(*  title:      HOL/Library/Topology_Euclidian_Space.thy

    Author:     Amine Chaieb, University of Cambridge

    Author:     Robert Himmelmann, TU Muenchen

*)

header {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_Space

imports SEQ Euclidean_Space Product_Vector

begin

subsection{* General notion of a topology *}

definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"

typedef (open) 'a topology = "{L::('a set) set. 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_def Collect_def] .

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 (metis mem_def set_ext)

    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}"

subsection{* 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 Collect_def mem_def

  by (metis mem_def subset_eq)+

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)"

  by (simp add: openin_clauses)

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

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 have ?rhs by auto }

  moreover

  {assume H: ?rhs

    then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"

      unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast

    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto

    have "\<Union> t`S = S" using t by auto

    with openin_Union[OF th0] have "openin U S" by simp }

  ultimately show ?thesis by blast

qed

subsection{* 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

subsection{* Subspace topology. *}

definition "subtopology U V = topology {S \<inter> V |S. openin U S}"

lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")

proof-

  have "{} \<in> ?L" by blast

  {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"

    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 "A \<inter> B \<in> ?L" by blast}

  moreover

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

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

      apply (rule set_ext)

      apply (simp add: Ball_def image_iff)

      by (metis mem_def)

    from K[unfolded th0 subset_image_iff]

    obtain Sk where Sk: "Sk \<subseteq> 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 mem_def)

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

  ultimately show ?thesis unfolding 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 simp add: Collect_def)

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)

subsection{* The universal Euclidean versions are what we use most of the time *}

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)

  by (auto simp add: mem_def subset_eq)

lemma topspace_euclidean: "topspace euclidean = UNIV"

  apply (simp add: topspace_def)

  apply (rule set_ext)

  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])

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: expand_set_eq) arith

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

  by (simp add: expand_set_eq)

subsection{* Topological properties of open balls *}

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 Collect_def Ball_def mem_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 expand_set_eq

  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{* 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 hence ?rhs unfolding openin_subtopology open_openin[symmetric]

      by (simp add: open_dist) blast}

  moreover

  {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"

    from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"

      by metis

    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"

    have oT: "open ?T" by auto

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

      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"

        apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto

        by (rule d [THEN conjunct1])

      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }

    moreover

    { fix y assume "y\<in>?T"

      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto

      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto

      assume "y\<in>U"

      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }

    ultimately have "S = ?T \<inter> U" by blast

    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}

  ultimately show ?thesis by blast

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{* 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{* Hausdorff and other separation properties *}

class t0_space =

  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"

class t1_space =

  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"

begin

subclass t0_space

proof

qed (fast dest: t1_space)

end

text {* T2 spaces are also known as Hausdorff spaces. *}

class t2_space =

  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"

begin

subclass t1_space

proof

qed (fast dest: hausdorff)

end

instance metric_space \<subseteq> t2_space

proof

  fix x y :: "'a::metric_space"

  assume xy: "x \<noteq> y"

  let ?U = "ball x (dist x y / 2)"

  let ?V = "ball y (dist x y / 2)"

  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y

               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith

  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"

    using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]

    by (auto simp add: expand_set_eq)

  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"

    by blast

qed

lemma separation_t2:

  fixes x y :: "'a::t2_space"

  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"

  using hausdorff[of x y] by blast

lemma separation_t1:

  fixes x y :: "'a::t1_space"

  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"

  using t1_space[of x y] by blast

lemma separation_t0:

  fixes x y :: "'a::t0_space"

  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"

  using t0_space[of x y] by blast

subsection{* Limit points *}

definition

  islimpt:: "'a::topological_space \<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_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)

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_def

  apply auto

  apply(erule_tac x="ball x e" in allE)

  apply auto

  apply(rule_tac x=y in bexI)

  apply (auto simp add: dist_commute)

  apply (simp add: open_dist, drule (1) bspec)

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

  done

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>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]

  by metis 

class perfect_space =

  (* FIXME: perfect_space should inherit from topological_space *)

  assumes islimpt_UNIV [simp, intro]: "(x::'a::metric_space) islimpt UNIV"

lemma perfect_choose_dist:

  fixes x :: "'a::perfect_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)

instance real :: perfect_space

apply default

apply (rule islimpt_approachable [THEN iffD2])

apply (clarify, rule_tac x="x + e/2" in bexI)

apply (auto simp add: dist_norm)

done

instance cart :: (perfect_space, finite) perfect_space

proof

  fix x :: "'a ^ 'b"

  {

    fix e :: real assume "0 < e"

    def a \<equiv> "x $ undefined"

    have "a islimpt UNIV" by (rule islimpt_UNIV)

    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"

      unfolding islimpt_approachable by auto

    def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"

    from `b \<noteq> a` have "y \<noteq> x"

      unfolding a_def y_def by (simp add: Cart_eq)

    from `dist b a < e` have "dist y x < e"

      unfolding dist_vector_def a_def y_def

      apply simp

      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])

      apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)

      done

    from `y \<noteq> x` and `dist y x < e`

    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto

  }

  then show "x islimpt UNIV" unfolding islimpt_approachable by blast

qed

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 insertCI insert_absorb mem_def)

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

  unfolding islimpt_def by auto

lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"

proof-

  let ?U = "UNIV :: 'n set"

  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"

  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"

    and xi: "x$i < 0"

    from xi have th0: "-x$i > 0" by arith

    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast

      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith

      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith

      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi

        apply (simp only: vector_component)

        by (rule th') auto

      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]

        apply (simp add: dist_norm) by norm

      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }

  then show ?thesis unfolding closed_limpt islimpt_approachable

    unfolding not_le[symmetric] by blast

qed

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 apply auto by ferrack

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 = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"

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

  apply (simp add: expand_set_eq interior_def)

  apply (subst (2) open_subopen) by (safe, blast+)

lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)

lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)

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

  apply (simp add: interior_def)

  apply (subst open_subopen) by blast

lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)

lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)

lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)

lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)

lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"

  by (metis equalityI interior_maximal interior_subset open_interior)

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

  apply (simp add: interior_def)

  by (metis open_contains_ball centre_in_ball open_ball subset_trans)

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

  by (metis interior_maximal interior_subset subset_trans)

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

  apply (rule equalityI, simp)

  apply (metis Int_lower1 Int_lower2 subset_interior)

  by (metis Int_mono interior_subset open_Int open_interior open_subset_interior)

lemma interior_limit_point [intro]:

  fixes x :: "'a::perfect_space"

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

proof-

  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"

    unfolding mem_interior subset_eq Ball_def mem_ball by blast

  {

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

    let ?m = "min d e"

    have mde2: "0 < ?m" using e(1) d(1) by simp

    from perfect_choose_dist [OF mde2, of x]

    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast

    then have "dist y x < e" "dist y x < d" by simp_all

    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)

    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"

      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast

  }

  then show ?thesis unfolding islimpt_approachable by blast

qed

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 subset_interior, blast)

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"

      unfolding interior_def by fast

    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 expand_set_eq 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

subsection{* Closure of a Set *}

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

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

proof-

  { fix x

    have "x\<in>- interior (- S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")

    proof

      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> - S)"

      assume "?lhs"

      hence *:"\<not> ?exT x"

        unfolding interior_def

        by simp

      { assume "\<not> ?rhs"

        hence False using *

          unfolding closure_def islimpt_def

          by blast

      }

      thus "?rhs"

        by blast

    next

      assume "?rhs" thus "?lhs"

        unfolding closure_def interior_def islimpt_def

        by blast

    qed

  }

  thus ?thesis

    by blast

qed

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

proof-

  { fix x

    have "x \<in> interior S \<longleftrightarrow> x \<in> - (closure (- S))"

      unfolding interior_def closure_def islimpt_def

      by auto

  }

  thus ?thesis

    by blast

qed

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

proof-

  have "closed (- interior (-S))" by blast

  thus ?thesis using closure_interior[of S] by simp

qed

lemma closure_hull: "closure S = closed hull S"

proof-

  have "S \<subseteq> closure S"

    unfolding closure_def

    by blast

  moreover

  have "closed (closure S)"

    using closed_closure[of S]

    by assumption

  moreover

  { fix t

    assume *:"S \<subseteq> t" "closed t"

    { fix x

      assume "x islimpt S"

      hence "x islimpt t" using *(1)

        using islimpt_subset[of x, of S, of t]

        by blast

    }

    with * have "closure S \<subseteq> t"

      unfolding closure_def

      using closed_limpt[of t]

      by auto

  }

  ultimately show ?thesis

    using hull_unique[of S, of "closure S", of closed]

    unfolding mem_def

    by simp

qed

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

  unfolding closure_hull

  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]

  by (metis mem_def subset_eq)

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

  using closure_eq[of S]

  by simp

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

  unfolding closure_hull

  using hull_hull[of closed S]

  by assumption

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

  unfolding closure_hull

  using hull_subset[of S closed]

  by assumption

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

  unfolding closure_hull

  using hull_mono[of S T closed]

  by assumption

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

  using hull_minimal[of S T closed]

  unfolding closure_hull mem_def

  by simp

lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"

  using hull_unique[of S T closed]

  unfolding closure_hull mem_def

  by simp

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

  using closed_empty closure_closed[of "{}"]

  by simp

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

  using closure_closed[of UNIV]

  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)"

proof-

  have "S = - (- S)"

    by auto

  thus ?thesis

    unfolding closure_interior

    by auto

qed

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

  unfolding closure_interior

  by blast

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 closure_empty)

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] by auto

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{* Common nets and The "within" modifier for nets. *}

definition

  at_infinity :: "'a::real_normed_vector net" where

  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"

definition

  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (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 nets. *}

lemma Rep_net_at_infinity:

  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"

unfolding at_infinity_def

apply (rule Abs_net_inverse')

apply (rule image_nonempty, simp)

apply (clarsimp, rename_tac r s)

apply (rule_tac x="max r s" in exI, auto)

done

lemma within_UNIV: "net within UNIV = net"

  by (simp add: Rep_net_inject [symmetric] Rep_net_within)

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

definition