z - a"]] unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 * by auto

  4734     hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }

  4735   hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto

  4736   thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto

  4737 qed

  4738

  4739 text {* Topological properties of linear functions. *}

  4740

  4741 lemma linear_lim_0:

  4742   assumes "bounded_linear f" shows "(f ---> 0) (at (0))"

  4743 proof-

  4744   interpret f: bounded_linear f by fact

  4745   have "(f ---> f 0) (at 0)"

  4746     using tendsto_ident_at by (rule f.tendsto)

  4747   thus ?thesis unfolding f.zero .

  4748 qed

  4749

  4750 lemma linear_continuous_at:

  4751   assumes "bounded_linear f"  shows "continuous (at a) f"

  4752   unfolding continuous_at using assms

  4753   apply (rule bounded_linear.tendsto)

  4754   apply (rule tendsto_ident_at)

  4755   done

  4756

  4757 lemma linear_continuous_within:

  4758   shows "bounded_linear f ==> continuous (at x within s) f"

  4759   using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

  4760

  4761 lemma linear_continuous_on:

  4762   shows "bounded_linear f ==> continuous_on s f"

  4763   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

  4764

  4765 text {* Also bilinear functions, in composition form. *}

  4766

  4767 lemma bilinear_continuous_at_compose:

  4768   shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h

  4769         ==> continuous (at x) (\<lambda>x. h (f x) (g x))"

  4770   unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

  4771

  4772 lemma bilinear_continuous_within_compose:

  4773   shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h

  4774         ==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"

  4775   unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

  4776

  4777 lemma bilinear_continuous_on_compose:

  4778   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h

  4779              ==> continuous_on s (\<lambda>x. h (f x) (g x))"

  4780   unfolding continuous_on_def

  4781   by (fast elim: bounded_bilinear.tendsto)

  4782

  4783 text {* Preservation of compactness and connectedness under continuous function. *}

  4784

  4785 lemma compact_eq_openin_cover:

  4786   "compact S \<longleftrightarrow>

  4787     (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4788       (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"

  4789 proof safe

  4790   fix C

  4791   assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"

  4792   hence "\<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}"

  4793     unfolding openin_open by force+

  4794   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"

  4795     by (rule compactE)

  4796   hence "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)"

  4797     by auto

  4798   thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4799 next

  4800   assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>

  4801         (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"

  4802   show "compact S"

  4803   proof (rule compactI)

  4804     fix C

  4805     let ?C = "image (\<lambda>T. S \<inter> T) C"

  4806     assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"

  4807     hence "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"

  4808       unfolding openin_open by auto

  4809     with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"

  4810       by metis

  4811     let ?D = "inv_into C (\<lambda>T. S \<inter> T)  D"

  4812     have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"

  4813     proof (intro conjI)

  4814       from D \<subseteq> ?C show "?D \<subseteq> C"

  4815         by (fast intro: inv_into_into)

  4816       from finite D show "finite ?D"

  4817         by (rule finite_imageI)

  4818       from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"

  4819         apply (rule subset_trans)

  4820         apply clarsimp

  4821         apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])

  4822         apply (erule rev_bexI, fast)

  4823         done

  4824     qed

  4825     thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..

  4826   qed

  4827 qed

  4828

  4829 lemma compact_continuous_image:

  4830   assumes "continuous_on s f" and "compact s"

  4831   shows "compact (f  s)"

  4832 using assms (* FIXME: long unstructured proof *)

  4833 unfolding continuous_on_open

  4834 unfolding compact_eq_openin_cover

  4835 apply clarify

  4836 apply (drule_tac x="image (\<lambda>t. {x \<in> s. f x \<in> t}) C" in spec)

  4837 apply (drule mp)

  4838 apply (rule conjI)

  4839 apply simp

  4840 apply clarsimp

  4841 apply (drule subsetD)

  4842 apply (erule imageI)

  4843 apply fast

  4844 apply (erule thin_rl)

  4845 apply clarify

  4846 apply (rule_tac x="image (inv_into C (\<lambda>t. {x \<in> s. f x \<in> t})) D" in exI)

  4847 apply (intro conjI)

  4848 apply clarify

  4849 apply (rule inv_into_into)

  4850 apply (erule (1) subsetD)

  4851 apply (erule finite_imageI)

  4852 apply (clarsimp, rename_tac x)

  4853 apply (drule (1) subsetD, clarify)

  4854 apply (drule (1) subsetD, clarify)

  4855 apply (rule rev_bexI)

  4856 apply assumption

  4857 apply (subgoal_tac "{x \<in> s. f x \<in> t} \<in> (\<lambda>t. {x \<in> s. f x \<in> t})  C")

  4858 apply (drule f_inv_into_f)

  4859 apply fast

  4860 apply (erule imageI)

  4861 done

  4862

  4863 lemma connected_continuous_image:

  4864   assumes "continuous_on s f"  "connected s"

  4865   shows "connected(f  s)"

  4866 proof-

  4867   { fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"

  4868     have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"

  4869       using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]

  4870       using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]

  4871       using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto

  4872     hence False using as(1,2)

  4873       using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }

  4874   thus ?thesis unfolding connected_clopen by auto

  4875 qed

  4876

  4877 text {* Continuity implies uniform continuity on a compact domain. *}

  4878

  4879 lemma compact_uniformly_continuous:

  4880   assumes f: "continuous_on s f" and s: "compact s"

  4881   shows "uniformly_continuous_on s f"

  4882   unfolding uniformly_continuous_on_def

  4883 proof (cases, safe)

  4884   fix e :: real assume "0 < e" "s \<noteq> {}"

  4885   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) } }"

  4886   let ?b = "(\<lambda>(y, d). ball y (d/2))"

  4887   have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"

  4888   proof safe

  4889     fix y assume "y \<in> s"

  4890     from continuous_open_in_preimage[OF f open_ball]

  4891     obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"

  4892       unfolding openin_subtopology open_openin by metis

  4893     then obtain d where "ball y d \<subseteq> T" "0 < d"

  4894       using 0 < e y \<in> s by (auto elim!: openE)

  4895     with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"

  4896       by (intro UN_I[of "(y, d)"]) auto

  4897   qed auto

  4898   with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"

  4899     by (rule compactE_image)

  4900   with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"

  4901     by (subst Min_gr_iff) auto

  4902   show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"

  4903   proof (rule, safe)

  4904     fix x x' assume in_s: "x' \<in> s" "x \<in> s"

  4905     with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"

  4906       by blast

  4907     moreover assume "dist x x' < Min (sndD) / 2"

  4908     ultimately have "dist y x' < d"

  4909       by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)

  4910     with D x in_s show  "dist (f x) (f x') < e"

  4911       by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)

  4912   qed (insert D, auto)

  4913 qed auto

  4914

  4915 text{* Continuity of inverse function on compact domain. *}

  4916

  4917 lemma continuous_on_inv:

  4918   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"

  4919   assumes "continuous_on s f"  "compact s"  "\<forall>x \<in> s. g (f x) = x"

  4920   shows "continuous_on (f  s) g"

  4921 unfolding continuous_on_topological

  4922 proof (clarsimp simp add: assms(3))

  4923   fix x :: 'a and B :: "'a set"

  4924   assume "x \<in> s" and "open B" and "x \<in> B"

  4925   have 1: "\<forall>x\<in>s. f x \<in> f  (s - B) \<longleftrightarrow> x \<in> s - B"

  4926     using assms(3) by (auto, metis)

  4927   have "continuous_on (s - B) f"

  4928     using continuous_on s f Diff_subset

  4929     by (rule continuous_on_subset)

  4930   moreover have "compact (s - B)"

  4931     using open B and compact s

  4932     unfolding Diff_eq by (intro compact_inter_closed closed_Compl)

  4933   ultimately have "compact (f  (s - B))"

  4934     by (rule compact_continuous_image)

  4935   hence "closed (f  (s - B))"

  4936     by (rule compact_imp_closed)

  4937   hence "open (- f  (s - B))"

  4938     by (rule open_Compl)

  4939   moreover have "f x \<in> - f  (s - B)"

  4940     using x \<in> s and x \<in> B by (simp add: 1)

  4941   moreover have "\<forall>y\<in>s. f y \<in> - f  (s - B) \<longrightarrow> y \<in> B"

  4942     by (simp add: 1)

  4943   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"

  4944     by fast

  4945 qed

  4946

  4947 text {* A uniformly convergent limit of continuous functions is continuous. *}

  4948

  4949 lemma continuous_uniform_limit:

  4950   fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"

  4951   assumes "\<not> trivial_limit F"

  4952   assumes "eventually (\<lambda>n. continuous_on s (f n)) F"

  4953   assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"

  4954   shows "continuous_on s g"

  4955 proof-

  4956   { fix x and e::real assume "x\<in>s" "e>0"

  4957     have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"

  4958       using e>0 assms(3)[THEN spec[where x="e/3"]] by auto

  4959     from eventually_happens [OF eventually_conj [OF this assms(2)]]

  4960     obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"

  4961       using assms(1) by blast

  4962     have "e / 3 > 0" using e>0 by auto

  4963     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"

  4964       using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast

  4965     { fix y assume "y \<in> s" and "dist y x < d"

  4966       hence "dist (f n y) (f n x) < e / 3"

  4967         by (rule d [rule_format])

  4968       hence "dist (f n y) (g x) < 2 * e / 3"

  4969         using dist_triangle [of "f n y" "g x" "f n x"]

  4970         using n(1)[THEN bspec[where x=x], OF x\<in>s]

  4971         by auto

  4972       hence "dist (g y) (g x) < e"

  4973         using n(1)[THEN bspec[where x=y], OF y\<in>s]

  4974         using dist_triangle3 [of "g y" "g x" "f n y"]

  4975         by auto }

  4976     hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"

  4977       using d>0 by auto }

  4978   thus ?thesis unfolding continuous_on_iff by auto

  4979 qed

  4980

  4981

  4982 subsection {* Topological stuff lifted from and dropped to R *}

  4983

  4984 lemma open_real:

  4985   fixes s :: "real set" shows

  4986  "open s \<longleftrightarrow>

  4987         (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")

  4988   unfolding open_dist dist_norm by simp

  4989

  4990 lemma islimpt_approachable_real:

  4991   fixes s :: "real set"

  4992   shows "x islimpt s \<longleftrightarrow> (\<forall>e>0.  \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"

  4993   unfolding islimpt_approachable dist_norm by simp

  4994

  4995 lemma closed_real:

  4996   fixes s :: "real set"

  4997   shows "closed s \<longleftrightarrow>

  4998         (\<forall>x. (\<forall>e>0.  \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)

  4999             --> x \<in> s)"

  5000   unfolding closed_limpt islimpt_approachable dist_norm by simp

  5001

  5002 lemma continuous_at_real_range:

  5003   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5004   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.

  5005         \<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"

  5006   unfolding continuous_at unfolding Lim_at

  5007   unfolding dist_nz[THEN sym] unfolding dist_norm apply auto

  5008   apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto

  5009   apply(erule_tac x=e in allE) by auto

  5010

  5011 lemma continuous_on_real_range:

  5012   fixes f :: "'a::real_normed_vector \<Rightarrow> real"

  5013   shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"

  5014   unfolding continuous_on_iff dist_norm by simp

  5015

  5016 text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

  5017

  5018 lemma compact_attains_sup:

  5019   fixes s :: "real set"

  5020   assumes "compact s"  "s \<noteq> {}"

  5021   shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"

  5022 proof-

  5023   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  5024   { fix e::real assume as: "\<forall>x\<in>s. x \<le> Sup s" "Sup s \<notin> s"  "0 < e" "\<forall>x'\<in>s. x' = Sup s \<or> \<not> Sup s - x' < e"

  5025     have "isLub UNIV s (Sup s)" using Sup[OF assms(2)] unfolding setle_def using as(1) by auto

  5026     moreover have "isUb UNIV s (Sup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto

  5027     ultimately have False using isLub_le_isUb[of UNIV s "Sup s" "Sup s - e"] using e>0 by auto  }

  5028   thus ?thesis using bounded_has_Sup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Sup s"]]

  5029     apply(rule_tac x="Sup s" in bexI) by auto

  5030 qed

  5031

  5032 lemma Inf:

  5033   fixes S :: "real set"

  5034   shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"

  5035 by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def)

  5036

  5037 lemma compact_attains_inf:

  5038   fixes s :: "real set"

  5039   assumes "compact s" "s \<noteq> {}"  shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"

  5040 proof-

  5041   from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto

  5042   { fix e::real assume as: "\<forall>x\<in>s. x \<ge> Inf s"  "Inf s \<notin> s"  "0 < e"

  5043       "\<forall>x'\<in>s. x' = Inf s \<or> \<not> abs (x' - Inf s) < e"

  5044     have "isGlb UNIV s (Inf s)" using Inf[OF assms(2)] unfolding setge_def using as(1) by auto

  5045     moreover

  5046     { fix x assume "x \<in> s"

  5047       hence *:"abs (x - Inf s) = x - Inf s" using as(1)[THEN bspec[where x=x]] by auto

  5048       have "Inf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) x\<in>s unfolding * by auto }

  5049     hence "isLb UNIV s (Inf s + e)" unfolding isLb_def and setge_def by auto

  5050     ultimately have False using isGlb_le_isLb[of UNIV s "Inf s" "Inf s + e"] using e>0 by auto  }

  5051   thus ?thesis using bounded_has_Inf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="Inf s"]]

  5052     apply(rule_tac x="Inf s" in bexI) by auto

  5053 qed

  5054

  5055 lemma continuous_attains_sup:

  5056   fixes f :: "'a::topological_space \<Rightarrow> real"

  5057   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  5058         ==> (\<exists>x \<in> s. \<forall>y \<in> s.  f y \<le> f x)"

  5059   using compact_attains_sup[of "f  s"]

  5060   using compact_continuous_image[of s f] by auto

  5061

  5062 lemma continuous_attains_inf:

  5063   fixes f :: "'a::topological_space \<Rightarrow> real"

  5064   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f

  5065         \<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"

  5066   using compact_attains_inf[of "f  s"]

  5067   using compact_continuous_image[of s f] by auto

  5068

  5069 lemma distance_attains_sup:

  5070   assumes "compact s" "s \<noteq> {}"

  5071   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"

  5072 proof (rule continuous_attains_sup [OF assms])

  5073   { fix x assume "x\<in>s"

  5074     have "(dist a ---> dist a x) (at x within s)"

  5075       by (intro tendsto_dist tendsto_const Lim_at_within tendsto_ident_at)

  5076   }

  5077   thus "continuous_on s (dist a)"

  5078     unfolding continuous_on ..

  5079 qed

  5080

  5081 text {* For \emph{minimal} distance, we only need closure, not compactness. *}

  5082

  5083 lemma distance_attains_inf:

  5084   fixes a :: "'a::heine_borel"

  5085   assumes "closed s"  "s \<noteq> {}"

  5086   shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"

  5087 proof-

  5088   from assms(2) obtain b where "b\<in>s" by auto

  5089   let ?B = "cball a (dist b a) \<inter> s"

  5090   have "b \<in> ?B" using b\<in>s by (simp add: dist_commute)

  5091   hence "?B \<noteq> {}" by auto

  5092   moreover

  5093   { fix x assume "x\<in>?B"

  5094     fix e::real assume "e>0"

  5095     { fix x' assume "x'\<in>?B" and as:"dist x' x < e"

  5096       from as have "\<bar>dist a x' - dist a x\<bar> < e"

  5097         unfolding abs_less_iff minus_diff_eq

  5098         using dist_triangle2 [of a x' x]

  5099         using dist_triangle [of a x x']

  5100         by arith

  5101     }

  5102     hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"

  5103       using e>0 by auto

  5104   }

  5105   hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"

  5106     unfolding continuous_on Lim_within dist_norm real_norm_def

  5107     by fast

  5108   moreover have "compact ?B"

  5109     using compact_cball[of a "dist b a"]

  5110     unfolding compact_eq_bounded_closed

  5111     using bounded_Int and closed_Int and assms(1) by auto

  5112   ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"

  5113     using continuous_attains_inf[of ?B "dist a"] by fastforce

  5114   thus ?thesis by fastforce

  5115 qed

  5116

  5117

  5118 subsection {* Pasted sets *}

  5119

  5120 lemma bounded_Times:

  5121   assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"

  5122 proof-

  5123   obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"

  5124     using assms [unfolded bounded_def] by auto

  5125   then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"

  5126     by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)

  5127   thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto

  5128 qed

  5129

  5130 lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"

  5131 by (induct x) simp

  5132

  5133 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"

  5134 unfolding seq_compact_def

  5135 apply clarify

  5136 apply (drule_tac x="fst \<circ> f" in spec)

  5137 apply (drule mp, simp add: mem_Times_iff)

  5138 apply (clarify, rename_tac l1 r1)

  5139 apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)

  5140 apply (drule mp, simp add: mem_Times_iff)

  5141 apply (clarify, rename_tac l2 r2)

  5142 apply (rule_tac x="(l1, l2)" in rev_bexI, simp)

  5143 apply (rule_tac x="r1 \<circ> r2" in exI)

  5144 apply (rule conjI, simp add: subseq_def)

  5145 apply (drule_tac r=r2 in lim_subseq [rotated], assumption)

  5146 apply (drule (1) tendsto_Pair) back

  5147 apply (simp add: o_def)

  5148 done

  5149

  5150 text {* Generalize to @{class topological_space} *}

  5151 lemma compact_Times:

  5152   fixes s :: "'a::metric_space set" and t :: "'b::metric_space set"

  5153   shows "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"

  5154   unfolding compact_eq_seq_compact_metric by (rule seq_compact_Times)

  5155

  5156 text{* Hence some useful properties follow quite easily. *}

  5157

  5158 lemma compact_scaling:

  5159   fixes s :: "'a::real_normed_vector set"

  5160   assumes "compact s"  shows "compact ((\<lambda>x. c *\<^sub>R x)  s)"

  5161 proof-

  5162   let ?f = "\<lambda>x. scaleR c x"

  5163   have *:"bounded_linear ?f" by (rule bounded_linear_scaleR_right)

  5164   show ?thesis using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]

  5165     using linear_continuous_at[OF *] assms by auto

  5166 qed

  5167

  5168 lemma compact_negations:

  5169   fixes s :: "'a::real_normed_vector set"

  5170   assumes "compact s"  shows "compact ((\<lambda>x. -x)  s)"

  5171   using compact_scaling [OF assms, of "- 1"] by auto

  5172

  5173 lemma compact_sums:

  5174   fixes s t :: "'a::real_normed_vector set"

  5175   assumes "compact s"  "compact t"  shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"

  5176 proof-

  5177   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z)  (s \<times> t)"

  5178     apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto

  5179   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"

  5180     unfolding continuous_on by (rule ballI) (intro tendsto_intros)

  5181   thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto

  5182 qed

  5183

  5184 lemma compact_differences:

  5185   fixes s t :: "'a::real_normed_vector set"

  5186   assumes "compact s" "compact t"  shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"

  5187 proof-

  5188   have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus  t)}"

  5189     apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5190   thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto

  5191 qed

  5192

  5193 lemma compact_translation:

  5194   fixes s :: "'a::real_normed_vector set"

  5195   assumes "compact s"  shows "compact ((\<lambda>x. a + x)  s)"

  5196 proof-

  5197   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x)  s" by auto

  5198   thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto

  5199 qed

  5200

  5201 lemma compact_affinity:

  5202   fixes s :: "'a::real_normed_vector set"

  5203   assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"

  5204 proof-

  5205   have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto

  5206   thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto

  5207 qed

  5208

  5209 text {* Hence we get the following. *}

  5210

  5211 lemma compact_sup_maxdistance:

  5212   fixes s :: "'a::real_normed_vector set"

  5213   assumes "compact s"  "s \<noteq> {}"

  5214   shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"

  5215 proof-

  5216   have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using s \<noteq> {} by auto

  5217   then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}"  "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"

  5218     using compact_differences[OF assms(1) assms(1)]

  5219     using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by auto

  5220   from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto

  5221   thus ?thesis using x(2)[unfolded x = a - b] by blast

  5222 qed

  5223

  5224 text {* We can state this in terms of diameter of a set. *}

  5225

  5226 definition "diameter s = (if s = {} then 0::real else Sup {norm(x - y) | x y. x \<in> s \<and> y \<in> s})"

  5227   (* TODO: generalize to class metric_space *)

  5228

  5229 lemma diameter_bounded:

  5230   assumes "bounded s"

  5231   shows "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  5232         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)"

  5233 proof-

  5234   let ?D = "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}"

  5235   obtain a where a:"\<forall>x\<in>s. norm x \<le> a" using assms[unfolded bounded_iff] by auto

  5236   { fix x y assume "x \<in> s" "y \<in> s"

  5237     hence "norm (x - y) \<le> 2 * a" using norm_triangle_ineq[of x "-y", unfolded norm_minus_cancel] a[THEN bspec[where x=x]] a[THEN bspec[where x=y]] by (auto simp add: field_simps)  }

  5238   note * = this

  5239   { fix x y assume "x\<in>s" "y\<in>s"  hence "s \<noteq> {}" by auto

  5240     have "norm(x - y) \<le> diameter s" unfolding diameter_def using s\<noteq>{} *[OF x\<in>s y\<in>s] x\<in>s y\<in>s

  5241       by simp (blast del: Sup_upper intro!: * Sup_upper) }

  5242   moreover

  5243   { fix d::real assume "d>0" "d < diameter s"

  5244     hence "s\<noteq>{}" unfolding diameter_def by auto

  5245     have "\<exists>d' \<in> ?D. d' > d"

  5246     proof(rule ccontr)

  5247       assume "\<not> (\<exists>d'\<in>{norm (x - y) |x y. x \<in> s \<and> y \<in> s}. d < d')"

  5248       hence "\<forall>d'\<in>?D. d' \<le> d" by auto (metis not_leE)

  5249       thus False using d < diameter s s\<noteq>{}

  5250         apply (auto simp add: diameter_def)

  5251         apply (drule Sup_real_iff [THEN [2] rev_iffD2])

  5252         apply (auto, force)

  5253         done

  5254     qed

  5255     hence "\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d" by auto  }

  5256   ultimately show "\<forall>x\<in>s. \<forall>y\<in>s. norm(x - y) \<le> diameter s"

  5257         "\<forall>d>0. d < diameter s --> (\<exists>x\<in>s. \<exists>y\<in>s. norm(x - y) > d)" by auto

  5258 qed

  5259

  5260 lemma diameter_bounded_bound:

  5261  "bounded s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s ==> norm(x - y) \<le> diameter s"

  5262   using diameter_bounded by blast

  5263

  5264 lemma diameter_compact_attained:

  5265   fixes s :: "'a::real_normed_vector set"

  5266   assumes "compact s"  "s \<noteq> {}"

  5267   shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"

  5268 proof-

  5269   have b:"bounded s" using assms(1) by (rule compact_imp_bounded)

  5270   then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto

  5271   hence "diameter s \<le> norm (x - y)"

  5272     unfolding diameter_def by clarsimp (rule Sup_least, fast+)

  5273   thus ?thesis

  5274     by (metis b diameter_bounded_bound order_antisym xys)

  5275 qed

  5276

  5277 text {* Related results with closure as the conclusion. *}

  5278

  5279 lemma closed_scaling:

  5280   fixes s :: "'a::real_normed_vector set"

  5281   assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"

  5282 proof(cases "s={}")

  5283   case True thus ?thesis by auto

  5284 next

  5285   case False

  5286   show ?thesis

  5287   proof(cases "c=0")

  5288     have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto

  5289     case True thus ?thesis apply auto unfolding * by auto

  5290   next

  5291     case False

  5292     { fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"

  5293       { fix n::nat have "scaleR (1 / c) (x n) \<in> s"

  5294           using as(1)[THEN spec[where x=n]]

  5295           using c\<noteq>0 by auto

  5296       }

  5297       moreover

  5298       { fix e::real assume "e>0"

  5299         hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto

  5300         then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"

  5301           using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto

  5302         hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"

  5303           unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]

  5304           using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }

  5305       hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto

  5306       ultimately have "l \<in> scaleR c  s"

  5307         using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]

  5308         unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }

  5309     thus ?thesis unfolding closed_sequential_limits by fast

  5310   qed

  5311 qed

  5312

  5313 lemma closed_negations:

  5314   fixes s :: "'a::real_normed_vector set"

  5315   assumes "closed s"  shows "closed ((\<lambda>x. -x)  s)"

  5316   using closed_scaling[OF assms, of "- 1"] by simp

  5317

  5318 lemma compact_closed_sums:

  5319   fixes s :: "'a::real_normed_vector set"

  5320   assumes "compact s"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5321 proof-

  5322   let ?S = "{x + y |x y. x \<in> s \<and> y \<in> t}"

  5323   { fix x l assume as:"\<forall>n. x n \<in> ?S"  "(x ---> l) sequentially"

  5324     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"

  5325       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto

  5326     obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"

  5327       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto

  5328     have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"

  5329       using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto

  5330     hence "l - l' \<in> t"

  5331       using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]

  5332       using f(3) by auto

  5333     hence "l \<in> ?S" using l' \<in> s apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto

  5334   }

  5335   thus ?thesis unfolding closed_sequential_limits by fast

  5336 qed

  5337

  5338 lemma closed_compact_sums:

  5339   fixes s t :: "'a::real_normed_vector set"

  5340   assumes "closed s"  "compact t"

  5341   shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"

  5342 proof-

  5343   have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto

  5344     apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto

  5345   thus ?thesis using compact_closed_sums[OF assms(2,1)] by simp

  5346 qed

  5347

  5348 lemma compact_closed_differences:

  5349   fixes s t :: "'a::real_normed_vector set"

  5350   assumes "compact s"  "closed t"

  5351   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5352 proof-

  5353   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"

  5354     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5355   thus ?thesis using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto

  5356 qed

  5357

  5358 lemma closed_compact_differences:

  5359   fixes s t :: "'a::real_normed_vector set"

  5360   assumes "closed s" "compact t"

  5361   shows "closed {x - y | x y. x \<in> s \<and> y \<in> t}"

  5362 proof-

  5363   have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"

  5364     apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto

  5365  thus ?thesis using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp

  5366 qed

  5367

  5368 lemma closed_translation:

  5369   fixes a :: "'a::real_normed_vector"

  5370   assumes "closed s"  shows "closed ((\<lambda>x. a + x)  s)"

  5371 proof-

  5372   have "{a + y |y. y \<in> s} = (op + a  s)" by auto

  5373   thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto

  5374 qed

  5375

  5376 lemma translation_Compl:

  5377   fixes a :: "'a::ab_group_add"

  5378   shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"

  5379   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

  5380

  5381 lemma translation_UNIV:

  5382   fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"

  5383   apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

  5384

  5385 lemma translation_diff:

  5386   fixes a :: "'a::ab_group_add"

  5387   shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"

  5388   by auto

  5389

  5390 lemma closure_translation:

  5391   fixes a :: "'a::real_normed_vector"

  5392   shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"

  5393 proof-

  5394   have *:"op + a  (- s) = - op + a  s"

  5395     apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto

  5396   show ?thesis unfolding closure_interior translation_Compl

  5397     using interior_translation[of a "- s"] unfolding * by auto

  5398 qed

  5399

  5400 lemma frontier_translation:

  5401   fixes a :: "'a::real_normed_vector"

  5402   shows "frontier((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (frontier s)"

  5403   unfolding frontier_def translation_diff interior_translation closure_translation by auto

  5404

  5405

  5406 subsection {* Separation between points and sets *}

  5407

  5408 lemma separate_point_closed:

  5409   fixes s :: "'a::heine_borel set"

  5410   shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"

  5411 proof(cases "s = {}")

  5412   case True

  5413   thus ?thesis by(auto intro!: exI[where x=1])

  5414 next

  5415   case False

  5416   assume "closed s" "a \<notin> s"

  5417   then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using s \<noteq> {} distance_attains_inf [of s a] by blast

  5418   with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s by blast

  5419 qed

  5420

  5421 lemma separate_compact_closed:

  5422   fixes s t :: "'a::heine_borel set"

  5423   assumes "compact s" and "closed t" and "s \<inter> t = {}"

  5424   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5425 proof - (* FIXME: long proof *)

  5426   let ?T = "\<Union>x\<in>s. { ball x (d / 2) | d. 0 < d \<and> (\<forall>y\<in>t. d \<le> dist x y) }"

  5427   note compact s

  5428   moreover have "\<forall>t\<in>?T. open t" by auto

  5429   moreover have "s \<subseteq> \<Union>?T"

  5430     apply auto

  5431     apply (rule rev_bexI, assumption)

  5432     apply (subgoal_tac "x \<notin> t")

  5433     apply (drule separate_point_closed [OF closed t])

  5434     apply clarify

  5435     apply (rule_tac x="ball x (d / 2)" in exI)

  5436     apply simp

  5437     apply fast

  5438     apply (cut_tac assms(3))

  5439     apply auto

  5440     done

  5441   ultimately obtain U where "U \<subseteq> ?T" and "finite U" and "s \<subseteq> \<Union>U"

  5442     by (rule compactE)

  5443   from finite U and U \<subseteq> ?T have "\<exists>d>0. \<forall>x\<in>\<Union>U. \<forall>y\<in>t. d \<le> dist x y"

  5444     apply (induct set: finite)

  5445     apply simp

  5446     apply (rule exI)

  5447     apply (rule zero_less_one)

  5448     apply clarsimp

  5449     apply (rename_tac y e)

  5450     apply (rule_tac x="min d (e / 2)" in exI)

  5451     apply simp

  5452     apply (subst ball_Un)

  5453     apply (rule conjI)

  5454     apply (intro ballI, rename_tac z)

  5455     apply (rule min_max.le_infI2)

  5456     apply (simp only: mem_ball)

  5457     apply (drule (1) bspec)

  5458     apply (cut_tac x=y and y=x and z=z in dist_triangle, arith)

  5459     apply simp

  5460     apply (intro ballI)

  5461     apply (rule min_max.le_infI1)

  5462     apply simp

  5463     done

  5464   with s \<subseteq> \<Union>U show ?thesis

  5465     by fast

  5466 qed

  5467

  5468 lemma separate_closed_compact:

  5469   fixes s t :: "'a::heine_borel set"

  5470   assumes "closed s" and "compact t" and "s \<inter> t = {}"

  5471   shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"

  5472 proof-

  5473   have *:"t \<inter> s = {}" using assms(3) by auto

  5474   show ?thesis using separate_compact_closed[OF assms(2,1) *]

  5475     apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)

  5476     by (auto simp add: dist_commute)

  5477 qed

  5478

  5479

  5480 subsection {* Intervals *}

  5481

  5482 lemma interval: fixes a :: "'a::ordered_euclidean_space" shows

  5483   "{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and

  5484   "{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"

  5485   by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5486

  5487 lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5488   "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"

  5489   "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"

  5490   using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

  5491

  5492 lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5493  "({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and

  5494  "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)

  5495 proof-

  5496   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"

  5497     hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto

  5498     hence "a\<bullet>i < b\<bullet>i" by auto

  5499     hence False using as by auto  }

  5500   moreover

  5501   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"

  5502     let ?x = "(1/2) *\<^sub>R (a + b)"

  5503     { fix i :: 'a assume i:"i\<in>Basis"

  5504       have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5505       hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"

  5506         by (auto simp: inner_add_left) }

  5507     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }

  5508   ultimately show ?th1 by blast

  5509

  5510   { fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"

  5511     hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto

  5512     hence "a\<bullet>i \<le> b\<bullet>i" by auto

  5513     hence False using as by auto  }

  5514   moreover

  5515   { assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"

  5516     let ?x = "(1/2) *\<^sub>R (a + b)"

  5517     { fix i :: 'a assume i:"i\<in>Basis"

  5518       have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto

  5519       hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"

  5520         by (auto simp: inner_add_left) }

  5521     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }

  5522   ultimately show ?th2 by blast

  5523 qed

  5524

  5525 lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows

  5526   "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and

  5527   "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"

  5528   unfolding interval_eq_empty[of a b] by fastforce+

  5529

  5530 lemma interval_sing:

  5531   fixes a :: "'a::ordered_euclidean_space"

  5532   shows "{a .. a} = {a}" and "{a<..<a} = {}"

  5533   unfolding set_eq_iff mem_interval eq_iff [symmetric]

  5534   by (auto intro: euclidean_eqI simp: ex_in_conv)

  5535

  5536 lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows

  5537  "(\<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}" and

  5538  "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and

  5539  "(\<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}" and

  5540  "(\<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}"

  5541   unfolding subset_eq[unfolded Ball_def] unfolding mem_interval

  5542   by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

  5543

  5544 lemma interval_open_subset_closed:

  5545   fixes a :: "'a::ordered_euclidean_space"

  5546   shows "{a<..<b} \<subseteq> {a .. b}"

  5547   unfolding subset_eq [unfolded Ball_def] mem_interval

  5548   by (fast intro: less_imp_le)

  5549

  5550 lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5551  "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and

  5552  "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and

  5553  "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and

  5554  "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)

  5555 proof-

  5556   show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)

  5557   show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)

  5558   { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5559     hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto

  5560     fix i :: 'a assume i:"i\<in>Basis"

  5561     (** TODO combine the following two parts as done in the HOL_light version. **)

  5562     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5563       assume as2: "a\<bullet>i > c\<bullet>i"

  5564       { fix j :: 'a assume j:"j\<in>Basis"

  5565         hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"

  5566           apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i

  5567           by (auto simp add: as2)  }

  5568       hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto

  5569       moreover

  5570       have "?x\<notin>{a .. b}"

  5571         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5572         using as(2)[THEN bspec[where x=i]] and as2 i

  5573         by auto

  5574       ultimately have False using as by auto  }

  5575     hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto

  5576     moreover

  5577     { let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"

  5578       assume as2: "b\<bullet>i < d\<bullet>i"

  5579       { fix j :: 'a assume "j\<in>Basis"

  5580         hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"

  5581           apply(cases "j=i") using as(2)[THEN bspec[where x=j]]

  5582           by (auto simp add: as2) }

  5583       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto

  5584       moreover

  5585       have "?x\<notin>{a .. b}"

  5586         unfolding mem_interval apply auto apply(rule_tac x=i in bexI)

  5587         using as(2)[THEN bspec[where x=i]] and as2 using i

  5588         by auto

  5589       ultimately have False using as by auto  }

  5590     hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto

  5591     ultimately

  5592     have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto

  5593   } note part1 = this

  5594   show ?th3

  5595     unfolding subset_eq and Ball_def and mem_interval

  5596     apply(rule,rule,rule,rule)

  5597     apply(rule part1)

  5598     unfolding subset_eq and Ball_def and mem_interval

  5599     prefer 4

  5600     apply auto

  5601     by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+

  5602   { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"

  5603     fix i :: 'a assume i:"i\<in>Basis"

  5604     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto

  5605     hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this

  5606   show ?th4 unfolding subset_eq and Ball_def and mem_interval

  5607     apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4

  5608     apply auto by(erule_tac x=xa in allE, simp)+

  5609 qed

  5610

  5611 lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5612  "{a .. b} \<inter> {c .. d} =  {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"

  5613   unfolding set_eq_iff and Int_iff and mem_interval by auto

  5614

  5615 lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows

  5616   "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and

  5617   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and

  5618   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and

  5619   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)

  5620 proof-

  5621   let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"

  5622   have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>

  5623       (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"

  5624     by blast

  5625   note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)

  5626   show ?th1 unfolding * by (intro **) auto

  5627   show ?th2 unfolding * by (intro **) auto

  5628   show ?th3 unfolding * by (intro **) auto

  5629   show ?th4 unfolding * by (intro **) auto

  5630 qed

  5631

  5632 (* Moved interval_open_subset_closed a bit upwards *)

  5633

  5634 lemma open_interval[intro]:

  5635   fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"

  5636 proof-

  5637   have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"

  5638     by (intro open_INT finite_lessThan ballI continuous_open_vimage allI

  5639       linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)

  5640   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"

  5641     by (auto simp add: eucl_less [where 'a='a])

  5642   finally show "open {a<..<b}" .

  5643 qed

  5644

  5645 lemma closed_interval[intro]:

  5646   fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"

  5647 proof-

  5648   have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"

  5649     by (intro closed_INT ballI continuous_closed_vimage allI

  5650       linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)

  5651   also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"

  5652     by (auto simp add: eucl_le [where 'a='a])

  5653   finally show "closed {a .. b}" .

  5654 qed

  5655

  5656 lemma interior_closed_interval [intro]:

  5657   fixes a b :: "'a::ordered_euclidean_space"

  5658   shows "interior {a..b} = {a<..<b}" (is "?L = ?R")

  5659 proof(rule subset_antisym)

  5660   show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval

  5661     by (rule interior_maximal)

  5662 next

  5663   { fix x assume "x \<in> interior {a..b}"

  5664     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..

  5665     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto

  5666     { fix i :: 'a assume i:"i\<in>Basis"

  5667       have "dist (x - (e / 2) *\<^sub>R i) x < e"

  5668            "dist (x + (e / 2) *\<^sub>R i) x < e"

  5669         unfolding dist_norm apply auto

  5670         unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto

  5671       hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"

  5672                      "(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"

  5673         using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]

  5674         and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]

  5675         unfolding mem_interval using i by blast+

  5676       hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"

  5677         using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }

  5678     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }

  5679   thus "?L \<subseteq> ?R" ..

  5680 qed

  5681

  5682 lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"

  5683 proof-

  5684   let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"

  5685   { fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"

  5686     { fix i :: 'a assume "i\<in>Basis"

  5687       hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }

  5688     hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto

  5689     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }

  5690   thus ?thesis unfolding interval and bounded_iff by auto

  5691 qed

  5692

  5693 lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows

  5694  "bounded {a .. b} \<and> bounded {a<..<b}"

  5695   using bounded_closed_interval[of a b]

  5696   using interval_open_subset_closed[of a b]

  5697   using bounded_subset[of "{a..b}" "{a<..<b}"]

  5698   by simp

  5699

  5700 lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows

  5701  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"

  5702   using bounded_interval[of a b] by auto

  5703

  5704 lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"

  5705   using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]

  5706   by (auto simp: compact_eq_seq_compact_metric)

  5707

  5708 lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"

  5709   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"

  5710 proof-

  5711   { fix i :: 'a assume "i\<in>Basis"

  5712     hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"

  5713       using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }

  5714   thus ?thesis unfolding mem_interval by auto

  5715 qed

  5716

  5717 lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"

  5718   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"

  5719   shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"

  5720 proof-

  5721   { fix i :: 'a assume i:"i\<in>Basis"

  5722     have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp

  5723     also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)

  5724       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all

  5725       using x unfolding mem_interval using i apply simp

  5726       using y unfolding mem_interval using i apply simp

  5727       done

  5728     finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto

  5729     moreover {

  5730     have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp

  5731  `