Theory Connected

theory Connected
imports Topology_Euclidean_Space
(*  Author:     L C Paulson, University of Cambridge
    Material split off from Topology_Euclidean_Space
*)

section ‹Connected Components, Homeomorphisms, Baire property, etc›

theory Connected
imports Topology_Euclidean_Space
begin

subsection%unimportant ‹More properties of closed balls, spheres, etc›

lemma mem_interior_cball: "x ∈ interior S ⟷ (∃e>0. cball x e ⊆ S)"
  apply (simp add: interior_def, safe)
  apply (force simp: 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 ⟷ 0 < e ∧ y ∈ cball x e"
  (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if ?lhs
  proof
    {
      assume "e ≤ 0"
      then have *: "ball x e = {}"
        using ball_eq_empty[of x e] by auto
      have False using ‹?lhs›
        unfolding * using islimpt_EMPTY[of y] by auto
    }
    then show "e > 0" by (metis not_less)
    show "y ∈ 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
  qed
  show ?lhs if ?rhs
  proof -
    from that have "e > 0" by auto
    {
      fix d :: real
      assume "d > 0"
      have "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
      proof (cases "d ≤ dist x y")
        case True
        then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
        proof (cases "x = y")
          case True
          then have False
            using ‹d ≤ dist x y› ‹d>0› by auto
          then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
            by auto
        next
          case False
          have "dist x (y - (d / (2 * dist y x)) *R (y - x)) =
            norm (x - y + (d / (2 * norm (y - x))) *R (y - x))"
            unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
            by auto
          also have "… = ¦- 1 + d / (2 * norm (x - y))¦ * norm (x - y)"
            using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
            unfolding scaleR_minus_left scaleR_one
            by (auto simp: norm_minus_commute)
          also have "… = ¦- norm (x - y) + d / 2¦"
            unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
            unfolding distrib_right using ‹x≠y›  by auto
          also have "… ≤ e - d/2" using ‹d ≤ dist x y› and ‹d>0› and ‹?rhs›
            by (auto simp: dist_norm)
          finally have "y - (d / (2 * dist y x)) *R (y - x) ∈ ball x e" using ‹d>0›
            by auto
          moreover
          have "(d / (2*dist y x)) *R (y - x) ≠ 0"
            using ‹x≠y›[unfolded dist_nz] ‹d>0› unfolding scaleR_eq_0_iff
            by (auto simp: dist_commute)
          moreover
          have "dist (y - (d / (2 * dist y x)) *R (y - x)) y < d"
            unfolding dist_norm
            apply simp
            unfolding norm_minus_cancel
            using ‹d > 0› ‹x≠y›[unfolded dist_nz] dist_commute[of x y]
            unfolding dist_norm
            apply auto
            done
          ultimately show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
            apply (rule_tac x = "y - (d / (2*dist y x)) *R (y - x)" in bexI)
            apply auto
            done
        qed
      next
        case False
        then have "d > dist x y" by auto
        show "∃x' ∈ ball x e. x' ≠ y ∧ dist x' y < d"
        proof (cases "x = y")
          case True
          obtain z where **: "z ≠ y" "dist z y < min e d"
            using perfect_choose_dist[of "min e d" y]
            using ‹d > 0› ‹e>0› by auto
          show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
            unfolding ‹x = y›
            using ‹z ≠ y› **
            apply (rule_tac x=z in bexI)
            apply (auto simp: dist_commute)
            done
        next
          case False
          then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
            using ‹d>0› ‹d > dist x y› ‹?rhs›
            apply (rule_tac x=x in bexI, auto)
            done
        qed
      qed
    }
    then show ?thesis
      unfolding mem_cball islimpt_approachable mem_ball by auto
  qed
qed

lemma closure_ball_lemma:
  fixes x y :: "'a::real_normed_vector"
  assumes "x ≠ y"
  shows "y islimpt ball x (dist x y)"
proof (rule islimptI)
  fix T
  assume "y ∈ T" "open T"
  then obtain r where "0 < r" "∀z. dist z y < r ⟶ z ∈ T"
    unfolding open_dist by fast
  (* choose point between x and y, within distance r of y. *)
  define k where "k = min 1 (r / (2 * dist x y))"
  define z where "z = 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)
  then have "z ∈ T"
    using ‹∀z. dist z y < r ⟶ z ∈ 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 "¦1 - k¦ * dist x y < 1 * dist x y", simp)
    apply (rule mult_strict_right_mono)
    apply (simp add: k_def ‹0 < r› ‹x ≠ y›)
    apply (simp add: ‹x ≠ y›)
    done
  then have "z ∈ ball x (dist x y)"
    by simp
  have "z ≠ y"
    unfolding z_def k_def using ‹x ≠ y› ‹0 < r›
    by (simp add: min_def)
  show "∃z∈ball x (dist x y). z ∈ T ∧ z ≠ y"
    using ‹z ∈ ball x (dist x y)› ‹z ∈ T› ‹z ≠ y›
    by fast
qed

lemma at_within_ball_bot_iff:
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
  shows "at x within ball y r = bot ⟷ (r=0 ∨ x ∉ cball y r)"
  unfolding trivial_limit_within 
apply (auto simp add:trivial_limit_within islimpt_ball )
by (metis le_less_trans less_eq_real_def zero_le_dist)

lemma closure_ball [simp]:
  fixes x :: "'a::real_normed_vector"
  shows "0 < e ⟹ 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, clarify)
  apply (rule closure_ball_lemma)
  apply simp
  done

(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball [simp]:
  fixes x :: "'a::{real_normed_vector, perfect_space}"
  shows "interior (cball x e) = ball x e"
proof (cases "e ≥ 0")
  case False note cs = this
  from cs have null: "ball x e = {}"
    using ball_empty[of e x] by auto
  moreover
  {
    fix y
    assume "y ∈ cball x e"
    then have False
      by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball subset_antisym subset_ball)
  }
  then have "cball x e = {}" by auto
  then have "interior (cball x e) = {}"
    using interior_empty by auto
  ultimately show ?thesis by blast
next
  case True note cs = this
  have "ball x e ⊆ cball x e"
    using ball_subset_cball by auto
  moreover
  {
    fix S y
    assume as: "S ⊆ cball x e" "open S" "y∈S"
    then obtain d where "d>0" and d: "∀x'. dist x' y < d ⟶ x' ∈ S"
      unfolding open_dist by blast
    then obtain xa where xa_y: "xa ≠ y" and xa: "dist xa y < d"
      using perfect_choose_dist [of d] by auto
    have "xa ∈ S"
      using d[THEN spec[where x = xa]]
      using xa by (auto simp: dist_commute)
    then have xa_cball: "xa ∈ cball x e"
      using as(1) by auto
    then have "y ∈ ball x e"
    proof (cases "x = y")
      case True
      then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
      then show "y ∈ ball x e"
        using ‹x = y › by simp
    next
      case False
      have "dist (y + (d / 2 / dist y x) *R (y - x)) y < d"
        unfolding dist_norm
        using ‹d>0› norm_ge_zero[of "y - x"] ‹x ≠ y› by auto
      then have *: "y + (d / 2 / dist y x) *R (y - x) ∈ cball x e"
        using d as(1)[unfolded subset_eq] by blast
      have "y - x ≠ 0" using ‹x ≠ y› by auto
      hence **:"d / (2 * norm (y - x)) > 0"
        unfolding zero_less_norm_iff[symmetric] using ‹d>0› by auto
      have "dist (y + (d / 2 / dist y x) *R (y - x)) x =
        norm (y + (d / (2 * norm (y - x))) *R y - (d / (2 * norm (y - x))) *R x - x)"
        by (auto simp: dist_norm algebra_simps)
      also have "… = norm ((1 + d / (2 * norm (y - x))) *R (y - x))"
        by (auto simp: algebra_simps)
      also have "… = ¦1 + d / (2 * norm (y - x))¦ * norm (y - x)"
        using ** by auto
      also have "… = (dist y x) + d/2"
        using ** by (auto simp: distrib_right dist_norm)
      finally have "e ≥ dist x y +d/2"
        using *[unfolded mem_cball] by (auto simp: dist_commute)
      then show "y ∈ ball x e"
        unfolding mem_ball using ‹d>0› by auto
    qed
  }
  then have "∀S ⊆ cball x e. open S ⟶ S ⊆ 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 interior_ball [simp]: "interior (ball x e) = ball x e"
  by (simp add: interior_open)

lemma frontier_ball [simp]:
  fixes a :: "'a::real_normed_vector"
  shows "0 < e ⟹ frontier (ball a e) = sphere a e"
  by (force simp: frontier_def)

lemma frontier_cball [simp]:
  fixes a :: "'a::{real_normed_vector, perfect_space}"
  shows "frontier (cball a e) = sphere a e"
  by (force simp: frontier_def)

lemma cball_eq_empty [simp]: "cball x e = {} ⟷ e < 0"
  apply (simp add: set_eq_iff not_le)
  apply (metis zero_le_dist dist_self order_less_le_trans)
  done

lemma cball_empty [simp]: "e < 0 ⟹ cball x e = {}"
  by simp

lemma cball_eq_sing:
  fixes x :: "'a::{metric_space,perfect_space}"
  shows "cball x e = {x} ⟷ e = 0"
proof (rule linorder_cases)
  assume e: "0 < e"
  obtain a where "a ≠ x" "dist a x < e"
    using perfect_choose_dist [OF e] by auto
  then have "a ≠ x" "dist x a ≤ e"
    by (auto simp: dist_commute)
  with e show ?thesis by (auto simp: set_eq_iff)
qed auto

lemma cball_sing:
  fixes x :: "'a::metric_space"
  shows "e = 0 ⟹ cball x e = {x}"
  by (auto simp: set_eq_iff)

lemma ball_divide_subset: "d ≥ 1 ⟹ ball x (e/d) ⊆ ball x e"
  apply (cases "e ≤ 0")
  apply (simp add: ball_empty divide_simps)
  apply (rule subset_ball)
  apply (simp add: divide_simps)
  done

lemma ball_divide_subset_numeral: "ball x (e / numeral w) ⊆ ball x e"
  using ball_divide_subset one_le_numeral by blast

lemma cball_divide_subset: "d ≥ 1 ⟹ cball x (e/d) ⊆ cball x e"
  apply (cases "e < 0")
  apply (simp add: divide_simps)
  apply (rule subset_cball)
  apply (metis div_by_1 frac_le not_le order_refl zero_less_one)
  done

lemma cball_divide_subset_numeral: "cball x (e / numeral w) ⊆ cball x e"
  using cball_divide_subset one_le_numeral by blast

lemma compact_cball[simp]:
  fixes x :: "'a::heine_borel"
  shows "compact (cball x e)"
  using compact_eq_bounded_closed bounded_cball closed_cball
  by blast

lemma compact_frontier_bounded[intro]:
  fixes S :: "'a::heine_borel set"
  shows "bounded S ⟹ compact (frontier S)"
  unfolding frontier_def
  using compact_eq_bounded_closed
  by blast

lemma compact_frontier[intro]:
  fixes S :: "'a::heine_borel set"
  shows "compact S ⟹ compact (frontier S)"
  using compact_eq_bounded_closed compact_frontier_bounded
  by blast

corollary compact_sphere [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "compact (sphere a r)"
using compact_frontier [of "cball a r"] by simp

corollary bounded_sphere [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "bounded (sphere a r)"
by (simp add: compact_imp_bounded)

corollary closed_sphere  [simp]:
  fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
  shows "closed (sphere a r)"
by (simp add: compact_imp_closed)

subsection%unimportant ‹Connectedness›

lemma connected_local:
 "connected S ⟷
  ¬ (∃e1 e2.
      openin (subtopology euclidean S) e1 ∧
      openin (subtopology euclidean S) e2 ∧
      S ⊆ e1 ∪ e2 ∧
      e1 ∩ e2 = {} ∧
      e1 ≠ {} ∧
      e2 ≠ {})"
  unfolding connected_def openin_open
  by safe blast+

lemma exists_diff:
  fixes P :: "'a set ⇒ bool"
  shows "(∃S. P (- S)) ⟷ (∃S. P S)"
    (is "?lhs ⟷ ?rhs")
proof -
  have ?rhs if ?lhs
    using that by blast
  moreover have "P (- (- S))" if "P S" for S
  proof -
    have "S = - (- S)" by simp
    with that show ?thesis by metis
  qed
  ultimately show ?thesis by metis
qed

lemma connected_clopen: "connected S ⟷
  (∀T. openin (subtopology euclidean S) T ∧
     closedin (subtopology euclidean S) T ⟶ T = {} ∨ T = S)" (is "?lhs ⟷ ?rhs")
proof -
  have "¬ connected S ⟷
    (∃e1 e2. open e1 ∧ open (- e2) ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
    unfolding connected_def openin_open closedin_closed
    by (metis double_complement)
  then have th0: "connected S ⟷
    ¬ (∃e2 e1. closed e2 ∧ open e1 ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
    (is " _ ⟷ ¬ (∃e2 e1. ?P e2 e1)")
    by (simp add: closed_def) metis
  have th1: "?rhs ⟷ ¬ (∃t' t. closed t'∧t = S∩t' ∧ t≠{} ∧ t≠S ∧ (∃t'. open t' ∧ t = S ∩ t'))"
    (is "_ ⟷ ¬ (∃t' t. ?Q t' t)")
    unfolding connected_def openin_open closedin_closed by auto
  have "(∃e1. ?P e2 e1) ⟷ (∃t. ?Q e2 t)" for e2
  proof -
    have "?P e2 e1 ⟷ (∃t. closed e2 ∧ t = S∩e2 ∧ open e1 ∧ t = S∩e1 ∧ t≠{} ∧ t ≠ S)" for e1
      by auto
    then show ?thesis
      by metis
  qed
  then have "∀e2. (∃e1. ?P e2 e1) ⟷ (∃t. ?Q e2 t)"
    by blast
  then show ?thesis
    by (simp add: th0 th1)
qed

lemma connected_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
  assumes "linear f" and "connected s"
  shows "connected (f ` s)"
using connected_continuous_image assms linear_continuous_on linear_conv_bounded_linear by blast

subsection ‹Connected components, considered as a connectedness relation or a set›

definition%important "connected_component s x y ≡ ∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t"

abbreviation "connected_component_set s x ≡ Collect (connected_component s x)"

lemma connected_componentI:
  "connected t ⟹ t ⊆ s ⟹ x ∈ t ⟹ y ∈ t ⟹ connected_component s x y"
  by (auto simp: connected_component_def)

lemma connected_component_in: "connected_component s x y ⟹ x ∈ s ∧ y ∈ s"
  by (auto simp: connected_component_def)

lemma connected_component_refl: "x ∈ s ⟹ connected_component s x x"
  by (auto simp: connected_component_def) (use connected_sing in blast)

lemma connected_component_refl_eq [simp]: "connected_component s x x ⟷ x ∈ s"
  by (auto simp: connected_component_refl) (auto simp: connected_component_def)

lemma connected_component_sym: "connected_component s x y ⟹ connected_component s y x"
  by (auto simp: connected_component_def)

lemma connected_component_trans:
  "connected_component s x y ⟹ connected_component s y z ⟹ connected_component s x z"
  unfolding connected_component_def
  by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)

lemma connected_component_of_subset:
  "connected_component s x y ⟹ s ⊆ t ⟹ connected_component t x y"
  by (auto simp: connected_component_def)

lemma connected_component_Union: "connected_component_set s x = ⋃{t. connected t ∧ x ∈ t ∧ t ⊆ s}"
  by (auto simp: connected_component_def)

lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
  by (auto simp: connected_component_Union intro: connected_Union)

lemma connected_iff_eq_connected_component_set:
  "connected s ⟷ (∀x ∈ s. connected_component_set s x = s)"
proof (cases "s = {}")
  case True
  then show ?thesis by simp
next
  case False
  then obtain x where "x ∈ s" by auto
  show ?thesis
  proof
    assume "connected s"
    then show "∀x ∈ s. connected_component_set s x = s"
      by (force simp: connected_component_def)
  next
    assume "∀x ∈ s. connected_component_set s x = s"
    then show "connected s"
      by (metis ‹x ∈ s› connected_connected_component)
  qed
qed

lemma connected_component_subset: "connected_component_set s x ⊆ s"
  using connected_component_in by blast

lemma connected_component_eq_self: "connected s ⟹ x ∈ s ⟹ connected_component_set s x = s"
  by (simp add: connected_iff_eq_connected_component_set)

lemma connected_iff_connected_component:
  "connected s ⟷ (∀x ∈ s. ∀y ∈ s. connected_component s x y)"
  using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)

lemma connected_component_maximal:
  "x ∈ t ⟹ connected t ⟹ t ⊆ s ⟹ t ⊆ (connected_component_set s x)"
  using connected_component_eq_self connected_component_of_subset by blast

lemma connected_component_mono:
  "s ⊆ t ⟹ connected_component_set s x ⊆ connected_component_set t x"
  by (simp add: Collect_mono connected_component_of_subset)

lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} ⟷ x ∉ s"
  using connected_component_refl by (fastforce simp: connected_component_in)

lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
  using connected_component_eq_empty by blast

lemma connected_component_eq:
  "y ∈ connected_component_set s x ⟹ (connected_component_set s y = connected_component_set s x)"
  by (metis (no_types, lifting)
      Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)

lemma closed_connected_component:
  assumes s: "closed s"
  shows "closed (connected_component_set s x)"
proof (cases "x ∈ s")
  case False
  then show ?thesis
    by (metis connected_component_eq_empty closed_empty)
next
  case True
  show ?thesis
    unfolding closure_eq [symmetric]
  proof
    show "closure (connected_component_set s x) ⊆ connected_component_set s x"
      apply (rule connected_component_maximal)
        apply (simp add: closure_def True)
       apply (simp add: connected_imp_connected_closure)
      apply (simp add: s closure_minimal connected_component_subset)
      done
  next
    show "connected_component_set s x ⊆ closure (connected_component_set s x)"
      by (simp add: closure_subset)
  qed
qed

lemma connected_component_disjoint:
  "connected_component_set s a ∩ connected_component_set s b = {} ⟷
    a ∉ connected_component_set s b"
  apply (auto simp: connected_component_eq)
  using connected_component_eq connected_component_sym
  apply blast
  done

lemma connected_component_nonoverlap:
  "connected_component_set s a ∩ connected_component_set s b = {} ⟷
    a ∉ s ∨ b ∉ s ∨ connected_component_set s a ≠ connected_component_set s b"
  apply (auto simp: connected_component_in)
  using connected_component_refl_eq
    apply blast
   apply (metis connected_component_eq mem_Collect_eq)
  apply (metis connected_component_eq mem_Collect_eq)
  done

lemma connected_component_overlap:
  "connected_component_set s a ∩ connected_component_set s b ≠ {} ⟷
    a ∈ s ∧ b ∈ s ∧ connected_component_set s a = connected_component_set s b"
  by (auto simp: connected_component_nonoverlap)

lemma connected_component_sym_eq: "connected_component s x y ⟷ connected_component s y x"
  using connected_component_sym by blast

lemma connected_component_eq_eq:
  "connected_component_set s x = connected_component_set s y ⟷
    x ∉ s ∧ y ∉ s ∨ x ∈ s ∧ y ∈ s ∧ connected_component s x y"
  apply (cases "y ∈ s", simp)
   apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)
  apply (cases "x ∈ s", simp)
   apply (metis connected_component_eq_empty)
  using connected_component_eq_empty
  apply blast
  done

lemma connected_iff_connected_component_eq:
  "connected s ⟷ (∀x ∈ s. ∀y ∈ s. connected_component_set s x = connected_component_set s y)"
  by (simp add: connected_component_eq_eq connected_iff_connected_component)

lemma connected_component_idemp:
  "connected_component_set (connected_component_set s x) x = connected_component_set s x"
  apply (rule subset_antisym)
   apply (simp add: connected_component_subset)
  apply (metis connected_component_eq_empty connected_component_maximal
      connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)
  done

lemma connected_component_unique:
  "⟦x ∈ c; c ⊆ s; connected c;
    ⋀c'. x ∈ c' ∧ c' ⊆ s ∧ connected c'
              ⟹ c' ⊆ c⟧
        ⟹ connected_component_set s x = c"
apply (rule subset_antisym)
apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)
by (simp add: connected_component_maximal)

lemma joinable_connected_component_eq:
  "⟦connected t; t ⊆ s;
    connected_component_set s x ∩ t ≠ {};
    connected_component_set s y ∩ t ≠ {}⟧
    ⟹ connected_component_set s x = connected_component_set s y"
apply (simp add: ex_in_conv [symmetric])
apply (rule connected_component_eq)
by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)


lemma Union_connected_component: "⋃(connected_component_set s ` s) = s"
  apply (rule subset_antisym)
  apply (simp add: SUP_least connected_component_subset)
  using connected_component_refl_eq
  by force


lemma complement_connected_component_unions:
    "s - connected_component_set s x =
     ⋃(connected_component_set s ` s - {connected_component_set s x})"
  apply (subst Union_connected_component [symmetric], auto)
  apply (metis connected_component_eq_eq connected_component_in)
  by (metis connected_component_eq mem_Collect_eq)

lemma connected_component_intermediate_subset:
        "⟦connected_component_set u a ⊆ t; t ⊆ u⟧
        ⟹ connected_component_set t a = connected_component_set u a"
  apply (case_tac "a ∈ u")
  apply (simp add: connected_component_maximal connected_component_mono subset_antisym)
  using connected_component_eq_empty by blast


subsection ‹The set of connected components of a set›

definition%important components:: "'a::topological_space set ⇒ 'a set set"
  where "components s ≡ connected_component_set s ` s"

lemma components_iff: "s ∈ components u ⟷ (∃x. x ∈ u ∧ s = connected_component_set u x)"
  by (auto simp: components_def)

lemma componentsI: "x ∈ u ⟹ connected_component_set u x ∈ components u"
  by (auto simp: components_def)

lemma componentsE:
  assumes "s ∈ components u"
  obtains x where "x ∈ u" "s = connected_component_set u x"
  using assms by (auto simp: components_def)

lemma Union_components [simp]: "⋃(components u) = u"
  apply (rule subset_antisym)
  using Union_connected_component components_def apply fastforce
  apply (metis Union_connected_component components_def set_eq_subset)
  done

lemma pairwise_disjoint_components: "pairwise (λX Y. X ∩ Y = {}) (components u)"
  apply (simp add: pairwise_def)
  apply (auto simp: components_iff)
  apply (metis connected_component_eq_eq connected_component_in)+
  done

lemma in_components_nonempty: "c ∈ components s ⟹ c ≠ {}"
    by (metis components_iff connected_component_eq_empty)

lemma in_components_subset: "c ∈ components s ⟹ c ⊆ s"
  using Union_components by blast

lemma in_components_connected: "c ∈ components s ⟹ connected c"
  by (metis components_iff connected_connected_component)

lemma in_components_maximal:
  "c ∈ components s ⟷
    c ≠ {} ∧ c ⊆ s ∧ connected c ∧ (∀d. d ≠ {} ∧ c ⊆ d ∧ d ⊆ s ∧ connected d ⟶ d = c)"
  apply (rule iffI)
   apply (simp add: in_components_nonempty in_components_connected)
   apply (metis (full_types) components_iff connected_component_eq_self connected_component_intermediate_subset connected_component_refl in_components_subset mem_Collect_eq rev_subsetD)
  apply (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
  done

lemma joinable_components_eq:
  "connected t ∧ t ⊆ s ∧ c1 ∈ components s ∧ c2 ∈ components s ∧ c1 ∩ t ≠ {} ∧ c2 ∩ t ≠ {} ⟹ c1 = c2"
  by (metis (full_types) components_iff joinable_connected_component_eq)

lemma closed_components: "⟦closed s; c ∈ components s⟧ ⟹ closed c"
  by (metis closed_connected_component components_iff)

lemma compact_components:
  fixes s :: "'a::heine_borel set"
  shows "⟦compact s; c ∈ components s⟧ ⟹ compact c"
by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)

lemma components_nonoverlap:
    "⟦c ∈ components s; c' ∈ components s⟧ ⟹ (c ∩ c' = {}) ⟷ (c ≠ c')"
  apply (auto simp: in_components_nonempty components_iff)
    using connected_component_refl apply blast
   apply (metis connected_component_eq_eq connected_component_in)
  by (metis connected_component_eq mem_Collect_eq)

lemma components_eq: "⟦c ∈ components s; c' ∈ components s⟧ ⟹ (c = c' ⟷ c ∩ c' ≠ {})"
  by (metis components_nonoverlap)

lemma components_eq_empty [simp]: "components s = {} ⟷ s = {}"
  by (simp add: components_def)

lemma components_empty [simp]: "components {} = {}"
  by simp

lemma connected_eq_connected_components_eq: "connected s ⟷ (∀c ∈ components s. ∀c' ∈ components s. c = c')"
  by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)

lemma components_eq_sing_iff: "components s = {s} ⟷ connected s ∧ s ≠ {}"
  apply (rule iffI)
  using in_components_connected apply fastforce
  apply safe
  using Union_components apply fastforce
   apply (metis components_iff connected_component_eq_self)
  using in_components_maximal
  apply auto
  done

lemma components_eq_sing_exists: "(∃a. components s = {a}) ⟷ connected s ∧ s ≠ {}"
  apply (rule iffI)
  using connected_eq_connected_components_eq apply fastforce
  apply (metis components_eq_sing_iff)
  done

lemma connected_eq_components_subset_sing: "connected s ⟷ components s ⊆ {s}"
  by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)

lemma connected_eq_components_subset_sing_exists: "connected s ⟷ (∃a. components s ⊆ {a})"
  by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)

lemma in_components_self: "s ∈ components s ⟷ connected s ∧ s ≠ {}"
  by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)

lemma components_maximal: "⟦c ∈ components s; connected t; t ⊆ s; c ∩ t ≠ {}⟧ ⟹ t ⊆ c"
  apply (simp add: components_def ex_in_conv [symmetric], clarify)
  by (meson connected_component_def connected_component_trans)

lemma exists_component_superset: "⟦t ⊆ s; s ≠ {}; connected t⟧ ⟹ ∃c. c ∈ components s ∧ t ⊆ c"
  apply (cases "t = {}", force)
  apply (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)
  done

lemma components_intermediate_subset: "⟦s ∈ components u; s ⊆ t; t ⊆ u⟧ ⟹ s ∈ components t"
  apply (auto simp: components_iff)
  apply (metis connected_component_eq_empty connected_component_intermediate_subset)
  done

lemma in_components_unions_complement: "c ∈ components s ⟹ s - c = ⋃(components s - {c})"
  by (metis complement_connected_component_unions components_def components_iff)

lemma connected_intermediate_closure:
  assumes cs: "connected s" and st: "s ⊆ t" and ts: "t ⊆ closure s"
  shows "connected t"
proof (rule connectedI)
  fix A B
  assume A: "open A" and B: "open B" and Alap: "A ∩ t ≠ {}" and Blap: "B ∩ t ≠ {}"
    and disj: "A ∩ B ∩ t = {}" and cover: "t ⊆ A ∪ B"
  have disjs: "A ∩ B ∩ s = {}"
    using disj st by auto
  have "A ∩ closure s ≠ {}"
    using Alap Int_absorb1 ts by blast
  then have Alaps: "A ∩ s ≠ {}"
    by (simp add: A open_Int_closure_eq_empty)
  have "B ∩ closure s ≠ {}"
    using Blap Int_absorb1 ts by blast
  then have Blaps: "B ∩ s ≠ {}"
    by (simp add: B open_Int_closure_eq_empty)
  then show False
    using cs [unfolded connected_def] A B disjs Alaps Blaps cover st
    by blast
qed

lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
proof (cases "connected_component_set s x = {}")
  case True
  then show ?thesis
    by (metis closedin_empty)
next
  case False
  then obtain y where y: "connected_component s x y"
    by blast
  have *: "connected_component_set s x ⊆ s ∩ closure (connected_component_set s x)"
    by (auto simp: closure_def connected_component_in)
  have "connected_component s x y ⟹ s ∩ closure (connected_component_set s x) ⊆ connected_component_set s x"
    apply (rule connected_component_maximal, simp)
    using closure_subset connected_component_in apply fastforce
    using * connected_intermediate_closure apply blast+
    done
  with y * show ?thesis
    by (auto simp: closedin_closed)
qed

lemma closedin_component:
   "C ∈ components s ⟹ closedin (subtopology euclidean s) C"
  using closedin_connected_component componentsE by blast


subsection ‹Intersecting chains of compact sets and the Baire property›

proposition bounded_closed_chain:
  fixes  :: "'a::heine_borel set set"
  assumes "B ∈ ℱ" "bounded B" and : "⋀S. S ∈ ℱ ⟹ closed S" and "{} ∉ ℱ"
      and chain: "⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
    shows "⋂ℱ ≠ {}"
proof -
  have "B ∩ ⋂ℱ ≠ {}"
  proof (rule compact_imp_fip)
    show "compact B" "⋀T. T ∈ ℱ ⟹ closed T"
      by (simp_all add: assms compact_eq_bounded_closed)
    show "⟦finite 𝒢; 𝒢 ⊆ ℱ⟧ ⟹ B ∩ ⋂𝒢 ≠ {}" for 𝒢
    proof (induction 𝒢 rule: finite_induct)
      case empty
      with assms show ?case by force
    next
      case (insert U 𝒢)
      then have "U ∈ ℱ" and ne: "B ∩ ⋂𝒢 ≠ {}" by auto
      then consider "B ⊆ U" | "U ⊆ B"
          using ‹B ∈ ℱ› chain by blast
        then show ?case
        proof cases
          case 1
          then show ?thesis
            using Int_left_commute ne by auto
        next
          case 2
          have "U ≠ {}"
            using ‹U ∈ ℱ› ‹{} ∉ ℱ› by blast
          moreover
          have False if "⋀x. x ∈ U ⟹ ∃Y∈𝒢. x ∉ Y"
          proof -
            have "⋀x. x ∈ U ⟹ ∃Y∈𝒢. Y ⊆ U"
              by (metis chain contra_subsetD insert.prems insert_subset that)
            then obtain Y where "Y ∈ 𝒢" "Y ⊆ U"
              by (metis all_not_in_conv ‹U ≠ {}›)
            moreover obtain x where "x ∈ ⋂𝒢"
              by (metis Int_emptyI ne)
            ultimately show ?thesis
              by (metis Inf_lower subset_eq that)
          qed
          with 2 show ?thesis
            by blast
        qed
      qed
  qed
  then show ?thesis by blast
qed

corollary compact_chain:
  fixes  :: "'a::heine_borel set set"
  assumes "⋀S. S ∈ ℱ ⟹ compact S" "{} ∉ ℱ"
          "⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
    shows "⋂ ℱ ≠ {}"
proof (cases "ℱ = {}")
  case True
  then show ?thesis by auto
next
  case False
  show ?thesis
    by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
qed

lemma compact_nest:
  fixes F :: "'a::linorder ⇒ 'b::heine_borel set"
  assumes F: "⋀n. compact(F n)" "⋀n. F n ≠ {}" and mono: "⋀m n. m ≤ n ⟹ F n ⊆ F m"
  shows "⋂range F ≠ {}"
proof -
  have *: "⋀S T. S ∈ range F ∧ T ∈ range F ⟹ S ⊆ T ∨ T ⊆ S"
    by (metis mono image_iff le_cases)
  show ?thesis
    apply (rule compact_chain [OF _ _ *])
    using F apply (blast intro: dest: *)+
    done
qed

text‹The Baire property of dense sets›
theorem Baire:
  fixes S::"'a::{real_normed_vector,heine_borel} set"
  assumes "closed S" "countable 𝒢"
      and ope: "⋀T. T ∈ 𝒢 ⟹ openin (subtopology euclidean S) T ∧ S ⊆ closure T"
 shows "S ⊆ closure(⋂𝒢)"
proof (cases "𝒢 = {}")
  case True
  then show ?thesis
    using closure_subset by auto
next
  let ?g = "from_nat_into 𝒢"
  case False
  then have gin: "?g n ∈ 𝒢" for n
    by (simp add: from_nat_into)
  show ?thesis
  proof (clarsimp simp: closure_approachable)
    fix x and e::real
    assume "x ∈ S" "0 < e"
    obtain TF where opeF: "⋀n. openin (subtopology euclidean S) (TF n)"
               and ne: "⋀n. TF n ≠ {}"
               and subg: "⋀n. S ∩ closure(TF n) ⊆ ?g n"
               and subball: "⋀n. closure(TF n) ⊆ ball x e"
               and decr: "⋀n. TF(Suc n) ⊆ TF n"
    proof -
      have *: "∃Y. (openin (subtopology euclidean S) Y ∧ Y ≠ {} ∧
                   S ∩ closure Y ⊆ ?g n ∧ closure Y ⊆ ball x e) ∧ Y ⊆ U"
        if opeU: "openin (subtopology euclidean S) U" and "U ≠ {}" and cloU: "closure U ⊆ ball x e" for U n
      proof -
        obtain T where T: "open T" "U = T ∩ S"
          using ‹openin (subtopology euclidean S) U› by (auto simp: openin_subtopology)
        with ‹U ≠ {}› have "T ∩ closure (?g n) ≠ {}"
          using gin ope by fastforce
        then have "T ∩ ?g n ≠ {}"
          using ‹open T› open_Int_closure_eq_empty by blast
        then obtain y where "y ∈ U" "y ∈ ?g n"
          using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
        moreover have "openin (subtopology euclidean S) (U ∩ ?g n)"
          using gin ope opeU by blast
        ultimately obtain d where U: "U ∩ ?g n ⊆ S" and "d > 0" and d: "ball y d ∩ S ⊆ U ∩ ?g n"
          by (force simp: openin_contains_ball)
        show ?thesis
        proof (intro exI conjI)
          show "openin (subtopology euclidean S) (S ∩ ball y (d/2))"
            by (simp add: openin_open_Int)
          show "S ∩ ball y (d/2) ≠ {}"
            using ‹0 < d› ‹y ∈ U› opeU openin_imp_subset by fastforce
          have "S ∩ closure (S ∩ ball y (d/2)) ⊆ S ∩ closure (ball y (d/2))"
            using closure_mono by blast
          also have "... ⊆ ?g n"
            using ‹d > 0› d by force
          finally show "S ∩ closure (S ∩ ball y (d/2)) ⊆ ?g n" .
          have "closure (S ∩ ball y (d/2)) ⊆ S ∩ ball y d"
          proof -
            have "closure (ball y (d/2)) ⊆ ball y d"
              using ‹d > 0› by auto
            then have "closure (S ∩ ball y (d/2)) ⊆ ball y d"
              by (meson closure_mono inf.cobounded2 subset_trans)
            then show ?thesis
              by (simp add: ‹closed S› closure_minimal)
          qed
          also have "...  ⊆ ball x e"
            using cloU closure_subset d by blast
          finally show "closure (S ∩ ball y (d/2)) ⊆ ball x e" .
          show "S ∩ ball y (d/2) ⊆ U"
            using ball_divide_subset_numeral d by blast
        qed
      qed
      let  = "λn X. openin (subtopology euclidean S) X ∧ X ≠ {} ∧
                      S ∩ closure X ⊆ ?g n ∧ closure X ⊆ ball x e"
      have "closure (S ∩ ball x (e / 2)) ⊆ closure(ball x (e/2))"
        by (simp add: closure_mono)
      also have "...  ⊆ ball x e"
        using ‹e > 0› by auto
      finally have "closure (S ∩ ball x (e / 2)) ⊆ ball x e" .
      moreover have"openin (subtopology euclidean S) (S ∩ ball x (e / 2))" "S ∩ ball x (e / 2) ≠ {}"
        using ‹0 < e› ‹x ∈ S› by auto
      ultimately obtain Y where Y: "?Φ 0 Y ∧ Y ⊆ S ∩ ball x (e / 2)"
            using * [of "S ∩ ball x (e/2)" 0] by metis
      show thesis
      proof (rule exE [OF dependent_nat_choice [of  "λn X Y. Y ⊆ X"]])
        show "∃x. ?Φ 0 x"
          using Y by auto
        show "∃Y. ?Φ (Suc n) Y ∧ Y ⊆ X" if "?Φ n X" for X n
          using that by (blast intro: *)
      qed (use that in metis)
    qed
    have "(⋂n. S ∩ closure (TF n)) ≠ {}"
    proof (rule compact_nest)
      show "⋀n. compact (S ∩ closure (TF n))"
        by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF ‹closed S›])
      show "⋀n. S ∩ closure (TF n) ≠ {}"
        by (metis Int_absorb1 opeF ‹closed S› closure_eq_empty closure_minimal ne openin_imp_subset)
      show "⋀m n. m ≤ n ⟹ S ∩ closure (TF n) ⊆ S ∩ closure (TF m)"
        by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
    qed
    moreover have "(⋂n. S ∩ closure (TF n)) ⊆ {y ∈ ⋂𝒢. dist y x < e}"
    proof (clarsimp, intro conjI)
      fix y
      assume "y ∈ S" and y: "∀n. y ∈ closure (TF n)"
      then show "∀T∈𝒢. y ∈ T"
        by (metis Int_iff from_nat_into_surj [OF ‹countable 𝒢›] set_mp subg)
      show "dist y x < e"
        by (metis y dist_commute mem_ball subball subsetCE)
    qed
    ultimately show "∃y ∈ ⋂𝒢. dist y x < e"
      by auto
  qed
qed

subsection%unimportant ‹Some theorems on sups and infs using the notion "bounded"›

lemma bounded_real: "bounded (S::real set) ⟷ (∃a. ∀x∈S. ¦x¦ ≤ a)"
  by (simp add: bounded_iff)

lemma bounded_imp_bdd_above: "bounded S ⟹ bdd_above (S :: real set)"
  by (auto simp: bounded_def bdd_above_def dist_real_def)
     (metis abs_le_D1 abs_minus_commute diff_le_eq)

lemma bounded_imp_bdd_below: "bounded S ⟹ bdd_below (S :: real set)"
  by (auto simp: bounded_def bdd_below_def dist_real_def)
     (metis abs_le_D1 add.commute diff_le_eq)

lemma bounded_inner_imp_bdd_above:
  assumes "bounded s"
    shows "bdd_above ((λx. x ∙ a) ` s)"
by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)

lemma bounded_inner_imp_bdd_below:
  assumes "bounded s"
    shows "bdd_below ((λx. x ∙ a) ` s)"
by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)

lemma bounded_has_Sup:
  fixes S :: "real set"
  assumes "bounded S"
    and "S ≠ {}"
  shows "∀x∈S. x ≤ Sup S"
    and "∀b. (∀x∈S. x ≤ b) ⟶ Sup S ≤ b"
proof
  show "∀b. (∀x∈S. x ≤ b) ⟶ Sup S ≤ b"
    using assms by (metis cSup_least)
qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

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 simp: bounded_imp_bdd_above sup_max cSup_insert_If)

lemma Sup_insert_finite:
  fixes S :: "'a::conditionally_complete_linorder set"
  shows "finite S ⟹ Sup (insert x S) = (if S = {} then x else max x (Sup S))"
by (simp add: cSup_insert sup_max)

lemma bounded_has_Inf:
  fixes S :: "real set"
  assumes "bounded S"
    and "S ≠ {}"
  shows "∀x∈S. x ≥ Inf S"
    and "∀b. (∀x∈S. x ≥ b) ⟶ Inf S ≥ b"
proof
  show "∀b. (∀x∈S. x ≥ b) ⟶ Inf S ≥ b"
    using assms by (metis cInf_greatest)
qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

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 simp: bounded_imp_bdd_below inf_min cInf_insert_If)

lemma Inf_insert_finite:
  fixes S :: "'a::conditionally_complete_linorder set"
  shows "finite S ⟹ Inf (insert x S) = (if S = {} then x else min x (Inf S))"
by (simp add: cInf_eq_Min)

lemma finite_imp_less_Inf:
  fixes a :: "'a::conditionally_complete_linorder"
  shows "⟦finite X; x ∈ X; ⋀x. x∈X ⟹ a < x⟧ ⟹ a < Inf X"
  by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)

lemma finite_less_Inf_iff:
  fixes a :: "'a :: conditionally_complete_linorder"
  shows "⟦finite X; X ≠ {}⟧ ⟹ a < Inf X ⟷ (∀x ∈ X. a < x)"
  by (auto simp: cInf_eq_Min)

lemma finite_imp_Sup_less:
  fixes a :: "'a::conditionally_complete_linorder"
  shows "⟦finite X; x ∈ X; ⋀x. x∈X ⟹ a > x⟧ ⟹ a > Sup X"
  by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)

lemma finite_Sup_less_iff:
  fixes a :: "'a :: conditionally_complete_linorder"
  shows "⟦finite X; X ≠ {}⟧ ⟹ a > Sup X ⟷ (∀x ∈ X. a > x)"
  by (auto simp: cSup_eq_Max)

proposition is_interval_compact:
   "is_interval S ∧ compact S ⟷ (∃a b. S = cbox a b)"   (is "?lhs = ?rhs")
proof (cases "S = {}")
  case True
  with empty_as_interval show ?thesis by auto
next
  case False
  show ?thesis
  proof
    assume L: ?lhs
    then have "is_interval S" "compact S" by auto
    define a where "a ≡ ∑i∈Basis. (INF x:S. x ∙ i) *R i"
    define b where "b ≡ ∑i∈Basis. (SUP x:S. x ∙ i) *R i"
    have 1: "⋀x i. ⟦x ∈ S; i ∈ Basis⟧ ⟹ (INF x:S. x ∙ i) ≤ x ∙ i"
      by (simp add: cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded L)
    have 2: "⋀x i. ⟦x ∈ S; i ∈ Basis⟧ ⟹ x ∙ i ≤ (SUP x:S. x ∙ i)"
      by (simp add: cSup_upper bounded_inner_imp_bdd_above compact_imp_bounded L)
    have 3: "x ∈ S" if inf: "⋀i. i ∈ Basis ⟹ (INF x:S. x ∙ i) ≤ x ∙ i"
                   and sup: "⋀i. i ∈ Basis ⟹ x ∙ i ≤ (SUP x:S. x ∙ i)" for x
    proof (rule mem_box_componentwiseI [OF ‹is_interval S›])
      fix i::'a
      assume i: "i ∈ Basis"
      have cont: "continuous_on S (λx. x ∙ i)"
        by (intro continuous_intros)
      obtain a where "a ∈ S" and a: "⋀y. y∈S ⟹ a ∙ i ≤ y ∙ i"
        using continuous_attains_inf [OF ‹compact S› False cont] by blast
      obtain b where "b ∈ S" and b: "⋀y. y∈S ⟹ y ∙ i ≤ b ∙ i"
        using continuous_attains_sup [OF ‹compact S› False cont] by blast
      have "a ∙ i ≤ (INF x:S. x ∙ i)"
        by (simp add: False a cINF_greatest)
      also have "… ≤ x ∙ i"
        by (simp add: i inf)
      finally have ai: "a ∙ i ≤ x ∙ i" .
      have "x ∙ i ≤ (SUP x:S. x ∙ i)"
        by (simp add: i sup)
      also have "(SUP x:S. x ∙ i) ≤ b ∙ i"
        by (simp add: False b cSUP_least)
      finally have bi: "x ∙ i ≤ b ∙ i" .
      show "x ∙ i ∈ (λx. x ∙ i) ` S"
        apply (rule_tac x="∑j∈Basis. (if j = i then x ∙ i else a ∙ j) *R j" in image_eqI)
        apply (simp add: i)
        apply (rule mem_is_intervalI [OF ‹is_interval S› ‹a ∈ S› ‹b ∈ S›])
        using i ai bi apply force
        done
    qed
    have "S = cbox a b"
      by (auto simp: a_def b_def mem_box intro: 1 2 3)
    then show ?rhs
      by blast
  next
    assume R: ?rhs
    then show ?lhs
      using compact_cbox is_interval_cbox by blast
  qed
qed

text ‹Hence some handy theorems on distance, diameter etc. of/from a set.›

lemma distance_attains_sup:
  assumes "compact s" "s ≠ {}"
  shows "∃x∈s. ∀y∈s. dist a y ≤ dist a x"
proof (rule continuous_attains_sup [OF assms])
  {
    fix x
    assume "x∈s"
    have "(dist a ⤏ dist a x) (at x within s)"
      by (intro tendsto_dist tendsto_const tendsto_ident_at)
  }
  then show "continuous_on s (dist a)"
    unfolding continuous_on ..
qed

text ‹For \emph{minimal} distance, we only need closure, not compactness.›

lemma distance_attains_inf:
  fixes a :: "'a::heine_borel"
  assumes "closed s" and "s ≠ {}"
  obtains x where "x∈s" "⋀y. y ∈ s ⟹ dist a x ≤ dist a y"
proof -
  from assms obtain b where "b ∈ s" by auto
  let ?B = "s ∩ cball a (dist b a)"
  have "?B ≠ {}" using ‹b ∈ s›
    by (auto simp: dist_commute)
  moreover have "continuous_on ?B (dist a)"
    by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_ident continuous_const)
  moreover have "compact ?B"
    by (intro closed_Int_compact ‹closed s› compact_cball)
  ultimately obtain x where "x ∈ ?B" "∀y∈?B. dist a x ≤ dist a y"
    by (metis continuous_attains_inf)
  with that show ?thesis by fastforce
qed


subsection%unimportant‹Relations among convergence and absolute convergence for power series›

lemma summable_imp_bounded:
  fixes f :: "nat ⇒ 'a::real_normed_vector"
  shows "summable f ⟹ bounded (range f)"
by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)

lemma summable_imp_sums_bounded:
   "summable f ⟹ bounded (range (λn. sum f {..<n}))"
by (auto simp: summable_def sums_def dest: convergent_imp_bounded)

lemma power_series_conv_imp_absconv_weak:
  fixes a:: "nat ⇒ 'a::{real_normed_div_algebra,banach}" and w :: 'a
  assumes sum: "summable (λn. a n * z ^ n)" and no: "norm w < norm z"
    shows "summable (λn. of_real(norm(a n)) * w ^ n)"
proof -
  obtain M where M: "⋀x. norm (a x * z ^ x) ≤ M"
    using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
  then have *: "summable (λn. norm (a n) * norm w ^ n)"
    by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
  show ?thesis
    apply (rule series_comparison_complex [of "(λn. of_real(norm(a n) * norm w ^ n))"])
    apply (simp only: summable_complex_of_real *)
    apply (auto simp: norm_mult norm_power)
    done
qed

subsection%unimportant ‹Bounded closed nest property (proof does not use Heine-Borel)›

lemma bounded_closed_nest:
  fixes S :: "nat ⇒ ('a::heine_borel) set"
  assumes "⋀n. closed (S n)"
      and "⋀n. S n ≠ {}"
      and "⋀m n. m ≤ n ⟹ S n ⊆ S m"
      and "bounded (S 0)"
  obtains a where "⋀n. a ∈ S n"
proof -
  from assms(2) obtain x where x: "∀n. x n ∈ S n"
    using choice[of "λn x. x ∈ S n"] by auto
  from assms(4,1) have "seq_compact (S 0)"
    by (simp add: bounded_closed_imp_seq_compact)
  then obtain l r where lr: "l ∈ S 0" "strict_mono r" "(x ∘ r) ⇢ l"
    using x and assms(3) unfolding seq_compact_def by blast
  have "∀n. l ∈ S n"
  proof
    fix n :: nat
    have "closed (S n)"
      using assms(1) by simp
    moreover have "∀i. (x ∘ r) i ∈ S i"
      using x and assms(3) and lr(2) [THEN seq_suble] by auto
    then have "∀i. (x ∘ r) (i + n) ∈ S n"
      using assms(3) by (fast intro!: le_add2)
    moreover have "(λi. (x ∘ r) (i + n)) ⇢ l"
      using lr(3) by (rule LIMSEQ_ignore_initial_segment)
    ultimately show "l ∈ S n"
      by (rule closed_sequentially)
  qed
  then show ?thesis 
    using that by blast
qed

text ‹Decreasing case does not even need compactness, just completeness.›

lemma decreasing_closed_nest:
  fixes S :: "nat ⇒ ('a::complete_space) set"
  assumes "⋀n. closed (S n)"
          "⋀n. S n ≠ {}"
          "⋀m n. m ≤ n ⟹ S n ⊆ S m"
          "⋀e. e>0 ⟹ ∃n. ∀x∈S n. ∀y∈S n. dist x y < e"
  obtains a where "⋀n. a ∈ S n"
proof -
  have "∀n. ∃x. x ∈ S n"
    using assms(2) by auto
  then have "∃t. ∀n. t n ∈ S n"
    using choice[of "λn x. x ∈ S n"] by auto
  then obtain t where t: "∀n. t n ∈ S n" by auto
  {
    fix e :: real
    assume "e > 0"
    then obtain N where N: "∀x∈S N. ∀y∈S N. dist x y < e"
      using assms(4) by blast
    {
      fix m n :: nat
      assume "N ≤ m ∧ N ≤ n"
      then have "t m ∈ S N" "t n ∈ S N"
        using assms(3) t unfolding  subset_eq t by blast+
      then have "dist (t m) (t n) < e"
        using N by auto
    }
    then have "∃N. ∀m n. N ≤ m ∧ N ≤ n ⟶ dist (t m) (t n) < e"
      by auto
  }
  then have "Cauchy t"
    unfolding cauchy_def by auto
  then obtain l where l:"(t ⤏ l) sequentially"
    using complete_UNIV unfolding complete_def by auto
  { fix n :: nat
    { fix e :: real
      assume "e > 0"
      then obtain N :: nat where N: "∀n≥N. dist (t n) l < e"
        using l[unfolded lim_sequentially] by auto
      have "t (max n N) ∈ S n"
        by (meson assms(3) contra_subsetD max.cobounded1 t)
      then have "∃y∈S n. dist y l < e"
        using N max.cobounded2 by blast
    }
    then have "l ∈ S n"
      using closed_approachable[of "S n" l] assms(1) by auto
  }
  then show ?thesis
    using that by blast
qed

text ‹Strengthen it to the intersection actually being a singleton.›

lemma decreasing_closed_nest_sing:
  fixes S :: "nat ⇒ 'a::complete_space set"
  assumes "⋀n. closed(S n)"
          "⋀n. S n ≠ {}"
          "⋀m n. m ≤ n ⟹ S n ⊆ S m"
          "⋀e. e>0 ⟹ ∃n. ∀x ∈ (S n). ∀ y∈(S n). dist x y < e"
  shows "∃a. ⋂(range S) = {a}"
proof -
  obtain a where a: "∀n. a ∈ S n"
    using decreasing_closed_nest[of S] using assms by auto
  { fix b
    assume b: "b ∈ ⋂(range S)"
    { fix e :: real
      assume "e > 0"
      then have "dist a b < e"
        using assms(4) and b and a by blast
    }
    then have "dist a b = 0"
      by (metis dist_eq_0_iff dist_nz less_le)
  }
  with a have "⋂(range S) = {a}"
    unfolding image_def by auto
  then show ?thesis ..
qed


subsection ‹Infimum Distance›

definition%important "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"

lemma bdd_below_image_dist[intro, simp]: "bdd_below (dist x ` A)"
  by (auto intro!: zero_le_dist)

lemma infdist_notempty: "A ≠ {} ⟹ infdist x A = (INF a:A. dist x a)"
  by (simp add: infdist_def)

lemma infdist_nonneg: "0 ≤ infdist x A"
  by (auto simp: infdist_def intro: cINF_greatest)

lemma infdist_le: "a ∈ A ⟹ infdist x A ≤ dist x a"
  by (auto intro: cINF_lower simp add: infdist_def)

lemma infdist_le2: "a ∈ A ⟹ dist x a ≤ d ⟹ infdist x A ≤ d"
  by (auto intro!: cINF_lower2 simp add: infdist_def)

lemma infdist_zero[simp]: "a ∈ A ⟹ infdist a A = 0"
  by (auto intro!: antisym infdist_nonneg infdist_le2)

lemma infdist_triangle: "infdist x A ≤ infdist y A + dist x y"
proof (cases "A = {}")
  case True
  then show ?thesis by (simp add: infdist_def)
next
  case False
  then obtain a where "a ∈ A" by auto
  have "infdist x A ≤ Inf {dist x y + dist y a |a. a ∈ A}"
  proof (rule cInf_greatest)
    from ‹A ≠ {}› show "{dist x y + dist y a |a. a ∈ A} ≠ {}"
      by simp
    fix d
    assume "d ∈ {dist x y + dist y a |a. a ∈ A}"
    then obtain a where d: "d = dist x y + dist y a" "a ∈ A"
      by auto
    show "infdist x A ≤ d"
      unfolding infdist_notempty[OF ‹A ≠ {}›]
    proof (rule cINF_lower2)
      show "a ∈ A" by fact
      show "dist x a ≤ d"
        unfolding d by (rule dist_triangle)
    qed simp
  qed
  also have "… = dist x y + infdist y A"
  proof (rule cInf_eq, safe)
    fix a
    assume "a ∈ A"
    then show "dist x y + infdist y A ≤ dist x y + dist y a"
      by (auto intro: infdist_le)
  next
    fix i
    assume inf: "⋀d. d ∈ {dist x y + dist y a |a. a ∈ A} ⟹ i ≤ d"
    then have "i - dist x y ≤ infdist y A"
      unfolding infdist_notempty[OF ‹A ≠ {}›] using ‹a ∈ A›
      by (intro cINF_greatest) (auto simp: field_simps)
    then show "i ≤ dist x y + infdist y A"
      by simp
  qed
  finally show ?thesis by simp
qed

lemma in_closure_iff_infdist_zero:
  assumes "A ≠ {}"
  shows "x ∈ closure A ⟷ infdist x A = 0"
proof
  assume "x ∈ closure A"
  show "infdist x A = 0"
  proof (rule ccontr)
    assume "infdist x A ≠ 0"
    with infdist_nonneg[of x A] have "infdist x A > 0"
      by auto
    then have "ball x (infdist x A) ∩ closure A = {}"
      apply auto
      apply (metis ‹x ∈ closure A› closure_approachable dist_commute infdist_le not_less)
      done
    then have "x ∉ closure A"
      by (metis ‹0 < infdist x A› centre_in_ball disjoint_iff_not_equal)
    then show False using ‹x ∈ closure A› by simp
  qed
next
  assume x: "infdist x A = 0"
  then obtain a where "a ∈ A"
    by atomize_elim (metis all_not_in_conv assms)
  show "x ∈ closure A"
    unfolding closure_approachable
    apply safe
  proof (rule ccontr)
    fix e :: real
    assume "e > 0"
    assume "¬ (∃y∈A. dist y x < e)"
    then have "infdist x A ≥ e" using ‹a ∈ A›
      unfolding infdist_def
      by (force simp: dist_commute intro: cINF_greatest)
    with x ‹e > 0› show False by auto
  qed
qed

lemma in_closed_iff_infdist_zero:
  assumes "closed A" "A ≠ {}"
  shows "x ∈ A ⟷ infdist x A = 0"
proof -
  have "x ∈ closure A ⟷ infdist x A = 0"
    by (rule in_closure_iff_infdist_zero) fact
  with assms show ?thesis by simp
qed

lemma infdist_pos_not_in_closed:
  assumes "closed S" "S ≠ {}" "x ∉ S"
  shows "infdist x S > 0"
using in_closed_iff_infdist_zero[OF assms(1) assms(2), of x] assms(3) infdist_nonneg le_less by fastforce

lemma
  infdist_attains_inf:
  fixes X::"'a::heine_borel set"
  assumes "closed X"
  assumes "X ≠ {}"
  obtains x where "x ∈ X" "infdist y X = dist y x"
proof -
  have "bdd_below (dist y ` X)"
    by auto
  from distance_attains_inf[OF assms, of y]
  obtain x where INF: "x ∈ X" "⋀z. z ∈ X ⟹ dist y x ≤ dist y z" by auto
  have "infdist y X = dist y x"
    by (auto simp: infdist_def assms
      intro!: antisym cINF_lower[OF _ ‹x ∈ X›] cINF_greatest[OF assms(2) INF(2)])
  with ‹x ∈ X› show ?thesis ..
qed


text ‹Every metric space is a T4 space:›

instance metric_space  t4_space
proof
  fix S T::"'a set" assume H: "closed S" "closed T" "S ∩ T = {}"
  consider "S = {}" | "T = {}" | "S ≠ {} ∧ T ≠ {}" by auto
  then show "∃U V. open U ∧ open V ∧ S ⊆ U ∧ T ⊆ V ∧ U ∩ V = {}"
  proof (cases)
    case 1
    show ?thesis
      apply (rule exI[of _ "{}"], rule exI[of _ UNIV]) using 1 by auto
  next
    case 2
    show ?thesis
      apply (rule exI[of _ UNIV], rule exI[of _ "{}"]) using 2 by auto
  next
    case 3
    define U where "U = (⋃x∈S. ball x ((infdist x T)/2))"
    have A: "open U" unfolding U_def by auto
    have "infdist x T > 0" if "x ∈ S" for x
      using H that 3 by (auto intro!: infdist_pos_not_in_closed)
    then have B: "S ⊆ U" unfolding U_def by auto
    define V where "V = (⋃x∈T. ball x ((infdist x S)/2))"
    have C: "open V" unfolding V_def by auto
    have "infdist x S > 0" if "x ∈ T" for x
      using H that 3 by (auto intro!: infdist_pos_not_in_closed)
    then have D: "T ⊆ V" unfolding V_def by auto

    have "(ball x ((infdist x T)/2)) ∩ (ball y ((infdist y S)/2)) = {}" if "x ∈ S" "y ∈ T" for x y
    proof (auto)
      fix z assume H: "dist x z * 2 < infdist x T" "dist y z * 2 < infdist y S"
      have "2 * dist x y ≤ 2 * dist x z + 2 * dist y z"
        using dist_triangle[of x y z] by (auto simp add: dist_commute)
      also have "... < infdist x T + infdist y S"
        using H by auto
      finally have "dist x y < infdist x T ∨ dist x y < infdist y S"
        by auto
      then show False
        using infdist_le[OF ‹x ∈ S›, of y] infdist_le[OF ‹y ∈ T›, of x] by (auto simp add: dist_commute)
    qed
    then have E: "U ∩ V = {}"
      unfolding U_def V_def by auto
    show ?thesis
      apply (rule exI[of _ U], rule exI[of _ V]) using A B C D E by auto
  qed
qed

lemma tendsto_infdist [tendsto_intros]:
  assumes f: "(f ⤏ l) F"
  shows "((λx. infdist (f x) A) ⤏ infdist l A) F"
proof (rule tendstoI)
  fix e ::real
  assume "e > 0"
  from tendstoD[OF f this]
  show "eventually (λx. dist (infdist (f x) A) (infdist l A) < e) F"
  proof (eventually_elim)
    fix x
    from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
    have "dist (infdist (f x) A) (infdist l A) ≤ dist (f x) l"
      by (simp add: dist_commute dist_real_def)
    also assume "dist (f x) l < e"
    finally show "dist (infdist (f x) A) (infdist l A) < e" .
  qed
qed

lemma continuous_infdist[continuous_intros]:
  assumes "continuous F f"
  shows "continuous F (λx. infdist (f x) A)"
  using assms unfolding continuous_def by (rule tendsto_infdist)

lemma compact_infdist_le:
  fixes A::"'a::heine_borel set"
  assumes "A ≠ {}"
  assumes "compact A"
  assumes "e > 0"
  shows "compact {x. infdist x A ≤ e}"
proof -
  from continuous_closed_vimage[of "{0..e}" "λx. infdist x A"]
    continuous_infdist[OF continuous_ident, of _ UNIV A]
  have "closed {x. infdist x A ≤ e}" by (auto simp: vimage_def infdist_nonneg)
  moreover
  from assms obtain x0 b where b: "⋀x. x ∈ A ⟹ dist x0 x ≤ b" "closed A"
    by (auto simp: compact_eq_bounded_closed bounded_def)
  {
    fix y
    assume le: "infdist y A ≤ e"
    from infdist_attains_inf[OF ‹closed A› ‹A ≠ {}›, of y]
    obtain z where z: "z ∈ A" "infdist y A = dist y z" by blast
    have "dist x0 y ≤ dist y z + dist x0 z"
      by (metis dist_commute dist_triangle)
    also have "dist y z ≤ e" using le z by simp
    also have "dist x0 z ≤ b" using b z by simp
    finally have "dist x0 y ≤ b + e" by arith
  } then
  have "bounded {x. infdist x A ≤ e}"
    by (auto simp: bounded_any_center[where a=x0] intro!: exI[where x="b + e"])
  ultimately show "compact {x. infdist x A ≤ e}"
    by (simp add: compact_eq_bounded_closed)
qed

subsection%unimportant ‹Equality of continuous functions on closure and related results›

lemma continuous_closedin_preimage_constant:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "continuous_on S f ⟹ closedin (subtopology euclidean S) {x ∈ S. f x = a}"
  using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

lemma continuous_closed_preimage_constant:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "continuous_on S f ⟹ closed S ⟹ closed {x ∈ S. f x = a}"
  using continuous_closed_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

lemma continuous_constant_on_closure:
  fixes f :: "_ ⇒ 'b::t1_space"
  assumes "continuous_on (closure S) f"
      and "⋀x. x ∈ S ⟹ f x = a"
      and "x ∈ closure S"
  shows "f x = a"
    using continuous_closed_preimage_constant[of "closure S" f a]
      assms closure_minimal[of S "{x ∈ closure S. f x = a}"] closure_subset
    unfolding subset_eq
    by auto

lemma image_closure_subset:
  assumes contf: "continuous_on (closure S) f"
    and "closed T"
    and "(f ` S) ⊆ T"
  shows "f ` (closure S) ⊆ T"
proof -
  have "S ⊆ {x ∈ closure S. f x ∈ T}"
    using assms(3) closure_subset by auto
  moreover have "closed (closure S ∩ f -` T)"
    using continuous_closed_preimage[OF contf] ‹closed T› by auto
  ultimately have "closure S = (closure S ∩ f -` T)"
    using closure_minimal[of S "(closure S ∩ f -` T)"] by auto
  then show ?thesis by auto
qed

lemma continuous_on_closure_norm_le:
  fixes f :: "'a::metric_space ⇒ 'b::real_normed_vector"
  assumes "continuous_on (closure s) f"
    and "∀y ∈ s. norm(f y) ≤ b"
    and "x ∈ (closure s)"
  shows "norm (f x) ≤ b"
proof -
  have *: "f ` s ⊆ cball 0 b"
    using assms(2)[unfolded mem_cball_0[symmetric]] by auto
  show ?thesis
    by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
qed

lemma isCont_indicator:
  fixes x :: "'a::t2_space"
  shows "isCont (indicator A :: 'a ⇒ real) x = (x ∉ frontier A)"
proof auto
  fix x
  assume cts_at: "isCont (indicator A :: 'a ⇒ real) x" and fr: "x ∈ frontier A"
  with continuous_at_open have 1: "∀V::real set. open V ∧ indicator A x ∈ V ⟶
    (∃U::'a set. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ V))" by auto
  show False
  proof (cases "x ∈ A")
    assume x: "x ∈ A"
    hence "indicator A x ∈ ({0<..<2} :: real set)" by simp
    hence "∃U. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ ({0<..<2} :: real set))"
      using 1 open_greaterThanLessThan by blast
    then guess U .. note U = this
    hence "∀y∈U. indicator A y > (0::real)"
      unfolding greaterThanLessThan_def by auto
    hence "U ⊆ A" using indicator_eq_0_iff by force
    hence "x ∈ interior A" using U interiorI by auto
    thus ?thesis using fr unfolding frontier_def by simp
  next
    assume x: "x ∉ A"
    hence "indicator A x ∈ ({-1<..<1} :: real set)" by simp
    hence "∃U. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ ({-1<..<1} :: real set))"
      using 1 open_greaterThanLessThan by blast
    then guess U .. note U = this
    hence "∀y∈U. indicator A y < (1::real)"
      unfolding greaterThanLessThan_def by auto
    hence "U ⊆ -A" by auto
    hence "x ∈ interior (-A)" using U interiorI by auto
    thus ?thesis using fr interior_complement unfolding frontier_def by auto
  qed
next
  assume nfr: "x ∉ frontier A"
  hence "x ∈ interior A ∨ x ∈ interior (-A)"
    by (auto simp: frontier_def closure_interior)
  thus "isCont ((indicator A)::'a ⇒ real) x"
  proof
    assume int: "x ∈ interior A"
    then obtain U where U: "open U" "x ∈ U" "U ⊆ A" unfolding interior_def by auto
    hence "∀y∈U. indicator A y = (1::real)" unfolding indicator_def by auto
    hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)
    thus ?thesis using U continuous_on_eq_continuous_at by auto
  next
    assume ext: "x ∈ interior (-A)"
    then obtain U where U: "open U" "x ∈ U" "U ⊆ -A" unfolding interior_def by auto
    then have "continuous_on U (indicator A)"
      using continuous_on_topological by (auto simp: subset_iff)
    thus ?thesis using U continuous_on_eq_continuous_at by auto
  qed
qed

subsection%unimportant ‹A function constant on a set›

definition constant_on  (infixl "(constant'_on)" 50)
  where "f constant_on A ≡ ∃y. ∀x∈A. f x = y"

lemma constant_on_subset: "⟦f constant_on A; B ⊆ A⟧ ⟹ f constant_on B"
  unfolding constant_on_def by blast

lemma injective_not_constant:
  fixes S :: "'a::{perfect_space} set"
  shows "⟦open S; inj_on f S; f constant_on S⟧ ⟹ S = {}"
unfolding constant_on_def
by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)

lemma constant_on_closureI:
  fixes f :: "_ ⇒ 'b::t1_space"
  assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
    shows "f constant_on (closure S)"
using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
by metis

subsection%unimportant‹Relating linear images to open/closed/interior/closure›

proposition open_surjective_linear_image:
  fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space"
  assumes "open A" "linear f" "surj f"
    shows "open(f ` A)"
unfolding open_dist
proof clarify
  fix x
  assume "x ∈ A"
  have "bounded (inv f ` Basis)"
    by (simp add: finite_imp_bounded)
  with bounded_pos obtain B where "B > 0" and B: "⋀x. x ∈ inv f ` Basis ⟹ norm x ≤ B"
    by metis
  obtain e where "e > 0" and e: "⋀z. dist z x < e ⟹ z ∈ A"
    by (metis open_dist ‹x ∈ A› ‹open A›)
  define δ where "δ ≡ e / B / DIM('b)"
  show "∃e>0. ∀y. dist y (f x) < e ⟶ y ∈ f ` A"
  proof (intro exI conjI)
    show "δ > 0"
      using ‹e > 0› ‹B > 0›  by (simp add: δ_def divide_simps)
    have "y ∈ f ` A" if "dist y (f x) * (B * real DIM('b)) < e" for y
    proof -
      define u where "u ≡ y - f x"
      show ?thesis
      proof (rule image_eqI)
        show "y = f (x + (∑i∈Basis. (u ∙ i) *R inv f i))"
          apply (simp add: linear_add linear_sum linear.scaleR ‹linear f› surj_f_inv_f ‹surj f›)
          apply (simp add: euclidean_representation u_def)
          done
        have "dist (x + (∑i∈Basis. (u ∙ i) *R inv f i)) x ≤ (∑i∈Basis. norm ((u ∙ i) *R inv f i))"
          by (simp add: dist_norm sum_norm_le)
        also have "... = (∑i∈Basis. ¦u ∙ i¦ * norm (inv f i))"
          by simp
        also have "... ≤ (∑i∈Basis. ¦u ∙ i¦) * B"
          by (simp add: B sum_distrib_right sum_mono mult_left_mono)
        also have "... ≤ DIM('b) * dist y (f x) * B"
          apply (rule mult_right_mono [OF sum_bounded_above])
          using ‹0 < B› by (auto simp: Basis_le_norm dist_norm u_def)
        also have "... < e"
          by (metis mult.commute mult.left_commute that)
        finally show "x + (∑i∈Basis. (u ∙ i) *R inv f i) ∈ A"
          by (rule e)
      qed
    qed
    then show "∀y. dist y (f x) < δ ⟶ y ∈ f ` A"
      using ‹e > 0› ‹B > 0›
      by (auto simp: δ_def divide_simps mult_less_0_iff)
  qed
qed

corollary open_bijective_linear_image_eq:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes "linear f" "bij f"
    shows "open(f ` A) ⟷ open A"
proof
  assume "open(f ` A)"
  then have "open(f -` (f ` A))"
    using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)
  then show "open A"
    by (simp add: assms bij_is_inj inj_vimage_image_eq)
next
  assume "open A"
  then show "open(f ` A)"
    by (simp add: assms bij_is_surj open_surjective_linear_image)
qed

corollary interior_bijective_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes "linear f" "bij f"
  shows "interior (f ` S) = f ` interior S"  (is "?lhs = ?rhs")
proof safe
  fix x
  assume x: "x ∈ ?lhs"
  then obtain T where "open T" and "x ∈ T" and "T ⊆ f ` S"
    by (metis interiorE)
  then show "x ∈ ?rhs"
    by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
next
  fix x
  assume x: "x ∈ interior S"
  then show "f x ∈ interior (f ` S)"
    by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
qed

lemma interior_injective_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
  assumes "linear f" "inj f"
   shows "interior(f ` S) = f ` (interior S)"
  by (simp add: linear_injective_imp_surjective assms bijI interior_bijective_linear_image)

lemma interior_surjective_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
  assumes "linear f" "surj f"
   shows "interior(f ` S) = f ` (interior S)"
  by (simp add: assms interior_injective_linear_image linear_surjective_imp_injective)

lemma interior_negations:
  fixes S :: "'a::euclidean_space set"
  shows "interior(uminus ` S) = image uminus (interior S)"
  by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)

text ‹Preservation of compactness and connectedness under continuous function.›

lemma compact_eq_openin_cover:
  "compact S ⟷
    (∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃C ⟶
      (∃D⊆C. finite D ∧ S ⊆ ⋃D))"
proof safe
  fix C
  assume "compact S" and "∀c∈C. openin (subtopology euclidean S) c" and "S ⊆ ⋃C"
  then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ ⋃{T. open T ∧ S ∩ T ∈ C}"
    unfolding openin_open by force+
  with ‹compact S› obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ ⋃D"
    by (meson compactE)
  then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ ⋃(image (λT. S ∩ T) D)"
    by auto
  then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
next
  assume 1: "∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃C ⟶
        (∃D⊆C. finite D ∧ S ⊆ ⋃D)"
  show "compact S"
  proof (rule compactI)
    fix C
    let ?C = "image (λT. S ∩ T) C"
    assume "∀t∈C. open t" and "S ⊆ ⋃C"
    then have "(∀c∈?C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃?C"
      unfolding openin_open by auto
    with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ ⋃D"
      by metis
    let ?D = "inv_into C (λT. S ∩ T) ` D"
    have "?D ⊆ C ∧ finite ?D ∧ S ⊆ ⋃?D"
    proof (intro conjI)
      from ‹D ⊆ ?C› show "?D ⊆ C"
        by (fast intro: inv_into_into)
      from ‹finite D› show "finite ?D"
        by (rule finite_imageI)
      from ‹S ⊆ ⋃D› show "S ⊆ ⋃?D"
        apply (rule subset_trans, clarsimp)
        apply (frule subsetD [OF ‹D ⊆ ?C›, THEN f_inv_into_f])
        apply (erule rev_bexI, fast)
        done
    qed
    then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
  qed
qed

subsection%unimportant‹ Theorems relating continuity and uniform continuity to closures›

lemma continuous_on_closure:
   "continuous_on (closure S) f ⟷
    (∀x e. x ∈ closure S ∧ 0 < e
           ⟶ (∃d. 0 < d ∧ (∀y. y ∈ S ∧ dist y x < d ⟶ dist (f y) (f x) < e)))"
   (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    unfolding continuous_on_iff  by (metis Un_iff closure_def)
next
  assume R [rule_format]: ?rhs
  show ?lhs
  proof
    fix x and e::real
    assume "0 < e" and x: "x ∈ closure S"
    obtain δ::real where "δ > 0"
                   and δ: "⋀y. ⟦y ∈ S; dist y x < δ⟧ ⟹ dist (f y) (f x) < e/2"
      using R [of x "e/2"] ‹0 < e› x by auto
    have "dist (f y) (f x) ≤ e" if y: "y ∈ closure S" and dyx: "dist y x < δ/2" for y
    proof -
      obtain δ'::real where "δ' > 0"
                      and δ': "⋀z. ⟦z ∈ S; dist z y < δ'⟧ ⟹ dist (f z) (f y) < e/2"
        using R [of y "e/2"] ‹0 < e› y by auto
      obtain z where "z ∈ S" and z: "dist z y < min δ' δ / 2"
        using closure_approachable y
        by (metis ‹0 < δ'› ‹0 < δ› divide_pos_pos min_less_iff_conj zero_less_numeral)
      have "dist (f z) (f y) < e/2"
        apply (rule δ' [OF ‹z ∈ S›])
        using z ‹0 < δ'› by linarith
      moreover have "dist (f z) (f x) < e/2"
        apply (rule δ [OF ‹z ∈ S›])
        using z ‹0 < δ›  dist_commute[of y z] dist_triangle_half_r [of y] dyx by auto
      ultimately show ?thesis
        by (metis dist_commute dist_triangle_half_l less_imp_le)
    qed
    then show "∃d>0. ∀x'∈closure S. dist x' x < d ⟶ dist (f x') (f x) ≤ e"
      by (rule_tac x="δ/2" in exI) (simp add: ‹δ > 0›)
  qed
qed

lemma continuous_on_closure_sequentially:
  fixes f :: "'a::metric_space ⇒ 'b :: metric_space"
  shows
   "continuous_on (closure S) f ⟷
    (∀x a. a ∈ closure S ∧ (∀n. x n ∈ S) ∧ x ⇢ a ⟶ (f ∘ x) ⇢ f a)"
   (is "?lhs = ?rhs")
proof -
  have "continuous_on (closure S) f ⟷
           (∀x ∈ closure S. continuous (at x within S) f)"
    by (force simp: continuous_on_closure continuous_within_eps_delta)
  also have "... = ?rhs"
    by (force simp: continuous_within_sequentially)
  finally show ?thesis .
qed

lemma uniformly_continuous_on_closure:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  assumes ucont: "uniformly_continuous_on S f"
      and cont: "continuous_on (closure S) f"
    shows "uniformly_continuous_on (closure S) f"
unfolding uniformly_continuous_on_def
proof (intro allI impI)
  fix e::real
  assume "0 < e"
  then obtain d::real
    where "d>0"
      and d: "⋀x x'. ⟦x∈S; x'∈S; dist x' x < d⟧ ⟹ dist (f x') (f x) < e/3"
    using ucont [unfolded uniformly_continuous_on_def, rule_format, of "e/3"] by auto
  show "∃d>0. ∀x∈closure S. ∀x'∈closure S. dist x' x < d ⟶ dist (f x') (f x) < e"
  proof (rule exI [where x="d/3"], clarsimp simp: ‹d > 0›)
    fix x y
    assume x: "x ∈ closure S" and y: "y ∈ closure S" and dyx: "dist y x * 3 < d"
    obtain d1::real where "d1 > 0"
           and d1: "⋀w. ⟦w ∈ closure S; dist w x < d1⟧ ⟹ dist (f w) (f x) < e/3"
      using cont [unfolded continuous_on_iff, rule_format, of "x" "e/3"] ‹0 < e› x by auto
     obtain x' where "x' ∈ S" and x': "dist x' x < min d1 (d / 3)"
        using closure_approachable [of x S]
        by (metis ‹0 < d1› ‹0 < d› divide_pos_pos min_less_iff_conj x zero_less_numeral)
    obtain d2::real where "d2 > 0"
           and d2: "∀w ∈ closure S. dist w y < d2 ⟶ dist (f w) (f y) < e/3"
      using cont [unfolded continuous_on_iff, rule_format, of "y" "e/3"] ‹0 < e› y by auto
     obtain y' where "y' ∈ S" and y': "dist y' y < min d2 (d / 3)"
        using closure_approachable [of y S]
        by (metis ‹0 < d2› ‹0 < d› divide_pos_pos min_less_iff_conj y zero_less_numeral)
     have "dist x' x < d/3" using x' by auto
     moreover have "dist x y < d/3"
       by (metis dist_commute dyx less_divide_eq_numeral1(1))
     moreover have "dist y y' < d/3"
       by (metis (no_types) dist_commute min_less_iff_conj y')
     ultimately have "dist x' y' < d/3 + d/3 + d/3"
       by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
     then have "dist x' y' < d" by simp
     then have "dist (f x') (f y') < e/3"
       by (rule d [OF ‹y' ∈ S› ‹x' ∈ S›])
     moreover have "dist (f x') (f x) < e/3" using ‹x' ∈ S› closure_subset x' d1
       by (simp add: closure_def)
     moreover have "dist (f y') (f y) < e/3" using ‹y' ∈ S› closure_subset y' d2
       by (simp add: closure_def)
     ultimately have "dist (f y) (f x) < e/3 + e/3 + e/3"
       by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
    then show "dist (f y) (f x) < e" by simp
  qed
qed

lemma uniformly_continuous_on_extension_at_closure:
  fixes f::"'a::metric_space ⇒ 'b::complete_space"
  assumes uc: "uniformly_continuous_on X f"
  assumes "x ∈ closure X"
  obtains l where "(f ⤏ l) (at x within X)"
proof -
  from assms obtain xs where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
    by (auto simp: closure_sequential)

  from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs]
  obtain l where l: "(λn. f (xs n)) ⇢ l"
    by atomize_elim (simp only: convergent_eq_Cauchy)

  have "(f ⤏ l) (at x within X)"
  proof (safe intro!: Lim_within_LIMSEQ)
    fix xs'
    assume "∀n. xs' n ≠ x ∧ xs' n ∈ X"
      and xs': "xs' ⇢ x"
    then have "xs' n ≠ x" "xs' n ∈ X" for n by auto

    from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF ‹xs' ⇢ x› ‹xs' _ ∈ X›]
    obtain l' where l': "(λn. f (xs' n)) ⇢ l'"
      by atomize_elim (simp only: convergent_eq_Cauchy)

    show "(λn. f (xs' n)) ⇢ l"
    proof (rule tendstoI)
      fix e::real assume "e > 0"
      define e' where "e' ≡ e / 2"
      have "e' > 0" using ‹e > 0› by (simp add: e'_def)

      have "∀F n in sequentially. dist (f (xs n)) l < e'"
        by (simp add: ‹0 < e'› l tendstoD)
      moreover
      from uc[unfolded uniformly_continuous_on_def, rule_format, OF ‹e' > 0›]
      obtain d where d: "d > 0" "⋀x x'. x ∈ X ⟹ x' ∈ X ⟹ dist x x' < d ⟹ dist (f x) (f x') < e'"
        by auto
      have "∀F n in sequentially. dist (xs n) (xs' n) < d"
        by (auto intro!: ‹0 < d› order_tendstoD tendsto_eq_intros xs xs')
      ultimately
      show "∀F n in sequentially. dist (f (xs' n)) l < e"
      proof eventually_elim
        case (elim n)
        have "dist (f (xs' n)) l ≤ dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l"
          by (metis dist_triangle dist_commute)
        also have "dist (f (xs n)) (f (xs' n)) < e'"
          by (auto intro!: d xs ‹xs' _ ∈ _› elim)
        also note ‹dist (f (xs n)) l < e'›
        also have "e' + e' = e" by (simp add: e'_def)
        finally show ?case by simp
      qed
    qed
  qed
  thus ?thesis ..
qed

lemma uniformly_continuous_on_extension_on_closure:
  fixes f::"'a::metric_space ⇒ 'b::complete_space"
  assumes uc: "uniformly_continuous_on X f"
  obtains g where "uniformly_continuous_on (closure X) g" "⋀x. x ∈ X ⟹ f x = g x"
    "⋀Y h x. X ⊆ Y ⟹ Y ⊆ closure X ⟹ continuous_on Y h ⟹ (⋀x. x ∈ X ⟹ f x = h x) ⟹ x ∈ Y ⟹ h x = g x"
proof -
  from uc have cont_f: "continuous_on X f"
    by (simp add: uniformly_continuous_imp_continuous)
  obtain y where y: "(f ⤏ y x) (at x within X)" if "x ∈ closure X" for x
    apply atomize_elim
    apply (rule choice)
    using uniformly_continuous_on_extension_at_closure[OF assms]
    by metis
  let ?g = "λx. if x ∈ X then f x else y x"

  have "uniformly_continuous_on (closure X) ?g"
    unfolding uniformly_continuous_on_def
  proof safe
    fix e::real assume "e > 0"
    define e' where "e' ≡ e / 3"
    have "e' > 0" using ‹e > 0› by (simp add: e'_def)
    from uc[unfolded uniformly_continuous_on_def, rule_format, OF ‹0 < e'›]
    obtain d where "d > 0" and d: "⋀x x'. x ∈ X ⟹ x' ∈ X ⟹ dist x' x < d ⟹ dist (f x') (f x) < e'"
      by auto
    define d' where "d' = d / 3"
    have "d' > 0" using ‹d > 0› by (simp add: d'_def)
    show "∃d>0. ∀x∈closure X. ∀x'∈closure X. dist x' x < d ⟶ dist (?g x') (?g x) < e"
    proof (safe intro!: exI[where x=d'] ‹d' > 0›)
      fix x x' assume x: "x ∈ closure X" and x': "x' ∈ closure X" and dist: "dist x' x < d'"
      then obtain xs xs' where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
        and xs': "xs' ⇢ x'" "⋀n. xs' n ∈ X"
        by (auto simp: closure_sequential)
      have "∀F n in sequentially. dist (xs' n) x' < d'"
        and "∀F n in sequentially. dist (xs n) x < d'"
        by (auto intro!: ‹0 < d'› order_tendstoD tendsto_eq_intros xs xs')
      moreover
      have "(λx. f (xs x)) ⇢ y x" if "x ∈ closure X" "x ∉ X" "xs ⇢ x" "⋀n. xs n ∈ X" for xs x
        using that not_eventuallyD
        by (force intro!: filterlim_compose[OF y[OF ‹x ∈ closure X›]] simp: filterlim_at)
      then have "(λx. f (xs' x)) ⇢ ?g x'" "(λx. f (xs x)) ⇢ ?g x"
        using x x'
        by (auto intro!: continuous_on_tendsto_compose[OF cont_f] simp: xs' xs)
      then have "∀F n in sequentially. dist (f (xs' n)) (?g x') < e'"
        "∀F n in sequentially. dist (f (xs n)) (?g x) < e'"
        by (auto intro!: ‹0 < e'› order_tendstoD tendsto_eq_intros)
      ultimately
      have "∀F n in sequentially. dist (?g x') (?g x) < e"
      proof eventually_elim
        case (elim n)
        have "dist (?g x') (?g x) ≤
          dist (f (xs' n)) (?g x') + dist (f (xs' n)) (f (xs n)) + dist (f (xs n)) (?g x)"
          by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le)
        also
        {
          have "dist (xs' n) (xs n) ≤ dist (xs' n) x' + dist x' x + dist (xs n) x"
            by (metis add.commute add_le_cancel_left  dist_triangle dist_triangle_le)
          also note ‹dist (xs' n) x' < d'›
          also note ‹dist x' x < d'›
          also note ‹dist (xs n) x < d'›
          finally have "dist (xs' n) (xs n) < d" by (simp add: d'_def)
        }
        with ‹xs _ ∈ X› ‹xs' _ ∈ X› have "dist (f (xs' n)) (f (xs n)) < e'"
          by (rule d)
        also note ‹dist (f (xs' n)) (?g x') < e'›
        also note ‹dist (f (xs n)) (?g x) < e'›
        finally show ?case by (simp add: e'_def)
      qed
      then show "dist (?g x') (?g x) < e" by simp
    qed
  qed
  moreover have "f x = ?g x" if "x ∈ X" for x using that by simp
  moreover
  {
    fix Y h x
    assume Y: "x ∈ Y" "X ⊆ Y" "Y ⊆ closure X" and cont_h: "continuous_on Y h"
      and extension: "(⋀x. x ∈ X ⟹ f x = h x)"
    {
      assume "x ∉ X"
      have "x ∈ closure X" using Y by auto
      then obtain xs where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
        by (auto simp: closure_sequential)
      from continuous_on_tendsto_compose[OF cont_h xs(1)] xs(2) Y
      have hx: "(λx. f (xs x)) ⇢ h x"
        by (auto simp: set_mp extension)
      then have "(λx. f (xs x)) ⇢ y x"
        using ‹x ∉ X› not_eventuallyD xs(2)
        by (force intro!: filterlim_compose[OF y[OF ‹x ∈ closure X›]] simp: filterlim_at xs)
      with hx have "h x = y x" by (rule LIMSEQ_unique)
    } then
    have "h x = ?g x"
      using extension by auto
  }
  ultimately show ?thesis ..
qed

lemma bounded_uniformly_continuous_image:
  fixes f :: "'a :: heine_borel ⇒ 'b :: heine_borel"
  assumes "uniformly_continuous_on S f" "bounded S"
  shows "bounded(f ` S)"
  by (metis (no_types, lifting) assms bounded_closure_image compact_closure compact_continuous_image compact_eq_bounded_closed image_cong uniformly_continuous_imp_continuous uniformly_continuous_on_extension_on_closure)

subsection%unimportant ‹Making a continuous function avoid some value in a neighbourhood›

lemma continuous_within_avoid:
  fixes f :: "'a::metric_space ⇒ 'b::t1_space"
  assumes "continuous (at x within s) f"
    and "f x ≠ a"
  shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"
proof -
  obtain U where "open U" and "f x ∈ U" and "a ∉ U"
    using t1_space [OF ‹f x ≠ a›] by fast
  have "(f ⤏ f x) (at x within s)"
    using assms(1) by (simp add: continuous_within)
  then have "eventually (λy. f y ∈ U) (at x within s)"
    using ‹open U› and ‹f x ∈ U›
    unfolding tendsto_def by fast
  then have "eventually (λy. f y ≠ a) (at x within s)"
    using ‹a ∉ U› by (fast elim: eventually_mono)
  then show ?thesis
    using ‹f x ≠ a› by (auto simp: dist_commute eventually_at)
qed

lemma continuous_at_avoid:
  fixes f :: "'a::metric_space ⇒ 'b::t1_space"
  assumes "continuous (at x) f"
    and "f x ≠ a"
  shows "∃e>0. ∀y. dist x y < e ⟶ f y ≠ a"
  using assms continuous_within_avoid[of x UNIV f a] by simp

lemma continuous_on_avoid:
  fixes f :: "'a::metric_space ⇒ 'b::t1_space"
  assumes "continuous_on s f"
    and "x ∈ s"
    and "f x ≠ a"
  shows "∃e>0. ∀y ∈ s. dist x y < e ⟶ f y ≠ a"
  using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],
    OF assms(2)] continuous_within_avoid[of x s f a]
  using assms(3)
  by auto

lemma continuous_on_open_avoid:
  fixes f :: "'a::metric_space ⇒ 'b::t1_space"
  assumes "continuous_on s f"
    and "open s"
    and "x ∈ s"
    and "f x ≠ a"
  shows "∃e>0. ∀y. dist x y < e ⟶ f y ≠ a"
  using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]
  using continuous_at_avoid[of x f a] assms(4)
  by auto

subsection%unimportant‹Quotient maps›

lemma quotient_map_imp_continuous_open:
  assumes T: "f ` S ⊆ T"
      and ope: "⋀U. U ⊆ T
              ⟹ (openin (subtopology euclidean S) (S ∩ f -` U) ⟷
                   openin (subtopology euclidean T) U)"
    shows "continuous_on S f"
proof -
  have [simp]: "S ∩ f -` f ` S = S" by auto
  show ?thesis
    using ope [OF T]
    apply (simp add: continuous_on_open)
    by (meson ope openin_imp_subset openin_trans)
qed

lemma quotient_map_imp_continuous_closed:
  assumes T: "f ` S ⊆ T"
      and ope: "⋀U. U ⊆ T
                  ⟹ (closedin (subtopology euclidean S) (S ∩ f -` U) ⟷
                       closedin (subtopology euclidean T) U)"
    shows "continuous_on S f"
proof -
  have [simp]: "S ∩ f -` f ` S = S" by auto
  show ?thesis
    using ope [OF T]
    apply (simp add: continuous_on_closed)
    by (metis (no_types, lifting) ope closedin_imp_subset closedin_trans)
qed

lemma open_map_imp_quotient_map:
  assumes contf: "continuous_on S f"
      and T: "T ⊆ f ` S"
      and ope: "⋀T. openin (subtopology euclidean S) T
                   ⟹ openin (subtopology euclidean (f ` S)) (f ` T)"
    shows "openin (subtopology euclidean S) (S ∩ f -` T) =
           openin (subtopology euclidean (f ` S)) T"
proof -
  have "T = f ` (S ∩ f -` T)"
    using T by blast
  then show ?thesis
    using "ope" contf continuous_on_open by metis
qed

lemma closed_map_imp_quotient_map:
  assumes contf: "continuous_on S f"
      and T: "T ⊆ f ` S"
      and ope: "⋀T. closedin (subtopology euclidean S) T
              ⟹ closedin (subtopology euclidean (f ` S)) (f ` T)"
    shows "openin (subtopology euclidean S) (S ∩ f -` T) ⟷
           openin (subtopology euclidean (f ` S)) T"
          (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have *: "closedin (subtopology euclidean S) (S - (S ∩ f -` T))"
    using closedin_diff by fastforce
  have [simp]: "(f ` S - f ` (S - (S ∩ f -` T))) = T"
    using T by blast
  show ?rhs
    using ope [OF *, unfolded closedin_def] by auto
next
  assume ?rhs
  with contf show ?lhs
    by (auto simp: continuous_on_open)
qed

lemma continuous_right_inverse_imp_quotient_map:
  assumes contf: "continuous_on S f" and imf: "f ` S ⊆ T"
      and contg: "continuous_on T g" and img: "g ` T ⊆ S"
      and fg [simp]: "⋀y. y ∈ T ⟹ f(g y) = y"
      and U: "U ⊆ T"
    shows "openin (subtopology euclidean S) (S ∩ f -` U) ⟷
           openin (subtopology euclidean T) U"
          (is "?lhs = ?rhs")
proof -
  have f: "⋀Z. openin (subtopology euclidean (f ` S)) Z ⟹
                openin (subtopology euclidean S) (S ∩ f -` Z)"
  and  g: "⋀Z. openin (subtopology euclidean (g ` T)) Z ⟹
                openin (subtopology euclidean T) (T ∩ g -` Z)"
    using contf contg by (auto simp: continuous_on_open)
  show ?thesis
  proof
    have "T ∩ g -` (g ` T ∩ (S ∩ f -` U)) = {x ∈ T. f (g x) ∈ U}"
      using imf img by blast
    also have "... = U"
      using U by auto
    finally have eq: "T ∩ g -` (g ` T ∩ (S ∩ f -` U)) = U" .
    assume ?lhs
    then have *: "openin (subtopology euclidean (g ` T)) (g ` T ∩ (S ∩ f -` U))"
      by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
    show ?rhs
      using g [OF *] eq by auto
  next
    assume rhs: ?rhs
    show ?lhs
      by (metis f fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
  qed
qed

lemma continuous_left_inverse_imp_quotient_map:
  assumes "continuous_on S f"
      and "continuous_on (f ` S) g"
      and  "⋀x. x ∈ S ⟹ g(f x) = x"
      and "U ⊆ f ` S"
    shows "openin (subtopology euclidean S) (S ∩ f -` U) ⟷
           openin (subtopology euclidean (f ` S)) U"
apply (rule continuous_right_inverse_imp_quotient_map)
using assms apply force+
done


text ‹Proving a function is constant by proving that a level set is open›

lemma continuous_levelset_openin_cases:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "connected s ⟹ continuous_on s f ⟹
        openin (subtopology euclidean s) {x ∈ s. f x = a}
        ⟹ (∀x ∈ s. f x ≠ a) ∨ (∀x ∈ s. f x = a)"
  unfolding connected_clopen
  using continuous_closedin_preimage_constant by auto

lemma continuous_levelset_openin:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "connected s ⟹ continuous_on s f ⟹
        openin (subtopology euclidean s) {x ∈ s. f x = a} ⟹
        (∃x ∈ s. f x = a)  ⟹ (∀x ∈ s. f x = a)"
  using continuous_levelset_openin_cases[of s f ]
  by meson

lemma continuous_levelset_open:
  fixes f :: "_ ⇒ 'b::t1_space"
  assumes "connected s"
    and "continuous_on s f"
    and "open {x ∈ s. f x = a}"
    and "∃x ∈ s.  f x = a"
  shows "∀x ∈ s. f x = a"
  using continuous_levelset_openin[OF assms(1,2), of a, unfolded openin_open]
  using assms (3,4)
  by fast

text ‹Some arithmetical combinations (more to prove).›

lemma open_scaling[intro]:
  fixes s :: "'a::real_normed_vector set"
  assumes "c ≠ 0"
    and "open s"
  shows "open((λx. c *R x) ` s)"
proof -
  {
    fix x
    assume "x ∈ s"
    then obtain e where "e>0"
      and e:"∀x'. dist x' x < e ⟶ x' ∈ s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
      by auto
    have "e * ¦c¦ > 0"
      using assms(1)[unfolded zero_less_abs_iff[symmetric]] ‹e>0› by auto
    moreover
    {
      fix y
      assume "dist y (c *R x) < e * ¦c¦"
      then have "norm ((1 / c) *R y - x) < e"
        unfolding dist_norm
        using norm_scaleR[of c "(1 / c) *R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
          assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
      then have "y ∈ ( *R) c ` s"
        using rev_image_eqI[of "(1 / c) *R y" s y "( *R) c"]
        using e[THEN spec[where x="(1 / c) *R y"]]
        using assms(1)
        unfolding dist_norm scaleR_scaleR
        by auto
    }
    ultimately have "∃e>0. ∀x'. dist x' (c *R x) < e ⟶ x' ∈ ( *R) c ` s"
      apply (rule_tac x="e * ¦c¦" in exI, auto)
      done
  }
  then show ?thesis unfolding open_dist by auto
qed

lemma minus_image_eq_vimage:
  fixes A :: "'a::ab_group_add set"
  shows "(λx. - x) ` A = (λx. - x) -` A"
  by (auto intro!: image_eqI [where f="λx. - x"])

lemma open_negations:
  fixes S :: "'a::real_normed_vector set"
  shows "open S ⟹ open ((λx. - x) ` S)"
  using open_scaling [of "- 1" S] by simp

lemma open_translation:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open((λx. a + x) ` S)"
proof -
  {
    fix x
    have "continuous (at x) (λx. x - a)"
      by (intro continuous_diff continuous_ident continuous_const)
  }
  moreover have "{x. x - a ∈ S} = (+) a ` S"
    by force
  ultimately show ?thesis
    by (metis assms continuous_open_vimage vimage_def)
qed

lemma open_neg_translation:
  fixes s :: "'a::real_normed_vector set"
  assumes "open s"
  shows "open((λx. a - x) ` s)"
  using open_translation[OF open_negations[OF assms], of a]
  by (auto simp: image_image)

lemma open_affinity:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"  "c ≠ 0"
  shows "open ((λx. a + c *R x) ` S)"
proof -
  have *: "(λx. a + c *R x) = (λx. a + x) ∘ (λx. c *R x)"
    unfolding o_def ..
  have "(+) a ` ( *R) c ` S = ((+) a ∘ ( *R) c) ` S"
    by auto
  then show ?thesis
    using assms open_translation[of "( *R) c ` S" a]
    unfolding *
    by auto
qed

lemma interior_translation:
  fixes S :: "'a::real_normed_vector set"
  shows "interior ((λx. a + x) ` S) = (λx. a + x) ` (interior S)"
proof (rule set_eqI, rule)
  fix x
  assume "x ∈ interior ((+) a ` S)"
  then obtain e where "e > 0" and e: "ball x e ⊆ (+) a ` S"
    unfolding mem_interior by auto
  then have "ball (x - a) e ⊆ S"
    unfolding subset_eq Ball_def mem_ball dist_norm
    by (auto simp: diff_diff_eq)
  then show "x ∈ (+) a ` interior S"
    unfolding image_iff
    apply (rule_tac x="x - a" in bexI)
    unfolding mem_interior
    using ‹e > 0›
    apply auto
    done
next
  fix x
  assume "x ∈ (+) a ` interior S"
  then obtain y e where "e > 0" and e: "ball y e ⊆ S" and y: "x = a + y"
    unfolding image_iff Bex_def mem_interior by auto
  {
    fix z
    have *: "a + y - z = y + a - z" by auto
    assume "z ∈ ball x e"
    then have "z - a ∈ S"
      using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
      unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
      by auto
    then have "z ∈ (+) a ` S"
      unfolding image_iff by (auto intro!: bexI[where x="z - a"])
  }
  then have "ball x e ⊆ (+) a ` S"
    unfolding subset_eq by auto
  then show "x ∈ interior ((+) a ` S)"
    unfolding mem_interior using ‹e > 0› by auto
qed

subsection ‹Continuity implies uniform continuity on a compact domain›

text‹From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of
J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)›

lemma Heine_Borel_lemma:
  assumes "compact S" and Ssub: "S ⊆ ⋃𝒢" and opn: "⋀G. G ∈ 𝒢 ⟹ open G"
  obtains e where "0 < e" "⋀x. x ∈ S ⟹ ∃G ∈ 𝒢. ball x e ⊆ G"
proof -
  have False if neg: "⋀e. 0 < e ⟹ ∃x ∈ S. ∀G ∈ 𝒢. ¬ ball x e ⊆ G"
  proof -
    have "∃x ∈ S. ∀G ∈ 𝒢. ¬ ball x (1 / Suc n) ⊆ G" for n
      using neg by simp
    then obtain f where "⋀n. f n ∈ S" and fG: "⋀G n. G ∈ 𝒢 ⟹ ¬ ball (f n) (1 / Suc n) ⊆ G"
      by metis
    then obtain l r where "l ∈ S" "strict_mono r" and to_l: "(f ∘ r) ⇢ l"
      using ‹compact S› compact_def that by metis
    then obtain G where "l ∈ G" "G ∈ 𝒢"
      using Ssub by auto
    then obtain e where "0 < e" and e: "⋀z. dist z l < e ⟹ z ∈ G"
      using opn open_dist by blast
    obtain N1 where N1: "⋀n. n ≥ N1 ⟹ dist (f (r n)) l < e/2"
      using to_l apply (simp add: lim_sequentially)
      using ‹0 < e› half_gt_zero that by blast
    obtain N2 where N2: "of_nat N2 > 2/e"
      using reals_Archimedean2 by blast
    obtain x where "x ∈ ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x ∉ G"
      using fG [OF ‹G ∈ 𝒢›, of "r (max N1 N2)"] by blast
    then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
      by simp
    also have "... ≤ 1 / real (Suc (max N1 N2))"
      apply (simp add: divide_simps del: max.bounded_iff)
      using ‹strict_mono r› seq_suble by blast
    also have "... ≤ 1 / real (Suc N2)"
      by (simp add: field_simps)
    also have "... < e/2"
      using N2 ‹0 < e› by (simp add: field_simps)
    finally have "dist (f (r (max N1 N2))) x < e / 2" .
    moreover have "dist (f (r (max N1 N2))) l < e/2"
      using N1 max.cobounded1 by blast
    ultimately have "dist x l < e"
      using dist_triangle_half_r by blast
    then show ?thesis
      using e ‹x ∉ G› by blast
  qed
  then show ?thesis
    by (meson that)
qed

lemma compact_uniformly_equicontinuous:
  assumes "compact S"
      and cont: "⋀x e. ⟦x ∈ S; 0 < e⟧
                        ⟹ ∃d. 0 < d ∧
                                (∀f ∈ ℱ. ∀x' ∈ S. dist x' x < d ⟶ dist (f x') (f x) < e)"
      and "0 < e"
  obtains d where "0 < d"
                  "⋀f x x'. ⟦f ∈ ℱ; x ∈ S; x' ∈ S; dist x' x < d⟧ ⟹ dist (f x') (f x) < e"
proof -
  obtain d where d_pos: "⋀x e. ⟦x ∈ S; 0 < e⟧ ⟹ 0 < d x e"
     and d_dist : "⋀x x' e f. ⟦dist x' x < d x e; x ∈ S; x' ∈ S; 0 < e; f ∈ ℱ⟧ ⟹ dist (f x') (f x) < e"
    using cont by metis
  let ?𝒢 = "((λx. ball x (d x (e / 2))) ` S)"
  have Ssub: "S ⊆ ⋃ ?𝒢"
    by clarsimp (metis d_pos ‹0 < e› dist_self half_gt_zero_iff)
  then obtain k where "0 < k" and k: "⋀x. x ∈ S ⟹ ∃G ∈ ?𝒢. ball x k ⊆ G"
    by (rule Heine_Borel_lemma [OF ‹compact S›]) auto
  moreover have "dist (f v) (f u) < e" if "f ∈ ℱ" "u ∈ S" "v ∈ S" "dist v u < k" for f u v
  proof -
    obtain G where "G ∈ ?𝒢" "u ∈ G" "v ∈ G"
      using k that
      by (metis ‹dist v u < k› ‹u ∈ S› ‹0 < k› centre_in_ball subsetD dist_commute mem_ball)
    then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w ∈ S"
      by auto
    with that d_dist have "dist (f w) (f v) < e/2"
      by (metis ‹0 < e› dist_commute half_gt_zero)
    moreover
    have "dist (f w) (f u) < e/2"
      using that d_dist w by (metis ‹0 < e› dist_commute divide_pos_pos zero_less_numeral)
    ultimately show ?thesis
      using dist_triangle_half_r by blast
  qed
  ultimately show ?thesis using that by blast
qed

corollary compact_uniformly_continuous:
  fixes f :: "'a :: metric_space ⇒ 'b :: metric_space"
  assumes f: "continuous_on S f" and S: "compact S"
  shows "uniformly_continuous_on S f"
  using f
    unfolding continuous_on_iff uniformly_continuous_on_def
    by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])

subsection%unimportant ‹Topological stuff about the set of Reals›

lemma open_real:
  fixes s :: "real set"
  shows "open s ⟷ (∀x ∈ s. ∃e>0. ∀x'. ¦x' - x¦ < e --> x' ∈ s)"
  unfolding open_dist dist_norm by simp

lemma islimpt_approachable_real:
  fixes s :: "real set"
  shows "x islimpt s ⟷ (∀e>0. ∃x'∈ s. x' ≠ x ∧ ¦x' - x¦ < e)"
  unfolding islimpt_approachable dist_norm by simp

lemma closed_real:
  fixes s :: "real set"
  shows "closed s ⟷ (∀x. (∀e>0.  ∃x' ∈ s. x' ≠ x ∧ ¦x' - x¦ < e) ⟶ x ∈ s)"
  unfolding closed_limpt islimpt_approachable dist_norm by simp

lemma continuous_at_real_range:
  fixes f :: "'a::real_normed_vector ⇒ real"
  shows "continuous (at x) f ⟷ (∀e>0. ∃d>0. ∀x'. norm(x' - x) < d --> ¦f x' - f x¦ < e)"
  unfolding continuous_at
  unfolding Lim_at
  unfolding dist_norm
  apply auto
  apply (erule_tac x=e in allE, auto)
  apply (rule_tac x=d in exI, auto)
  apply (erule_tac x=x' in allE, auto)
  apply (erule_tac x=e in allE, auto)
  done

lemma continuous_on_real_range:
  fixes f :: "'a::real_normed_vector ⇒ real"
  shows "continuous_on s f ⟷
    (∀x ∈ s. ∀e>0. ∃d>0. (∀x' ∈ s. norm(x' - x) < d ⟶ ¦f x' - f x¦ < e))"
  unfolding continuous_on_iff dist_norm by simp


subsection%unimportant ‹Cartesian products›

lemma bounded_Times:
  assumes "bounded s" "bounded t"
  shows "bounded (s × t)"
proof -
  obtain x y a b where "∀z∈s. dist x z ≤ a" "∀z∈t. dist y z ≤ b"
    using assms [unfolded bounded_def] by auto
  then have "∀z∈s × t. dist (x, y) z ≤ sqrt (a2 + b2)"
    by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
  then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
qed

lemma mem_Times_iff: "x ∈ A × B ⟷ fst x ∈ A ∧ snd x ∈ B"
  by (induct x) simp

lemma seq_compact_Times: "seq_compact s ⟹ seq_compact t ⟹ seq_compact (s × t)"
  unfolding seq_compact_def
  apply clarify
  apply (drule_tac x="fst ∘ f" in spec)
  apply (drule mp, simp add: mem_Times_iff)
  apply (clarify, rename_tac l1 r1)
  apply (drule_tac x="snd ∘ f ∘ r1" in spec)
  apply (drule mp, simp add: mem_Times_iff)
  apply (clarify, rename_tac l2 r2)
  apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
  apply (rule_tac x="r1 ∘ r2" in exI)
  apply (rule conjI, simp add: strict_mono_def)
  apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
  apply (drule (1) tendsto_Pair) back
  apply (simp add: o_def)
  done

lemma compact_Times:
  assumes "compact s" "compact t"
  shows "compact (s × t)"
proof (rule compactI)
  fix C
  assume C: "∀t∈C. open t" "s × t ⊆ ⋃C"
  have "∀x∈s. ∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ ⋃d)"
  proof
    fix x
    assume "x ∈ s"
    have "∀y∈t. ∃a b c. c ∈ C ∧ open a ∧ open b ∧ x ∈ a ∧ y ∈ b ∧ a × b ⊆ c" (is "∀y∈t. ?P y")
    proof
      fix y
      assume "y ∈ t"
      with ‹x ∈ s› C obtain c where "c ∈ C" "(x, y) ∈ c" "open c" by auto
      then show "?P y" by (auto elim!: open_prod_elim)
    qed
    then obtain a b c where b: "⋀y. y ∈ t ⟹ open (b y)"
      and c: "⋀y. y ∈ t ⟹ c y ∈ C ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y"
      by metis
    then have "∀y∈t. open (b y)" "t ⊆ (⋃y∈t. b y)" by auto
    with compactE_image[OF ‹compact t›] obtain D where D: "D ⊆ t" "finite D" "t ⊆ (⋃y∈D. b y)"
      by metis
    moreover from D c have "(⋂y∈D. a y) × t ⊆ (⋃y∈D. c y)"
      by (fastforce simp: subset_eq)
    ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ ⋃d)"
      using c by (intro exI[of _ "c`D"] exI[of _ "⋂(a`D)"] conjI) (auto intro!: open_INT)
  qed
  then obtain a d where a: "⋀x. x∈s ⟹ open (a x)" "s ⊆ (⋃x∈s. a x)"
    and d: "⋀x. x ∈ s ⟹ d x ⊆ C ∧ finite (d x) ∧ a x × t ⊆ ⋃d x"
    unfolding subset_eq UN_iff by metis
  moreover
  from compactE_image[OF ‹compact s› a]
  obtain e where e: "e ⊆ s" "finite e" and s: "s ⊆ (⋃x∈e. a x)"
    by auto
  moreover
  {
    from s have "s × t ⊆ (⋃x∈e. a x × t)"
      by auto
    also have "… ⊆ (⋃x∈e. ⋃d x)"
      using d ‹e ⊆ s› by (intro UN_mono) auto
    finally have "s × t ⊆ (⋃x∈e. ⋃d x)" .
  }
  ultimately show "∃C'⊆C. finite C' ∧ s × t ⊆ ⋃C'"
    by (intro exI[of _ "(⋃x∈e. d x)"]) (auto simp: subset_eq)
qed

text‹Hence some useful properties follow quite easily.›

lemma compact_scaling:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((λx. c *R x) ` s)"
proof -
  let ?f = "λx. scaleR c x"
  have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
  show ?thesis
    using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
    using linear_continuous_at[OF *] assms
    by auto
qed

lemma compact_negations:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((λx. - x) ` s)"
  using compact_scaling [OF assms, of "- 1"] by auto

lemma compact_sums:
  fixes s t :: "'a::real_normed_vector set"
  assumes "compact s"
    and "compact t"
  shows "compact {x + y | x y. x ∈ s ∧ y ∈ t}"
proof -
  have *: "{x + y | x y. x ∈ s ∧ y ∈ t} = (λz. fst z + snd z) ` (s × t)"
    apply auto
    unfolding image_iff
    apply (rule_tac x="(xa, y)" in bexI)
    apply auto
    done
  have "continuous_on (s × t) (λz. fst z + snd z)"
    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
  then show ?thesis
    unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed

lemma compact_differences:
  fixes s t :: "'a::real_normed_vector set"
  assumes "compact s"
    and "compact t"
  shows "compact {x - y | x y. x ∈ s ∧ y ∈ t}"
proof-
  have "{x - y | x y. x∈s ∧ y ∈ t} =  {x + y | x y. x ∈ s ∧ y ∈ (uminus ` t)}"
    apply auto
    apply (rule_tac x= xa in exI, auto)
    done
  then show ?thesis
    using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed

lemma compact_translation:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((λx. a + x) ` s)"
proof -
  have "{x + y |x y. x ∈ s ∧ y ∈ {a}} = (λx. a + x) ` s"
    by auto
  then show ?thesis
    using compact_sums[OF assms compact_sing[of a]] by auto
qed

lemma compact_affinity:
  fixes s :: "'a::real_normed_vector set"
  assumes "compact s"
  shows "compact ((λx. a + c *R x) ` s)"
proof -
  have "(+) a ` ( *R) c ` s = (λx. a + c *R x) ` s"
    by auto
  then show ?thesis
    using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed

text ‹Hence we get the following.›

lemma compact_sup_maxdistance:
  fixes s :: "'a::metric_space set"
  assumes "compact s"
    and "s ≠ {}"
  shows "∃x∈s. ∃y∈s. ∀u∈s. ∀v∈s. dist u v ≤ dist x y"
proof -
  have "compact (s × s)"
    using ‹compact s› by (intro compact_Times)
  moreover have "s × s ≠ {}"
    using ‹s ≠ {}› by auto
  moreover have "continuous_on (s × s) (λx. dist (fst x) (snd x))"
    by (intro continuous_at_imp_continuous_on ballI continuous_intros)
  ultimately show ?thesis
    using continuous_attains_sup[of "s × s" "λx. dist (fst x) (snd x)"] by auto
qed


subsection ‹The diameter of a set›

definition%important diameter :: "'a::metric_space set ⇒ real" where
  "diameter S = (if S = {} then 0 else SUP (x,y):S×S. dist x y)"

lemma diameter_empty [simp]: "diameter{} = 0"
  by (auto simp: diameter_def)

lemma diameter_singleton [simp]: "diameter{x} = 0"
  by (auto simp: diameter_def)

lemma diameter_le:
  assumes "S ≠ {} ∨ 0 ≤ d"
      and no: "⋀x y. ⟦x ∈ S; y ∈ S⟧ ⟹ norm(x - y) ≤ d"
    shows "diameter S ≤ d"
using assms
  by (auto simp: dist_norm diameter_def intro: cSUP_least)

lemma diameter_bounded_bound:
  fixes s :: "'a :: metric_space set"
  assumes s: "bounded s" "x ∈ s" "y ∈ s"
  shows "dist x y ≤ diameter s"
proof -
  from s obtain z d where z: "⋀x. x ∈ s ⟹ dist z x ≤ d"
    unfolding bounded_def by auto
  have "bdd_above (case_prod dist ` (s×s))"
  proof (intro bdd_aboveI, safe)
    fix a b
    assume "a ∈ s" "b ∈ s"
    with z[of a] z[of b] dist_triangle[of a b z]
    show "dist a b ≤ 2 * d"
      by (simp add: dist_commute)
  qed
  moreover have "(x,y) ∈ s×s" using s by auto
  ultimately have "dist x y ≤ (SUP (x,y):s×s. dist x y)"
    by (rule cSUP_upper2) simp
  with ‹x ∈ s› show ?thesis
    by (auto simp: diameter_def)
qed

lemma diameter_lower_bounded:
  fixes s :: "'a :: metric_space set"
  assumes s: "bounded s"
    and d: "0 < d" "d < diameter s"
  shows "∃x∈s. ∃y∈s. d < dist x y"
proof (rule ccontr)
  assume contr: "¬ ?thesis"
  moreover have "s ≠ {}"
    using d by (auto simp: diameter_def)
  ultimately have "diameter s ≤ d"
    by (auto simp: not_less diameter_def intro!: cSUP_least)
  with ‹d < diameter s› show False by auto
qed

lemma diameter_bounded:
  assumes "bounded s"
  shows "∀x∈s. ∀y∈s. dist x y ≤ diameter s"
    and "∀d>0. d < diameter s ⟶ (∃x∈s. ∃y∈s. dist x y > d)"
  using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
  by auto

lemma bounded_two_points:
  "bounded S ⟷ (∃e. ∀x∈S. ∀y∈S. dist x y ≤ e)"
  apply (rule iffI)
  subgoal using diameter_bounded(1) by auto
  subgoal using bounded_any_center[of S] by meson
  done

lemma diameter_compact_attained:
  assumes "compact s"
    and "s ≠ {}"
  shows "∃x∈s. ∃y∈s. dist x y = diameter s"
proof -
  have b: "bounded s" using assms(1)
    by (rule compact_imp_bounded)
  then obtain x y where xys: "x∈s" "y∈s"
    and xy: "∀u∈s. ∀v∈s. dist u v ≤ dist x y"
    using compact_sup_maxdistance[OF assms] by auto
  then have "diameter s ≤ dist x y"
    unfolding diameter_def
    apply clarsimp
    apply (rule cSUP_least, fast+)
    done
  then show ?thesis
    by (metis b diameter_bounded_bound order_antisym xys)
qed

lemma diameter_ge_0:
  assumes "bounded S"  shows "0 ≤ diameter S"
  by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)

lemma diameter_subset:
  assumes "S ⊆ T" "bounded T"
  shows "diameter S ≤ diameter T"
proof (cases "S = {} ∨ T = {}")
  case True
  with assms show ?thesis
    by (force simp: diameter_ge_0)
next
  case False
  then have "bdd_above ((λx. case x of (x, xa) ⇒ dist x xa) ` (T × T))"
    using ‹bounded T› diameter_bounded_bound by (force simp: bdd_above_def)
  with False ‹S ⊆ T› show ?thesis
    apply (simp add: diameter_def)
    apply (rule cSUP_subset_mono, auto)
    done
qed

lemma diameter_closure:
  assumes "bounded S"
  shows "diameter(closure S) = diameter S"
proof (rule order_antisym)
  have "False" if "diameter S < diameter (closure S)"
  proof -
    define d where "d = diameter(closure S) - diameter(S)"
    have "d > 0"
      using that by (simp add: d_def)
    then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
      by simp
    have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
      by (simp add: d_def divide_simps)
     have bocl: "bounded (closure S)"
      using assms by blast
    moreover have "0 ≤ diameter S"
      using assms diameter_ge_0 by blast
    ultimately obtain x y where "x ∈ closure S" "y ∈ closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
      using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] ‹d > 0› d_def by auto
    then obtain x' y' where x'y': "x' ∈ S" "dist x' x < d/4" "y' ∈ S" "dist y' y < d/4"
      using closure_approachable
      by (metis ‹0 < d› zero_less_divide_iff zero_less_numeral)
    then have "dist x' y' ≤ diameter S"
      using assms diameter_bounded_bound by blast
    with x'y' have "dist x y ≤ d / 4 + diameter S + d / 4"
      by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)
    then show ?thesis
      using xy d_def by linarith
  qed
  then show "diameter (closure S) ≤ diameter S"
    by fastforce
  next
    show "diameter S ≤ diameter (closure S)"
      by (simp add: assms bounded_closure closure_subset diameter_subset)
qed

lemma diameter_cball [simp]:
  fixes a :: "'a::euclidean_space"
  shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
proof -
  have "diameter(cball a r) = 2*r" if "r ≥ 0"
  proof (rule order_antisym)
    show "diameter (cball a r) ≤ 2*r"
    proof (rule diameter_le)
      fix x y assume "x ∈ cball a r" "y ∈ cball a r"
      then have "norm (x - a) ≤ r" "norm (a - y) ≤ r"
        by (auto simp: dist_norm norm_minus_commute)
      then have "norm (x - y) ≤ r+r"
        using norm_diff_triangle_le by blast
      then show "norm (x - y) ≤ 2*r" by simp
    qed (simp add: that)
    have "2*r = dist (a + r *R (SOME i. i ∈ Basis)) (a - r *R (SOME i. i ∈ Basis))"
      apply (simp add: dist_norm)
      by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
    also have "... ≤ diameter (cball a r)"
      apply (rule diameter_bounded_bound)
      using that by (auto simp: dist_norm)
    finally show "2*r ≤ diameter (cball a r)" .
  qed
  then show ?thesis by simp
qed

lemma diameter_ball [simp]:
  fixes a :: "'a::euclidean_space"
  shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
proof -
  have "diameter(ball a r) = 2*r" if "r > 0"
    by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
  then show ?thesis
    by (simp add: diameter_def)
qed

lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
proof -
  have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
  then show ?thesis
    by simp
qed

lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
proof -
  have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
    by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
  then show ?thesis
    by simp
qed

proposition Lebesgue_number_lemma:
  assumes "compact S" "𝒞 ≠ {}" "S ⊆ ⋃𝒞" and ope: "⋀B. B ∈ 𝒞 ⟹ open B"
  obtains δ where "0 < δ" "⋀T. ⟦T ⊆ S; diameter T < δ⟧ ⟹ ∃B ∈ 𝒞. T ⊆ B"
proof (cases "S = {}")
  case True
  then show ?thesis
    by (metis ‹𝒞 ≠ {}› zero_less_one empty_subsetI equals0I subset_trans that)
next
  case False
  { fix x assume "x ∈ S"
    then obtain C where C: "x ∈ C" "C ∈ 𝒞"
      using ‹S ⊆ ⋃𝒞› by blast
    then obtain r where r: "r>0" "ball x (2*r) ⊆ C"
      by (metis mult.commute mult_2_right not_le ope openE field_sum_of_halves zero_le_numeral zero_less_mult_iff)
    then have "∃r C. r > 0 ∧ ball x (2*r) ⊆ C ∧ C ∈ 𝒞"
      using C by blast
  }
  then obtain r where r: "⋀x. x ∈ S ⟹ r x > 0 ∧ (∃C ∈ 𝒞. ball x (2*r x) ⊆ C)"
    by metis
  then have "S ⊆ (⋃x ∈ S. ball x (r x))"
    by auto
  then obtain 𝒯 where "finite 𝒯" "S ⊆ ⋃𝒯" and 𝒯: "𝒯 ⊆ (λx. ball x (r x)) ` S"
    by (rule compactE [OF ‹compact S›]) auto
  then obtain S0 where "S0 ⊆ S" "finite S0" and S0: "𝒯 = (λx. ball x (r x)) ` S0"
    by (meson finite_subset_image)
  then have "S0 ≠ {}"
    using False ‹S ⊆ ⋃𝒯› by auto
  define δ where "δ = Inf (r ` S0)"
  have "δ > 0"
    using ‹finite S0› ‹S0 ⊆ S› ‹S0 ≠ {}› r by (auto simp: δ_def finite_less_Inf_iff)
  show ?thesis
  proof
    show "0 < δ"
      by (simp add: ‹0 < δ›)
    show "∃B ∈ 𝒞. T ⊆ B" if "T ⊆ S" and dia: "diameter T < δ" for T
    proof (cases "T = {}")
      case True
      then show ?thesis
        using ‹𝒞 ≠ {}› by blast
    next
      case False
      then obtain y where "y ∈ T" by blast
      then have "y ∈ S"
        using ‹T ⊆ S› by auto
      then obtain x where "x ∈ S0" and x: "y ∈ ball x (r x)"
        using ‹S ⊆ ⋃𝒯› S0 that by blast
      have "ball y δ ⊆ ball y (r x)"
        by (metis δ_def ‹S0 ≠ {}› ‹finite S0› ‹x ∈ S0› empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
      also have "... ⊆ ball x (2*r x)"
        by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)
      finally obtain C where "C ∈ 𝒞" "ball y δ ⊆ C"
        by (meson r ‹S0 ⊆ S› ‹x ∈ S0› dual_order.trans subsetCE)
      have "bounded T"
        using ‹compact S› bounded_subset compact_imp_bounded ‹T ⊆ S› by blast
      then have "T ⊆ ball y δ"
        using ‹y ∈ T› dia diameter_bounded_bound by fastforce
      then show ?thesis
        apply (rule_tac x=C in bexI)
        using ‹ball y δ ⊆ C› ‹C ∈ 𝒞› by auto
    qed
  qed
qed

lemma diameter_cbox:
  fixes a b::"'a::euclidean_space"
  shows "(∀i ∈ Basis. a ∙ i ≤ b ∙ i) ⟹ diameter (cbox a b) = dist a b"
  by (force simp: diameter_def intro!: cSup_eq_maximum L2_set_mono
     simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)

subsection ‹Separation between points and sets›

proposition separate_point_closed:
  fixes s :: "'a::heine_borel set"
  assumes "closed s" and "a ∉ s"
  shows "∃d>0. ∀x∈s. d ≤ dist a x"
proof (cases "s = {}")
  case True
  then show ?thesis by(auto intro!: exI[where x=1])
next
  case False
  from assms obtain x where "x∈s" "∀y∈s. dist a x ≤ dist a y"
    using ‹s ≠ {}› by (blast intro: distance_attains_inf [of s a])
  with ‹x∈s› show ?thesis using dist_pos_lt[of a x] and‹a ∉ s›
    by blast
qed

proposition separate_compact_closed:
  fixes s t :: "'a::heine_borel set"
  assumes "compact s"
    and t: "closed t" "s ∩ t = {}"
  shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof cases
  assume "s ≠ {} ∧ t ≠ {}"
  then have "s ≠ {}" "t ≠ {}" by auto
  let ?inf = "λx. infdist x t"
  have "continuous_on s ?inf"
    by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)
  then obtain x where x: "x ∈ s" "∀y∈s. ?inf x ≤ ?inf y"
    using continuous_attains_inf[OF ‹compact s› ‹s ≠ {}›] by auto
  then have "0 < ?inf x"
    using t ‹t ≠ {}› in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
  moreover have "∀x'∈s. ∀y∈t. ?inf x ≤ dist x' y"
    using x by (auto intro: order_trans infdist_le)
  ultimately show ?thesis by auto
qed (auto intro!: exI[of _ 1])

proposition separate_closed_compact:
  fixes s t :: "'a::heine_borel set"
  assumes "closed s"
    and "compact t"
    and "s ∩ t = {}"
  shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof -
  have *: "t ∩ s = {}"
    using assms(3) by auto
  show ?thesis
    using separate_compact_closed[OF assms(2,1) *] by (force simp: dist_commute)
qed

proposition compact_in_open_separated:
  fixes A::"'a::heine_borel set"
  assumes "A ≠ {}"
  assumes "compact A"
  assumes "open B"
  assumes "A ⊆ B"
  obtains e where "e > 0" "{x. infdist x A ≤ e} ⊆ B"
proof atomize_elim
  have "closed (- B)" "compact A" "- B ∩ A = {}"
    using assms by (auto simp: open_Diff compact_eq_bounded_closed)
  from separate_closed_compact[OF this]
  obtain d'::real where d': "d'>0" "⋀x y. x ∉ B ⟹ y ∈ A ⟹ d' ≤ dist x y"
    by auto
  define d where "d = d' / 2"
  hence "d>0" "d < d'" using d' by auto
  with d' have d: "⋀x y. x ∉ B ⟹ y ∈ A ⟹ d < dist x y"
    by force
  show "∃e>0. {x. infdist x A ≤ e} ⊆ B"
  proof (rule ccontr)
    assume "∄e. 0 < e ∧ {x. infdist x A ≤ e} ⊆ B"
    with ‹d > 0› obtain x where x: "infdist x A ≤ d" "x ∉ B"
      by auto
    from assms have "closed A" "A ≠ {}" by (auto simp: compact_eq_bounded_closed)
    from infdist_attains_inf[OF this]
    obtain y where y: "y ∈ A" "infdist x A = dist x y"
      by auto
    have "dist x y ≤ d" using x y by simp
    also have "… < dist x y" using y d x by auto
    finally show False by simp
  qed
qed


subsection%unimportant ‹Compact sets and the closure operation›

lemma closed_scaling:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "closed ((λx. c *R x) ` S)"
proof (cases "c = 0")
  case True then show ?thesis
    by (auto simp: image_constant_conv)
next
  case False
  from assms have "closed ((λx. inverse c *R x) -` S)"
    by (simp add: continuous_closed_vimage)
  also have "(λx. inverse c *R x) -` S = (λx. c *R x) ` S"
    using ‹c ≠ 0› by (auto elim: image_eqI [rotated])
  finally show ?thesis .
qed

lemma closed_negations:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "closed ((λx. -x) ` S)"
  using closed_scaling[OF assms, of "- 1"] by simp

lemma compact_closed_sums:
  fixes S :: "'a::real_normed_vector set"
  assumes "compact S" and "closed T"
  shows "closed (⋃x∈ S. ⋃y ∈ T. {x + y})"
proof -
  let ?S = "{x + y |x y. x ∈ S ∧ y ∈ T}"
  {
    fix x l
    assume as: "∀n. x n ∈ ?S"  "(x ⤏ l) sequentially"
    from as(1) obtain f where f: "∀n. x n = fst (f n) + snd (f n)"  "∀n. fst (f n) ∈ S"  "∀n. snd (f n) ∈ T"
      using choice[of "λn y. x n = (fst y) + (snd y) ∧ fst y ∈ S ∧ snd y ∈ T"] by auto
    obtain l' r where "l'∈S" and r: "strict_mono r" and lr: "(((λn. fst (f n)) ∘ r) ⤏ l') sequentially"
      using assms(1)[unfolded compact_def, THEN spec[where x="λ n. fst (f n)"]] using f(2) by auto
    have "((λn. snd (f (r n))) ⤏ l - l') sequentially"
      using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
      unfolding o_def
      by auto
    then have "l - l' ∈ T"
      using assms(2)[unfolded closed_sequential_limits,
        THEN spec[where x="λ n. snd (f (r n))"],
        THEN spec[where x="l - l'"]]
      using f(3)
      by auto
    then have "l ∈ ?S"
      using ‹l' ∈ S›
      apply auto
      apply (rule_tac x=l' in exI)
      apply (rule_tac x="l - l'" in exI, auto)
      done
  }
  moreover have "?S = (⋃x∈ S. ⋃y ∈ T. {x + y})"
    by force
  ultimately show ?thesis
    unfolding closed_sequential_limits
    by (metis (no_types, lifting))
qed

lemma closed_compact_sums:
  fixes S T :: "'a::real_normed_vector set"
  assumes "closed S" "compact T"
  shows "closed (⋃x∈ S. ⋃y ∈ T. {x + y})"
proof -
  have "(⋃x∈ T. ⋃y ∈ S. {x + y}) = (⋃x∈ S. ⋃y ∈ T. {x + y})"
    by auto
  then show ?thesis
    using compact_closed_sums[OF assms(2,1)] by simp
qed

lemma compact_closed_differences:
  fixes S T :: "'a::real_normed_vector set"
  assumes "compact S" "closed T"
  shows "closed (⋃x∈ S. ⋃y ∈ T. {x - y})"
proof -
  have "(⋃x∈ S. ⋃y ∈ uminus ` T. {x + y}) = (⋃x∈ S. ⋃y ∈ T. {x - y})"
    by force
  then show ?thesis
    using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
qed

lemma closed_compact_differences:
  fixes S T :: "'a::real_normed_vector set"
  assumes "closed S" "compact T"
  shows "closed (⋃x∈ S. ⋃y ∈ T. {x - y})"
proof -
  have "(⋃x∈ S. ⋃y ∈ uminus ` T. {x + y}) = {x - y |x y. x ∈ S ∧ y ∈ T}"
    by auto
 then show ?thesis
  using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed

lemma closed_translation:
  fixes a :: "'a::real_normed_vector"
  assumes "closed S"
  shows "closed ((λx. a + x) ` S)"
proof -
  have "(⋃x∈ {a}. ⋃y ∈ S. {x + y}) = ((+) a ` S)" by auto
  then show ?thesis
    using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed

lemma closure_translation:
  fixes a :: "'a::real_normed_vector"
  shows "closure ((λx. a + x) ` s) = (λx. a + x) ` (closure s)"
proof -
  have *: "(+) a ` (- s) = - (+) a ` s"
    apply auto
    unfolding image_iff
    apply (rule_tac x="x - a" in bexI, auto)
    done
  show ?thesis
    unfolding closure_interior translation_Compl
    using interior_translation[of a "- s"]
    unfolding *
    by auto
qed

lemma frontier_translation:
  fixes a :: "'a::real_normed_vector"
  shows "frontier((λx. a + x) ` s) = (λx. a + x) ` (frontier s)"
  unfolding frontier_def translation_diff interior_translation closure_translation
  by auto

lemma sphere_translation:
  fixes a :: "'n::euclidean_space"
  shows "sphere (a+c) r = (+) a ` sphere c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done

lemma cball_translation:
  fixes a :: "'n::euclidean_space"
  shows "cball (a+c) r = (+) a ` cball c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done

lemma ball_translation:
  fixes a :: "'n::euclidean_space"
  shows "ball (a+c) r = (+) a ` ball c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done


subsection%unimportant ‹Closure of halfspaces and hyperplanes›

lemma continuous_on_closed_Collect_le:
  fixes f g :: "'a::t2_space ⇒ real"
  assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
  shows "closed {x ∈ s. f x ≤ g x}"
proof -
  have "closed ((λx. g x - f x) -` {0..} ∩ s)"
    using closed_real_atLeast continuous_on_diff [OF g f]
    by (simp add: continuous_on_closed_vimage [OF s])
  also have "((λx. g x - f x) -` {0..} ∩ s) = {x∈s. f x ≤ g x}"
    by auto
  finally show ?thesis .
qed

lemma continuous_at_inner: "continuous (at x) (inner a)"
  unfolding continuous_at by (intro tendsto_intros)

lemma closed_halfspace_le: "closed {x. inner a x ≤ b}"
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_halfspace_ge: "closed {x. inner a x ≥ b}"
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_hyperplane: "closed {x. inner a x = b}"
  by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x∙i ≤ a}"
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x∙i ≥ a}"
  by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_interval_left:
  fixes b :: "'a::euclidean_space"
  shows "closed {x::'a. ∀i∈Basis. x∙i ≤ b∙i}"
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma closed_interval_right:
  fixes a :: "'a::euclidean_space"
  shows "closed {x::'a. ∀i∈Basis. a∙i ≤ x∙i}"
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)

lemma continuous_le_on_closure:
  fixes a::real
  assumes f: "continuous_on (closure s) f"
      and x: "x ∈ closure(s)"
      and xlo: "⋀x. x ∈ s ==> f(x) ≤ a"
    shows "f(x) ≤ a"
    using image_closure_subset [OF f]
  using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
  by force

lemma continuous_ge_on_closure:
  fixes a::real
  assumes f: "continuous_on (closure s) f"
      and x: "x ∈ closure(s)"
      and xlo: "⋀x. x ∈ s ==> f(x) ≥ a"
    shows "f(x) ≥ a"
  using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms
  by force

lemma Lim_component_le:
  fixes f :: "'a ⇒ 'b::euclidean_space"
  assumes "(f ⤏ l) net"
    and "¬ (trivial_limit net)"
    and "eventually (λx. f(x)∙i ≤ b) net"
  shows "l∙i ≤ b"
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])

lemma Lim_component_ge:
  fixes f :: "'a ⇒ 'b::euclidean_space"
  assumes "(f ⤏ l) net"
    and "¬ (trivial_limit net)"
    and "eventually (λx. b ≤ (f x)∙i) net"
  shows "b ≤ l∙i"
  by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])

lemma Lim_component_eq:
  fixes f :: "'a ⇒ 'b::euclidean_space"
  assumes net: "(f ⤏ l) net" "¬ trivial_limit net"
    and ev:"eventually (λx. f(x)∙i = b) net"
  shows "l∙i = b"
  using ev[unfolded order_eq_iff eventually_conj_iff]
  using Lim_component_ge[OF net, of b i]
  using Lim_component_le[OF net, of i b]
  by auto

text ‹Limits relative to a union.›

lemma eventually_within_Un:
  "eventually P (at x within (s ∪ t)) ⟷
    eventually P (at x within s) ∧ eventually P (at x within t)"
  unfolding eventually_at_filter
  by (auto elim!: eventually_rev_mp)

lemma Lim_within_union:
 "(f ⤏ l) (at x within (s ∪ t)) ⟷
  (f ⤏ l) (at x within s) ∧ (f ⤏ l) (at x within t)"
  unfolding tendsto_def
  by (auto simp: eventually_within_Un)

lemma Lim_topological:
  "(f ⤏ l) net ⟷
    trivial_limit net ∨ (∀S. open S ⟶ l ∈ S ⟶ eventually (λx. f x ∈ S) net)"
  unfolding tendsto_def trivial_limit_eq by auto

text ‹Continuity relative to a union.›

lemma continuous_on_Un_local:
    "⟦closedin (subtopology euclidean (s ∪ t)) s; closedin (subtopology euclidean (s ∪ t)) t;
      continuous_on s f; continuous_on t f⟧
     ⟹ continuous_on (s ∪ t) f"
  unfolding continuous_on closedin_limpt
  by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)

lemma continuous_on_cases_local:
     "⟦closedin (subtopology euclidean (s ∪ t)) s; closedin (subtopology euclidean (s ∪ t)) t;
       continuous_on s f; continuous_on t g;
       ⋀x. ⟦x ∈ s ∧ ~P x ∨ x ∈ t ∧ P x⟧ ⟹ f x = g x⟧
      ⟹ continuous_on (s ∪ t) (λx. if P x then f x else g x)"
  by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)

lemma continuous_on_cases_le:
  fixes h :: "'a :: topological_space ⇒ real"
  assumes "continuous_on {t ∈ s. h t ≤ a} f"
      and "continuous_on {t ∈ s. a ≤ h t} g"
      and h: "continuous_on s h"
      and "⋀t. ⟦t ∈ s; h t = a⟧ ⟹ f t = g t"
    shows "continuous_on s (λt. if h t ≤ a then f(t) else g(t))"
proof -
  have s: "s = (s ∩ h -` atMost a) ∪ (s ∩ h -` atLeast a)"
    by force
  have 1: "closedin (subtopology euclidean s) (s ∩ h -` atMost a)"
    by (rule continuous_closedin_preimage [OF h closed_atMost])
  have 2: "closedin (subtopology euclidean s) (s ∩ h -` atLeast a)"
    by (rule continuous_closedin_preimage [OF h closed_atLeast])
  have eq: "s ∩ h -` {..a} = {t ∈ s. h t ≤ a}" "s ∩ h -` {a..} = {t ∈ s. a ≤ h t}"
    by auto
  show ?thesis
    apply (rule continuous_on_subset [of s, OF _ order_refl])
    apply (subst s)
    apply (rule continuous_on_cases_local)
    using 1 2 s assms apply (auto simp: eq)
    done
qed

lemma continuous_on_cases_1:
  fixes s :: "real set"
  assumes "continuous_on {t ∈ s. t ≤ a} f"
      and "continuous_on {t ∈ s. a ≤ t} g"
      and "a ∈ s ⟹ f a = g a"
    shows "continuous_on s (λt. if t ≤ a then f(t) else g(t))"
using assms
by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])

subsubsection‹Some more convenient intermediate-value theorem formulations›

lemma connected_ivt_hyperplane:
  assumes "connected S" and xy: "x ∈ S" "y ∈ S" and b: "inner a x ≤ b" "b ≤ inner a y"
  shows "∃z ∈ S. inner a z = b"
proof (rule ccontr)
  assume as:"¬ (∃z∈S. inner a z = b)"
  let ?A = "{x. inner a x < b}"
  let ?B = "{x. inner a x > b}"
  have "open ?A" "open ?B"
    using open_halfspace_lt and open_halfspace_gt by auto
  moreover have "?A ∩ ?B = {}" by auto
  moreover have "S ⊆ ?A ∪ ?B" using as by auto
  ultimately show False
    using ‹connected S›[unfolded connected_def not_ex,
      THEN spec[where x="?A"], THEN spec[where x="?B"]]
    using xy b by auto
qed

lemma connected_ivt_component:
  fixes x::"'a::euclidean_space"
  shows "connected S ⟹ x ∈ S ⟹ y ∈ S ⟹ x∙k ≤ a ⟹ a ≤ y∙k ⟹ (∃z∈S.  z∙k = a)"
  using connected_ivt_hyperplane[of S x y "k::'a" a]
  by (auto simp: inner_commute)

lemma image_affinity_cbox: fixes m::real
  fixes a b c :: "'a::euclidean_space"
  shows "(λx. m *R x + c) ` cbox a b =
    (if cbox a b = {} then {}
     else (if 0 ≤ m then cbox (m *R a + c) (m *R b + c)
     else cbox (m *R b + c) (m *R a + c)))"
proof (cases "m = 0")
  case True
  {
    fix x
    assume "∀i∈Basis. x ∙ i ≤ c ∙ i" "∀i∈Basis. c ∙ i ≤ x ∙ i"
    then have "x = c"
      by (simp add: dual_order.antisym euclidean_eqI)
  }
  moreover have "c ∈ cbox (m *R a + c) (m *R b + c)"
    unfolding True by (auto simp: cbox_sing)
  ultimately show ?thesis using True by (auto simp: cbox_def)
next
  case False
  {
    fix y
    assume "∀i∈Basis. a ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ b ∙ i" "m > 0"
    then have "∀i∈Basis. (m *R a + c) ∙ i ≤ (m *R y + c) ∙ i" and "∀i∈Basis. (m *R y + c) ∙ i ≤ (m *R b + c) ∙ i"
      by (auto simp: inner_distrib)
  }
  moreover
  {
    fix y
    assume "∀i∈Basis. a ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ b ∙ i" "m < 0"
    then have "∀i∈Basis. (m *R b + c) ∙ i ≤ (m *R y + c) ∙ i" and "∀i∈Basis. (m *R y + c) ∙ i ≤ (m *R a + c) ∙ i"
      by (auto simp: mult_left_mono_neg inner_distrib)
  }
  moreover
  {
    fix y
    assume "m > 0" and "∀i∈Basis. (m *R a + c) ∙ i ≤ y ∙ i" and "∀i∈Basis. y ∙ i ≤ (m *R b + c) ∙ i"
    then have "y ∈ (λx. m *R x + c) ` cbox a b"
      unfolding image_iff Bex_def mem_box
      apply (intro exI[where x="(1 / m) *R (y - c)"])
      apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
      done
  }
  moreover
  {
    fix y
    assume "∀i∈Basis. (m *R b + c) ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ (m *R a + c) ∙ i" "m < 0"
    then have "y ∈ (λx. m *R x + c) ` cbox a b"
      unfolding image_iff Bex_def mem_box
      apply (intro exI[where x="(1 / m) *R (y - c)"])
      apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
      done
  }
  ultimately show ?thesis using False by (auto simp: cbox_def)
qed

lemma image_smult_cbox:"(λx. m *R (x::_::euclidean_space)) ` cbox a b =
  (if cbox a b = {} then {} else if 0 ≤ m then cbox (m *R a) (m *R b) else cbox (m *R b) (m *R a))"
  using image_affinity_cbox[of m 0 a b] by auto

lemma islimpt_greaterThanLessThan1:
  fixes a b::"'a::{linorder_topology, dense_order}"
  assumes "a < b"
  shows  "a islimpt {a<..<b}"
proof (rule islimptI)
  fix T
  assume "open T" "a ∈ T"
  from open_right[OF this ‹a < b›]
  obtain c where c: "a < c" "{a..<c} ⊆ T" by auto
  with assms dense[of a "min c b"]
  show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ a"
    by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
      not_le order.strict_implies_order subset_eq)
qed

lemma islimpt_greaterThanLessThan2:
  fixes a b::"'a::{linorder_topology, dense_order}"
  assumes "a < b"
  shows  "b islimpt {a<..<b}"
proof (rule islimptI)
  fix T
  assume "open T" "b ∈ T"
  from open_left[OF this ‹a < b›]
  obtain c where c: "c < b" "{c<..b} ⊆ T" by auto
  with assms dense[of "max a c" b]
  show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ b"
    by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
      not_le order.strict_implies_order subset_eq)
qed

lemma closure_greaterThanLessThan[simp]:
  fixes a b::"'a::{linorder_topology, dense_order}"
  shows "a < b ⟹ closure {a <..< b} = {a .. b}" (is "_ ⟹ ?l = ?r")
proof
  have "?l ⊆ closure ?r"
    by (rule closure_mono) auto
  thus "closure {a<..<b} ⊆ {a..b}" by simp
qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
  islimpt_greaterThanLessThan2)

lemma closure_greaterThan[simp]:
  fixes a b::"'a::{no_top, linorder_topology, dense_order}"
  shows "closure {a<..} = {a..}"
proof -
  from gt_ex obtain b where "a < b" by auto
  hence "{a<..} = {a<..<b} ∪ {b..}" by auto
  also have "closure … = {a..}" using ‹a < b› unfolding closure_Un
    by auto
  finally show ?thesis .
qed

lemma closure_lessThan[simp]:
  fixes b::"'a::{no_bot, linorder_topology, dense_order}"
  shows "closure {..<b} = {..b}"
proof -
  from lt_ex obtain a where "a < b" by auto
  hence "{..<b} = {a<..<b} ∪ {..a}" by auto
  also have "closure … = {..b}" using ‹a < b› unfolding closure_Un
    by auto
  finally show ?thesis .
qed

lemma closure_atLeastLessThan[simp]:
  fixes a b::"'a::{linorder_topology, dense_order}"
  assumes "a < b"
  shows "closure {a ..< b} = {a .. b}"
proof -
  from assms have "{a ..< b} = {a} ∪ {a <..< b}" by auto
  also have "closure … = {a .. b}" unfolding closure_Un
    by (auto simp: assms less_imp_le)
  finally show ?thesis .
qed

lemma closure_greaterThanAtMost[simp]:
  fixes a b::"'a::{linorder_topology, dense_order}"
  assumes "a < b"
  shows "closure {a <.. b} = {a .. b}"
proof -
  from assms have "{a <.. b} = {b} ∪ {a <..< b}" by auto
  also have "closure … = {a .. b}" unfolding closure_Un
    by (auto simp: assms less_imp_le)
  finally show ?thesis .
qed


subsection ‹Homeomorphisms›

definition%important "homeomorphism s t f g ⟷
  (∀x∈s. (g(f x) = x)) ∧ (f ` s = t) ∧ continuous_on s f ∧
  (∀y∈t. (f(g y) = y)) ∧ (g ` t = s) ∧ continuous_on t g"

lemma homeomorphismI [intro?]:
  assumes "continuous_on S f" "continuous_on T g"
          "f ` S ⊆ T" "g ` T ⊆ S" "⋀x. x ∈ S ⟹ g(f x) = x" "⋀y. y ∈ T ⟹ f(g y) = y"
    shows "homeomorphism S T f g"
  using assms by (force simp: homeomorphism_def)

lemma homeomorphism_translation:
  fixes a :: "'a :: real_normed_vector"
  shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)

lemma homeomorphism_ident: "homeomorphism T T (λa. a) (λa. a)"
  by (rule homeomorphismI) (auto simp: continuous_on_id)

lemma homeomorphism_compose:
  assumes "homeomorphism S T f g" "homeomorphism T U h k"
    shows "homeomorphism S U (h o f) (g o k)"
  using assms
  unfolding homeomorphism_def
  by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])

lemma homeomorphism_symD: "homeomorphism S t f g ⟹ homeomorphism t S g f"
  by (simp add: homeomorphism_def)

lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
  by (force simp: homeomorphism_def)

definition homeomorphic :: "'a::topological_space set ⇒ 'b::topological_space set ⇒ bool"
    (infixr "homeomorphic" 60)
  where "s homeomorphic t ≡ (∃f g. homeomorphism s t f g)"

lemma homeomorphic_empty [iff]:
     "S homeomorphic {} ⟷ S = {}" "{} homeomorphic S ⟷ S = {}"
  by (auto simp: homeomorphic_def homeomorphism_def)

lemma homeomorphic_refl: "s homeomorphic s"
  unfolding homeomorphic_def homeomorphism_def
  using continuous_on_id
  apply (rule_tac x = "(λx. x)" in exI)
  apply (rule_tac x = "(λx. x)" in exI)
  apply blast
  done

lemma homeomorphic_sym: "s homeomorphic t ⟷ t homeomorphic s"
  unfolding homeomorphic_def homeomorphism_def
  by blast

lemma homeomorphic_trans [trans]:
  assumes "S homeomorphic T"
      and "T homeomorphic U"
    shows "S homeomorphic U"
  using assms
  unfolding homeomorphic_def
by (metis homeomorphism_compose)

lemma homeomorphic_minimal:
  "s homeomorphic t ⟷
    (∃f g. (∀x∈s. f(x) ∈ t ∧ (g(f(x)) = x)) ∧
           (∀y∈t. g(y) ∈ s ∧ (f(g(y)) = y)) ∧
           continuous_on s f ∧ continuous_on t g)"
   (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (fastforce simp: homeomorphic_def homeomorphism_def)
next
  assume ?rhs
  then show ?lhs
    apply clarify
    unfolding homeomorphic_def homeomorphism_def
    by (metis equalityI image_subset_iff subsetI)
 qed

lemma homeomorphicI [intro?]:
   "⟦f ` S = T; g ` T = S;
     continuous_on S f; continuous_on T g;
     ⋀x. x ∈ S ⟹ g(f(x)) = x;
     ⋀y. y ∈ T ⟹ f(g(y)) = y⟧ ⟹ S homeomorphic T"
unfolding homeomorphic_def homeomorphism_def by metis

lemma homeomorphism_of_subsets:
   "⟦homeomorphism S T f g; S' ⊆ S; T'' ⊆ T; f ` S' = T'⟧
    ⟹ homeomorphism S' T' f g"
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
by (metis subsetD imageI)

lemma homeomorphism_apply1: "⟦homeomorphism S T f g; x ∈ S⟧ ⟹ g(f x) = x"
  by (simp add: homeomorphism_def)

lemma homeomorphism_apply2: "⟦homeomorphism S T f g; x ∈ T⟧ ⟹ f(g x) = x"
  by (simp add: homeomorphism_def)

lemma homeomorphism_image1: "homeomorphism S T f g ⟹ f ` S = T"
  by (simp add: homeomorphism_def)

lemma homeomorphism_image2: "homeomorphism S T f g ⟹ g ` T = S"
  by (simp add: homeomorphism_def)

lemma homeomorphism_cont1: "homeomorphism S T f g ⟹ continuous_on S f"
  by (simp add: homeomorphism_def)

lemma homeomorphism_cont2: "homeomorphism S T f g ⟹ continuous_on T g"
  by (simp add: homeomorphism_def)

lemma continuous_on_no_limpt:
   "(⋀x. ¬ x islimpt S) ⟹ continuous_on S f"
  unfolding continuous_on_def
  by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)

lemma continuous_on_finite:
  fixes S :: "'a::t1_space set"
  shows "finite S ⟹ continuous_on S f"
by (metis continuous_on_no_limpt islimpt_finite)

lemma homeomorphic_finite:
  fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
  assumes "finite T"
  shows "S homeomorphic T ⟷ finite S ∧ finite T ∧ card S = card T" (is "?lhs = ?rhs")
proof
  assume "S homeomorphic T"
  with assms show ?rhs
    apply (auto simp: homeomorphic_def homeomorphism_def)
     apply (metis finite_imageI)
    by (metis card_image_le finite_imageI le_antisym)
next
  assume R: ?rhs
  with finite_same_card_bij obtain h where "bij_betw h S T"
    by auto
  with R show ?lhs
    apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
    apply (rule_tac x=h in exI)
    apply (rule_tac x="inv_into S h" in exI)
    apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
    apply (metis bij_betw_def bij_betw_inv_into)
    done
qed

text ‹Relatively weak hypotheses if a set is compact.›

lemma homeomorphism_compact:
  fixes f :: "'a::topological_space ⇒ 'b::t2_space"
  assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
  shows "∃g. homeomorphism s t f g"
proof -
  define g where "g x = (SOME y. y∈s ∧ f y = x)" for x
  have g: "∀x∈s. g (f x) = x"
    using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
  {
    fix y
    assume "y ∈ t"
    then obtain x where x:"f x = y" "x∈s"
      using assms(3) by auto
    then have "g (f x) = x" using g by auto
    then have "f (g y) = y" unfolding x(1)[symmetric] by auto
  }
  then have g':"∀x∈t. f (g x) = x" by auto
  moreover
  {
    fix x
    have "x∈s ⟹ x ∈ g ` t"
      using g[THEN bspec[where x=x]]
      unfolding image_iff
      using assms(3)
      by (auto intro!: bexI[where x="f x"])
    moreover
    {
      assume "x∈g ` t"
      then obtain y where y:"y∈t" "g y = x" by auto
      then obtain x' where x':"x'∈s" "f x' = y"
        using assms(3) by auto
      then have "x ∈ s"
        unfolding g_def
        using someI2[of "λb. b∈s ∧ f b = y" x' "λx. x∈s"]
        unfolding y(2)[symmetric] and g_def
        by auto
    }
    ultimately have "x∈s ⟷ x ∈ g ` t" ..
  }
  then have "g ` t = s" by auto
  ultimately show ?thesis
    unfolding homeomorphism_def homeomorphic_def
    apply (rule_tac x=g in exI)
    using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
    apply auto
    done
qed

lemma homeomorphic_compact:
  fixes f :: "'a::topological_space ⇒ 'b::t2_space"
  shows "compact s ⟹ continuous_on s f ⟹ (f ` s = t) ⟹ inj_on f s ⟹ s homeomorphic t"
  unfolding homeomorphic_def by (metis homeomorphism_compact)

text‹Preservation of topological properties.›

lemma homeomorphic_compactness: "s homeomorphic t ⟹ (compact s ⟷ compact t)"
  unfolding homeomorphic_def homeomorphism_def
  by (metis compact_continuous_image)

text‹Results on translation, scaling etc.›

lemma homeomorphic_scaling:
  fixes s :: "'a::real_normed_vector set"
  assumes "c ≠ 0"
  shows "s homeomorphic ((λx. c *R x) ` s)"
  unfolding homeomorphic_minimal
  apply (rule_tac x="λx. c *R x" in exI)
  apply (rule_tac x="λx. (1 / c) *R x" in exI)
  using assms
  apply (auto simp: continuous_intros)
  done

lemma homeomorphic_translation:
  fixes s :: "'a::real_normed_vector set"
  shows "s homeomorphic ((λx. a + x) ` s)"
  unfolding homeomorphic_minimal
  apply (rule_tac x="λx. a + x" in exI)
  apply (rule_tac x="λx. -a + x" in exI)
  using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
    continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
  apply auto
  done

lemma homeomorphic_affinity:
  fixes s :: "'a::real_normed_vector set"
  assumes "c ≠ 0"
  shows "s homeomorphic ((λx. a + c *R x) ` s)"
proof -
  have *: "(+) a ` ( *R) c ` s = (λx. a + c *R x) ` s" by auto
  show ?thesis
    using homeomorphic_trans
    using homeomorphic_scaling[OF assms, of s]
    using homeomorphic_translation[of "(λx. c *R x) ` s" a]
    unfolding *
    by auto
qed

lemma homeomorphic_balls:
  fixes a b ::"'a::real_normed_vector"
  assumes "0 < d"  "0 < e"
  shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
    and "(cball a d) homeomorphic (cball b e)" (is ?cth)
proof -
  show ?th unfolding homeomorphic_minimal
    apply(rule_tac x="λx. b + (e/d) *R (x - a)" in exI)
    apply(rule_tac x="λx. a + (d/e) *R (x - b)" in exI)
    using assms
    apply (auto intro!: continuous_intros
      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
    done
  show ?cth unfolding homeomorphic_minimal
    apply(rule_tac x="λx. b + (e/d) *R (x - a)" in exI)
    apply(rule_tac x="λx. a + (d/e) *R (x - b)" in exI)
    using assms
    apply (auto intro!: continuous_intros
      simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
    done
qed

lemma homeomorphic_spheres:
  fixes a b ::"'a::real_normed_vector"
  assumes "0 < d"  "0 < e"
  shows "(sphere a d) homeomorphic (sphere b e)"
unfolding homeomorphic_minimal
    apply(rule_tac x="λx. b + (e/d) *R (x - a)" in exI)
    apply(rule_tac x="λx. a + (d/e) *R (x - b)" in exI)
    using assms
    apply (auto intro!: continuous_intros
      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
    done

lemma homeomorphic_ball01_UNIV:
  "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
  (is "?B homeomorphic ?U")
proof
  have "x ∈ (λz. z /R (1 - norm z)) ` ball 0 1" for x::'a
    apply (rule_tac x="x /R (1 + norm x)" in image_eqI)
     apply (auto simp: divide_simps)
    using norm_ge_zero [of x] apply linarith+
    done
  then show "(λz::'a. z /R (1 - norm z)) ` ?B = ?U"
    by blast
  have "x ∈ range (λz. (1 / (1 + norm z)) *R z)" if "norm x < 1" for x::'a
    apply (rule_tac x="x /R (1 - norm x)" in image_eqI)
    using that apply (auto simp: divide_simps)
    done
  then show "(λz::'a. z /R (1 + norm z)) ` ?U = ?B"
    by (force simp: divide_simps dest: add_less_zeroD)
  show "continuous_on (ball 0 1) (λz. z /R (1 - norm z))"
    by (rule continuous_intros | force)+
  show "continuous_on UNIV (λz. z /R (1 + norm z))"
    apply (intro continuous_intros)
    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
    done
  show "⋀x. x ∈ ball 0 1 ⟹
         x /R (1 - norm x) /R (1 + norm (x /R (1 - norm x))) = x"
    by (auto simp: divide_simps)
  show "⋀y. y /R (1 + norm y) /R (1 - norm (y /R (1 + norm y))) = y"
    apply (auto simp: divide_simps)
    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
    done
qed

proposition homeomorphic_ball_UNIV:
  fixes a ::"'a::real_normed_vector"
  assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
  using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast


text ‹Connectedness is invariant under homeomorphisms.›

lemma homeomorphic_connectedness:
  assumes "s homeomorphic t"
  shows "connected s ⟷ connected t"
using assms unfolding homeomorphic_def homeomorphism_def by (metis connected_continuous_image)


subsection%unimportant‹Inverse function property for open/closed maps›

lemma continuous_on_inverse_open_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "⋀x. x ∈ S ⟹ g (f x) = x"
    and oo: "⋀U. openin (subtopology euclidean S) U ⟹ openin (subtopology euclidean T) (f ` U)"
  shows "continuous_on T g"
proof -
  from imf injf have gTS: "g ` T = S"
    by force
  from imf injf have fU: "U ⊆ S ⟹ (f ` U) = T ∩ g -` U" for U
    by force
  show ?thesis
    by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
qed

lemma continuous_on_inverse_closed_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "⋀x. x ∈ S ⟹ g(f x) = x"
    and oo: "⋀U. closedin (subtopology euclidean S) U ⟹ closedin (subtopology euclidean T) (f ` U)"
  shows "continuous_on T g"
proof -
  from imf injf have gTS: "g ` T = S"
    by force
  from imf injf have fU: "U ⊆ S ⟹ (f ` U) = T ∩ g -` U" for U
    by force
  show ?thesis
    by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
qed

lemma homeomorphism_injective_open_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "inj_on f S"
    and oo: "⋀U. openin (subtopology euclidean S) U ⟹ openin (subtopology euclidean T) (f ` U)"
  obtains g where "homeomorphism S T f g"
proof
  have "continuous_on T (inv_into S f)"
    by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
  with imf injf contf show "homeomorphism S T f (inv_into S f)"
    by (auto simp: homeomorphism_def)
qed

lemma homeomorphism_injective_closed_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "inj_on f S"
    and oo: "⋀U. closedin (subtopology euclidean S) U ⟹ closedin (subtopology euclidean T) (f ` U)"
  obtains g where "homeomorphism S T f g"
proof
  have "continuous_on T (inv_into S f)"
    by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
  with imf injf contf show "homeomorphism S T f (inv_into S f)"
    by (auto simp: homeomorphism_def)
qed

lemma homeomorphism_imp_open_map:
  assumes hom: "homeomorphism S T f g"
    and oo: "openin (subtopology euclidean S) U"
  shows "openin (subtopology euclidean T) (f ` U)"
proof -
  from hom oo have [simp]: "f ` U = T ∩ g -` U"
    using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
  from hom have "continuous_on T g"
    unfolding homeomorphism_def by blast
  moreover have "g ` T = S"
    by (metis hom homeomorphism_def)
  ultimately show ?thesis
    by (simp add: continuous_on_open oo)
qed

lemma homeomorphism_imp_closed_map:
  assumes hom: "homeomorphism S T f g"
    and oo: "closedin (subtopology euclidean S) U"
  shows "closedin (subtopology euclidean T) (f ` U)"
proof -
  from hom oo have [simp]: "f ` U = T ∩ g -` U"
    using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
  from hom have "continuous_on T g"
    unfolding homeomorphism_def by blast
  moreover have "g ` T = S"
    by (metis hom homeomorphism_def)
  ultimately show ?thesis
    by (simp add: continuous_on_closed oo)
qed


subsection ‹"Isometry" (up to constant bounds) of injective linear map etc›

lemma cauchy_isometric:
  assumes e: "e > 0"
    and s: "subspace s"
    and f: "bounded_linear f"
    and normf: "∀x∈s. norm (f x) ≥ e * norm x"
    and xs: "∀n. x n ∈ s"
    and cf: "Cauchy (f ∘ x)"
  shows "Cauchy x"
proof -
  interpret f: bounded_linear f by fact
  have "∃N. ∀n≥N. norm (x n - x N) < d" if "d > 0" for d :: real
  proof -
    from that obtain N where N: "∀n≥N. norm (f (x n) - f (x N)) < e * d"
      using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
      by auto
    have "norm (x n - x N) < d" if "n ≥ N" for n
    proof -
      have "e * norm (x n - x N) ≤ norm (f (x n - x N))"
        using subspace_diff[OF s, of "x n" "x N"]
        using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
        using normf[THEN bspec[where x="x n - x N"]]
        by auto
      also have "norm (f (x n - x N)) < e * d"
        using ‹N ≤ n› N unfolding f.diff[symmetric] by auto
      finally show ?thesis
        using ‹e>0› by simp
    qed
    then show ?thesis by auto
  qed
  then show ?thesis
    by (simp add: Cauchy_altdef2 dist_norm)
qed

lemma complete_isometric_image:
  assumes "0 < e"
    and s: "subspace s"
    and f: "bounded_linear f"
    and normf: "∀x∈s. norm(f x) ≥ e * norm(x)"
    and cs: "complete s"
  shows "complete (f ` s)"
proof -
  have "∃l∈f ` s. (g ⤏ l) sequentially"
    if as:"∀n::nat. g n ∈ f ` s" and cfg:"Cauchy g" for g
  proof -
    from that obtain x where "∀n. x n ∈ s ∧ g n = f (x n)"
      using choice[of "λ n xa. xa ∈ s ∧ g n = f xa"] by auto
    then have x: "∀n. x n ∈ s" "∀n. g n = f (x n)" by auto
    then have "f ∘ x = g" by (simp add: fun_eq_iff)
    then obtain l where "l∈s" and l:"(x ⤏ l) sequentially"
      using cs[unfolded complete_def, THEN spec[where x=x]]
      using cauchy_isometric[OF ‹0 < e› s f normf] and cfg and x(1)
      by auto
    then show ?thesis
      using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
      by (auto simp: ‹f ∘ x = g›)
  qed
  then show ?thesis
    unfolding complete_def by auto
qed

proposition injective_imp_isometric:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes s: "closed s" "subspace s"
    and f: "bounded_linear f" "∀x∈s. f x = 0 ⟶ x = 0"
  shows "∃e>0. ∀x∈s. norm (f x) ≥ e * norm x"
proof (cases "s ⊆ {0::'a}")
  case True
  have "norm x ≤ norm (f x)" if "x ∈ s" for x
  proof -
    from True that have "x = 0" by auto
    then show ?thesis by simp
  qed
  then show ?thesis
    by (auto intro!: exI[where x=1])
next
  case False
  interpret f: bounded_linear f by fact
  from False obtain a where a: "a ≠ 0" "a ∈ s"
    by auto
  from False have "s ≠ {}"
    by auto
  let ?S = "{f x| x. x ∈ s ∧ norm x = norm a}"
  let ?S' = "{x::'a. x∈s ∧ norm x = norm a}"
  let ?S'' = "{x::'a. norm x = norm a}"

  have "?S'' = frontier (cball 0 (norm a))"
    by (simp add: sphere_def dist_norm)
  then have "compact ?S''" by (metis compact_cball compact_frontier)
  moreover have "?S' = s ∩ ?S''" by auto
  ultimately have "compact ?S'"
    using closed_Int_compact[of s ?S''] using s(1) by auto
  moreover have *:"f ` ?S' = ?S" by auto
  ultimately have "compact ?S"
    using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
  then have "closed ?S"
    using compact_imp_closed by auto
  moreover from a have "?S ≠ {}" by auto
  ultimately obtain b' where "b'∈?S" "∀y∈?S. norm b' ≤ norm y"
    using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
  then obtain b where "b∈s"
    and ba: "norm b = norm a"
    and b: "∀x∈{x ∈ s. norm x = norm a}. norm (f b) ≤ norm (f x)"
    unfolding *[symmetric] unfolding image_iff by auto

  let ?e = "norm (f b) / norm b"
  have "norm b > 0"
    using ba and a and norm_ge_zero by auto
  moreover have "norm (f b) > 0"
    using f(2)[THEN bspec[where x=b], OF ‹b∈s›]
    using ‹norm b >0› by simp
  ultimately have "0 < norm (f b) / norm b" by simp
  moreover
  have "norm (f b) / norm b * norm x ≤ norm (f x)" if "x∈s" for x
  proof (cases "x = 0")
    case True
    then show "norm (f b) / norm b * norm x ≤ norm (f x)"
      by auto
  next
    case False
    with ‹a ≠ 0› have *: "0 < norm a / norm x"
      unfolding zero_less_norm_iff[symmetric] by simp
    have "∀x∈s. c *R x ∈ s" for c
      using s[unfolded subspace_def] by simp
    with ‹x ∈ s› ‹x ≠ 0› have "(norm a / norm x) *R x ∈ {x ∈ s. norm x = norm a}"
      by simp
    with ‹x ≠ 0› ‹a ≠ 0› show "norm (f b) / norm b * norm x ≤ norm (f x)"
      using b[THEN bspec[where x="(norm a / norm x) *R x"]]
      unfolding f.scaleR and ba
      by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
  qed
  ultimately show ?thesis by auto
qed

proposition closed_injective_image_subspace:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes "subspace s" "bounded_linear f" "∀x∈s. f x = 0 ⟶ x = 0" "closed s"
  shows "closed(f ` s)"
proof -
  obtain e where "e > 0" and e: "∀x∈s. e * norm x ≤ norm (f x)"
    using injective_imp_isometric[OF assms(4,1,2,3)] by auto
  show ?thesis
    using complete_isometric_image[OF ‹e>0› assms(1,2) e] and assms(4)
    unfolding complete_eq_closed[symmetric] by auto
qed


subsection%unimportant ‹Some properties of a canonical subspace›

lemma subspace_substandard: "subspace {x::'a::euclidean_space. (∀i∈Basis. P i ⟶ x∙i = 0)}"
  by (auto simp: subspace_def inner_add_left)

lemma closed_substandard: "closed {x::'a::euclidean_space. ∀i∈Basis. P i ⟶ x∙i = 0}"
  (is "closed ?A")
proof -
  let ?D = "{i∈Basis. P i}"
  have "closed (⋂i∈?D. {x::'a. x∙i = 0})"
    by (simp add: closed_INT closed_Collect_eq continuous_on_inner
        continuous_on_const continuous_on_id)
  also have "(⋂i∈?D. {x::'a. x∙i = 0}) = ?A"
    by auto
  finally show "closed ?A" .
qed

lemma dim_substandard:
  assumes d: "d ⊆ Basis"
  shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0} = card d" (is "dim ?A = _")
proof (rule dim_unique)
  from d show "d ⊆ ?A"
    by (auto simp: inner_Basis)
  from d show "independent d"
    by (rule independent_mono [OF independent_Basis])
  have "x ∈ span d" if "∀i∈Basis. i ∉ d ⟶ x ∙ i = 0" for x
  proof -
    have "finite d"
      by (rule finite_subset [OF d finite_Basis])
    then have "(∑i∈d. (x ∙ i) *R i) ∈ span d"
      by (simp add: span_sum span_clauses)
    also have "(∑i∈d. (x ∙ i) *R i) = (∑i∈Basis. (x ∙ i) *R i)"
      by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
    finally show "x ∈ span d"
      by (simp only: euclidean_representation)
  qed
  then show "?A ⊆ span d" by auto
qed simp

text ‹Hence closure and completeness of all subspaces.›
lemma ex_card:
  assumes "n ≤ card A"
  shows "∃S⊆A. card S = n"
proof (cases "finite A")
  case True
  from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
  moreover from f ‹n ≤ card A› have "{..< n} ⊆ {..< card A}" "inj_on f {..< n}"
    by (auto simp: bij_betw_def intro: subset_inj_on)
  ultimately have "f ` {..< n} ⊆ A" "card (f ` {..< n}) = n"
    by (auto simp: bij_betw_def card_image)
  then show ?thesis by blast
next
  case False
  with ‹n ≤ card A› show ?thesis by force
qed

lemma closed_subspace:
  fixes s :: "'a::euclidean_space set"
  assumes "subspace s"
  shows "closed s"
proof -
  have "dim s ≤ card (Basis :: 'a set)"
    using dim_subset_UNIV by auto
  with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d ⊆ Basis"
    by auto
  let ?t = "{x::'a. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
  have "∃f. linear f ∧ f ` {x::'a. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} = s ∧
      inj_on f {x::'a. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
    using dim_substandard[of d] t d assms
    by (intro subspace_isomorphism[OF subspace_substandard[of "λi. i ∉ d"]]) (auto simp: inner_Basis)
  then obtain f where f:
      "linear f"
      "f ` {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} = s"
      "inj_on f {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
    by blast
  interpret f: bounded_linear f
    using f by (simp add: linear_conv_bounded_linear)
  have "x ∈ ?t ⟹ f x = 0 ⟹ x = 0" for x
    using f.zero d f(3)[THEN inj_onD, of x 0] by auto
  moreover have "closed ?t" by (rule closed_substandard)
  moreover have "subspace ?t" by (rule subspace_substandard)
  ultimately show ?thesis
    using closed_injective_image_subspace[of ?t f]
    unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
qed

lemma complete_subspace: "subspace s ⟹ complete s"
  for s :: "'a::euclidean_space set"
  using complete_eq_closed closed_subspace by auto

lemma closed_span [iff]: "closed (span s)"
  for s :: "'a::euclidean_space set"
  by (simp add: closed_subspace subspace_span)

lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
  for s :: "'a::euclidean_space set"
proof -
  have "?dc ≤ ?d"
    using closure_minimal[OF span_superset, of s]
    using closed_subspace[OF subspace_span, of s]
    using dim_subset[of "closure s" "span s"]
    by simp
  then show ?thesis
    using dim_subset[OF closure_subset, of s]
    by simp
qed


subsection%unimportant ‹Affine transformations of intervals›

lemma real_affinity_le: "0 < m ⟹ m * x + c ≤ y ⟷ x ≤ inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_le_affinity: "0 < m ⟹ y ≤ m * x + c ⟷ inverse m * y + - (c / m) ≤ x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_affinity_lt: "0 < m ⟹ m * x + c < y ⟷ x < inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_lt_affinity: "0 < m ⟹ y < m * x + c ⟷ inverse m * y + - (c / m) < x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_affinity_eq: "m ≠ 0 ⟹ m * x + c = y ⟷ x = inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_eq_affinity: "m ≠ 0 ⟹ y = m * x + c  ⟷ inverse m * y + - (c / m) = x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)


subsection ‹Banach fixed point theorem (not really topological ...)›

theorem banach_fix:
  assumes s: "complete s" "s ≠ {}"
    and c: "0 ≤ c" "c < 1"
    and f: "f ` s ⊆ s"
    and lipschitz: "∀x∈s. ∀y∈s. dist (f x) (f y) ≤ c * dist x y"
  shows "∃!x∈s. f x = x"
proof -
  from c have "1 - c > 0" by simp

  from s(2) obtain z0 where z0: "z0 ∈ s" by blast
  define z where "z n = (f ^^ n) z0" for n
  with f z0 have z_in_s: "z n ∈ s" for n :: nat
    by (induct n) auto
  define d where "d = dist (z 0) (z 1)"

  have fzn: "f (z n) = z (Suc n)" for n
    by (simp add: z_def)
  have cf_z: "dist (z n) (z (Suc n)) ≤ (c ^ n) * d" for n :: nat
  proof (induct n)
    case 0
    then show ?case
      by (simp add: d_def)
  next
    case (Suc m)
    with ‹0 ≤ c› have "c * dist (z m) (z (Suc m)) ≤ c ^ Suc m * d"
      using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
    then show ?case
      using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
      by (simp add: fzn mult_le_cancel_left)
  qed

  have cf_z2: "(1 - c) * dist (z m) (z (m + n)) ≤ (c ^ m) * d * (1 - c ^ n)" for n m :: nat
  proof (induct n)
    case 0
    show ?case by simp
  next
    case (Suc k)
    from c have "(1 - c) * dist (z m) (z (m + Suc k)) ≤
        (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
      by (simp add: dist_triangle)
    also from c cf_z[of "m + k"] have "… ≤ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
      by simp
    also from Suc have "… ≤ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
      by (simp add: field_simps)
    also have "… = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
      by (simp add: power_add field_simps)
    also from c have "… ≤ (c ^ m) * d * (1 - c ^ Suc k)"
      by (simp add: field_simps)
    finally show ?case by simp
  qed

  have "∃N. ∀m n. N ≤ m ∧ N ≤ n ⟶ dist (z m) (z n) < e" if "e > 0" for e
  proof (cases "d = 0")
    case True
    from ‹1 - c > 0› have "(1 - c) * x ≤ 0 ⟷ x ≤ 0" for x
      by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
    with c cf_z2[of 0] True have "z n = z0" for n
      by (simp add: z_def)
    with ‹e > 0› show ?thesis by simp
  next
    case False
    with zero_le_dist[of "z 0" "z 1"] have "d > 0"
      by (metis d_def less_le)
    with ‹1 - c > 0› ‹e > 0› have "0 < e * (1 - c) / d"
      by simp
    with c obtain N where N: "c ^ N < e * (1 - c) / d"
      using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
    have *: "dist (z m) (z n) < e" if "m > n" and as: "m ≥ N" "n ≥ N" for m n :: nat
    proof -
      from c ‹n ≥ N› have *: "c ^ n ≤ c ^ N"
        using power_decreasing[OF ‹n≥N›, of c] by simp
      from c ‹m > n› have "1 - c ^ (m - n) > 0"
        using power_strict_mono[of c 1 "m - n"] by simp
      with ‹d > 0› ‹0 < 1 - c› have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
        by simp
      from cf_z2[of n "m - n"] ‹m > n›
      have "dist (z m) (z n) ≤ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
        by (simp add: pos_le_divide_eq[OF ‹1 - c > 0›] mult.commute dist_commute)
      also have "… ≤ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
        using mult_right_mono[OF * order_less_imp_le[OF **]]
        by (simp add: mult.assoc)
      also have "… < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
        using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
      also from c ‹d > 0› ‹1 - c > 0› have "… = e * (1 - c ^ (m - n))"
        by simp
      also from c ‹1 - c ^ (m - n) > 0› ‹e > 0› have "… ≤ e"
        using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
      finally show ?thesis by simp
    qed
    have "dist (z n) (z m) < e" if "N ≤ m" "N ≤ n" for m n :: nat
    proof (cases "n = m")
      case True
      with ‹e > 0› show ?thesis by simp
    next
      case False
      with *[of n m] *[of m n] and that show ?thesis
        by (auto simp: dist_commute nat_neq_iff)
    qed
    then show ?thesis by auto
  qed
  then have "Cauchy z"
    by (simp add: cauchy_def)
  then obtain x where "x∈s" and x:"(z ⤏ x) sequentially"
    using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto

  define e where "e = dist (f x) x"
  have "e = 0"
  proof (rule ccontr)
    assume "e ≠ 0"
    then have "e > 0"
      unfolding e_def using zero_le_dist[of "f x" x]
      by (metis dist_eq_0_iff dist_nz e_def)
    then obtain N where N:"∀n≥N. dist (z n) x < e / 2"
      using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
    then have N':"dist (z N) x < e / 2" by auto
    have *: "c * dist (z N) x ≤ dist (z N) x"
      unfolding mult_le_cancel_right2
      using zero_le_dist[of "z N" x] and c
      by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
    have "dist (f (z N)) (f x) ≤ c * dist (z N) x"
      using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
      using z_in_s[of N] ‹x∈s›
      using c
      by auto
    also have "… < e / 2"
      using N' and c using * by auto
    finally show False
      unfolding fzn
      using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
      unfolding e_def
      by auto
  qed
  then have "f x = x" by (auto simp: e_def)
  moreover have "y = x" if "f y = y" "y ∈ s" for y
  proof -
    from ‹x ∈ s› ‹f x = x› that have "dist x y ≤ c * dist x y"
      using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp
    with c and zero_le_dist[of x y] have "dist x y = 0"
      by (simp add: mult_le_cancel_right1)
    then show ?thesis by simp
  qed
  ultimately show ?thesis
    using ‹x∈s› by blast
qed

lemma banach_fix_type:
  fixes f::"'a::complete_space⇒'a"
  assumes c:"0 ≤ c" "c < 1"
      and lipschitz:"∀x. ∀y. dist (f x) (f y) ≤ c * dist x y"
  shows "∃!x. (f x = x)"
  using assms banach_fix[OF complete_UNIV UNIV_not_empty assms(1,2) subset_UNIV, of f]
  by auto


subsection ‹Edelstein fixed point theorem›

theorem edelstein_fix:
  fixes s :: "'a::metric_space set"
  assumes s: "compact s" "s ≠ {}"
    and gs: "(g ` s) ⊆ s"
    and dist: "∀x∈s. ∀y∈s. x ≠ y ⟶ dist (g x) (g y) < dist x y"
  shows "∃!x∈s. g x = x"
proof -
  let ?D = "(λx. (x, x)) ` s"
  have D: "compact ?D" "?D ≠ {}"
    by (rule compact_continuous_image)
       (auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)

  have "⋀x y e. x ∈ s ⟹ y ∈ s ⟹ 0 < e ⟹ dist y x < e ⟹ dist (g y) (g x) < e"
    using dist by fastforce
  then have "continuous_on s g"
    by (auto simp: continuous_on_iff)
  then have cont: "continuous_on ?D (λx. dist ((g ∘ fst) x) (snd x))"
    unfolding continuous_on_eq_continuous_within
    by (intro continuous_dist ballI continuous_within_compose)
       (auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)

  obtain a where "a ∈ s" and le: "⋀x. x ∈ s ⟹ dist (g a) a ≤ dist (g x) x"
    using continuous_attains_inf[OF D cont] by auto

  have "g a = a"
  proof (rule ccontr)
    assume "g a ≠ a"
    with ‹a ∈ s› gs have "dist (g (g a)) (g a) < dist (g a) a"
      by (intro dist[rule_format]) auto
    moreover have "dist (g a) a ≤ dist (g (g a)) (g a)"
      using ‹a ∈ s› gs by (intro le) auto
    ultimately show False by auto
  qed
  moreover have "⋀x. x ∈ s ⟹ g x = x ⟹ x = a"
    using dist[THEN bspec[where x=a]] ‹g a = a› and ‹a∈s› by auto
  ultimately show "∃!x∈s. g x = x"
    using ‹a ∈ s› by blast
qed


lemma cball_subset_cball_iff:
  fixes a :: "'a :: euclidean_space"
  shows "cball a r ⊆ cball a' r' ⟷ dist a a' + r ≤ r' ∨ r < 0"
    (is "?lhs ⟷ ?rhs")
proof
  assume ?lhs
  then show ?rhs
  proof (cases "r < 0")
    case True
    then show ?rhs by simp
  next
    case False
    then have [simp]: "r ≥ 0" by simp
    have "norm (a - a') + r ≤ r'"
    proof (cases "a = a'")
      case True
      then show ?thesis
        using subsetD [where c = "a + r *R (SOME i. i ∈ Basis)", OF ‹?lhs›] subsetD [where c = a, OF ‹?lhs›]
        by (force simp: SOME_Basis dist_norm)
    next
      case False
      have "norm (a' - (a + (r / norm (a - a')) *R (a - a'))) = norm (a' - a - (r / norm (a - a')) *R (a - a'))"
        by (simp add: algebra_simps)
      also have "... = norm ((-1 - (r / norm (a - a'))) *R (a - a'))"
        by (simp add: algebra_simps)
      also from ‹a ≠ a'› have "... = ¦- norm (a - a') - r¦"
        by (simp add: abs_mult_pos field_simps)
      finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *R (a - a'))) = ¦norm (a - a') + r¦"
        by linarith
      from ‹a ≠ a'› show ?thesis
        using subsetD [where c = "a' + (1 + r / norm(a - a')) *R (a - a')", OF ‹?lhs›]
        by (simp add: dist_norm scaleR_add_left)
    qed
    then show ?rhs
      by (simp add: dist_norm)
  qed
next
  assume ?rhs
  then show ?lhs
    by (auto simp: ball_def dist_norm)
      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
qed

lemma cball_subset_ball_iff: "cball a r ⊆ ball a' r' ⟷ dist a a' + r < r' ∨ r < 0"
  (is "?lhs ⟷ ?rhs")
  for a :: "'a::euclidean_space"
proof
  assume ?lhs
  then show ?rhs
  proof (cases "r < 0")
    case True then
    show ?rhs by simp
  next
    case False
    then have [simp]: "r ≥ 0" by simp
    have "norm (a - a') + r < r'"
    proof (cases "a = a'")
      case True
      then show ?thesis
        using subsetD [where c = "a + r *R (SOME i. i ∈ Basis)", OF ‹?lhs›] subsetD [where c = a, OF ‹?lhs›]
        by (force simp: SOME_Basis dist_norm)
    next
      case False
      have False if "norm (a - a') + r ≥ r'"
      proof -
        from that have "¦r' - norm (a - a')¦ ≤ r"
          by (simp split: abs_split)
            (metis ‹0 ≤ r› ‹?lhs› centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
        then show ?thesis
          using subsetD [where c = "a + (r' / norm(a - a') - 1) *R (a - a')", OF ‹?lhs›] ‹a ≠ a'›
          by (simp add: dist_norm field_simps)
            (simp add: diff_divide_distrib scaleR_left_diff_distrib)
      qed
      then show ?thesis by force
    qed
    then show ?rhs by (simp add: dist_norm)
  qed
next
  assume ?rhs
  then show ?lhs
    by (auto simp: ball_def dist_norm)
      (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
qed

lemma ball_subset_cball_iff: "ball a r ⊆ cball a' r' ⟷ dist a a' + r ≤ r' ∨ r ≤ 0"
  (is "?lhs = ?rhs")
  for a :: "'a::euclidean_space"
proof (cases "r ≤ 0")
  case True
  then show ?thesis
    using dist_not_less_zero less_le_trans by force
next
  case False
  show ?thesis
  proof
    assume ?lhs
    then have "(cball a r ⊆ cball a' r')"
      by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
    with False show ?rhs
      by (fastforce iff: cball_subset_cball_iff)
  next
    assume ?rhs
    with False show ?lhs
      using ball_subset_cball cball_subset_cball_iff by blast
  qed
qed

lemma ball_subset_ball_iff:
  fixes a :: "'a :: euclidean_space"
  shows "ball a r ⊆ ball a' r' ⟷ dist a a' + r ≤ r' ∨ r ≤ 0"
        (is "?lhs = ?rhs")
proof (cases "r ≤ 0")
  case True then show ?thesis
    using dist_not_less_zero less_le_trans by force
next
  case False show ?thesis
  proof
    assume ?lhs
    then have "0 < r'"
      by (metis (no_types) False ‹?lhs› centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
    then have "(cball a r ⊆ cball a' r')"
      by (metis False‹?lhs› closure_ball closure_mono not_less)
    then show ?rhs
      using False cball_subset_cball_iff by fastforce
  next
  assume ?rhs then show ?lhs
    apply (auto simp: ball_def)
    apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
    using dist_not_less_zero order.strict_trans2 apply blast
    done
  qed
qed


lemma ball_eq_ball_iff:
  fixes x :: "'a :: euclidean_space"
  shows "ball x d = ball y e ⟷ d ≤ 0 ∧ e ≤ 0 ∨ x=y ∧ d=e"
        (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
  proof (cases "d ≤ 0 ∨ e ≤ 0")
    case True
      with ‹?lhs› show ?rhs
        by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
  next
    case False
    with ‹?lhs› show ?rhs
      apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
      done
  qed
next
  assume ?rhs then show ?lhs
    by (auto simp: set_eq_subset ball_subset_ball_iff)
qed

lemma cball_eq_cball_iff:
  fixes x :: "'a :: euclidean_space"
  shows "cball x d = cball y e ⟷ d < 0 ∧ e < 0 ∨ x=y ∧ d=e"
        (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
  proof (cases "d < 0 ∨ e < 0")
    case True
      with ‹?lhs› show ?rhs
        by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
  next
    case False
    with ‹?lhs› show ?rhs
      apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
      done
  qed
next
  assume ?rhs then show ?lhs
    by (auto simp: set_eq_subset cball_subset_cball_iff)
qed

lemma ball_eq_cball_iff:
  fixes x :: "'a :: euclidean_space"
  shows "ball x d = cball y e ⟷ d ≤ 0 ∧ e < 0" (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
    using ‹?lhs› ball_eq_empty cball_eq_empty apply blast+
    done
next
  assume ?rhs then show ?lhs by auto
qed

lemma cball_eq_ball_iff:
  fixes x :: "'a :: euclidean_space"
  shows "cball x d = ball y e ⟷ d < 0 ∧ e ≤ 0"
  using ball_eq_cball_iff by blast

lemma finite_ball_avoid:
  fixes S :: "'a :: euclidean_space set"
  assumes "open S" "finite X" "p ∈ S"
  shows "∃e>0. ∀w∈ball p e. w∈S ∧ (w≠p ⟶ w∉X)"
proof -
  obtain e1 where "0 < e1" and e1_b:"ball p e1 ⊆ S"
    using open_contains_ball_eq[OF ‹open S›] assms by auto
  obtain e2 where "0 < e2" and "∀x∈X. x ≠ p ⟶ e2 ≤ dist p x"
    using finite_set_avoid[OF ‹finite X›,of p] by auto
  hence "∀w∈ball p (min e1 e2). w∈S ∧ (w≠p ⟶ w∉X)" using e1_b by auto
  thus "∃e>0. ∀w∈ball p e. w ∈ S ∧ (w ≠ p ⟶ w ∉ X)" using ‹e2>0› ‹e1>0›
    apply (rule_tac x="min e1 e2" in exI)
    by auto
qed

lemma finite_cball_avoid:
  fixes S :: "'a :: euclidean_space set"
  assumes "open S" "finite X" "p ∈ S"
  shows "∃e>0. ∀w∈cball p e. w∈S ∧ (w≠p ⟶ w∉X)"
proof -
  obtain e1 where "e1>0" and e1: "∀w∈ball p e1. w∈S ∧ (w≠p ⟶ w∉X)"
    using finite_ball_avoid[OF assms] by auto
  define e2 where "e2 ≡ e1/2"
  have "e2>0" and "e2 < e1" unfolding e2_def using ‹e1>0› by auto
  then have "cball p e2 ⊆ ball p e1" by (subst cball_subset_ball_iff,auto)
  then show "∃e>0. ∀w∈cball p e. w ∈ S ∧ (w ≠ p ⟶ w ∉ X)" using ‹e2>0› e1 by auto
qed

subsection‹Various separability-type properties›

lemma univ_second_countable:
  obtains  :: "'a::euclidean_space set set"
  where "countable ℬ" "⋀C. C ∈ ℬ ⟹ open C"
       "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
by (metis ex_countable_basis topological_basis_def)

lemma subset_second_countable:
  obtains  :: "'a:: euclidean_space set set"
    where "countable ℬ"
          "{} ∉ ℬ"
          "⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
          "⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
proof -
  obtain  :: "'a set set"
    where "countable ℬ"
      and opeB: "⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
      and :    "⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
  proof -
    obtain 𝒞 :: "'a set set"
      where "countable 𝒞" and ope: "⋀C. C ∈ 𝒞 ⟹ open C"
        and 𝒞: "⋀S. open S ⟹ ∃U. U ⊆ 𝒞 ∧ S = ⋃U"
      by (metis univ_second_countable that)
    show ?thesis
    proof
      show "countable ((λC. S ∩ C) ` 𝒞)"
        by (simp add: ‹countable 𝒞›)
      show "⋀C. C ∈ (∩) S ` 𝒞 ⟹ openin (subtopology euclidean S) C"
        using ope by auto
      show "⋀T. openin (subtopology euclidean S) T ⟹ ∃𝒰⊆(∩) S ` 𝒞. T = ⋃𝒰"
        by (metis 𝒞 image_mono inf_Sup openin_open)
    qed
  qed
  show ?thesis
  proof
    show "countable (ℬ - {{}})"
      using ‹countable ℬ› by blast
    show "⋀C. ⟦C ∈ ℬ - {{}}⟧ ⟹ openin (subtopology euclidean S) C"
      by (simp add: ‹⋀C. C ∈ ℬ ⟹ openin (subtopology euclidean S) C›)
    show "∃𝒰⊆ℬ - {{}}. T = ⋃𝒰" if "openin (subtopology euclidean S) T" for T
      using  [OF that]
      apply clarify
      apply (rule_tac x="𝒰 - {{}}" in exI, auto)
        done
  qed auto
qed

lemma univ_second_countable_sequence:
  obtains B :: "nat ⇒ 'a::euclidean_space set"
    where "inj B" "⋀n. open(B n)" "⋀S. open S ⟹ ∃k. S = ⋃{B n |n. n ∈ k}"
proof -
  obtain  :: "'a set set"
  where "countable ℬ"
    and opn: "⋀C. C ∈ ℬ ⟹ open C"
    and Un: "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
    using univ_second_countable by blast
  have *: "infinite (range (λn. ball (0::'a) (inverse(Suc n))))"
    apply (rule Infinite_Set.range_inj_infinite)
    apply (simp add: inj_on_def ball_eq_ball_iff)
    done
  have "infinite ℬ"
  proof
    assume "finite ℬ"
    then have "finite (Union ` (Pow ℬ))"
      by simp
    then have "finite (range (λn. ball (0::'a) (inverse(Suc n))))"
      apply (rule rev_finite_subset)
      by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
    with * show False by simp
  qed
  obtain f :: "nat ⇒ 'a set" where "ℬ = range f" "inj f"
    by (blast intro: countable_as_injective_image [OF ‹countable ℬ› ‹infinite ℬ›])
  have *: "∃k. S = ⋃{f n |n. n ∈ k}" if "open S" for S
    using Un [OF that]
    apply clarify
    apply (rule_tac x="f-`U" in exI)
    using ‹inj f› ‹ℬ = range f› apply force
    done
  show ?thesis
    apply (rule that [OF ‹inj f› _ *])
    apply (auto simp: ‹ℬ = range f› opn)
    done
qed

proposition separable:
  fixes S :: "'a:: euclidean_space set"
  obtains T where "countable T" "T ⊆ S" "S ⊆ closure T"
proof -
  obtain  :: "'a:: euclidean_space set set"
    where "countable ℬ"
      and "{} ∉ ℬ"
      and ope: "⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
      and if_ope: "⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
    by (meson subset_second_countable)
  then obtain f where f: "⋀C. C ∈ ℬ ⟹ f C ∈ C"
    by (metis equals0I)
  show ?thesis
  proof
    show "countable (f ` ℬ)"
      by (simp add: ‹countable ℬ›)
    show "f ` ℬ ⊆ S"
      using ope f openin_imp_subset by blast
    show "S ⊆ closure (f ` ℬ)"
    proof (clarsimp simp: closure_approachable)
      fix x and e::real
      assume "x ∈ S" "0 < e"
      have "openin (subtopology euclidean S) (S ∩ ball x e)"
        by (simp add: openin_Int_open)
      with if_ope obtain 𝒰 where  𝒰: "𝒰 ⊆ ℬ" "S ∩ ball x e = ⋃𝒰"
        by meson
      show "∃C ∈ ℬ. dist (f C) x < e"
      proof (cases "𝒰 = {}")
        case True
        then show ?thesis
          using ‹0 < e›  𝒰 ‹x ∈ S› by auto
      next
        case False
        then obtain C where "C ∈ 𝒰" by blast
        show ?thesis
        proof
          show "dist (f C) x < e"
            by (metis Int_iff Union_iff 𝒰 ‹C ∈ 𝒰› dist_commute f mem_ball subsetCE)
          show "C ∈ ℬ"
            using ‹𝒰 ⊆ ℬ› ‹C ∈ 𝒰› by blast
        qed
      qed
    qed
  qed
qed

proposition Lindelof:
  fixes  :: "'a::euclidean_space set set"
  assumes : "⋀S. S ∈ ℱ ⟹ open S"
  obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
  obtain  :: "'a set set"
    where "countable ℬ" "⋀C. C ∈ ℬ ⟹ open C"
      and : "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
    using univ_second_countable by blast
  define 𝒟 where "𝒟 ≡ {S. S ∈ ℬ ∧ (∃U. U ∈ ℱ ∧ S ⊆ U)}"
  have "countable 𝒟"
    apply (rule countable_subset [OF _ ‹countable ℬ›])
    apply (force simp: 𝒟_def)
    done
  have "⋀S. ∃U. S ∈ 𝒟 ⟶ U ∈ ℱ ∧ S ⊆ U"
    by (simp add: 𝒟_def)
  then obtain G where G: "⋀S. S ∈ 𝒟 ⟶ G S ∈ ℱ ∧ S ⊆ G S"
    by metis
  have "⋃ℱ ⊆ ⋃𝒟"
    unfolding 𝒟_def by (blast dest:  )
  moreover have "⋃𝒟 ⊆ ⋃ℱ"
    using 𝒟_def by blast
  ultimately have eq1: "⋃ℱ = ⋃𝒟" ..
  have eq2: "⋃𝒟 = UNION 𝒟 G"
    using G eq1 by auto
  show ?thesis
    apply (rule_tac ℱ' = "G ` 𝒟" in that)
    using G ‹countable 𝒟›  apply (auto simp: eq1 eq2)
    done
qed

lemma Lindelof_openin:
  fixes  :: "'a::euclidean_space set set"
  assumes "⋀S. S ∈ ℱ ⟹ openin (subtopology euclidean U) S"
  obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
  have "⋀S. S ∈ ℱ ⟹ ∃T. open T ∧ S = U ∩ T"
    using assms by (simp add: openin_open)
  then obtain tf where tf: "⋀S. S ∈ ℱ ⟹ open (tf S) ∧ (S = U ∩ tf S)"
    by metis
  have [simp]: "⋀ℱ'. ℱ' ⊆ ℱ ⟹ ⋃ℱ' = U ∩ ⋃(tf ` ℱ')"
    using tf by fastforce
  obtain 𝒢 where "countable 𝒢 ∧ 𝒢 ⊆ tf ` ℱ" "⋃𝒢 = UNION ℱ tf"
    using tf by (force intro: Lindelof [of "tf ` ℱ"])
  then obtain ℱ' where ℱ': "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
    by (clarsimp simp add: countable_subset_image)
  then show ?thesis ..
qed

lemma countable_disjoint_open_subsets:
  fixes  :: "'a::euclidean_space set set"
  assumes "⋀S. S ∈ ℱ ⟹ open S" and pw: "pairwise disjnt ℱ"
    shows "countable ℱ"
proof -
  obtain ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
    by (meson assms Lindelof)
  with pw have "ℱ ⊆ insert {} ℱ'"
    by (fastforce simp add: pairwise_def disjnt_iff)
  then show ?thesis
    by (simp add: ‹countable ℱ'› countable_subset)
qed

lemma countable_disjoint_nonempty_interior_subsets:
  fixes  :: "'a::euclidean_space set set"
  assumes pw: "pairwise disjnt ℱ" and int: "⋀S. ⟦S ∈ ℱ; interior S = {}⟧ ⟹ S = {}"
  shows "countable ℱ"
proof (rule countable_image_inj_on)
  have "disjoint (interior ` ℱ)"
    using pw by (simp add: disjoint_image_subset interior_subset)
  then show "countable (interior ` ℱ)"
    by (auto intro: countable_disjoint_open_subsets)
  show "inj_on interior ℱ"
    using pw apply (clarsimp simp: inj_on_def pairwise_def)
    apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)
    done
qed

lemma closedin_compact:
   "⟦compact S; closedin (subtopology euclidean S) T⟧ ⟹ compact T"
by (metis closedin_closed compact_Int_closed)

lemma closedin_compact_eq:
  fixes S :: "'a::t2_space set"
  shows
   "compact S
         ⟹ (closedin (subtopology euclidean S) T ⟷
              compact T ∧ T ⊆ S)"
by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)

lemma continuous_imp_closed_map:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  assumes "closedin (subtopology euclidean S) U"
          "continuous_on S f" "f ` S = T" "compact S"
    shows "closedin (subtopology euclidean T) (f ` U)"
  by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)

lemma continuous_imp_quotient_map:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  assumes "continuous_on S f" "f ` S = T" "compact S" "U ⊆ T"
    shows "openin (subtopology euclidean S) (S ∩ f -` U) ⟷
           openin (subtopology euclidean T) U"
  by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)


lemma open_map_restrict:
  assumes opeU: "openin (subtopology euclidean (S ∩ f -` T')) U"
    and oo: "⋀U. openin (subtopology euclidean S) U ⟹ openin (subtopology euclidean T) (f ` U)"
    and "T' ⊆ T"
  shows "openin (subtopology euclidean T') (f ` U)"
proof -
  obtain V where "open V" "U = S ∩ f -` T' ∩ V"
    using opeU by (auto simp: openin_open)
  with oo [of "S ∩ V"] ‹T' ⊆ T› show ?thesis
    by (fastforce simp add: openin_open)
qed

lemma closed_map_restrict:
  assumes cloU: "closedin (subtopology euclidean (S ∩ f -` T')) U"
    and cc: "⋀U. closedin (subtopology euclidean S) U ⟹ closedin (subtopology euclidean T) (f ` U)"
    and "T' ⊆ T"
  shows "closedin (subtopology euclidean T') (f ` U)"
proof -
  obtain V where "closed V" "U = S ∩ f -` T' ∩ V"
    using cloU by (auto simp: closedin_closed)
  with cc [of "S ∩ V"] ‹T' ⊆ T› show ?thesis
    by (fastforce simp add: closedin_closed)
qed

lemma connected_monotone_quotient_preimage:
  assumes "connected T"
      and contf: "continuous_on S f" and fim: "f ` S = T"
      and opT: "⋀U. U ⊆ T
                 ⟹ openin (subtopology euclidean S) (S ∩ f -` U) ⟷
                     openin (subtopology euclidean T) U"
      and connT: "⋀y. y ∈ T ⟹ connected (S ∩ f -` {y})"
    shows "connected S"
proof (rule connectedI)
  fix U V
  assume "open U" and "open V" and "U ∩ S ≠ {}" and "V ∩ S ≠ {}"
    and "U ∩ V ∩ S = {}" and "S ⊆ U ∪ V"
  moreover
  have disjoint: "f ` (S ∩ U) ∩ f ` (S ∩ V) = {}"
  proof -
    have False if "y ∈ f ` (S ∩ U) ∩ f ` (S ∩ V)" for y
    proof -
      have "y ∈ T"
        using fim that by blast
      show ?thesis
        using connectedD [OF connT [OF ‹y ∈ T›] ‹open U› ‹open V›]
              ‹S ⊆ U ∪ V› ‹U ∩ V ∩ S = {}› that by fastforce
    qed
    then show ?thesis by blast
  qed
  ultimately have UU: "(S ∩ f -` f ` (S ∩ U)) = S ∩ U" and VV: "(S ∩ f -` f ` (S ∩ V)) = S ∩ V"
    by auto
  have opeU: "openin (subtopology euclidean T) (f ` (S ∩ U))"
    by (metis UU ‹open U› fim image_Int_subset le_inf_iff opT openin_open_Int)
  have opeV: "openin (subtopology euclidean T) (f ` (S ∩ V))"
    by (metis opT fim VV ‹open V› openin_open_Int image_Int_subset inf.bounded_iff)
  have "T ⊆ f ` (S ∩ U) ∪ f ` (S ∩ V)"
    using ‹S ⊆ U ∪ V› fim by auto
  then show False
    using ‹connected T› disjoint opeU opeV ‹U ∩ S ≠ {}› ‹V ∩ S ≠ {}›
    by (auto simp: connected_openin)
qed

lemma connected_open_monotone_preimage:
  assumes contf: "continuous_on S f" and fim: "f ` S = T"
    and ST: "⋀C. openin (subtopology euclidean S) C ⟹ openin (subtopology euclidean T) (f ` C)"
    and connT: "⋀y. y ∈ T ⟹ connected (S ∩ f -` {y})"
    and "connected C" "C ⊆ T"
  shows "connected (S ∩ f -` C)"
proof -
  have contf': "continuous_on (S ∩ f -` C) f"
    by (meson contf continuous_on_subset inf_le1)
  have eqC: "f ` (S ∩ f -` C) = C"
    using ‹C ⊆ T› fim by blast
  show ?thesis
  proof (rule connected_monotone_quotient_preimage [OF ‹connected C› contf' eqC])
    show "connected (S ∩ f -` C ∩ f -` {y})" if "y ∈ C" for y
    proof -
      have "S ∩ f -` C ∩ f -` {y} = S ∩ f -` {y}"
        using that by blast
      moreover have "connected (S ∩ f -` {y})"
        using ‹C ⊆ T› connT that by blast
      ultimately show ?thesis
        by metis
    qed
    have "⋀U. openin (subtopology euclidean (S ∩ f -` C)) U
               ⟹ openin (subtopology euclidean C) (f ` U)"
      using open_map_restrict [OF _ ST ‹C ⊆ T›] by metis
    then show "⋀D. D ⊆ C
          ⟹ openin (subtopology euclidean (S ∩ f -` C)) (S ∩ f -` C ∩ f -` D) =
              openin (subtopology euclidean C) D"
      using open_map_imp_quotient_map [of "(S ∩ f -` C)" f] contf' by (simp add: eqC)
  qed
qed


lemma connected_closed_monotone_preimage:
  assumes contf: "continuous_on S f" and fim: "f ` S = T"
    and ST: "⋀C. closedin (subtopology euclidean S) C ⟹ closedin (subtopology euclidean T) (f ` C)"
    and connT: "⋀y. y ∈ T ⟹ connected (S ∩ f -` {y})"
    and "connected C" "C ⊆ T"
  shows "connected (S ∩ f -` C)"
proof -
  have contf': "continuous_on (S ∩ f -` C) f"
    by (meson contf continuous_on_subset inf_le1)
  have eqC: "f ` (S ∩ f -` C) = C"
    using ‹C ⊆ T› fim by blast
  show ?thesis
  proof (rule connected_monotone_quotient_preimage [OF ‹connected C› contf' eqC])
    show "connected (S ∩ f -` C ∩ f -` {y})" if "y ∈ C" for y
    proof -
      have "S ∩ f -` C ∩ f -` {y} = S ∩ f -` {y}"
        using that by blast
      moreover have "connected (S ∩ f -` {y})"
        using ‹C ⊆ T› connT that by blast
      ultimately show ?thesis
        by metis
    qed
    have "⋀U. closedin (subtopology euclidean (S ∩ f -` C)) U
               ⟹ closedin (subtopology euclidean C) (f ` U)"
      using closed_map_restrict [OF _ ST ‹C ⊆ T›] by metis
    then show "⋀D. D ⊆ C
          ⟹ openin (subtopology euclidean (S ∩ f -` C)) (S ∩ f -` C ∩ f -` D) =
              openin (subtopology euclidean C) D"
      using closed_map_imp_quotient_map [of "(S ∩ f -` C)" f] contf' by (simp add: eqC)
  qed
qed



subsection‹A couple of lemmas about components (see Newman IV, 3.3 and 3.4)›


lemma connected_Un_clopen_in_complement:
  fixes S U :: "'a::metric_space set"
  assumes "connected S" "connected U" "S ⊆ U" 
      and opeT: "openin (subtopology euclidean (U - S)) T" 
      and cloT: "closedin (subtopology euclidean (U - S)) T"
    shows "connected (S ∪ T)"
proof -
  have *: "⟦⋀x y. P x y ⟷ P y x; ⋀x y. P x y ⟹ S ⊆ x ∨ S ⊆ y;
            ⋀x y. ⟦P x y; S ⊆ x⟧ ⟹ False⟧ ⟹ ~(∃x y. (P x y))" for P
    by metis
  show ?thesis
    unfolding connected_closedin_eq
  proof (rule *)
    fix H1 H2
    assume H: "closedin (subtopology euclidean (S ∪ T)) H1 ∧ 
               closedin (subtopology euclidean (S ∪ T)) H2 ∧
               H1 ∪ H2 = S ∪ T ∧ H1 ∩ H2 = {} ∧ H1 ≠ {} ∧ H2 ≠ {}"
    then have clo: "closedin (subtopology euclidean S) (S ∩ H1)"
                   "closedin (subtopology euclidean S) (S ∩ H2)"
      by (metis Un_upper1 closedin_closed_subset inf_commute)+
    have Seq: "S ∩ (H1 ∪ H2) = S"
      by (simp add: H)
    have "S ∩ ((S ∪ T) ∩ H1) ∪ S ∩ ((S ∪ T) ∩ H2) = S"
      using Seq by auto
    moreover have "H1 ∩ (S ∩ ((S ∪ T) ∩ H2)) = {}"
      using H by blast
    ultimately have "S ∩ H1 = {} ∨ S ∩ H2 = {}"
      by (metis (no_types) H Int_assoc ‹S ∩ (H1 ∪ H2) = S› ‹connected S›
          clo Seq connected_closedin inf_bot_right inf_le1)
    then show "S ⊆ H1 ∨ S ⊆ H2"
      using H ‹connected S› unfolding connected_closedin by blast
  next
    fix H1 H2
    assume H: "closedin (subtopology euclidean (S ∪ T)) H1 ∧
               closedin (subtopology euclidean (S ∪ T)) H2 ∧
               H1 ∪ H2 = S ∪ T ∧ H1 ∩ H2 = {} ∧ H1 ≠ {} ∧ H2 ≠ {}" 
       and "S ⊆ H1"
    then have H2T: "H2 ⊆ T"
      by auto
    have "T ⊆ U"
      using Diff_iff opeT openin_imp_subset by auto
    with ‹S ⊆ U› have Ueq: "U = (U - S) ∪ (S ∪ T)" 
      by auto
    have "openin (subtopology euclidean ((U - S) ∪ (S ∪ T))) H2"
    proof (rule openin_subtopology_Un)
      show "openin (subtopology euclidean (S ∪ T)) H2"
        using ‹H2 ⊆ T› apply (auto simp: openin_closedin_eq)
        by (metis Diff_Diff_Int Diff_disjoint Diff_partition Diff_subset H Int_absorb1 Un_Diff)
      then show "openin (subtopology euclidean (U - S)) H2"
        by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)
    qed
    moreover have "closedin (subtopology euclidean ((U - S) ∪ (S ∪ T))) H2"
    proof (rule closedin_subtopology_Un)
      show "closedin (subtopology euclidean (U - S)) H2"
        using H H2T cloT closedin_subset_trans 
        by (blast intro: closedin_subtopology_Un closedin_trans)
    qed (simp add: H)
    ultimately
    have H2: "H2 = {} ∨ H2 = U"
      using Ueq ‹connected U› unfolding connected_clopen by metis   
    then have "H2 ⊆ S"
      by (metis Diff_partition H Un_Diff_cancel Un_subset_iff ‹H2 ⊆ T› assms(3) inf.orderE opeT openin_imp_subset)
    moreover have "T ⊆ H2 - S"
      by (metis (no_types) H2 H opeT openin_closedin_eq topspace_euclidean_subtopology)
    ultimately show False
      using H ‹S ⊆ H1› by blast
  qed blast
qed


proposition component_diff_connected:
  fixes S :: "'a::metric_space set"
  assumes "connected S" "connected U" "S ⊆ U" and C: "C ∈ components (U - S)"
  shows "connected(U - C)"
  using ‹connected S› unfolding connected_closedin_eq not_ex de_Morgan_conj
proof clarify
  fix H3 H4 
  assume clo3: "closedin (subtopology euclidean (U - C)) H3" 
    and clo4: "closedin (subtopology euclidean (U - C)) H4" 
    and "H3 ∪ H4 = U - C" and "H3 ∩ H4 = {}" and "H3 ≠ {}" and "H4 ≠ {}"
    and * [rule_format]:
    "∀H1 H2. ¬ closedin (subtopology euclidean S) H1 ∨
                      ¬ closedin (subtopology euclidean S) H2 ∨
                      H1 ∪ H2 ≠ S ∨ H1 ∩ H2 ≠ {} ∨ ¬ H1 ≠ {} ∨ ¬ H2 ≠ {}"
  then have "H3 ⊆ U-C" and ope3: "openin (subtopology euclidean (U - C)) (U - C - H3)"
    and "H4 ⊆ U-C" and ope4: "openin (subtopology euclidean (U - C)) (U - C - H4)"
    by (auto simp: closedin_def)
  have "C ≠ {}" "C ⊆ U-S" "connected C"
    using C in_components_nonempty in_components_subset in_components_maximal by blast+
  have cCH3: "connected (C ∪ H3)"
  proof (rule connected_Un_clopen_in_complement [OF ‹connected C› ‹connected U› _ _ clo3])
    show "openin (subtopology euclidean (U - C)) H3"
      apply (simp add: openin_closedin_eq ‹H3 ⊆ U - C›)
      apply (simp add: closedin_subtopology)
      by (metis Diff_cancel Diff_triv Un_Diff clo4 ‹H3 ∩ H4 = {}› ‹H3 ∪ H4 = U - C› closedin_closed inf_commute sup_bot.left_neutral)
  qed (use clo3 ‹C ⊆ U - S› in auto)
  have cCH4: "connected (C ∪ H4)"
  proof (rule connected_Un_clopen_in_complement [OF ‹connected C› ‹connected U› _ _ clo4])
    show "openin (subtopology euclidean (U - C)) H4"
      apply (simp add: openin_closedin_eq ‹H4 ⊆ U - C›)
      apply (simp add: closedin_subtopology)
      by (metis Diff_cancel Int_commute Un_Diff Un_Diff_Int ‹H3 ∩ H4 = {}› ‹H3 ∪ H4 = U - C› clo3 closedin_closed)
  qed (use clo4 ‹C ⊆ U - S› in auto)
  have "closedin (subtopology euclidean S) (S ∩ H3)" "closedin (subtopology euclidean S) (S ∩ H4)"
    using clo3 clo4 ‹S ⊆ U› ‹C ⊆ U - S› by (auto simp: closedin_closed)
  moreover have "S ∩ H3 ≠ {}"      
    using components_maximal [OF C cCH3] ‹C ≠ {}› ‹C ⊆ U - S› ‹H3 ≠ {}› ‹H3 ⊆ U - C› by auto
  moreover have "S ∩ H4 ≠ {}"
    using components_maximal [OF C cCH4] ‹C ≠ {}› ‹C ⊆ U - S› ‹H4 ≠ {}› ‹H4 ⊆ U - C› by auto
  ultimately show False
    using * [of "S ∩ H3" "S ∩ H4"] ‹H3 ∩ H4 = {}› ‹C ⊆ U - S› ‹H3 ∪ H4 = U - C› ‹S ⊆ U› 
    by auto
qed

subsection%unimportant‹ Finite intersection property›

text‹Also developed in HOL's toplogical spaces theory, but the Heine-Borel type class isn't available there.›

lemma closed_imp_fip:
  fixes S :: "'a::heine_borel set"
  assumes "closed S"
      and T: "T ∈ ℱ" "bounded T"
      and clof: "⋀T. T ∈ ℱ ⟹ closed T"
      and none: "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ S ∩ ⋂ℱ' ≠ {}"
    shows "S ∩ ⋂ℱ ≠ {}"
proof -
  have "compact (S ∩ T)"
    using ‹closed S› clof compact_eq_bounded_closed T by blast
  then have "(S ∩ T) ∩ ⋂ℱ ≠ {}"
    apply (rule compact_imp_fip)
     apply (simp add: clof)
    by (metis Int_assoc complete_lattice_class.Inf_insert finite_insert insert_subset none ‹T ∈ ℱ›)
  then show ?thesis by blast
qed

lemma closed_imp_fip_compact:
  fixes S :: "'a::heine_borel set"
  shows
   "⟦closed S; ⋀T. T ∈ ℱ ⟹ compact T;
     ⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ S ∩ ⋂ℱ' ≠ {}⟧
        ⟹ S ∩ ⋂ℱ ≠ {}"
by (metis Inf_greatest closed_imp_fip compact_eq_bounded_closed empty_subsetI finite.emptyI inf.orderE)

lemma closed_fip_heine_borel:
  fixes  :: "'a::heine_borel set set"
  assumes "closed S" "T ∈ ℱ" "bounded T"
      and "⋀T. T ∈ ℱ ⟹ closed T"
      and "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ ⋂ℱ' ≠ {}"
    shows "⋂ℱ ≠ {}"
proof -
  have "UNIV ∩ ⋂ℱ ≠ {}"
    using assms closed_imp_fip [OF closed_UNIV] by auto
  then show ?thesis by simp
qed

lemma compact_fip_heine_borel:
  fixes  :: "'a::heine_borel set set"
  assumes clof: "⋀T. T ∈ ℱ ⟹ compact T"
      and none: "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ ⋂ℱ' ≠ {}"
    shows "⋂ℱ ≠ {}"
by (metis InterI all_not_in_conv clof closed_fip_heine_borel compact_eq_bounded_closed none)

lemma compact_sequence_with_limit:
  fixes f :: "nat ⇒ 'a::heine_borel"
  shows "(f ⤏ l) sequentially ⟹ compact (insert l (range f))"
apply (simp add: compact_eq_bounded_closed, auto)
apply (simp add: convergent_imp_bounded)
by (simp add: closed_limpt islimpt_insert sequence_unique_limpt)


subsection%unimportant‹Componentwise limits and continuity›

text‹But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}›
lemma Euclidean_dist_upper: "i ∈ Basis ⟹ dist (x ∙ i) (y ∙ i) ≤ dist x y"
  by (metis (no_types) member_le_L2_set euclidean_dist_l2 finite_Basis)

text‹But is the premise @{term ‹i ∈ Basis›} really necessary?›
lemma open_preimage_inner:
  assumes "open S" "i ∈ Basis"
    shows "open {x. x ∙ i ∈ S}"
proof (rule openI, simp)
  fix x
  assume x: "x ∙ i ∈ S"
  with assms obtain e where "0 < e" and e: "ball (x ∙ i) e ⊆ S"
    by (auto simp: open_contains_ball_eq)
  have "∃e>0. ball (y ∙ i) e ⊆ S" if dxy: "dist x y < e / 2" for y
  proof (intro exI conjI)
    have "dist (x ∙ i) (y ∙ i) < e / 2"
      by (meson ‹i ∈ Basis› dual_order.trans Euclidean_dist_upper not_le that)
    then have "dist (x ∙ i) z < e" if "dist (y ∙ i) z < e / 2" for z
      by (metis dist_commute dist_triangle_half_l that)
    then have "ball (y ∙ i) (e / 2) ⊆ ball (x ∙ i) e"
      using mem_ball by blast
      with e show "ball (y ∙ i) (e / 2) ⊆ S"
        by (metis order_trans)
  qed (simp add: ‹0 < e›)
  then show "∃e>0. ball x e ⊆ {s. s ∙ i ∈ S}"
    by (metis (no_types, lifting) ‹0 < e› ‹open S› half_gt_zero_iff mem_Collect_eq mem_ball open_contains_ball_eq subsetI)
qed

proposition tendsto_componentwise_iff:
  fixes f :: "_ ⇒ 'b::euclidean_space"
  shows "(f ⤏ l) F ⟷ (∀i ∈ Basis. ((λx. (f x ∙ i)) ⤏ (l ∙ i)) F)"
         (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding tendsto_def
    apply clarify
    apply (drule_tac x="{s. s ∙ i ∈ S}" in spec)
    apply (auto simp: open_preimage_inner)
    done
next
  assume R: ?rhs
  then have "⋀e. e > 0 ⟹ ∀i∈Basis. ∀F x in F. dist (f x ∙ i) (l ∙ i) < e"
    unfolding tendsto_iff by blast
  then have R': "⋀e. e > 0 ⟹ ∀F x in F. ∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e"
      by (simp add: eventually_ball_finite_distrib [symmetric])
  show ?lhs
  unfolding tendsto_iff
  proof clarify
    fix e::real
    assume "0 < e"
    have *: "L2_set (λi. dist (f x ∙ i) (l ∙ i)) Basis < e"
             if "∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e / real DIM('b)" for x
    proof -
      have "L2_set (λi. dist (f x ∙ i) (l ∙ i)) Basis ≤ sum (λi. dist (f x ∙ i) (l ∙ i)) Basis"
        by (simp add: L2_set_le_sum)
      also have "... < DIM('b) * (e / real DIM('b))"
        apply (rule sum_bounded_above_strict)
        using that by auto
      also have "... = e"
        by (simp add: field_simps)
      finally show "L2_set (λi. dist (f x ∙ i) (l ∙ i)) Basis < e" .
    qed
    have "∀F x in F. ∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e / DIM('b)"
      apply (rule R')
      using ‹0 < e› by simp
    then show "∀F x in F. dist (f x) l < e"
      apply (rule eventually_mono)
      apply (subst euclidean_dist_l2)
      using * by blast
  qed
qed


corollary continuous_componentwise:
   "continuous F f ⟷ (∀i ∈ Basis. continuous F (λx. (f x ∙ i)))"
by (simp add: continuous_def tendsto_componentwise_iff [symmetric])

corollary continuous_on_componentwise:
  fixes S :: "'a :: t2_space set"
  shows "continuous_on S f ⟷ (∀i ∈ Basis. continuous_on S (λx. (f x ∙ i)))"
  apply (simp add: continuous_on_eq_continuous_within)
  using continuous_componentwise by blast

lemma linear_componentwise_iff:
     "(linear f') ⟷ (∀i∈Basis. linear (λx. f' x ∙ i))"
  apply (auto simp: linear_iff inner_left_distrib)
   apply (metis inner_left_distrib euclidean_eq_iff)
  by (metis euclidean_eqI inner_scaleR_left)

lemma bounded_linear_componentwise_iff:
     "(bounded_linear f') ⟷ (∀i∈Basis. bounded_linear (λx. f' x ∙ i))"
     (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    by (simp add: bounded_linear_inner_left_comp)
next
  assume ?rhs
  then have "(∀i∈Basis. ∃K. ∀x. ¦f' x ∙ i¦ ≤ norm x * K)" "linear f'"
    by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
  then obtain F where F: "⋀i x. i ∈ Basis ⟹ ¦f' x ∙ i¦ ≤ norm x * F i"
    by metis
  have "norm (f' x) ≤ norm x * sum F Basis" for x
  proof -
    have "norm (f' x) ≤ (∑i∈Basis. ¦f' x ∙ i¦)"
      by (rule norm_le_l1)
    also have "... ≤ (∑i∈Basis. norm x * F i)"
      by (metis F sum_mono)
    also have "... = norm x * sum F Basis"
      by (simp add: sum_distrib_left)
    finally show ?thesis .
  qed
  then show ?lhs
    by (force simp: bounded_linear_def bounded_linear_axioms_def ‹linear f'›)
qed

subsection%unimportant‹Pasting functions together›

subsubsection%unimportant‹on open sets›

lemma pasting_lemma:
  fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
  assumes clo: "⋀i. i ∈ I ⟹ openin (subtopology euclidean S) (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
      and g: "⋀x. x ∈ S ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
    shows "continuous_on S g"
proof (clarsimp simp: continuous_openin_preimage_eq)
  fix U :: "'b set"
  assume "open U"
  have S: "⋀i. i ∈ I ⟹ (T i) ⊆ S"
    using clo openin_imp_subset by blast
  have *: "(S ∩ g -` U) = (⋃i ∈ I. T i ∩ f i -` U)"
    using S f g by fastforce
  show "openin (subtopology euclidean S) (S ∩ g -` U)"
    apply (subst *)
    apply (rule openin_Union, clarify)
    using ‹open U› clo cont continuous_openin_preimage_gen openin_trans by blast
qed

lemma pasting_lemma_exists:
  fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
  assumes S: "S ⊆ (⋃i ∈ I. T i)"
      and clo: "⋀i. i ∈ I ⟹ openin (subtopology euclidean S) (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    obtains g where "continuous_on S g" "⋀x i. ⟦i ∈ I; x ∈ S ∩ T i⟧ ⟹ g x = f i x"
proof
  show "continuous_on S (λx. f (SOME i. i ∈ I ∧ x ∈ T i) x)"
    apply (rule pasting_lemma [OF clo cont])
     apply (blast intro: f)+
    apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
    done
next
  fix x i
  assume "i ∈ I" "x ∈ S ∩ T i"
  then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
    by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed

subsubsection%unimportant‹Likewise on closed sets, with a finiteness assumption›

lemma pasting_lemma_closed:
  fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
  assumes "finite I"
      and clo: "⋀i. i ∈ I ⟹ closedin (subtopology euclidean S) (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
      and g: "⋀x. x ∈ S ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
    shows "continuous_on S g"
proof (clarsimp simp: continuous_closedin_preimage_eq)
  fix U :: "'b set"
  assume "closed U"
  have S: "⋀i. i ∈ I ⟹ (T i) ⊆ S"
    using clo closedin_imp_subset by blast
  have *: "(S ∩ g -` U) = (⋃i ∈ I. T i ∩ f i -` U)"
    using S f g by fastforce
  show "closedin (subtopology euclidean S) (S ∩ g -` U)"
    apply (subst *)
    apply (rule closedin_Union)
    using ‹finite I› apply simp
    apply (blast intro: ‹closed U› continuous_closedin_preimage cont clo closedin_trans)
    done
qed

lemma pasting_lemma_exists_closed:
  fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
  assumes "finite I"
      and S: "S ⊆ (⋃i ∈ I. T i)"
      and clo: "⋀i. i ∈ I ⟹ closedin (subtopology euclidean S) (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    obtains g where "continuous_on S g" "⋀x i. ⟦i ∈ I; x ∈ S ∩ T i⟧ ⟹ g x = f i x"
proof
  show "continuous_on S (λx. f (SOME i. i ∈ I ∧ x ∈ T i) x)"
    apply (rule pasting_lemma_closed [OF ‹finite I› clo cont])
     apply (blast intro: f)+
    apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
    done
next
  fix x i
  assume "i ∈ I" "x ∈ S ∩ T i"
  then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
    by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed

lemma tube_lemma:
  assumes "compact K"
  assumes "open W"
  assumes "{x0} × K ⊆ W"
  shows "∃X0. x0 ∈ X0 ∧ open X0 ∧ X0 × K ⊆ W"
proof -
  {
    fix y assume "y ∈ K"
    then have "(x0, y) ∈ W" using assms by auto
    with ‹open W›
    have "∃X0 Y. open X0 ∧ open Y ∧ x0 ∈ X0 ∧ y ∈ Y ∧ X0 × Y ⊆ W"
      by (rule open_prod_elim) blast
  }
  then obtain X0 Y where
    *: "∀y ∈ K. open (X0 y) ∧ open (Y y) ∧ x0 ∈ X0 y ∧ y ∈ Y y ∧ X0 y × Y y ⊆ W"
    by metis
  from * have "∀t∈Y ` K. open t" "K ⊆ ⋃(Y ` K)" by auto
  with ‹compact K› obtain CC where CC: "CC ⊆ Y ` K" "finite CC" "K ⊆ ⋃CC"
    by (meson compactE)
  then obtain c where c: "⋀C. C ∈ CC ⟹ c C ∈ K ∧ C = Y (c C)"
    by (force intro!: choice)
  with * CC show ?thesis
    by (force intro!: exI[where x="⋂C∈CC. X0 (c C)"]) (* SLOW *)
qed

lemma continuous_on_prod_compactE:
  fixes fx::"'a::topological_space × 'b::topological_space ⇒ 'c::metric_space"
    and e::real
  assumes cont_fx: "continuous_on (U × C) fx"
  assumes "compact C"
  assumes [intro]: "x0 ∈ U"
  notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
  assumes "e > 0"
  obtains X0 where "x0 ∈ X0" "open X0"
    "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof -
  define psi where "psi = (λ(x, t). dist (fx (x, t)) (fx (x0, t)))"
  define W0 where "W0 = {(x, t) ∈ U × C. psi (x, t) < e}"
  have W0_eq: "W0 = psi -` {..<e} ∩ U × C"
    by (auto simp: vimage_def W0_def)
  have "open {..<e}" by simp
  have "continuous_on (U × C) psi"
    by (auto intro!: continuous_intros simp: psi_def split_beta')
  from this[unfolded continuous_on_open_invariant, rule_format, OF ‹open {..<e}›]
  obtain W where W: "open W" "W ∩ U × C = W0 ∩ U × C"
    unfolding W0_eq by blast
  have "{x0} × C ⊆ W ∩ U × C"
    unfolding W
    by (auto simp: W0_def psi_def ‹0 < e›)
  then have "{x0} × C ⊆ W" by blast
  from tube_lemma[OF ‹compact C› ‹open W› this]
  obtain X0 where X0: "x0 ∈ X0" "open X0" "X0 × C ⊆ W"
    by blast

  have "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
  proof safe
    fix x assume x: "x ∈ X0" "x ∈ U"
    fix t assume t: "t ∈ C"
    have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
      by (auto simp: psi_def)
    also
    {
      have "(x, t) ∈ X0 × C"
        using t x
        by auto
      also note ‹… ⊆ W›
      finally have "(x, t) ∈ W" .
      with t x have "(x, t) ∈ W ∩ U × C"
        by blast
      also note ‹W ∩ U × C = W0 ∩ U × C›
      finally  have "psi (x, t) < e"
        by (auto simp: W0_def)
    }
    finally show "dist (fx (x, t)) (fx (x0, t)) ≤ e" by simp
  qed
  from X0(1,2) this show ?thesis ..
qed


subsection%unimportant‹Constancy of a function from a connected set into a finite, disconnected or discrete set›

text‹Still missing: versions for a set that is smaller than R, or countable.›

lemma continuous_disconnected_range_constant:
  assumes S: "connected S"
      and conf: "continuous_on S f"
      and fim: "f ` S ⊆ t"
      and cct: "⋀y. y ∈ t ⟹ connected_component_set t y = {y}"
    shows "f constant_on S"
proof (cases "S = {}")
  case True then show ?thesis
    by (simp add: constant_on_def)
next
  case False
  { fix x assume "x ∈ S"
    then have "f ` S ⊆ {f x}"
    by (metis connected_continuous_image conf connected_component_maximal fim image_subset_iff rev_image_eqI S cct)
  }
  with False show ?thesis
    unfolding constant_on_def by blast
qed

lemma discrete_subset_disconnected:
  fixes S :: "'a::topological_space set"
  fixes t :: "'b::real_normed_vector set"
  assumes conf: "continuous_on S f"
      and no: "⋀x. x ∈ S ⟹ ∃e>0. ∀y. y ∈ S ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)"
   shows "f ` S ⊆ {y. connected_component_set (f ` S) y = {y}}"
proof -
  { fix x assume x: "x ∈ S"
    then obtain e where "e>0" and ele: "⋀y. ⟦y ∈ S; f y ≠ f x⟧ ⟹ e ≤ norm (f y - f x)"
      using conf no [OF x] by auto
    then have e2: "0 ≤ e / 2"
      by simp
    have "f y = f x" if "y ∈ S" and ccs: "f y ∈ connected_component_set (f ` S) (f x)" for y
      apply (rule ccontr)
      using connected_closed [of "connected_component_set (f ` S) (f x)"] ‹e>0›
      apply (simp add: del: ex_simps)
      apply (drule spec [where x="cball (f x) (e / 2)"])
      apply (drule spec [where x="- ball(f x) e"])
      apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)
        apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
       using centre_in_cball connected_component_refl_eq e2 x apply blast
      using ccs
      apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF ‹y ∈ S›])
      done
    moreover have "connected_component_set (f ` S) (f x) ⊆ f ` S"
      by (auto simp: connected_component_in)
    ultimately have "connected_component_set (f ` S) (f x) = {f x}"
      by (auto simp: x)
  }
  with assms show ?thesis
    by blast
qed

lemma finite_implies_discrete:
  fixes S :: "'a::topological_space set"
  assumes "finite (f ` S)"
  shows "(∀x ∈ S. ∃e>0. ∀y. y ∈ S ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x))"
proof -
  have "∃e>0. ∀y. y ∈ S ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)" if "x ∈ S" for x
  proof (cases "f ` S - {f x} = {}")
    case True
    with zero_less_numeral show ?thesis
      by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
  next
    case False
    then obtain z where z: "z ∈ S" "f z ≠ f x"
      by blast
    have finn: "finite {norm (z - f x) |z. z ∈ f ` S - {f x}}"
      using assms by simp
    then have *: "0 < Inf{norm(z - f x) | z. z ∈ f ` S - {f x}}"
      apply (rule finite_imp_less_Inf)
      using z apply force+
      done
    show ?thesis
      by (force intro!: * cInf_le_finite [OF finn])
  qed
  with assms show ?thesis
    by blast
qed

text‹This proof requires the existence of two separate values of the range type.›
lemma finite_range_constant_imp_connected:
  assumes "⋀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
              ⟦continuous_on S f; finite(f ` S)⟧ ⟹ f constant_on S"
    shows "connected S"
proof -
  { fix t u
    assume clt: "closedin (subtopology euclidean S) t"
       and clu: "closedin (subtopology euclidean S) u"
       and tue: "t ∩ u = {}" and tus: "t ∪ u = S"
    have conif: "continuous_on S (λx. if x ∈ t then 0 else 1)"
      apply (subst tus [symmetric])
      apply (rule continuous_on_cases_local)
      using clt clu tue
      apply (auto simp: tus continuous_on_const)
      done
    have fi: "finite ((λx. if x ∈ t then 0 else 1) ` S)"
      by (rule finite_subset [of _ "{0,1}"]) auto
    have "t = {} ∨ u = {}"
      using assms [OF conif fi] tus [symmetric]
      by (auto simp: Ball_def constant_on_def) (metis IntI empty_iff one_neq_zero tue)
  }
  then show ?thesis
    by (simp add: connected_closedin_eq)
qed

lemma continuous_disconnected_range_constant_eq:
      "(connected S ⟷
           (∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
            ∀t. continuous_on S f ∧ f ` S ⊆ t ∧ (∀y ∈ t. connected_component_set t y = {y})
            ⟶ f constant_on S))" (is ?thesis1)
  and continuous_discrete_range_constant_eq:
      "(connected S ⟷
         (∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
          continuous_on S f ∧
          (∀x ∈ S. ∃e. 0 < e ∧ (∀y. y ∈ S ∧ (f y ≠ f x) ⟶ e ≤ norm(f y - f x)))
          ⟶ f constant_on S))" (is ?thesis2)
  and continuous_finite_range_constant_eq:
      "(connected S ⟷
         (∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
          continuous_on S f ∧ finite (f ` S)
          ⟶ f constant_on S))" (is ?thesis3)
proof -
  have *: "⋀s t u v. ⟦s ⟹ t; t ⟹ u; u ⟹ v; v ⟹ s⟧
    ⟹ (s ⟷ t) ∧ (s ⟷ u) ∧ (s ⟷ v)"
    by blast
  have "?thesis1 ∧ ?thesis2 ∧ ?thesis3"
    apply (rule *)
    using continuous_disconnected_range_constant apply metis
    apply clarify
    apply (frule discrete_subset_disconnected; blast)
    apply (blast dest: finite_implies_discrete)
    apply (blast intro!: finite_range_constant_imp_connected)
    done
  then show ?thesis1 ?thesis2 ?thesis3
    by blast+
qed

lemma continuous_discrete_range_constant:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_algebra_1"
  assumes S: "connected S"
      and "continuous_on S f"
      and "⋀x. x ∈ S ⟹ ∃e>0. ∀y. y ∈ S ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)"
    shows "f constant_on S"
  using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms by blast

lemma continuous_finite_range_constant:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_algebra_1"
  assumes "connected S"
      and "continuous_on S f"
      and "finite (f ` S)"
    shows "f constant_on S"
  using assms continuous_finite_range_constant_eq  by blast



subsection%unimportant ‹Continuous Extension›

definition clamp :: "'a::euclidean_space ⇒ 'a ⇒ 'a ⇒ 'a" where
  "clamp a b x = (if (∀i∈Basis. a ∙ i ≤ b ∙ i)
    then (∑i∈Basis. (if x∙i < a∙i then a∙i else if x∙i ≤ b∙i then x∙i else b∙i) *R i)
    else a)"

lemma clamp_in_interval[simp]:
  assumes "⋀i. i ∈ Basis ⟹ a ∙ i ≤ b ∙ i"
  shows "clamp a b x ∈ cbox a b"
  unfolding clamp_def
  using box_ne_empty(1)[of a b] assms by (auto simp: cbox_def)

lemma clamp_cancel_cbox[simp]:
  fixes x a b :: "'a::euclidean_space"
  assumes x: "x ∈ cbox a b"
  shows "clamp a b x = x"
  using assms
  by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a])

lemma clamp_empty_interval:
  assumes "i ∈ Basis" "a ∙ i > b ∙ i"
  shows "clamp a b = (λ_. a)"
  using assms
  by (force simp: clamp_def[abs_def] split: if_splits intro!: ext)

lemma dist_clamps_le_dist_args:
  fixes x :: "'a::euclidean_space"
  shows "dist (clamp a b y) (clamp a b x) ≤ dist y x"
proof cases
  assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
  then have "(∑i∈Basis. (dist (clamp a b y ∙ i) (clamp a b x ∙ i))2) ≤
    (∑i∈Basis. (dist (y ∙ i) (x ∙ i))2)"
    by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
  then show ?thesis
    by (auto intro: real_sqrt_le_mono
      simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] L2_set_def)
qed (auto simp: clamp_def)

lemma clamp_continuous_at:
  fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
    and x :: 'a
  assumes f_cont: "continuous_on (cbox a b) f"
  shows "continuous (at x) (λx. f (clamp a b x))"
proof cases
  assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
  show ?thesis
    unfolding continuous_at_eps_delta
  proof safe
    fix x :: 'a
    fix e :: real
    assume "e > 0"
    moreover have "clamp a b x ∈ cbox a b"
      by (simp add: clamp_in_interval le)
    moreover note f_cont[simplified continuous_on_iff]
    ultimately
    obtain d where d: "0 < d"
      "⋀x'. x' ∈ cbox a b ⟹ dist x' (clamp a b x) < d ⟹ dist (f x') (f (clamp a b x)) < e"
      by force
    show "∃d>0. ∀x'. dist x' x < d ⟶
      dist (f (clamp a b x')) (f (clamp a b x)) < e"
      using le
      by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])
  qed
qed (auto simp: clamp_empty_interval)

lemma clamp_continuous_on:
  fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
  assumes f_cont: "continuous_on (cbox a b) f"
  shows "continuous_on S (λx. f (clamp a b x))"
  using assms
  by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)

lemma clamp_bounded:
  fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
  assumes bounded: "bounded (f ` (cbox a b))"
  shows "bounded (range (λx. f (clamp a b x)))"
proof cases
  assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
  from bounded obtain c where f_bound: "∀x∈f ` cbox a b. dist undefined x ≤ c"
    by (auto simp: bounded_any_center[where a=undefined])
  then show ?thesis
    by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]
        simp: bounded_any_center[where a=undefined])
qed (auto simp: clamp_empty_interval image_def)


definition ext_cont :: "('a::euclidean_space ⇒ 'b::metric_space) ⇒ 'a ⇒ 'a ⇒ 'a ⇒ 'b"
  where "ext_cont f a b = (λx. f (clamp a b x))"

lemma ext_cont_cancel_cbox[simp]:
  fixes x a b :: "'a::euclidean_space"
  assumes x: "x ∈ cbox a b"
  shows "ext_cont f a b x = f x"
  using assms
  unfolding ext_cont_def
  by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])

lemma continuous_on_ext_cont[continuous_intros]:
  "continuous_on (cbox a b) f ⟹ continuous_on S (ext_cont f a b)"
  by (auto intro!: clamp_continuous_on simp: ext_cont_def)

end