Theory Topology_Euclidean_Space

theory Topology_Euclidean_Space
imports Elementary_Normed_Spaces Linear_Algebra Norm_Arith
(*  Author:     L C Paulson, University of Cambridge
    Author:     Amine Chaieb, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Brian Huffman, Portland State University
*)

chapter ‹Vector Analysis›

theory Topology_Euclidean_Space
  imports
    Elementary_Normed_Spaces
    Linear_Algebra
    Norm_Arith
begin

section ‹Elementary Topology in Euclidean Space›

lemma euclidean_dist_l2:
  fixes x y :: "'a :: euclidean_space"
  shows "dist x y = L2_set (λi. dist (x ∙ i) (y ∙ i)) Basis"
  unfolding dist_norm norm_eq_sqrt_inner L2_set_def
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

lemma norm_nth_le: "norm (x ∙ i) ≤ norm x" if "i ∈ Basis"
proof -
  have "(x ∙ i)2 = (∑i∈{i}. (x ∙ i)2)"
    by simp
  also have "… ≤ (∑i∈Basis. (x ∙ i)2)"
    by (intro sum_mono2) (auto simp: that)
  finally show ?thesis
    unfolding norm_conv_dist euclidean_dist_l2[of x] L2_set_def
    by (auto intro!: real_le_rsqrt)
qed

subsection ‹Continuity of the representation WRT an orthogonal basis›

lemma orthogonal_Basis: "pairwise orthogonal Basis"
  by (simp add: inner_not_same_Basis orthogonal_def pairwise_def)

lemma representation_bound:
  fixes B :: "'N::real_inner set"
  assumes "finite B" "independent B" "b ∈ B" and orth: "pairwise orthogonal B"
  obtains m where "m > 0" "⋀x. x ∈ span B ⟹ ¦representation B x b¦ ≤ m * norm x"
proof 
  fix x
  assume x: "x ∈ span B"
  have "b ≠ 0"
    using ‹independent B› ‹b ∈ B› dependent_zero by blast
  have [simp]: "b ∙ b' = (if b' = b then (norm b)2 else 0)"
    if "b ∈ B" "b' ∈ B" for b b'
    using orth by (simp add: orthogonal_def pairwise_def norm_eq_sqrt_inner that)
  have "norm x = norm (∑b∈B. representation B x b *R b)"
    using real_vector.sum_representation_eq [OF ‹independent B› x ‹finite B› order_refl]
    by simp
  also have "… = sqrt ((∑b∈B. representation B x b *R b) ∙ (∑b∈B. representation B x b *R b))"
    by (simp add: norm_eq_sqrt_inner)
  also have "… = sqrt (∑b∈B. (representation B x b *R b) ∙ (representation B x b *R b))"
    using ‹finite B›
    by (simp add: inner_sum_left inner_sum_right if_distrib [of "λx. _ * x"] cong: if_cong sum.cong_simp)
  also have "… = sqrt (∑b∈B. (norm (representation B x b *R b))2)"
    by (simp add: mult.commute mult.left_commute power2_eq_square)
  also have "… = sqrt (∑b∈B. (representation B x b)2 * (norm b)2)"
    by (simp add: norm_mult power_mult_distrib)
  finally have "norm x = sqrt (∑b∈B. (representation B x b)2 * (norm b)2)" .
  moreover
  have "sqrt ((representation B x b)2 * (norm b)2) ≤ sqrt (∑b∈B. (representation B x b)2 * (norm b)2)"
    using ‹b ∈ B› ‹finite B› by (auto intro: member_le_sum)
  then have "¦representation B x b¦ ≤ (1 / norm b) * sqrt (∑b∈B. (representation B x b)2 * (norm b)2)"
    using ‹b ≠ 0› by (simp add: divide_simps real_sqrt_mult del: real_sqrt_le_iff)
  ultimately show "¦representation B x b¦ ≤ (1 / norm b) * norm x"
    by simp
next
  show "0 < 1 / norm b"
    using ‹independent B› ‹b ∈ B› dependent_zero by auto
qed 

lemma continuous_on_representation:
  fixes B :: "'N::euclidean_space set"
  assumes "finite B" "independent B" "b ∈ B" "pairwise orthogonal B" 
  shows "continuous_on (span B) (λx. representation B x b)"
proof
  show "∃d>0. ∀x'∈span B. dist x' x < d ⟶ dist (representation B x' b) (representation B x b) ≤ e"
    if "e > 0" "x ∈ span B" for x e
  proof -
    obtain m where "m > 0" and m: "⋀x. x ∈ span B ⟹ ¦representation B x b¦ ≤ m * norm x"
      using assms representation_bound by blast
    show ?thesis
      unfolding dist_norm
    proof (intro exI conjI ballI impI)
      show "e/m > 0"
        by (simp add: ‹e > 0› ‹m > 0›)
      show "norm (representation B x' b - representation B x b) ≤ e"
        if x': "x' ∈ span B" and less: "norm (x'-x) < e/m" for x' 
      proof -
        have "¦representation B (x'-x) b¦ ≤ m * norm (x'-x)"
          using m [of "x'-x"] ‹x ∈ span B› span_diff x' by blast
        also have "… < e"
          by (metis ‹m > 0› less mult.commute pos_less_divide_eq)
        finally have "¦representation B (x'-x) b¦ ≤ e" by simp
        then show ?thesis
          by (simp add: ‹x ∈ span B› ‹independent B› representation_diff x')
      qed
    qed
  qed
qed

subsection✐‹tag unimportant›‹Balls in Euclidean Space›

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 subsetD)
    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

lemma dim_cball:
  assumes "e > 0"
  shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
proof -
  {
    fix x :: "'n::euclidean_space"
    define y where "y = (e / norm x) *R x"
    then have "y ∈ cball 0 e"
      using assms by auto
    moreover have *: "x = (norm x / e) *R y"
      using y_def assms by simp
    moreover from * have "x = (norm x/e) *R y"
      by auto
    ultimately have "x ∈ span (cball 0 e)"
      using span_scale[of y "cball 0 e" "norm x/e"]
        span_superset[of "cball 0 e"]
      by (simp add: span_base)
  }
  then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
    by auto
  then show ?thesis
    using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
qed


subsection ‹Boxes›

abbreviation One :: "'a::euclidean_space"
  where "One ≡ ∑Basis"

lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
proof -
  have "dependent (Basis :: 'a set)"
    apply (simp add: dependent_finite)
    apply (rule_tac x="λi. 1" in exI)
    using SOME_Basis apply (auto simp: assms)
    done
  with independent_Basis show False by force
qed

corollary One_neq_0[iff]: "One ≠ 0"
  by (metis One_non_0)

corollary Zero_neq_One[iff]: "0 ≠ One"
  by (metis One_non_0)

definition✐‹tag important› (in euclidean_space) eucl_less (infix "<e" 50)
  where "eucl_less a b ⟷ (∀i∈Basis. a ∙ i < b ∙ i)"

definition✐‹tag important› box_eucl_less: "box a b = {x. a <e x ∧ x <e b}"
definition✐‹tag important› "cbox a b = {x. ∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i}"

lemma box_def: "box a b = {x. ∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i}"
  and in_box_eucl_less: "x ∈ box a b ⟷ a <e x ∧ x <e b"
  and mem_box: "x ∈ box a b ⟷ (∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i)"
    "x ∈ cbox a b ⟷ (∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i)"
  by (auto simp: box_eucl_less eucl_less_def cbox_def)

lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b × cbox c d"
  by (force simp: cbox_def Basis_prod_def)

lemma cbox_Pair_iff [iff]: "(x, y) ∈ cbox (a, c) (b, d) ⟷ x ∈ cbox a b ∧ y ∈ cbox c d"
  by (force simp: cbox_Pair_eq)

lemma cbox_Complex_eq: "cbox (Complex a c) (Complex b d) = (λ(x,y). Complex x y) ` (cbox a b × cbox c d)"
  apply (auto simp: cbox_def Basis_complex_def)
  apply (rule_tac x = "(Re x, Im x)" in image_eqI)
  using complex_eq by auto

lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} ⟷ cbox a b = {} ∨ cbox c d = {}"
  by (force simp: cbox_Pair_eq)

lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
  by auto

lemma mem_box_real[simp]:
  "(x::real) ∈ box a b ⟷ a < x ∧ x < b"
  "(x::real) ∈ cbox a b ⟷ a ≤ x ∧ x ≤ b"
  by (auto simp: mem_box)

lemma box_real[simp]:
  fixes a b:: real
  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
  by auto

lemma box_Int_box:
  fixes a :: "'a::euclidean_space"
  shows "box a b ∩ box c d =
    box (∑i∈Basis. max (a∙i) (c∙i) *R i) (∑i∈Basis. min (b∙i) (d∙i) *R i)"
  unfolding set_eq_iff and Int_iff and mem_box by auto

lemma rational_boxes:
  fixes x :: "'a::euclidean_space"
  assumes "e > 0"
  shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e"
proof -
  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
  then have e: "e' > 0"
    using assms by (auto simp: DIM_positive)
  have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i")
  proof
    fix i
    from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e
    show "?th i" by auto
  qed
  from choice[OF this] obtain a where
    a: "∀xa. a xa ∈ ℚ ∧ a xa < x ∙ xa ∧ x ∙ xa - a xa < e'" ..
  have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i")
  proof
    fix i
    from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e
    show "?th i" by auto
  qed
  from choice[OF this] obtain b where
    b: "∀xa. b xa ∈ ℚ ∧ x ∙ xa < b xa ∧ b xa - x ∙ xa < e'" ..
  let ?a = "∑i∈Basis. a i *R i" and ?b = "∑i∈Basis. b i *R i"
  show ?thesis
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
    fix y :: 'a
    assume *: "y ∈ box ?a ?b"
    have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))2)"
      unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
    also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"
    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
      fix i :: "'a"
      assume i: "i ∈ Basis"
      have "a i < y∙i ∧ y∙i < b i"
        using * i by (auto simp: box_def)
      moreover have "a i < x∙i" "x∙i - a i < e'"
        using a by auto
      moreover have "x∙i < b i" "b i - x∙i < e'"
        using b by auto
      ultimately have "¦x∙i - y∙i¦ < 2 * e'"
        by auto
      then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))"
        unfolding e'_def by (auto simp: dist_real_def)
      then have "(dist (x ∙ i) (y ∙ i))2 < (e/sqrt (real (DIM('a))))2"
        by (rule power_strict_mono) auto
      then show "(dist (x ∙ i) (y ∙ i))2 < e2 / real DIM('a)"
        by (simp add: power_divide)
    qed auto
    also have "… = e"
      using ‹0 < e› by simp
    finally show "y ∈ ball x e"
      by (auto simp: ball_def)
  qed (insert a b, auto simp: box_def)
qed

lemma open_UNION_box:
  fixes M :: "'a::euclidean_space set"
  assumes "open M"
  defines "a' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. fst (f i) *R i)"
  defines "b' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. snd (f i) *R i)"
  defines "I ≡ {f∈Basis →E ℚ × ℚ. box (a' f) (b' f) ⊆ M}"
  shows "M = (⋃f∈I. box (a' f) (b' f))"
proof -
  have "x ∈ (⋃f∈I. box (a' f) (b' f))" if "x ∈ M" for x
  proof -
    obtain e where e: "e > 0" "ball x e ⊆ M"
      using openE[OF ‹open M› ‹x ∈ M›] by auto
    moreover obtain a b where ab:
      "x ∈ box a b"
      "∀i ∈ Basis. a ∙ i ∈ ℚ"
      "∀i∈Basis. b ∙ i ∈ ℚ"
      "box a b ⊆ ball x e"
      using rational_boxes[OF e(1)] by metis
    ultimately show ?thesis
       by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"])
          (auto simp: euclidean_representation I_def a'_def b'_def)
  qed
  then show ?thesis by (auto simp: I_def)
qed

corollary open_countable_Union_open_box:
  fixes S :: "'a :: euclidean_space set"
  assumes "open S"
  obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = box a b" "⋃𝒟 = S"
proof -
  let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *R i)"
  let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *R i)"
  let ?I = "{f∈Basis →E ℚ × ℚ. box (?a f) (?b f) ⊆ S}"
  let ?𝒟 = "(λf. box (?a f) (?b f)) ` ?I"
  show ?thesis
  proof
    have "countable ?I"
      by (simp add: countable_PiE countable_rat)
    then show "countable ?𝒟"
      by blast
    show "⋃?𝒟 = S"
      using open_UNION_box [OF assms] by metis
  qed auto
qed

lemma rational_cboxes:
  fixes x :: "'a::euclidean_space"
  assumes "e > 0"
  shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ cbox a b ∧ cbox a b ⊆ ball x e"
proof -
  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
  then have e: "e' > 0"
    using assms by auto
  have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i")
  proof
    fix i
    from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e
    show "?th i" by auto
  qed
  from choice[OF this] obtain a where
    a: "∀u. a u ∈ ℚ ∧ a u < x ∙ u ∧ x ∙ u - a u < e'" ..
  have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i")
  proof
    fix i
    from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e
    show "?th i" by auto
  qed
  from choice[OF this] obtain b where
    b: "∀u. b u ∈ ℚ ∧ x ∙ u < b u ∧ b u - x ∙ u < e'" ..
  let ?a = "∑i∈Basis. a i *R i" and ?b = "∑i∈Basis. b i *R i"
  show ?thesis
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
    fix y :: 'a
    assume *: "y ∈ cbox ?a ?b"
    have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))2)"
      unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
    also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"
    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
      fix i :: "'a"
      assume i: "i ∈ Basis"
      have "a i ≤ y∙i ∧ y∙i ≤ b i"
        using * i by (auto simp: cbox_def)
      moreover have "a i < x∙i" "x∙i - a i < e'"
        using a by auto
      moreover have "x∙i < b i" "b i - x∙i < e'"
        using b by auto
      ultimately have "¦x∙i - y∙i¦ < 2 * e'"
        by auto
      then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))"
        unfolding e'_def by (auto simp: dist_real_def)
      then have "(dist (x ∙ i) (y ∙ i))2 < (e/sqrt (real (DIM('a))))2"
        by (rule power_strict_mono) auto
      then show "(dist (x ∙ i) (y ∙ i))2 < e2 / real DIM('a)"
        by (simp add: power_divide)
    qed auto
    also have "… = e"
      using ‹0 < e› by simp
    finally show "y ∈ ball x e"
      by (auto simp: ball_def)
  next
    show "x ∈ cbox (∑i∈Basis. a i *R i) (∑i∈Basis. b i *R i)"
      using a b less_imp_le by (auto simp: cbox_def)
  qed (use a b cbox_def in auto)
qed

lemma open_UNION_cbox:
  fixes M :: "'a::euclidean_space set"
  assumes "open M"
  defines "a' ≡ λf. (∑(i::'a)∈Basis. fst (f i) *R i)"
  defines "b' ≡ λf. (∑(i::'a)∈Basis. snd (f i) *R i)"
  defines "I ≡ {f∈Basis →E ℚ × ℚ. cbox (a' f) (b' f) ⊆ M}"
  shows "M = (⋃f∈I. cbox (a' f) (b' f))"
proof -
  have "x ∈ (⋃f∈I. cbox (a' f) (b' f))" if "x ∈ M" for x
  proof -
    obtain e where e: "e > 0" "ball x e ⊆ M"
      using openE[OF ‹open M› ‹x ∈ M›] by auto
    moreover obtain a b where ab: "x ∈ cbox a b" "∀i ∈ Basis. a ∙ i ∈ ℚ"
                                  "∀i ∈ Basis. b ∙ i ∈ ℚ" "cbox a b ⊆ ball x e"
      using rational_cboxes[OF e(1)] by metis
    ultimately show ?thesis
       by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"])
          (auto simp: euclidean_representation I_def a'_def b'_def)
  qed
  then show ?thesis by (auto simp: I_def)
qed

corollary open_countable_Union_open_cbox:
  fixes S :: "'a :: euclidean_space set"
  assumes "open S"
  obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = cbox a b" "⋃𝒟 = S"
proof -
  let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *R i)"
  let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *R i)"
  let ?I = "{f∈Basis →E ℚ × ℚ. cbox (?a f) (?b f) ⊆ S}"
  let ?𝒟 = "(λf. cbox (?a f) (?b f)) ` ?I"
  show ?thesis
  proof
    have "countable ?I"
      by (simp add: countable_PiE countable_rat)
    then show "countable ?𝒟"
      by blast
    show "⋃?𝒟 = S"
      using open_UNION_cbox [OF assms] by metis
  qed auto
qed

lemma box_eq_empty:
  fixes a :: "'a::euclidean_space"
  shows "(box a b = {} ⟷ (∃i∈Basis. b∙i ≤ a∙i))" (is ?th1)
    and "(cbox a b = {} ⟷ (∃i∈Basis. b∙i < a∙i))" (is ?th2)
proof -
  {
    fix i x
    assume i: "i∈Basis" and as:"b∙i ≤ a∙i" and x:"x∈box a b"
    then have "a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i"
      unfolding mem_box by (auto simp: box_def)
    then have "a∙i < b∙i" by auto
    then have False using as by auto
  }
  moreover
  {
    assume as: "∀i∈Basis. ¬ (b∙i ≤ a∙i)"
    let ?x = "(1/2) *R (a + b)"
    {
      fix i :: 'a
      assume i: "i ∈ Basis"
      have "a∙i < b∙i"
        using as[THEN bspec[where x=i]] i by auto
      then have "a∙i < ((1/2) *R (a+b)) ∙ i" "((1/2) *R (a+b)) ∙ i < b∙i"
        by (auto simp: inner_add_left)
    }
    then have "box a b ≠ {}"
      using mem_box(1)[of "?x" a b] by auto
  }
  ultimately show ?th1 by blast

  {
    fix i x
    assume i: "i ∈ Basis" and as:"b∙i < a∙i" and x:"x∈cbox a b"
    then have "a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i"
      unfolding mem_box by auto
    then have "a∙i ≤ b∙i" by auto
    then have False using as by auto
  }
  moreover
  {
    assume as:"∀i∈Basis. ¬ (b∙i < a∙i)"
    let ?x = "(1/2) *R (a + b)"
    {
      fix i :: 'a
      assume i:"i ∈ Basis"
      have "a∙i ≤ b∙i"
        using as[THEN bspec[where x=i]] i by auto
      then have "a∙i ≤ ((1/2) *R (a+b)) ∙ i" "((1/2) *R (a+b)) ∙ i ≤ b∙i"
        by (auto simp: inner_add_left)
    }
    then have "cbox a b ≠ {}"
      using mem_box(2)[of "?x" a b] by auto
  }
  ultimately show ?th2 by blast
qed

lemma box_ne_empty:
  fixes a :: "'a::euclidean_space"
  shows "cbox a b ≠ {} ⟷ (∀i∈Basis. a∙i ≤ b∙i)"
  and "box a b ≠ {} ⟷ (∀i∈Basis. a∙i < b∙i)"
  unfolding box_eq_empty[of a b] by fastforce+

lemma
  fixes a :: "'a::euclidean_space"
  shows cbox_sing [simp]: "cbox a a = {a}"
    and box_sing [simp]: "box a a = {}"
  unfolding set_eq_iff mem_box eq_iff [symmetric]
  by (auto intro!: euclidean_eqI[where 'a='a])
     (metis all_not_in_conv nonempty_Basis)

lemma subset_box_imp:
  fixes a :: "'a::euclidean_space"
  shows "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ cbox c d ⊆ cbox a b"
    and "(∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i) ⟹ cbox c d ⊆ box a b"
    and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ cbox a b"
     and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ box a b"
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box
  by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

lemma box_subset_cbox:
  fixes a :: "'a::euclidean_space"
  shows "box a b ⊆ cbox a b"
  unfolding subset_eq [unfolded Ball_def] mem_box
  by (fast intro: less_imp_le)

lemma subset_box:
  fixes a :: "'a::euclidean_space"
  shows "cbox c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th1)
    and "cbox c d ⊆ box a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i)" (is ?th2)
    and "box c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th3)
    and "box c d ⊆ box a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th4)
proof -
  let ?lesscd = "∀i∈Basis. c∙i < d∙i"
  let ?lerhs = "∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
  show ?th1 ?th2
    by (fastforce simp: mem_box)+
  have acdb: "a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
    if i: "i ∈ Basis" and box: "box c d ⊆ cbox a b" and cd: "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i
  proof -
    have "box c d ≠ {}"
      using that
      unfolding box_eq_empty by force
    { let ?x = "(∑j∈Basis. (if j=i then ((min (a∙j) (d∙j))+c∙j)/2 else (c∙j+d∙j)/2) *R j)::'a"
      assume *: "a∙i > c∙i"
      then have "c ∙ j < ?x ∙ j ∧ ?x ∙ j < d ∙ j" if "j ∈ Basis" for j
        using cd that by (fastforce simp add: i *)
      then have "?x ∈ box c d"
        unfolding mem_box by auto
      moreover have "?x ∉ cbox a b"
        using i cd * by (force simp: mem_box)
      ultimately have False using box by auto
    }
    then have "a∙i ≤ c∙i" by force
    moreover
    { let ?x = "(∑j∈Basis. (if j=i then ((max (b∙j) (c∙j))+d∙j)/2 else (c∙j+d∙j)/2) *R j)::'a"
      assume *: "b∙i < d∙i"
      then have "d ∙ j > ?x ∙ j ∧ ?x ∙ j > c ∙ j" if "j ∈ Basis" for j
        using cd that by (fastforce simp add: i *)
      then have "?x ∈ box c d"
        unfolding mem_box by auto
      moreover have "?x ∉ cbox a b"
        using i cd * by (force simp: mem_box)
      ultimately have False using box by auto
    }
    then have "b∙i ≥ d∙i" by (rule ccontr) auto
    ultimately show ?thesis by auto
  qed
  show ?th3
    using acdb by (fastforce simp add: mem_box)
  have acdb': "a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
    if "i ∈ Basis" "box c d ⊆ box a b" "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i
      using box_subset_cbox[of a b] that acdb by auto
  show ?th4
    using acdb' by (fastforce simp add: mem_box)
qed

lemma eq_cbox: "cbox a b = cbox c d ⟷ cbox a b = {} ∧ cbox c d = {} ∨ a = c ∧ b = d"
      (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have "cbox a b ⊆ cbox c d" "cbox c d ⊆ cbox a b"
    by auto
  then show ?rhs
    by (force simp: subset_box box_eq_empty intro: antisym euclidean_eqI)
next
  assume ?rhs
  then show ?lhs
    by force
qed

lemma eq_cbox_box [simp]: "cbox a b = box c d ⟷ cbox a b = {} ∧ box c d = {}"
  (is "?lhs ⟷ ?rhs")
proof
  assume L: ?lhs
  then have "cbox a b ⊆ box c d" "box c d ⊆ cbox a b"
    by auto
  then show ?rhs
    apply (simp add: subset_box)
    using L box_ne_empty box_sing apply (fastforce simp add:)
    done
qed force

lemma eq_box_cbox [simp]: "box a b = cbox c d ⟷ box a b = {} ∧ cbox c d = {}"
  by (metis eq_cbox_box)

lemma eq_box: "box a b = box c d ⟷ box a b = {} ∧ box c d = {} ∨ a = c ∧ b = d"
  (is "?lhs ⟷ ?rhs")
proof
  assume L: ?lhs
  then have "box a b ⊆ box c d" "box c d ⊆ box a b"
    by auto
  then show ?rhs
    apply (simp add: subset_box)
    using box_ne_empty(2) L
    apply auto
     apply (meson euclidean_eqI less_eq_real_def not_less)+
    done
qed force

lemma subset_box_complex:
   "cbox a b ⊆ cbox c d ⟷
      (Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
   "cbox a b ⊆ box c d ⟷
      (Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a > Re c ∧ Im a > Im c ∧ Re b < Re d ∧ Im b < Im d"
   "box a b ⊆ cbox c d ⟷
      (Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
   "box a b ⊆ box c d ⟷
      (Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
  by (subst subset_box; force simp: Basis_complex_def)+

lemma Int_interval:
  fixes a :: "'a::euclidean_space"
  shows "cbox a b ∩ cbox c d =
    cbox (∑i∈Basis. max (a∙i) (c∙i) *R i) (∑i∈Basis. min (b∙i) (d∙i) *R i)"
  unfolding set_eq_iff and Int_iff and mem_box
  by auto

lemma disjoint_interval:
  fixes a::"'a::euclidean_space"
  shows "cbox a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i < c∙i ∨ b∙i < c∙i ∨ d∙i < a∙i))" (is ?th1)
    and "cbox a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th2)
    and "box a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i < c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th3)
    and "box a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th4)
proof -
  let ?z = "(∑i∈Basis. (((max (a∙i) (c∙i)) + (min (b∙i) (d∙i))) / 2) *R i)::'a"
  have **: "⋀P Q. (⋀i :: 'a. i ∈ Basis ⟹ Q ?z i ⟹ P i) ⟹
      (⋀i x :: 'a. i ∈ Basis ⟹ P i ⟹ Q x i) ⟹ (∀x. ∃i∈Basis. Q x i) ⟷ (∃i∈Basis. P i)"
    by blast
  note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
  show ?th1 unfolding * by (intro **) auto
  show ?th2 unfolding * by (intro **) auto
  show ?th3 unfolding * by (intro **) auto
  show ?th4 unfolding * by (intro **) auto
qed

lemma UN_box_eq_UNIV: "(⋃i::nat. box (- (real i *R One)) (real i *R One)) = UNIV"
proof -
  have "¦x ∙ b¦ < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)"
    if [simp]: "b ∈ Basis" for x b :: 'a
  proof -
    have "¦x ∙ b¦ ≤ real_of_int ⌈¦x ∙ b¦⌉"
      by (rule le_of_int_ceiling)
    also have "… ≤ real_of_int ⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉"
      by (auto intro!: ceiling_mono)
    also have "… < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)"
      by simp
    finally show ?thesis .
  qed
  then have "∃n::nat. ∀b∈Basis. ¦x ∙ b¦ < real n" for x :: 'a
    by (metis order.strict_trans reals_Archimedean2)
  moreover have "⋀x b::'a. ⋀n::nat.  ¦x ∙ b¦ < real n ⟷ - real n < x ∙ b ∧ x ∙ b < real n"
    by auto
  ultimately show ?thesis
    by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
qed

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 swap_continuous:
  assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
    shows "continuous_on (cbox (c,a) (d,b)) (λ(x, y). f y x)"
proof -
  have "(λ(x, y). f y x) = (λ(x, y). f x y) ∘ prod.swap"
    by auto
  then show ?thesis
    apply (rule ssubst)
    apply (rule continuous_on_compose)
    apply (simp add: split_def)
    apply (rule continuous_intros | simp add: assms)+
    done
qed


subsection ‹General Intervals›

definition✐‹tag important› "is_interval (s::('a::euclidean_space) set) ⟷
  (∀a∈s. ∀b∈s. ∀x. (∀i∈Basis. ((a∙i ≤ x∙i ∧ x∙i ≤ b∙i) ∨ (b∙i ≤ x∙i ∧ x∙i ≤ a∙i))) ⟶ x ∈ s)"

lemma is_interval_1:
  "is_interval (s::real set) ⟷ (∀a∈s. ∀b∈s. ∀ x. a ≤ x ∧ x ≤ b ⟶ x ∈ s)"
  unfolding is_interval_def by auto

lemma is_interval_Int: "is_interval X ⟹ is_interval Y ⟹ is_interval (X ∩ Y)"
  unfolding is_interval_def
  by blast

lemma is_interval_cbox [simp]: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
  and is_interval_box [simp]: "is_interval (box a b)" (is ?th2)
  unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
  by (meson order_trans le_less_trans less_le_trans less_trans)+

lemma is_interval_empty [iff]: "is_interval {}"
  unfolding is_interval_def  by simp

lemma is_interval_univ [iff]: "is_interval UNIV"
  unfolding is_interval_def  by simp

lemma mem_is_intervalI:
  assumes "is_interval s"
    and "a ∈ s" "b ∈ s"
    and "⋀i. i ∈ Basis ⟹ a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i ∨ b ∙ i ≤ x ∙ i ∧ x ∙ i ≤ a ∙ i"
  shows "x ∈ s"
  by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

lemma interval_subst:
  fixes S::"'a::euclidean_space set"
  assumes "is_interval S"
    and "x ∈ S" "y j ∈ S"
    and "j ∈ Basis"
  shows "(∑i∈Basis. (if i = j then y i ∙ i else x ∙ i) *R i) ∈ S"
  by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

lemma mem_box_componentwiseI:
  fixes S::"'a::euclidean_space set"
  assumes "is_interval S"
  assumes "⋀i. i ∈ Basis ⟹ x ∙ i ∈ ((λx. x ∙ i) ` S)"
  shows "x ∈ S"
proof -
  from assms have "∀i ∈ Basis. ∃s ∈ S. x ∙ i = s ∙ i"
    by auto
  with finite_Basis obtain s and bs::"'a list"
    where s: "⋀i. i ∈ Basis ⟹ x ∙ i = s i ∙ i" "⋀i. i ∈ Basis ⟹ s i ∈ S"
      and bs: "set bs = Basis" "distinct bs"
    by (metis finite_distinct_list)
  from nonempty_Basis s obtain j where j: "j ∈ Basis" "s j ∈ S"
    by blast
  define y where
    "y = rec_list (s j) (λj _ Y. (∑i∈Basis. (if i = j then s i ∙ i else Y ∙ i) *R i))"
  have "x = (∑i∈Basis. (if i ∈ set bs then s i ∙ i else s j ∙ i) *R i)"
    using bs by (auto simp: s(1)[symmetric] euclidean_representation)
  also have [symmetric]: "y bs = …"
    using bs(2) bs(1)[THEN equalityD1]
    by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
  also have "y bs ∈ S"
    using bs(1)[THEN equalityD1]
    apply (induct bs)
     apply (auto simp: y_def j)
    apply (rule interval_subst[OF assms(1)])
      apply (auto simp: s)
    done
  finally show ?thesis .
qed

lemma cbox01_nonempty [simp]: "cbox 0 One ≠ {}"
  by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)

lemma box01_nonempty [simp]: "box 0 One ≠ {}"
  by (simp add: box_ne_empty inner_Basis inner_sum_left)

lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
  using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast

lemma interval_subset_is_interval:
  assumes "is_interval S"
  shows "cbox a b ⊆ S ⟷ cbox a b = {} ∨ a ∈ S ∧ b ∈ S" (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs  using box_ne_empty(1) mem_box(2) by fastforce
next
  assume ?rhs
  have "cbox a b ⊆ S" if "a ∈ S" "b ∈ S"
    using assms unfolding is_interval_def
    apply (clarsimp simp add: mem_box)
    using that by blast
  with ‹?rhs› show ?lhs
    by blast
qed

lemma is_real_interval_union:
  "is_interval (X ∪ Y)"
  if X: "is_interval X" and Y: "is_interval Y" and I: "(X ≠ {} ⟹ Y ≠ {} ⟹ X ∩ Y ≠ {})"
  for X Y::"real set"
proof -
  consider "X ≠ {}" "Y ≠ {}" | "X = {}" | "Y = {}" by blast
  then show ?thesis
  proof cases
    case 1
    then obtain r where "r ∈ X ∨ X ∩ Y = {}" "r ∈ Y ∨ X ∩ Y = {}"
      by blast
    then show ?thesis
      using I 1 X Y unfolding is_interval_1
      by (metis (full_types) Un_iff le_cases)
  qed (use that in auto)
qed

lemma is_interval_translationI:
  assumes "is_interval X"
  shows "is_interval ((+) x ` X)"
  unfolding is_interval_def
proof safe
  fix b d e
  assume "b ∈ X" "d ∈ X"
    "∀i∈Basis. (x + b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + d) ∙ i ∨
       (x + d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + b) ∙ i"
  hence "e - x ∈ X"
    by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "e - x"])
      (auto simp: algebra_simps)
  thus "e ∈ (+) x ` X" by force
qed

lemma is_interval_uminusI:
  assumes "is_interval X"
  shows "is_interval (uminus ` X)"
  unfolding is_interval_def
proof safe
  fix b d e
  assume "b ∈ X" "d ∈ X"
    "∀i∈Basis. (- b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- d) ∙ i ∨
       (- d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- b) ∙ i"
  hence "- e ∈ X"
    by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "- e"])
      (auto simp: algebra_simps)
  thus "e ∈ uminus ` X" by force
qed

lemma is_interval_uminus[simp]: "is_interval (uminus ` x) = is_interval x"
  using is_interval_uminusI[of x] is_interval_uminusI[of "uminus ` x"]
  by (auto simp: image_image)

lemma is_interval_neg_translationI:
  assumes "is_interval X"
  shows "is_interval ((-) x ` X)"
proof -
  have "(-) x ` X = (+) x ` uminus ` X"
    by (force simp: algebra_simps)
  also have "is_interval …"
    by (metis is_interval_uminusI is_interval_translationI assms)
  finally show ?thesis .
qed

lemma is_interval_translation[simp]:
  "is_interval ((+) x ` X) = is_interval X"
  using is_interval_neg_translationI[of "(+) x ` X" x]
  by (auto intro!: is_interval_translationI simp: image_image)

lemma is_interval_minus_translation[simp]:
  shows "is_interval ((-) x ` X) = is_interval X"
proof -
  have "(-) x ` X = (+) x ` uminus ` X"
    by (force simp: algebra_simps)
  also have "is_interval … = is_interval X"
    by simp
  finally show ?thesis .
qed

lemma is_interval_minus_translation'[simp]:
  shows "is_interval ((λx. x - c) ` X) = is_interval X"
  using is_interval_translation[of "-c" X]
  by (metis image_cong uminus_add_conv_diff)


subsection✐‹tag unimportant› ‹Bounded Projections›

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)


subsection✐‹tag unimportant› ‹Structural rules for pointwise continuity›

lemma continuous_infnorm[continuous_intros]:
  "continuous F f ⟹ continuous F (λx. infnorm (f x))"
  unfolding continuous_def by (rule tendsto_infnorm)

lemma continuous_inner[continuous_intros]:
  assumes "continuous F f"
    and "continuous F g"
  shows "continuous F (λx. inner (f x) (g x))"
  using assms unfolding continuous_def by (rule tendsto_inner)


subsection✐‹tag unimportant› ‹Structural rules for setwise continuity›

lemma continuous_on_infnorm[continuous_intros]:
  "continuous_on s f ⟹ continuous_on s (λx. infnorm (f x))"
  unfolding continuous_on by (fast intro: tendsto_infnorm)

lemma continuous_on_inner[continuous_intros]:
  fixes g :: "'a::topological_space ⇒ 'b::real_inner"
  assumes "continuous_on s f"
    and "continuous_on s g"
  shows "continuous_on s (λx. inner (f x) (g x))"
  using bounded_bilinear_inner assms
  by (rule bounded_bilinear.continuous_on)


subsection✐‹tag unimportant› ‹Openness of halfspaces.›

lemma open_halfspace_lt: "open {x. inner a x < b}"
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)

lemma open_halfspace_gt: "open {x. inner a x > b}"
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)

lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x∙i < a}"
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)

lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x∙i > a}"
  by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)

lemma eucl_less_eq_halfspaces:
  fixes a :: "'a::euclidean_space"
  shows "{x. x <e a} = (⋂i∈Basis. {x. x ∙ i < a ∙ i})"
        "{x. a <e x} = (⋂i∈Basis. {x. a ∙ i < x ∙ i})"
  by (auto simp: eucl_less_def)

lemma open_Collect_eucl_less[simp, intro]:
  fixes a :: "'a::euclidean_space"
  shows "open {x. x <e a}" "open {x. a <e x}"
  by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)

subsection✐‹tag unimportant› ‹Closure of halfspaces and hyperplanes›

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)


subsection✐‹tag unimportant›‹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)


subsection ‹Limit Component Bounds›

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

lemma open_box[intro]: "open (box a b)"
proof -
  have "open (⋂i∈Basis. ((∙) i) -` {a ∙ i <..< b ∙ i})"
    by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
  also have "(⋂i∈Basis. ((∙) i) -` {a ∙ i <..< b ∙ i}) = box a b"
    by (auto simp: box_def inner_commute)
  finally show ?thesis .
qed

lemma closed_cbox[intro]:
  fixes a b :: "'a::euclidean_space"
  shows "closed (cbox a b)"
proof -
  have "closed (⋂i∈Basis. (λx. x∙i) -` {a∙i .. b∙i})"
    by (intro closed_INT ballI continuous_closed_vimage allI
      linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
  also have "(⋂i∈Basis. (λx. x∙i) -` {a∙i .. b∙i}) = cbox a b"
    by (auto simp: cbox_def)
  finally show "closed (cbox a b)" .
qed

lemma interior_cbox [simp]:
  fixes a b :: "'a::euclidean_space"
  shows "interior (cbox a b) = box a b" (is "?L = ?R")
proof(rule subset_antisym)
  show "?R ⊆ ?L"
    using box_subset_cbox open_box
    by (rule interior_maximal)
  {
    fix x
    assume "x ∈ interior (cbox a b)"
    then obtain s where s: "open s" "x ∈ s" "s ⊆ cbox a b" ..
    then obtain e where "e>0" and e:"∀x'. dist x' x < e ⟶ x' ∈ cbox a b"
      unfolding open_dist and subset_eq by auto
    {
      fix i :: 'a
      assume i: "i ∈ Basis"
      have "dist (x - (e / 2) *R i) x < e"
        and "dist (x + (e / 2) *R i) x < e"
        unfolding dist_norm
        apply auto
        unfolding norm_minus_cancel
        using norm_Basis[OF i] ‹e>0›
        apply auto
        done
      then have "a ∙ i ≤ (x - (e / 2) *R i) ∙ i" and "(x + (e / 2) *R i) ∙ i ≤ b ∙ i"
        using e[THEN spec[where x="x - (e/2) *R i"]]
          and e[THEN spec[where x="x + (e/2) *R i"]]
        unfolding mem_box
        using i
        by blast+
      then have "a ∙ i < x ∙ i" and "x ∙ i < b ∙ i"
        using ‹e>0› i
        by (auto simp: inner_diff_left inner_Basis inner_add_left)
    }
    then have "x ∈ box a b"
      unfolding mem_box by auto
  }
  then show "?L ⊆ ?R" ..
qed

lemma bounded_cbox [simp]:
  fixes a :: "'a::euclidean_space"
  shows "bounded (cbox a b)"
proof -
  let ?b = "∑i∈Basis. ¦a∙i¦ + ¦b∙i¦"
  {
    fix x :: "'a"
    assume "⋀i. i∈Basis ⟹ a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i"
    then have "(∑i∈Basis. ¦x ∙ i¦) ≤ ?b"
      by (force simp: intro!: sum_mono)
    then have "norm x ≤ ?b"
      using norm_le_l1[of x] by auto
  }
  then show ?thesis
    unfolding cbox_def bounded_iff by force
qed

lemma bounded_box [simp]:
  fixes a :: "'a::euclidean_space"
  shows "bounded (box a b)"
  using bounded_cbox[of a b] box_subset_cbox[of a b] bounded_subset[of "cbox a b" "box a b"]
  by simp

lemma not_interval_UNIV [simp]:
  fixes a :: "'a::euclidean_space"
  shows "cbox a b ≠ UNIV" "box a b ≠ UNIV"
  using bounded_box[of a b] bounded_cbox[of a b] by force+

lemma not_interval_UNIV2 [simp]:
  fixes a :: "'a::euclidean_space"
  shows "UNIV ≠ cbox a b" "UNIV ≠ box a b"
  using bounded_box[of a b] bounded_cbox[of a b] by force+

lemma box_midpoint:
  fixes a :: "'a::euclidean_space"
  assumes "box a b ≠ {}"
  shows "((1/2) *R (a + b)) ∈ box a b"
proof -
  have "a ∙ i < ((1 / 2) *R (a + b)) ∙ i ∧ ((1 / 2) *R (a + b)) ∙ i < b ∙ i" if "i ∈ Basis" for i
    using assms that by (auto simp: inner_add_left box_ne_empty)
  then show ?thesis unfolding mem_box by auto
qed

lemma open_cbox_convex:
  fixes x :: "'a::euclidean_space"
  assumes x: "x ∈ box a b"
    and y: "y ∈ cbox a b"
    and e: "0 < e" "e ≤ 1"
  shows "(e *R x + (1 - e) *R y) ∈ box a b"
proof -
  {
    fix i :: 'a
    assume i: "i ∈ Basis"
    have "a ∙ i = e * (a ∙ i) + (1 - e) * (a ∙ i)"
      unfolding left_diff_distrib by simp
    also have "… < e * (x ∙ i) + (1 - e) * (y ∙ i)"
    proof (rule add_less_le_mono)
      show "e * (a ∙ i) < e * (x ∙ i)"
        using ‹0 < e› i mem_box(1) x by auto
      show "(1 - e) * (a ∙ i) ≤ (1 - e) * (y ∙ i)"
        by (meson diff_ge_0_iff_ge ‹e ≤ 1› i mem_box(2) mult_left_mono y)
    qed
    finally have "a ∙ i < (e *R x + (1 - e) *R y) ∙ i"
      unfolding inner_simps by auto
    moreover
    {
      have "b ∙ i = e * (b∙i) + (1 - e) * (b∙i)"
        unfolding left_diff_distrib by simp
      also have "… > e * (x ∙ i) + (1 - e) * (y ∙ i)"
      proof (rule add_less_le_mono)
        show "e * (x ∙ i) < e * (b ∙ i)"
          using ‹0 < e› i mem_box(1) x by auto
        show "(1 - e) * (y ∙ i) ≤ (1 - e) * (b ∙ i)"
          by (meson diff_ge_0_iff_ge ‹e ≤ 1› i mem_box(2) mult_left_mono y)
      qed
      finally have "(e *R x + (1 - e) *R y) ∙ i < b ∙ i"
        unfolding inner_simps by auto
    }
    ultimately have "a ∙ i < (e *R x + (1 - e) *R y) ∙ i ∧ (e *R x + (1 - e) *R y) ∙ i < b ∙ i"
      by auto
  }
  then show ?thesis
    unfolding mem_box by auto
qed

lemma closure_cbox [simp]: "closure (cbox a b) = cbox a b"
  by (simp add: closed_cbox)

lemma closure_box [simp]:
  fixes a :: "'a::euclidean_space"
   assumes "box a b ≠ {}"
  shows "closure (box a b) = cbox a b"
proof -
  have ab: "a <e b"
    using assms by (simp add: eucl_less_def box_ne_empty)
  let ?c = "(1 / 2) *R (a + b)"
  {
    fix x
    assume as:"x ∈ cbox a b"
    define f where [abs_def]: "f n = x + (inverse (real n + 1)) *R (?c - x)" for n
    {
      fix n
      assume fn: "f n <e b ⟶ a <e f n ⟶ f n = x" and xc: "x ≠ ?c"
      have *: "0 < inverse (real n + 1)" "inverse (real n + 1) ≤ 1"
        unfolding inverse_le_1_iff by auto
      have "(inverse (real n + 1)) *R ((1 / 2) *R (a + b)) + (1 - inverse (real n + 1)) *R x =
        x + (inverse (real n + 1)) *R (((1 / 2) *R (a + b)) - x)"
        by (auto simp: algebra_simps)
      then have "f n <e b" and "a <e f n"
        using open_cbox_convex[OF box_midpoint[OF assms] as *]
        unfolding f_def by (auto simp: box_def eucl_less_def)
      then have False
        using fn unfolding f_def using xc by auto
    }
    moreover
    {
      assume "¬ (f ⤏ x) sequentially"
      {
        fix e :: real
        assume "e > 0"
        then obtain N :: nat where N: "inverse (real (N + 1)) < e"
          using reals_Archimedean by auto
        have "inverse (real n + 1) < e" if "N ≤ n" for n
          by (auto intro!: that le_less_trans [OF _ N])
        then have "∃N::nat. ∀n≥N. inverse (real n + 1) < e" by auto
      }
      then have "((λn. inverse (real n + 1)) ⤏ 0) sequentially"
        unfolding lim_sequentially by(auto simp: dist_norm)
      then have "(f ⤏ x) sequentially"
        unfolding f_def
        using tendsto_add[OF tendsto_const, of "λn::nat. (inverse (real n + 1)) *R ((1 / 2) *R (a + b) - x)" 0 sequentially x]
        using tendsto_scaleR [OF _ tendsto_const, of "λn::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *R (a + b) - x)"]
        by auto
    }
    ultimately have "x ∈ closure (box a b)"
      using as box_midpoint[OF assms]
      unfolding closure_def islimpt_sequential
      by (cases "x=?c") (auto simp: in_box_eucl_less)
  }
  then show ?thesis
    using closure_minimal[OF box_subset_cbox, of a b] by blast
qed

lemma bounded_subset_box_symmetric:
  fixes S :: "('a::euclidean_space) set"
  assumes "bounded S"
  obtains a where "S ⊆ box (-a) a"
proof -
  obtain b where "b>0" and b: "∀x∈S. norm x ≤ b"
    using assms[unfolded bounded_pos] by auto
  define a :: 'a where "a = (∑i∈Basis. (b + 1) *R i)"
  have "(-a)∙i < x∙i" and "x∙i < a∙i" if "x ∈ S" and i: "i ∈ Basis" for x i
    using b Basis_le_norm[OF i, of x] that by (auto simp: a_def)
  then have "S ⊆ box (-a) a"
    by (auto simp: simp add: box_def)
  then show ?thesis ..
qed

lemma bounded_subset_cbox_symmetric:
  fixes S :: "('a::euclidean_space) set"
  assumes "bounded S"
  obtains a where "S ⊆ cbox (-a) a"
proof -
  obtain a where "S ⊆ box (-a) a"
    using bounded_subset_box_symmetric[OF assms] by auto
  then show ?thesis
    by (meson box_subset_cbox dual_order.trans that)
qed

lemma frontier_cbox:
  fixes a b :: "'a::euclidean_space"
  shows "frontier (cbox a b) = cbox a b - box a b"
  unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] ..

lemma frontier_box:
  fixes a b :: "'a::euclidean_space"
  shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
proof (cases "box a b = {}")
  case True
  then show ?thesis
    using frontier_empty by auto
next
  case False
  then show ?thesis
    unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box]
    by auto
qed

lemma Int_interval_mixed_eq_empty:
  fixes a :: "'a::euclidean_space"
   assumes "box c d ≠ {}"
  shows "box a b ∩ cbox c d = {} ⟷ box a b ∩ box c d = {}"
  unfolding closure_box[OF assms, symmetric]
  unfolding open_Int_closure_eq_empty[OF open_box] ..

subsection ‹Class Instances›

lemma compact_lemma:
  fixes f :: "nat ⇒ 'a::euclidean_space"
  assumes "bounded (range f)"
  shows "∀d⊆Basis. ∃l::'a. ∃ r.
    strict_mono r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) ∙ i) (l ∙ i) < e) sequentially)"
  by (rule compact_lemma_general[where unproj="λe. ∑i∈Basis. e i *R i"])
     (auto intro!: assms bounded_linear_inner_left bounded_linear_image
       simp: euclidean_representation)

instance✐‹tag important› euclidean_space  heine_borel
proof
  fix f :: "nat ⇒ 'a"
  assume f: "bounded (range f)"
  then obtain l::'a and r where r: "strict_mono r"
    and l: "∀e>0. eventually (λn. ∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e) sequentially"
    using compact_lemma [OF f] by blast
  {
    fix e::real
    assume "e > 0"
    hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)
    with l have "eventually (λn. ∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e / (real_of_nat DIM('a))) sequentially"
      by simp
    moreover
    {
      fix n
      assume n: "∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e / (real_of_nat DIM('a))"
      have "dist (f (r n)) l ≤ (∑i∈Basis. dist (f (r n) ∙ i) (l ∙ i))"
        apply (subst euclidean_dist_l2)
        using zero_le_dist
        apply (rule L2_set_le_sum)
        done
      also have "… < (∑i∈(Basis::'a set). e / (real_of_nat DIM('a)))"
        apply (rule sum_strict_mono)
        using n
        apply auto
        done
      finally have "dist (f (r n)) l < e"
        by auto
    }
    ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially"
      by (rule eventually_mono)
  }
  then have *: "((f ∘ r) ⤏ l) sequentially"
    unfolding o_def tendsto_iff by simp
  with r show "∃l r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
    by auto
qed

instance✐‹tag important› euclidean_space  banach ..

instance euclidean_space  second_countable_topology
proof
  define a where "a f = (∑i∈Basis. fst (f i) *R i)" for f :: "'a ⇒ real × real"
  then have a: "⋀f. (∑i∈Basis. fst (f i) *R i) = a f"
    by simp
  define b where "b f = (∑i∈Basis. snd (f i) *R i)" for f :: "'a ⇒ real × real"
  then have b: "⋀f. (∑i∈Basis. snd (f i) *R i) = b f"
    by simp
  define B where "B = (λf. box (a f) (b f)) ` (Basis →E (ℚ × ℚ))"

  have "Ball B open" by (simp add: B_def open_box)
  moreover have "(∀A. open A ⟶ (∃B'⊆B. ⋃B' = A))"
  proof safe
    fix A::"'a set"
    assume "open A"
    show "∃B'⊆B. ⋃B' = A"
      apply (rule exI[of _ "{b∈B. b ⊆ A}"])
      apply (subst (3) open_UNION_box[OF ‹open A›])
      apply (auto simp: a b B_def)
      done
  qed
  ultimately
  have "topological_basis B"
    unfolding topological_basis_def by blast
  moreover
  have "countable B"
    unfolding B_def
    by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)
  ultimately show "∃B::'a set set. countable B ∧ open = generate_topology B"
    by (blast intro: topological_basis_imp_subbasis)
qed

instance euclidean_space  polish_space ..


subsection ‹Compact Boxes›

lemma compact_cbox [simp]:
  fixes a :: "'a::euclidean_space"
  shows "compact (cbox a b)"
  using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b]
  by (auto simp: compact_eq_seq_compact_metric)

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


subsection✐‹tag 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✐‹tag 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)


subsection ‹Separability›

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::{metric_space, second_countable_topology} set"
  obtains T where "countable T" "T ⊆ S" "S ⊆ closure T"
proof -
  obtain  :: "'a set set"
    where "countable ℬ"
      and "{} ∉ ℬ"
      and ope: "⋀C. C ∈ ℬ ⟹ openin(top_of_set S) C"
      and if_ope: "⋀T. openin(top_of_set 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 (top_of_set 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


subsection✐‹tag unimportant› ‹Diameter›

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

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✐‹tag unimportant›‹Relating linear images to open/closed/interior/closure/connected›

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)

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✐‹tag unimportant› ‹"Isometry" (up to constant bounds) of Injective Linear Map›

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✐‹tag unimportant› ‹Some properties of a canonical subspace›

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 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 ‹Set Distance›

lemma setdist_compact_closed:
  fixes A :: "'a::heine_borel set"
  assumes A: "compact A" and B: "closed B"
    and "A ≠ {}" "B ≠ {}"
  shows "∃x ∈ A. ∃y ∈ B. dist x y = setdist A B"
proof -
  obtain x where "x ∈ A" "setdist A B = infdist x B"
    by (metis A assms(3) setdist_attains_inf setdist_sym)
  moreover
  obtain y where"y ∈ B" "infdist x B = dist x y"
    using B ‹B ≠ {}› infdist_attains_inf by blast
  ultimately show ?thesis
    using ‹x ∈ A› ‹y ∈ B› by auto
qed

lemma setdist_closed_compact:
  fixes S :: "'a::heine_borel set"
  assumes S: "closed S" and T: "compact T"
      and "S ≠ {}" "T ≠ {}"
    shows "∃x ∈ S. ∃y ∈ T. dist x y = setdist S T"
  using setdist_compact_closed [OF T S ‹T ≠ {}› ‹S ≠ {}›]
  by (metis dist_commute setdist_sym)

lemma setdist_eq_0_compact_closed:
  assumes S: "compact S" and T: "closed T"
    shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ S ∩ T ≠ {}"
proof (cases "S = {} ∨ T = {}")
  case True
  then show ?thesis
    by force
next
  case False
  then show ?thesis
    by (metis S T disjoint_iff_not_equal in_closed_iff_infdist_zero setdist_attains_inf setdist_eq_0I setdist_sym)
qed

corollary setdist_gt_0_compact_closed:
  assumes S: "compact S" and T: "closed T"
    shows "setdist S T > 0 ⟷ (S ≠ {} ∧ T ≠ {} ∧ S ∩ T = {})"
  using setdist_pos_le [of S T] setdist_eq_0_compact_closed [OF assms] by linarith

lemma setdist_eq_0_closed_compact:
  assumes S: "closed S" and T: "compact T"
    shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ S ∩ T ≠ {}"
  using setdist_eq_0_compact_closed [OF T S]
  by (metis Int_commute setdist_sym)

lemma setdist_eq_0_bounded:
  fixes S :: "'a::heine_borel set"
  assumes "bounded S ∨ bounded T"
  shows "setdist S T = 0 ⟷ S = {} ∨ T = {} ∨ closure S ∩ closure T ≠ {}"
proof (cases "S = {} ∨ T = {}")
  case False
  then show ?thesis
    using setdist_eq_0_compact_closed [of "closure S" "closure T"]
          setdist_eq_0_closed_compact [of "closure S" "closure T"] assms
    by (force simp:  bounded_closure compact_eq_bounded_closed)
qed force

lemma setdist_eq_0_sing_1:
  "setdist {x} S = 0 ⟷ S = {} ∨ x ∈ closure S"
  by (metis in_closure_iff_infdist_zero infdist_def infdist_eq_setdist)

lemma setdist_eq_0_sing_2:
  "setdist S {x} = 0 ⟷ S = {} ∨ x ∈ closure S"
  by (metis setdist_eq_0_sing_1 setdist_sym)

lemma setdist_neq_0_sing_1:
  "⟦setdist {x} S = a; a ≠ 0⟧ ⟹ S ≠ {} ∧ x ∉ closure S"
  by (metis setdist_closure_2 setdist_empty2 setdist_eq_0I singletonI)

lemma setdist_neq_0_sing_2:
  "⟦setdist S {x} = a; a ≠ 0⟧ ⟹ S ≠ {} ∧ x ∉ closure S"
  by (simp add: setdist_neq_0_sing_1 setdist_sym)

lemma setdist_sing_in_set:
   "x ∈ S ⟹ setdist {x} S = 0"
  by (simp add: setdist_eq_0I)

lemma setdist_eq_0_closed:
   "closed S ⟹ (setdist {x} S = 0 ⟷ S = {} ∨ x ∈ S)"
by (simp add: setdist_eq_0_sing_1)

lemma setdist_eq_0_closedin:
  shows "⟦closedin (top_of_set U) S; x ∈ U⟧
         ⟹ (setdist {x} S = 0 ⟷ S = {} ∨ x ∈ S)"
  by (auto simp: closedin_limpt setdist_eq_0_sing_1 closure_def)

lemma setdist_gt_0_closedin:
  shows "⟦closedin (top_of_set U) S; x ∈ U; S ≠ {}; x ∉ S⟧
         ⟹ setdist {x} S > 0"
  using less_eq_real_def setdist_eq_0_closedin by fastforce

no_notation
  eucl_less (infix "<e" 50)

end