(*  title:      HOL/Library/Topology_Euclidian_Space.thy

     Author:     Amine Chaieb, University of Cambridge

     Author:     Robert Himmelmann, TU Muenchen

     Author:     Brian Huffman, Portland State University

 *)

 header {* Elementary topology in Euclidean space. *}

 theory Topology_Euclidean_Space

 imports

   Complex_Main

   "~~/src/HOL/Library/Diagonal_Subsequence"

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

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

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

   Linear_Algebra

   Norm_Arith

 begin

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

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

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

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

   apply (frule isGlb_isLb)

   apply (frule_tac x = y in isGlb_isLb)

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

   done

 lemma countable_PiE: 

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

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

 subsection {* Topological Basis *}

 context topological_space

 begin

 definition "topological_basis B =

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

 lemma topological_basis_iff:

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

   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'))"

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

 proof safe

   fix O' and x::'a

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

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

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

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

 next

   assume H: ?rhs

   show "topological_basis B" using assms unfolding topological_basis_def

   proof safe

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

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

       by (force intro: bchoice simp: Bex_def)

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

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

   qed

 qed

 lemma topological_basisI:

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

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

   shows "topological_basis B"

   using assms by (subst topological_basis_iff) auto

 lemma topological_basisE:

   fixes O'

   assumes "topological_basis B"

   assumes "open O'"

   assumes "x \<in> O'"

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

 proof atomize_elim

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

   with topological_basis_iff assms

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

 qed

 lemma topological_basis_open:

   assumes "topological_basis B"

   assumes "X \<in> B"

   shows "open X"

   using assms

   by (simp add: topological_basis_def)

 lemma basis_dense:

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

   assumes "topological_basis B"

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

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

 proof (intro allI impI)

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

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

   guess B' . note B' = this

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

 qed

 end

 lemma topological_basis_prod:

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

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

   unfolding topological_basis_def

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

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

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

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

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

     from open_prod_elim[OF `open S` this]

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

       by (metis mem_Sigma_iff)

     moreover from topological_basisE[OF A a] guess A0 .

     moreover from topological_basisE[OF B b] guess B0 .

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

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

   qed auto

 qed (metis A B topological_basis_open open_Times)

 subsection {* Countable Basis *}

 locale countable_basis =

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

   assumes is_basis: "topological_basis B"

   assumes countable_basis: "countable B"

 begin

 lemma open_countable_basis_ex:

   assumes "open X"

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

   using assms countable_basis is_basis unfolding topological_basis_def by blast

 lemma open_countable_basisE:

   assumes "open X"

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

   using assms open_countable_basis_ex by (atomize_elim) simp

 lemma countable_dense_exists:

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

 proof -

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

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

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

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

 qed

 lemma countable_dense_setE:

   obtains D :: "'a set"

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

   using countable_dense_exists by blast

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

   therefore basis which is closed under union. *}

 definition union_closed_basis::"'a set set" where

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

 lemma basis_union_closed_basis: "topological_basis union_closed_basis"

 proof (rule topological_basisI)

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

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

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

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

 next

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

   thus "open B'"

     using topological_basis_open[OF is_basis]

     by (auto simp: union_closed_basis_def)

 qed

 lemma countable_union_closed_basis: "countable union_closed_basis"

   unfolding union_closed_basis_def using countable_basis by simp

 lemmas open_union_closed_basis = topological_basis_open[OF basis_union_closed_basis]

 lemma union_closed_basis_ex:

  assumes X: "X \<in> union_closed_basis"

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

 proof -

   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)

   thus ?thesis by auto

 qed

 lemma union_closed_basisE:

   assumes "X \<in> union_closed_basis"

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

 lemma union_closed_basisI:

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

   shows "X \<in> union_closed_basis"

 proof -

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

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

 qed

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

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

 lemma union_basisI[intro]:

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

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

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

 lemma open_imp_Union_of_incseq:

   assumes "open X"

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

 proof -

   from open_countable_basis_ex[OF `open X`]

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

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

   show ?thesis

   proof cases

     assume "B' \<noteq> {}"

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

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

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

     moreover

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

     proof safe

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

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

         by (metis `countable B'` from_nat_into_surj)

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

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

     next

       fix x n

       assume "x \<in> S n"

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

         by (simp add: S_def)

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

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

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

     qed

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

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

     ultimately show ?thesis by auto

   qed (auto simp: B')

 qed

 lemma open_incseqE:

   assumes "open X"

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

   using open_imp_Union_of_incseq assms by atomize_elim

 end

 class first_countable_topology = topological_space +

   assumes first_countable_basis:

     "\<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))"

 lemma (in first_countable_topology) countable_basis_at_decseq:

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

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

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

 proof atomize_elim

   from first_countable_basis[of x] obtain A

     where "countable A"

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

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

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

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

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

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

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

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

     fix i

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

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

   next

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

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

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

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

       by (auto simp: F_def)

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

       by (auto simp: eventually_sequentially)

   qed

 qed

 lemma (in first_countable_topology) first_countable_basisE:

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

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

   using first_countable_basis[of x]

   by atomize_elim auto

 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology

 proof

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

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

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

   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))"

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

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

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

       unfolding mem_Times_iff by (auto intro: open_Times)

   next

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

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

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

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

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

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

   qed (simp add: A B)

 qed

 instance metric_space \<subseteq> first_countable_topology

 proof

   fix x :: 'a

   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))"

   proof (intro exI, safe)

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

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

       by (auto simp: open_dist dist_commute subset_eq)

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

     moreover

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

       by (auto simp: inverse_eq_divide)

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

       by auto

   qed (auto intro: open_ball)

 qed

 class second_countable_topology = topological_space +

   assumes ex_countable_basis:

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

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

   using someI_ex[OF ex_countable_basis] by unfold_locales safe

 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology

 proof

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

     using ex_countable_basis by auto

   moreover

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

     using ex_countable_basis by auto

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

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

 qed

 instance second_countable_topology \<subseteq> first_countable_topology

 proof

   fix x :: 'a

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

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

     using countable_basis is_basis

     by (auto simp: countable_basis is_basis)

   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))"

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

        (fastforce simp: topological_space_class.topological_basis_def)

 qed

 subsection {* Polish spaces *}

 text {* Textbooks define Polish spaces as completely metrizable.

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

 class polish_space = complete_space + second_countable_topology

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

 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))"

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

   morphisms "openin" "topology"

   unfolding istopology_def by blast

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

   using openin[of U] by blast

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

   using topology_inverse[unfolded mem_Collect_eq] .

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

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

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

 proof-

   { assume "T1=T2"

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

   moreover

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

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

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

     hence "T1 = T2" unfolding openin_inverse .

   }

   ultimately show ?thesis by blast

 qed

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

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

 subsubsection {* Main properties of open sets *}

 lemma openin_clauses:

   fixes U :: "'a topology"

   shows "openin U {}"

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

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

   using openin[of U] unfolding istopology_def mem_Collect_eq

   by fast+

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

   unfolding topspace_def by blast

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

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

   using openin_clauses by simp

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

   using openin_clauses by simp

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

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

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

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

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

 proof

   assume ?lhs

   then show ?rhs by auto

 next

   assume H: ?rhs

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

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

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

   finally show "openin U S" .

 qed

 subsubsection {* Closed sets *}

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

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

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

 lemma closedin_topspace[intro,simp]:

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

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

   by (auto simp add: Diff_Un closedin_def)

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

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

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

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

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

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

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

   apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)

   apply (metis openin_subset subset_eq)

   done

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

   by (simp add: openin_closedin_eq)

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

 proof-

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

     by (auto simp add: topspace_def openin_subset)

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

 qed

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

 proof-

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

     by (auto simp add: topspace_def )

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

 qed

 subsubsection {* Subspace topology *}

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

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

   (is "istopology ?L")

 proof-

   have "?L {}" by blast

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

     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

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

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

   moreover

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

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

       apply (rule set_eqI)

       apply (simp add: Ball_def image_iff)

       by metis

     from K[unfolded th0 subset_image_iff]

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

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

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

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

   ultimately show ?thesis

     unfolding subset_eq mem_Collect_eq istopology_def by blast

 qed

 lemma openin_subtopology:

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

   unfolding subtopology_def topology_inverse'[OF istopology_subtopology]

   by auto

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

   by (auto simp add: topspace_def openin_subtopology)

 lemma closedin_subtopology:

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

   unfolding closedin_def topspace_subtopology

   apply (simp add: openin_subtopology)

   apply (rule iffI)

   apply clarify

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

   by auto

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

   unfolding openin_subtopology

   apply (rule iffI, clarify)

   apply (frule openin_subset[of U])  apply blast

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

   apply auto

   done

 lemma subtopology_superset:

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

   shows "subtopology U V = U"

 proof-

   {fix S

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

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

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

     moreover

     {assume S: "openin U S"

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

         using openin_subset[OF S] UV by auto}

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

   then show ?thesis unfolding topology_eq openin_subtopology by blast

 qed

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

   by (simp add: subtopology_superset)

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

   by (simp add: subtopology_superset)

 subsubsection {* The standard Euclidean topology *}

 definition

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

   "euclidean = topology open"

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

   unfolding euclidean_def

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

   apply (rule topology_inverse[symmetric])

   apply (auto simp add: istopology_def)

   done

 lemma topspace_euclidean: "topspace euclidean = UNIV"

   apply (simp add: topspace_def)

   apply (rule set_eqI)

   by (auto simp add: open_openin[symmetric])

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

   by (simp add: topspace_euclidean topspace_subtopology)

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

   by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

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

   by (simp add: open_openin openin_subopen[symmetric])

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

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

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

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

   by (auto simp add: openin_open)

 lemma open_openin_trans[trans]:

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

   by (metis Int_absorb1  openin_open_Int)

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

   by (auto simp add: openin_open)

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

   by (simp add: closedin_subtopology closed_closedin Int_ac)

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

   by (metis closedin_closed)

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

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

   apply auto

   apply (frule closedin_closed_Int[of T S])

   by simp

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

   by (auto simp add: closedin_closed)

 lemma openin_euclidean_subtopology_iff:

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

   shows "openin (subtopology euclidean U) S

   \<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")

 proof

   assume ?lhs thus ?rhs unfolding openin_open open_dist by blast

 next

   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}"

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

     unfolding T_def

     apply clarsimp

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

     apply (clarsimp simp add: less_diff_eq)

     apply (erule rev_bexI)

     apply (rule_tac x=d in exI, clarify)

     apply (erule le_less_trans [OF dist_triangle])

     done

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

     unfolding T_def

     apply auto

     apply (drule (1) bspec, erule rev_bexI)

     apply auto

     done

   from 1 2 show ?lhs

     unfolding openin_open open_dist by fast

 qed

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

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

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

   unfolding open_openin openin_open by blast

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

   by (auto simp add: openin_open intro: openin_trans)

 lemma closedin_trans[trans]:

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

            closedin (subtopology euclidean U) T

            ==> closedin (subtopology euclidean U) S"

   by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)

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

   by (auto simp add: closedin_closed intro: closedin_trans)

 subsection {* Open and closed balls *}

 definition

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

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

 definition

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

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

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

   by (simp add: ball_def)

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

   by (simp add: cball_def)

 lemma mem_ball_0:

   fixes x :: "'a::real_normed_vector"

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

   by (simp add: dist_norm)

 lemma mem_cball_0:

   fixes x :: "'a::real_normed_vector"

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

   by (simp add: dist_norm)

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

   by simp

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

   by simp

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

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

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

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

   by (simp add: set_eq_iff) arith

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

   by (simp add: set_eq_iff)

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

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

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

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

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

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

   unfolding open_dist ball_def mem_Collect_eq Ball_def

   unfolding dist_commute

   apply clarify

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

   using dist_triangle_alt[where z=x]

   apply (clarsimp simp add: diff_less_iff)

   apply atomize

   apply (erule_tac x="y" in allE)

   apply (erule_tac x="xa" in allE)

   by arith

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

   unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

 lemma openE[elim?]:

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

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

   using assms unfolding open_contains_ball by auto

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

   by (metis open_contains_ball subset_eq centre_in_ball)

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

   unfolding mem_ball set_eq_iff

   apply (simp add: not_less)

   by (metis zero_le_dist order_trans dist_self)

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

 lemma euclidean_dist_l2:

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

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

   unfolding dist_norm norm_eq_sqrt_inner setL2_def

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

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

 lemma rational_boxes:

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

   assumes "0 < e"

   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"

 proof -

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

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

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

   proof

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

   qed

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

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

   proof

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

   qed

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

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

   show ?thesis

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

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

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

       unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)

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

     proof (rule real_sqrt_less_mono, rule setsum_strict_mono)

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

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

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

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

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

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

         unfolding e'_def by (auto simp: dist_real_def)

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

         by (rule power_strict_mono) auto

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

         by (simp add: power_divide)

     qed auto

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

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

   qed (insert a b, auto simp: box_def)

 qed

 lemma open_UNION_box:

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

   assumes "open M" 

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

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

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

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

 proof safe

   fix x assume "x \<in> M"

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

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

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

     "\<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"

     using rational_boxes[OF e(1)] by metis

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

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

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

 qed (auto simp: I_def)

 subsection{* Connectedness *}

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

 lemma connected_local:

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

                  openin (subtopology euclidean S) e1 \<and>

                  openin (subtopology euclidean S) e2 \<and>

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

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

                  ~(e1 = {}) \<and>

                  ~(e2 = {}))"

 unfolding connected_def openin_open by (safe, blast+)

 lemma exists_diff:

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

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

 proof-

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

   moreover

   {fix S assume H: "P S"

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

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

   ultimately show ?thesis by metis

 qed

 lemma connected_clopen: "connected S \<longleftrightarrow>

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

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

 proof-

   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> {})"

     unfolding connected_def openin_open closedin_closed

     apply (subst exists_diff) by blast

   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> {})"

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

   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'))"

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

     unfolding connected_def openin_open closedin_closed by auto

   {fix e2

     {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)"

         by auto}

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

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

   then show ?thesis unfolding th0 th1 by simp

 qed

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

   by (simp add: connected_def)

 subsection{* Limit points *}

 definition (in topological_space)

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

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

 lemma islimptI:

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

   shows "x islimpt S"

   using assms unfolding islimpt_def by auto

 lemma islimptE:

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

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

   using assms unfolding islimpt_def by auto

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

   unfolding islimpt_def eventually_at_topological by auto

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

   unfolding islimpt_def by fast

 lemma islimpt_approachable:

   fixes x :: "'a::metric_space"

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

   unfolding islimpt_iff_eventually eventually_at by fast

 lemma islimpt_approachable_le:

   fixes x :: "'a::metric_space"

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

   unfolding islimpt_approachable

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

     THEN arg_cong [where f=Not]]

   by (simp add: Bex_def conj_commute conj_left_commute)

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

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

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

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

   unfolding islimpt_UNIV_iff by (rule not_open_singleton)

 lemma perfect_choose_dist:

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

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

 using islimpt_UNIV [of x]

 by (simp add: islimpt_approachable)

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

   unfolding closed_def

   apply (subst open_subopen)

   apply (simp add: islimpt_def subset_eq)

   by (metis ComplE ComplI)

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

   unfolding islimpt_def by auto

 lemma finite_set_avoid:

   fixes a :: "'a::metric_space"

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

 proof(induct rule: finite_induct[OF fS])

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

 next

   case (2 x F)

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

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

   moreover

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

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

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

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

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

   ultimately show ?case by blast

 qed

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

   by (simp add: islimpt_iff_eventually eventually_conj_iff)

 lemma discrete_imp_closed:

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

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

   shows "closed S"

 proof-

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

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

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

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

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

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

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

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

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

     have False by (auto simp add: dist_commute)}

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

 qed

 subsection {* Interior of a Set *}

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

 lemma interiorI [intro?]:

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

   shows "x \<in> interior S"

   using assms unfolding interior_def by fast

 lemma interiorE [elim?]:

   assumes "x \<in> interior S"

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

   using assms unfolding interior_def by fast

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

   by (simp add: interior_def open_Union)

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

   by (auto simp add: interior_def)

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

   by (auto simp add: interior_def)

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

   by (intro equalityI interior_subset interior_maximal subset_refl)

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

   by (metis open_interior interior_open)

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

   by (metis interior_maximal interior_subset subset_trans)

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

   using open_empty by (rule interior_open)

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

   using open_UNIV by (rule interior_open)

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

   using open_interior by (rule interior_open)

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

   by (auto simp add: interior_def)

 lemma interior_unique:

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

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

   shows "interior S = T"

   by (intro equalityI assms interior_subset open_interior interior_maximal)

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

   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1

     Int_lower2 interior_maximal interior_subset open_Int open_interior)

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

   using open_contains_ball_eq [where S="interior S"]

   by (simp add: open_subset_interior)

 lemma interior_limit_point [intro]:

   fixes x :: "'a::perfect_space"

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

   using x islimpt_UNIV [of x]

   unfolding interior_def islimpt_def

   apply (clarsimp, rename_tac T T')

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

   apply (auto simp add: open_Int)

   done

 lemma interior_closed_Un_empty_interior:

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

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

 proof

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

     by (rule interior_mono, rule Un_upper1)

 next

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

   proof

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

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

     show "x \<in> interior S"

     proof (rule ccontr)

       assume "x \<notin> interior S"

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

         unfolding interior_def by fast

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

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

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

       show "False" unfolding interior_def by fast

     qed

   qed

 qed

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

 proof (rule interior_unique)

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

     by (intro Sigma_mono interior_subset)

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

     by (intro open_Times open_interior)

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

   proof (safe)

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

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

       using `open T` unfolding open_prod_def by fast

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

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

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

       by (auto intro: interiorI)

   qed

 qed

 subsection {* Closure of a Set *}

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

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

   unfolding interior_def closure_def islimpt_def by auto

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

   unfolding interior_closure by simp

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

   unfolding closure_interior by (simp add: closed_Compl)

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

   unfolding closure_def by simp

 lemma closure_hull: "closure S = closed hull S"

   unfolding hull_def closure_interior interior_def by auto

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

   unfolding closure_hull using closed_Inter by (rule hull_eq)

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

   unfolding closure_eq .

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

   unfolding closure_hull by (rule hull_hull)

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

   unfolding closure_hull by (rule hull_mono)

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

   unfolding closure_hull by (rule hull_minimal)

 lemma closure_unique:

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

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

   shows "closure S = T"

   using assms unfolding closure_hull by (rule hull_unique)

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

   using closed_empty by (rule closure_closed)

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

   using closed_UNIV by (rule closure_closed)

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

   unfolding closure_interior by simp

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

   using closure_empty closure_subset[of S]

   by blast

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

   using closure_eq[of S] closure_subset[of S]

   by simp

 lemma open_inter_closure_eq_empty:

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

   using open_subset_interior[of S "- T"]

   using interior_subset[of "- T"]

   unfolding closure_interior

   by auto

 lemma open_inter_closure_subset:

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

 proof

   fix x

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

   { assume *:"x islimpt T"

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

     proof (rule islimptI)

       fix A

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

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

         by (simp_all add: open_Int)

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

         by (rule islimptE)

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

         by simp_all

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

     qed

   }

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

     unfolding closure_def

     by blast

 qed

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

   unfolding closure_interior by simp

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

   unfolding closure_interior by simp

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

 proof (rule closure_unique)

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

     by (intro Sigma_mono closure_subset)

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

     by (intro closed_Times closed_closure)

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

     apply (simp add: closed_def open_prod_def, clarify)

     apply (rule ccontr)

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

     apply (simp add: closure_interior interior_def)

     apply (drule_tac x=C in spec)

     apply (drule_tac x=D in spec)

     apply auto

     done

 qed

 subsection {* Frontier (aka boundary) *}

 definition "frontier S = closure S - interior S"

 lemma frontier_closed: "closed(frontier S)"

   by (simp add: frontier_def closed_Diff)

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

   by (auto simp add: frontier_def interior_closure)

 lemma frontier_straddle:

   fixes a :: "'a::metric_space"

   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))"

   unfolding frontier_def closure_interior

   by (auto simp add: mem_interior subset_eq ball_def)

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

   by (metis frontier_def closure_closed Diff_subset)

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

   by (simp add: frontier_def)

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

 proof-

   { assume "frontier S \<subseteq> S"

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

     hence "closed S" using closure_subset_eq by auto

   }

   thus ?thesis using frontier_subset_closed[of S] ..

 qed

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

   by (auto simp add: frontier_def closure_complement interior_complement)

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

   using frontier_complement frontier_subset_eq[of "- S"]

   unfolding open_closed by auto

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

 definition

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

     (infixr "indirection" 70) where

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

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

 lemma trivial_limit_within:

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

 proof

   assume "trivial_limit (at a within S)"

   thus "\<not> a islimpt S"

     unfolding trivial_limit_def

     unfolding eventually_within eventually_at_topological

     unfolding islimpt_def

     apply (clarsimp simp add: set_eq_iff)

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

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

     done

 next

   assume "\<not> a islimpt S"

   thus "trivial_limit (at a within S)"

     unfolding trivial_limit_def

     unfolding eventually_within eventually_at_topological

     unfolding islimpt_def

     apply clarsimp

     apply (rule_tac x=T in exI)

     apply auto

     done

 qed

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

   using trivial_limit_within [of a UNIV] by simp

 lemma trivial_limit_at:

   fixes a :: "'a::perfect_space"

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

   by (rule at_neq_bot)

 lemma trivial_limit_at_infinity:

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

   unfolding trivial_limit_def eventually_at_infinity

   apply clarsimp

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

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

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

   apply (drule_tac x=UNIV in spec, simp)

   done

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

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

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

 unfolding eventually_at dist_nz by auto

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

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

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

   by (rule eventually_within_less)

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

   unfolding trivial_limit_def

   by (auto elim: eventually_rev_mp)

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

   by simp

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

   by (simp add: filter_eq_iff)

 text{* Combining theorems for "eventually" *}

 lemma eventually_rev_mono:

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

 using eventually_mono [of P Q] by fast

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

   by (simp add: eventually_False)

 subsection {* Limits *}

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

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

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

 lemma Lim:

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

         trivial_limit net \<or>

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

   unfolding tendsto_iff trivial_limit_eq by auto

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

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

            (\<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)"

   by (auto simp add: tendsto_iff eventually_within_le)

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

         (\<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)"

   by (auto simp add: tendsto_iff eventually_within)

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

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

   by (auto simp add: tendsto_iff eventually_at)

 lemma Lim_at_infinity:

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

   by (auto simp add: tendsto_iff eventually_at_infinity)

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

   by (rule topological_tendstoI, auto elim: eventually_rev_mono)

 text{* The expected monotonicity property. *}

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

   unfolding tendsto_def Limits.eventually_within by simp

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

   unfolding tendsto_def Limits.eventually_within

   by (auto elim!: eventually_elim1)

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

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

   using assms unfolding tendsto_def Limits.eventually_within

   apply clarify

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

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

   apply (auto elim: eventually_elim2)

   done

 lemma Lim_Un_univ:

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

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

   by (metis Lim_Un within_UNIV)

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

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

   (* FIXME: rename *)

   unfolding tendsto_def Limits.eventually_within

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

   by (auto elim!: eventually_elim1)

 lemma eventually_within_interior:

   assumes "x \<in> interior S"

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

 proof-

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

   { assume "?lhs"

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

       unfolding Limits.eventually_within Limits.eventually_at_topological

       by auto

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

       by auto

     then have "?rhs"

       unfolding Limits.eventually_at_topological by auto

   } moreover

   { assume "?rhs" hence "?lhs"

       unfolding Limits.eventually_within

       by (auto elim: eventually_elim1)

   } ultimately

   show "?thesis" ..

 qed

 lemma at_within_interior:

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

   by (simp add: filter_eq_iff eventually_within_interior)

 lemma at_within_open:

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

   by (simp only: at_within_interior interior_open)

 lemma Lim_within_open:

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

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

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

   using assms by (simp only: at_within_open)

 lemma Lim_within_LIMSEQ:

   fixes a :: "'a::metric_space"

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

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

   using assms unfolding tendsto_def [where l=L]

   by (simp add: sequentially_imp_eventually_within)

 lemma Lim_right_bound:

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

   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"

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

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

 proof cases

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

 next

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

   show ?thesis

   proof (rule Lim_within_LIMSEQ, safe)

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

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

     proof (rule LIMSEQ_I, rule ccontr)

       fix r :: real assume "0 < r"

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

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

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

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

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

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

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

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

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

         by (auto simp: not_less field_simps)

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

       show False by auto

     qed

   qed

 qed

 text{* Another limit point characterization. *}

 lemma islimpt_sequential:

   fixes x :: "'a::first_countable_topology"

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

     (is "?lhs = ?rhs")

 proof

   assume ?lhs

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

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

   { fix n

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

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

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

       unfolding f_def by (rule someI_ex)

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

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

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

   proof (rule topological_tendstoI)

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

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

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

   qed

   ultimately show ?rhs by fast

 next

   assume ?rhs

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

   show ?lhs

     unfolding islimpt_def

   proof safe

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

     from lim[THEN topological_tendstoD, OF this] f

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

       unfolding eventually_sequentially by auto

   qed

 qed

 lemma Lim_inv: (* TODO: delete *)

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

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

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

   unfolding o_def using assms by (rule tendsto_inverse)

 lemma Lim_null:

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

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

   by (simp add: Lim dist_norm)

 lemma Lim_null_comparison:

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

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

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

 proof (rule metric_tendsto_imp_tendsto)

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

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

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

 qed

 lemma Lim_transform_bound:

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

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

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

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

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

   by (rule Lim_null_comparison)

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

 lemma Lim_in_closed_set:

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

   shows "l \<in> S"

 proof (rule ccontr)

   assume "l \<notin> S"

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

     by (simp_all add: open_Compl)

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

     by (rule topological_tendstoD)

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

     by (rule eventually_elim2) simp

   with assms(3) show "False"

     by (simp add: eventually_False)

 qed

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

 lemma Lim_dist_ubound:

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

   shows "dist a l <= e"

 proof-

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

   proof (rule Lim_in_closed_set)

     show "closed {..e}" by simp

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

     show "\<not> trivial_limit net" by fact

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

   qed

   thus ?thesis by simp

 qed

 lemma Lim_norm_ubound:

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

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

   shows "norm(l) <= e"

 proof-

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

   proof (rule Lim_in_closed_set)

     show "closed {..e}" by simp

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

     show "\<not> trivial_limit net" by fact

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

   qed

   thus ?thesis by simp

 qed

 lemma Lim_norm_lbound:

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

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

   shows "e \<le> norm l"

 proof-

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

   proof (rule Lim_in_closed_set)

     show "closed {e..}" by simp

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

     show "\<not> trivial_limit net" by fact

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

   qed

   thus ?thesis by simp

 qed

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

 lemma tendsto_Lim:

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

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

   unfolding Lim_def using tendsto_unique[of net f] by auto

 text{* Limit under bilinear function *}

 lemma Lim_bilinear:

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

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

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

 by (rule bounded_bilinear.tendsto)

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

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

   unfolding id_def by (rule tendsto_ident_at_within)

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

   unfolding id_def by (rule tendsto_ident_at)

 lemma Lim_at_zero:

   fixes a :: "'a::real_normed_vector"

   fixes l :: "'b::topological_space"

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

   using LIM_offset_zero LIM_offset_zero_cancel ..

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

 definition

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

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

 lemma netlimit_within:

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

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

 unfolding netlimit_def

 apply (rule some_equality)

 apply (rule Lim_at_within)

 apply (rule tendsto_ident_at)

 apply (erule tendsto_unique [OF assms])

 apply (rule Lim_at_within)

 apply (rule tendsto_ident_at)

 done

 lemma netlimit_at:

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

   shows "netlimit (at a) = a"

   using netlimit_within [of a UNIV] by simp

 lemma lim_within_interior:

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

   by (simp add: at_within_interior)

 lemma netlimit_within_interior:

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

   assumes "x \<in> interior S"

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

 using assms by (simp add: at_within_interior netlimit_at)

 text{* Transformation of limit. *}

 lemma Lim_transform:

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

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

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

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

 lemma Lim_transform_eventually:

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

   apply (rule topological_tendstoI)

   apply (drule (2) topological_tendstoD)

   apply (erule (1) eventually_elim2, simp)

   done

 lemma Lim_transform_within:

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

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

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

 proof (rule Lim_transform_eventually)

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

     unfolding eventually_within

     using assms(1,2) by auto

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

 qed

 lemma Lim_transform_at:

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

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

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

 proof (rule Lim_transform_eventually)

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

     unfolding eventually_at

     using assms(1,2) by auto

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

 qed

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

 lemma Lim_transform_away_within:

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

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

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

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

 proof (rule Lim_transform_eventually)

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

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

     unfolding Limits.eventually_within eventually_at_topological

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

 qed

 lemma Lim_transform_away_at:

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

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

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

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

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

   by simp

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

 lemma Lim_transform_within_open:

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

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

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

 proof (rule Lim_transform_eventually)

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

     unfolding eventually_at_topological

     using assms(1,2,3) by auto

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

 qed

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

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

 lemma Lim_cong_within(*[cong add]*):

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

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

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

   unfolding tendsto_def Limits.eventually_within eventually_at_topological

   using assms by simp

 lemma Lim_cong_at(*[cong add]*):

   assumes "a = b" "x = y"

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

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

   unfolding tendsto_def eventually_at_topological

   using assms by simp

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

 lemma closure_sequential:

   fixes l :: "'a::first_countable_topology"

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

 proof

   assume "?lhs" moreover

   { assume "l \<in> S"

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

   } moreover

   { assume "l islimpt S"

     hence "?rhs" unfolding islimpt_sequential by auto

   } ultimately

   show "?rhs" unfolding closure_def by auto

 next

   assume "?rhs"

   thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto

 qed

 lemma closed_sequential_limits:

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

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

   unfolding closed_limpt

   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]

   by metis

 lemma closure_approachable:

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

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

   apply (auto simp add: closure_def islimpt_approachable)

   by (metis dist_self)

 lemma closed_approachable:

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

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

   by (metis closure_closed closure_approachable)

 subsection {* Infimum Distance *}

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

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

   by (simp add: infdist_def)

 lemma infdist_nonneg:

   shows "0 \<le> infdist x A"

   using assms by (auto simp add: infdist_def)

 lemma infdist_le:

   assumes "a \<in> A"

   assumes "d = dist x a"

   shows "infdist x A \<le> d"

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

 lemma infdist_zero[simp]:

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

 proof -

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

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

 qed

 lemma infdist_triangle:

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

 proof cases

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

 next

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

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

   proof

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

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

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

     show "infdist x A \<le> d"

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

     proof (rule Inf_lower2)

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

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

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

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

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

     qed

   qed

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

   proof (rule Inf_eq, safe)

     fix a assume "a \<in> A"

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

   next

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

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

       by (intro Inf_greatest) (auto simp: field_simps)

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

   qed

   finally show ?thesis by simp

 qed

 lemma

   in_closure_iff_infdist_zero:

   assumes "A \<noteq> {}"

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

 proof

   assume "x \<in> closure A"

   show "infdist x A = 0"

   proof (rule ccontr)

     assume "infdist x A \<noteq> 0"

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

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

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

         eucl_less_not_refl euclidean_trans(2) infdist_le)

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

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

   qed

 next

   assume x: "infdist x A = 0"

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

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

   proof (safe, rule ccontr)

     fix e::real assume "0 < e"

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

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

       unfolding infdist_def

       by (force simp: dist_commute)

     with x `0 < e` show False by auto

   qed

 qed

 lemma

   in_closed_iff_infdist_zero:

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

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

 proof -

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

     by (rule in_closure_iff_infdist_zero) fact

   with assms show ?thesis by simp

 qed

 lemma tendsto_infdist [tendsto_intros]:

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

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

 proof (rule tendstoI)

   fix e ::real assume "0 < e"

   from tendstoD[OF f this]

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

   proof (eventually_elim)

     fix x

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

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

       by (simp add: dist_commute dist_real_def)

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

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

   qed

 qed

 text{* Some other lemmas about sequences. *}

 lemma sequentially_offset:

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

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

   using assms unfolding eventually_sequentially by (metis trans_le_add1)

 lemma seq_offset:

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

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

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

 lemma seq_offset_neg:

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

   apply (rule topological_tendstoI)

   apply (drule (2) topological_tendstoD)

   apply (simp only: eventually_sequentially)

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

   apply metis

   by arith

 lemma seq_offset_rev:

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

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

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

   using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

 subsection {* More properties of closed balls *}

 lemma closed_cball: "closed (cball x e)"

 unfolding cball_def closed_def

 unfolding Collect_neg_eq [symmetric] not_le

 apply (clarsimp simp add: open_dist, rename_tac y)

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

 apply (rename_tac x')

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

 apply simp

 done

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

 proof-

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

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

   } moreover

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

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

   } ultimately

   show ?thesis unfolding open_contains_ball by auto

 qed

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

   by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

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

   apply (simp add: interior_def, safe)

   apply (force simp add: open_contains_cball)

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

   apply (simp add: subset_trans [OF ball_subset_cball])

   done

 lemma islimpt_ball:

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

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

 proof

   assume "?lhs"

   { assume "e \<le> 0"

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

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

   }

   hence "e > 0" by (metis not_less)

   moreover

   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

   ultimately show "?rhs" by auto

 next

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

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

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

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

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

       proof(cases "x=y")

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

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

       next

         case False

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

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

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

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

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

           unfolding scaleR_minus_left scaleR_one

           by (auto simp add: norm_minus_commute)

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

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

           unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto

         also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)

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

         moreover

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

           using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)

         moreover

         have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel

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

           unfolding dist_norm by auto

         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

       qed

     next

       case False hence "d > dist x y" by auto

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

       proof(cases "x=y")

         case True

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

           using perfect_choose_dist[of "min e d" y]

           using `d > 0` `e>0` by auto

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

           unfolding `x = y`

           using `z \<noteq> y` **

           by (rule_tac x=z in bexI, auto simp add: dist_commute)

       next

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

           using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)

       qed

     qed  }

   thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto

 qed

 lemma closure_ball_lemma:

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

   assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"

 proof (rule islimptI)

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

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

     unfolding open_dist by fast

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

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

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

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

     unfolding z_def by (simp add: algebra_simps)

   have "dist z y < r"

     unfolding z_def k_def using `0 < r`

     by (simp add: dist_norm min_def)

   hence "z \<in> T" using `\<forall>z. dist z y < r \<longrightarrow> z \<in> T` by simp

   have "dist x z < dist x y"

     unfolding z_def2 dist_norm

     apply (simp add: norm_minus_commute)

     apply (simp only: dist_norm [symmetric])

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

     apply (rule mult_strict_right_mono)

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

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

     done

   hence "z \<in> ball x (dist x y)" by simp

   have "z \<noteq> y"

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

     by (simp add: min_def)

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

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

     by fast

 qed

 lemma closure_ball:

   fixes x :: "'a::real_normed_vector"

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

 apply (rule equalityI)

 apply (rule closure_minimal)

 apply (rule ball_subset_cball)

 apply (rule closed_cball)

 apply (rule subsetI, rename_tac y)

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

 apply (erule disjE)

 apply (rule subsetD [OF closure_subset], simp)

 apply (simp add: closure_def)

 apply clarify

 apply (rule closure_ball_lemma)

 apply (simp add: zero_less_dist_iff)

 done

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

 lemma interior_cball:

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

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

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

   case False note cs = this

   from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover

   { fix y assume "y \<in> cball x e"

     hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }

   hence "cball x e = {}" by auto

   hence "interior (cball x e) = {}" using interior_empty by auto

   ultimately show ?thesis by blast

 next

   case True note cs = this

   have "ball x e \<subseteq> cball x e" using ball_subset_cball by auto moreover

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

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

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

       using perfect_choose_dist [of d] by auto

     have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)

     hence xa_cball:"xa \<in> cball x e" using as(1) by auto

     hence "y \<in> ball x e" proof(cases "x = y")

       case True

       hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)

       thus "y \<in> ball x e" using `x = y ` by simp

     next

       case False

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

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

       hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast

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

       hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]

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

       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)"

         by (auto simp add: dist_norm algebra_simps)

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

         by (auto simp add: algebra_simps)

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

         using ** by auto

       also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: distrib_right dist_norm)

       finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)

       thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto

     qed  }

   hence "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e" by auto

   ultimately show ?thesis using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto

 qed

 lemma frontier_ball:

   fixes a :: "'a::real_normed_vector"

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

   apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)

   apply (simp add: set_eq_iff)

   by arith

 lemma frontier_cball:

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

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

   apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)

   apply (simp add: set_eq_iff)

   by arith

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

   apply (simp add: set_eq_iff not_le)

   by (metis zero_le_dist dist_self order_less_le_trans)

 lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

 lemma cball_eq_sing:

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

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

 proof (rule linorder_cases)

   assume e: "0 < e"

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

     using perfect_choose_dist [OF e] by auto

   hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)

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

 qed auto

 lemma cball_sing:

   fixes x :: "'a::metric_space"

   shows "e = 0 ==> cball x e = {x}"

   by (auto simp add: set_eq_iff)

 subsection {* Boundedness *}

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

 definition (in metric_space)

   bounded :: "'a set \<Rightarrow> bool" where

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

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

 unfolding bounded_def

 apply safe

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

 apply (drule (1) bspec)

 apply (erule order_trans [OF dist_triangle add_left_mono])

 apply auto

 done

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

 unfolding bounded_any_center [where a=0]

 by (simp add: dist_norm)

 lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"

   unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)

   using assms by auto

 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)

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

   by (metis bounded_def subset_eq)

 lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"

   by (metis bounded_subset interior_subset)

 lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"

 proof-

   from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto

   { fix y assume "y \<in> closure S"

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

       unfolding closure_sequential by auto

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

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

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

     have "dist x y \<le> a"

       apply (rule Lim_dist_ubound [of sequentially f])

       apply (rule trivial_limit_sequentially)

       apply (rule f(2))

       apply fact

       done

   }

   thus ?thesis unfolding bounded_def by auto

 qed

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

   apply (simp add: bounded_def)

   apply (rule_tac x=x in exI)

   apply (rule_tac x=e in exI)

   apply auto

   done

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

   by (metis ball_subset_cball bounded_cball bounded_subset)

 lemma finite_imp_bounded[intro]:

   fixes S :: "'a::metric_space set" assumes "finite S" shows "bounded S"

 proof-

   { fix a and F :: "'a set" assume as:"bounded F"

     then obtain x e where "\<forall>y\<in>F. dist x y \<le> e" unfolding bounded_def by auto

     hence "\<forall>y\<in>(insert a F). dist x y \<le> max e (dist x a)" by auto

     hence "bounded (insert a F)" unfolding bounded_def by (intro exI)

   }

   thus ?thesis using finite_induct[of S bounded]  using bounded_empty assms by auto

 qed

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

   apply (auto simp add: bounded_def)

   apply (rename_tac x y r s)

   apply (rule_tac x=x in exI)

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

   apply (rule ballI, rename_tac z, safe)

   apply (drule (1) bspec, simp)

   apply (drule (1) bspec)

   apply (rule min_max.le_supI2)

   apply (erule order_trans [OF dist_triangle add_left_mono])

   done

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

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

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

   apply (simp add: bounded_iff)

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

   by metis arith

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

   by (metis Int_lower1 Int_lower2 bounded_subset)

 lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"

 apply (metis Diff_subset bounded_subset)

 done

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

   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)

 lemma not_bounded_UNIV[simp, intro]:

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

 proof(auto simp add: bounded_pos not_le)

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

     using perfect_choose_dist [OF zero_less_one] by fast

   fix b::real  assume b: "b >0"

   have b1: "b +1 \<ge> 0" using b by simp

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

     by (simp add: norm_sgn)

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

 qed

 lemma bounded_linear_image:

   assumes "bounded S" "bounded_linear f"

   shows "bounded(f ` S)"

 proof-

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

   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)

   { fix x assume "x\<in>S"

     hence "norm x \<le> b" using b by auto

     hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)

       by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)

   }

   thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)

     using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)

 qed

 lemma bounded_scaling:

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

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

   apply (rule bounded_linear_image, assumption)

   apply (rule bounded_linear_scaleR_right)

   done

 lemma bounded_translation:

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

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

 proof-

   from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto

   { fix x assume "x\<in>S"

     hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto

   }

   thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]

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

 qed

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

 lemma bounded_real:

   fixes S :: "real set"

   shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

   by (simp add: bounded_iff)

 lemma bounded_has_Sup:

   fixes S :: "real set"

   assumes "bounded S" "S \<noteq> {}"

   shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"

 proof

   fix x assume "x\<in>S"

   thus "x \<le> Sup S"

     by (metis SupInf.Sup_upper abs_le_D1 assms(1) bounded_real)

 next

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

     by (metis SupInf.Sup_least)

 qed

 lemma Sup_insert:

   fixes S :: "real set"

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

 by auto (metis Int_absorb Sup_insert_nonempty assms bounded_has_Sup(1) disjoint_iff_not_equal) 

 lemma Sup_insert_finite:

   fixes S :: "real set"

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

   apply (rule Sup_insert)

   apply (rule finite_imp_bounded)

   by simp

 lemma bounded_has_Inf:

   fixes S :: "real set"

   assumes "bounded S"  "S \<noteq> {}"

   shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"

 proof

   fix x assume "x\<in>S"

   from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto

   thus "x \<ge> Inf S" using `x\<in>S`

     by (metis Inf_lower_EX abs_le_D2 minus_le_iff)

 next

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

     by (metis SupInf.Inf_greatest)

 qed

 lemma Inf_insert:

   fixes S :: "real set"

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

 by auto (metis Int_absorb Inf_insert_nonempty bounded_has_Inf(1) disjoint_iff_not_equal)

 lemma Inf_insert_finite:

   fixes S :: "real set"

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

   by (rule Inf_insert, rule finite_imp_bounded, simp)

 subsection {* Compactness *}

 subsubsection{* Open-cover compactness *}

 definition compact :: "'a::topological_space set \<Rightarrow> bool" where

   compact_eq_heine_borel: -- "This name is used for backwards compatibility"

     "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))"

 lemma compactI:

   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'"

   shows "compact s"

   unfolding compact_eq_heine_borel using assms by metis

 lemma compactE:

   assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"

   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"

   using assms unfolding compact_eq_heine_borel by metis

 lemma compactE_image:

   assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"

   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"

   using assms unfolding ball_simps[symmetric] SUP_def

   by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])

 subsubsection {* Bolzano-Weierstrass property *}

 lemma heine_borel_imp_bolzano_weierstrass:

   assumes "compact s" "infinite t"  "t \<subseteq> s"

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

 proof(rule ccontr)

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

   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

     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

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

     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

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

   { fix x y assume "x\<in>t" "y\<in>t" "f x = f y"

     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

     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  }

   hence "inj_on f t" unfolding inj_on_def by simp

   hence "infinite (f ` t)" using assms(2) using finite_imageD by auto

   moreover

   { fix x assume "x\<in>t" "f x \<notin> g"

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

     then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto

     hence "y = x" using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto

     hence False using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto  }

   hence "f ` t \<subseteq> g" by auto

   ultimately show False using g(2) using finite_subset by auto

 qed

 lemma acc_point_range_imp_convergent_subsequence:

   fixes l :: "'a :: first_countable_topology"

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

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

 proof -

   from countable_basis_at_decseq[of l] guess A . note A = this

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

   { fix n i

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

       using l A by auto

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

       unfolding ex_in_conv by (intro notI) simp

     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"

       by auto

     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"

       by (auto simp: not_le)

     then have "i < s n i" "f (s n i) \<in> A (Suc n)"

       unfolding s_def by (auto intro: someI2_ex) }

   note s = this

   def r \<equiv> "nat_rec (s 0 0) s"

   have "subseq r"

     by (auto simp: r_def s subseq_Suc_iff)

   moreover

   have "(\<lambda>n. f (r n)) ----> l"

   proof (rule topological_tendstoI)

     fix S assume "open S" "l \<in> S"

     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto

     moreover

     { fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"

         by (cases i) (simp_all add: r_def s) }

     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)

     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"

       by eventually_elim auto

   qed

   ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

     by (auto simp: convergent_def comp_def)

 qed

 lemma sequence_infinite_lemma:

   fixes f :: "nat \<Rightarrow> 'a::t1_space"

   assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"

   shows "infinite (range f)"

 proof

   assume "finite (range f)"

   hence "closed (range f)" by (rule finite_imp_closed)

   hence "open (- range f)" by (rule open_Compl)

   from assms(1) have "l \<in> - range f" by auto

   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"

     using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)

   thus False unfolding eventually_sequentially by auto

 qed

 lemma closure_insert:

   fixes x :: "'a::t1_space"

   shows "closure (insert x s) = insert x (closure s)"

 apply (rule closure_unique)

 apply (rule insert_mono [OF closure_subset])

 apply (rule closed_insert [OF closed_closure])

 apply (simp add: closure_minimal)

 done

 lemma islimpt_insert:

   fixes x :: "'a::t1_space"

   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"

 proof

   assume *: "x islimpt (insert a s)"

   show "x islimpt s"

   proof (rule islimptI)

     fix t assume t: "x \<in> t" "open t"

     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"

     proof (cases "x = a")

       case True

       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"

         using * t by (rule islimptE)

       with `x = a` show ?thesis by auto

     next

       case False

       with t have t': "x \<in> t - {a}" "open (t - {a})"

         by (simp_all add: open_Diff)

       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"

         using * t' by (rule islimptE)

       thus ?thesis by auto

     qed

   qed

 next

   assume "x islimpt s" thus "x islimpt (insert a s)"

     by (rule islimpt_subset) auto

 qed

 lemma islimpt_finite:

   fixes x :: "'a::t1_space"

   shows "finite s \<Longrightarrow> \<not> x islimpt s"

 by (induct set: finite, simp_all add: islimpt_insert)

 lemma islimpt_union_finite:

   fixes x :: "'a::t1_space"

   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

 by (simp add: islimpt_Un islimpt_finite)

 lemma islimpt_eq_acc_point:

   fixes l :: "'a :: t1_space"

   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"

 proof (safe intro!: islimptI)

   fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"

   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"

     by (auto intro: finite_imp_closed)

   then show False

     by (rule islimptE) auto

 next

   fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"

   then have "infinite (T \<inter> S - {l})" by auto

   then have "\<exists>x. x \<in> (T \<inter> S - {l})"

     unfolding ex_in_conv by (intro notI) simp

   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"

     by auto

 qed

 lemma islimpt_range_imp_convergent_subsequence:

   fixes l :: "'a :: {t1_space, first_countable_topology}"

   assumes l: "l islimpt (range f)"

   shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"

   using l unfolding islimpt_eq_acc_point

   by (rule acc_point_range_imp_convergent_subsequence)

 lemma sequence_unique_limpt:

   fixes f :: "nat \<Rightarrow> 'a::t2_space"

   assumes "(f ---> l) sequentially" and "l' islimpt (range f)"

   shows "l' = l"

 proof (rule ccontr)

   assume "l' \<noteq> l"

   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"

     using hausdorff [OF `l' \<noteq> l`] by auto

   have "eventually (\<lambda>n. f n \<in> t) sequentially"

     using assms(1) `open t` `l \<in> t` by (rule topological_tendstoD)

   then obtain N where "\<forall>n\<ge>N. f n \<in> t"

     unfolding eventually_sequentially by auto

   have "UNIV = {..<N} \<union> {N..}" by auto

   hence "l' islimpt (f ` ({..<N} \<union> {N..}))" using assms(2) by simp

   hence "l' islimpt (f ` {..<N} \<union> f ` {N..})" by (simp add: image_Un)

   hence "l' islimpt (f ` {N..})" by (simp add: islimpt_union_finite)

   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"

     using `l' \<in> s` `open s` by (rule islimptE)

   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto

   with `\<forall>n\<ge>N. f n \<in> t` have "f n \<in> s \<inter> t" by simp

   with `s \<inter> t = {}` show False by simp

 qed

 lemma bolzano_weierstrass_imp_closed:

   fixes s :: "'a::{first_countable_topology, t2_space} set"

   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

   shows "closed s"

 proof-

   { fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"

     hence "l \<in> s"

     proof(cases "\<forall>n. x n \<noteq> l")

       case False thus "l\<in>s" using as(1) by auto

     next

       case True note cas = this

       with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto

       then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto

       thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto

     qed  }

   thus ?thesis unfolding closed_sequential_limits by fast

 qed

 lemma compact_imp_closed:

   fixes s :: "'a::t2_space set"

   assumes "compact s" shows "closed s"

 unfolding closed_def

 proof (rule openI)

   fix y assume "y \<in> - s"

   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"

   note `compact s`

   moreover have "\<forall>u\<in>?C. open u" by simp

   moreover have "s \<subseteq> \<Union>?C"

   proof

     fix x assume "x \<in> s"

     with `y \<in> - s` have "x \<noteq> y" by clarsimp

     hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"

       by (rule hausdorff)

     with `x \<in> s` show "x \<in> \<Union>?C"

       unfolding eventually_nhds by auto

   qed

   ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"

     by (rule compactE)

   from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto

   with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"

     by (simp add: eventually_Ball_finite)

   with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"

     by (auto elim!: eventually_mono [rotated])

   thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"

     by (simp add: eventually_nhds subset_eq)

 qed

 lemma compact_imp_bounded:

   assumes "compact U" shows "bounded U"

 proof -

   have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto

   then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"

     by (elim compactE_image)

   def d \<equiv> "SOME d. d \<in> D"

   show "bounded U"

     unfolding bounded_def

   proof (intro exI, safe)

     fix x assume "x \<in> U"

     with D obtain d' where "d' \<in> D" "x \<in> ball d' 1" by auto

     moreover have "dist d x \<le> dist d d' + dist d' x"

       using dist_triangle[of d x d'] by (simp add: dist_commute)

     moreover

     from `x\<in>U` D have "d \<in> D"

       unfolding d_def by (rule_tac someI_ex) auto

     ultimately

     show "dist d x \<le> Max ((\<lambda>d'. dist d d' + 1) ` D)"

       using D by (subst Max_ge_iff) (auto intro!: bexI[of _ d'])

   qed

 qed

 text{* In particular, some common special cases. *}

 lemma compact_empty[simp]:

  "compact {}"

   unfolding compact_eq_heine_borel

   by auto

 lemma compact_union [intro]:

   assumes "compact s" "compact t" shows " compact (s \<union> t)"

 proof (rule compactI)

   fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"

   from * `compact s` obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"

     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

   moreover from * `compact t` obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"

     unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis

   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"

     by (auto intro!: exI[of _ "s' \<union> t'"])

 qed

 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"

   by (induct set: finite) auto

 lemma compact_UN [intro]:

   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"

   unfolding SUP_def by (rule compact_Union) auto

 lemma compact_inter_closed [intro]:

   assumes "compact s" and "closed t"

   shows "compact (s \<inter> t)"

 proof (rule compactI)

   fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"

   from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto

   moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto

   ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"

     using `compact s` unfolding compact_eq_heine_borel by auto

   then guess D ..

   then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"

     by (intro exI[of _ "D - {-t}"]) auto

 qed

 lemma closed_inter_compact [intro]:

   assumes "closed s" and "compact t"

   shows "compact (s \<inter> t)"

   using compact_inter_closed [of t s] assms

   by (simp add: Int_commute)