(*  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 SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith

 begin

 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 (open) '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"])

   by auto

 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 [simp]:

   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 [simp]:

   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_cball[simp]: "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 centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)

 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

 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_finite:

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

   assumes fS: "finite S" shows "\<not> a islimpt S"

   unfolding islimpt_approachable

   using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)

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

   apply (rule iffI)

   defer

   apply (metis Un_upper1 Un_upper2 islimpt_subset)

   unfolding islimpt_def

   apply (rule ccontr, clarsimp, rename_tac A B)

   apply (drule_tac x="A \<inter> B" in spec)

   apply (auto simp add: open_Int)

   done

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

 proof

   assume "?lhs"

   { fix e::real

     assume "e > 0"

     let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"

     { assume "a\<in>S"

       have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto

       moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`

         unfolding frontier_closures closure_def islimpt_def using `e>0`

         by (auto, erule_tac x="ball a e" in allE, auto)

       ultimately have ?rhse by auto

     }

     moreover

     { assume "a\<notin>S"

       hence ?rhse using `?lhs`

         unfolding frontier_closures closure_def islimpt_def

         using open_ball[of a e] `e > 0`

           by simp (metis centre_in_ball mem_ball open_ball) 

     }

     ultimately have ?rhse by auto

   }

   thus ?rhs by auto

 next

   assume ?rhs

   moreover

   { fix T assume "a\<notin>S" and

     as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"

     from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto

     then obtain e where "e>0" "ball a e \<subseteq> T" by auto

     then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto

     have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"

       using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto

   }

   hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto

   moreover

   { fix T assume "a \<in> T"  "open T" "a\<in>S"

     then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto

     obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto

     hence "\<exists>y\<in>- S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto

   }

   hence "a islimpt (- S) \<or> a\<notin>S" unfolding islimpt_def by auto

   ultimately show ?lhs unfolding frontier_closures using closure_def[of "- S"] by auto

 qed

 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

   at_infinity :: "'a::real_normed_vector filter" where

   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"

 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{* Prove That They are all filters. *}

 lemma eventually_at_infinity:

   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"

 unfolding at_infinity_def

 proof (rule eventually_Abs_filter, rule is_filter.intro)

   fix P Q :: "'a \<Rightarrow> bool"

   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"

   then obtain r s where

     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto

   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp

   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..

 qed auto

 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 add: within_UNIV)

 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: "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)"

 unfolding eventually_within eventually_at dist_nz by auto

 lemma eventually_within_le: "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)" (is "?lhs = ?rhs")

 unfolding eventually_within

 by auto (metis Rats_dense_in_nn_real order_le_less_trans order_refl) 

 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"

   unfolding trivial_limit_def by (auto elim: eventually_rev_mp)

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

   unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)

 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_sequentially:

  "(S ---> l) sequentially \<longleftrightarrow>

           (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"

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

 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::metric_space"

   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

   then obtain f where f:"\<forall>y. y>0 \<longrightarrow> f y \<in> S \<and> f y \<noteq> x \<and> dist (f y) x < y"

     unfolding islimpt_approachable

     using choice[of "\<lambda>e y. e>0 \<longrightarrow> y\<in>S \<and> y\<noteq>x \<and> dist y x < e"] by auto

   let ?I = "\<lambda>n. inverse (real (Suc n))"

   have "\<forall>n. f (?I n) \<in> S - {x}" using f by simp

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

   proof (rule metric_tendsto_imp_tendsto)

     show "?I ----> 0"

       by (rule LIMSEQ_inverse_real_of_nat)

     show "eventually (\<lambda>n. dist (f (?I n)) x \<le> dist (?I n) 0) sequentially"

       by (simp add: norm_conv_dist [symmetric] less_imp_le f)

   qed

   ultimately show ?rhs by fast

 next

   assume ?rhs

   then obtain f::"nat\<Rightarrow>'a"  where f:"(\<forall>n. f n \<in> S - {x})" "(\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f n) x < e)" unfolding Lim_sequentially by auto

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

     then obtain N where "dist (f N) x < e" using f(2) by auto

     moreover have "f N\<in>S" "f N \<noteq> x" using f(1) by auto

     ultimately have "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" by auto

   }

   thus ?lhs unfolding islimpt_approachable by auto

 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 tendsto_def Limits.eventually_within eventually_at_topological

   by auto

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

 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)

 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"

   apply (subst within_UNIV[symmetric])

   using netlimit_within[of a UNIV]

   by (simp add: within_UNIV)

 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 (auto simp add: within_UNIV)

 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::metric_space"

   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::metric_space 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)

 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 left_distrib 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: left_distrib 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_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!: add 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)

 (* 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

 subsection {* Equivalent versions of compactness *}

 subsubsection{* Sequential compactness *}

 definition

   compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)

   "compact S \<longleftrightarrow>

    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>

        (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

 lemma compactI:

   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"

   shows "compact S"

   unfolding compact_def using assms by fast

 lemma compactE:

   assumes "compact S" "\<forall>n. f n \<in> S"

   obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"

   using assms unfolding compact_def by fast

 text {*

   A metric space (or topological vector space) is said to have the

   Heine-Borel property if every closed and bounded subset is compact.

 *}

 class heine_borel = metric_space +

   assumes bounded_imp_convergent_subsequence:

     "bounded s \<Longrightarrow> \<forall>n. f n \<in> s

       \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

 lemma bounded_closed_imp_compact:

   fixes s::"'a::heine_borel set"

   assumes "bounded s" and "closed s" shows "compact s"

 proof (unfold compact_def, clarify)

   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

   obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

     using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto

   from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp

   have "l \<in> s" using `closed s` fr l

     unfolding closed_sequential_limits by blast

   show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

     using `l \<in> s` r l by blast

 qed

 lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"

 proof(induct n)

   show "0 \<le> r 0" by auto

 next

   fix n assume "n \<le> r n"

   moreover have "r n < r (Suc n)"

     using assms [unfolded subseq_def] by auto

   ultimately show "Suc n \<le> r (Suc n)" by auto

 qed

 lemma eventually_subseq:

   assumes r: "subseq r"

   shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"

 unfolding eventually_sequentially

 by (metis subseq_bigger [OF r] le_trans)

 lemma lim_subseq:

   "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"

 unfolding tendsto_def eventually_sequentially o_def

 by (metis subseq_bigger le_trans)

 lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"

   unfolding Ex1_def

   apply (rule_tac x="nat_rec e f" in exI)

   apply (rule conjI)+

 apply (rule def_nat_rec_0, simp)

 apply (rule allI, rule def_nat_rec_Suc, simp)

 apply (rule allI, rule impI, rule ext)

 apply (erule conjE)

 apply (induct_tac x)

 apply simp

 apply (erule_tac x="n" in allE)

 apply (simp)

 done

 lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"

   assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"

   shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N.  abs(s n - l) < e"

 proof-

   have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto

   then obtain t where t:"isLub UNIV (range s) t" using reals_complete[of "range s" ] by auto

   { fix e::real assume "e>0" and as:"\<forall>N. \<exists>n\<ge>N. \<not> \<bar>s n - t\<bar> < e"

     { fix n::nat

       obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto

       have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto

       with n have "s N \<le> t - e" using `e>0` by auto

       hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto  }

     hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto

     hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto  }

   thus ?thesis by blast

 qed

 lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"

   assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"

   shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"

   using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]

   unfolding monoseq_def incseq_def

   apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]

   unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto

 (* TODO: merge this lemma with the ones above *)

 lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real"

   assumes "bounded {s n| n::nat. True}"  "\<forall>n. (s n) \<le>(s(Suc n))"

   shows "\<exists>l. (s ---> l) sequentially"

 proof-

   obtain a where a:"\<forall>n. \<bar> (s n)\<bar> \<le>  a" using assms(1)[unfolded bounded_iff] by auto

   { fix m::nat

     have "\<And> n. n\<ge>m \<longrightarrow>  (s m) \<le> (s n)"

       apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE)

       apply(case_tac "m \<le> na") unfolding not_less_eq_eq by(auto simp add: not_less_eq_eq)  }

   hence "\<forall>m n. m \<le> n \<longrightarrow> (s m) \<le> (s n)" by auto

   then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar> (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a]

     unfolding monoseq_def by auto

   thus ?thesis unfolding Lim_sequentially apply(rule_tac x="l" in exI)

     unfolding dist_norm  by auto

 qed

 lemma compact_real_lemma:

   assumes "\<forall>n::nat. abs(s n) \<le> b"

   shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"

 proof-

   obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"

     using seq_monosub[of s] by auto

   thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms

     unfolding tendsto_iff dist_norm eventually_sequentially by auto

 qed

 instance real :: heine_borel

 proof

   fix s :: "real set" and f :: "nat \<Rightarrow> real"

   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

   then obtain b where b: "\<forall>n. abs (f n) \<le> b"

     unfolding bounded_iff by auto

   obtain l :: real and r :: "nat \<Rightarrow> nat" where

     r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"

     using compact_real_lemma [OF b] by auto

   thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

     by auto

 qed

 lemma bounded_component: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $$ i) ` s)"

   apply (erule bounded_linear_image)

   apply (rule bounded_linear_euclidean_component)

   done

 lemma compact_lemma:

   fixes f :: "nat \<Rightarrow> 'a::euclidean_space"

   assumes "bounded s" and "\<forall>n. f n \<in> s"

   shows "\<forall>d. \<exists>l::'a. \<exists> r. subseq r \<and>

         (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"

 proof

   fix d'::"nat set" def d \<equiv> "d' \<inter> {..<DIM('a)}"

   have "finite d" "d\<subseteq>{..<DIM('a)}" unfolding d_def by auto

   hence "\<exists>l::'a. \<exists>r. subseq r \<and>

       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"

   proof(induct d) case empty thus ?case unfolding subseq_def by auto

   next case (insert k d) have k[intro]:"k<DIM('a)" using insert by auto

     have s': "bounded ((\<lambda>x. x $$ k) ` s)" using `bounded s` by (rule bounded_component)

     obtain l1::"'a" and r1 where r1:"subseq r1" and

       lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially"

       using insert(3) using insert(4) by auto

     have f': "\<forall>n. f (r1 n) $$ k \<in> (\<lambda>x. x $$ k) ` s" using `\<forall>n. f n \<in> s` by simp

     obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $$ k) ---> l2) sequentially"

       using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto

     def r \<equiv> "r1 \<circ> r2" have r:"subseq r"

       using r1 and r2 unfolding r_def o_def subseq_def by auto

     moreover

     def l \<equiv> "(\<chi>\<chi> i. if i = k then l2 else l1$$i)::'a"

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

       from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $$ i) (l1 $$ i) < e) sequentially" by blast

       from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $$ k) l2 < e) sequentially" by (rule tendstoD)

       from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $$ i) (l1 $$ i) < e) sequentially"

         by (rule eventually_subseq)

       have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $$ i) (l $$ i) < e) sequentially"

         using N1' N2 apply(rule eventually_elim2) unfolding l_def r_def o_def

         using insert.prems by auto

     }

     ultimately show ?case by auto

   qed

   thus "\<exists>l::'a. \<exists>r. subseq r \<and>

       (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d'. dist (f (r n) $$ i) (l $$ i) < e) sequentially)"

     apply safe apply(rule_tac x=l in exI,rule_tac x=r in exI) apply safe

     apply(erule_tac x=e in allE) unfolding d_def eventually_sequentially apply safe 

     apply(rule_tac x=N in exI) apply safe apply(erule_tac x=n in allE,safe)

     apply(erule_tac x=i in ballE) 

   proof- fix i and r::"nat=>nat" and n::nat and e::real and l::'a

     assume "i\<in>d'" "i \<notin> d' \<inter> {..<DIM('a)}" and e:"e>0"

     hence *:"i\<ge>DIM('a)" by auto

     thus "dist (f (r n) $$ i) (l $$ i) < e" using e by auto

   qed

 qed

 instance euclidean_space \<subseteq> heine_borel

 proof

   fix s :: "'a set" and f :: "nat \<Rightarrow> 'a"

   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

   then obtain l::'a and r where r: "subseq r"

     and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $$ i) (l $$ i) < e) sequentially"

     using compact_lemma [OF s f] by blast

   let ?d = "{..<DIM('a)}"

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

     hence "0 < e / (real_of_nat (card ?d))"

       using DIM_positive using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto

     with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))) sequentially"

       by simp

     moreover

     { fix n assume n: "\<forall>i. dist (f (r n) $$ i) (l $$ i) < e / (real_of_nat (card ?d))"

       have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $$ i) (l $$ i))"

         apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)

       also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"

         apply(rule setsum_strict_mono) using n by auto

       finally have "dist (f (r n)) l < e" unfolding setsum_constant

         using DIM_positive[where 'a='a] by auto

     }

     ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"

       by (rule eventually_elim1)

   }

   hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp

   with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto

 qed

 lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"

 unfolding bounded_def

 apply clarify

 apply (rule_tac x="a" in exI)

 apply (rule_tac x="e" in exI)

 apply clarsimp

 apply (drule (1) bspec)

 apply (simp add: dist_Pair_Pair)

 apply (erule order_trans [OF real_sqrt_sum_squares_ge1])

 done

 lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"

 unfolding bounded_def

 apply clarify

 apply (rule_tac x="b" in exI)

 apply (rule_tac x="e" in exI)

 apply clarsimp

 apply (drule (1) bspec)

 apply (simp add: dist_Pair_Pair)

 apply (erule order_trans [OF real_sqrt_sum_squares_ge2])

 done

 instance prod :: (heine_borel, heine_borel) heine_borel

 proof

   fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"

   assume s: "bounded s" and f: "\<forall>n. f n \<in> s"

   from s have s1: "bounded (fst ` s)" by (rule bounded_fst)

   from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp

   obtain l1 r1 where r1: "subseq r1"

     and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"

     using bounded_imp_convergent_subsequence [OF s1 f1]

     unfolding o_def by fast

   from s have s2: "bounded (snd ` s)" by (rule bounded_snd)

   from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp

   obtain l2 r2 where r2: "subseq r2"

     and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"

     using bounded_imp_convergent_subsequence [OF s2 f2]

     unfolding o_def by fast

   have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"

     using lim_subseq [OF r2 l1] unfolding o_def .

   have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"

     using tendsto_Pair [OF l1' l2] unfolding o_def by simp

   have r: "subseq (r1 \<circ> r2)"

     using r1 r2 unfolding subseq_def by simp

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

     using l r by fast

 qed

 subsubsection{* Completeness *}

 lemma cauchy_def:

   "Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"

 unfolding Cauchy_def by blast

 definition

   complete :: "'a::metric_space set \<Rightarrow> bool" where

   "complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f

                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"

 lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")

 proof-

   { assume ?rhs

     { fix e::real

       assume "e>0"

       with `?rhs` obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"

         by (erule_tac x="e/2" in allE) auto

       { fix n m

         assume nm:"N \<le> m \<and> N \<le> n"

         hence "dist (s m) (s n) < e" using N

           using dist_triangle_half_l[of "s m" "s N" "e" "s n"]

           by blast

       }

       hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"

         by blast

     }

     hence ?lhs

       unfolding cauchy_def

       by blast

   }

   thus ?thesis

     unfolding cauchy_def

     using dist_triangle_half_l

     by blast

 qed

 lemma convergent_imp_cauchy:

  "(s ---> l) sequentially ==> Cauchy s"

 proof(simp only: cauchy_def, rule, rule)

   fix e::real assume "e>0" "(s ---> l) sequentially"

   then obtain N::nat where N:"\<forall>n\<ge>N. dist (s n) l < e/2" unfolding Lim_sequentially by(erule_tac x="e/2" in allE) auto

   thus "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"  using dist_triangle_half_l[of _ l e _] by (rule_tac x=N in exI) auto

 qed

 lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"

 proof-

   from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto

   hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto

   moreover

   have "bounded (s ` {0..N})" using finite_imp_bounded[of "s ` {1..N}"] by auto

   then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"

     unfolding bounded_any_center [where a="s N"] by auto

   ultimately show "?thesis"

     unfolding bounded_any_center [where a="s N"]

     apply(rule_tac x="max a 1" in exI) apply auto

     apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto

 qed

 lemma compact_imp_complete: assumes "compact s" shows "complete s"

 proof-

   { fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"

     from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast

     note lr' = subseq_bigger [OF lr(2)]

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

       from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto

       from lr(3)[unfolded Lim_sequentially, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using `e>0` by auto

       { fix n::nat assume n:"n \<ge> max N M"

         have "dist ((f \<circ> r) n) l < e/2" using n M by auto

         moreover have "r n \<ge> N" using lr'[of n] n by auto

         hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto

         ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }

       hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }

     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `l\<in>s` unfolding Lim_sequentially by auto  }

   thus ?thesis unfolding complete_def by auto

 qed

 instance heine_borel < complete_space

 proof

   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

   hence "bounded (range f)"

     by (rule cauchy_imp_bounded)

   hence "compact (closure (range f))"

     using bounded_closed_imp_compact [of "closure (range f)"] by auto

   hence "complete (closure (range f))"

     by (rule compact_imp_complete)

   moreover have "\<forall>n. f n \<in> closure (range f)"

     using closure_subset [of "range f"] by auto

   ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"

     using `Cauchy f` unfolding complete_def by auto

   then show "convergent f"

     unfolding convergent_def by auto

 qed

 lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"

 proof(simp add: complete_def, rule, rule)

   fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"

   hence "convergent f" by (rule Cauchy_convergent)

   thus "\<exists>l. f ----> l" unfolding convergent_def .  

 qed

 lemma complete_imp_closed: assumes "complete s" shows "closed s"

 proof -

   { fix x assume "x islimpt s"

     then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"

       unfolding islimpt_sequential by auto

     then obtain l where l: "l\<in>s" "(f ---> l) sequentially"

       using `complete s`[unfolded complete_def] using convergent_imp_cauchy[of f x] by auto

     hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto

   }

   thus "closed s" unfolding closed_limpt by auto

 qed

 lemma complete_eq_closed:

   fixes s :: "'a::complete_space set"

   shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")

 proof

   assume ?lhs thus ?rhs by (rule complete_imp_closed)

 next

   assume ?rhs

   { fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"

     then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto

     hence "\<exists>l\<in>s. (f ---> l) sequentially" using `?rhs`[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }

   thus ?lhs unfolding complete_def by auto

 qed

 lemma convergent_eq_cauchy:

   fixes s :: "nat \<Rightarrow> 'a::complete_space"

   shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s" (is "?lhs = ?rhs")

 proof

   assume ?lhs then obtain l where "(s ---> l) sequentially" by auto

   thus ?rhs using convergent_imp_cauchy by auto

 next

   assume ?rhs thus ?lhs using complete_univ[unfolded complete_def, THEN spec[where x=s]] by auto

 qed

 lemma convergent_imp_bounded:

   fixes s :: "nat \<Rightarrow> 'a::metric_space"

   shows "(s ---> l) sequentially ==> bounded (s ` (UNIV::(nat set)))"

   using convergent_imp_cauchy[of s]

   using cauchy_imp_bounded[of s]

   unfolding image_def

   by auto

 subsubsection{* Total boundedness *}

 fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where

   "helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"

 declare helper_1.simps[simp del]

 lemma compact_imp_totally_bounded:

   assumes "compact s"

   shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e) ` k))"

 proof(rule, rule, rule ccontr)

   fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k)"

   def x \<equiv> "helper_1 s e"

   { fix n

     have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"

     proof(induct_tac rule:nat_less_induct)

       fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"

       assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"

       have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x ` {0 ..< n}" in allE) using as by auto

       then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)" unfolding subset_eq by auto

       have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]

         apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto

       thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto

     qed }

   hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+

   then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto

   from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto

   then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto

   show False

     using N[THEN spec[where x=N], THEN spec[where x="N+1"]]

     using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]

     using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto

 qed

 subsubsection{* Heine-Borel theorem *}

 text {* Following Burkill \& Burkill vol. 2. *}

 lemma heine_borel_lemma: fixes s::"'a::metric_space set"

   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"

   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"

 proof(rule ccontr)

   assume "\<not> (\<exists>e>0. \<forall>x\<in>s. \<exists>b\<in>t. ball x e \<subseteq> b)"

   hence cont:"\<forall>e>0. \<exists>x\<in>s. \<forall>xa\<in>t. \<not> (ball x e \<subseteq> xa)" by auto

   { fix n::nat

     have "1 / real (n + 1) > 0" by auto

     hence "\<exists>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> (ball x (inverse (real (n+1))) \<subseteq> xa))" using cont unfolding Bex_def by auto }

   hence "\<forall>n::nat. \<exists>x. x \<in> s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)" by auto

   then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"

     using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto

   then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"

     using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto

   obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto

   then obtain e where "e>0" and e:"\<forall>z. dist z l < e \<longrightarrow> z\<in>b"

     using assms(3)[THEN bspec[where x=b]] unfolding open_dist by auto

   then obtain N1 where N1:"\<forall>n\<ge>N1. dist ((f \<circ> r) n) l < e / 2"

     using lr[unfolded Lim_sequentially, THEN spec[where x="e/2"]] by auto

   obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto

   have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"

     apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2

     using subseq_bigger[OF r, of "N1 + N2"] by auto

   def x \<equiv> "(f (r (N1 + N2)))"

   have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def

     using f[THEN spec[where x="r (N1 + N2)"]] using `b\<in>t` by auto

   have "\<exists>y\<in>ball x (inverse (real (r (N1 + N2) + 1))). y\<notin>b" apply(rule ccontr) using x by auto

   then obtain y where y:"y \<in> ball x (inverse (real (r (N1 + N2) + 1)))" "y \<notin> b" by auto

   have "dist x l < e/2" using N1 unfolding x_def o_def by auto

   hence "dist y l < e" using y N2' using dist_triangle[of y l x]by (auto simp add:dist_commute)

   thus False using e and `y\<notin>b` by auto

 qed

 lemma compact_imp_heine_borel: "compact s ==> (\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f)

                \<longrightarrow> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f')))"

 proof clarify

   fix f assume "compact s" " \<forall>t\<in>f. open t" "s \<subseteq> \<Union>f"

   then obtain e::real where "e>0" and "\<forall>x\<in>s. \<exists>b\<in>f. ball x e \<subseteq> b" using heine_borel_lemma[of s f] by auto

   hence "\<forall>x\<in>s. \<exists>b. b\<in>f \<and> ball x e \<subseteq> b" by auto

   hence "\<exists>bb. \<forall>x\<in>s. bb x \<in>f \<and> ball x e \<subseteq> bb x" using bchoice[of s "\<lambda>x b. b\<in>f \<and> ball x e \<subseteq> b"] by auto

   then obtain  bb where bb:"\<forall>x\<in>s. (bb x) \<in> f \<and> ball x e \<subseteq> (bb x)" by blast

   from `compact s` have  "\<exists> k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" using compact_imp_totally_bounded[of s] `e>0` by auto

   then obtain k where k:"finite k" "k \<subseteq> s" "s \<subseteq> \<Union>(\<lambda>x. ball x e) ` k" by auto

   have "finite (bb ` k)" using k(1) by auto

   moreover

   { fix x assume "x\<in>s"

     hence "x\<in>\<Union>(\<lambda>x. ball x e) ` k" using k(3)  unfolding subset_eq by auto

     hence "\<exists>X\<in>bb ` k. x \<in> X" using bb k(2) by blast

     hence "x \<in> \<Union>(bb ` k)" using  Union_iff[of x "bb ` k"] by auto

   }

   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<subseteq> \<Union>f'" using bb k(2) by (rule_tac x="bb ` k" in exI) auto

 qed

 subsubsection {* Bolzano-Weierstrass property *}

 lemma heine_borel_imp_bolzano_weierstrass:

   assumes "\<forall>f. (\<forall>t \<in> f. open t) \<and> s \<subseteq> (\<Union> f) --> (\<exists>f'. f' \<subseteq> f \<and> finite f' \<and> s \<subseteq> (\<Union> f'))"

           "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)[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

 subsubsection {* Complete the chain of compactness variants *}

 lemma islimpt_range_imp_convergent_subsequence:

   fixes f :: "nat \<Rightarrow> 'a::metric_space"

   assumes "l islimpt (range f)"

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

 proof (intro exI conjI)

   have *: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. 0 < dist (f n) l \<and> dist (f n) l < e"

     using assms unfolding islimpt_def

     by (drule_tac x="ball l e" in spec)

        (auto simp add: zero_less_dist_iff dist_commute)

   def t \<equiv> "\<lambda>e. LEAST n. 0 < dist (f n) l \<and> dist (f n) l < e"

   have f_t_neq: "\<And>e. 0 < e \<Longrightarrow> 0 < dist (f (t e)) l"

     unfolding t_def by (rule LeastI2_ex [OF * conjunct1])

   have f_t_closer: "\<And>e. 0 < e \<Longrightarrow> dist (f (t e)) l < e"

     unfolding t_def by (rule LeastI2_ex [OF * conjunct2])

   have t_le: "\<And>n e. 0 < dist (f n) l \<Longrightarrow> dist (f n) l < e \<Longrightarrow> t e \<le> n"

     unfolding t_def by (simp add: Least_le)

   have less_tD: "\<And>n e. n < t e \<Longrightarrow> 0 < dist (f n) l \<Longrightarrow> e \<le> dist (f n) l"

     unfolding t_def by (drule not_less_Least) simp

   have t_antimono: "\<And>e e'. 0 < e \<Longrightarrow> e \<le> e' \<Longrightarrow> t e' \<le> t e"

     apply (rule t_le)

     apply (erule f_t_neq)

     apply (erule (1) less_le_trans [OF f_t_closer])

     done

   have t_dist_f_neq: "\<And>n. 0 < dist (f n) l \<Longrightarrow> t (dist (f n) l) \<noteq> n"

     by (drule f_t_closer) auto

   have t_less: "\<And>e. 0 < e \<Longrightarrow> t e < t (dist (f (t e)) l)"

     apply (subst less_le)

     apply (rule conjI)

     apply (rule t_antimono)

     apply (erule f_t_neq)

     apply (erule f_t_closer [THEN less_imp_le])

     apply (rule t_dist_f_neq [symmetric])

     apply (erule f_t_neq)

     done

   have dist_f_t_less':

     "\<And>e e'. 0 < e \<Longrightarrow> 0 < e' \<Longrightarrow> t e \<le> t e' \<Longrightarrow> dist (f (t e')) l < e"

     apply (simp add: le_less)

     apply (erule disjE)

     apply (rule less_trans)

     apply (erule f_t_closer)

     apply (rule le_less_trans)

     apply (erule less_tD)

     apply (erule f_t_neq)

     apply (erule f_t_closer)

     apply (erule subst)

     apply (erule f_t_closer)

     done

   def r \<equiv> "nat_rec (t 1) (\<lambda>_ x. t (dist (f x) l))"

   have r_simps: "r 0 = t 1" "\<And>n. r (Suc n) = t (dist (f (r n)) l)"

     unfolding r_def by simp_all

   have f_r_neq: "\<And>n. 0 < dist (f (r n)) l"

     by (induct_tac n) (simp_all add: r_simps f_t_neq)

   show "subseq r"

     unfolding subseq_Suc_iff

     apply (rule allI)

     apply (case_tac n)

     apply (simp_all add: r_simps)

     apply (rule t_less, rule zero_less_one)

     apply (rule t_less, rule f_r_neq)

     done

   show "((f \<circ> r) ---> l) sequentially"

     unfolding Lim_sequentially o_def

     apply (clarify, rule_tac x="t e" in exI, clarify)

     apply (drule le_trans, rule seq_suble [OF `subseq r`])

     apply (case_tac n, simp_all add: r_simps dist_f_t_less' f_r_neq)

     done

 qed

 lemma finite_range_imp_infinite_repeats:

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

   assumes "finite (range f)"

   shows "\<exists>k. infinite {n. f n = k}"

 proof -

   { fix A :: "'a set" assume "finite A"

     hence "\<And>f. infinite {n. f n \<in> A} \<Longrightarrow> \<exists>k. infinite {n. f n = k}"

     proof (induct)

       case empty thus ?case by simp

     next

       case (insert x A)

      show ?case

       proof (cases "finite {n. f n = x}")

         case True

         with `infinite {n. f n \<in> insert x A}`

         have "infinite {n. f n \<in> A}" by simp

         thus "\<exists>k. infinite {n. f n = k}" by (rule insert)

       next

         case False thus "\<exists>k. infinite {n. f n = k}" ..

       qed

     qed

   } note H = this

   from assms show "\<exists>k. infinite {n. f n = k}"

     by (rule H) simp

 qed

 lemma bolzano_weierstrass_imp_compact:

   fixes s :: "'a::metric_space set"

   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

   shows "compact s"

 proof -

   { fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

     have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

     proof (cases "finite (range f)")

       case True

       hence "\<exists>l. infinite {n. f n = l}"

         by (rule finite_range_imp_infinite_repeats)

       then obtain l where "infinite {n. f n = l}" ..

       hence "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> {n. f n = l})"

         by (rule infinite_enumerate)

       then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto

       hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"

         unfolding o_def by (simp add: fr tendsto_const)

       hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

         by - (rule exI)

       from f have "\<forall>n. f (r n) \<in> s" by simp

       hence "l \<in> s" by (simp add: fr)

       thus "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

         by (rule rev_bexI) fact

     next

       case False

       with f assms have "\<exists>x\<in>s. x islimpt (range f)" by auto

       then obtain l where "l \<in> s" "l islimpt (range f)" ..

       have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

         using `l islimpt (range f)`

         by (rule islimpt_range_imp_convergent_subsequence)

       with `l \<in> s` show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..

     qed

   }

   thus ?thesis unfolding compact_def by auto

 qed

 primrec helper_2::"(real \<Rightarrow> 'a::metric_space) \<Rightarrow> nat \<Rightarrow> 'a" where

   "helper_2 beyond 0 = beyond 0" |

   "helper_2 beyond (Suc n) = beyond (dist undefined (helper_2 beyond n) + 1 )"

 lemma bolzano_weierstrass_imp_bounded: fixes s::"'a::metric_space set"

   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"

   shows "bounded s"

 proof(rule ccontr)

   assume "\<not> bounded s"

   then obtain beyond where "\<forall>a. beyond a \<in>s \<and> \<not> dist undefined (beyond a) \<le> a"

     unfolding bounded_any_center [where a=undefined]

     apply simp using choice[of "\<lambda>a x. x\<in>s \<and> \<not> dist undefined x \<le> a"] by auto

   hence beyond:"\<And>a. beyond a \<in>s" "\<And>a. dist undefined (beyond a) > a"

     unfolding linorder_not_le by auto

   def x \<equiv> "helper_2 beyond"

   { fix m n ::nat assume "m<n"

     hence "dist undefined (x m) + 1 < dist undefined (x n)"

     proof(induct n)

       case 0 thus ?case by auto

     next

       case (Suc n)

       have *:"dist undefined (x n) + 1 < dist undefined (x (Suc n))"

         unfolding x_def and helper_2.simps

         using beyond(2)[of "dist undefined (helper_2 beyond n) + 1"] by auto

       thus ?case proof(cases "m < n")

         case True thus ?thesis using Suc and * by auto

       next

         case False hence "m = n" using Suc(2) by auto

         thus ?thesis using * by auto

       qed

     qed  } note * = this

   { fix m n ::nat assume "m\<noteq>n"

     have "1 < dist (x m) (x n)"

     proof(cases "m<n")

       case True

       hence "1 < dist undefined (x n) - dist undefined (x m)" using *[of m n] by auto

       thus ?thesis using dist_triangle [of undefined "x n" "x m"] by arith

     next

       case False hence "n<m" using `m\<noteq>n` by auto

       hence "1 < dist undefined (x m) - dist undefined (x n)" using *[of n m] by auto

       thus ?thesis using dist_triangle2 [of undefined "x m" "x n"] by arith

     qed  } note ** = this

   { fix a b assume "x a = x b" "a \<noteq> b"

     hence False using **[of a b] by auto  }

   hence "inj x" unfolding inj_on_def by auto

   moreover

   { fix n::nat

     have "x n \<in> s"

     proof(cases "n = 0")

       case True thus ?thesis unfolding x_def using beyond by auto

     next

       case False then obtain z where "n = Suc z" using not0_implies_Suc by auto

       thus ?thesis unfolding x_def using beyond by auto

     qed  }

   ultimately have "infinite (range x) \<and> range x \<subseteq> s" unfolding x_def using range_inj_infinite[of "helper_2 beyond"] using beyond(1) by auto

   then obtain l where "l\<in>s" and l:"l islimpt range x" using assms[THEN spec[where x="range x"]] by auto

   then obtain y where "x y \<noteq> l" and y:"dist (x y) l < 1/2" unfolding islimpt_approachable apply(erule_tac x="1/2" in allE) by auto

   then obtain z where "x z \<noteq> l" and z:"dist (x z) l < dist (x y) l" using l[unfolded islimpt_approachable, THEN spec[where x="dist (x y) l"]]

     unfolding dist_nz by auto

   show False using y and z and dist_triangle_half_l[of "x y" l 1 "x z"] and **[of y z] by auto

 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)