Theory Topology_Euclidean_Space

(*  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 "  (iBasis. (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

subsectiontag unimportant› ‹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 (bB. 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 ((bB. representation B x b *R b)  (bB. representation B x b *R b))"
    by (simp add: norm_eq_sqrt_inner)
  also have " = sqrt (bB. (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 (bB. (norm (representation B x b *R b))2)"
    by (simp add: mult.commute mult.left_commute power2_eq_square)
  also have " = sqrt (bB. (representation B x b)2 * (norm b)2)"
    by (simp add: norm_mult power_mult_distrib)
  finally have "norm x = sqrt (bB. (representation B x b)2 * (norm b)2)" .
  moreover
  have "sqrt ((representation B x b)2 * (norm b)2)  sqrt (bB. (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 (bB. (representation B x b)2 * (norm b)2)"
    using b  0 by (simp add: field_split_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

subsectiontag 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 ((-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: divide_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
qed metric

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 (smt (verit, best) 0  r ?lhs ball_subset_cball cball_subset_cball_iff dist_norm order_trans)
        then show ?thesis
          using subsetD [where c = "a + (r' / norm(a - a') - 1) *R (a - a')", OF ?lhs] a  a'
          apply (simp add: dist_norm)
          apply (simp add: scaleR_left_diff_distrib)
          apply (simp add: field_simps)
          done
      qed
      then show ?thesis by force
    qed
    then show ?rhs by (simp add: dist_norm)
  qed
next
  assume ?rhs
  then show ?lhs
    by metric
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
    by metric
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
      by metric
  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
    by metric
next
  case False show ?thesis
  proof
    assume ?lhs
    then have "0 < r'"
      using False by metric
    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
  qed metric
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"
  by (smt (verit, del_insts) ball_empty ball_subset_cball_iff dist_norm norm_pths(2))

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"
  by (smt (verit, ccfv_SIG) cball_empty cball_subset_cball_iff dist_norm norm_pths(2) zero_le_dist)

lemma ball_eq_cball_iff:
  fixes x :: "'a :: euclidean_space"
  shows "ball x d = cball y e  d  0  e < 0" (is "?lhs = ?rhs")
  by (smt (verit) ball_eq_empty ball_subset_cball_iff cball_eq_empty cball_subset_ball_iff order.refl)

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. wball p e. wS  (wp  wX)"
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 "xX. x  p  e2  dist p x"
    using finite_set_avoid[OF finite X,of p] by auto
  hence "wball p (min e1 e2). wS  (wp  wX)" using e1_b by auto
  thus "e>0. wball p e. w  S  (w  p  w  X)" 
    using e2>0 e1>0 by (rule_tac x="min e1 e2" in exI) auto
qed

lemma finite_cball_avoid:
  fixes S :: "'a :: euclidean_space set"
  assumes "open S" "finite X" "p  S"
  shows "e>0. wcball p e. wS  (wp  wX)"
proof -
  obtain e1 where "e1>0" and e1: "wball p e1. wS  (wp  wX)"
    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. wcball 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)
qed


subsection ‹Boxes›

abbreviationtag important› 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

corollarytag unimportant› One_neq_0[iff]: "One  0"
  by (metis One_non_0)

corollarytag unimportant› Zero_neq_One[iff]: "0  One"
  by (metis One_non_0)

definitiontag important› (in euclidean_space) eucl_less (infix "<e" 50) where 
"eucl_less a b  (iBasis. a  i < b  i)"

definitiontag important› box_eucl_less: "box a b = {x. a <e x  x <e b}"
definitiontag important› "cbox a b = {x. iBasis. a  i  x  i  x  i  b  i}"

lemma box_def: "box a b = {x. iBasis. 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  (iBasis. a  i < x  i  x  i < b  i)"
    "x  cbox a b  (iBasis. 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)"
  by (force simp: cbox_def Basis_complex_def)

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 (iBasis. max (ai) (ci) *R i) (iBasis. min (bi) (di) *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. (iBasis. a  i    b  i  )  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)
  have "y. y    y < x  i  x  i - y < e'" for i
    using Rats_dense_in_real[of "x  i - e'" "x  i"] e by force
  then obtain a where
    a: "u. a u    a u < x  u  x  u - a u < e'" by metis
  have "y. y    x  i < y  y - x  i < e'" for i
    using Rats_dense_in_real[of "x  i" "x  i + e'"] e by force
  then obtain b where
    b: "u. b u    x  u < b u  b u - x  u < e'" by metis
  let ?a = "iBasis. a i *R i" and ?b = "iBasis. 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 (iBasis. (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 < yi  yi < b i"
        using * i by (auto simp: box_def)
      moreover have "a i < xi" "xi - a i < e'" "xi < b i" "b i - xi < e'"
        using a b by auto
      ultimately have "¦xi - yi¦ < 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 (use a b in 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  {fBasis E  × . box (a' f) (b' f)  M}"
  shows "M = (fI. box (a' f) (b' f))"
proof -
  have "x  (fI. 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  "
      "iBasis. b  i  "
      "box a b  ball x e"
      using rational_boxes[OF e(1)] by metis
    ultimately show ?thesis
       by (intro UN_I[of "λiBasis. (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 = "{fBasis E  × . 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. (iBasis. a  i    b  i  )  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 "y. y    y < x  i  x  i - y < e'" for i
    using Rats_dense_in_real[of "x  i - e'" "x  i"] e by force
  then obtain a where
    a: "u. a u    a u < x  u  x  u - a u < e'" by metis
  have "y. y    x  i < y  y - x  i < e'" for i
    using Rats_dense_in_real[of "x  i" "x  i + e'"] e by force
  then obtain b where
    b: "u. b u    x  u < b u  b u - x  u < e'" by metis
  let ?a = "iBasis. a i *R i" and ?b = "iBasis. 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 (iBasis. (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  yi  yi  b i"
        using * i by (auto simp: cbox_def)
      moreover have "a i < xi" "xi - a i < e'" "xi < b i" "b i - xi < e'"
        using a b by auto
      ultimately have "¦xi - yi¦ < 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 (iBasis. a i *R i) (iBasis. 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  {fBasis E  × . cbox (a' f) (b' f)  M}"
  shows "M = (fI. cbox (a' f) (b' f))"
proof -
  have "x  (fI. 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  "
                                  "i  Basis. b  i  " "cbox a b  ball x e"
      using rational_cboxes[OF e(1)] by metis
    ultimately show ?thesis
       by (intro UN_I[of "λiBasis. (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 = "{fBasis E  × . 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 = {}  (iBasis. bi  ai))" (is ?th1)
    and "(cbox a b = {}  (iBasis. bi < ai))" (is ?th2)
proof -
  have False if "i  Basis" and "bi  ai" and "x  box a b" for i x
    by (smt (verit, ccfv_SIG) mem_box(1) that)
  moreover
  { assume as: "iBasis. ¬ (bi  ai)"
    let ?x = "(1/2) *R (a + b)"
    { fix i :: 'a
      assume i: "i  Basis"
      have "ai < bi"
        using as i by fastforce
      then have "ai < ((1/2) *R (a+b))  i" "((1/2) *R (a+b))  i < bi"
        by (auto simp: inner_add_left)
    }
    then have "box a b  {}"
      by (metis (no_types, opaque_lifting) emptyE mem_box(1))
  }
  ultimately show ?th1 by blast

  have False if "iBasis" and "bi < ai" and "x  cbox a b" for i x
    using mem_box(2) that by force
  moreover
  have "cbox a b  {}" if "iBasis. ¬ (bi < ai)"
    by (metis emptyE linorder_linear mem_box(2) order.strict_iff_not that)
  ultimately show ?th2 by blast
qed

lemma box_ne_empty:
  fixes a :: "'a::euclidean_space"
  shows "cbox a b  {}  (iBasis. ai  bi)"
  and "box a b  {}  (iBasis. ai < bi)"
  unfolding box_eq_empty[of a b] by fastforce+

lemma
  fixes a :: "'a::euclidean_space"
  shows cbox_idem [simp]: "cbox a a = {a}"
    and box_idem [simp]: "box a a = {}"
  unfolding set_eq_iff mem_box eq_iff [symmetric] using euclidean_eq_iff by fastforce+

lemma subset_box_imp:
  fixes a :: "'a::euclidean_space"
  shows "(iBasis. ai  ci  di  bi)  cbox c d  cbox a b"
    and "(iBasis. ai < ci  di < bi)  cbox c d  box a b"
    and "(iBasis. ai  ci  di  bi)  box c d  cbox a b"
     and "(iBasis. ai  ci  di  bi)  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  (iBasis. ci  di)  (iBasis. ai  ci  di  bi)" (is ?th1)
    and "cbox c d  box a b  (iBasis. ci  di)  (iBasis. ai < ci  di < bi)" (is ?th2)
    and "box c d  cbox a b  (iBasis. ci < di)  (iBasis. ai  ci  di  bi)" (is ?th3)
    and "box c d  box a b  (iBasis. ci < di)  (iBasis. ai  ci  di  bi)" (is ?th4)
proof -
  let ?lesscd = "iBasis. ci < di"
  let ?lerhs = "iBasis. ai  ci  di  bi"
  show ?th1 ?th2
    by (fastforce simp: mem_box)+
  have acdb: "ai  ci  di  bi"
    if i: "i  Basis" and box: "box c d  cbox a b" and cd: "i. i  Basis  ci < di" for i
  proof -
    have "box c d  {}"
      using that
      unfolding box_eq_empty by force
    { let ?x = "(jBasis. (if j=i then ((min (aj) (dj))+cj)/2 else (cj+dj)/2) *R j)::'a"
      assume *: "ai > ci"
      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 "ai  ci" by force
    moreover
    { let ?x = "(jBasis. (if j=i then ((max (bj) (cj))+dj)/2 else (cj+dj)/2) *R j)::'a"
      assume *: "bi < di"
      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 "bi  di" by (rule ccontr) auto
    ultimately show ?thesis by auto
  qed
  show ?th3
    using acdb by (fastforce simp add: mem_box)
  have acdb': "ai  ci  di  bi"
    if "i  Basis" "box c d  box a b" "i. i  Basis  ci < di" 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)
qed auto

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
  with L subset_box show ?rhs
    by (smt (verit) SOME_Basis box_ne_empty(1))
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
    unfolding subset_box by (smt (verit) box_ne_empty(2) euclidean_eq_iff)+
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 in_cbox_complex_iff:
  "x  cbox a b  Re x  {Re a..Re b}  Im x  {Im a..Im b}"
  by (cases x; cases a; cases b) (auto simp: cbox_Complex_eq)

lemma cbox_complex_of_real: "cbox (complex_of_real x) (complex_of_real y) = complex_of_real ` {x..y}"
proof -
  have "(x  Re z  Re z  y  Im z = 0) = (z  complex_of_real ` {x..y})" for z
    by (cases z) (simp add: complex_eq_cancel_iff2 image_iff)
  then show ?thesis
    by (auto simp: in_cbox_complex_iff)
qed

lemma box_Complex_eq:
  "box (Complex a c) (Complex b d) = (λ(x,y). Complex x y) ` (box a b × box c d)"
  by (auto simp: box_def Basis_complex_def image_iff complex_eq_iff)

lemma in_box_complex_iff:
  "x  box a b  Re x  {Re a<..<Re b}  Im x  {Im a<..<Im b}"
  by (cases x; cases a; cases b) (auto simp: box_Complex_eq)

lemma box_complex_of_real [simp]: "box (complex_of_real x) (complex_of_real y) = {}"
  by (auto simp: in_box_complex_iff)

lemma Int_interval:
  fixes a :: "'a::euclidean_space"
  shows "cbox a b  cbox c d =
    cbox (iBasis. max (ai) (ci) *R i) (iBasis. min (bi) (di) *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 = {}  (iBasis. (bi < ai  di < ci  bi < ci  di < ai))" (is ?th1)
    and "cbox a b  box c d = {}  (iBasis. (bi < ai  di  ci  bi  ci  di  ai))" (is ?th2)
    and "box a b  cbox c d = {}  (iBasis. (bi  ai  di < ci  bi  ci  di  ai))" (is ?th3)
    and "box a b  box c d = {}  (iBasis. (bi  ai  di  ci  bi  ci  di  ai))" (is ?th4)
proof -
  let ?z = "(iBasis. (((max (ai) (ci)) + (min (bi) (di))) / 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. iBasis. Q x i)  (iBasis. 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. bBasis. ¦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 "iBasis. x  i  c  i" "iBasis. 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
  ultimately show ?thesis using True by (auto simp: cbox_def)
next
  case False
  {
    fix y
    assume "iBasis. a  i  y  i" "iBasis. y  i  b  i" "m > 0"
    then have "iBasis. (m *R a + c)  i  (m *R y + c)  i" 
          and "iBasis. (m *R y + c)  i  (m *R b + c)  i"
      by (auto simp: inner_distrib)
  }
  moreover
  {
    fix y
    assume "iBasis. a  i  y  i" "iBasis. y  i  b  i" "m < 0"
    then have "iBasis. (m *R b + c)  i  (m *R y + c)  i"
         and  "iBasis. (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 "iBasis. (m *R a + c)  i  y  i"
      and  "iBasis. 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 "iBasis. (m *R b + c)  i  y  i" "iBasis. 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
    by (metis assms continuous_on_compose continuous_on_swap swap_cbox_Pair)
qed

lemma open_contains_cbox:
  fixes x :: "'a :: euclidean_space"
  assumes "open A" "x  A"
  obtains a b where "cbox a b  A" "x  box a b" "iBasis. a  i < b  i"
proof -
  from assms obtain R where R: "R > 0" "ball x R  A"
    by (auto simp: open_contains_ball)
  define r :: real where "r = R / (2 * sqrt DIM('a))"
  from R > 0 have [simp]: "r > 0" by (auto simp: r_def)
  define d :: 'a where "d = r *R Topology_Euclidean_Space.One"
  have "cbox (x - d) (x + d)  A"
  proof safe
    fix y assume y: "y  cbox (x - d) (x + d)"
    have "dist x y = sqrt (iBasis. (dist (x  i) (y  i))2)"
      by (subst euclidean_dist_l2) (auto simp: L2_set_def)
    also from y have "sqrt (iBasis. (dist (x  i) (y  i))2)  sqrt (i(Basis::'a set). r2)"
      by (intro real_sqrt_le_mono sum_mono power_mono)
         (auto simp: dist_norm d_def cbox_def algebra_simps)
    also have " = sqrt (DIM('a) * r2)" by simp
    also have "DIM('a) * r2 = (R / 2) ^ 2"
      by (simp add: r_def power_divide)
    also have "sqrt  = R / 2"
      using R > 0 by simp
    also from R > 0 have " < R" by simp
    finally have "y  ball x R" by simp
    with R show "y  A" by blast
  qed
  thus ?thesis
    using that[of "x - d" "x + d"] by (auto simp: algebra_simps d_def box_def)
qed

lemma open_contains_box:
  fixes x :: "'a :: euclidean_space"
  assumes "open A" "x  A"
  obtains a b where "box a b  A" "x  box a b" "iBasis. a  i < b  i"
  by (meson assms box_subset_cbox dual_order.trans open_contains_cbox)

lemma inner_image_box:
  assumes "(i :: 'a :: euclidean_space)  Basis"
  assumes "iBasis. a  i < b  i"
  shows   "(λx. x  i) ` box a b = {a  i<..<b  i}"
proof safe
  fix x assume x: "x  {a  i<..<b  i}"
  let ?y = "(jBasis. (if i = j then x else (a + b)  j / 2) *R j)"
  from x assms have "?y  i  (λx. x  i) ` box a b"
    by (intro imageI) (auto simp: box_def algebra_simps)
  also have "?y  i = (jBasis. (if i = j then x else (a + b)  j / 2) * (j  i))"
    by (simp add: inner_sum_left)
  also have " = (jBasis. if i = j then x else 0)"
    by (intro sum.cong) (auto simp: inner_not_same_Basis assms)
  also have " = x" using assms by simp
  finally show "x  (λx. x  i) ` box a b"  .
qed (insert assms, auto simp: box_def)

lemma inner_image_cbox:
  assumes "(i :: 'a :: euclidean_space)  Basis"
  assumes "iBasis. a  i  b  i"
  shows   "(λx. x  i) ` cbox a b = {a  i..b  i}"
proof safe
  fix x assume x: "x  {a  i..b  i}"
  let ?y = "(jBasis. (if i = j then x else a  j) *R j)"
  from x assms have "?y  i  (λx. x  i) ` cbox a b"
    by (intro imageI) (auto simp: cbox_def)
  also have "?y  i = (jBasis. (if i = j then x else a  j) * (j  i))"
    by (simp add: inner_sum_left)
  also have " = (jBasis. if i = j then x else 0)"
    by (intro sum.cong) (auto simp: inner_not_same_Basis assms)
  also have " = x" using assms by simp
  finally show "x  (λx. x  i) ` cbox a b"  .
qed (insert assms, auto simp: cbox_def)

subsection ‹General Intervals›

definitiontag important› "is_interval (s::('a::euclidean_space) set) 
  (as. bs. x. (iBasis. ((ai  xi  xi  bi)  (bi  xi  xi  ai)))  x  s)"

lemma is_interval_1:
  "is_interval (s::real set)  (as. bs.  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"
  using assms is_interval_def by force

lemma interval_subst:
  fixes S::"'a::euclidean_space set"
  assumes "is_interval S"
    and "x  S" "y j  S"
    and "j  Basis"
  shows "(iBasis. (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. (iBasis. (if i = j then s i  i else Y  i) *R i))"
  have "x = (iBasis. (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]
  proof (induction bs)
    case Nil
    then show ?case
      by (simp add: j y_def)
  next
    case (Cons a bs)
    then show ?case
      using interval_subst[OF assms(1)] s by (simp add: y_def)
  qed
  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 that 
    by (force simp: mem_box intro: mem_is_intervalI)
  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"
    "iBasis. (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"
    "iBasis. (- b)  i  e  i  e  i  (- d)  i 
       (- d)  i  e  i  e  i  (- b)  i"
  hence "- e  X"
    by (smt (verit, ccfv_threshold) assms inner_minus_left mem_is_intervalI)
  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)

lemma is_interval_cball_1[intro, simp]: "is_interval (cball a b)" for a b::real
  by (simp add: cball_eq_atLeastAtMost is_interval_def)

lemma is_interval_ball_real: "is_interval (ball a b)" for a b::real
  by (simp add: ball_eq_greaterThanLessThan is_interval_def)


subsectiontag 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)


subsectiontag 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)


subsectiontag 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)


subsectiontag unimportant› ‹Openness of halfspaces.›

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

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

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

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

lemma eucl_less_eq_halfspaces:
  fixes a :: "'a::euclidean_space"
  shows "{x. x <e a} = (iBasis. {x. x  i < a  i})"
        "{x. a <e x} = (iBasis. {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)

subsectiontag unimportant› ‹Closure and Interior 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)

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

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

lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. xi  a}"
  by (simp add: closed_Collect_le continuous_on_inner)

lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. xi  a}"
  by (simp add: closed_Collect_le continuous_on_inner)

lemma closed_interval_left:
  fixes b :: "'a::euclidean_space"
  shows "closed {x::'a. iBasis. xi  bi}"
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner)

lemma closed_interval_right:
  fixes a :: "'a::euclidean_space"
  shows "closed {x::'a. iBasis. ai  xi}"
  by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner)

lemma interior_halfspace_le [simp]:
  assumes "a  0"
    shows "interior {x. a  x  b} = {x. a  x < b}"
proof -
  have *: "a  x < b" if x: "x  S" and S: "S  {x. a  x  b}" and "open S" for S x
  proof -
    obtain e where "e>0" and e: "cball x e  S"
      using open S open_contains_cball x by blast
    then have "x + (e / norm a) *R a  cball x e"
      by (simp add: dist_norm)
    then have "x + (e / norm a) *R a  S"
      using e by blast
    then have "x + (e / norm a) *R a  {x. a  x  b}"
      using S by blast
    moreover have "e * (a  a) / norm a > 0"
      by (simp add: 0 < e assms)
    ultimately show ?thesis
      by (simp add: algebra_simps)
  qed
  show ?thesis
    by (rule interior_unique) (auto simp: open_halfspace_lt *)
qed

lemma interior_halfspace_ge [simp]:
   "a  0  interior {x. a  x  b} = {x. a  x > b}"
using interior_halfspace_le [of "-a" "-b"] by simp

lemma closure_halfspace_lt [simp]:
  assumes "a  0"
    shows "closure {x. a  x < b} = {x. a  x  b}"
proof -
  have [simp]: "-{x. a  x < b} = {x. a  x  b}"
    by force
  then show ?thesis
    using interior_halfspace_ge [of a b] assms
    by (force simp: closure_interior)
qed

lemma closure_halfspace_gt [simp]:
   "a  0  closure {x. a  x > b} = {x. a  x  b}"
using closure_halfspace_lt [of "-a" "-b"] by simp

lemma interior_hyperplane [simp]:
  assumes "a  0"
    shows "interior {x. a  x = b} = {}"
proof -
  have [simp]: "{x. a  x = b} = {x. a  x  b}  {x. a  x  b}"
    by force
  then show ?thesis
    by (auto simp: assms)
qed

lemma frontier_halfspace_le:
  assumes "a  0  b  0"
    shows "frontier {x. a  x  b} = {x. a  x = b}"
proof (cases "a = 0")
  case True with assms show ?thesis by simp
next
  case False then show ?thesis
    by (force simp: frontier_def closed_halfspace_le)
qed

lemma frontier_halfspace_ge:
  assumes "a  0  b  0"
    shows "frontier {x. a  x  b} = {x. a  x = b}"
proof (cases "a = 0")
  case True with assms show ?thesis by simp
next
  case False then show ?thesis
    by (force simp: frontier_def closed_halfspace_ge)
qed

lemma frontier_halfspace_lt:
  assumes "a  0  b  0"
    shows "frontier {x. a  x < b} = {x. a  x = b}"
proof (cases "a = 0")
  case True with assms show ?thesis by simp
next
  case False then show ?thesis
    by (force simp: frontier_def interior_open open_halfspace_lt)
qed

lemma frontier_halfspace_gt:
  assumes "a  0  b  0"
    shows "frontier {x. a  x > b} = {x. a  x = b}"
proof (cases "a = 0")
  case True with assms show ?thesis by simp
next
  case False then show ?thesis
    by (force simp: frontier_def interior_open open_halfspace_gt)
qed

subsectiontag 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:"¬ (zS. 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 unfolding connected_def
    by (smt (verit, del_insts) as b disjoint_iff empty_iff mem_Collect_eq xy)
qed

lemma connected_ivt_component:
  fixes x::"'a::euclidean_space"
  shows "connected S  x  S  y  S  xk  a  a  yk  (zS.  zk = 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 "li  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  li"
  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 "li = 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 (iBasis. ((∙) i) -` {a  i <..< b  i})"
    by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
  also have "(iBasis. ((∙) 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 (iBasis. (λx. xi) -` {ai .. bi})"
    by (intro closed_INT ballI continuous_closed_vimage allI
      linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
  also have "(iBasis. (λx. xi) -` {ai .. bi}) = 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"
         using norm_Basis[OF i] e>0 by (auto simp: dist_norm)
      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 = "iBasis. ¦ai¦ + ¦bi¦"
  {
    fix x :: "'a"
    assume "i. iBasis  a  i  x  i  x  i  b  i"
    then have "(iBasis. ¦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)"
  by (metis bounded_cbox bounded_interior interior_cbox)

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)"
      by (smt (verit, best) e i mem_box mult_le_cancel_left_pos mult_left_mono x y)
    finally have "a  i < (e *R x + (1 - e) *R y)  i"
      unfolding inner_simps by auto
    moreover
    {
      have "b  i = e * (bi) + (1 - e) * (bi)"
        unfolding left_diff_distrib by simp
      also have " > e * (x  i) + (1 - e) * (y  i)"
        by (smt (verit, best) e i mem_box mult_le_cancel_left_pos mult_left_mono x y)
      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
    {
      have "N::nat. nN. inverse (real n + 1) < ε" if "ε > 0" for ε
          using reals_Archimedean [of ε] that
          by (metis inverse_inverse_eq inverse_less_imp_less nat_le_real_less order_less_trans 
                  reals_Archimedean2)
      then have "(λn. inverse (real n + 1))  0"
        unfolding lim_sequentially by(auto simp: dist_norm)
      then have "f  x"
        unfolding f_def
        using tendsto_add[OF tendsto_const, of "λn. (inverse (real n + 1)) *R ((1 / 2) *R (a + b) - x)" 0 sequentially x]
        using tendsto_scaleR [OF _ tendsto_const, of "λn. 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: "xS. norm x  b"
    using assms[unfolded bounded_pos] by auto
  define a :: 'a where "a = (iBasis. (b + 1) *R i)"
  have "(-a)i < xi" and "xi < ai" 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"
  by (meson assms bounded_subset_box_symmetric box_subset_cbox order.trans)

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)"
  by (simp add: frontier_def interior_open open_box)

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 "dBasis. l::'a.  r.
    strict_mono r  (e>0. eventually (λn. id. dist (f (r n)  i) (l  i) < e) sequentially)"
  by (rule compact_lemma_general[where unproj="λe. iBasis. e i *R i"])
     (auto intro!: assms bounded_linear_inner_left bounded_linear_image
       simp: euclidean_representation)

instancetag 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. iBasis. 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)
    with l have "eventually (λn. iBasis. dist (f (r n)  i) (l  i) < e / (real_of_nat DIM('a))) sequentially"
      by simp
    moreover
    { fix n
      assume n: "iBasis. dist (f (r n)  i) (l  i) < e / (real_of_nat DIM('a))"
      have "dist (f (r n)) l  (iBasis. dist (f (r n)  i) (l  i))"
        using L2_set_le_sum [OF zero_le_dist] by (subst euclidean_dist_l2)
      also have " < (i(Basis::'a set). e / (real_of_nat DIM('a)))"
        by (meson eucl.finite_Basis n nonempty_Basis sum_strict_mono)
      finally have "dist (f (r n)) l < e"
        by auto
    }
    ultimately have "F n in sequentially. dist (f (r n)) l < e"
      by (rule eventually_mono)
  }
  then have *: "(f  r)  l"
    unfolding o_def tendsto_iff by simp
  with r show "l r. strict_mono r  (f  r)  l"
    by auto
qed

instancetag important› euclidean_space  banach ..

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

  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"
      using open_UNION_box[OF open A]
      by (smt (verit, ccfv_threshold) B_def a b image_iff mem_Collect_eq subsetI)
  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  iBasis. (INF xS. x  i) *R i"
    define b where "b  iBasis. (SUP xS. x  i) *R i"
    have 1: "x i. x  S; i  Basis  (INF xS. 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 xS. 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 xS. x  i)  x  i"
                   and sup: "i. i  Basis  x  i  (SUP xS. 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. yS  a  i  y  i"
        using continuous_attains_inf [OF compact S False cont] by blast
      obtain b where "b  S" and b: "y. yS  y  i  b  i"
        using continuous_attains_sup [OF compact S False cont] by blast
      have "a  i  (INF xS. 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 xS. x  i)"
        by (simp add: i sup)
      also have "(SUP xS. 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="jBasis. (((∙)a)(i := x  j))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


subsectiontag 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 termi  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
    by (smt (verit) eventually_elim2 mem_Collect_eq open_preimage_inner)
next
  assume R: ?rhs
  then have "e. e > 0  iBasis. 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. iBasis. 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 "iBasis. 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))"
        by (meson DIM_positive sum_bounded_above_strict that)
      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. iBasis. dist (f x  i) (l  i) < e / DIM('b)"
      by (simp add: R' 0 < e)
    then show "F x in F. dist (f x) l < e"
      by eventually_elim (metis (full_types) "*" euclidean_dist_l2)
  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)))"
  by (metis continuous_componentwise continuous_on_eq_continuous_within)

lemma linear_componentwise_iff:
     "linear f'  (iBasis. linear (λx. f' x  i))" (is "?lhs  ?rhs")
proof
  show "?lhs  ?rhs"
    by (simp add: Real_Vector_Spaces.linear_iff inner_left_distrib)
  show "?rhs  ?lhs"
    by (simp add: linear_iff) (metis euclidean_eqI inner_left_distrib inner_scaleR_left)
qed

lemma bounded_linear_componentwise_iff:
     "(bounded_linear f')  (iBasis. bounded_linear (λx. f' x  i))"
     (is "?lhs = ?rhs")
proof
  assume ?rhs
  then have "(iBasis. 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)  (iBasis. ¦f' x  i¦)"
      by (rule norm_le_l1)
    also have "...  (iBasis. 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 (simp add: bounded_linear_inner_left_comp)

subsectiontag unimportant› ‹Continuous Extension›

definition clamp :: "'a::euclidean_space  'a  'a  'a" where
  "clamp a b x = (if (iBasis. a  i  b  i)
    then (iBasis. (if xi < ai then ai else if xi  bi then xi else bi) *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: "(iBasis. a  i  b  i)"
  then have "(iBasis. (dist (clamp a b y  i) (clamp a b x  i))2) 
    (iBasis. (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: "(iBasis. 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: 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: "(iBasis. a  i  b  i)"
  from bounded obtain c where f_bound: "xf ` cbox a b. dist undefined x  c"
    by (auto simp: bounded_any_center[where a=undefined])
  then show ?thesis
    by (metis bounded bounded_subset clamp_in_interval image_mono image_subsetI le range_composition)
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 by (simp add: ext_cont_def)

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 "
    and opn: "C. C    open C"
    and Un: "S. open S  U. U    S = U"
    using univ_second_countable by blast
  have *: "infinite (range (λn. ball (0::'a) (inverse(Suc n))))"
    by (simp add: inj_on_def ball_eq_ball_iff Infinite_Set.range_inj_infinite)
  have "infinite "
  proof
    assume "finite "
    then have "finite (Union ` (Pow ))"
      by simp
    moreover have "range (λn. ball 0 (inverse (real (Suc n))))   ` Pow "
      by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
    ultimately show False
      by (metis finite_subset *)
  qed
  obtain f :: "nat  'a set" where " = range f" "inj f"
    by (blast intro: countable_as_injective_image [OF countable  infinite ])
  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  = range f apply force
    done
  show ?thesis
    using "*"  = range f inj f opn that by force
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 "
      and "{}  "
      and ope: "C. C    openin(top_of_set S) C"
      and if_ope: "T. openin(top_of_set S) T  𝒰. 𝒰    T = 𝒰"
    by (meson subset_second_countable)
  then obtain f where f: "C. C    f C  C"
    by (metis equals0I)
  show ?thesis
  proof
    show "countable (f ` )"
      by (simp add: countable )
    show "f `   S"
      using ope f openin_imp_subset by blast
    show "S  closure (f ` )"
    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  " "S  ball x e = 𝒰"
        by meson
      show "C  . 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 show ?thesis
          by (metis IntI Union_iff 𝒰 0 < e x  S dist_commute dist_self f inf_le2 mem_ball subset_eq)
      qed
    qed
  qed
qed


subsectiontag 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))"
      using 0  r that by (simp add: dist_norm flip: scaleR_2)
    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)"
    using atLeastAtMost_eq_cball by blast
  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)"
    using greaterThanLessThan_eq_ball by blast
  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)


subsectiontag 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 field_split_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 + (iBasis. (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 + (iBasis. (u  i) *R inv f i)) x  (iBasis. norm ((u  i) *R inv f i))"
          by (simp add: dist_norm sum_norm_le)
        also have "... = (iBasis. ¦u  i¦ * norm (inv f i))"
          by simp
        also have "...  (iBasis. ¦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 + (iBasis. (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 field_split_simps)
  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 show "open A"
    by (metis assms bij_is_inj continuous_open_vimage inj_vimage_image_eq linear_continuous_at linear_linear)
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" 
  by (smt (verit) assms bij_is_inj inj_image_subset_iff interior_maximal interior_subset 
      open_bijective_linear_image_eq open_interior subset_antisym subset_imageE)

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


subsectiontag 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" "xs. f x = 0  x = 0"
  shows "e>0. xs. 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. xs  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 "bs"
    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 bs]
    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 "xs" 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 "xs. 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" "xs. f x = 0  x = 0" "closed s"
  shows "closed(f ` s)"
proof -
  obtain e where "e > 0" and e: "xs. e * norm x  norm (f x)"
    using assms injective_imp_isometric by blast
  with assms show ?thesis
    by (meson complete_eq_closed complete_isometric_image)
qed
                               

lemma closure_bounded_linear_image_subset:
  assumes f: "bounded_linear f"
  shows "f ` closure S  closure (f ` S)"
  using linear_continuous_on [OF f] closed_closure closure_subset
  by (rule image_closure_subset)

lemma closure_linear_image_subset:
  fixes f :: "'m::euclidean_space  'n::real_normed_vector"
  assumes "linear f"
  shows "f ` (closure S)  closure (f ` S)"
  using assms unfolding linear_conv_bounded_linear
  by (rule closure_bounded_linear_image_subset)

lemma closed_injective_linear_image:
    fixes f :: "'a::euclidean_space  'b::euclidean_space"
    assumes S: "closed S" and f: "linear f" "inj f"
    shows "closed (f ` S)"
proof -
  obtain g where g: "linear g" "g  f = id"
    using linear_injective_left_inverse [OF f] by blast
  then have confg: "continuous_on (range f) g"
    using linear_continuous_on linear_conv_bounded_linear by blast
  have [simp]: "g ` f ` S = S"
    using g by (simp add: image_comp)
  have cgf: "closed (g ` f ` S)"
    by (simp add: g  f = id S image_comp)
  have [simp]: "(range f  g -` S) = f ` S"
    using g unfolding o_def id_def image_def by auto metis+
  show ?thesis
  proof (rule closedin_closed_trans [of "range f"])
    show "closedin (top_of_set (range f)) (f ` S)"
      using continuous_closedin_preimage [OF confg cgf] by simp
    show "closed (range f)"
      using closed_injective_image_subspace f linear_conv_bounded_linear 
          linear_injective_0 subspace_UNIV by blast
  qed
qed

lemma closed_injective_linear_image_eq:
    fixes f :: "'a::euclidean_space  'b::euclidean_space"
    assumes f: "linear f" "inj f"
      shows "(closed(image f s)  closed s)"
  by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)

lemma closure_injective_linear_image:
    fixes f :: "'a::euclidean_space  'b::euclidean_space"
    shows "linear f; inj f  f ` (closure S) = closure (f ` S)"
  by (simp add: closed_injective_linear_image closure_linear_image_subset 
        closure_minimal closure_subset image_mono subset_antisym)

lemma closure_bounded_linear_image:
  fixes f :: "'a::euclidean_space  'b::euclidean_space"
  assumes "linear f" "bounded S"
    shows "f ` (closure S) = closure (f ` S)"  (is "?lhs = ?rhs")
proof
  show "?lhs  ?rhs"
    using assms closure_linear_image_subset by blast
  show "?rhs  ?lhs"
    using assms by (meson closure_minimal closure_subset compact_closure compact_eq_bounded_closed
                      compact_continuous_image image_mono linear_continuous_on linear_linear)
qed

lemma closure_scaleR:
  fixes S :: "'a::real_normed_vector set"
  shows "((*R) c) ` (closure S) = closure (((*R) c) ` S)"  (is "?lhs = ?rhs")
proof
  show "?lhs  ?rhs"
    using bounded_linear_scaleR_right by (rule closure_bounded_linear_image_subset)
  show "?rhs  ?lhs"
    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
qed


subsectiontag unimportant› ‹Some properties of a canonical subspace›

lemma closed_substandard: "closed {x::'a::euclidean_space. iBasis. P i  xi = 0}"
  (is "closed ?A")
proof -
  let ?D = "{iBasis. P i}"
  have "closed (i?D. {x::'a. xi = 0})"
    by (simp add: closed_INT closed_Collect_eq continuous_on_inner)
  also have "(i?D. {x::'a. xi = 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 obtain_subset_with_card_n 
  obtain d :: "'a set" where cd: "card d = dim S" and d: "d  Basis"
    by metis
  let ?t = "{x::'a. iBasis. i  d  xi = 0}"
  have "f. linear f  f ` {x::'a. iBasis. i  d  x  i = 0} = S 
      inj_on f {x::'a. iBasis. i  d  x  i = 0}"
    using dim_substandard[of d] cd 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. iBasis. i  d  x  i = 0} = S"
      "inj_on f {x. iBasis. 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
  then show ?thesis
    using closed_injective_image_subspace[of ?t f] closed_substandard subspace_substandard
    using f(2) f.bounded_linear_axioms by force
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)

lemma dim_closure [simp]: "dim (closure S) = dim S" (is "?dc = ?d")
  for S :: "'a::euclidean_space set"
  by (metis closed_span closure_minimal closure_subset dim_eq_span span_eq_dim span_superset subset_le_dim)


subsection ‹Set Distance›

lemma setdist_compact_closed:
  fixes A :: "'a::heine_borel set"
  assumes "compact A" "closed B"
    and "A  {}" "B  {}"
  shows "x  A. y  B. dist x y = setdist A B"
  by (metis assms infdist_attains_inf setdist_attains_inf setdist_sym)

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 False
  then show ?thesis
    by (metis S T disjoint_iff in_closed_iff_infdist_zero setdist_attains_inf setdist_eq_0I setdist_sym)
qed auto

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