Impressum Starlike.thy

(* Title:      HOL/Analysis/Starlike.thy
  Author: L C Paulson, University of Cambridge
  Author: Robert Himmelmann, TU Muenchen
  Author: Bogdan Grechuk, University of Edinburgh
  Author: Armin Heller, TU Muenchen
  Author: Johannes Hoelzl, TU Muenchen
*)
chapter ‹Unsorted›

theory Starlike
  imports
    Convex_Euclidean_Space
    Line_Segment
begin

lemma affine_hull_closed_segment [simp]:
     "affine hull (closed_segment a b) = affine hull {a,b}"
  by (simp add: segment_convex_hull)

lemma affine_hull_open_segment [simp]:
    fixes a :: "'a::euclidean_space"
    shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)

lemma rel_interior_closure_convex_segment:
  fixes S :: "_::euclidean_space set"
  assumes "convex S" "a ∈ rel_interior S" "b ∈ closure S"
    shows "open_segment a b ⊆ rel_interior S"
proof
  fix x
  have [simp]: "(1 - u) *🪙R a + u *🪙R b = b - (1 - u) *🪙R (b - a)" for u
    by (simp add: algebra_simps)
  assume "x ∈ open_segment a b"
  then show "x ∈ rel_interior S"
    unfolding closed_segment_def open_segment_def  using assms
    by (auto intro: rel_interior_closure_convex_shrink)
qed

lemma convex_hull_insert_segments:
   "convex hull (insert a S) =
    (if S = {} then {a} else ∪x ∈ convex hull S. closed_segment a x)"
  by (force simp add: convex_hull_insert_alt in_segment)

lemma Int_convex_hull_insert_rel_exterior:
  fixes z :: "'a::euclidean_space"
  assumes "convex C" "T ⊆ C" and z: "z ∈ rel_interior C" and dis: "disjnt S (rel_interior C)"
  shows "S ∩ (convex hull (insert z T)) = S ∩ (convex hull T)" (is "?lhs = ?rhs")
proof
  have *: "T = {} ==> z ∉ S"
    using dis z by (auto simp add: disjnt_def)
  { fix x y
    assume "x ∈ S" and y: "y ∈ convex hull T" and "x ∈ closed_segment z y"
    have "y ∈ closure C"
      by (metis y ‹convex C› ‹T ⊆ C› closure_subset contra_subsetD convex_hull_eq hull_mono)
    moreover have "x ∉ rel_interior C"
      by (meson ‹x ∈ S› dis disjnt_iff)
    moreover have "x ∈ open_segment z y ∪ {z, y}"
      using ‹x ∈ closed_segment z y› closed_segment_eq_open by blast
    ultimately have "x ∈ convex hull T"
      using rel_interior_closure_convex_segment [OF ‹convex C› z]
      using y z by blast
  }
  with * show "?lhs ⊆ ?rhs"
    by (auto simp add: convex_hull_insert_segments)
  show "?rhs ⊆ ?lhs"
    by (meson hull_mono inf_mono subset_insertI subset_refl)
qed

subsection🍋‹tag unimportant› ‹Shrinking towards the interior of a convex set›

lemma mem_interior_convex_shrink:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S"
    and "c ∈ interior S"
    and "x ∈ S"
    and "0 < e"
    and "e ≤ 1"
  shows "x - e *🪙R (x - c) ∈ interior S"
proof -
  obtain d where "d > 0" and d: "ball c d ⊆ S"
    using assms(2) unfolding mem_interior by auto
  show ?thesis
    unfolding mem_interior
  proof (intro exI subsetI conjI)
    fix y
    assume "y ∈ ball (x - e *🪙R (x - c)) (e*d)"
    then have as: "dist (x - e *🪙R (x - c)) y < e * d"
      by simp
    have *: "y = (1 - (1 - e)) *🪙R ((1 / e) *🪙R y - ((1 - e) / e) *🪙R x) + (1 - e) *🪙R x"
      using ‹e > 0› by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
    have "c - ((1 / e) *🪙R y - ((1 - e) / e) *🪙R x) = (1 / e) *🪙R (e *🪙R c - y + (1 - e) *🪙R x)"
      using ‹e > 0›
      by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
    then have "dist c ((1 / e) *🪙R y - ((1 - e) / e) *🪙R x) = ∣1/e∣ * norm (e *🪙R c - y + (1 - e) *🪙R x)"
      by (simp add: dist_norm)
    also have "… = ∣1/e∣ * norm (x - e *🪙R (x - c) - y)"
      by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
    also have "… < d"
      using as[unfolded dist_norm] and ‹e > 0›
      by (auto simp add:pos_divide_less_eq[OF ‹e > 0›] mult.commute)
    finally have "(1 - (1 - e)) *🪙R ((1 / e) *🪙R y - ((1 - e) / e) *🪙R x) + (1 - e) *🪙R x ∈ S"
      using assms(3-5) d
      by (intro convexD_alt [OF ‹convex S›]) (auto intro: convexD_alt [OF ‹convex S›])
    with ‹e > 0› show "y ∈ S"
      by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
  qed (use ‹e>0› ‹d>0› in auto)
qed

lemma mem_interior_closure_convex_shrink:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S"
    and "c ∈ interior S"
    and "x ∈ closure S"
    and "0 < e"
    and "e ≤ 1"
  shows "x - e *🪙R (x - c) ∈ interior S"
proof -
  obtain d where "d > 0" and d: "ball c d ⊆ S"
    using assms(2) unfolding mem_interior by auto
  have "∃y∈S. norm (y - x) * (1 - e) < e * d"
  proof (cases "x ∈ S")
    case True
    then show ?thesis
      using ‹e > 0› ‹d > 0› by force
  next
    case False
    then have x: "x islimpt S"
      using assms(3)[unfolded closure_def] by auto
    show ?thesis
    proof (cases "e = 1")
      case True
      obtain y where "y ∈ S" "y ≠ x" "dist y x < 1"
        using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
      then show ?thesis
        using True ‹0 🚫› by auto
    next
      case False
      then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
        using ‹e ≤ 1› ‹e > 0› ‹d > 0› by auto
      then obtain y where "y ∈ S" "y ≠ x" "dist y x < e * d / (1 - e)"
        using islimpt_approachable x by blast
      then have "norm (y - x) * (1 - e) < e * d"
        by (metis "*" dist_norm mult_imp_div_pos_le not_less)
      then show ?thesis
        using ‹y ∈ S› by blast
    qed
  qed
  then obtain y where "y ∈ S" and y: "norm (y - x) * (1 - e) < e * d"
    by auto
  define z where "z = c + ((1 - e) / e) *🪙R (x - y)"
  have *: "x - e *🪙R (x - c) = y - e *🪙R (y - z)"
    unfolding z_def using ‹e > 0›
    by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
  have "(1 - e) * norm (x - y) / e < d"
    using y ‹0 🚫› by (simp add: field_simps norm_minus_commute)
  then have "z ∈ interior (ball c d)"
    using ‹0 🚫› ‹e ≤ 1› by (simp add: interior_open[OF open_ball] z_def dist_norm)
  then have "z ∈ interior S"
    using d interiorI interior_ball by blast
  then show ?thesis
    unfolding * using mem_interior_convex_shrink ‹y ∈ S› assms by blast
qed

lemma in_interior_closure_convex_segment:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and a: "a ∈ interior S" and b: "b ∈ closure S"
  shows "open_segment a b ⊆ interior S"
proof -
  { fix u::real
    assume u: "0 < u" "u < 1"
    have "(1 - u) *🪙R a + u *🪙R b = b - (1 - u) *🪙R (b - a)"
      by (simp add: algebra_simps)
    also have "... ∈ interior S" using mem_interior_closure_convex_shrink [OF assms] u
      by simp
    finally have "(1 - u) *🪙R a + u *🪙R b ∈ interior S" .
  }
  then show ?thesis
    by (clarsimp simp: in_segment)
qed

lemma convex_closure_interior:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and int: "interior S ≠ {}"
  shows "closure(interior S) = closure S"
proof -
  obtain a where a: "a ∈ interior S"
    using int by auto
  have "closure S ⊆ closure(interior S)"
  proof
    fix x
    assume x: "x ∈ closure S"
    show "x ∈ closure (interior S)"
    proof (cases "x=a")
      case True
      then show ?thesis
        using ‹a ∈ interior S› closure_subset by blast
    next
      case False
      { fix e::real
        assume xnotS: "x ∉ interior S" and "0 < e"
        have "∃x'∈interior S. x' ≠ x ∧ dist x' x < e"
        proof (intro bexI conjI)
          show "x - min (e/2 / norm (x - a)) 1 *🪙R (x - a) ≠ x"
            using False ‹0 🚫› by (auto simp: algebra_simps min_def)
          show "dist (x - min (e/2 / norm (x - a)) 1 *🪙R (x - a)) x < e"
            using ‹0 🚫› by (auto simp: dist_norm min_def)
          show "x - min (e/2 / norm (x - a)) 1 *🪙R (x - a) ∈ interior S"
            using ‹0 🚫› False
            by (auto simp add: min_def a intro: mem_interior_closure_convex_shrink [OF ‹convex S› a x])
        qed
      }
      then show ?thesis
        by (auto simp add: closure_def islimpt_approachable)
    qed
  qed
  then show ?thesis
    by (simp add: closure_mono interior_subset subset_antisym)
qed

lemma openin_subset_relative_interior:
  fixes S :: "'a::euclidean_space set"
  shows "openin (top_of_set (affine hull T)) S ==> (S ⊆ rel_interior T) = (S ⊆ T)"
  by (meson order.trans rel_interior_maximal rel_interior_subset)

lemma conic_hull_eq_span_affine_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "0 ∈ rel_interior S"
  shows "conic hull S = span S ∧ conic hull S = affine hull S"
proof -
  obtain ε where "ε>0" and ε: "cball 0 ε ∩ affine hull S ⊆ S"
    using assms mem_rel_interior_cball by blast
  have *: "affine hull S = span S"
    by (meson affine_hull_span_0 assms hull_inc mem_rel_interior_cball)
  moreover
  have "conic hull S ⊆ span S"
    by (simp add: hull_minimal span_superset)
  moreover
  { fix x
    assume "x ∈ affine hull S"
    have "x ∈ conic hull S"
    proof (cases "x=0")
      case True
      then show ?thesis
        using ‹x ∈ affine hull S› by auto
    next
      case False
      then have "(ε / norm x) *🪙R x ∈ cball 0 ε ∩ affine hull S"
        using ‹0 🚫ε› ‹x ∈ affine hull S› * span_mul by fastforce
      then have "(ε / norm x) *🪙R x ∈ S"
        by (meson ε subsetD)
      then have "∃c xa. x = c *🪙R xa ∧ 0 ≤ c ∧ xa ∈ S"
        by (smt (verit, del_insts) ‹0 🚫ε› divide_nonneg_nonneg eq_vector_fraction_iff norm_eq_zero norm_ge_zero)
      then show ?thesis
        by (simp add: conic_hull_explicit)
    qed
  }
  then have "affine hull S ⊆ conic hull S"
    by auto
  ultimately show ?thesis
    by blast
qed

lemma conic_hull_eq_span:
  fixes S :: "'a::euclidean_space set"
  assumes "0 ∈ rel_interior S"
  shows "conic hull S = span S"
  by (simp add: assms conic_hull_eq_span_affine_hull)

lemma conic_hull_eq_affine_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "0 ∈ rel_interior S"
  shows "conic hull S = affine hull S"
  using assms conic_hull_eq_span_affine_hull by blast

lemma conic_hull_eq_span_eq:
  fixes S :: "'a::euclidean_space set"
  shows "0 ∈ rel_interior(conic hull S) ⟷ conic hull S = span S" (is "?lhs = ?rhs")
proof
  show "?lhs ==> ?rhs"
    by (metis conic_hull_eq_span conic_span hull_hull hull_minimal hull_subset span_eq)
  show "?rhs ==> ?lhs"
  by (metis rel_interior_affine subspace_affine subspace_span)
qed

lemma aff_dim_psubset:
   "(affine hull S) ⊂ (affine hull T) ==> aff_dim S < aff_dim T"
  by (metis aff_dim_affine_hull aff_dim_empty aff_dim_subset affine_affine_hull affine_dim_equal order_less_le)

lemma aff_dim_eq_full_gen:
   "S ⊆ T ==> (aff_dim S = aff_dim T ⟷ affine hull S = affine hull T)"
  by (smt (verit, del_insts) aff_dim_affine_hull2 aff_dim_psubset hull_mono psubsetI)

lemma aff_dim_eq_full:
  fixes S :: "'n::euclidean_space set"
  shows "aff_dim S = (DIM('n)) ⟷ affine hull S = UNIV"
  by (metis aff_dim_UNIV aff_dim_affine_hull affine_hull_UNIV)

lemma closure_convex_Int_superset:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "interior S ≠ {}" "interior S ⊆ closure T"
  shows "closure(S ∩ T) = closure S"
proof -
  have "closure S ⊆ closure(interior S)"
    by (simp add: convex_closure_interior assms)
  also have "... ⊆ closure (S ∩ T)"
    using interior_subset [of S] assms
    by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
  finally show ?thesis
    by (simp add: closure_mono dual_order.antisym)
qed


subsection🍋‹tag unimportant› ‹Some obvious but surprisingly hard simplex lemmas›

lemma simplex:
  assumes "finite S"
    and "0 ∉ S"
  shows "convex hull (insert 0 S) = {y. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S ≤ 1 ∧ sum (λx. u x *🪙R x) S = y}"
proof -
  { fix x and u :: "'a ==> real"
    assume "∀x∈S. 0 ≤ u x" "sum u S ≤ 1"
    then have "∃v. 0 ≤ v 0 ∧ (∀x∈S. 0 ≤ v x) ∧ v 0 + sum v S = 1 ∧ (∑x∈S. v x *🪙R x) = (∑x∈S. u x *🪙R x)"
      by (rule_tac x="λx. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
  }
  then show ?thesis by (auto simp: convex_hull_finite set_eq_iff assms)
qed

lemma substd_simplex:
  assumes d: "d ⊆ Basis"
  shows "convex hull (insert 0 d) =
    {x. (∀i∈Basis. 0 ≤ x∙i) ∧ (∑i∈d. x∙i) ≤ 1 ∧ (∀i∈Basis. i ∉ d ⟶ x∙i = 0)}"
  (is "convex hull (insert 0 ?p) = ?s")
proof -
  let ?D = d
  have "0 ∉ ?p"
    using assms by (auto simp: image_def)
  from d have "finite d"
    by (blast intro: finite_subset finite_Basis)
  show ?thesis
    unfolding simplex[OF ‹finite d› ‹0 ∉ ?p›]
  proof (intro set_eqI; safe)
    fix u :: "'a ==> real"
    assume as: "∀x∈?D. 0 ≤ u x" "sum u ?D ≤ 1" 
    let ?x = "(∑x∈?D. u x *🪙R x)"
    have ind: "∀i∈Basis. i ∈ d ⟶ u i = ?x ∙ i"
      and notind: "(∀i∈Basis. i ∉ d ⟶ ?x ∙ i = 0)"
      using substdbasis_expansion_unique[OF assms] by blast+
    then have **: "sum u ?D = sum ((∙) ?x) ?D"
      using assms by (meson subset_iff sum.cong)
    show "0 ≤ ?x ∙ i" if "i ∈ Basis" for i
      using as(1) ind notind that by fastforce
    show "sum ((∙) ?x) ?D ≤ 1"
      using "**" as(2) by linarith
    show "?x ∙ i = 0" if "i ∈ Basis" "i ∉ d" for i
      using notind that by blast
  next
    fix x 
    assume "∀i∈Basis. 0 ≤ x ∙ i" "sum ((∙) x) ?D ≤ 1" "(∀i∈Basis. i ∉ d ⟶ x ∙ i = 0)"
    with d show "∃u. (∀x∈?D. 0 ≤ u x) ∧ sum u ?D ≤ 1 ∧ (∑x∈?D. u x *🪙R x) = x"
      unfolding substdbasis_expansion_unique[OF assms] 
      by (rule_tac x="inner x" in exI) auto
  qed
qed

lemma std_simplex:
  "convex hull (insert 0 Basis) =
    {x::'a::euclidean_space. (∀i∈Basis. 0 ≤ x∙i) ∧ sum (λi. x∙i) Basis ≤ 1}"
  using substd_simplex[of Basis] by auto

lemma interior_std_simplex:
  "interior (convex hull (insert 0 Basis)) =
    {x::'a::euclidean_space. (∀i∈Basis. 0 < x∙i) ∧ sum (λi. x∙i) Basis < 1}"
  unfolding set_eq_iff mem_interior std_simplex
proof (intro allI iffI CollectI; clarify)
  fix x :: 'a
  fix e
  assume "e > 0" and as: "ball x e ⊆ {x. (∀i∈Basis. 0 ≤ x ∙ i) ∧ sum ((∙) x) Basis ≤ 1}"
  show "(∀i∈Basis. 0 < x ∙ i) ∧ sum ((∙) x) Basis < 1"
  proof (intro strip conjI)
    fix i :: 'a
    assume i: "i ∈ Basis"
    then show "0 < x ∙ i"
      using as[THEN subsetD[where c="x - (e/2) *🪙R i"]] and ‹e > 0›
      by (force simp add: inner_simps)
  next
    obtain i::'a where i: "i ∈ Basis"
      using nonempty_Basis by blast
    have **: "dist x (x + (e/2) *🪙R i) < e" using ‹e > 0›
      unfolding dist_norm
      by (auto intro!: mult_strict_left_mono simp: i)
    have "∧i. i ∈ Basis ==> (x + (e/2) *🪙R i) ∙ i = x∙i + (if i = i then e/2 else 0)"
      by (auto simp: inner_simps)
    then have *: "sum ((∙) (x + (e/2) *🪙R i)) Basis = sum (λj. x∙j + (if j = i then e/2 else 0)) Basis"
      using i by (auto simp: inner_Basis inner_left_distrib intro!: sum.cong)
    have "sum ((∙) x) Basis < sum ((∙) (x + (e/2) *🪙R i)) Basis"
      using ‹e > 0› DIM_positive by (auto simp: i sum.distrib *)
    also have "… ≤ 1"
      using ** as by force
    finally show "sum ((∙) x) Basis < 1" by auto
  qed 
next
  fix x :: 'a
  assume as: "∀i∈Basis. 0 < x ∙ i" "sum ((∙) x) Basis < 1"
  obtain a :: 'b where "a ∈ UNIV" using UNIV_witness ..
  let ?d = "(1 - sum ((∙) x) Basis) / real (DIM('a))"
  show "∃e>0. ball x e ⊆ {x. (∀i∈Basis. 0 ≤ x ∙ i) ∧ sum ((∙) x) Basis ≤ 1}"
  proof (intro exI conjI subsetI CollectI)
    fix y
    assume y: "y ∈ ball x (min (Min ((∙) x ` Basis)) ?d)"
    have "sum ((∙) y) Basis ≤ sum (λi. x∙i + ?d) Basis"
    proof (rule sum_mono)
      fix i :: 'a
      assume i: "i ∈ Basis"
      have "∣y∙i - x∙i∣ ≤ norm (y - x)"
        by (metis Basis_le_norm i inner_commute inner_diff_right)
      also have "... < ?d"
        using y by (simp add: dist_norm norm_minus_commute)
      finally have "∣y∙i - x∙i∣ < ?d" .
      then show "y ∙ i ≤ x ∙ i + ?d" by auto
    qed
    also have "… ≤ 1"
      unfolding sum.distrib sum_constant
      by (auto simp add: Suc_le_eq)
    finally show "sum ((∙) y) Basis ≤ 1" .
    show "(∀i∈Basis. 0 ≤ y ∙ i)"
    proof (intro strip)
      fix i :: 'a
      assume i: "i ∈ Basis"
      have "norm (x - y) < Min (((∙) x) ` Basis)"
        using y by (auto simp: dist_norm less_eq_real_def)
      also have "... ≤ x∙i"
        using i by auto
      finally have "norm (x - y) < x∙i" .
      then show "0 ≤ y∙i"
        using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
        by (auto simp: inner_simps)
    qed
  next
    have "Min (((∙) x) ` Basis) > 0"
      using as by simp
    moreover have "?d > 0"
      using as by (auto simp: Suc_le_eq)
    ultimately show "0 < min (Min ((∙) x ` Basis)) ((1 - sum ((∙) x) Basis) / real DIM('a))"
      by linarith
  qed 
qed

lemma interior_std_simplex_nonempty:
  obtains a :: "'a::euclidean_space" where
    "a ∈ interior(convex hull (insert 0 Basis))"
proof -
  let ?D = "Basis :: 'a set"
  let ?a = "sum (λb::'a. inverse (2 * real DIM('a)) *🪙R b) Basis"
  {
    fix i :: 'a
    assume i: "i ∈ Basis"
    have "?a ∙ i = inverse (2 * real DIM('a))"
      by (rule trans[of _ "sum (λj. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
         (simp_all add: sum.If_cases i) }
  note ** = this
  show ?thesis
  proof
    show "?a ∈ interior(convex hull (insert 0 Basis))"
      unfolding interior_std_simplex mem_Collect_eq
    proof safe
      fix i :: 'a
      assume i: "i ∈ Basis"
      show "0 < ?a ∙ i"
        unfolding **[OF i] by (auto simp add: Suc_le_eq)
    next
      have "sum ((∙) ?a) ?D = sum (λi. inverse (2 * real DIM('a))) ?D"
        by simp
      also have "… < 1"
        unfolding sum_constant divide_inverse[symmetric]
        by (auto simp add: field_simps)
      finally show "sum ((∙) ?a) ?D < 1" by auto
    qed
  qed
qed

lemma rel_interior_substd_simplex:
  assumes D: "D ⊆ Basis"
  shows "rel_interior (convex hull (insert 0 D)) =
         {x::'a::euclidean_space. (∀i∈D. 0 < x∙i) ∧ (∑i∈D. x∙i) < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0)}"
     (is "_ = ?s")
proof -
  have "finite D"
    using D finite_Basis finite_subset by blast
  show ?thesis
  proof (cases "D = {}")
    case True
    then show ?thesis
      using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
  next
    case False
    have h0: "affine hull (convex hull (insert 0 D)) =
              {x::'a::euclidean_space. (∀i∈Basis. i ∉ D ⟶ x∙i = 0)}"
      using affine_hull_convex_hull affine_hull_substd_basis assms by auto
    have aux: "∧x::'a. ∀i∈Basis. (∀i∈D. 0 ≤ x∙i) ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0) ⟶ 0 ≤ x∙i"
      by auto
    {
      fix x :: "'a::euclidean_space"
      assume x: "x ∈ rel_interior (convex hull (insert 0 D))"
      then obtain e where "e > 0" and
        "ball x e ∩ {xa. (∀i∈Basis. i ∉ D ⟶ xa∙i = 0)} ⊆ convex hull (insert 0 D)"
        using mem_rel_interior_ball[of x "convex hull (insert 0 D)"] h0 by auto
      then have as: "∧y. [dist x y < e ∧ (∀i∈Basis. i ∉ D ⟶ y∙i = 0)] ==>
                            (∀i∈D. 0 ≤ y ∙ i) ∧ sum ((∙) y) D ≤ 1"
        using assms by (force simp: substd_simplex)
      have x0: "(∀i∈Basis. i ∉ D ⟶ x∙i = 0)"
        using x rel_interior_subset  substd_simplex[OF assms] by auto
      have "(∀i∈D. 0 < x ∙ i) ∧ sum ((∙) x) D < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x∙i = 0)"
      proof (intro conjI ballI)
        fix i :: 'a
        assume "i ∈ D"
        then have "∀j∈D. 0 ≤ (x - (e/2) *🪙R i) ∙ j"
          using D ‹e > 0› x0
          by (intro as[THEN conjunct1]) (force simp: dist_norm inner_simps inner_Basis)
        then show "0 < x ∙ i"
          using ‹e > 0› ‹i ∈ D› D  by (force simp: inner_simps inner_Basis)
      next
        obtain a where a: "a ∈ D"
          using ‹D ≠ {}› by auto
        then have **: "dist x (x + (e/2) *🪙R a) < e"
          using ‹e > 0› norm_Basis[of a] D by (auto simp: dist_norm)
        have "∧i. i ∈ Basis ==> (x + (e/2) *🪙R a) ∙ i = x∙i + (if i = a then e/2 else 0)"
          using a D by (auto simp: inner_simps inner_Basis)
        then have *: "sum ((∙) (x + (e/2) *🪙R a)) D = sum (λi. x∙i + (if a = i then e/2 else 0)) D"
          using D by (intro sum.cong) auto
        have "a ∈ Basis"
          using ‹a ∈ D› D by auto
        then have h1: "(∀i∈Basis. i ∉ D ⟶ (x + (e/2) *🪙R a) ∙ i = 0)"
          using x0 D ‹a∈D› by (auto simp add: inner_add_left inner_Basis)
        have "sum ((∙) x) D < sum ((∙) (x + (e/2) *🪙R a)) D"
          using ‹e > 0› ‹a ∈ D› ‹finite D› by (auto simp add: * sum.distrib)
        also have "… ≤ 1"
          using ** h1 as[rule_format, of "x + (e/2) *🪙R a"]
          by auto
        finally show "sum ((∙) x) D < 1" "∧i. i∈Basis ==> i ∉ D ⟶ x∙i = 0"
          using x0 by auto
      qed
    }
    moreover
    {
      fix x :: "'a::euclidean_space"
      assume as: "x ∈ ?s"
      have "∀i. 0 < x∙i ∨ 0 = x∙i ⟶ 0 ≤ x∙i"
        by auto
      moreover have "∀i. i ∈ D ∨ i ∉ D" by auto
      ultimately
      have "∀i. (∀i∈D. 0 < x∙i) ∧ (∀i. i ∉ D ⟶ x∙i = 0) ⟶ 0 ≤ x∙i"
        by metis
      then have h2: "x ∈ convex hull (insert 0 D)"
        using as assms by (force simp add: substd_simplex)
      obtain a where a: "a ∈ D"
        using ‹D ≠ {}› by auto
      define d where "d ≡ (1 - sum ((∙) x) D) / real (card D)"
      have "∃e>0. ball x e ∩ {x. ∀i∈Basis. i ∉ D ⟶ x ∙ i = 0} ⊆ convex hull insert 0 D"
        unfolding substd_simplex[OF assms]
      proof (intro exI; safe)
        have "0 < card D" using ‹D ≠ {}› ‹finite D›
          by (simp add: card_gt_0_iff)
        have "Min (((∙) x) ` D) > 0"
          using as ‹D ≠ {}› ‹finite D› by (simp)
        moreover have "d > 0" 
          using as ‹0 🚫 D› by (auto simp: d_def)
        ultimately show "min (Min (((∙) x) ` D)) d > 0"
          by auto
        fix y :: 'a
        assume y2: "∀i∈Basis. i ∉ D ⟶ y∙i = 0"
        assume "y ∈ ball x (min (Min ((∙) x ` D)) d)"
        then have y: "dist x y < min (Min ((∙) x ` D)) d"
          by auto
        have "sum ((∙) y) D ≤ sum (λi. x∙i + d) D"
        proof (rule sum_mono)
          fix i
          assume "i ∈ D"
          with D have i: "i ∈ Basis"
            by auto
          have "∣y∙i - x∙i∣ ≤ norm (y - x)"
            by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
          also have "... < d"
            by (metis dist_norm min_less_iff_conj norm_minus_commute y)
          finally have "∣y∙i - x∙i∣ < d" .
          then show "y ∙ i ≤ x ∙ i + d" by auto
        qed
        also have "… ≤ 1"
          unfolding sum.distrib sum_constant d_def using ‹0 🚫 D›
          by auto
        finally show "sum ((∙) y) D ≤ 1" .

        fix i :: 'a
        assume i: "i ∈ Basis"
        then show "0 ≤ y∙i"
        proof (cases "i∈D")
          case True
          have "norm (x - y) < x∙i"
            using y Min_gr_iff[of "(∙) x ` D" "norm (x - y)"] ‹0 🚫 D› ‹i ∈ D›
            by (simp add: dist_norm card_gt_0_iff)
          then show "0 ≤ y∙i"
            using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
            by (auto simp: inner_simps)
        qed (use y2 in auto)
      qed
      then have "x ∈ rel_interior (convex hull (insert 0 D))"
        using h0 h2 rel_interior_ball by force
    }
    ultimately have
      "∧x. x ∈ rel_interior (convex hull insert 0 D) ⟷
        x ∈ {x. (∀i∈D. 0 < x ∙ i) ∧ sum ((∙) x) D < 1 ∧ (∀i∈Basis. i ∉ D ⟶ x ∙ i = 0)}"
      by blast
    then show ?thesis by (rule set_eqI)
  qed
qed

lemma rel_interior_substd_simplex_nonempty:
  assumes "D ≠ {}"
    and "D ⊆ Basis"
  obtains a :: "'a::euclidean_space"
    where "a ∈ rel_interior (convex hull (insert 0 D))"
proof -
  let ?a = "(∑b∈D. b /🪙R (2 * real (card D)))"
  have "finite D"
    using assms finite_Basis infinite_super by blast
  then have d1: "0 < real (card D)"
    using ‹D ≠ {}› by auto
  {
    fix i
    assume "i ∈ D"
    have "?a ∙ i = (∑j∈D. if i = j then inverse (2 * real (card D)) else 0)"
      unfolding inner_sum_left
      using ‹i ∈ D› by (auto simp: inner_Basis subsetD[OF assms(2)] intro: sum.cong)
    also have "... = inverse (2 * real (card D))"
      using ‹i ∈ D› ‹finite D› by auto
    finally have "?a ∙ i = inverse (2 * real (card D))" .
  }
  note ** = this
  show ?thesis
  proof
    show "?a ∈ rel_interior (convex hull (insert 0 D))"
      unfolding rel_interior_substd_simplex[OF assms(2)] 
    proof safe
      fix i
      assume "i ∈ D"
      have "0 < inverse (2 * real (card D))"
        using d1 by auto
      also have "… = ?a ∙ i" using **[of i] ‹i ∈ D›
        by auto
      finally show "0 < ?a ∙ i" by auto
    next
      have "sum ((∙) ?a) D = sum (λi. inverse (2 * real (card D))) D"
        by (rule sum.cong [OF refl **]) 
      also have "… < 1"
        unfolding sum_constant divide_real_def[symmetric]
        by (auto simp add: field_simps)
      finally show "sum ((∙) ?a) D < 1" by auto
    next
      fix i
      assume "i ∈ Basis" and "i ∉ D"
      have "?a ∈ span D"
      proof (rule span_sum[of D "(λb. b /🪙R (2 * real (card D)))" D])
        {
          fix x :: "'a::euclidean_space"
          assume "x ∈ D"
          then have "x ∈ span D"
            using span_base[of _ "D"] by auto
          then have "x /🪙R (2 * real (card D)) ∈ span D"
            using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
        }
        then show "∧x. x∈D ==> x /🪙R (2 * real (card D)) ∈ span D"
          by auto
      qed
      then show "?a ∙ i = 0 "
        using ‹i ∉ D› unfolding span_substd_basis[OF assms(2)] using ‹i ∈ Basis› by auto
    qed
  qed
qed

subsection🍋‹tag unimportant› ‹Relative interior of convex set›

lemma rel_interior_convex_nonempty_aux:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "0 ∈ S"
  shows "rel_interior S ≠ {}"
proof (cases "S = {0}")
  case True
  then show ?thesis using rel_interior_sing by auto
next
  case False
  obtain B where B: "independent B ∧ B ≤ S ∧ S ≤ span B ∧ card B = dim S"
    using basis_exists[of S] by metis
  then have "B ≠ {}"
    using B assms ‹S ≠ {0}› span_empty by auto
  have "insert 0 B ≤ span B"
    using subspace_span[of B] subspace_0[of "span B"]
      span_superset by auto
  then have "span (insert 0 B) ≤ span B"
    using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
  then have "convex hull insert 0 B ≤ span B"
    using convex_hull_subset_span[of "insert 0 B"] by auto
  then have "span (convex hull insert 0 B) ≤ span B"
    using span_span[of B]
      span_mono[of "convex hull insert 0 B" "span B"] by blast
  then have *: "span (convex hull insert 0 B) = span B"
    using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  then have "span (convex hull insert 0 B) = span S"
    using B span_mono[of B S] span_mono[of S "span B"]
      span_span[of B] by auto
  moreover have "0 ∈ affine hull (convex hull insert 0 B)"
    using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
  ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
    using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
      assms hull_subset[of S]
    by auto
  obtain d and f :: "'n ==> 'n" where
    fd: "card d = card B" "linear f" "f ` B = d"
      "f ` span B = {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = (0::real)} ∧ inj_on f (span B)"
    and d: "d ⊆ Basis"
    using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
  then have "bounded_linear f"
    using linear_conv_bounded_linear by auto
  have "d ≠ {}"
    using fd B ‹B ≠ {}› by auto
  have "insert 0 d = f ` (insert 0 B)"
    using fd linear_0 by auto
  then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
    using convex_hull_linear_image[of f "(insert 0 d)"]
      convex_hull_linear_image[of f "(insert 0 B)"] ‹linear f›
    by auto
  moreover have "rel_interior (f ` (convex hull insert 0 B)) = f ` rel_interior (convex hull insert 0 B)"
  proof (rule rel_interior_injective_on_span_linear_image[OF ‹bounded_linear f›])
    show "inj_on f (span (convex hull insert 0 B))"
      using fd * by auto
  qed
  ultimately have "rel_interior (convex hull insert 0 B) ≠ {}"
    using rel_interior_substd_simplex_nonempty[OF ‹d ≠ {}› d] by fastforce
  moreover have "convex hull (insert 0 B) ⊆ S"
    using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
  ultimately show ?thesis
    using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
qed

lemma rel_interior_eq_empty:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior S = {} ⟷ S = {}"
proof -
  {
    assume "S ≠ {}"
    then obtain a where "a ∈ S" by auto
    then have "0 ∈ (+) (-a) ` S"
      using assms exI[of "(λx. x ∈ S ∧ - a + x = 0)" a] by auto
    then have "rel_interior ((+) (-a) ` S) ≠ {}"
      using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
        convex_translation[of S "-a"] assms
      by auto
    then have "rel_interior S ≠ {}"
      using rel_interior_translation [of "- a"] by simp
  }
  then show ?thesis by auto
qed

lemma interior_simplex_nonempty:
  fixes S :: "'N :: euclidean_space set"
  assumes "independent S" "finite S" "card S = DIM('N)"
  obtains a where "a ∈ interior (convex hull (insert 0 S))"
proof -
  have "affine hull (insert 0 S) = UNIV"
    by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
         assms(1) assms(3) dim_eq_card_independent)
  moreover have "rel_interior (convex hull insert 0 S) ≠ {}"
    using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
  ultimately have "interior (convex hull insert 0 S) ≠ {}"
    by (simp add: rel_interior_interior)
  with that show ?thesis
    by auto
qed

lemma convex_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "convex (rel_interior S)"
proof -
  {
    fix x y and u :: real
    assume assm: "x ∈ rel_interior S" "y ∈ rel_interior S" "0 ≤ u" "u ≤ 1"
    then have "x ∈ S"
      using rel_interior_subset by auto
    have "x - u *🪙R (x-y) ∈ rel_interior S"
    proof (cases "0 = u")
      case False
      then have "0 < u" using assm by auto
      then show ?thesis
        using assm rel_interior_convex_shrink[of S y x u] assms ‹x ∈ S› by auto
    next
      case True
      then show ?thesis using assm by auto
    qed
    then have "(1 - u) *🪙R x + u *🪙R y ∈ rel_interior S"
      by (simp add: algebra_simps)
  }
  then show ?thesis
    unfolding convex_alt by auto
qed

lemma convex_closure_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "closure (rel_interior S) = closure S"
proof -
  have h1: "closure (rel_interior S) ≤ closure S"
    using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
  show ?thesis
  proof (cases "S = {}")
    case False
    then obtain a where a: "a ∈ rel_interior S"
      using rel_interior_eq_empty assms by auto
    { fix x
      assume x: "x ∈ closure S"
      {
        assume "x = a"
        then have "x ∈ closure (rel_interior S)"
          using a unfolding closure_def by auto
      }
      moreover
      {
        assume "x ≠ a"
         {
           fix e :: real
           assume "e > 0"
           define e1 where "e1 = min 1 (e/norm (x - a))"
           then have e1: "e1 > 0" "e1 ≤ 1" "e1 * norm (x - a) ≤ e"
             using ‹x ≠ a› ‹e > 0› le_divide_eq[of e1 e "norm (x - a)"]
             by simp_all
           then have *: "x - e1 *🪙R (x - a) ∈ rel_interior S"
             using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
             by auto
           have "∃y. y ∈ rel_interior S ∧ y ≠ x ∧ dist y x ≤ e"
             using "*" ‹x ≠ a› e1 by force
        }
        then have "x islimpt rel_interior S"
          unfolding islimpt_approachable_le by auto
        then have "x ∈ closure(rel_interior S)"
          unfolding closure_def by auto
      }
      ultimately have "x ∈ closure(rel_interior S)" by auto
    }
    then show ?thesis using h1 by auto
  qed auto
qed

lemma empty_interior_subset_hyperplane_aux:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "0 ∈ S" and empty_int: "interior S = {}"
  shows "∃a b. a≠0 ∧ S ⊆ {x. a ∙ x = b}"
proof -
  have False if "∧a. a = 0 ∨ (∀b. ∃T ∈ S. a ∙ T ≠ b)"
  proof -
    have rel_int: "rel_interior S ≠ {}"
      using assms rel_interior_eq_empty by auto
    moreover 
    have "dim S ≠ dim (UNIV::'a set)"
      by (metis aff_dim_zero affine_hull_UNIV ‹0 ∈ S› dim_UNIV empty_int hull_inc rel_int rel_interior_interior)
    then obtain a where "a ≠ 0" and a: "span S ⊆ {x. a ∙ x = 0}"
      using lowdim_subset_hyperplane
      by (metis dim_UNIV dim_subset_UNIV order_less_le)
    have "span UNIV = span S"
      by (metis span_base span_not_UNIV_orthogonal that)
    then have "UNIV ⊆ affine hull S"
      by (simp add: ‹0 ∈ S› hull_inc affine_hull_span_0)
    ultimately show False
      using ‹rel_interior S ≠ {}› empty_int rel_interior_interior by blast
  qed
  then show ?thesis
    by blast
qed

lemma empty_interior_subset_hyperplane:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and int: "interior S = {}"
  obtains a b where "a ≠ 0" "S ⊆ {x. a ∙ x = b}"
proof (cases "S = {}")
  case True
  then show ?thesis
    using that by blast
next
  case False
  then obtain u where "u ∈ S"
    by blast
  have "∃a b. a ≠ 0 ∧ (λx. x - u) ` S ⊆ {x. a ∙ x = b}"
  proof (rule empty_interior_subset_hyperplane_aux)
    show "convex ((λx. x - u) ` S)"
      using ‹convex S› by force
    show "0 ∈ (λx. x - u) ` S"
      by (simp add: ‹u ∈ S›)
    show "interior ((λx. x - u) ` S) = {}"
      by (simp add: int interior_translation_subtract)
  qed
  then obtain a b where "a ≠ 0" and ab: "(λx. x - u) ` S ⊆ {x. a ∙ x = b}"
    by metis
  then have "S ⊆ {x. a ∙ x = b + (a ∙ u)}"
    using ab by (auto simp: algebra_simps)
  then show ?thesis
    using ‹a ≠ 0› that by auto
qed

lemma rel_interior_same_affine_hull:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "affine hull (rel_interior S) = affine hull S"
  by (metis assms closure_same_affine_hull convex_closure_rel_interior)

lemma rel_interior_aff_dim:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "aff_dim (rel_interior S) = aff_dim S"
  by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)

lemma rel_interior_rel_interior:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior (rel_interior S) = rel_interior S"
proof -
  have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
    using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
  then show ?thesis
    using rel_interior_def by auto
qed

lemma rel_interior_rel_open:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_open (rel_interior S)"
  unfolding rel_open_def using rel_interior_rel_interior assms by auto

lemma convex_rel_interior_closure_aux:
  fixes x y z :: "'n::euclidean_space"
  assumes "0 < a" "0 < b" "(a + b) *🪙R z = a *🪙R x + b *🪙R y"
  obtains e where "0 < e" "e < 1" "z = y - e *🪙R (y - x)"
proof -
  define e where "e = a / (a + b)"
  have "z = (1 / (a + b)) *🪙R ((a + b) *🪙R z)"
    using assms  by (simp add: eq_vector_fraction_iff)
  also have "… = (1 / (a + b)) *🪙R (a *🪙R x + b *🪙R y)"
    using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *🪙R z" "a *🪙R x + b *🪙R y"]
    by auto
  also have "… = y - e *🪙R (y-x)"
    using e_def assms
    by (simp add: divide_simps vector_fraction_eq_iff) (simp add: algebra_simps)
  finally have "z = y - e *🪙R (y-x)"
    by auto
  moreover have "e > 0" "e < 1" using e_def assms by auto
  ultimately show ?thesis using that[of e] by auto
qed

lemma convex_rel_interior_closure:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "rel_interior (closure S) = rel_interior S"
proof (cases "S = {}")
  case True
  then show ?thesis
    using assms rel_interior_eq_empty by auto
next
  case False
  have "rel_interior (closure S) 🪙 rel_interior S"
    using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
    by auto
  moreover
  {
    fix z
    assume z: "z ∈ rel_interior (closure S)"
    obtain x where x: "x ∈ rel_interior S"
      using ‹S ≠ {}› assms rel_interior_eq_empty by auto
    have "z ∈ rel_interior S"
    proof (cases "x = z")
      case True
      then show ?thesis using x by auto
    next
      case False
      obtain e where e: "e > 0" "cball z e ∩ affine hull closure S ≤ closure S"
        using z rel_interior_cball[of "closure S"] by auto
      hence *: "0 < e/norm(z-x)" using e False by auto
      define y where "y = z + (e/norm(z-x)) *🪙R (z-x)"
      have yball: "y ∈ cball z e"
        using y_def dist_norm[of z y] e by auto
      have "x ∈ affine hull closure S"
        using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
      moreover have "z ∈ affine hull closure S"
        using z rel_interior_subset hull_subset[of "closure S"] by blast
      ultimately have "y ∈ affine hull closure S"
        using y_def affine_affine_hull[of "closure S"]
          mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
      then have "y ∈ closure S" using e yball by auto
      have "(1 + (e/norm(z-x))) *🪙R z = (e/norm(z-x)) *🪙R x + y"
        using y_def by (simp add: algebra_simps)
      then obtain e1 where "0 < e1" "e1 < 1" "z = y - e1 *🪙R (y - x)"
        using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
        by (auto simp add: algebra_simps)
      then show ?thesis
        using rel_interior_closure_convex_shrink assms x ‹y ∈ closure S›
        by fastforce
    qed
  }
  ultimately show ?thesis by auto
qed

lemma convex_interior_closure:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "interior (closure S) = interior S"
  using closure_aff_dim[of S] interior_rel_interior_gen[of S]
    interior_rel_interior_gen[of "closure S"]
    convex_rel_interior_closure[of S] assms
  by auto

lemma open_subset_closure_of_interval:
  assumes "open U" "is_interval S"
  shows "U ⊆ closure S ⟷ U ⊆ interior S"
  by (metis assms convex_interior_closure is_interval_convex open_subset_interior)

lemma closure_eq_rel_interior_eq:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
  shows "closure S1 = closure S2 ⟷ rel_interior S1 = rel_interior S2"
  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)

lemma closure_eq_between:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
  shows "closure S1 = closure S2 ⟷ rel_interior S1 ≤ S2 ∧ S2 ⊆ closure S1"
  (is "?A ⟷ ?B")
proof
  assume ?A
  then show ?B
    by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
next
  assume ?B
  then have "closure S1 ⊆ closure S2"
    by (metis assms(1) convex_closure_rel_interior closure_mono)
  moreover from ‹?B› have "closure S1 🪙 closure S2"
    by (metis closed_closure closure_minimal)
  ultimately show ?A ..
qed

lemma open_inter_closure_rel_interior:
  fixes S A :: "'n::euclidean_space set"
  assumes "convex S"
    and "open A"
  shows "A ∩ closure S = {} ⟷ A ∩ rel_interior S = {}"
  by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)

lemma rel_interior_open_segment:
  fixes a :: "'a :: euclidean_space"
  shows "rel_interior(open_segment a b) = open_segment a b"
proof (cases "a = b")
  case True then show ?thesis by auto
next
  case False then
  have "open_segment a b = affine hull {a, b} ∩ ball ((a + b) /🪙R 2) (norm (b - a) / 2)"
    by (simp add: open_segment_as_ball)
  then show ?thesis
    unfolding rel_interior_eq openin_open
    by (metis Elementary_Metric_Spaces.open_ball False affine_hull_open_segment)
qed

lemma rel_interior_closed_segment:
  fixes a :: "'a :: euclidean_space"
  shows "rel_interior(closed_segment a b) =
         (if a = b then {a} else open_segment a b)"
proof (cases "a = b")
  case True then show ?thesis by auto
next
  case False then show ?thesis
    by simp
       (metis closure_open_segment convex_open_segment convex_rel_interior_closure
              rel_interior_open_segment)
qed

lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment

subsection‹The relative frontier of a set›

definition🍋‹tag important› "rel_frontier S = closure S - rel_interior S"

lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
  by (simp add: rel_frontier_def)

lemma rel_frontier_eq_empty:
    fixes S :: "'n::euclidean_space set"
    shows "rel_frontier S = {} ⟷ affine S"
  unfolding rel_frontier_def
  using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])

lemma rel_frontier_sing [simp]:
    fixes a :: "'n::euclidean_space"
    shows "rel_frontier {a} = {}"
  by (simp add: rel_frontier_def)

lemma rel_frontier_affine_hull:
  fixes S :: "'a::euclidean_space set"
  shows "rel_frontier S ⊆ affine hull S"
using closure_affine_hull rel_frontier_def by fastforce

lemma rel_frontier_cball [simp]:
    fixes a :: "'n::euclidean_space"
    shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
proof (cases rule: linorder_cases [of r 0])
  case less then show ?thesis
    by (force simp: sphere_def)
next
  case equal then show ?thesis by simp
next
  case greater then show ?thesis
    by simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
qed

lemma rel_frontier_translation:
  fixes a :: "'a::euclidean_space"
  shows "rel_frontier((λx. a + x) ` S) = (λx. a + x) ` (rel_frontier S)"
  by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)

lemma rel_frontier_nonempty_interior:
  fixes S :: "'n::euclidean_space set"
  shows "interior S ≠ {} ==> rel_frontier S = frontier S"
  by (metis frontier_def interior_rel_interior_gen rel_frontier_def)

lemma rel_frontier_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "affine hull S = UNIV ==> rel_frontier S = frontier S"
  by (simp add: frontier_def rel_frontier_def rel_interior_interior)

lemma closest_point_in_rel_frontier:
   "[closed S; S ≠ {}; x ∈ affine hull S - rel_interior S]
   ==> closest_point S x ∈ rel_frontier S"
  by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)

lemma closed_rel_frontier [iff]:
  fixes S :: "'n::euclidean_space set"
  shows "closed (rel_frontier S)"
proof -
  have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
    by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
  show ?thesis
  proof (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
    show "closedin (top_of_set (affine hull S)) (rel_frontier S)"
      by (simp add: "*" rel_frontier_def)
  qed simp
qed

lemma closed_rel_boundary:
  fixes S :: "'n::euclidean_space set"
  shows "closed S ==> closed(S - rel_interior S)"
  by (metis closed_rel_frontier closure_closed rel_frontier_def)

lemma compact_rel_boundary:
  fixes S :: "'n::euclidean_space set"
  shows "compact S ==> compact(S - rel_interior S)"
  by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)

lemma bounded_rel_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "bounded S ==> bounded(rel_frontier S)"
by (simp add: bounded_closure bounded_diff rel_frontier_def)

lemma compact_rel_frontier_bounded:
  fixes S :: "'n::euclidean_space set"
  shows "bounded S ==> compact(rel_frontier S)"
using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast

lemma compact_rel_frontier:
  fixes S :: "'n::euclidean_space set"
  shows "compact S ==> compact(rel_frontier S)"
by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)

lemma convex_same_rel_interior_closure:
  fixes S :: "'n::euclidean_space set"
  shows "[convex S; convex T]
         ==> rel_interior S = rel_interior T ⟷ closure S = closure T"
by (simp add: closure_eq_rel_interior_eq)

lemma convex_same_rel_interior_closure_straddle:
  fixes S :: "'n::euclidean_space set"
  shows "[convex S; convex T]
         ==> rel_interior S = rel_interior T ⟷
             rel_interior S ⊆ T ∧ T ⊆ closure S"
by (simp add: closure_eq_between convex_same_rel_interior_closure)

lemma convex_rel_frontier_aff_dim:
  fixes S1 S2 :: "'n::euclidean_space set"
  assumes "convex S1"
    and "convex S2"
    and "S2 ≠ {}"
    and "S1 ≤ rel_frontier S2"
  shows "aff_dim S1 < aff_dim S2"
proof -
  have "S1 ⊆ closure S2"
    using assms unfolding rel_frontier_def by auto
  then have *: "affine hull S1 ⊆ affine hull S2"
    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
  then have "aff_dim S1 ≤ aff_dim S2"
    using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
      aff_dim_subset[of "affine hull S1" "affine hull S2"]
    by auto
  moreover
  {
    assume eq: "aff_dim S1 = aff_dim S2"
    then have "S1 ≠ {}"
      using aff_dim_empty[of S1] aff_dim_empty[of S2] ‹S2 ≠ {}› by auto
    have **: "affine hull S1 = affine hull S2"
      by (simp_all add: * eq ‹S1 ≠ {}› affine_dim_equal)
    obtain a where a: "a ∈ rel_interior S1"
      using ‹S1 ≠ {}› rel_interior_eq_empty assms by auto
    obtain T where T: "open T" "a ∈ T ∩ S1" "T ∩ affine hull S1 ⊆ S1"
       using mem_rel_interior[of a S1] a by auto
    then have "a ∈ T ∩ closure S2"
      using a assms unfolding rel_frontier_def by auto
    then obtain b where b: "b ∈ T ∩ rel_interior S2"
      using open_inter_closure_rel_interior[of S2 T] assms T by auto
    then have "b ∈ affine hull S1"
      using rel_interior_subset hull_subset[of S2] ** by auto
    then have "b ∈ S1"
      using T b by auto
    then have False
      using b assms unfolding rel_frontier_def by auto
  }
  ultimately show ?thesis
    using less_le by auto
qed

lemma convex_rel_interior_if:
  fixes S ::  "'n::euclidean_space set"
  assumes "convex S"
    and "z ∈ rel_interior S"
  shows "∀x∈affine hull S. ∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *🪙R x + e *🪙R z ∈ S)"
proof -
  obtain e1 where e1: "e1 > 0 ∧ cball z e1 ∩ affine hull S ⊆ S"
    using mem_rel_interior_cball[of z S] assms by auto
  {
    fix x
    assume x: "x ∈ affine hull S"
    {
      assume "x ≠ z"
      define m where "m = 1 + e1/norm(x-z)"
      hence "m > 1" using e1 ‹x ≠ z› by auto
      {
        fix e
        assume e: "e > 1 ∧ e ≤ m"
        have "z ∈ affine hull S"
          using assms rel_interior_subset hull_subset[of S] by auto
        then have *: "(1 - e)*🪙R x + e *🪙R z ∈ affine hull S"
          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
          by auto
        have "norm (z + e *🪙R x - (x + e *🪙R z)) = norm ((e - 1) *🪙R (x - z))"
          by (simp add: algebra_simps)
        also have "… = (e - 1) * norm (x-z)"
          using norm_scaleR e by auto
        also have "… ≤ (m - 1) * norm (x - z)"
          using e mult_right_mono[of _ _ "norm(x-z)"] by auto
        also have "… = (e1 / norm (x - z)) * norm (x - z)"
          using m_def by auto
        also have "… = e1"
          using ‹x ≠ z› e1 by simp
        finally have **: "norm (z + e *🪙R x - (x + e *🪙R z)) ≤ e1"
          by auto
        have "(1 - e)*🪙R x+ e *🪙R z ∈ cball z e1"
          using m_def **
          unfolding cball_def dist_norm
          by (auto simp add: algebra_simps)
        then have "(1 - e) *🪙R x+ e *🪙R z ∈ S"
          using e * e1 by auto
      }
      then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *🪙R x + e *🪙R z ∈ S )"
        using ‹m> 1 › by auto
    }
    moreover
    {
      assume "x = z"
      define m where "m = 1 + e1"
      then have "m > 1"
        using e1 by auto
      {
        fix e
        assume e: "e > 1 ∧ e ≤ m"
        then have "(1 - e) *🪙R x + e *🪙R z ∈ S"
          using e1 x ‹x = z› by (auto simp add: algebra_simps)
        then have "(1 - e) *🪙R x + e *🪙R z ∈ S"
          using e by auto
      }
      then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *🪙R x + e *🪙R z ∈ S)"
        using ‹m > 1› by auto
    }
    ultimately have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e) *🪙R x + e *🪙R z ∈ S )"
      by blast
  }
  then show ?thesis by auto
qed

lemma convex_rel_interior_if2:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  assumes "z ∈ rel_interior S"
  shows "∀x∈affine hull S. ∃e. e > 1 ∧ (1 - e)*🪙R x + e *🪙R z ∈ S"
  using convex_rel_interior_if[of S z] assms by auto

lemma convex_rel_interior_only_if:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S ≠ {}"
  assumes "∀x∈S. ∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S"
  shows "z ∈ rel_interior S"
proof -
  obtain x where x: "x ∈ rel_interior S"
    using rel_interior_eq_empty assms by auto
  then have "x ∈ S"
    using rel_interior_subset by auto
  then obtain e where e: "e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S"
    using assms by auto
  define y where [abs_def]: "y = (1 - e) *🪙R x + e *🪙R z"
  then have "y ∈ S" using e by auto
  define e1 where "e1 = 1/e"
  then have "0 < e1 ∧ e1 < 1" using e by auto
  then have "z =y - (1 - e1) *🪙R (y - x)"
    using e1_def y_def by (auto simp add: algebra_simps)
  then show ?thesis
    using rel_interior_convex_shrink[of S x y "1-e1"] ‹0 🚫 ∧ e1 🚫› ‹y ∈ S› x assms
    by auto
qed

lemma convex_rel_interior_iff:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S ≠ {}"
  shows "z ∈ rel_interior S ⟷ (∀x∈S. ∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S)"
  using assms hull_subset[of S "affine"]
    convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
  by auto

lemma convex_rel_interior_iff2:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
    and "S ≠ {}"
  shows "z ∈ rel_interior S ⟷ (∀x∈affine hull S. ∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S)"
  using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
  by auto

lemma convex_interior_iff:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "z ∈ interior S ⟷ (∀x. ∃e. e > 0 ∧ z + e *🪙R x ∈ S)"
proof (cases "aff_dim S = int DIM('n)")
  case False
  { assume "z ∈ interior S"
    then have False
      using False interior_rel_interior_gen[of S] by auto }
  moreover
  { assume r: "∀x. ∃e. e > 0 ∧ z + e *🪙R x ∈ S"
    { fix x
      obtain e1 where e1: "e1 > 0 ∧ z + e1 *🪙R (x - z) ∈ S"
        using r by auto
      obtain e2 where e2: "e2 > 0 ∧ z + e2 *🪙R (z - x) ∈ S"
        using r by auto
      define x1 where [abs_def]: "x1 = z + e1 *🪙R (x - z)"
      then have x1: "x1 ∈ affine hull S"
        using e1 hull_subset[of S] by auto
      define x2 where [abs_def]: "x2 = z + e2 *🪙R (z - x)"
      then have x2: "x2 ∈ affine hull S"
        using e2 hull_subset[of S] by auto
      have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
        using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
      then have "z = (e2/(e1+e2)) *🪙R x1 + (e1/(e1+e2)) *🪙R x2"
        by (simp add: x1_def x2_def algebra_simps) (simp add: "*" pth_8)
      then have z: "z ∈ affine hull S"
        using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
          x1 x2 affine_affine_hull[of S] *
        by auto
      have "x1 - x2 = (e1 + e2) *🪙R (x - z)"
        using x1_def x2_def by (auto simp add: algebra_simps)
      then have "x = z+(1/(e1+e2)) *🪙R (x1-x2)"
        using e1 e2 by simp
      then have "x ∈ affine hull S"
        using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
          x1 x2 z affine_affine_hull[of S]
        by auto
    }
    then have "affine hull S = UNIV"
      by auto
    then have "aff_dim S = int DIM('n)"
      using aff_dim_affine_hull[of S] by (simp)
    then have False
      using False by auto
  }
  ultimately show ?thesis by auto
next
  case True
  then have "S ≠ {}"
    using aff_dim_empty[of S] by auto
  have *: "affine hull S = UNIV"
    using True affine_hull_UNIV by auto
  {
    assume "z ∈ interior S"
    then have "z ∈ rel_interior S"
      using True interior_rel_interior_gen[of S] by auto
    then have **: "∀x. ∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S"
      using convex_rel_interior_iff2[of S z] assms ‹S ≠ {}› * by auto
    fix x
    obtain e1 where e1: "e1 > 1" "(1 - e1) *🪙R (z - x) + e1 *🪙R z ∈ S"
      using **[rule_format, of "z-x"] by auto
    define e where [abs_def]: "e = e1 - 1"
    then have "(1 - e1) *🪙R (z - x) + e1 *🪙R z = z + e *🪙R x"
      by (simp add: algebra_simps)
    then have "e > 0" "z + e *🪙R x ∈ S"
      using e1 e_def by auto
    then have "∃e. e > 0 ∧ z + e *🪙R x ∈ S"
      by auto
  }
  moreover
  {
    assume r: "∀x. ∃e. e > 0 ∧ z + e *🪙R x ∈ S"
    {
      fix x
      obtain e1 where e1: "e1 > 0" "z + e1 *🪙R (z - x) ∈ S"
        using r[rule_format, of "z-x"] by auto
      define e where "e = e1 + 1"
      then have "z + e1 *🪙R (z - x) = (1 - e) *🪙R x + e *🪙R z"
        by (simp add: algebra_simps)
      then have "e > 1" "(1 - e)*🪙R x + e *🪙R z ∈ S"
        using e1 e_def by auto
      then have "∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ S" by auto
    }
    then have "z ∈ rel_interior S"
      using convex_rel_interior_iff2[of S z] assms ‹S ≠ {}› by auto
    then have "z ∈ interior S"
      using True interior_rel_interior_gen[of S] by auto
  }
  ultimately show ?thesis by auto
qed


subsubsection🍋‹tag unimportant› ‹Relative interior and closure under common operations›

lemma rel_interior_inter_aux: "∩{rel_interior S |S. S ∈ I} ⊆ ∩I"
proof -
  { fix y
    assume "y ∈ ∩{rel_interior S |S. S ∈ I}"
    then have y: "∀S ∈ I. y ∈ rel_interior S"
      by auto
    { fix S
      assume "S ∈ I"
      then have "y ∈ S"
        using rel_interior_subset y by auto
    }
    then have "y ∈ ∩I" by auto
  }
  then show ?thesis by auto
qed

lemma convex_closure_rel_interior_Int:
  assumes "∧S. S∈F ==> convex (S :: 'n::euclidean_space set)"
    and "∩(rel_interior ` F) ≠ {}"
  shows "∩(closure ` F) ⊆ closure (∩(rel_interior ` F))"
proof -
  obtain x where x: "∀S∈F. x ∈ rel_interior S"
    using assms by auto
  show ?thesis
  proof
    fix y
    assume y: "y ∈ ∩ (closure ` F)"
    show "y ∈ closure (∩(rel_interior ` F))"
    proof (cases "y=x")
      case True
      with closure_subset x show ?thesis 
        by fastforce
    next
      case False
      show ?thesis
      proof (clarsimp simp: closure_approachable_le)
        fix ε :: real
        assume e: "ε > 0"
        define e1 where "e1 = min 1 (ε/norm (y - x))"
        then have e1: "e1 > 0" "e1 ≤ 1" "e1 * norm (y - x) ≤ ε"
          using ‹y ≠ x› ‹ε > 0› le_divide_eq[of e1 ε "norm (y - x)"]
          by simp_all
        define z where "z = y - e1 *🪙R (y - x)"
        {
          fix S
          assume "S ∈ F"
          then have "z ∈ rel_interior S"
            using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
            by auto
        }
        then have *: "z ∈ ∩(rel_interior ` F)"
          by auto
        show "∃x∈∩ (rel_interior ` F). dist x y ≤ ε"
          using ‹y ≠ x› z_def * e1 e dist_norm[of z y]
          by force
      qed
    qed
  qed
qed


lemma closure_Inter_convex:
  fixes F :: "'n::euclidean_space set set"
  assumes "∧S. S ∈ F ==> convex S" and "∩(rel_interior ` F) ≠ {}"
  shows "closure(∩F) = ∩(closure ` F)"
proof -
  have "∩(closure ` F) ≤ closure (∩(rel_interior ` F))"
    by (meson assms convex_closure_rel_interior_Int)
  moreover
  have "closure (∩(rel_interior ` F)) ⊆ closure (∩F)"
    using rel_interior_inter_aux closure_mono[of "∩(rel_interior ` F)" "∩F"]
    by auto
  ultimately show ?thesis
    using closure_Int[of F] by blast
qed

lemma closure_Inter_convex_open:
    "(∧S::'n::euclidean_space set. S ∈ F ==> convex S ∧ open S)
        ==> closure(∩F) = (if ∩F = {} then {} else ∩(closure ` F))"
  by (simp add: closure_Inter_convex rel_interior_open)

lemma convex_Inter_rel_interior_same_closure:
  fixes F :: "'n::euclidean_space set set"
  assumes "∧S. S ∈ F ==> convex S"
    and "∩(rel_interior ` F) ≠ {}"
  shows "closure (∩(rel_interior ` F)) = closure (∩F)"
proof -
  have "∩(closure ` F) ⊆ closure (∩(rel_interior ` F))"
    by (meson assms convex_closure_rel_interior_Int)
  moreover
  have "closure (∩(rel_interior ` F)) ⊆ closure (∩F)"
    by (metis Setcompr_eq_image closure_mono rel_interior_inter_aux)
  ultimately show ?thesis
    by (simp add: assms closure_Inter_convex)
qed

lemma convex_rel_interior_Inter:
  fixes F :: "'n::euclidean_space set set"
  assumes "∧S. S ∈ F ==> convex S"
    and "∩(rel_interior ` F) ≠ {}"
  shows "rel_interior (∩F) ⊆ ∩(rel_interior ` F)"
proof -
  have "convex (∩F)"
    using assms convex_Inter by auto
  moreover
  have "convex (∩(rel_interior ` F))"
    using assms by (metis convex_rel_interior convex_INT)
  ultimately
  have "rel_interior (∩(rel_interior ` F)) = rel_interior (∩F)"
    using convex_Inter_rel_interior_same_closure assms
      closure_eq_rel_interior_eq[of "∩(rel_interior ` F)" "∩F"]
    by blast
  then show ?thesis
    using rel_interior_subset[of "∩(rel_interior ` F)"] by auto
qed

lemma convex_rel_interior_finite_Inter:
  fixes F :: "'n::euclidean_space set set"
  assumes "∧S. S ∈ F ==> convex S"
    and "∩(rel_interior ` F) ≠ {}"
    and "finite F"
  shows "rel_interior (∩F) = ∩(rel_interior ` F)"
proof -
  have "∩F ≠ {}"
    using assms rel_interior_inter_aux[of F] by auto
  have "convex (∩F)"
    using convex_Inter assms by auto
  show ?thesis
  proof (cases "F = {}")
    case True
    then show ?thesis
      using Inter_empty rel_interior_UNIV by auto
  next
    case False
    {
      fix z
      assume z: "z ∈ ∩(rel_interior ` F)"
      {
        fix x
        assume x: "x ∈ ∩F"
        {
          fix S
          assume S: "S ∈ F"
          then have "z ∈ rel_interior S" "x ∈ S"
            using z x by auto
          then have "∃m. m > 1 ∧ (∀e. e > 1 ∧ e ≤ m ⟶ (1 - e)*🪙R x + e *🪙R z ∈ S)"
            using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
        }
        then obtain mS where
          mS: "∀S∈F. mS S > 1 ∧ (∀e. e > 1 ∧ e ≤ mS S ⟶ (1 - e) *🪙R x + e *🪙R z ∈ S)" by metis
        define e where "e = Min (mS ` F)"
        then have "e ∈ mS ` F" using assms ‹F ≠ {}› by simp
        then have "e > 1" using mS by auto
        moreover have "∀S∈F. e ≤ mS S"
          using e_def assms by auto
        ultimately have "∃e > 1. (1 - e) *🪙R x + e *🪙R z ∈ ∩F"
          using mS by auto
      }
      then have "z ∈ rel_interior (∩F)"
        using convex_rel_interior_iff[of "∩F" z] ‹∩F ≠ {}› ‹convex (∩F)› by auto
    }
    then show ?thesis
      using convex_rel_interior_Inter[of F] assms by auto
  qed
qed

lemma closure_Int_convex:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "convex T"
  assumes "rel_interior S ∩ rel_interior T ≠ {}"
  shows "closure (S ∩ T) = closure S ∩ closure T"
  using closure_Inter_convex[of "{S,T}"] assms by auto

lemma convex_rel_interior_inter_two:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "convex T"
    and "rel_interior S ∩ rel_interior T ≠ {}"
  shows "rel_interior (S ∩ T) = rel_interior S ∩ rel_interior T"
  using convex_rel_interior_finite_Inter[of "{S,T}"] assms by auto

lemma convex_affine_closure_Int:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "affine T"
    and "rel_interior S ∩ T ≠ {}"
  shows "closure (S ∩ T) = closure S ∩ T"
  by (metis affine_imp_convex assms closure_Int_convex rel_interior_affine rel_interior_eq_closure)

lemma connected_component_1_gen:
  fixes S :: "'a :: euclidean_space set"
  assumes "DIM('a) = 1"
  shows "connected_component S a b ⟷ closed_segment a b ⊆ S"
unfolding connected_component_def
by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
            ends_in_segment connected_convex_1_gen)

lemma connected_component_1:
  fixes S :: "real set"
  shows "connected_component S a b ⟷ closed_segment a b ⊆ S"
by (simp add: connected_component_1_gen)

lemma convex_affine_rel_interior_Int:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "affine T"
    and "rel_interior S ∩ T ≠ {}"
  shows "rel_interior (S ∩ T) = rel_interior S ∩ T"
  by (simp add: affine_imp_convex assms convex_rel_interior_inter_two rel_interior_affine)

lemma convex_affine_rel_frontier_Int:
   fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "affine T"
    and "interior S ∩ T ≠ {}"
  shows "rel_frontier(S ∩ T) = frontier S ∩ T"
using assms
  unfolding rel_frontier_def  frontier_def
  using convex_affine_closure_Int convex_affine_rel_interior_Int rel_interior_nonempty_interior by fastforce

lemma rel_interior_convex_Int_affine:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "affine T" "interior S ∩ T ≠ {}"
  shows "rel_interior(S ∩ T) = interior S ∩ T"
  by (metis Int_emptyI assms convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)

lemma subset_rel_interior_convex:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
    and "convex T"
    and "S ≤ closure T"
    and "¬ S ⊆ rel_frontier T"
  shows "rel_interior S ⊆ rel_interior T"
proof -
  have *: "S ∩ closure T = S"
    using assms by auto
  have "¬ rel_interior S ⊆ rel_frontier T"
    using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
      closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
    by auto
  then have "rel_interior S ∩ rel_interior (closure T) ≠ {}"
    using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
    by auto
  then have "rel_interior S ∩ rel_interior T = rel_interior (S ∩ closure T)"
    using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
      convex_rel_interior_closure[of T]
    by auto
  also have "… = rel_interior S"
    using * by auto
  finally show ?thesis
    by auto
qed

lemma rel_interior_convex_linear_image:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
    and "convex S"
  shows "f ` (rel_interior S) = rel_interior (f ` S)"
proof (cases "S = {}")
  case True
  then show ?thesis
    using assms by auto
next
  case False
  interpret linear f by fact
  have *: "f ` (rel_interior S) ⊆ f ` S"
    unfolding image_mono using rel_interior_subset by auto
  have "f ` S ⊆ f ` (closure S)"
    unfolding image_mono using closure_subset by auto
  also have "… = f ` (closure (rel_interior S))"
    using convex_closure_rel_interior assms by auto
  also have "… ⊆ closure (f ` (rel_interior S))"
    using closure_linear_image_subset assms by auto
  finally have "closure (f ` S) = closure (f ` rel_interior S)"
    using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
      closure_mono[of "f ` rel_interior S" "f ` S"] *
    by auto
  then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
    using assms convex_rel_interior
      linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
      convex_linear_image[of _ "rel_interior S"]
      closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
    by auto
  then have "rel_interior (f ` S) ⊆ f ` rel_interior S"
    using rel_interior_subset by auto
  moreover
  {
    fix z
    assume "z ∈ f ` rel_interior S"
    then obtain z1 where z1: "z1 ∈ rel_interior S" "f z1 = z" by auto
    {
      fix x
      assume "x ∈ f ` S"
      then obtain x1 where x1: "x1 ∈ S" "f x1 = x" by auto
      then obtain e where e: "e > 1" "(1 - e) *🪙R x1 + e *🪙R z1 ∈ S"
        using convex_rel_interior_iff[of S z1] ‹convex S› x1 z1 by auto
      moreover have "f ((1 - e) *🪙R x1 + e *🪙R z1) = (1 - e) *🪙R x + e *🪙R z"
        using x1 z1 by (simp add: linear_add linear_scale ‹linear f›)
      ultimately have "(1 - e) *🪙R x + e *🪙R z ∈ f ` S"
        using imageI[of "(1 - e) *🪙R x1 + e *🪙R z1" S f] by auto
      then have "∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ f ` S"
        using e by auto
    }
    then have "z ∈ rel_interior (f ` S)"
      using convex_rel_interior_iff[of "f ` S" z] ‹convex S› ‹linear f›
        ‹S ≠ {}› convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
      by auto
  }
  ultimately show ?thesis by auto
qed

lemma rel_interior_convex_linear_preimage:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
    and "convex S"
    and "f -` (rel_interior S) ≠ {}"
  shows "rel_interior (f -` S) = f -` (rel_interior S)"
proof -
  interpret linear f by fact
  have "S ≠ {}"
    using assms by auto
  have nonemp: "f -` S ≠ {}"
    by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
  then have "S ∩ (range f) ≠ {}"
    by auto
  have conv: "convex (f -` S)"
    using convex_linear_vimage assms by auto
  then have "convex (S ∩ range f)"
    by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
  {
    fix z
    assume "z ∈ f -` (rel_interior S)"
    then have z: "f z ∈ rel_interior S"
      by auto
    {
      fix x
      assume "x ∈ f -` S"
      then have "f x ∈ S" by auto
      then obtain e where e: "e > 1" "(1 - e) *🪙R f x + e *🪙R f z ∈ S"
        using convex_rel_interior_iff[of S "f z"] z assms ‹S ≠ {}› by auto
      moreover have "(1 - e) *🪙R f x + e *🪙R f z = f ((1 - e) *🪙R x + e *🪙R z)"
        using ‹linear f› by (simp add: linear_iff)
      ultimately have "∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R z ∈ f -` S"
        using e by auto
    }
    then have "z ∈ rel_interior (f -` S)"
      using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
  }
  moreover
  {
    fix z
    assume z: "z ∈ rel_interior (f -` S)"
    {
      fix x
      assume "x ∈ S ∩ range f"
      then obtain y where y: "f y = x" "y ∈ f -` S" by auto
      then obtain e where e: "e > 1" "(1 - e) *🪙R y + e *🪙R z ∈ f -` S"
        using convex_rel_interior_iff[of "f -` S" z] z conv by auto
      moreover have "(1 - e) *🪙R x + e *🪙R f z = f ((1 - e) *🪙R y + e *🪙R z)"
        using ‹linear f› y by (simp add: linear_iff)
      ultimately have "∃e. e > 1 ∧ (1 - e) *🪙R x + e *🪙R f z ∈ S ∩ range f"
        using e by auto
    }
    then have "f z ∈ rel_interior (S ∩ range f)"
      using ‹convex (S ∩ (range f))› ‹S ∩ range f ≠ {}›
        convex_rel_interior_iff[of "S ∩ (range f)" "f z"]
      by auto
    moreover have "affine (range f)"
      by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
    ultimately have "f z ∈ rel_interior S"
      using convex_affine_rel_interior_Int[of S "range f"] assms by auto
    then have "z ∈ f -` (rel_interior S)"
      by auto
  }
  ultimately show ?thesis by auto
qed

lemma rel_interior_Times:
  fixes S :: "'n::euclidean_space set"
    and T :: "'m::euclidean_space set"
  assumes "convex S"
    and "convex T"
  shows "rel_interior (S × T) = rel_interior S × rel_interior T"
proof (cases "S = {} ∨ T = {}")
  case True
  then show ?thesis 
    by auto
next
  case False
  then have "S ≠ {}" "T ≠ {}"
    by auto
  then have ri: "rel_interior S ≠ {}" "rel_interior T ≠ {}"
    using rel_interior_eq_empty assms by auto
  then have "fst -` rel_interior S ≠ {}"
    using fst_vimage_eq_Times[of "rel_interior S"] by auto
  then have "rel_interior ((fst :: 'n * 'm ==> 'n) -` S) = fst -` rel_interior S"
    using linear_fst ‹convex S› rel_interior_convex_linear_preimage[of fst S] by auto
  then have s: "rel_interior (S × (UNIV :: 'm set)) = rel_interior S × UNIV"
    by (simp add: fst_vimage_eq_Times)
  from ri have "snd -` rel_interior T ≠ {}"
    using snd_vimage_eq_Times[of "rel_interior T"] by auto
  then have "rel_interior ((snd :: 'n * 'm ==> 'm) -` T) = snd -` rel_interior T"
    using linear_snd ‹convex T› rel_interior_convex_linear_preimage[of snd T] by auto
  then have t: "rel_interior ((UNIV :: 'n set) × T) = UNIV × rel_interior T"
    by (simp add: snd_vimage_eq_Times)
  from s t have *: "rel_interior (S × (UNIV :: 'm set)) ∩ rel_interior ((UNIV :: 'n set) × T) =
      rel_interior S × rel_interior T" by auto
  have "S × T = S × (UNIV :: 'm set) ∩ (UNIV :: 'n set) × T"
    by auto
  then have "rel_interior (S × T) = rel_interior ((S × (UNIV :: 'm set)) ∩ ((UNIV :: 'n set) × T))"
    by auto
  also have "… = rel_interior (S × (UNIV :: 'm set)) ∩ rel_interior ((UNIV :: 'n set) × T)"
    using * ri assms convex_Times
    by (subst convex_rel_interior_inter_two) auto
  finally show ?thesis using * by auto
qed

lemma rel_interior_scaleR:
  fixes S :: "'n::euclidean_space set"
  assumes "c ≠ 0"
  shows "((*🪙R) c) ` (rel_interior S) = rel_interior (((*🪙R) c) ` S)"
  using rel_interior_injective_linear_image[of "((*🪙R) c)" S]
    linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
  by auto

lemma rel_interior_convex_scaleR:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
  by (metis assms linear_scaleR rel_interior_convex_linear_image)
 
 lemma convex_rel_open_scaleR:
  fixes S :: "'n::euclidean_space set"
  assumes "convex S"
  and "rel_open S"
  shows "convex (((*🪙R) c) ` S) ∧ rel_open (((*🪙R) c) ` S)"
  by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 
 lemma convex_rel_open_finite_Inter:
  fixes F :: "'n::euclidean_space set set"
  assumes "∧S. S ∈ F ==> convex S ∧ rel_open S"
  and "finite F"
  shows "convex (∩F) ∧ rel_open (∩F)"
 proof (cases "∩{rel_interior S |S. S ∈ F} = {}")
  case True
  then have "∩F = {}"
  using assms unfolding rel_open_def by auto
  then show ?thesis
  unfolding rel_open_def by auto
 next
  case False
  then have "rel_open (∩F)"
  using assms convex_rel_interior_finite_Inter[of F] by (force simp: rel_open_def)
  then show ?thesis
  using convex_Inter assms by auto
 qed
 
 lemma convex_rel_open_linear_image:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
  and "convex S"
  and "rel_open S"
  shows "convex (f ` S) ∧ rel_open (f ` S)"
  by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
 
 lemma convex_rel_open_linear_preimage:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
  and "convex S"
  and "rel_open S"
  shows "convex (f -` S) ∧ rel_open (f -` S)"
 proof (cases "f -` (rel_interior S) = {}")
  case True
  then have "f -` S = {}"
  using assms unfolding rel_open_def by auto
  then show ?thesis
  unfolding rel_open_def by auto
 next
  case False
  then have "rel_open (f -` S)"
  using assms unfolding rel_open_def
  using rel_interior_convex_linear_preimage[of f S]
  by auto
  then show ?thesis
  using convex_linear_vimage assms
  by auto
 qed
 
 lemma rel_interior_projection:
  fixes S :: "('m::euclidean_space × 'n::euclidean_space) set"
  and f :: "'m::euclidean_space ==> 'n::euclidean_space set"
  assumes "convex S"
  and "f = (λy. {z. (y, z) ∈ S})"
  shows "(y, z) ∈ rel_interior S ⟷ (y ∈ rel_interior {y. (f y ≠ {})} ∧ z ∈ rel_interior (f y))"
 proof -
  {
  fix y
  assume "y ∈ {y. f y ≠ {}}"
  then obtain z where "(y, z) ∈ S"
  using assms by auto
  then have "∃x. x ∈ S ∧ y = fst x"
  by auto
  then obtain x where "x ∈ S" "y = fst x"
  by blast
  then have "y ∈ fst ` S"
  unfolding image_def by auto
  }
  then have "fst ` S = {y. f y ≠ {}}"
  unfolding fst_def using assms by auto
  then have h1: "fst ` rel_interior S = rel_interior {y. f y ≠ {}}"
  using rel_interior_convex_linear_image[of fst S] assms linear_fst by auto
  {
  fix y
  assume "y ∈ rel_interior {y. f y ≠ {}}"
  then have "y ∈ fst ` rel_interior S"
  using h1 by auto
  then have *: "rel_interior S ∩ fst -` {y} ≠ {}"
  by auto
  moreover have aff: "affine (fst -` {y})"
  unfolding affine_alt by (simp add: algebra_simps)
  ultimately have **: "rel_interior (S ∩ fst -` {y}) = rel_interior S ∩ fst -` {y}"
  using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
  have conv: "convex (S ∩ fst -` {y})"
  using convex_Int assms aff affine_imp_convex by auto
  {
  fix x
  assume "x ∈ f y"
  then have "(y, x) ∈ S ∩ (fst -` {y})"
  using assms by auto
  moreover have "x = snd (y, x)" by auto
  ultimately have "x ∈ snd ` (S ∩ fst -` {y})"
  by blast
  }
  then have "snd ` (S ∩ fst -` {y}) = f y"
  using assms by auto
  then have ***: "rel_interior (f y) = snd ` rel_interior (S ∩ fst -` {y})"
  using rel_interior_convex_linear_image[of snd "S ∩ fst -` {y}"] linear_snd conv
  by auto
  {
  fix z
  assume "z ∈ rel_interior (f y)"
  then have "z ∈ snd ` rel_interior (S ∩ fst -` {y})"
  using *** by auto
  moreover have "{y} = fst ` rel_interior (S ∩ fst -` {y})"
  using * ** rel_interior_subset by auto
  ultimately have "(y, z) ∈ rel_interior (S ∩ fst -` {y})"
  by force
  then have "(y,z) ∈ rel_interior S"
  using ** by auto
  }
  moreover
  {
  fix z
  assume "(y, z) ∈ rel_interior S"
  then have "(y, z) ∈ rel_interior (S ∩ fst -` {y})"
  using ** by auto
  then have "z ∈ snd ` rel_interior (S ∩ fst -` {y})"
  by (metis Range_iff snd_eq_Range)
  then have "z ∈ rel_interior (f y)"
  using *** by auto
  }
  ultimately have "∧z. (y, z) ∈ rel_interior S ⟷ z ∈ rel_interior (f y)"
  by auto
  }
  then have h2: "∧y z. y ∈ rel_interior {t. f t ≠ {}} ==>
  (y, z) ∈ rel_interior S ⟷ z ∈ rel_interior (f y)"
  by auto
  {
  fix y z
  assume asm: "(y, z) ∈ rel_interior S"
  then have "y ∈ fst ` rel_interior S"
  by (metis Domain_iff fst_eq_Domain)
  then have "y ∈ rel_interior {t. f t ≠ {}}"
  using h1 by auto
  then have "y ∈ rel_interior {t. f t ≠ {}}" and "(z ∈ rel_interior (f y))"
  using h2 asm by auto
  }
  then show ?thesis using h2 by blast
 qed
 
 lemma rel_frontier_Times:
  fixes S :: "'n::euclidean_space set"
  and T :: "'m::euclidean_space set"
  assumes "convex S"
  and "convex T"
  shows "rel_frontier S × rel_frontier T ⊆ rel_frontier (S × T)"
  by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
 
 
 subsubsection🍋‹tag unimportant› ‹Relative interior of convex cone›
 
 lemma cone_rel_interior:
  fixes S :: "'m::euclidean_space set"
  assumes "cone S"
  shows "cone ({0} ∪ rel_interior S)"
 proof (cases "S = {}")
  case True
  then show ?thesis
  by (simp add: cone_0)
 next
  case False
  then have *: "0 ∈ S ∧ (∀c. c > 0 ⟶ (*🪙R) c ` S = S)"
  using cone_iff[of S] assms by auto
  then have *: "0 ∈ ({0} ∪ rel_interior S)"
  and "∀c. c > 0 ⟶ (*🪙R) c ` ({0} ∪ rel_interior S) = ({0} ∪ rel_interior S)"
  by (auto simp add: rel_interior_scaleR)
  then show ?thesis
  using cone_iff[of "{0} ∪ rel_interior S"] by auto
 qed
 
 lemma rel_interior_convex_cone_aux:
  fixes S :: "'m::euclidean_space set"
  assumes "convex S"
  shows "(c, x) ∈ rel_interior (cone hull ({(1 :: real)} × S)) ⟷
  c > 0 ∧ x ∈ (((*🪙R) c) ` (rel_interior S))"
 proof (cases "S = {}")
  case True
  then show ?thesis
  by (simp add: cone_hull_empty)
 next
  case False
  then obtain s where "s ∈ S" by auto
  have conv: "convex ({(1 :: real)} × S)"
  using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
  by auto
  define f where "f y = {z. (y, z) ∈ cone hull ({1 :: real} × S)}" for y
  then have *: "(c, x) ∈ rel_interior (cone hull ({(1 :: real)} × S)) =
  (c ∈ rel_interior {y. f y ≠ {}} ∧ x ∈ rel_interior (f c))"
  using convex_cone_hull[of "{(1 :: real)} × S"] conv
  by (subst rel_interior_projection) auto
  {
  fix y :: real
  assume "y ≥ 0"
  then have "y *🪙R (1,s) ∈ cone hull ({1 :: real} × S)"
  using cone_hull_expl[of "{(1 :: real)} × S"] ‹s ∈ S› by auto
  then have "f y ≠ {}"
  using f_def by auto
  }
  then have "{y. f y ≠ {}} = {0..}"
  using f_def cone_hull_expl[of "{1 :: real} × S"] by auto
  then have **: "rel_interior {y. f y ≠ {}} = {0🚫}"
  using rel_interior_real_semiline by auto
  {
  fix c :: real
  assume "c > 0"
  then have "f c = ((*🪙R) c ` S)"
  using f_def cone_hull_expl[of "{1 :: real} × S"] by auto
  then have "rel_interior (f c) = (*🪙R) c ` rel_interior S"
  using rel_interior_convex_scaleR[of S c] assms by auto
  }
  then show ?thesis using * ** by auto
 qed
 
 lemma rel_interior_convex_cone:
  fixes S :: "'m::euclidean_space set"
  assumes "convex S"
  shows "rel_interior (cone hull ({1 :: real} × S)) =
  {(c, c *🪙R x) | c x. c > 0 ∧ x ∈ rel_interior S}"
  (is "?lhs = ?rhs")
 proof -
  {
  fix z
  assume "z ∈ ?lhs"
  have *: "z = (fst z, snd z)"
  by auto
  then have "z ∈ ?rhs"
  using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms ‹z ∈ ?lhs› by fastforce
  }
  moreover
  {
  fix z
  assume "z ∈ ?rhs"
  then have "z ∈ ?lhs"
  using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
  by auto
  }
  ultimately show ?thesis by blast
 qed
 
 lemma convex_hull_finite_union:
  assumes "finite I"
  assumes "∀i∈I. convex (S i) ∧ (S i) ≠ {}"
  shows "convex hull (∪(S ` I)) =
  {sum (λi. c i *🪙R s i) I | c s. (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)}"
  (is "?lhs = ?rhs")
 proof -
  have "?lhs 🪙 ?rhs"
  proof
  fix x
  assume "x ∈ ?rhs"
  then obtain c s where *: "sum (λi. c i *🪙R s i) I = x" "sum c I = 1"
  "(∀i∈I. c i ≥ 0) ∧ (∀i∈I. s i ∈ S i)" by auto
  then have "∀i∈I. s i ∈ convex hull (∪(S ` I))"
  using hull_subset[of "∪(S ` I)" convex] by auto
  then show "x ∈ ?lhs"
  unfolding *(1)[symmetric]
  using * assms convex_convex_hull
  by (subst convex_sum) auto
  qed
  {
  fix i
  assume "i ∈ I"
  with assms have "∃p. p ∈ S i" by auto
  }
  then obtain p where p: "∀i∈I. p i ∈ S i" by metis
  {
  fix i
  assume "i ∈ I"
  {
  fix x
  assume "x ∈ S i"
  define c where "c j = (if j = i then 1::real else 0)" for j
  then have *: "sum c I = 1"
  using ‹finite I› ‹i ∈ I› sum.delta[of I i "λj::'a. 1::real"]
  by auto
  define s where "s j = (if j = i then x else p j)" for j
  then have "∀j. c j *🪙R s j = (if j = i then x else 0)"
  using c_def by (auto simp add: algebra_simps)
  then have "x = sum (λi. c i *🪙R s i) I"
  using s_def c_def ‹finite I› ‹i ∈ I› sum.delta[of I i "λj::'a. x"]
  by auto
  moreover have "(∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)"
  using * c_def s_def p ‹x ∈ S i› by auto
  ultimately have "x ∈ ?rhs"
  by force
  }
  then have "?rhs 🪙 S i" by auto
  }
  then have *: "?rhs 🪙 ∪(S ` I)" by auto
 
  {
  fix u v :: real
  assume uv: "u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1"
  fix x y
  assume xy: "x ∈ ?rhs ∧ y ∈ ?rhs"
  from xy obtain c s where
  xc: "x = sum (λi. c i *🪙R s i) I ∧ (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ S i)"
  by auto
  from xy obtain d t where
  yc: "y = sum (λi. d i *🪙R t i) I ∧ (∀i∈I. d i ≥ 0) ∧ sum d I = 1 ∧ (∀i∈I. t i ∈ S i)"
  by auto
  define e where "e i = u * c i + v * d i" for i
  have ge0: "∀i∈I. e i ≥ 0"
  using e_def xc yc uv by simp
  have "sum (λi. u * c i) I = u * sum c I"
  by (simp add: sum_distrib_left)
  moreover have "sum (λi. v * d i) I = v * sum d I"
  by (simp add: sum_distrib_left)
  ultimately have sum1: "sum e I = 1"
  using e_def xc yc uv by (simp add: sum.distrib)
  define q where "q i = (if e i = 0 then p i else (u * c i / e i) *🪙R s i + (v * d i / e i) *🪙R t i)"
  for i
  {
  fix i
  assume i: "i ∈ I"
  have "q i ∈ S i"
  proof (cases "e i = 0")
  case True
  then show ?thesis using i p q_def by auto
  next
  case False
  then show ?thesis
  using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
  mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
  assms q_def e_def i False xc yc uv
  by (auto simp del: mult_nonneg_nonneg)
  qed
  }
  then have qs: "∀i∈I. q i ∈ S i" by auto
  {
  fix i
  assume i: "i ∈ I"
  have "(u * c i) *🪙R s i + (v * d i) *🪙R t i = e i *🪙R q i"
  proof (cases "e i = 0")
  case True
  have ge: "u * (c i) ≥ 0 ∧ v * d i ≥ 0"
  using xc yc uv i by simp
  moreover from ge have "u * c i ≤ 0 ∧ v * d i ≤ 0"
  using True e_def i by simp
  ultimately have "u * c i = 0 ∧ v * d i = 0" by auto
  with True show ?thesis by auto
  next
  case False
  then have "(u * (c i)/(e i))*🪙R (s i)+(v * (d i)/(e i))*🪙R (t i) = q i"
  using q_def by auto
  then have "e i *🪙R ((u * (c i)/(e i))*🪙R (s i)+(v * (d i)/(e i))*🪙R (t i))
  = (e i) *🪙R (q i)" by auto
  with False show ?thesis by (simp add: algebra_simps)
  qed
  }
  then have *: "∀i∈I. (u * c i) *🪙R s i + (v * d i) *🪙R t i = e i *🪙R q i"
  by auto
  have "u *🪙R x + v *🪙R y = sum (λi. (u * c i) *🪙R s i + (v * d i) *🪙R t i) I"
  using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
  also have "… = sum (λi. e i *🪙R q i) I"
  using * by auto
  finally have "u *🪙R x + v *🪙R y = sum (λi. (e i) *🪙R (q i)) I"
  by auto
  then have "u *🪙R x + v *🪙R y ∈ ?rhs"
  using ge0 sum1 qs by auto
  }
  then have "convex ?rhs" unfolding convex_def by auto
  then show ?thesis
  using ‹?lhs 🪙 ?rhs› * hull_minimal[of "∪(S ` I)" ?rhs convex]
  by blast
 qed
 
 lemma convex_hull_union_two:
  fixes S T :: "'m::euclidean_space set"
  assumes "convex S"
  and "S ≠ {}"
  and "convex T"
  and "T ≠ {}"
  shows "convex hull (S ∪ T) =
  {u *🪙R s + v *🪙R t | u v s t. u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1 ∧ s ∈ S ∧ t ∈ T}"
  (is "?lhs = ?rhs")
 proof
  define I :: "nat set" where "I = {1, 2}"
  define s where "s i = (if i = (1::nat) then S else T)" for i
  have "∪(s ` I) = S ∪ T"
  using s_def I_def by auto
  then have "convex hull (∪(s ` I)) = convex hull (S ∪ T)"
  by auto
  moreover have "convex hull ∪(s ` I) =
  {∑ i∈I. c i *🪙R sa i | c sa. (∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. sa i ∈ s i)}"
  using assms s_def I_def
  by (subst convex_hull_finite_union) auto
  moreover have
  "{∑i∈I. c i *🪙R sa i | c sa. (∀i∈I. 0 ≤ c i) ∧ sum c I = 1 ∧ (∀i∈I. sa i ∈ s i)} ≤ ?rhs"
  using s_def I_def by auto
  ultimately show "?lhs ⊆ ?rhs" by auto
  {
  fix x
  assume "x ∈ ?rhs"
  then obtain u v s t where *: "x = u *🪙R s + v *🪙R t ∧ u ≥ 0 ∧ v ≥ 0 ∧ u + v = 1 ∧ s ∈ S ∧ t ∈ T"
  by auto
  then have "x ∈ convex hull {s, t}"
  using convex_hull_2[of s t] by auto
  then have "x ∈ convex hull (S ∪ T)"
  using * hull_mono[of "{s, t}" "S ∪ T"] by auto
  }
  then show "?lhs 🪙 ?rhs" by blast
 qed
 
 proposition ray_to_rel_frontier:
  fixes a :: "'a::real_inner"
  assumes "bounded S"
  and a: "a ∈ rel_interior S"
  and aff: "(a + l) ∈ affine hull S"
  and "l ≠ 0"
  obtains d where "0 🚫" "(a + d *🪙R l) ∈ rel_frontier S"
  "∧e. [0 ≤ e; e 🚫] ==> (a + e *🪙R l) ∈ rel_interior S"
 proof -
  have aaff: "a ∈ affine hull S"
  by (meson a hull_subset rel_interior_subset rev_subsetD)
  let ?D = "{d. 0 🚫 ∧ a + d *🪙R l ∉ rel_interior S}"
  obtain B where "B > 0" and B: "S ⊆ ball a B"
  using bounded_subset_ballD [OF ‹bounded S›] by blast
  have "a + (B / norm l) *🪙R l ∉ ball a B"
  by (simp add: dist_norm ‹l ≠ 0›)
  with B have "a + (B / norm l) *🪙R l ∉ rel_interior S"
  using rel_interior_subset subsetCE by blast
  with ‹B > 0› ‹l ≠ 0› have nonMT: "?D ≠ {}"
  using divide_pos_pos zero_less_norm_iff by fastforce
  have bdd: "bdd_below ?D"
  by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
  have relin_Ex: "∧x. x ∈ rel_interior S ==>
  ∃e>0. ∀x'∈affine hull S. dist x' x 🚫 ⟶ x' ∈ rel_interior S"
  using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
  define d where "d = Inf ?D"
  obtain ε where "0 🚫ε" and ε: "∧🪙. [0 ≤ 🪙; 🪙 🚫ε] ==> (a + 🪙 *🪙R l) ∈ rel_interior S"
  proof -
  obtain e where "e>0"
  and e: "∧x'. x' ∈ affine hull S ==> dist x' a 🚫 ==> x' ∈ rel_interior S"
  using relin_Ex a by blast
  show thesis
  proof (rule_tac ε = "e / norm l" in that)
  show "0 🚫 / norm l" by (simp add: ‹0 🚫› ‹l ≠ 0›)
  next
  show "a + 🪙 *🪙R l ∈ rel_interior S" if "0 ≤ 🪙" "🪙 🚫 / norm l" for 🪙
  proof (rule e)
  show "a + 🪙 *🪙R l ∈ affine hull S"
  by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
  show "dist (a + 🪙 *🪙R l) a 🚫"
  using that by (simp add: ‹l ≠ 0› dist_norm pos_less_divide_eq)
  qed
  qed
  qed
  have inint: "∧e. [0 ≤ e; e 🚫] ==> a + e *🪙R l ∈ rel_interior S"
  unfolding d_def using cInf_lower [OF _ bdd]
  by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
  have "ε ≤ d"
  unfolding d_def
  using ε dual_order.strict_implies_order le_less_linear
  by (blast intro: cInf_greatest [OF nonMT])
  with ‹0 🚫ε› have "0 🚫" by simp
  have "a + d *🪙R l ∉ rel_interior S"
  proof
  assume adl: "a + d *🪙R l ∈ rel_interior S"
  obtain e where "e > 0"
  and e: "∧x'. x' ∈ affine hull S ==> dist x' (a + d *🪙R l) 🚫 ==> x' ∈ rel_interior S"
  using relin_Ex adl by blast
  have "d + e / norm l ≤ x"
  if "0 🚫" and nonrel: "a + x *🪙R l ∉ rel_interior S" for x
  proof (cases "x 🚫")
  case True with inint nonrel ‹0 🚫›
  show ?thesis by auto
  next
  case False
  then have dle: "x 🚫 + e / norm l ==> dist (a + x *🪙R l) (a + d *🪙R l) ??"
  by (simp add: field_simps ‹l ≠ 0›)
  have ain: "a + x *🪙R l ∈ affine hull S"
  by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
  show ?thesis
  using e [OF ain] nonrel dle by force
  qed
  then
  have "d + e / norm l ≤ Inf {d. 0 🚫 ∧ a + d *🪙R l ∉ rel_interior S}"
  by (force simp add: intro: cInf_greatest [OF nonMT])
  then show False
  using ‹0 🚫› ‹l ≠ 0› by (simp add: d_def [symmetric] field_simps)
  qed
  moreover
  have "∃y∈S. dist y (a + d *🪙R l) 🚫🪙" if "0 🚫🪙" for 🪙::real
  proof -
  have 1: "a + (d - min d (🪙 / 2 / norm l)) *🪙R l ∈ S"
  proof (rule subsetD [OF rel_interior_subset inint])
  show "d - min d (🪙 / 2 / norm l) 🚫"
  using ‹l ≠ 0› ‹0 🚫› ‹0 🚫🪙› by auto
  qed auto
  have "norm l * min d (🪙 / (norm l * 2)) ≤ norm l * (🪙 / (norm l * 2))"
  by (metis min_def mult_left_mono norm_ge_zero order_refl)
  also have "... 🚫🪙"
  using ‹l ≠ 0› ‹0 🚫🪙› by (simp add: field_simps)
  finally have 2: "norm l * min d (🪙 / (norm l * 2)) 🚫🪙" .
  show ?thesis
  using 1 2 ‹0 🚫› ‹0 🚫🪙›
  by (rule_tac x="a + (d - min d (🪙 / 2 / norm l)) *🪙R l" in bexI) (auto simp: algebra_simps)
  qed
  then have "a + d *🪙R l ∈ closure S"
  by (auto simp: closure_approachable)
  ultimately have infront: "a + d *🪙R l ∈ rel_frontier S"
  by (simp add: rel_frontier_def)
  show ?thesis
  by (rule that [OF ‹0 🚫› infront inint])
 qed
 
 corollary ray_to_frontier:
  fixes a :: "'a::euclidean_space"
  assumes "bounded S"
  and a: "a ∈ interior S"
  and "l ≠ 0"
  obtains d where "0 🚫" "(a + d *🪙R l) ∈ frontier S"
  "∧e. [0 ≤ e; e 🚫] ==> (a + e *🪙R l) ∈ interior S"
 proof -
  have 🍋: "interior S = rel_interior S"
  using a rel_interior_nonempty_interior by auto
  then have "a ∈ rel_interior S"
  using a by simp
  moreover have "a + l ∈ affine hull S"
  using a affine_hull_nonempty_interior by blast
  ultimately show thesis
  by (metis 🍋 ‹bounded S› ‹l ≠ 0› frontier_def ray_to_rel_frontier rel_frontier_def that)
 qed
 
 
 lemma segment_to_rel_frontier_aux:
  fixes x :: "'a::euclidean_space"
  assumes "convex S" "bounded S" and x: "x ∈ rel_interior S" and y: "y ∈ S" and xy: "x ≠ y"
  obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
  "open_segment x z ⊆ rel_interior S"
 proof -
  have "x + (y - x) ∈ affine hull S"
  using hull_inc [OF y] by auto
  then obtain d where "0 🚫" and df: "(x + d *🪙R (y-x)) ∈ rel_frontier S"
  and di: "∧e. [0 ≤ e; e 🚫] ==> (x + e *🪙R (y-x)) ∈ rel_interior S"
  by (rule ray_to_rel_frontier [OF ‹bounded S› x]) (use xy in auto)
  show ?thesis
  proof
  show "x + d *🪙R (y - x) ∈ rel_frontier S"
  by (simp add: df)
  next
  have "open_segment x y ⊆ rel_interior S"
  using rel_interior_closure_convex_segment [OF ‹convex S› x] closure_subset y by blast
  moreover have "x + d *🪙R (y - x) ∈ open_segment x y" if "d 🚫"
  using xy ‹0 🚫› that by (force simp: in_segment algebra_simps)
  ultimately have "1 ≤ d"
  using df rel_frontier_def by fastforce
  moreover have "x = (1 / d) *🪙R x + ((d - 1) / d) *🪙R x"
  by (metis ‹0 🚫› add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
  ultimately show "y ∈ closed_segment x (x + d *🪙R (y - x))"
  unfolding in_segment
  by (rule_tac x="1/d" in exI) (auto simp: algebra_simps)
  next
  show "open_segment x (x + d *🪙R (y - x)) ⊆ rel_interior S"
  proof (rule rel_interior_closure_convex_segment [OF ‹convex S› x])
  show "x + d *🪙R (y - x) ∈ closure S"
  using df rel_frontier_def by auto
  qed
  qed
 qed
 
 lemma segment_to_rel_frontier:
  fixes x :: "'a::euclidean_space"
  assumes S: "convex S" "bounded S" and x: "x ∈ rel_interior S"
  and y: "y ∈ S" and xy: "¬(x = y ∧ S = {x})"
  obtains z where "z ∈ rel_frontier S" "y ∈ closed_segment x z"
  "open_segment x z ⊆ rel_interior S"
 proof (cases "x=y")
  case True
  with xy have "S ≠ {x}"
  by blast
  with True show ?thesis
  by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
 next
  case False
  then show ?thesis
  using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
 proposition rel_frontier_not_sing:
  fixes a :: "'a::euclidean_space"
  assumes "bounded S"
  shows "rel_frontier S ≠ {a}"
 proof (cases "S = {}")
  case True then show ?thesis by simp
 next
  case False
  then obtain z where "z ∈ S"
  by blast
  then show ?thesis
  proof (cases "S = {z}")
  case True then show ?thesis by simp
  next
  case False
  then obtain w where "w ∈ S" "w ≠ z"
  using ‹z ∈ S› by blast
  show ?thesis
  proof
  assume "rel_frontier S = {a}"
  then consider "w ∉ rel_frontier S" | "z ∉ rel_frontier S"
  using ‹w ≠ z› by auto
  then show False
  proof cases
  case 1
  then have w: "w ∈ rel_interior S"
  using ‹w ∈ S› closure_subset rel_frontier_def by fastforce
  have "w + (w - z) ∈ affine hull S"
  by (metis ‹w ∈ S› ‹z ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
  then obtain e where "0 🚫" "(w + e *🪙R (w - z)) ∈ rel_frontier S"
  using ‹w ≠ z› ‹z ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq w)
  moreover obtain d where "0 🚫" "(w + d *🪙R (z - w)) ∈ rel_frontier S"
  using ray_to_rel_frontier [OF ‹bounded S› w, of "1 *🪙R (z - w)"] ‹w ≠ z› ‹z ∈ S›
  by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
  ultimately have "d *🪙R (z - w) = e *🪙R (w - z)"
  using ‹rel_frontier S = {a}› by force
  moreover have "e ≠ -d "
  using ‹0 🚫› ‹0 🚫› by force
  ultimately show False
  by (metis (no_types, lifting) ‹w ≠ z› eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
  next
  case 2
  then have z: "z ∈ rel_interior S"
  using ‹z ∈ S› closure_subset rel_frontier_def by fastforce
  have "z + (z - w) ∈ affine hull S"
  by (metis ‹z ∈ S› ‹w ∈ S› affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
  then obtain e where "0 🚫" "(z + e *🪙R (z - w)) ∈ rel_frontier S"
  using ‹w ≠ z› ‹w ∈ S› by (metis assms ray_to_rel_frontier right_minus_eq z)
  moreover obtain d where "0 🚫" "(z + d *🪙R (w - z)) ∈ rel_frontier S"
  using ray_to_rel_frontier [OF ‹bounded S› z, of "1 *🪙R (w - z)"] ‹w ≠ z› ‹w ∈ S›
  by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
  ultimately have "d *🪙R (w - z) = e *🪙R (z - w)"
  using ‹rel_frontier S = {a}› by force
  moreover have "e ≠ -d "
  using ‹0 🚫› ‹0 🚫› by force
  ultimately show False
  by (metis (no_types, lifting) ‹w ≠ z› eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
  qed
  qed
  qed
 qed
 
 
 subsection🍋‹tag unimportant› ‹Convexity on direct sums›
 
 lemma closure_sum:
  fixes S T :: "'a::real_normed_vector set"
  shows "closure S + closure T ⊆ closure (S + T)"
  unfolding set_plus_image closure_Times [symmetric] split_def
  by (intro closure_bounded_linear_image_subset bounded_linear_add
  bounded_linear_fst bounded_linear_snd)
 
 lemma fst_snd_linear: "linear (λ(x,y). x + y)"
  unfolding linear_iff by (simp add: algebra_simps)
 
 lemma rel_interior_sum:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
  and "convex T"
  shows "rel_interior (S + T) = rel_interior S + rel_interior T"
 proof -
  have "rel_interior S + rel_interior T = (λ(x,y). x + y) ` (rel_interior S × rel_interior T)"
  by (simp add: set_plus_image)
  also have "… = (λ(x,y). x + y) ` rel_interior (S × T)"
  using rel_interior_Times assms by auto
  also have "… = rel_interior (S + T)"
  using fst_snd_linear convex_Times assms
  rel_interior_convex_linear_image[of "(λ(x,y). x + y)" "S × T"]
  by (auto simp add: set_plus_image)
  finally show ?thesis ..
 qed
 
 lemma rel_interior_sum_gen:
  fixes S :: "'a ==> 'n::euclidean_space set"
  assumes "∧i. i∈I ==> convex (S i)"
  shows "rel_interior (sum S I) = sum (λi. rel_interior (S i)) I"
  using rel_interior_sum rel_interior_sing[of "0"] assms
  by (subst sum_set_cond_linear[of convex], auto simp add: convex_set_plus)
 
 lemma convex_rel_open_direct_sum:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
  and "rel_open S"
  and "convex T"
  and "rel_open T"
  shows "convex (S × T) ∧ rel_open (S × T)"
  by (metis assms convex_Times rel_interior_Times rel_open_def)
 
 lemma convex_rel_open_sum:
  fixes S T :: "'n::euclidean_space set"
  assumes "convex S"
  and "rel_open S"
  and "convex T"
  and "rel_open T"
  shows "convex (S + T) ∧ rel_open (S + T)"
  by (metis assms convex_set_plus rel_interior_sum rel_open_def)
 
 lemma convex_hull_finite_union_cones:
  assumes "finite I"
  and "I ≠ {}"
  assumes "∧i. i∈I ==> convex (S i) ∧ cone (S i) ∧ S i ≠ {}"
  shows "convex hull (∪(S ` I)) = sum S I"
  (is "?lhs = ?rhs")
 proof -
  {
  fix x
  assume "x ∈ ?lhs"
  then obtain c xs where
  x: "x = sum (λi. c i *🪙R xs i) I ∧ (∀i∈I. c i ≥ 0) ∧ sum c I = 1 ∧ (∀i∈I. xs i ∈ S i)"
  using convex_hull_finite_union[of I S] assms by auto
  define s where "s i = c i *🪙R xs i" for i
  have "∀i∈I. s i ∈ S i"
  using s_def x assms by (simp add: mem_cone)
  moreover have "x = sum s I" using x s_def by auto
  ultimately have "x ∈ ?rhs"
  using set_sum_alt[of I S] assms by auto
  }
  moreover
  {
  fix x
  assume "x ∈ ?rhs"
  then obtain s where x: "x = sum s I ∧ (∀i∈I. s i ∈ S i)"
  using set_sum_alt[of I S] assms by auto
  define xs where "xs i = of_nat(card I) *🪙R s i" for i
  then have "x = sum (λi. ((1 :: real) / of_nat(card I)) *🪙R xs i) I"
  using x assms by auto
  moreover have "∀i∈I. xs i ∈ S i"
  using x xs_def assms by (simp add: cone_def)
  moreover have "∀i∈I. (1 :: real) / of_nat (card I) ≥ 0"
  by auto
  moreover have "sum (λi. (1 :: real) / of_nat (card I)) I = 1"
  using assms by auto
  ultimately have "x ∈ ?lhs"
  using assms
  apply (simp add: convex_hull_finite_union[of I S])
  by (rule_tac x = "(λi. 1 / (card I))" in exI) auto
  }
  ultimately show ?thesis by auto
 qed
 
 lemma convex_hull_union_cones_two:
  fixes S T :: "'m::euclidean_space set"
  assumes "convex S"
  and "cone S"
  and "S ≠ {}"
  assumes "convex T"
  and "cone T"
  and "T ≠ {}"
  shows "convex hull (S ∪ T) = S + T"
 proof -
  define I :: "nat set" where "I = {1, 2}"
  define A where "A i = (if i = (1::nat) then S else T)" for i
  have "∪(A ` I) = S ∪ T"
  using A_def I_def by auto
  then have "convex hull (∪(A ` I)) = convex hull (S ∪ T)"
  by auto
  moreover have "convex hull ∪(A ` I) = sum A I"
  using A_def I_def
  by (metis assms convex_hull_finite_union_cones empty_iff finite.emptyI finite.insertI insertI1)
  moreover have "sum A I = S + T"
  using A_def I_def by (force simp add: set_plus_def)
  ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_hull_union:
  fixes S :: "'a ==> 'n::euclidean_space set"
  assumes "finite I"
  and "∀i∈I. convex (S i) ∧ S i ≠ {}"
  shows "rel_interior (convex hull (∪(S ` I))) =
  {sum (λi. c i *🪙R s i) I | c s. (∀i∈I. c i > 0) ∧ sum c I = 1 ∧
  (∀i∈I. s i ∈ rel_interior(S i))}"
  (is "?lhs = ?rhs")
 proof (cases "I = {}")
  case True
  then show ?thesis
  using convex_hull_empty by auto
 next
  case False
  define C0 where "C0 = convex hull (∪(S ` I))"
  have "∀i∈I. C0 ≥ S i"
  unfolding C0_def using hull_subset[of "∪(S ` I)"] by auto
  define K0 where "K0 = cone hull ({1 :: real} × C0)"
  define K where "K i = cone hull ({1 :: real} × S i)" for i
  have "∀i∈I. K i ≠ {}"
  unfolding K_def using assms
  by (simp add: cone_hull_empty_iff[symmetric])
  have convK: "∀i∈I. convex (K i)"
  unfolding K_def
  by (simp add: assms(2) convex_Times convex_cone_hull)
  have "K0 🪙 K i" if "i ∈ I" for i
  unfolding K0_def K_def
  by (simp add: Sigma_mono ‹∀i∈I. S i ⊆ C0› hull_mono that)
  then have "K0 🪙 ∪(K ` I)" by auto
  moreover have "convex K0"
  unfolding K0_def by (simp add: C0_def convex_Times convex_cone_hull)
  ultimately have geq: "K0 🪙 convex hull (∪(K ` I))"
  using hull_minimal[of _ "K0" "convex"] by blast
  have "∀i∈I. K i 🪙 {1 :: real} × S i"
  using K_def by (simp add: hull_subset)
  then have "∪(K ` I) 🪙 {1 :: real} × ∪(S ` I)"
  by auto
  then have "convex hull ∪(K ` I) 🪙 convex hull ({1 :: real} × ∪(S ` I))"
  by (simp add: hull_mono)
  then have "convex hull ∪(K ` I) 🪙 {1 :: real} × C0"
  unfolding C0_def
  using convex_hull_Times[of "{(1 :: real)}" "∪(S ` I)"] convex_hull_singleton
  by auto
  moreover have "cone (convex hull (∪(K ` I)))"
  by (simp add: K_def cone_Union cone_cone_hull cone_convex_hull)
  ultimately have "convex hull (∪(K ` I)) 🪙 K0"
  unfolding K0_def
  using hull_minimal[of _ "convex hull (∪(K ` I))" "cone"]
  by blast
  then have "K0 = convex hull (∪(K ` I))"
  using geq by auto
  also have "… = sum K I"
  using assms False ‹∀i∈I. K i ≠ {}› cone_hull_eq convK
  by (intro convex_hull_finite_union_cones; fastforce simp: K_def)
  finally have "K0 = sum K I" by auto
  then have *: "rel_interior K0 = sum (λi. (rel_interior (K i))) I"
  using rel_interior_sum_gen[of I K] convK by auto
  {
  fix x
  assume "x ∈ ?lhs"
  then have "(1::real, x) ∈ rel_interior K0"
  using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
  by auto
  then obtain k where k: "(1::real, x) = sum k I ∧ (∀i∈I. k i ∈ rel_interior (K i))"
  using ‹finite I› * set_sum_alt[of I "λi. rel_interior (K i)"] by auto
  {
  fix i
  assume "i ∈ I"
  then have "convex (S i) ∧ k i ∈ rel_interior (cone hull {1} × S i)"
  using k K_def assms by auto
  then have "∃ci si. k i = (ci, ci *🪙R si) ∧ 0 🚫 ∧ si ∈ rel_interior (S i)"
  using rel_interior_convex_cone[of "S i"] by auto
  }
  then obtain c s where cs: "∀i∈I. k i = (c i, c i *🪙R s i) ∧ 0 🚫 i ∧ s i ∈ rel_interior (S i)"
  by metis
  then have "x = (∑i∈I. c i *🪙R s i) ∧ sum c I = 1"
  using k by (simp add: sum_prod)
  then have "x ∈ ?rhs"
  using k cs by auto
  }
  moreover
  {
  fix x
  assume "x ∈ ?rhs"
  then obtain c s where cs: "x = sum (λi. c i *🪙R s i) I ∧
  (∀i∈I. c i > 0) ∧ sum c I = 1 ∧ (∀i∈I. s i ∈ rel_interior (S i))"
  by auto
  define k where "k i = (c i, c i *🪙R s i)" for i
  {
  fix i assume "i ∈ I"
  then have "k i ∈ rel_interior (K i)"
  using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
  by auto
  }
  then have "(1, x) ∈ rel_interior K0"
  using * set_sum_alt[of I "(λi. rel_interior (K i))"] assms cs
  by (simp add: k_def) (metis (mono_tags, lifting) sum_prod)
  then have "x ∈ ?lhs"
  using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
  by auto
  }
  ultimately show ?thesis by blast
 qed
 
 
 lemma convex_le_Inf_differential:
  fixes f :: "real ==> real"
  assumes "convex_on I f"
  and "x ∈ interior I"
  and "y ∈ I"
  shows "f y ≥ f x + Inf ((λt. (f x - f t) / (x - t)) ` ({x🚫} ∩ I)) * (y - x)"
  (is "_ ≥ _ + Inf (?F x) * (y - x)")
 proof (cases rule: linorder_cases)
  assume "x 🚫"
  moreover
  have "open (interior I)" by auto
  from openE[OF this ‹x ∈ interior I›]
  obtain e where e: "0 🚫" "ball x e ⊆ interior I" .
  moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
  ultimately have "x 🚫" "t 🚫" "t ∈ ball x e"
  by (auto simp: dist_real_def field_simps split: split_min)
  with ‹x ∈ interior I› e interior_subset[of I] have "t ∈ I" "x ∈ I" by auto
 
  define K where "K = x - e / 2"
  with ‹0 🚫› have "K ∈ ball x e" "K 🚫"
  by (auto simp: dist_real_def)
  then have "K ∈ I"
  using ‹interior I ⊆ I› e(2) by blast
 
  have "Inf (?F x) ≤ (f x - f y) / (x - y)"
  proof (intro bdd_belowI cInf_lower2)
  show "(f x - f t) / (x - t) ∈ ?F x"
  using ‹t ∈ I› ‹x 🚫› by auto
  show "(f x - f t) / (x - t) ≤ (f x - f y) / (x - y)"
  using ‹convex_on I f› ‹x ∈ I› ‹y ∈ I› ‹x 🚫› ‹t 🚫›
  by (rule convex_on_slope_le)
  next
  fix y
  assume "y ∈ ?F x"
  with order_trans[OF convex_on_slope_le[OF ‹convex_on I f› ‹K ∈ I› _ ‹K 🚫› _]]
  show "(f K - f x) / (K - x) ≤ y" by auto
  qed
  then show ?thesis
  using ‹x 🚫› by (simp add: field_simps)
 next
  assume "y 🚫"
  moreover
  have "open (interior I)" by auto
  from openE[OF this ‹x ∈ interior I›]
  obtain e where e: "0 🚫" "ball x e ⊆ interior I" .
  moreover define t where "t = x + e / 2"
  ultimately have "x 🚫" "t ∈ ball x e"
  by (auto simp: dist_real_def field_simps)
  with ‹x ∈ interior I› e interior_subset[of I] have "t ∈ I" "x ∈ I" by auto
 
  have "(f x - f y) / (x - y) ≤ Inf (?F x)"
  proof (rule cInf_greatest)
  have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
  using ‹y 🚫› by (auto simp: field_simps)
  also
  fix z
  assume "z ∈ ?F x"
  with order_trans[OF convex_on_slope_le[OF ‹convex_on I f› ‹y ∈ I› _ ‹y 🚫›]]
  have "(f y - f x) / (y - x) ≤ z"
  by auto
  finally show "(f x - f y) / (x - y) ≤ z" .
  next
  have "x + e / 2 ∈ ball x e"
  using e by (auto simp: dist_real_def)
  with e interior_subset[of I] have "x + e / 2 ∈ {x🚫} ∩ I"
  by auto
  then show "?F x ≠ {}"
  by blast
  qed
  then show ?thesis
  using ‹y 🚫› by (simp add: field_simps)
 qed simp
 
 subsection🍋‹tag unimportant›\‹Explicit formulas for interior and relative interior of convex hull›
 
 lemma at_within_cbox_finite:
  assumes "x ∈ box a b" "x ∉ S" "finite S"
  shows "(at x within cbox a b - S) = at x"
 proof -
  have "interior (cbox a b - S) = box a b - S"
  using ‹finite S› by (simp add: interior_diff finite_imp_closed)
  then show ?thesis
  using at_within_interior assms by fastforce
 qed
 
 lemma affine_independent_convex_affine_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S" "T ⊆ S"
  shows "convex hull T = affine hull T ∩ convex hull S"
 proof -
  have fin: "finite S" "finite T" using assms aff_independent_finite finite_subset by auto
  have "convex hull T ⊆ affine hull T"
  using convex_hull_subset_affine_hull by blast
  moreover have "convex hull T ⊆ convex hull S"
  using assms hull_mono by blast
  moreover have "affine hull T ∩ convex hull S ⊆ convex hull T"
  proof -
  have 0: "∧u. sum u S = 0 ==> (∀v∈S. u v = 0) ∨ (∑v∈S. u v *🪙R v) ≠ 0"
  using affine_dependent_explicit_finite assms(1) fin(1) by auto
  show ?thesis
  proof (clarsimp simp add: affine_hull_finite fin)
  fix u
  assume S: "(∑v∈T. u v *🪙R v) ∈ convex hull S"
  and T1: "sum u T = 1"
  then obtain v where v: "∀x∈S. 0 ≤ v x" "sum v S = 1" "(∑x∈S. v x *🪙R x) = (∑v∈T. u v *🪙R v)"
  by (auto simp add: convex_hull_finite fin)
  { fix x
  assume"x ∈ T"
  then have S: "S = (S - T) ∪ T" 🍋 ‹split into separate cases›
  using assms by auto
  have [simp]: "(∑x∈T. v x *🪙R x) + (∑x∈S - T. v x *🪙R x) = (∑x∈T. u x *🪙R x)"
  "sum v T + sum v (S - T) = 1"
  using v fin S
  by (auto simp: sum.union_disjoint [symmetric] Un_commute)
  have "(∑x∈S. if x ∈ T then v x - u x else v x) = 0"
  "(∑x∈S. (if x ∈ T then v x - u x else v x) *🪙R x) = 0"
  using v fin T1
  by (subst S, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
  } note [simp] = this
  have "(∀x∈T. 0 ≤ u x)"
  using 0 [of "λx. if x ∈ T then v x - u x else v x"] ‹T ⊆ S› v(1) by fastforce
  then show "(∑v∈T. u v *🪙R v) ∈ convex hull T"
  using 0 [of "λx. if x ∈ T then v x - u x else v x"] ‹T ⊆ S› T1
  by (fastforce simp add: convex_hull_finite fin)
  qed
  qed
  ultimately show ?thesis
  by blast
 qed
 
 lemma affine_independent_span_eq:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S" "card S = Suc (DIM ('a))"
  shows "affine hull S = UNIV"
 proof (cases "S = {}")
  case True then show ?thesis
  using assms by simp
 next
  case False
  then obtain a T where T: "a ∉ T" "S = insert a T"
  by blast
  then have fin: "finite T" using assms
  by (metis finite_insert aff_independent_finite)
  have "UNIV ⊆ (+) a ` span ((λx. x - a) ` T)"
  proof (intro card_ge_dim_independent Fun.vimage_subsetD)
  show "independent ((λx. x - a) ` T)"
  using T affine_dependent_iff_dependent assms(1) by auto
  show "dim ((+) a -` UNIV) ≤ card ((λx. x - a) ` T)"
  using assms T fin by (auto simp: card_image inj_on_def)
  qed (use surj_plus in auto)
  then show ?thesis
  using T(2) affine_hull_insert_span_gen equalityI by fastforce
 qed
 
 lemma affine_independent_span_gt:
  fixes S :: "'a::euclidean_space set"
  assumes ind: "¬ affine_dependent S" and dim: "DIM ('a) 🚫 S"
  shows "affine hull S = UNIV"
 proof (intro affine_independent_span_eq [OF ind] antisym)
  show "card S ≤ Suc DIM('a)"
  using aff_independent_finite affine_dependent_biggerset ind by fastforce
  show "Suc DIM('a) ≤ card S"
  using Suc_leI dim by blast
 qed
 
 lemma empty_interior_affine_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "finite S" and dim: "card S ≤ DIM ('a)"
  shows "interior(affine hull S) = {}"
  using assms
 proof (induct S rule: finite_induct)
  case (insert x S)
  then have "dim (span ((λy. y - x) ` S)) 🚫('a)"
  by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
  then show ?case
  by (simp add: empty_interior_lowdim affine_hull_insert_span_gen interior_translation)
 qed auto
 
 lemma empty_interior_convex_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "finite S" and dim: "card S ≤ DIM ('a)"
  shows "interior(convex hull S) = {}"
  by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
  interior_mono empty_interior_affine_hull [OF assms])
 
 lemma explicit_subset_rel_interior_convex_hull:
  fixes S :: "'a::euclidean_space set"
  shows "finite S
  ==> {y. ∃u. (∀x ∈ S. 0 🚫 x ∧ u x 🚫) ∧ sum u S = 1 ∧ sum (λx. u x *🪙R x) S = y}
  ⊆ rel_interior (convex hull S)"
  by (force simp add: rel_interior_convex_hull_union [where S="λx. {x}" and I=S, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
  fixes S :: "'a::euclidean_space set"
  shows "finite S
  ==> {y. ∃u. (∀x ∈ S. 0 🚫 x) ∧ sum u S = 1 ∧ sum (λx. u x *🪙R x) S = y}
  ⊆ rel_interior (convex hull S)"
  by (force simp add: rel_interior_convex_hull_union [where S="λx. {x}" and I=S, simplified])
 
 lemma rel_interior_convex_hull_explicit:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "rel_interior(convex hull S) =
  {y. ∃u. (∀x ∈ S. 0 🚫 x) ∧ sum u S = 1 ∧ sum (λx. u x *🪙R x) S = y}"
  (is "?lhs = ?rhs")
 proof
  show "?rhs ≤ ?lhs"
  by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
 next
  show "?lhs ≤ ?rhs"
  proof (cases "∃a. S = {a}")
  case True then show "?lhs ≤ ?rhs"
  by force
  next
  case False
  have fs: "finite S"
  using assms by (simp add: aff_independent_finite)
  { fix a b and d::real
  assume ab: "a ∈ S" "b ∈ S" "a ≠ b"
  then have S: "S = (S - {a,b}) ∪ {a,b}" 🍋 ‹split into separate cases›
  by auto
  have "(∑x∈S. if x = a then - d else if x = b then d else 0) = 0"
  "(∑x∈S. (if x = a then - d else if x = b then d else 0) *🪙R x) = d *🪙R b - d *🪙R a"
  using ab fs
  by (subst S, subst sum.union_disjoint, auto)+
  } note [simp] = this
  { fix y
  assume y: "y ∈ convex hull S" "y ∉ ?rhs"
  have *: False if
  ua: "∀x∈S. 0 ≤ u x" "sum u S = 1" "¬ 0 🚫 a" "a ∈ S"
  and yT: "y = (∑x∈S. u x *🪙R x)" "y ∈ T" "open T"
  and sb: "T ∩ affine hull S ⊆ {w. ∃u. (∀x∈S. 0 ≤ u x) ∧ sum u S = 1 ∧ (∑x∈S. u x *🪙R x) = w}"
  for u T a
  proof -
  have ua0: "u a = 0"
  using ua by auto
  obtain b where b: "b∈S" "a ≠ b"
  using ua False by auto
  obtain e where e: "0 🚫" "ball (∑x∈S. u x *🪙R x) e ⊆ T"
  using yT by (auto elim: openE)
  with b obtain d where d: "0 🚫" "norm(d *🪙R (a-b)) 🚫"
  by (auto intro: that [of "e / 2 / norm(a-b)"])
  have "(∑x∈S. u x *🪙R x) ∈ affine hull S"
  using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
  then have "(∑x∈S. u x *🪙R x) - d *🪙R (a - b) ∈ affine hull S"
  using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
  then have "y - d *🪙R (a - b) ∈ T ∩ affine hull S"
  using d e yT by auto
  then obtain v where v: "∀x∈S. 0 ≤ v x"
  "sum v S = 1"
  "(∑x∈S. v x *🪙R x) = (∑x∈S. u x *🪙R x) - d *🪙R (a - b)"
  using subsetD [OF sb] yT
  by auto
  have aff: "∧u. sum u S = 0 ==> (∀v∈S. u v = 0) ∨ (∑v∈S. u v *🪙R v) ≠ 0"
  using assms by (simp add: affine_dependent_explicit_finite fs)
  show False
  using ua b d v aff [of "λx. (v x - u x) - (if x = a then -d else if x = b then d else 0)"]
  by (auto simp: algebra_simps sum_subtractf sum.distrib)
  qed
  have "y ∉ rel_interior (convex hull S)"
  using y convex_hull_finite [OF fs] *
  apply simp
  by (metis (no_types, lifting) IntD1 affine_hull_convex_hull mem_rel_interior)
  } with rel_interior_subset show "?lhs ≤ ?rhs"
  by blast
  qed
 qed
 
 lemma interior_convex_hull_explicit_minimal:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows
  "interior(convex hull S) =
  (if card(S) ≤ DIM('a) then {}
  else {y. ∃u. (∀x ∈ S. 0 🚫 x) ∧ sum u S = 1 ∧ (∑x∈S. u x *🪙R x) = y})"
  (is "_ = (if _ then _ else ?rhs)")
 proof -
  { assume S: "¬ card S ≤ DIM('a)"
  have "interior (convex hull S) = rel_interior(convex hull S)"
  using assms S by (simp add: affine_independent_span_gt rel_interior_interior)
  then have "interior(convex hull S) = ?rhs"
  by (simp add: assms S rel_interior_convex_hull_explicit)
  }
  then show ?thesis
  by (auto simp: aff_independent_finite empty_interior_convex_hull assms)
 qed
 
 lemma interior_convex_hull_explicit:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows
  "interior(convex hull S) =
  (if card(S) ≤ DIM('a) then {}
  else {y. ∃u. (∀x ∈ S. 0 🚫 x ∧ u x 🚫) ∧ sum u S = 1 ∧ (∑x∈S. u x *🪙R x) = y})"
 proof -
  { fix u :: "'a ==> real" and a
  assume "card Basis 🚫 S" and u: "∧x. x∈S ==> 0 🚫 x" "sum u S = 1" and a: "a ∈ S"
  then have cs: "Suc 0 🚫 S"
  by (metis DIM_positive less_trans_Suc)
  obtain b where b: "b ∈ S" "a ≠ b"
  proof (cases "S ≤ {a}")
  case True
  then show thesis
  using cs subset_singletonD by fastforce
  qed blast
  have "u a + u b ≤ sum u {a,b}"
  using a b by simp
  also have "... ≤ sum u S"
  using a b u
  by (intro Groups_Big.sum_mono2) (auto simp: less_imp_le aff_independent_finite assms)
  finally have "u a 🚫"
  using ‹b ∈ S› u by fastforce
  } note [simp] = this
  show ?thesis
  using assms by (force simp add: not_le interior_convex_hull_explicit_minimal)
 qed
 
 lemma interior_closed_segment_ge2:
  fixes a :: "'a::euclidean_space"
  assumes "2 ≤ DIM('a)"
  shows "interior(closed_segment a b) = {}"
 using assms unfolding segment_convex_hull
 proof -
  have "card {a, b} ≤ DIM('a)"
  using assms
  by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
  then show "interior (convex hull {a, b}) = {}"
  by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
 qed
 
 lemma interior_open_segment:
  fixes a :: "'a::euclidean_space"
  shows "interior(open_segment a b) =
  (if 2 ≤ DIM('a) then {} else open_segment a b)"
 proof (cases "2 ≤ DIM('a)")
  case True
  then have "interior (open_segment a b) = {}"
  using interior_closed_segment_ge2 interior_mono segment_open_subset_closed by blast
  with True show ?thesis
  by auto
 next
  case ge2: False
  have "interior (open_segment a b) = open_segment a b"
  proof (cases "a = b")
  case True then show ?thesis by auto
  next
  case False
  with ge2 have "affine hull (open_segment a b) = UNIV"
  by (simp add: False affine_independent_span_gt)
  then show "interior (open_segment a b) = open_segment a b"
  using rel_interior_interior rel_interior_open_segment by blast
  qed
  with ge2 show ?thesis
  by auto
 qed
 
 lemma interior_closed_segment:
  fixes a :: "'a::euclidean_space"
  shows "interior(closed_segment a b) =
  (if 2 ≤ DIM('a) then {} else open_segment a b)"
 proof (cases "a = b")
  case True then show ?thesis by simp
 next
  case False
  then have "closure (open_segment a b) = closed_segment a b"
  by simp
  then show ?thesis
  by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
 qed
 
 lemmas interior_segment = interior_closed_segment interior_open_segment
 
 lemma closed_segment_eq [simp]:
  fixes a :: "'a::euclidean_space"
  shows "closed_segment a b = closed_segment c d ⟷ {a,b} = {c,d}"
 proof
  assume abcd: "closed_segment a b = closed_segment c d"
  show "{a,b} = {c,d}"
  proof (cases "a=b ∨ c=d")
  case True with abcd show ?thesis by force
  next
  case False
  then have neq: "a ≠ b ∧ c ≠ d" by force
  have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
  using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
  have "b ∈ {c, d}"
  proof -
  have "insert b (closed_segment c d) = closed_segment c d"
  using abcd by blast
  then show ?thesis
  by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
  qed
  moreover have "a ∈ {c, d}"
  by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
  ultimately show "{a, b} = {c, d}"
  using neq by fastforce
  qed
 next
  assume "{a,b} = {c,d}"
  then show "closed_segment a b = closed_segment c d"
  by (simp add: segment_convex_hull)
 qed
 
 lemma closed_open_segment_eq [simp]:
  fixes a :: "'a::euclidean_space"
  shows "closed_segment a b ≠ open_segment c d"
 by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
 
 lemma open_closed_segment_eq [simp]:
  fixes a :: "'a::euclidean_space"
  shows "open_segment a b ≠ closed_segment c d"
 using closed_open_segment_eq by blast
 
 lemma open_segment_eq [simp]:
  fixes a :: "'a::euclidean_space"
  shows "open_segment a b = open_segment c d ⟷ a = b ∧ c = d ∨ {a,b} = {c,d}"
  (is "?lhs = ?rhs")
 proof
  assume abcd: ?lhs
  show ?rhs
  proof (cases "a=b ∨ c=d")
  case True with abcd show ?thesis
  using finite_open_segment by fastforce
  next
  case False
  then have a2: "a ≠ b ∧ c ≠ d" by force
  with abcd show ?rhs
  unfolding open_segment_def
  by (metis (no_types) abcd closed_segment_eq closure_open_segment)
  qed
 next
  assume ?rhs
  then show ?lhs
  by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
 qed
 
 subsection🍋‹tag unimportant›\‹Similar results for closure and (relative or absolute) frontier›
 
 lemma closure_convex_hull [simp]:
  fixes S :: "'a::euclidean_space set"
  shows "compact S ==> closure(convex hull S) = convex hull S"
  by (simp add: compact_imp_closed compact_convex_hull)
 
 lemma rel_frontier_convex_hull_explicit:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "rel_frontier(convex hull S) =
  {y. ∃u. (∀x ∈ S. 0 ≤ u x) ∧ (∃x ∈ S. u x = 0) ∧ sum u S = 1 ∧ sum (λx. u x *🪙R x) S = y}"
 proof -
  have fs: "finite S"
  using assms by (simp add: aff_independent_finite)
  have "∧u y v.
  [y ∈ S; u y = 0; sum u S = 1; ∀x∈S. 0 🚫 x;
  sum v S = 1; (∑x∈S. v x *🪙R x) = (∑x∈S. u x *🪙R x)]
  ==> ∃u. sum u S = 0 ∧ (∃v∈S. u v ≠ 0) ∧ (∑v∈S. u v *🪙R v) = 0"
  apply (rule_tac x = "λx. u x - v x" in exI)
  apply (force simp: sum_subtractf scaleR_diff_left)
  done
  then show ?thesis
  using fs assms
  apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit)
  apply (auto simp: convex_hull_finite)
  apply (metis less_eq_real_def)
  by (simp add: affine_dependent_explicit_finite)
 qed
 
 lemma frontier_convex_hull_explicit:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "frontier(convex hull S) =
  {y. ∃u. (∀x ∈ S. 0 ≤ u x) ∧ (DIM ('a) 🚫 S ⟶ (∃x ∈ S. u x = 0)) ∧
  sum u S = 1 ∧ sum (λx. u x *🪙R x) S = y}"
 proof -
  have fs: "finite S"
  using assms by (simp add: aff_independent_finite)
  show ?thesis
  proof (cases "DIM ('a) 🚫 S")
  case True
  with assms fs show ?thesis
  by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
  interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
  next
  case False
  then have "card S ≤ DIM ('a)"
  by linarith
  then show ?thesis
  using assms fs
  apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
  apply (simp add: convex_hull_finite)
  done
  qed
 qed
 
 lemma rel_frontier_convex_hull_cases:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "rel_frontier(convex hull S) = ∪{convex hull (S - {x}) |x. x ∈ S}"
 proof -
  have fs: "finite S"
  using assms by (simp add: aff_independent_finite)
  { fix u a
  have "∀x∈S. 0 ≤ u x ==> a ∈ S ==> u a = 0 ==> sum u S = 1 ==>
  ∃x v. x ∈ S ∧
  (∀x∈S - {x}. 0 ≤ v x) ∧
  sum v (S - {x}) = 1 ∧ (∑x∈S - {x}. v x *🪙R x) = (∑x∈S. u x *🪙R x)"
  apply (rule_tac x=a in exI)
  apply (rule_tac x=u in exI)
  apply (simp add: Groups_Big.sum_diff1 fs)
  done }
  moreover
  { fix a u
  have "a ∈ S ==> ∀x∈S - {a}. 0 ≤ u x ==> sum u (S - {a}) = 1 ==>
  ∃v. (∀x∈S. 0 ≤ v x) ∧
  (∃x∈S. v x = 0) ∧ sum v S = 1 ∧ (∑x∈S. v x *🪙R x) = (∑x∈S - {a}. u x *🪙R x)"
  apply (rule_tac x="λx. if x = a then 0 else u x" in exI)
  apply (auto simp: sum.If_cases Diff_eq if_smult fs)
  done }
  ultimately show ?thesis
  using assms
  apply (simp add: rel_frontier_convex_hull_explicit)
  apply (auto simp add: convex_hull_finite fs Union_SetCompr_eq)
  done
 qed
 
 lemma frontier_convex_hull_eq_rel_frontier:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "frontier(convex hull S) =
  (if card S ≤ DIM ('a) then convex hull S else rel_frontier(convex hull S))"
  using assms
  unfolding rel_frontier_def frontier_def
  by (simp add: affine_independent_span_gt rel_interior_interior
  finite_imp_compact empty_interior_convex_hull aff_independent_finite)
 
 lemma frontier_convex_hull_cases:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent S"
  shows "frontier(convex hull S) =
  (if card S ≤ DIM ('a) then convex hull S else ∪{convex hull (S - {x}) |x. x ∈ S})"
 by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
 
 lemma in_frontier_convex_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "finite S" "card S ≤ Suc (DIM ('a))" "x ∈ S"
  shows "x ∈ frontier(convex hull S)"
 proof (cases "affine_dependent S")
  case True
  with assms obtain y where "y ∈ S" and y: "y ∈ affine hull (S - {y})"
  by (auto simp: affine_dependent_def)
  moreover have "x ∈ closure (convex hull S)"
  by (meson closure_subset hull_inc subset_eq ‹x ∈ S›)
  moreover have "x ∉ interior (convex hull S)"
  using assms
  by (metis Suc_mono affine_hull_convex_hull affine_hull_nonempty_interior ‹y ∈ S› y card.remove empty_iff empty_interior_affine_hull finite_Diff hull_redundant insert_Diff interior_UNIV not_less)
  ultimately show ?thesis
  unfolding frontier_def by blast
 next
  case False
  { assume "card S = Suc (card Basis)"
  then have cs: "Suc 0 🚫 S"
  by (simp)
  with subset_singletonD have "∃y ∈ S. y ≠ x"
  by (cases "S ≤ {x}") fastforce+
  } note [dest!] = this
  show ?thesis using assms
  unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
  by (auto simp: le_Suc_eq hull_inc)
 qed
 
 lemma not_in_interior_convex_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "finite S" "card S ≤ Suc (DIM ('a))" "x ∈ S"
  shows "x ∉ interior(convex hull S)"
 using in_frontier_convex_hull [OF assms]
 by (metis Diff_iff frontier_def)
 
 lemma interior_convex_hull_eq_empty:
  fixes S :: "'a::euclidean_space set"
  assumes "card S = Suc (DIM ('a))"
  shows "interior(convex hull S) = {} ⟷ affine_dependent S"
 proof
  show "affine_dependent S ==> interior (convex hull S) = {}"
  proof (clarsimp simp: affine_dependent_def)
  fix a b
  assume "b ∈ S" "b ∈ affine hull (S - {b})"
  then have "interior(affine hull S) = {}" using assms
  by (metis DIM_positive One_nat_def Suc_mono card.remove card.infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
  then show "interior (convex hull S) = {}"
  using affine_hull_nonempty_interior by fastforce
  qed
 next
  show "interior (convex hull S) = {} ==> affine_dependent S"
  by (metis affine_hull_convex_hull affine_hull_empty affine_independent_span_eq assms convex_convex_hull empty_not_UNIV rel_interior_eq_empty rel_interior_interior)
 qed
 
 
 subsection ‹Coplanarity, and collinearity in terms of affine hull›
 
 definition🍋‹tag important› coplanar where
  "coplanar S ≡ ∃u v w. S ⊆ affine hull {u,v,w}"
 
 lemma collinear_affine_hull:
  "collinear S ⟷ (∃u v. S ⊆ affine hull {u,v})"
 proof (cases "S={}")
  case True then show ?thesis
  by simp
 next
  case False
  then obtain x where x: "x ∈ S" by auto
  { fix u
  assume *: "∧x y. [x∈S; y∈S] ==> ∃c. x - y = c *🪙R u"
  have "∧y c. x - y = c *🪙R u ==> ∃a b. y = a *🪙R x + b *🪙R (x + u) ∧ a + b = 1"
  by (rule_tac x="1+c" in exI, rule_tac x="-c" in exI, simp add: algebra_simps)
  then have "∃u v. S ⊆ {a *🪙R u + b *🪙R v |a b. a + b = 1}"
  using * [OF x] by (rule_tac x=x in exI, rule_tac x="x+u" in exI, force)
  } moreover
  { fix u v x y
  assume *: "S ⊆ {a *🪙R u + b *🪙R v |a b. a + b = 1}"
  have "∃c. x - y = c *🪙R (v-u)" if "x∈S" "y∈S"
  proof -
  obtain a r where "a + r = 1" "x = a *🪙R u + r *🪙R v"
  using "*" ‹x ∈ S› by blast
  moreover
  obtain b s where "b + s = 1" "y = b *🪙R u + s *🪙R v"
  using "*" ‹y ∈ S› by blast
  ultimately have "x - y = (r-s) *🪙R (v-u)"
  by (simp add: algebra_simps) (metis scaleR_left.add)
  then show ?thesis
  by blast
  qed
  } ultimately
  show ?thesis
  unfolding collinear_def affine_hull_2
  by blast
 qed
 
 lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
  by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
 
 lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
  unfolding open_segment_def
  by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
  convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
 
 lemma collinear_between_cases:
  fixes c :: "'a::euclidean_space"
  shows "collinear {a,b,c} ⟷ between (b,c) a ∨ between (c,a) b ∨ between (a,b) c"
  (is "?lhs = ?rhs")
 proof
  assume ?lhs
  then obtain u v where uv: "∧x. x ∈ {a, b, c} ==> ∃c. x = u + c *🪙R v"
  by (auto simp: collinear_alt)
  show ?rhs
  using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
 next
  assume ?rhs
  then show ?lhs
  unfolding between_mem_convex_hull
  by (metis (no_types, opaque_lifting) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
 qed
 
 
 lemma subset_continuous_image_segment_1:
  fixes f :: "'a::euclidean_space ==> real"
  assumes "continuous_on (closed_segment a b) f"
  shows "closed_segment (f a) (f b) ⊆ image f (closed_segment a b)"
 by (metis connected_segment convex_contains_segment ends_in_segment imageI
  is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
 
 lemma continuous_injective_image_segment_1:
  fixes f :: "'a::euclidean_space ==> real"
  assumes contf: "continuous_on (closed_segment a b) f"
  and injf: "inj_on f (closed_segment a b)"
  shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
 proof
  show "closed_segment (f a) (f b) ⊆ f ` closed_segment a b"
  by (metis subset_continuous_image_segment_1 contf)
  show "f ` closed_segment a b ⊆ closed_segment (f a) (f b)"
  proof (cases "a = b")
  case True
  then show ?thesis by auto
  next
  case False
  then have fnot: "f a ≠ f b"
  using inj_onD injf by fastforce
  moreover
  have "f a ∉ open_segment (f c) (f b)" if c: "c ∈ closed_segment a b" for c
  proof (clarsimp simp add: open_segment_def)
  assume fa: "f a ∈ closed_segment (f c) (f b)"
  moreover have "closed_segment (f c) (f b) ⊆ f ` closed_segment c b"
  by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
  ultimately have "f a ∈ f ` closed_segment c b"
  by blast
  then have a: "a ∈ closed_segment c b"
  by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
  have cb: "closed_segment c b ⊆ closed_segment a b"
  by (simp add: closed_segment_subset that)
  show "f a = f c"
  proof (rule between_antisym)
  show "between (f c, f b) (f a)"
  by (simp add: between_mem_segment fa)
  show "between (f a, f b) (f c)"
  by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
  qed
  qed
  moreover
  have "f b ∉ open_segment (f a) (f c)" if c: "c ∈ closed_segment a b" for c
  proof (clarsimp simp add: open_segment_def fnot eq_commute)
  assume fb: "f b ∈ closed_segment (f a) (f c)"
  moreover have "closed_segment (f a) (f c) ⊆ f ` closed_segment a c"
  by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
  ultimately have "f b ∈ f ` closed_segment a c"
  by blast
  then have b: "b ∈ closed_segment a c"
  by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
  have ca: "closed_segment a c ⊆ closed_segment a b"
  by (simp add: closed_segment_subset that)
  show "f b = f c"
  proof (rule between_antisym)
  show "between (f c, f a) (f b)"
  by (simp add: between_commute between_mem_segment fb)
  show "between (f b, f a) (f c)"
  by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
  qed
  qed
  ultimately show ?thesis
  by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
  qed
 qed
 
 lemma continuous_injective_image_open_segment_1:
  fixes f :: "'a::euclidean_space ==> real"
  assumes contf: "continuous_on (closed_segment a b) f"
  and injf: "inj_on f (closed_segment a b)"
  shows "f ` (open_segment a b) = open_segment (f a) (f b)"
 proof -
  have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
  by (metis (no_types, opaque_lifting) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
  also have "... = open_segment (f a) (f b)"
  using continuous_injective_image_segment_1 [OF assms]
  by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
  finally show ?thesis .
 qed
 
 lemma collinear_imp_coplanar:
  "collinear s ==> coplanar s"
 by (metis collinear_affine_hull coplanar_def insert_absorb2)
 
 lemma collinear_small:
  assumes "finite s" "card s ≤ 2"
  shows "collinear s"
 proof -
  have "card s = 0 ∨ card s = 1 ∨ card s = 2"
  using assms by linarith
  then show ?thesis using assms
  using card_eq_SucD numeral_2_eq_2 by (force simp: card_1_singleton_iff)
 qed
 
 lemma coplanar_small:
  assumes "finite s" "card s ≤ 3"
  shows "coplanar s"
 proof -
  consider "card s ≤ 2" | "card s = Suc (Suc (Suc 0))"
  using assms by linarith
  then show ?thesis
  proof cases
  case 1
  then show ?thesis
  by (simp add: ‹finite s› collinear_imp_coplanar collinear_small)
  next
  case 2
  then show ?thesis
  using hull_subset [of "{_,_,_}"]
  by (fastforce simp: coplanar_def dest!: card_eq_SucD)
  qed
 qed
 
 lemma coplanar_empty: "coplanar {}"
  by (simp add: coplanar_small)
 
 lemma coplanar_sing: "coplanar {a}"
  by (simp add: coplanar_small)
 
 lemma coplanar_2: "coplanar {a,b}"
  by (auto simp: card_insert_if coplanar_small)
 
 lemma coplanar_3: "coplanar {a,b,c}"
  by (auto simp: card_insert_if coplanar_small)
 
 lemma collinear_affine_hull_collinear: "collinear(affine hull s) ⟷ collinear s"
  unfolding collinear_affine_hull
  by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) ⟷ coplanar s"
  unfolding coplanar_def
  by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_linear_image:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "coplanar S" "linear f" shows "coplanar(f ` S)"
 proof -
  { fix u v w
  assume "S ⊆ affine hull {u, v, w}"
  then have "f ` S ⊆ f ` (affine hull {u, v, w})"
  by (simp add: image_mono)
  then have "f ` S ⊆ affine hull (f ` {u, v, w})"
  by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
  } then
  show ?thesis
  by auto (meson assms(1) coplanar_def)
 qed
 
 lemma coplanar_translation_imp:
  assumes "coplanar S" shows "coplanar ((λx. a + x) ` S)"
 proof -
  obtain u v w where "S ⊆ affine hull {u,v,w}"
  by (meson assms coplanar_def)
  then have "(+) a ` S ⊆ affine hull {u + a, v + a, w + a}"
  using affine_hull_translation [of a "{u,v,w}" for u v w]
  by (force simp: add.commute)
  then show ?thesis
  unfolding coplanar_def by blast
 qed
 
 lemma coplanar_translation_eq: "coplanar((λx. a + x) ` S) ⟷ coplanar S"
  by (metis (no_types) coplanar_translation_imp translation_galois)
 
 lemma coplanar_linear_image_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "linear f" "inj f" shows "coplanar(f ` S) = coplanar S"
 proof
  assume "coplanar S"
  then show "coplanar (f ` S)"
  using assms(1) coplanar_linear_image by blast
 next
  obtain g where g: "linear g" "g ∘ f = id"
  using linear_injective_left_inverse [OF assms]
  by blast
  assume "coplanar (f ` S)"
  then show "coplanar S"
  by (metis coplanar_linear_image g(1) g(2) id_apply image_comp image_id)
 qed
 
 lemma coplanar_subset: "[coplanar t; S ⊆ t] ==> coplanar S"
  by (meson coplanar_def order_trans)
 
 lemma affine_hull_3_imp_collinear: "c ∈ affine hull {a,b} ==> collinear {a,b,c}"
  by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
 
 lemma collinear_3_imp_in_affine_hull:
  assumes "collinear {a,b,c}" "a ≠ b" shows "c ∈ affine hull {a,b}"
 proof -
  obtain u x y where "b - a = y *🪙R u" "c - a = x *🪙R u"
  using assms unfolding collinear_def by auto
  with ‹a ≠ b› have "∃v. c = (1 - x / y) *🪙R a + v *🪙R b ∧ 1 - x / y + v = 1"
  by (simp add: algebra_simps)
  then show ?thesis
  by (simp add: hull_inc mem_affine)
 qed
 
 lemma collinear_3_affine_hull:
  assumes "a ≠ b"
  shows "collinear {a,b,c} ⟷ c ∈ affine hull {a,b}"
  using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
 
 lemma collinear_3_eq_affine_dependent:
  "collinear{a,b,c} ⟷ a = b ∨ a = c ∨ b = c ∨ affine_dependent {a,b,c}"
 proof (cases "a = b ∨ a = c ∨ b = c")
  case True
  then show ?thesis
  by (auto simp: insert_commute)
 next
  case False
  then have "collinear{a,b,c}" if "affine_dependent {a,b,c}"
  using that unfolding affine_dependent_def
  by (auto simp: insert_Diff_if; metis affine_hull_3_imp_collinear insert_commute)
  moreover
  have "affine_dependent {a,b,c}" if "collinear{a,b,c}"
  using False that by (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
  ultimately
  show ?thesis
  using False by blast
 qed
 
 lemma affine_dependent_imp_collinear_3:
  "affine_dependent {a,b,c} ==> collinear{a,b,c}"
  by (simp add: collinear_3_eq_affine_dependent)
 
 lemma collinear_3: "NO_MATCH 0 x ==> collinear {x,y,z} ⟷ collinear {0, x-y, z-y}"
  by (auto simp add: collinear_def)
 
 lemma collinear_3_expand:
  "collinear{a,b,c} ⟷ a = c ∨ (∃u. b = u *🪙R a + (1 - u) *🪙R c)"
 proof -
  have "collinear{a,b,c} = collinear{a,c,b}"
  by (simp add: insert_commute)
  also have "... = collinear {0, a - c, b - c}"
  by (simp add: collinear_3)
  also have "... ⟷ (a = c ∨ b = c ∨ (∃ca. b - c = ca *🪙R (a - c)))"
  by (simp add: collinear_lemma)
  also have "... ⟷ a = c ∨ (∃u. b = u *🪙R a + (1 - u) *🪙R c)"
  by (cases "a = c ∨ b = c") (auto simp: algebra_simps)
  finally show ?thesis .
 qed
 
 lemma collinear_aff_dim: "collinear S ⟷ aff_dim S ≤ 1"
 proof
  assume "collinear S"
  then obtain u and v :: "'a" where "aff_dim S ≤ aff_dim {u,v}"
  by (metis ‹collinear S› aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
  then show "aff_dim S ≤ 1"
  using order_trans by fastforce
 next
  assume "aff_dim S ≤ 1"
  then have le1: "aff_dim (affine hull S) ≤ 1"
  by simp
  obtain B where "B ⊆ S" and B: "¬ affine_dependent B" "affine hull S = affine hull B"
  using affine_basis_exists [of S] by auto
  then have "finite B" "card B ≤ 2"
  using B le1 by (auto simp: affine_independent_iff_card)
  then have "collinear B"
  by (rule collinear_small)
  then show "collinear S"
  by (metis ‹affine hull S = affine hull B› collinear_affine_hull_collinear)
 qed
 
 lemma collinear_midpoint: "collinear{a, midpoint a b, b}"
 proof -
  have 🍋: "[a ≠ midpoint a b; b - midpoint a b ≠ - 1 *🪙R (a - midpoint a b)] ==> b = midpoint a b"
  by (simp add: algebra_simps)
  show ?thesis
  by (auto simp: collinear_3 collinear_lemma intro: 🍋)
 qed
 
 lemma midpoint_collinear:
  fixes a b c :: "'a::real_normed_vector"
  assumes "a ≠ c"
  shows "b = midpoint a c ⟷ collinear{a,b,c} ∧ dist a b = dist b c"
 proof -
  have *: "a - (u *🪙R a + (1 - u) *🪙R c) = (1 - u) *🪙R (a - c)"
  "u *🪙R a + (1 - u) *🪙R c - c = u *🪙R (a - c)"
  "∣1 - u∣ = ∣u∣ ⟷ u = 1/2" for u::real
  by (auto simp: algebra_simps)
  have "b = midpoint a c ==> collinear{a,b,c}"
  using collinear_midpoint by blast
  moreover have "b = midpoint a c ⟷ dist a b = dist b c" if "collinear{a,b,c}"
  proof -
  consider "a = c" | u where "b = u *🪙R a + (1 - u) *🪙R c"
  using ‹collinear {a,b,c}› unfolding collinear_3_expand by blast
  then show ?thesis
  proof cases
  case 2
  with assms have "dist a b = dist b c ==> b = midpoint a c"
  by (simp add: dist_norm * midpoint_def scaleR_add_right del: divide_const_simps)
  then show ?thesis
  by (auto simp: dist_midpoint)
  qed (use assms in auto)
  qed
  ultimately show ?thesis by blast
 qed
 
 lemma between_imp_collinear:
  fixes x :: "'a :: euclidean_space"
  assumes "between (a,b) x"
  shows "collinear {a,x,b}"
 proof (cases "x = a ∨ x = b ∨ a = b")
  case True with assms show ?thesis
  by (auto simp: dist_commute)
 next
  case False
  then have False if "∧c. b - x ≠ c *🪙R (a - x)"
  using that [of "-(norm(b - x) / norm(x - a))"] assms
  by (simp add: between_norm vector_add_divide_simps flip: real_vector.scale_minus_right)
  then show ?thesis
  by (auto simp: collinear_3 collinear_lemma)
 qed
 
 lemma midpoint_between:
  fixes a b :: "'a::euclidean_space"
  shows "b = midpoint a c ⟷ between (a,c) b ∧ dist a b = dist b c"
 proof (cases "a = c")
  case False
  show ?thesis
  using False between_imp_collinear between_midpoint(1) midpoint_collinear by blast
 qed (auto simp: dist_commute)
 
 lemma collinear_triples:
  assumes "a ≠ b"
  shows "collinear(insert a (insert b S)) ⟷ (∀x ∈ S. collinear{a,b,x})"
  (is "?lhs = ?rhs")
 proof safe
  fix x
  assume ?lhs and "x ∈ S"
  then show "collinear {a, b, x}"
  using collinear_subset by force
 next
  assume ?rhs
  then have "∀x ∈ S. collinear{a,x,b}"
  by (simp add: insert_commute)
  then have *: "∃u. x = u *🪙R a + (1 - u) *🪙R b" if "x ∈ insert a (insert b S)" for x
  using that assms collinear_3_expand by fastforce+
  have "∃c. x - y = c *🪙R (b - a)"
  if x: "x ∈ insert a (insert b S)" and y: "y ∈ insert a (insert b S)" for x y
  proof -
  obtain u v where "x = u *🪙R a + (1 - u) *🪙R b" "y = v *🪙R a + (1 - v) *🪙R b"
  using "*" x y by presburger
  then have "x - y = (v - u) *🪙R (b - a)"
  by (simp add: scale_left_diff_distrib scale_right_diff_distrib)
  then show ?thesis ..
  qed
  then show ?lhs
  unfolding collinear_def by metis
 qed
 
 lemma collinear_4_3:
  assumes "a ≠ b"
  shows "collinear {a,b,c,d} ⟷ collinear{a,b,c} ∧ collinear{a,b,d}"
  using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
 
 lemma collinear_3_trans:
  assumes "collinear{a,b,c}" "collinear{b,c,d}" "b ≠ c"
  shows "collinear{a,b,d}"
 proof -
  have "collinear{b,c,a,d}"
  by (metis (full_types) assms collinear_4_3 insert_commute)
  then show ?thesis
  by (simp add: collinear_subset)
 qed
 
 lemma affine_hull_2_alt:
  fixes a b :: "'a::real_vector"
  shows "affine hull {a,b} = range (λu. a + u *🪙R (b - a))"
 proof -
  have 1: "u *🪙R a + v *🪙R b = a + v *🪙R (b - a)" if "u + v = 1" for u v
  using that
  by (simp add: algebra_simps flip: scaleR_add_left)
  have 2: "a + u *🪙R (b - a) = (1 - u) *🪙R a + u *🪙R b" for u
  by (auto simp: algebra_simps)
  show ?thesis
  by (force simp add: affine_hull_2 dest: 1 intro!: 2)
 qed
 
 lemma interior_convex_hull_3_minimal:
  fixes a :: "'a::euclidean_space"
  assumes "¬ collinear{a,b,c}" and 2: "DIM('a) = 2"
  shows "interior(convex hull {a,b,c}) =
  {v. ∃x y z. 0 🚫 ∧ 0 🚫 ∧ 0 🚫 ∧ x + y + z = 1 ∧ x *🪙R a + y *🪙R b + z *🪙R c = v}"
  (is "?lhs = ?rhs")
 proof
  have abc: "a ≠ b" "a ≠ c" "b ≠ c" "¬ affine_dependent {a, b, c}"
  using assms by (auto simp: collinear_3_eq_affine_dependent)
  with 2 show "?lhs ⊆ ?rhs"
  by (fastforce simp add: interior_convex_hull_explicit_minimal)
  show "?rhs ⊆ ?lhs"
  using abc 2
  apply (clarsimp simp add: interior_convex_hull_explicit_minimal)
  subgoal for x y z
  by (rule_tac x="λr. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI) auto
  done
 qed
 
 
 subsection🍋‹tag unimportant›\‹Basic lemmas about hyperplanes and halfspaces›
 
 lemma halfspace_Int_eq:
  "{x. a ∙ x ≤ b} ∩ {x. b ≤ a ∙ x} = {x. a ∙ x = b}"
  "{x. b ≤ a ∙ x} ∩ {x. a ∙ x ≤ b} = {x. a ∙ x = b}"
  by auto
 
 lemma hyperplane_eq_Ex:
  assumes "a ≠ 0" obtains x where "a ∙ x = b"
  by (rule_tac x = "(b / (a ∙ a)) *🪙R a" in that) (simp add: assms)
 
 lemma hyperplane_eq_empty:
  "{x. a ∙ x = b} = {} ⟷ a = 0 ∧ b ≠ 0"
  using hyperplane_eq_Ex
  by (metis (mono_tags, lifting) empty_Collect_eq inner_zero_left)
 
 lemma hyperplane_eq_UNIV:
  "{x. a ∙ x = b} = UNIV ⟷ a = 0 ∧ b = 0"
 proof -
  have "a = 0 ∧ b = 0" if "UNIV ⊆ {x. a ∙ x = b}"
  using subsetD [OF that, where c = "((b+1) / (a ∙ a)) *🪙R a"]
  by (simp add: field_split_simps split: if_split_asm)
  then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_lt:
  "{x. a ∙ x 🚫} = {} ⟷ a = 0 ∧ b ≤ 0"
 proof -
  have "a = 0 ∧ b ≤ 0" if "{x. a ∙ x 🚫} ⊆ {}"
  using subsetD [OF that, where c = "((b-1) / (a ∙ a)) *🪙R a"]
  by (force simp add: field_split_simps split: if_split_asm)
  then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_gt:
  "{x. a ∙ x > b} = {} ⟷ a = 0 ∧ b ≥ 0"
  using halfspace_eq_empty_lt [of "-a" "-b"]
  by simp
 
 lemma halfspace_eq_empty_le:
  "{x. a ∙ x ≤ b} = {} ⟷ a = 0 ∧ b 🚫"
 proof -
  have "a = 0 ∧ b 🚫" if "{x. a ∙ x ≤ b} ⊆ {}"
  using subsetD [OF that, where c = "((b-1) / (a ∙ a)) *🪙R a"]
  by (force simp add: field_split_simps split: if_split_asm)
  then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_ge:
  "{x. a ∙ x ≥ b} = {} ⟷ a = 0 ∧ b > 0"
  using halfspace_eq_empty_le [of "-a" "-b"] by simp
 
 subsection🍋‹tag unimportant›\‹Use set distance for an easy proof of separation properties›
 
 proposition🍋‹tag unimportant› separation_closures:
  fixes S :: "'a::euclidean_space set"
  assumes "S ∩ closure T = {}" "T ∩ closure S = {}"
  obtains U V where "U ∩ V = {}" "open U" "open V" "S ⊆ U" "T ⊆ V"
 proof (cases "S = {} ∨ T = {}")
  case True with that show ?thesis by auto
 next
  case False
  define f where "f ≡ λx. setdist {x} T - setdist {x} S"
  have contf: "continuous_on UNIV f"
  unfolding f_def by (intro continuous_intros continuous_on_setdist)
  show ?thesis
  proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x 🚫}" in that)
  show "{x. 0 🚫 x} ∩ {x. f x 🚫} = {}"
  by auto
  show "open {x. 0 🚫 x}"
  by (simp add: open_Collect_less contf)
  show "open {x. f x 🚫}"
  by (simp add: open_Collect_less contf)
  have "∧x. x ∈ S ==> setdist {x} T ≠ 0" "∧x. x ∈ T ==> setdist {x} S ≠ 0"
  by (meson False assms disjoint_iff setdist_eq_0_sing_1)+
  then show "S ⊆ {x. 0 🚫 x}" "T ⊆ {x. f x 🚫}"
  using less_eq_real_def by (fastforce simp add: f_def setdist_sing_in_set)+
  qed
 qed
 
 lemma separation_normal:
  fixes S :: "'a::euclidean_space set"
  assumes "closed S" "closed T" "S ∩ T = {}"
  obtains U V where "open U" "open V" "S ⊆ U" "T ⊆ V" "U ∩ V = {}"
 using separation_closures [of S T]
 by (metis assms closure_closed disjnt_def inf_commute)
 
 lemma separation_normal_local:
  fixes S :: "'a::euclidean_space set"
  assumes US: "closedin (top_of_set U) S"
  and UT: "closedin (top_of_set U) T"
  and "S ∩ T = {}"
  obtains S' T' where "openin (top_of_set U) S'"
  "openin (top_of_set U) T'"
  "S ⊆ S'" "T ⊆ T'" "S' ∩ T' = {}"
 proof (cases "S = {} ∨ T = {}")
  case True with that show ?thesis
  using UT US by (blast dest: closedin_subset)
 next
  case False
  define f where "f ≡ λx. setdist {x} T - setdist {x} S"
  have contf: "continuous_on U f"
  unfolding f_def by (intro continuous_intros)
  show ?thesis
  proof (rule_tac S' = "(U ∩ f -` {0🚫})" and T' = "(U ∩ f -` {..🚫)" in that)
  show "(U ∩ f -` {0🚫}) ∩ (U ∩ f -` {..🚫) = {}"
  by auto
  show "openin (top_of_set U) (U ∩ f -` {0🚫})"
  by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
  next
  show "openin (top_of_set U) (U ∩ f -` {..🚫)"
  by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
  next
  have "S ⊆ U" "T ⊆ U"
  using closedin_imp_subset assms by blast+
  then show "S ⊆ U ∩ f -` {0🚫}" "T ⊆ U ∩ f -` {..🚫"
  using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
  qed
 qed
 
 lemma separation_normal_compact:
  fixes S :: "'a::euclidean_space set"
  assumes "compact S" "closed T" "S ∩ T = {}"
  obtains U V where "open U" "compact(closure U)" "open V" "S ⊆ U" "T ⊆ V" "U ∩ V = {}"
 proof -
  have "closed S" "bounded S"
  using assms by (auto simp: compact_eq_bounded_closed)
  then obtain r where "r>0" and r: "S ⊆ ball 0 r"
  by (auto dest!: bounded_subset_ballD)
  have **: "closed (T ∪ - ball 0 r)" "S ∩ (T ∪ - ball 0 r) = {}"
  using assms r by blast+
  then obtain U V where UV: "open U" "open V" "S ⊆ U" "T ∪ - ball 0 r ⊆ V" "U ∩ V = {}"
  by (meson ‹closed S› separation_normal)
  then have "compact(closure U)"
  by (meson bounded_ball bounded_subset compact_closure compl_le_swap2 disjoint_eq_subset_Compl le_sup_iff)
  with UV show thesis
  using that by auto
 qed
 
 subsection‹Connectedness of the intersection of a chain›
 
 proposition connected_chain:
  fixes F :: "'a :: euclidean_space set set"
  assumes cc: "∧S. S ∈ F ==> compact S ∧ connected S"
  and linear: "∧S T. S ∈ F ∧ T ∈ F ==> S ⊆ T ∨ T ⊆ S"
  shows "connected(∩F)"
 proof (cases "F = {}")
  case True then show ?thesis
  by auto
 next
  case False
  then have cf: "compact(∩F)"
  by (simp add: cc compact_Inter)
  have False if AB: "closed A" "closed B" "A ∩ B = {}"
  and ABeq: "A ∪ B = ∩F" and "A ≠ {}" "B ≠ {}" for A B
  proof -
  obtain U V where "open U" "open V" "A ⊆ U" "B ⊆ V" "U ∩ V = {}"
  using separation_normal [OF AB] by metis
  obtain K where "K ∈ F" "compact K"
  using cc False by blast
  then obtain N where "open N" and "K ⊆ N"
  by blast
  let ?C = "insert (U ∪ V) ((λS. N - S) ` F)"
  obtain D where "D ⊆ ?C" "finite D" "K ⊆ ∪D"
  proof (rule compactE [OF ‹compact K›])
  show "K ⊆ ∪(insert (U ∪ V) ((-) N ` F))"
  using ‹K ⊆ N› ABeq ‹A ⊆ U› ‹B ⊆ V› by auto
  show "∧B. B ∈ insert (U ∪ V) ((-) N ` F) ==> open B"
  by (auto simp: ‹open U› ‹open V› open_Un ‹open N› cc compact_imp_closed open_Diff)
  qed
  then have "finite(D - {U ∪ V})"
  by blast
  moreover have "D - {U ∪ V} ⊆ (λS. N - S) ` F"
  using ‹D ⊆ ?C› by blast
  ultimately obtain G where "G ⊆ F" "finite G" and Deq: "D - {U ∪ V} = (λS. N-S) ` G"
  using finite_subset_image by metis
  obtain J where "J ∈ F" and J: "(∪S∈G. N - S) ⊆ N - J"
  proof (cases "G = {}")
  case True
  with ‹F ≠ {}› that show ?thesis
  by auto
  next
  case False
  have "∧S T. [S ∈ G; T ∈ G] ==> S ⊆ T ∨ T ⊆ S"
  by (meson ‹G ⊆ F› in_mono local.linear)
  with ‹finite G› ‹G ≠ {}›
  have "∃J ∈ G. (∪S∈G. N - S) ⊆ N - J"
  proof induction
  case (insert X H)
  show ?case
  proof (cases "H = {}")
  case True then show ?thesis by auto
  next
  case False
  then have "∧S T. [S ∈ H; T ∈ H] ==> S ⊆ T ∨ T ⊆ S"
  by (simp add: insert.prems)
  with insert.IH False obtain J where "J ∈ H" and J: "(∪Y∈H. N - Y) ⊆ N - J"
  by metis
  have "N - J ⊆ N - X ∨ N - X ⊆ N - J"
  by (meson Diff_mono ‹J ∈ H› insert.prems(2) insert_iff order_refl)
  then show ?thesis
  proof
  assume "N - J ⊆ N - X" with J show ?thesis
  by auto
  next
  assume "N - X ⊆ N - J"
  with J have "N - X ∪ ∪ ((-) N ` H) ⊆ N - J"
  by auto
  with ‹J ∈ H› show ?thesis
  by blast
  qed
  qed
  qed simp
  with ‹G ⊆ F› show ?thesis by (blast intro: that)
  qed
  have "K ⊆ ∪(insert (U ∪ V) (D - {U ∪ V}))"
  using ‹K ⊆ ∪D› by auto
  also have "... ⊆ (U ∪ V) ∪ (N - J)"
  by (metis (no_types, opaque_lifting) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
  finally have "J ∩ K ⊆ U ∪ V"
  by blast
  moreover have "connected(J ∩ K)"
  by (metis Int_absorb1 ‹J ∈ F› ‹K ∈ F› cc inf.orderE local.linear)
  moreover have "U ∩ (J ∩ K) ≠ {}"
  using ABeq ‹J ∈ F› ‹K ∈ F› ‹A ≠ {}› ‹A ⊆ U› by blast
  moreover have "V ∩ (J ∩ K) ≠ {}"
  using ABeq ‹J ∈ F› ‹K ∈ F› ‹B ≠ {}› ‹B ⊆ V› by blast
  ultimately show False
  using connectedD [of "J ∩ K" U V] ‹open U› ‹open V› ‹U ∩ V = {}› by auto
  qed
  with cf show ?thesis
  by (auto simp: connected_closed_set compact_imp_closed)
 qed
 
 lemma connected_chain_gen:
  fixes F :: "'a :: euclidean_space set set"
  assumes X: "X ∈ F" "compact X"
  and cc: "∧T. T ∈ F ==> closed T ∧ connected T"
  and linear: "∧S T. S ∈ F ∧ T ∈ F ==> S ⊆ T ∨ T ⊆ S"
  shows "connected(∩F)"
 proof -
  have "∩F = (∩T∈F. X ∩ T)"
  using X by blast
  moreover have "connected (∩T∈F. X ∩ T)"
  proof (rule connected_chain)
  show "∧T. T ∈ (∩) X ` F ==> compact T ∧ connected T"
  using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
  show "∧S T. S ∈ (∩) X ` F ∧ T ∈ (∩) X ` F ==> S ⊆ T ∨ T ⊆ S"
  using local.linear by blast
  qed
  ultimately show ?thesis
  by metis
 qed
 
 lemma connected_nest:
  fixes S :: "'a::linorder ==> 'b::euclidean_space set"
  assumes S: "∧n. compact(S n)" "∧n. connected(S n)"
  and nest: "∧m n. m ≤ n ==> S n ⊆ S m"
  shows "connected(∩ (range S))"
 proof (rule connected_chain)
  show "∧A T. A ∈ range S ∧ T ∈ range S ==> A ⊆ T ∨ T ⊆ A"
  by (metis image_iff le_cases nest)
 qed (use S in blast)
 
 lemma connected_nest_gen:
  fixes S :: "'a::linorder ==> 'b::euclidean_space set"
  assumes S: "∧n. closed(S n)" "∧n. connected(S n)" "compact(S k)"
  and nest: "∧m n. m ≤ n ==> S n ⊆ S m"
  shows "connected(∩ (range S))"
 proof (rule connected_chain_gen [of "S k"])
  show "∧A T. A ∈ range S ∧ T ∈ range S ==> A ⊆ T ∨ T ⊆ A"
  by (metis imageE le_cases nest)
 qed (use S in auto)
 
 subsection‹Proper maps, including projections out of compact sets›
 
 lemma finite_indexed_bound:
  assumes A: "finite A" "∧x. x ∈ A ==> ∃n::'a::linorder. P x n"
  shows "∃m. ∀x ∈ A. ∃k≤m. P x k"
 using A
 proof (induction A)
  case empty then show ?case by force
 next
  case (insert a A)
  then obtain m n where "∀x ∈ A. ∃k≤m. P x k" "P a n"
  by force
  then show ?case
  by (metis dual_order.trans insert_iff le_cases)
 qed
 
 proposition proper_map:
  fixes f :: "'a::heine_borel ==> 'b::heine_borel"
  assumes "closedin (top_of_set S) K"
  and com: "∧U. [U ⊆ T; compact U] ==> compact (S ∩ f -` U)"
  and "f ` S ⊆ T"
  shows "closedin (top_of_set T) (f ` K)"
 proof -
  have "K ⊆ S"
  using assms closedin_imp_subset by metis
  obtain C where "closed C" and Keq: "K = S ∩ C"
  using assms by (auto simp: closedin_closed)
  have *: "y ∈ f ` K" if "y ∈ T" and y: "y islimpt f ` K" for y
  proof -
  obtain h where "∀n. (∃x∈K. h n = f x) ∧ h n ≠ y" "inj h" and hlim: "(h ---> y) sequentially"
  using ‹y ∈ T› y by (force simp: limpt_sequential_inj)
  then obtain X where X: "∧n. X n ∈ K ∧ h n = f (X n) ∧ h n ≠ y"
  by metis
  then have fX: "∧n. f (X n) = h n"
  by metis
  define Ψ where "Ψ ≡ λn. {a ∈ K. f a ∈ insert y (range (λi. f (X (n + i))))}"
  have "compact (C ∩ (S ∩ f -` insert y (range (λi. f(X(n + i))))))" for n
  proof (intro closed_Int_compact [OF ‹closed C› com] compact_sequence_with_limit)
  show "insert y (range (λi. f (X (n + i)))) ⊆ T"
  using X ‹K ⊆ S› ‹f ` S ⊆ T› ‹y ∈ T› by blast
  show "(λi. f (X (n + i))) <---- y"
  by (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
  qed
  then have comf: "compact (Ψ n)" for n
  by (simp add: Keq Int_def Ψ_def conj_commute)
  have ne: "∩F ≠ {}"
  if "finite F"
  and F: "∧t. t ∈ F ==> (∃n. t = Ψ n)"
  for F
  proof -
  obtain m where m: "∧t. t ∈ F ==> ∃k≤m. t = Ψ k"
  by (rule exE [OF finite_indexed_bound [OF ‹finite F› F]], force+)
  have "X m ∈ ∩F"
  using X le_Suc_ex by (fastforce simp: Ψ_def dest: m)
  then show ?thesis by blast
  qed
  have "(∩n. Ψ n) ≠ {}"
  proof (rule compact_fip_Heine_Borel)
  show "∧F'. [finite F'; F' ⊆ range Ψ] ==> ∩ F' ≠ {}"
  by (meson ne rangeE subset_eq)
  qed (use comf in blast)
  then obtain x where "x ∈ K" "∧n. (f x = y ∨ (∃u. f x = h (n + u)))"
  by (force simp add: Ψ_def fX)
  then show ?thesis
  unfolding image_iff by (metis ‹inj h› le_add1 not_less_eq_eq rangeI range_ex1_eq)
  qed
  with assms closedin_subset show ?thesis
  by (force simp: closedin_limpt)
 qed
 
 subsection ‹Closure of conic hulls›
 proposition closedin_conic_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "compact T" "0 ∉ T" "T ⊆ S"
  shows "closedin (top_of_set (conic hull S)) (conic hull T)"
 proof -
  have **: "compact ({0..} × T ∩ (λz. fst z *🪙R snd z) -` K)" (is "compact ?L")
  if "K ⊆ (λz. (fst z) *🪙R snd z) ` ({0..} × S)" "compact K" for K
  proof -
  obtain r where "r > 0" and r: "∧x. x ∈ K ==> norm x ≤ r"
  by (metis ‹compact K› bounded_normE compact_imp_bounded)
  show ?thesis
  unfolding compact_eq_bounded_closed
  proof
  have "bounded ({0..r / setdist{0}T} × T)"
  by (simp add: assms(1) bounded_Times compact_imp_bounded)
  moreover have "?L ⊆ ({0..r / setdist{0}T} × T)"
  proof clarsimp
  fix a b
  assume "a *🪙R b ∈ K" and "b ∈ T" and "0 ≤ a"
  have "setdist {0} T ≠ 0"
  using ‹b ∈ T› assms compact_imp_closed setdist_eq_0_closed by auto
  then have T0: "setdist {0} T > 0"
  using less_eq_real_def by fastforce
  then have "a * setdist {0} T ≤ r"
  by (smt (verit, ccfv_SIG) ‹0 ≤ a› ‹a *🪙R b ∈ K› ‹b ∈ T› dist_0_norm mult_mono' norm_scaleR r setdist_le_dist singletonI)
  with T0 ‹r>0› show "a ≤ r / setdist {0} T"
  by (simp add: divide_simps)
  qed
  ultimately show "bounded ?L"
  by (meson bounded_subset)
  show "closed ?L"
  proof (rule continuous_closed_preimage)
  show "continuous_on ({0..} × T) (λz. fst z *🪙R snd z)"
  by (intro continuous_intros)
  show "closed ({0::real..} × T)"
  by (simp add: assms(1) closed_Times compact_imp_closed)
  show "closed K"
  by (simp add: compact_imp_closed that(2))
  qed
  qed
  qed
  show ?thesis
  unfolding conic_hull_as_image
  proof (rule proper_map)
  show "compact ({0..} × T ∩ (λz. fst z *🪙R snd z) -` K)" (is "compact ?L")
  if "K ⊆ (λz. (fst z) *🪙R snd z) ` ({0..} × S)" "compact K" for K
  proof -
  obtain r where "r > 0" and r: "∧x. x ∈ K ==> norm x ≤ r"
  by (metis ‹compact K› bounded_normE compact_imp_bounded)
  show ?thesis
  unfolding compact_eq_bounded_closed
  proof
  have "bounded ({0..r / setdist{0}T} × T)"
  by (simp add: assms(1) bounded_Times compact_imp_bounded)
  moreover have "?L ⊆ ({0..r / setdist{0}T} × T)"
  proof clarsimp
  fix a b
  assume "a *🪙R b ∈ K" and "b ∈ T" and "0 ≤ a"
  have "setdist {0} T ≠ 0"
  using ‹b ∈ T› assms compact_imp_closed setdist_eq_0_closed by auto
  then have T0: "setdist {0} T > 0"
  using less_eq_real_def by fastforce
  then have "a * setdist {0} T ≤ r"
  by (smt (verit, ccfv_SIG) ‹0 ≤ a› ‹a *🪙R b ∈ K› ‹b ∈ T› dist_0_norm mult_mono' norm_scaleR r setdist_le_dist singletonI)
  with T0 ‹r>0› show "a ≤ r / setdist {0} T"
  by (simp add: divide_simps)
  qed
  ultimately show "bounded ?L"
  by (meson bounded_subset)
  show "closed ?L"
  proof (rule continuous_closed_preimage)
  show "continuous_on ({0..} × T) (λz. fst z *🪙R snd z)"
  by (intro continuous_intros)
  show "closed ({0::real..} × T)"
  by (simp add: assms(1) closed_Times compact_imp_closed)
  show "closed K"
  by (simp add: compact_imp_closed that(2))
  qed
  qed
  qed
  show "(λz. fst z *🪙R snd z) ` ({0::real..} × T) ⊆ (λz. fst z *🪙R snd z) ` ({0..} × S)"
  using ‹T ⊆ S› by force
  qed auto
 qed
 
 lemma closed_conic_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "0 ∈ rel_interior S ∨ compact S ∧ 0 ∉ S"
  shows "closed(conic hull S)"
  using assms
 proof
  assume "0 ∈ rel_interior S"
  then show "closed (conic hull S)"
  by (simp add: conic_hull_eq_span)
 next
  assume "compact S ∧ 0 ∉ S"
  then have "closedin (top_of_set UNIV) (conic hull S)"
  using closedin_conic_hull by force
  then show "closed (conic hull S)"
  by simp
 qed
 
 lemma conic_closure:
  fixes S :: "'a::euclidean_space set"
  shows "conic S ==> conic(closure S)"
  by (meson Convex.cone_def cone_closure conic_def)
 
 lemma closure_conic_hull:
  fixes S :: "'a::euclidean_space set"
  assumes "0 ∈ rel_interior S ∨ bounded S ∧ ~(0 ∈ closure S)"
  shows "closure(conic hull S) = conic hull (closure S)"
  using assms
 proof
  assume "0 ∈ rel_interior S"
  then show "closure (conic hull S) = conic hull closure S"
  by (metis closed_affine_hull closure_closed closure_same_affine_hull closure_subset conic_hull_eq_affine_hull subsetD subset_rel_interior)
 next
  have "∧x. x ∈ conic hull closure S ==> x ∈ closure (conic hull S)"
  by (metis (no_types, opaque_lifting) closure_mono conic_closure conic_conic_hull subset_eq subset_hull)
  moreover
  assume "bounded S ∧ 0 ∉ closure S"
  then have "∧x. x ∈ closure (conic hull S) ==> x ∈ conic hull closure S"
  by (metis closed_conic_hull closure_Un_frontier closure_closed closure_mono compact_closure hull_Un_subset le_sup_iff subsetD)
  ultimately show "closure (conic hull S) = conic hull closure S"
  by blast
 qed
 
 lemma compact_continuous_image_eq:
  fixes f :: "'a::heine_borel ==> 'b::heine_borel"
  assumes f: "inj_on f S"
  shows "continuous_on S f ⟷ (∀T. compact T ∧ T ⊆ S ⟶ compact(f ` T))"
  (is "?lhs = ?rhs")
 proof
  assume ?lhs then show ?rhs
  by (metis continuous_on_subset compact_continuous_image)
 next
  assume RHS: ?rhs
  obtain g where gf: "∧x. x ∈ S ==> g (f x) = x"
  by (metis inv_into_f_f f)
  then have *: "(S ∩ f -` U) = g ` U" if "U ⊆ f ` S" for U
  using that by fastforce
  have gfim: "g ` f ` S ⊆ S" using gf by auto
  have **: "compact (f ` S ∩ g -` C)" if C: "C ⊆ S" "compact C" for C
  proof -
  obtain h where "h C ∈ C ∧ h C ∉ S ∨ compact (f ` C)"
  by (force simp: C RHS)
  moreover have "f ` C = (f ` S ∩ g -` C)"
  using C gf by auto
  ultimately show ?thesis
  using C by auto
  qed
  show ?lhs
  using proper_map [OF _ _ gfim] **
  by (simp add: continuous_on_closed * closedin_imp_subset)
 qed
 
 subsection🍋‹tag unimportant›\‹Trivial fact: convexity equals connectedness for collinear sets›
 
 lemma convex_connected_collinear:
  fixes S :: "'a::euclidean_space set"
  assumes "collinear S"
  shows "convex S ⟷ connected S"
 proof
  assume "convex S"
  then show "connected S"
  using convex_connected by blast
 next
  assume S: "connected S"
  show "convex S"
  proof (cases "S = {}")
  case True
  then show ?thesis by simp
  next
  case False
  then obtain a where "a ∈ S" by auto
  have "collinear (affine hull S)"
  by (simp add: assms collinear_affine_hull_collinear)
  then obtain z where "z ≠ 0" "∧x. x ∈ affine hull S ==> ∃c. x - a = c *🪙R z"
  by (meson ‹a ∈ S› collinear hull_inc)
  then obtain f where f: "∧x. x ∈ affine hull S ==> x - a = f x *🪙R z"
  by metis
  then have inj_f: "inj_on f (affine hull S)"
  by (metis diff_add_cancel inj_onI)
  have diff: "x - y = (f x - f y) *🪙R z" if x: "x ∈ affine hull S" and y: "y ∈ affine hull S" for x y
  proof -
  have "f x *🪙R z = x - a"
  by (simp add: f hull_inc x)
  moreover have "f y *🪙R z = y - a"
  by (simp add: f hull_inc y)
  ultimately show ?thesis
  by (simp add: scaleR_left.diff)
  qed
  have cont_f: "continuous_on (affine hull S) f"
  proof (clarsimp simp: dist_norm continuous_on_iff diff)
  show "∧x e. 0 🚫 ==> ∃d>0. ∀y ∈ affine hull S. ∣f y - f x∣ * norm z 🚫 ⟶ ∣f y - f x∣ 🚫"
  by (metis ‹z ≠ 0› mult_pos_pos mult_less_cancel_right_pos zero_less_norm_iff)
  qed
  then have conn_fS: "connected (f ` S)"
  by (meson S connected_continuous_image continuous_on_subset hull_subset)
  show ?thesis
  proof (clarsimp simp: convex_contains_segment)
  fix x y z
  assume "x ∈ S" "y ∈ S" "z ∈ closed_segment x y"
  have False if "z ∉ S"
  proof -
  have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
  proof (rule continuous_injective_image_segment_1)
  show "continuous_on (closed_segment x y) f"
  by (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
  show "inj_on f (closed_segment x y)"
  by (meson ‹x ∈ S› ‹y ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
  qed
  then have fz: "f z ∈ closed_segment (f x) (f y)"
  using ‹z ∈ closed_segment x y› by blast
  have "z ∈ affine hull S"
  by (meson ‹x ∈ S› ‹y ∈ S› ‹z ∈ closed_segment x y› convex_affine_hull convex_contains_segment hull_inc subset_eq)
  then have fz_notin: "f z ∉ f ` S"
  using hull_subset inj_f inj_onD that by fastforce
  moreover have "{..🚫z} ∩ f ` S ≠ {}" "{f z🚫} ∩ f ` S ≠ {}"
  proof -
  consider "f x ≤ f z ∧ f z ≤ f y" | "f y ≤ f z ∧ f z ≤ f x"
  using fz
  by (auto simp add: closed_segment_eq_real_ivl split: if_split_asm)
  then have "{..🚫z} ∩ f ` {x,y} ≠ {} ∧ {f z🚫} ∩ f ` {x,y} ≠ {}"
  by cases (use fz_notin ‹x ∈ S› ‹y ∈ S› in ‹auto simp: image_iff›)
  then show "{..🚫z} ∩ f ` S ≠ {}" "{f z🚫} ∩ f ` S ≠ {}"
  using ‹x ∈ S› ‹y ∈ S› by blast+
  qed
  ultimately show False
  using connectedD [OF conn_fS, of "{..🚫z}" "{f z🚫}"] by force
  qed
  then show "z ∈ S" by meson
  qed
  qed
 qed
 
 lemma compact_convex_collinear_segment_alt:
  fixes S :: "'a::euclidean_space set"
  assumes "S ≠ {}" "compact S" "connected S" "collinear S"
  obtains a b where "S = closed_segment a b"
 proof -
  obtain ξ where "ξ ∈ S" using ‹S ≠ {}› by auto
  have "collinear (affine hull S)"
  by (simp add: assms collinear_affine_hull_collinear)
  then obtain z where "z ≠ 0" "∧x. x ∈ affine hull S ==> ∃c. x - ξ = c *🪙R z"
  by (meson ‹ξ ∈ S› collinear hull_inc)
  then obtain f where f: "∧x. x ∈ affine hull S ==> x - ξ = f x *🪙R z"
  by metis
  let ?g = "λr. r *🪙R z + ξ"
  have gf: "?g (f x) = x" if "x ∈ affine hull S" for x
  by (metis diff_add_cancel f that)
  then have inj_f: "inj_on f (affine hull S)"
  by (metis inj_onI)
  have diff: "x - y = (f x - f y) *🪙R z" if x: "x ∈ affine hull S" and y: "y ∈ affine hull S" for x y
  proof -
  have "f x *🪙R z = x - ξ"
  by (simp add: f hull_inc x)
  moreover have "f y *🪙R z = y - ξ"
  by (simp add: f hull_inc y)
  ultimately show ?thesis
  by (simp add: scaleR_left.diff)
  qed
  have cont_f: "continuous_on (affine hull S) f"
  proof (clarsimp simp: dist_norm continuous_on_iff diff)
  show "∧x e. 0 🚫 ==> ∃d>0. ∀y ∈ affine hull S. ∣f y - f x∣ * norm z 🚫 ⟶ ∣f y - f x∣ 🚫"
  by (metis ‹z ≠ 0› mult_pos_pos mult_less_cancel_right_pos zero_less_norm_iff)
  qed
  then have "connected (f ` S)"
  by (meson ‹connected S› connected_continuous_image continuous_on_subset hull_subset)
  moreover have "compact (f ` S)"
  by (meson ‹compact S› compact_continuous_image_eq cont_f hull_subset inj_f)
  ultimately obtain x y where "f ` S = {x..y}"
  by (meson connected_compact_interval_1)
  then have fS_eq: "f ` S = closed_segment x y"
  using ‹S ≠ {}› closed_segment_eq_real_ivl by auto
  obtain a b where "a ∈ S" "f a = x" "b ∈ S" "f b = y"
  by (metis (full_types) ends_in_segment fS_eq imageE)
  have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
  proof (rule continuous_injective_image_segment_1)
  show "continuous_on (closed_segment a b) f"
  by (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
  show "inj_on f (closed_segment a b)"
  by (meson ‹a ∈ S› ‹b ∈ S› convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
  qed
  then have "f ` (closed_segment a b) = f ` S"
  by (simp add: ‹f a = x› ‹f b = y› fS_eq)
  then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
  by simp
  moreover have "(λx. f x *🪙R z + ξ) ` closed_segment a b = closed_segment a b"
  unfolding image_def using ‹a ∈ S› ‹b ∈ S›
  by (safe; metis (mono_tags, lifting) convex_affine_hull convex_contains_segment gf hull_subset subsetCE)
  ultimately have "closed_segment a b = S"
  using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
  then show ?thesis
  using that by blast
 qed
 
 lemma compact_convex_collinear_segment:
  fixes S :: "'a::euclidean_space set"
  assumes "S ≠ {}" "compact S" "convex S" "collinear S"
  obtains a b where "S = closed_segment a b"
  using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
 
 
 lemma proper_map_from_compact:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes contf: "continuous_on S f" and imf: "f ∈ S → T" and "compact S"
  "closedin (top_of_set T) K"
  shows "compact (S ∩ f -` K)"
 by (rule closedin_compact [OF ‹compact S›] continuous_closedin_preimage_gen assms)+
 
 lemma proper_map_fst:
  assumes "compact T" "K ⊆ S" "compact K"
  shows "compact (S × T ∩ fst -` K)"
 proof -
  have "(S × T ∩ fst -` K) = K × T"
  using assms by auto
  then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_fst:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "compact T" "closedin (top_of_set (S × T)) c"
  shows "closedin (top_of_set S) (fst ` c)"
 proof -
  have *: "fst ` (S × T) ⊆ S"
  by auto
  show ?thesis
  using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
 qed
 
 lemma proper_map_snd:
  assumes "compact S" "K ⊆ T" "compact K"
  shows "compact (S × T ∩ snd -` K)"
 proof -
  have "(S × T ∩ snd -` K) = S × K"
  using assms by auto
  then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_snd:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "compact S" "closedin (top_of_set (S × T)) c"
  shows "closedin (top_of_set T) (snd ` c)"
 proof -
  have *: "snd ` (S × T) ⊆ T"
  by auto
  show ?thesis
  using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
 qed
 
 lemma closedin_compact_projection:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes "compact S" and clo: "closedin (top_of_set (S × T)) U"
  shows "closedin (top_of_set T) {y. ∃x. x ∈ S ∧ (x, y) ∈ U}"
 proof -
  have "U ⊆ S × T"
  by (metis clo closedin_imp_subset)
  then have "{y. ∃x. x ∈ S ∧ (x, y) ∈ U} = snd ` U"
  by force
  moreover have "closedin (top_of_set T) (snd ` U)"
  by (rule closed_map_snd [OF assms])
  ultimately show ?thesis
  by simp
 qed
 
 
 lemma closed_compact_projection:
  fixes S :: "'a::euclidean_space set"
  and T :: "('a * 'b::euclidean_space) set"
  assumes "compact S" and clo: "closed T"
  shows "closed {y. ∃x. x ∈ S ∧ (x, y) ∈ T}"
 proof -
  have *: "{y. ∃x. x ∈ S ∧ Pair x y ∈ T} = {y. ∃x. x ∈ S ∧ Pair x y ∈ ((S × UNIV) ∩ T)}"
  by auto
  show ?thesis
  unfolding *
  by (intro clo closedin_closed_Int closedin_closed_trans [OF _ closed_UNIV] closedin_compact_projection [OF ‹compact S›])
 qed
 
 subsubsection🍋‹tag unimportant›\‹Representing affine hull as a finite intersection of hyperplanes›
 
 proposition🍋‹tag unimportant› affine_hull_convex_Int_nonempty_interior:
  fixes S :: "'a::real_normed_vector set"
  assumes "convex S" "S ∩ interior T ≠ {}"
  shows "affine hull (S ∩ T) = affine hull S"
 proof
  show "affine hull (S ∩ T) ⊆ affine hull S"
  by (simp add: hull_mono)
 next
  obtain a where "a ∈ S" "a ∈ T" and at: "a ∈ interior T"
  using assms interior_subset by blast
  then obtain e where "e > 0" and e: "cball a e ⊆ T"
  using mem_interior_cball by blast
  have *: "x ∈ (+) a ` span ((λx. x - a) ` (S ∩ T))" if "x ∈ S" for x
  proof (cases "x = a")
  case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
  by blast
  next
  case False
  define k where "k = min (1/2) (e / norm (x-a))"
  have k: "0 🚫" "k 🚫"
  using ‹e > 0› False by (auto simp: k_def)
  then have xa: "(x-a) = inverse k *🪙R k *🪙R (x-a)"
  by simp
  have "e / norm (x - a) ≥ k"
  using k_def by linarith
  then have "a + k *🪙R (x - a) ∈ cball a e"
  using ‹0 🚫› False
  by (simp add: dist_norm) (simp add: field_simps)
  then have T: "a + k *🪙R (x - a) ∈ T"
  using e by blast
  have S: "a + k *🪙R (x - a) ∈ S"
  using k ‹a ∈ S› convexD [OF ‹convex S› ‹a ∈ S› ‹x ∈ S›, of "1-k" k]
  by (simp add: algebra_simps)
  have "inverse k *🪙R k *🪙R (x-a) ∈ span ((λx. x - a) ` (S ∩ T))"
  by (intro span_mul [OF span_base] image_eqI [where x = "a + k *🪙R (x - a)"]) (auto simp: S T)
  with xa image_iff show ?thesis by fastforce
  qed
  have "S ⊆ affine hull (S ∩ T)"
  by (force simp: * ‹a ∈ S› ‹a ∈ T› hull_inc affine_hull_span_gen [of a])
  then show "affine hull S ⊆ affine hull (S ∩ T)"
  by (simp add: subset_hull)
 qed
 
 corollary affine_hull_convex_Int_open:
  fixes S :: "'a::real_normed_vector set"
  assumes "convex S" "open T" "S ∩ T ≠ {}"
  shows "affine hull (S ∩ T) = affine hull S"
  using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
 
 corollary affine_hull_affine_Int_nonempty_interior:
  fixes S :: "'a::real_normed_vector set"
  assumes "affine S" "S ∩ interior T ≠ {}"
  shows "affine hull (S ∩ T) = affine hull S"
  by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
 
 corollary affine_hull_affine_Int_open:
  fixes S :: "'a::real_normed_vector set"
  assumes "affine S" "open T" "S ∩ T ≠ {}"
  shows "affine hull (S ∩ T) = affine hull S"
  by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
 
 corollary affine_hull_convex_Int_openin:
  fixes S :: "'a::real_normed_vector set"
  assumes "convex S" "openin (top_of_set (affine hull S)) T" "S ∩ T ≠ {}"
  shows "affine hull (S ∩ T) = affine hull S"
  using assms unfolding openin_open
  by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
  fixes S :: "'a::real_normed_vector set"
  assumes "openin (top_of_set (affine hull T)) S" "S ≠ {}"
  shows "affine hull S = affine hull T"
  using assms unfolding openin_open
  by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
 
 corollary affine_hull_open:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S" "S ≠ {}"
  shows "affine hull S = UNIV"
  by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
 
 lemma aff_dim_convex_Int_nonempty_interior:
  fixes S :: "'a::euclidean_space set"
  shows "[convex S; S ∩ interior T ≠ {}] ==> aff_dim(S ∩ T) = aff_dim S"
  using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
 
 lemma aff_dim_convex_Int_open:
  fixes S :: "'a::euclidean_space set"
  shows "[convex S; open T; S ∩ T ≠ {}] ==> aff_dim(S ∩ T) = aff_dim S"
  using aff_dim_convex_Int_nonempty_interior interior_eq by blast
 
 lemma affine_hull_Diff:
  fixes S:: "'a::real_normed_vector set"
  assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F ⊂ S"
  shows "affine hull (S - F) = affine hull S"
 proof -
  have clo: "closedin (top_of_set S) F"
  using assms finite_imp_closedin by auto
  moreover have "S - F ≠ {}"
  using assms by auto
  ultimately show ?thesis
  by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
 qed
 
 lemma affine_hull_halfspace_lt:
  fixes a :: "'a::euclidean_space"
  shows "affine hull {x. a ∙ x 🚫} = (if a = 0 ∧ r ≤ 0 then {} else UNIV)"
 using halfspace_eq_empty_lt [of a r]
 by (simp add: open_halfspace_lt affine_hull_open)
 
 lemma affine_hull_halfspace_le:
  fixes a :: "'a::euclidean_space"
  shows "affine hull {x. a ∙ x ≤ r} = (if a = 0 ∧ r 🚫 then {} else UNIV)"
 proof (cases "a = 0")
  case True then show ?thesis by simp
 next
  case False
  then have "affine hull closure {x. a ∙ x 🚫} = UNIV"
  using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
  moreover have "{x. a ∙ x 🚫} ⊆ {x. a ∙ x ≤ r}"
  by (simp add: Collect_mono)
  ultimately show ?thesis using False antisym_conv hull_mono top_greatest
  by (metis affine_hull_halfspace_lt)
 qed
 
 lemma affine_hull_halfspace_gt:
  fixes a :: "'a::euclidean_space"
  shows "affine hull {x. a ∙ x > r} = (if a = 0 ∧ r ≥ 0 then {} else UNIV)"
 using halfspace_eq_empty_gt [of r a]
 by (simp add: open_halfspace_gt affine_hull_open)
 
 lemma affine_hull_halfspace_ge:
  fixes a :: "'a::euclidean_space"
  shows "affine hull {x. a ∙ x ≥ r} = (if a = 0 ∧ r > 0 then {} else UNIV)"
 using affine_hull_halfspace_le [of "-a" "-r"] by simp
 
 lemma aff_dim_halfspace_lt:
  fixes a :: "'a::euclidean_space"
  shows "aff_dim {x. a ∙ x 🚫} =
  (if a = 0 ∧ r ≤ 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
 
 lemma aff_dim_halfspace_le:
  fixes a :: "'a::euclidean_space"
  shows "aff_dim {x. a ∙ x ≤ r} =
  (if a = 0 ∧ r 🚫 then -1 else DIM('a))"
 proof -
  have "int (DIM('a)) = aff_dim (UNIV::'a set)"
  by (simp)
  then have "aff_dim (affine hull {x. a ∙ x ≤ r}) = DIM('a)" if "(a = 0 ⟶ r ≥ 0)"
  using that by (simp add: affine_hull_halfspace_le not_less)
  then show ?thesis
  by (force)
 qed
 
 lemma aff_dim_halfspace_gt:
  fixes a :: "'a::euclidean_space"
  shows "aff_dim {x. a ∙ x > r} =
  (if a = 0 ∧ r ≥ 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
 
 lemma aff_dim_halfspace_ge:
  fixes a :: "'a::euclidean_space"
  shows "aff_dim {x. a ∙ x ≥ r} =
  (if a = 0 ∧ r > 0 then -1 else DIM('a))"
 using aff_dim_halfspace_le [of "-a" "-r"] by simp
 
 proposition aff_dim_eq_hyperplane:
  fixes S :: "'a::euclidean_space set"
  shows "aff_dim S = DIM('a) - 1 ⟷ (∃a b. a ≠ 0 ∧ affine hull S = {x. a ∙ x = b})"
  (is "?lhs = ?rhs")
 proof (cases "S = {}")
  case True then show ?thesis
  by (auto simp: dest: hyperplane_eq_Ex)
 next
  case False
  then obtain c where "c ∈ S" by blast
  show ?thesis
  proof (cases "c = 0")
  case True
  have "?lhs ⟷ (∃a. a ≠ 0 ∧ span ((λx. x - c) ` S) = {x. a ∙ x = 0})"
  by (simp add: aff_dim_eq_dim [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane del: One_nat_def)
  also have "... ⟷ ?rhs"
  using span_zero [of S] True ‹c ∈ S› affine_hull_span_0 hull_inc
  by (fastforce simp add: affine_hull_span_gen [of c] ‹c = 0›)
  finally show ?thesis .
  next
  case False
  have xc_im: "x ∈ (+) c ` {y. a ∙ y = 0}" if "a ∙ x = a ∙ c" for a x
  proof -
  have "∃y. a ∙ y = 0 ∧ c + y = x"
  by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
  then show "x ∈ (+) c ` {y. a ∙ y = 0}"
  by blast
  qed
  have 2: "span ((λx. x - c) ` S) = {x. a ∙ x = 0}"
  if "(+) c ` span ((λx. x - c) ` S) = {x. a ∙ x = b}" for a b
  proof -
  have "b = a ∙ c"
  using span_0 that by fastforce
  with that have "(+) c ` span ((λx. x - c) ` S) = {x. a ∙ x = a ∙ c}"
  by simp
  then have "span ((λx. x - c) ` S) = (λx. x - c) ` {x. a ∙ x = a ∙ c}"
  by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
  also have "... = {x. a ∙ x = 0}"
  by (force simp: inner_distrib inner_diff_right
  intro: image_eqI [where x="x+c" for x])
  finally show ?thesis .
  qed
  have "?lhs = (∃a. a ≠ 0 ∧ span ((λx. x - c) ` S) = {x. a ∙ x = 0})"
  by (simp add: aff_dim_eq_dim [of c] ‹c ∈ S› hull_inc dim_eq_hyperplane del: One_nat_def)
  also have "... = ?rhs"
  by (fastforce simp add: affine_hull_span_gen [of c] ‹c ∈ S› hull_inc inner_distrib intro: xc_im intro!: 2)
  finally show ?thesis .
  qed
 qed
 
 corollary aff_dim_hyperplane [simp]:
  fixes a :: "'a::euclidean_space"
  shows "a ≠ 0 ==> aff_dim {x. a ∙ x = r} = DIM('a) - 1"
 by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
 
 subsection🍋‹tag unimportant›\‹Some stepping theorems›
 
 lemma aff_dim_insert:
  fixes a :: "'a::euclidean_space"
  shows "aff_dim (insert a S) = (if a ∈ affine hull S then aff_dim S else aff_dim S + 1)"
 proof (cases "S = {}")
  case True then show ?thesis
  by simp
 next
  case False
  then obtain x s' where S: "S = insert x s'" "x ∉ s'"
  by (meson Set.set_insert all_not_in_conv)
  show ?thesis using S
  by (force simp add: affine_hull_insert_span_gen span_zero insert_commute [of a] aff_dim_eq_dim [of x] dim_insert)
 qed
 
 lemma affine_dependent_choose:
  fixes a :: "'a :: euclidean_space"
  assumes "¬(affine_dependent S)"
  shows "affine_dependent(insert a S) ⟷ a ∉ S ∧ a ∈ affine hull S"
  (is "?lhs = ?rhs")
 proof safe
  assume "affine_dependent (insert a S)" and "a ∈ S"
  then show "False"
  using ‹a ∈ S› assms insert_absorb by fastforce
 next
  assume lhs: "affine_dependent (insert a S)"
  then have "a ∉ S"
  by (metis (no_types) assms insert_absorb)
  moreover have "finite S"
  using affine_independent_iff_card assms by blast
  moreover have "aff_dim (insert a S) ≠ int (card S)"
  using ‹finite S› affine_independent_iff_card ‹a ∉ S› lhs by fastforce
  ultimately show "a ∈ affine hull S"
  by (metis aff_dim_affine_independent aff_dim_insert assms)
 next
  assume "a ∉ S" and "a ∈ affine hull S"
  show "affine_dependent (insert a S)"
  by (simp add: ‹a ∈ affine hull S› ‹a ∉ S› affine_dependent_def)
 qed
 
 lemma affine_independent_insert:
  fixes a :: "'a :: euclidean_space"
  shows "[¬ affine_dependent S; a ∉ affine hull S] ==> ¬ affine_dependent(insert a S)"
  by (simp add: affine_dependent_choose)
 
 lemma subspace_bounded_eq_trivial:
  fixes S :: "'a::real_normed_vector set"
  assumes "subspace S"
  shows "bounded S ⟷ S = {0}"
 proof -
  have "False" if "bounded S" "x ∈ S" "x ≠ 0" for x
  proof -
  obtain B where B: "∧y. y ∈ S ==> norm y 🚫" "B > 0"
  using ‹bounded S› by (force simp: bounded_pos_less)
  have "(B / norm x) *🪙R x ∈ S"
  using assms subspace_mul ‹x ∈ S› by auto
  moreover have "norm ((B / norm x) *🪙R x) = B"
  using that B by (simp add: algebra_simps)
  ultimately show False using B by force
  qed
  then have "bounded S ==> S = {0}"
  using assms subspace_0 by fastforce
  then show ?thesis
  by blast
 qed
 
 lemma affine_bounded_eq_trivial:
  fixes S :: "'a::real_normed_vector set"
  assumes "affine S"
  shows "bounded S ⟷ S = {} ∨ (∃a. S = {a})"
 proof (cases "S = {}")
  case True then show ?thesis
  by simp
 next
  case False
  then obtain b where "b ∈ S" by blast
  with False assms
  have "bounded S ==> S = {b}"
  using affine_diffs_subspace [OF assms ‹b ∈ S›]
  by (metis (no_types, lifting) ab_group_add_class.ab_left_minus bounded_translation image_empty image_insert subspace_bounded_eq_trivial translation_invert)
  then show ?thesis by auto
 qed
 
 lemma affine_bounded_eq_lowdim:
  fixes S :: "'a::euclidean_space set"
  assumes "affine S"
  shows "bounded S ⟷ aff_dim S ≤ 0"
 proof
  show "aff_dim S ≤ 0 ==> bounded S"
  by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
 qed (use affine_bounded_eq_trivial assms in fastforce)
 
 
 lemma bounded_hyperplane_eq_trivial_0:
  fixes a :: "'a::euclidean_space"
  assumes "a ≠ 0"
  shows "bounded {x. a ∙ x = 0} ⟷ DIM('a) = 1"
 proof
  assume "bounded {x. a ∙ x = 0}"
  then have 0: "aff_dim {x. a ∙ x = 0} ≤ 0"
  by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
  with assms 0
  have "int DIM('a) = 1"
  using order_neq_le_trans by fastforce
  then show "DIM('a) = 1" by auto
 next
  assume "DIM('a) = 1"
  then show "bounded {x. a ∙ x = 0}"
  by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
 qed
 
 lemma bounded_hyperplane_eq_trivial:
  fixes a :: "'a::euclidean_space"
  shows "bounded {x. a ∙ x = r} ⟷ (if a = 0 then r ≠ 0 else DIM('a) = 1)"
 proof -
  { assume "r ≠ 0" "a ≠ 0"
  have "aff_dim {x. y ∙ x = 0} = aff_dim {x. a ∙ x = r}" if "y ≠ 0" for y::'a
  by (metis that ‹a ≠ 0› aff_dim_hyperplane)
  then have "bounded {x. a ∙ x = r} = (DIM('a) = Suc 0)"
  by (metis One_nat_def ‹a ≠ 0› affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
  }
  then show ?thesis
  by (auto simp: bounded_hyperplane_eq_trivial_0)
 qed
 
 subsection🍋‹tag unimportant›\‹General case without assuming closure and getting non-strict separation›
 
 proposition🍋‹tag unimportant› separating_hyperplane_closed_point_inset:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "closed S" "S ≠ {}" "z ∉ S"
  obtains a b where "a ∈ S" "(a - z) ∙ z 🚫" "∧x. x ∈ S ==> b 🚫a - z) ∙ x"
 proof -
  obtain y where "y ∈ S" and y: "∧u. u ∈ S ==> dist z y ≤ dist z u"
  using distance_attains_inf [of S z] assms by auto
  then have *: "(y - z) ∙ z 🚫y - z) ∙ z + (norm (y - z))🪙2 / 2"
  using ‹y ∈ S› ‹z ∉ S› by auto
  show ?thesis
  proof (rule that [OF ‹y ∈ S› *])
  fix x
  assume "x ∈ S"
  have yz: "0 🚫y - z) ∙ (y - z)"
  using ‹y ∈ S› ‹z ∉ S› by auto
  { assume 0: "0 🚫(z - y) ∙ (x - y))"
  with any_closest_point_dot [OF ‹convex S› ‹closed S›]
  have False
  using y ‹x ∈ S› ‹y ∈ S› not_less by blast
  }
  then have "0 ≤ ((y - z) ∙ (x - y))"
  by (force simp: not_less inner_diff_left)
  with yz have "0 🚫 * ((y - z) ∙ (x - y)) + (y - z) ∙ (y - z)"
  by (simp add: algebra_simps)
  then show "(y - z) ∙ z + (norm (y - z))🪙2 / 2 🚫y - z) ∙ x"
  by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
  qed
 qed
 
 lemma separating_hyperplane_closed_0_inset:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "closed S" "S ≠ {}" "0 ∉ S"
  obtains a b where "a ∈ S" "a ≠ 0" "0 🚫" "∧x. x ∈ S ==> a ∙ x > b"
  using separating_hyperplane_closed_point_inset [OF assms] by simp (metis ‹0 ∉ S›)
 
 
 proposition🍋‹tag unimportant› separating_hyperplane_set_0_inspan:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "S ≠ {}" "0 ∉ S"
  obtains a where "a ∈ span S" "a ≠ 0" "∧x. x ∈ S ==> 0 ≤ a ∙ x"
 proof -
  define k where [abs_def]: "k c = {x. 0 ≤ c ∙ x}" for c :: 'a
  have "span S ∩ frontier (cball 0 1) ∩ ∩f' ≠ {}"
  if f': "finite f'" "f' ⊆ k ` S" for f'
  proof -
  obtain C where "C ⊆ S" "finite C" and C: "f' = k ` C"
  using finite_subset_image [OF f'] by blast
  obtain a where "a ∈ S" "a ≠ 0"
  using ‹S ≠ {}› ‹0 ∉ S› ex_in_conv by blast
  then have "norm (a /🪙R (norm a)) = 1"
  by simp
  moreover have "a /🪙R (norm a) ∈ span S"
  by (simp add: ‹a ∈ S› span_scale span_base)
  ultimately have ass: "a /🪙R (norm a) ∈ span S ∩ sphere 0 1"
  by simp
  show ?thesis
  proof (cases "C = {}")
  case True with C ass show ?thesis
  by auto
  next
  case False
  have "closed (convex hull C)"
  using ‹finite C› compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
  moreover have "convex hull C ≠ {}"
  by (simp add: False)
  moreover have "0 ∉ convex hull C"
  by (metis ‹C ⊆ S› ‹convex S› ‹0 ∉ S› convex_hull_subset hull_same insert_absorb insert_subset)
  ultimately obtain a b
  where "a ∈ convex hull C" "a ≠ 0" "0 🚫"
  and ab: "∧x. x ∈ convex hull C ==> a ∙ x > b"
  using separating_hyperplane_closed_0_inset by blast
  then have "a ∈ S"
  by (metis ‹C ⊆ S› assms(1) subsetCE subset_hull)
  moreover have "norm (a /🪙R (norm a)) = 1"
  using ‹a ≠ 0› by simp
  moreover have "a /🪙R (norm a) ∈ span S"
  by (simp add: ‹a ∈ S› span_scale span_base)
  ultimately have ass: "a /🪙R (norm a) ∈ span S ∩ sphere 0 1"
  by simp
  have "∧x. [a ≠ 0; x ∈ C] ==> 0 ≤ x ∙ a"
  using ab ‹0 🚫› by (metis hull_inc inner_commute less_eq_real_def less_trans)
  then have aa: "a /🪙R (norm a) ∈ (∩c∈C. {x. 0 ≤ c ∙ x})"
  by (auto simp add: field_split_simps)
  show ?thesis
  unfolding C k_def
  using ass aa Int_iff empty_iff by force
  qed
  qed
  moreover have "∧T. T ∈ k ` S ==> closed T"
  using closed_halfspace_ge k_def by blast
  ultimately have "(span S ∩ frontier(cball 0 1)) ∩ (∩ (k ` S)) ≠ {}"
  by (metis compact_imp_fip closed_Int_compact closed_span compact_cball compact_frontier)
  then show ?thesis
  unfolding set_eq_iff k_def
  by simp (metis inner_commute norm_eq_zero that zero_neq_one)
 qed
 
 
 lemma separating_hyperplane_set_point_inaff:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "S ≠ {}" and zno: "z ∉ S"
  obtains a b where "(z + a) ∈ affine hull (insert z S)"
  and "a ≠ 0" and "a ∙ z ≤ b"
  and "∧x. x ∈ S ==> a ∙ x ≥ b"
 proof -
  from separating_hyperplane_set_0_inspan [of "image (λx. -z + x) S"]
  have "convex ((+) (- z) ` S)"
  using ‹convex S› by simp
  moreover have "(+) (- z) ` S ≠ {}"
  by (simp add: ‹S ≠ {}›)
  moreover have "0 ∉ (+) (- z) ` S"
  using zno by auto
  ultimately obtain a where "a ∈ span ((+) (- z) ` S)" "a ≠ 0"
  and a: "∧x. x ∈ ((+) (- z) ` S) ==> 0 ≤ a ∙ x"
  using separating_hyperplane_set_0_inspan [of "image (λx. -z + x) S"]
  by blast
  then have szx: "∧x. x ∈ S ==> a ∙ z ≤ a ∙ x"
  by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
  moreover
  have "z + a ∈ affine hull insert z S"
  using ‹a ∈ span ((+) (- z) ` S)› affine_hull_insert_span_gen by blast
  ultimately show ?thesis
  using ‹a ≠ 0› szx that by auto
 qed
 
 proposition🍋‹tag unimportant› supporting_hyperplane_rel_boundary:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "x ∈ S" and xno: "x ∉ rel_interior S"
  obtains a where "a ≠ 0"
  and "∧y. y ∈ S ==> a ∙ x ≤ a ∙ y"
  and "∧y. y ∈ rel_interior S ==> a ∙ x 🚫 ∙ y"
 proof -
  obtain a b where aff: "(x + a) ∈ affine hull (insert x (rel_interior S))"
  and "a ≠ 0" and "a ∙ x ≤ b"
  and ageb: "∧u. u ∈ (rel_interior S) ==> a ∙ u ≥ b"
  using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
  by (auto simp: rel_interior_eq_empty convex_rel_interior)
  have le_ay: "a ∙ x ≤ a ∙ y" if "y ∈ S" for y
  proof -
  have con: "continuous_on (closure (rel_interior S)) ((∙) a)"
  by (rule continuous_intros continuous_on_subset | blast)+
  have y: "y ∈ closure (rel_interior S)"
  using ‹convex S› closure_def convex_closure_rel_interior ‹y ∈ S›
  by fastforce
  show ?thesis
  using continuous_ge_on_closure [OF con y] ageb ‹a ∙ x ≤ b›
  by fastforce
  qed
  have 3: "a ∙ x 🚫 ∙ y" if "y ∈ rel_interior S" for y
  proof -
  obtain e where "0 🚫" "y ∈ S" and e: "cball y e ∩ affine hull S ⊆ S"
  using ‹y ∈ rel_interior S› by (force simp: rel_interior_cball)
  define y' where "y' = y - (e / norm a) *🪙R ((x + a) - x)"
  have "y' ∈ cball y e"
  unfolding y'_def using ‹0 🚫› by force
  moreover have "y' ∈ affine hull S"
  unfolding y'_def
  by (metis ‹x ∈ S› ‹y ∈ S› ‹convex S› aff affine_affine_hull hull_redundant
  rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
  ultimately have "y' ∈ S"
  using e by auto
  have "a ∙ x ≤ a ∙ y"
  using le_ay ‹a ≠ 0› ‹y ∈ S› by blast
  moreover have "a ∙ x ≠ a ∙ y"
  using le_ay [OF ‹y' ∈ S›] ‹a ≠ 0› ‹0 🚫› not_le
  by (fastforce simp add: y'_def inner_diff dot_square_norm power2_eq_square)
  ultimately show ?thesis by force
  qed
  show ?thesis
  by (rule that [OF ‹a ≠ 0› le_ay 3])
 qed
 
 lemma supporting_hyperplane_relative_frontier:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "x ∈ closure S" "x ∉ rel_interior S"
  obtains a where "a ≠ 0"
  and "∧y. y ∈ closure S ==> a ∙ x ≤ a ∙ y"
  and "∧y. y ∈ rel_interior S ==> a ∙ x 🚫 ∙ y"
 using supporting_hyperplane_rel_boundary [of "closure S" x]
 by (metis assms convex_closure convex_rel_interior_closure)
 
 
 subsection🍋‹tag unimportant›\‹ Some results on decomposing convex hulls: intersections, simplicial subdivision›
 
 lemma
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent(S ∪ T)"
  shows convex_hull_Int_subset: "convex hull S ∩ convex hull T ⊆ convex hull (S ∩ T)" (is ?C)
  and affine_hull_Int_subset: "affine hull S ∩ affine hull T ⊆ affine hull (S ∩ T)" (is ?A)
 proof -
  have [simp]: "finite S" "finite T"
  using aff_independent_finite assms by blast+
  have "sum u (S ∩ T) = 1 ∧
  (∑v∈S ∩ T. u v *🪙R v) = (∑v∈S. u v *🪙R v)"
  if [simp]: "sum u S = 1"
  "sum v T = 1"
  and eq: "(∑x∈T. v x *🪙R x) = (∑x∈S. u x *🪙R x)" for u v
  proof -
  define f where "f x = (if x ∈ S then u x else 0) - (if x ∈ T then v x else 0)" for x
  have "sum f (S ∪ T) = 0"
  by (simp add: f_def sum_Un sum_subtractf flip: sum.inter_restrict)
  moreover have "(∑x∈(S ∪ T). f x *🪙R x) = 0"
  by (simp add: eq f_def sum_Un scaleR_left_diff_distrib sum_subtractf if_smult flip: sum.inter_restrict cong: if_cong)
  ultimately have "∧v. v ∈ S ∪ T ==> f v = 0"
  using aff_independent_finite assms unfolding affine_dependent_explicit
  by blast
  then have u [simp]: "∧x. x ∈ S ==> u x = (if x ∈ T then v x else 0)"
  by (simp add: f_def) presburger
  have "sum u (S ∩ T) = sum u S"
  by (simp add: sum.inter_restrict)
  then have "sum u (S ∩ T) = 1"
  using that by linarith
  moreover have "(∑v∈S ∩ T. u v *🪙R v) = (∑v∈S. u v *🪙R v)"
  by (auto simp: sum.inter_restrict intro: sum.cong)
  ultimately show ?thesis
  by force
  qed
  then show ?A ?C
  by (auto simp: convex_hull_finite affine_hull_finite)
 qed
 
 
 proposition🍋‹tag unimportant› affine_hull_Int:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent(S ∪ T)"
  shows "affine hull (S ∩ T) = affine hull S ∩ affine hull T"
  by (simp add: affine_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition🍋‹tag unimportant› convex_hull_Int:
  fixes S :: "'a::euclidean_space set"
  assumes "¬ affine_dependent(S ∪ T)"
  shows "convex hull (S ∩ T) = convex hull S ∩ convex hull T"
  by (simp add: convex_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition🍋‹tag unimportant›
  fixes S :: "'a::euclidean_space set set"
  assumes "¬ affine_dependent (∪S)"
  shows affine_hull_Inter: "affine hull (∩S) = (∩T∈S. affine hull T)" (is "?A")
  and convex_hull_Inter: "convex hull (∩S) = (∩T∈S. convex hull T)" (is "?C")
 proof -
  have "finite S"
  using aff_independent_finite assms finite_UnionD by blast
  then have "?A ∧ ?C" using assms
  proof (induction S rule: finite_induct)
  case empty then show ?case by auto
  next
  case (insert T F)
  then show ?case
  proof (cases "F={}")
  case True then show ?thesis by simp
  next
  case False
  with "insert.prems" have [simp]: "¬ affine_dependent (T ∪ ∩F)"
  by (auto intro: affine_dependent_subset)
  have [simp]: "¬ affine_dependent (∪F)"
  using affine_independent_subset insert.prems by fastforce
  show ?thesis
  by (simp add: affine_hull_Int convex_hull_Int insert.IH)
  qed
  qed
  then show "?A" "?C"
  by auto
 qed
 
 proposition🍋‹tag unimportant› in_convex_hull_exchange_unique:
  fixes S :: "'a::euclidean_space set"
  assumes naff: "¬ affine_dependent S" and a: "a ∈ convex hull S"
  and S: "T ⊆ S" "T' ⊆ S"
  and x: "x ∈ convex hull (insert a T)"
  and x': "x ∈ convex hull (insert a T')"
  shows "x ∈ convex hull (insert a (T ∩ T'))"
 proof (cases "a ∈ S")
  case True
  then have "¬ affine_dependent (insert a T ∪ insert a T')"
  using affine_dependent_subset assms by auto
  then have "x ∈ convex hull (insert a T ∩ insert a T')"
  by (metis IntI convex_hull_Int x x')
  then show ?thesis
  by simp
 next
  case False
  then have anot: "a ∉ T" "a ∉ T'"
  using assms by auto
  have [simp]: "finite S"
  by (simp add: aff_independent_finite assms)
  then obtain b where b0: "∧s. s ∈ S ==> 0 ≤ b s"
  and b1: "sum b S = 1" and aeq: "a = (∑s∈S. b s *🪙R s)"
  using a by (auto simp: convex_hull_finite)
  have fin [simp]: "finite T" "finite T'"
  using assms infinite_super ‹finite S› by blast+
  then obtain c c' where c0: "∧t. t ∈ insert a T ==> 0 ≤ c t"
  and c1: "sum c (insert a T) = 1"
  and xeq: "x = (∑t ∈ insert a T. c t *🪙R t)"
  and c'0: "∧t. t ∈ insert a T' ==> 0 ≤ c' t"
  and c'1: "sum c' (insert a T') = 1"
  and x'eq: "x = (∑t ∈ insert a T'. c' t *🪙R t)"
  using x x' by (auto simp: convex_hull_finite)
  with fin anot
  have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
  and wsumT: "(∑t ∈ T. c t *🪙R t) = x - c a *🪙R a"
  by simp_all
  have wsumT': "(∑t ∈ T'. c' t *🪙R t) = x - c' a *🪙R a"
  using x'eq fin anot by simp
  define cc where "cc ≡ λx. if x ∈ T then c x else 0"
  define cc' where "cc' ≡ λx. if x ∈ T' then c' x else 0"
  define dd where "dd ≡ λx. cc x - cc' x + (c a - c' a) * b x"
  have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
  unfolding cc_def cc'_def using S
  by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
  have wsumSS: "(∑t ∈ S. cc t *🪙R t) = x - c a *🪙R a" "(∑t ∈ S. cc' t *🪙R t) = x - c' a *🪙R a"
  unfolding cc_def cc'_def using S
  by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
  have sum_dd0: "sum dd S = 0"
  unfolding dd_def using S
  by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
  algebra_simps sum_distrib_right [symmetric] b1)
  have "(∑v∈S. (b v * x) *🪙R v) = x *🪙R (∑v∈S. b v *🪙R v)" for x
  by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
  then have *: "(∑v∈S. (b v * x) *🪙R v) = x *🪙R a" for x
  using aeq by blast
  have "(∑v ∈ S. dd v *🪙R v) = 0"
  unfolding dd_def using S
  by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
  then have dd0: "dd v = 0" if "v ∈ S" for v
  using naff [unfolded affine_dependent_explicit not_ex, rule_format, of S dd]
  using that sum_dd0 by force
  consider "c' a ≤ c a" | "c a ≤ c' a" by linarith
  then show ?thesis
  proof cases
  case 1
  then have "sum cc S ≤ sum cc' S"
  by (simp add: sumSS')
  then have le: "cc x ≤ cc' x" if "x ∈ S" for x
  using dd0 [OF that] 1 b0 mult_left_mono that
  by (fastforce simp add: dd_def algebra_simps)
  have cc0: "cc x = 0" if "x ∈ S" "x ∉ T ∩ T'" for x
  using le [OF ‹x ∈ S›] that c0
  by (force simp: cc_def cc'_def split: if_split_asm)
  have ge0: "∀x∈T ∩ T'. 0 ≤ (cc(a := c a)) x"
  by (simp add: c0 cc_def)
  have "sum (cc(a := c a)) (insert a (T ∩ T')) = c a + sum (cc(a := c a)) (T ∩ T')"
  by (simp add: anot)
  also have "... = c a + sum (cc(a := c a)) S"
  using ‹T ⊆ S› False cc0 cc_def ‹a ∉ S› by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
  also have "... = c a + (1 - c a)"
  by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS'(1))
  finally have 1: "sum (cc(a := c a)) (insert a (T ∩ T')) = 1"
  by simp
  have "(∑x∈insert a (T ∩ T'). (cc(a := c a)) x *🪙R x) = c a *🪙R a + (∑x ∈ T ∩ T'. (cc(a := c a)) x *🪙R x)"
  by (simp add: anot)
  also have "... = c a *🪙R a + (∑x ∈ S. (cc(a := c a)) x *🪙R x)"
  using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
  also have "... = c a *🪙R a + x - c a *🪙R a"
  by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
  finally have self: "(∑x∈insert a (T ∩ T'). (cc(a := c a)) x *🪙R x) = x"
  by simp
  show ?thesis
  by (force simp: convex_hull_finite c0 intro!: ge0 1 self exI [where x = "cc(a := c a)"])
  next
  case 2
  then have "sum cc' S ≤ sum cc S"
  by (simp add: sumSS')
  then have le: "cc' x ≤ cc x" if "x ∈ S" for x
  using dd0 [OF that] 2 b0 mult_left_mono that
  by (fastforce simp add: dd_def algebra_simps)
  have cc0: "cc' x = 0" if "x ∈ S" "x ∉ T ∩ T'" for x
  using le [OF ‹x ∈ S›] that c'0
  by (force simp: cc_def cc'_def split: if_split_asm)
  have ge0: "∀x∈T ∩ T'. 0 ≤ (cc'(a := c' a)) x"
  by (simp add: c'0 cc'_def)
  have "sum (cc'(a := c' a)) (insert a (T ∩ T')) = c' a + sum (cc'(a := c' a)) (T ∩ T')"
  by (simp add: anot)
  also have "... = c' a + sum (cc'(a := c' a)) S"
  using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
  also have "... = c' a + (1 - c' a)"
  by (metis ‹a ∉ S› fun_upd_other sum.cong sumSS')
  finally have 1: "sum (cc'(a := c' a)) (insert a (T ∩ T')) = 1"
  by simp
  have "(∑x∈insert a (T ∩ T'). (cc'(a := c' a)) x *🪙R x) = c' a *🪙R a + (∑x ∈ T ∩ T'. (cc'(a := c' a)) x *🪙R x)"
  by (simp add: anot)
  also have "... = c' a *🪙R a + (∑x ∈ S. (cc'(a := c' a)) x *🪙R x)"
  using ‹T ⊆ S› False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
  also have "... = c a *🪙R a + x - c a *🪙R a"
  by (simp add: wsumSS ‹a ∉ S› if_smult sum_delta_notmem)
  finally have self: "(∑x∈insert a (T ∩ T'). (cc'(a := c' a)) x *🪙R x) = x"
  by simp
  show ?thesis
  by (force simp: convex_hull_finite c'0 intro!: ge0 1 self exI [where x = "cc'(a := c' a)"])
  qed
 qed
 
 corollary🍋‹tag unimportant› convex_hull_exchange_Int:
  fixes a :: "'a::euclidean_space"
  assumes "¬ affine_dependent S" "a ∈ convex hull S" "T ⊆ S" "T' ⊆ S"
  shows "(convex hull (insert a T)) ∩ (convex hull (insert a T')) =
  convex hull (insert a (T ∩ T'))" (is "?lhs = ?rhs")
 proof (rule subset_antisym)
  show "?lhs ⊆ ?rhs"
  using in_convex_hull_exchange_unique assms by blast
  show "?rhs ⊆ ?lhs"
  by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
 qed
 
 lemma Int_closed_segment:
  fixes b :: "'a::euclidean_space"
  assumes "b ∈ closed_segment a c ∨ ¬ collinear{a,b,c}"
  shows "closed_segment a b ∩ closed_segment b c = {b}"
 proof (cases "c = a")
  case True
  then show ?thesis
  using assms collinear_3_eq_affine_dependent by fastforce
 next
  case False
  from assms show ?thesis
  proof
  assume "b ∈ closed_segment a c"
  moreover have "¬ affine_dependent {a, c}"
  by (simp)
  ultimately show ?thesis
  using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
  by (simp add: segment_convex_hull insert_commute)
  next
  assume ncoll: "¬ collinear {a, b, c}"
  have False if "closed_segment a b ∩ closed_segment b c ≠ {b}"
  proof -
  have "b ∈ closed_segment a b" and "b ∈ closed_segment b c"
  by auto
  with that obtain d where "b ≠ d" "d ∈ closed_segment a b" "d ∈ closed_segment b c"
  by force
  then have d: "collinear {a, d, b}" "collinear {b, d, c}"
  by (auto simp: between_mem_segment between_imp_collinear)
  have "collinear {a, b, c}"
  by (metis (full_types) ‹b ≠ d› collinear_3_trans d insert_commute)
  with ncoll show False ..
  qed
  then show ?thesis
  by blast
  qed
 qed
 
 lemma affine_hull_finite_intersection_hyperplanes:
  fixes S :: "'a::euclidean_space set"
  obtains F where
  "finite F"
  "of_nat (card F) + aff_dim S = DIM('a)"
  "affine hull S = ∩F"
  "∧h. h ∈ F ==> ∃a b. a ≠ 0 ∧ h = {x. a ∙ x = b}"
 proof -
  obtain b where "b ⊆ S"
  and indb: "¬ affine_dependent b"
  and eq: "affine hull S = affine hull b"
  using affine_basis_exists by blast
  obtain c where indc: "¬ affine_dependent c" and "b ⊆ c"
  and affc: "affine hull c = UNIV"
  by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
  then have "finite c"
  by (simp add: aff_independent_finite)
  then have fbc: "finite b" "card b ≤ card c"
  using ‹b ⊆ c› infinite_super by (auto simp: card_mono)
  have imeq: "(λx. affine hull x) ` ((λa. c - {a}) ` (c - b)) = ((λa. affine hull (c - {a})) ` (c - b))"
  by blast
  have card_cb: "(card (c - b)) + aff_dim S = DIM('a)"
  proof -
  have aff: "aff_dim (UNIV::'a set) = aff_dim c"
  by (metis aff_dim_affine_hull affc)
  have "aff_dim b = aff_dim S"
  by (metis (no_types) aff_dim_affine_hull eq)
  then have "int (card b) = 1 + aff_dim S"
  by (simp add: aff_dim_affine_independent indb)
  then show ?thesis
  using fbc aff
  by (simp add: ‹¬ affine_dependent c› ‹b ⊆ c› aff_dim_affine_independent card_Diff_subset of_nat_diff)
  qed
  show ?thesis
  proof (cases "c = b")
  case True show ?thesis
  proof
  show "int (card {}) + aff_dim S = int DIM('a)"
  using True card_cb by auto
  show "affine hull S = ∩ {}"
  using True affc eq by blast
  qed auto
  next
  case False
  have ind: "¬ affine_dependent (∪a∈c - b. c - {a})"
  by (rule affine_independent_subset [OF indc]) auto
  have *: "1 + aff_dim (c - {t}) = int (DIM('a))" if t: "t ∈ c" for t
  proof -
  have "insert t c = c"
  using t by blast
  then show ?thesis
  by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
  qed
  let ?F = "(λx. affine hull x) ` ((λa. c - {a}) ` (c - b))"
  show ?thesis
  proof
  have "card ((λa. affine hull (c - {a})) ` (c - b)) = card (c - b)"
  proof (rule card_image)
  show "inj_on (λa. affine hull (c - {a})) (c - b)"
  unfolding inj_on_def
  by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
  qed
  then show "int (card ?F) + aff_dim S = int DIM('a)"
  by (simp add: imeq card_cb)
  show "affine hull S = ∩ ?F"
  by (metis Diff_eq_empty_iff INT_simps(4) UN_singleton order_refl ‹b ⊆ c› False eq double_diff affine_hull_Inter [OF ind])
  have "∧a. [a ∈ c; a ∉ b] ==> aff_dim (c - {a}) = int (DIM('a) - Suc 0)"
  by (metis "*" DIM_ge_Suc0 One_nat_def add_diff_cancel_left' int_ops(2) of_nat_diff)
  then show "∧h. h ∈ ?F ==> ∃a b. a ≠ 0 ∧ h = {x. a ∙ x = b}"
  by (auto simp only: One_nat_def aff_dim_eq_hyperplane [symmetric])
  qed (use ‹finite c› in auto)
  qed
 qed
 
 lemma affine_hyperplane_sums_eq_UNIV_0:
  fixes S :: "'a :: euclidean_space set"
  assumes "affine S"
  and "0 ∈ S" and "w ∈ S"
  and "a ∙ w ≠ 0"
  shows "{x + y| x y. x ∈ S ∧ a ∙ y = 0} = UNIV"
 proof -
  have "subspace S"
  by (simp add: assms subspace_affine)
  have span1: "span {y. a ∙ y = 0} ⊆ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
  using ‹0 ∈ S› add.left_neutral by (intro span_mono) force
  have "w ∉ span {y. a ∙ y = 0}"
  using ‹a ∙ w ≠ 0› span_induct subspace_hyperplane by auto
  moreover have "w ∈ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
  using ‹w ∈ S›
  by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
  ultimately have span2: "span {y. a ∙ y = 0} ≠ span {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
  by blast
  have "a ≠ 0" using assms inner_zero_left by blast
  then have "DIM('a) - 1 = dim {y. a ∙ y = 0}"
  by (simp add: dim_hyperplane)
  also have "... 🚫 {x + y |x y. x ∈ S ∧ a ∙ y = 0}"
  using span1 span2 by (blast intro: dim_psubset)
  finally have "DIM('a) - 1 🚫 {x + y |x y. x ∈ S ∧ a ∙ y = 0}" .
  then have DD: "dim {x + y |x y. x ∈ S ∧ a ∙ y = 0} = DIM('a)"
  using antisym dim_subset_UNIV lowdim_subset_hyperplane not_le by fastforce
  have subs: "subspace {x + y| x y. x ∈ S ∧ a ∙ y = 0}"
  using subspace_sums [OF ‹subspace S› subspace_hyperplane] by simp
  moreover have "span {x + y| x y. x ∈ S ∧ a ∙ y = 0} = UNIV"
  using DD dim_eq_full by blast
  ultimately show ?thesis
  by (simp add: subs) (metis (lifting) span_eq_iff subs)
 qed
 
 proposition🍋‹tag unimportant› affine_hyperplane_sums_eq_UNIV:
  fixes S :: "'a :: euclidean_space set"
  assumes "affine S"
  and "S ∩ {v. a ∙ v = b} ≠ {}"
  and "S - {v. a ∙ v = b} ≠ {}"
  shows "{x + y| x y. x ∈ S ∧ a ∙ y = b} = UNIV"
 proof (cases "a = 0")
  case True with assms show ?thesis
  by (auto simp: if_splits)
 next
  case False
  obtain c where "c ∈ S" and c: "a ∙ c = b"
  using assms by force
  with affine_diffs_subspace [OF ‹affine S›]
  have "subspace ((+) (- c) ` S)" by blast
  then have aff: "affine ((+) (- c) ` S)"
  by (simp add: subspace_imp_affine)
  have 0: "0 ∈ (+) (- c) ` S"
  by (simp add: ‹c ∈ S›)
  obtain d where "d ∈ S" and "a ∙ d ≠ b" and dc: "d-c ∈ (+) (- c) ` S"
  using assms by auto
  then have adc: "a ∙ (d - c) ≠ 0"
  by (simp add: c inner_diff_right)
  define U where "U ≡ {x + y |x y. x ∈ (+) (- c) ` S ∧ a ∙ y = 0}"
  have "u + v ∈ (+) (c+c) ` U"
  if "u ∈ S" "b = a ∙ v" for u v
  proof
  show "u + v = c + c + (u + v - c - c)"
  by (simp add: algebra_simps)
  have "∃x y. u + v - c - c = x + y ∧ (∃xa∈S. x = xa - c) ∧ a ∙ y = 0"
  proof (intro exI conjI)
  show "u + v - c - c = (u-c) + (v-c)" "a ∙ (v - c) = 0"
  by (simp_all add: algebra_simps c that)
  qed (use that in auto)
  then show "u + v - c - c ∈ U"
  by (auto simp: U_def image_def)
  qed
  moreover have "[a ∙ v = 0; u ∈ S]
  ==> ∃x ya. v + (u + c) = x + ya ∧ x ∈ S ∧ a ∙ ya = b" for v u
  by (metis add.left_commute c inner_right_distrib pth_d)
  ultimately have "{x + y |x y. x ∈ S ∧ a ∙ y = b} = (+) (c+c) ` U"
  by (fastforce simp: algebra_simps U_def)
  also have "... = range ((+) (c + c))"
  by (simp only: U_def affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
  also have "... = UNIV"
  by simp
  finally show ?thesis .
 qed
 
 lemma aff_dim_sums_Int_0:
  assumes "affine S"
  and "affine T"
  and "0 ∈ S" "0 ∈ T"
  shows "aff_dim {x + y| x y. x ∈ S ∧ y ∈ T} = (aff_dim S + aff_dim T) - aff_dim(S ∩ T)"
 proof -
  have "0 ∈ {x + y |x y. x ∈ S ∧ y ∈ T}"
  using assms by force
  then have 0: "0 ∈ affine hull {x + y |x y. x ∈ S ∧ y ∈ T}"
  by (metis (lifting) hull_inc)
  have sub: "subspace S" "subspace T"
  using assms by (auto simp: subspace_affine)
  show ?thesis
  using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
 qed
 
 proposition aff_dim_sums_Int:
  assumes "affine S"
  and "affine T"
  and "S ∩ T ≠ {}"
  shows "aff_dim {x + y| x y. x ∈ S ∧ y ∈ T} = (aff_dim S + aff_dim T) - aff_dim(S ∩ T)"
 proof -
  obtain a where a: "a ∈ S" "a ∈ T" using assms by force
  have aff: "affine ((+) (-a) ` S)" "affine ((+) (-a) ` T)"
  using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
  have zero: "0 ∈ ((+) (-a) ` S)" "0 ∈ ((+) (-a) ` T)"
  using a assms by auto
  have "{x + y |x y. x ∈ (+) (- a) ` S ∧ y ∈ (+) (- a) ` T} =
  (+) (- 2 *🪙R a) ` {x + y| x y. x ∈ S ∧ y ∈ T}"
  by (force simp: algebra_simps scaleR_2)
  moreover have "(+) (- a) ` S ∩ (+) (- a) ` T = (+) (- a) ` (S ∩ T)"
  by auto
  ultimately show ?thesis
  using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
  by (metis (lifting))
 qed
 
 lemma aff_dim_affine_Int_hyperplane:
  fixes a :: "'a::euclidean_space"
  assumes "affine S"
  shows "aff_dim(S ∩ {x. a ∙ x = b}) =
  (if S ∩ {v. a ∙ v = b} = {} then - 1
  else if S ⊆ {v. a ∙ v = b} then aff_dim S
  else aff_dim S - 1)"
 proof (cases "a = 0")
  case True with assms show ?thesis
  by auto
 next
  case False
  then have "aff_dim (S ∩ {x. a ∙ x = b}) = aff_dim S - 1"
  if "x ∈ S" "a ∙ x ≠ b" and non: "S ∩ {v. a ∙ v = b} ≠ {}" for x
  proof -
  have [simp]: "{x + y| x y. x ∈ S ∧ a ∙ y = b} = UNIV"
  using affine_hyperplane_sums_eq_UNIV [OF assms non] that by blast
  show ?thesis
  using aff_dim_sums_Int [OF assms affine_hyperplane non]
  by (simp add: of_nat_diff False)
  qed
  then show ?thesis
  by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
 qed
 
 lemma aff_dim_lt_full:
  fixes S :: "'a::euclidean_space set"
  shows "aff_dim S 🚫('a) ⟷ (affine hull S ≠ UNIV)"
 by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
 
 lemma aff_dim_openin:
  fixes S :: "'a::euclidean_space set"
  assumes ope: "openin (top_of_set T) S" and "affine T" "S ≠ {}"
  shows "aff_dim S = aff_dim T"
 proof -
  show ?thesis
  proof (rule order_antisym)
  show "aff_dim S ≤ aff_dim T"
  by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
  next
  obtain a where "a ∈ S"
  using ‹S ≠ {}› by blast
  have "S ⊆ T"
  using ope openin_imp_subset by auto
  then have "a ∈ T"
  using ‹a ∈ S› by auto
  then have subT': "subspace ((λx. - a + x) ` T)"
  using affine_diffs_subspace ‹affine T› by auto
  then obtain B where Bsub: "B ⊆ ((λx. - a + x) ` T)" and po: "pairwise orthogonal B"
  and eq1: "∧x. x ∈ B ==> norm x = 1" and "independent B"
  and cardB: "card B = dim ((λx. - a + x) ` T)"
  and spanB: "span B = ((λx. - a + x) ` T)"
  by (rule orthonormal_basis_subspace) auto
  obtain e where "0 🚫" and e: "cball a e ∩ T ⊆ S"
  by (meson ‹a ∈ S› openin_contains_cball ope)
  have "aff_dim T = aff_dim ((λx. - a + x) ` T)"
  by (metis aff_dim_translation_eq)
  also have "... = dim ((λx. - a + x) ` T)"
  using aff_dim_subspace subT' by blast
  also have "... = card B"
  by (simp add: cardB)
  also have "... = card ((λx. e *🪙R x) ` B)"
  using ‹0 🚫› by (force simp: inj_on_def card_image)
  also have "... ≤ dim ((λx. - a + x) ` S)"
  proof -
  have e': "cball 0 e ∩ (λx. x - a) ` T ⊆ (λx. x - a) ` S"
  using e by (auto simp: dist_norm norm_minus_commute subset_eq)
  have "(λx. e *🪙R x) ` B ⊆ cball 0 e ∩ (λx. x - a) ` T"
  using Bsub ‹0 🚫› eq1 subT' ‹a ∈ T› by (auto simp: subspace_def)
  then have "(λx. e *🪙R x) ` B ⊆ (λx. x - a) ` S"
  using e' by blast
  moreover
  have "inj_on ((*🪙R) e) (span B)"
  using ‹0 🚫› inj_on_def by fastforce
  then have "independent ((λx. e *🪙R x) ` B)"
  using linear_scale_self ‹independent B› linear_dependent_inj_imageD by blast
  ultimately show ?thesis
  by (auto simp: intro!: independent_card_le_dim)
  qed
  also have "... = aff_dim S"
  using ‹a ∈ S› aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
  finally show "aff_dim T ≤ aff_dim S" .
  qed
 qed
 
 lemma dim_openin:
  fixes S :: "'a::euclidean_space set"
  assumes ope: "openin (top_of_set T) S" and "subspace T" "S ≠ {}"
  shows "dim S = dim T"
 proof (rule order_antisym)
  show "dim S ≤ dim T"
  by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
 next
  have "dim T = aff_dim S"
  using aff_dim_openin
  by (metis aff_dim_subspace ‹subspace T› ‹S ≠ {}› ope subspace_affine)
  also have "... ≤ dim S"
  by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
  subspace_span)
  finally show "dim T ≤ dim S" by simp
 qed
 
 subsection‹Lower-dimensional affine subsets are nowhere dense›
 
 proposition dense_complement_subspace:
  fixes S :: "'a :: euclidean_space set"
  assumes dim_less: "dim T 🚫 S" and "subspace S" shows "closure(S - T) = S"
 proof -
  have "closure(S - U) = S" if "dim U 🚫 S" "U ⊆ S" for U
  proof -
  have "span U ⊂ span S"
  by (metis neq_iff psubsetI span_eq_dim span_mono that)
  then obtain a where "a ≠ 0" "a ∈ span S" and a: "∧y. y ∈ span U ==> orthogonal a y"
  using orthogonal_to_subspace_exists_gen by metis
  show ?thesis
  proof
  have "closed S"
  by (simp add: ‹subspace S› closed_subspace)
  then show "closure (S - U) ⊆ S"
  by (simp add: closure_minimal)
  show "S ⊆ closure (S - U)"
  proof (clarsimp simp: closure_approachable)
  fix x and e::real
  assume "x ∈ S" "0 🚫"
  show "∃y∈S - U. dist y x 🚫"
  proof (cases "x ∈ U")
  case True
  let ?y = "x + (e/2 / norm a) *🪙R a"
  show ?thesis
  proof
  show "dist ?y x 🚫"
  using ‹0 🚫› by (simp add: dist_norm)
  next
  have "?y ∈ S"
  by (metis ‹a ∈ span S› ‹x ∈ S› assms(2) span_eq_iff subspace_add subspace_scale)
  moreover have "?y ∉ U"
  proof -
  have "e/2 / norm a ≠ 0"
  using ‹0 🚫› ‹a ≠ 0› by auto
  then show ?thesis
  by (metis True ‹a ≠ 0› a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
  qed
  ultimately show "?y ∈ S - U" by blast
  qed
  next
  case False
  with ‹0 🚫› ‹x ∈ S› show ?thesis by force
  qed
  qed
  qed
  qed
  moreover have "S - S ∩ T = S-T"
  by blast
  moreover have "dim (S ∩ T) 🚫 S"
  by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
  ultimately show ?thesis
  by force
 qed
 
 corollary🍋‹tag unimportant› dense_complement_affine:
  fixes S :: "'a :: euclidean_space set"
  assumes less: "aff_dim T 🚫 S" and "affine S" shows "closure(S - T) = S"
 proof (cases "S ∩ T = {}")
  case True
  then show ?thesis
  by (metis Diff_triv affine_hull_eq ‹affine S› closure_same_affine_hull closure_subset hull_subset subset_antisym)
 next
  case False
  then obtain z where z: "z ∈ S ∩ T" by blast
  then have "subspace ((+) (- z) ` S)"
  by (meson IntD1 affine_diffs_subspace ‹affine S›)
  moreover have "int (dim ((+) (- z) ` T)) 🚫 (dim ((+) (- z) ` S))"
 thm aff_dim_eq_dim
  using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
  ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
  by (simp add: dense_complement_subspace)
  then show ?thesis
  by (metis closure_translation translation_diff translation_invert)
 qed
 
 corollary🍋‹tag unimportant› dense_complement_openin_affine_hull:
  fixes S :: "'a :: euclidean_space set"
  assumes less: "aff_dim T 🚫 S"
  and ope: "openin (top_of_set (affine hull S)) S"
  shows "closure(S - T) = closure S"
 proof -
  have "affine hull S - T ⊆ affine hull S"
  by blast
  then have "closure (S ∩ closure (affine hull S - T)) = closure (S ∩ (affine hull S - T))"
  by (rule closure_openin_Int_closure [OF ope])
  then show ?thesis
  by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
 qed
 
 corollary🍋‹tag unimportant› dense_complement_convex:
  fixes S :: "'a :: euclidean_space set"
  assumes "aff_dim T 🚫 S" "convex S"
  shows "closure(S - T) = closure S"
 proof
  show "closure (S - T) ⊆ closure S"
  by (simp add: closure_mono)
  have "closure (rel_interior S - T) = closure (rel_interior S)"
  by (simp add: assms dense_complement_openin_affine_hull openin_rel_interior rel_interior_aff_dim rel_interior_same_affine_hull)
  then show "closure S ⊆ closure (S - T)"
  by (metis Diff_mono ‹convex S› closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
 qed
 
 corollary🍋‹tag unimportant› dense_complement_convex_closed:
  fixes S :: "'a :: euclidean_space set"
  assumes "aff_dim T 🚫 S" "convex S" "closed S"
  shows "closure(S - T) = S"
  by (simp add: assms dense_complement_convex)
 
 
 subsection🍋‹tag unimportant›\‹Parallel slices, etc›
 
 text‹ If we take a slice out of a set, we can do it perpendicularly,
  with the normal vector to the slice parallel to the affine hull.›
 
 proposition🍋‹tag unimportant› affine_parallel_slice:
  fixes S :: "'a :: euclidean_space set"
  assumes "affine S"
  and "S ∩ {x. a ∙ x ≤ b} ≠ {}"
  and "¬ (S ⊆ {x. a ∙ x ≤ b})"
  obtains a' b' where "a' ≠ 0"
  "S ∩ {x. a' ∙ x ≤ b'} = S ∩ {x. a ∙ x ≤ b}"
  "S ∩ {x. a' ∙ x = b'} = S ∩ {x. a ∙ x = b}"
  "∧w. w ∈ S ==> (w + a') ∈ S"
 proof (cases "S ∩ {x. a ∙ x = b} = {}")
  case True
  then obtain u v where "u ∈ S" "v ∈ S" "a ∙ u ≤ b" "a ∙ v > b"
  using assms by (auto simp: not_le)
  define 🪙 where "🪙 = u + ((b - a ∙ u) / (a ∙ v - a ∙ u)) *🪙R (v - u)"
  have "🪙 ∈ S"
  by (simp add: 🪙_def ‹u ∈ S› ‹v ∈ S› ‹affine S› mem_affine_3_minus)
  moreover have "a ∙ 🪙 = b"
  using ‹a ∙ u ≤ b› ‹b 🚫 ∙ v›
  by (simp add: 🪙_def algebra_simps) (simp add: field_simps)
  ultimately have False
  using True by force
  then show ?thesis ..
 next
  case False
  then obtain z where "z ∈ S" and z: "a ∙ z = b"
  using assms by auto
  with affine_diffs_subspace [OF ‹affine S›]
  have sub: "subspace ((+) (- z) ` S)" by blast
  then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
  by (auto simp: subspace_imp_affine)
  obtain a' a'' where a': "a' ∈ span ((+) (- z) ` S)" and a: "a = a' + a''"
  and "∧w. w ∈ span ((+) (- z) ` S) ==> orthogonal a'' w"
  using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
  then have "∧w. w ∈ S ==> a'' ∙ (w-z) = 0"
  by (simp add: span_base orthogonal_def)
  then have a'': "∧w. w ∈ S ==> a'' ∙ w = (a - a') ∙ z"
  by (simp add: a inner_diff_right)
  then have ba'': "∧w. w ∈ S ==> a'' ∙ w = b - a' ∙ z"
  by (simp add: inner_diff_left z)
  show ?thesis
  proof (cases "a' = 0")
  case True
  with a assms True a'' diff_zero less_irrefl show ?thesis
  by auto
  next
  case False
  show ?thesis
  proof
  show "S ∩ {x. a' ∙ x ≤ a' ∙ z} = S ∩ {x. a ∙ x ≤ b}"
  "S ∩ {x. a' ∙ x = a' ∙ z} = S ∩ {x. a ∙ x = b}"
  by (auto simp: a ba'' inner_left_distrib)
  have "∧w. w ∈ (+) (- z) ` S ==> (w + a') ∈ (+) (- z) ` S"
  by (metis subspace_add a' span_eq_iff sub)
  then show "∧w. w ∈ S ==> (w + a') ∈ S"
  by fastforce
  qed (use False in auto)
  qed
 qed
 
 lemma diffs_affine_hull_span:
  assumes "a ∈ S"
  shows "(λx. x - a) ` (affine hull S) = span ((λx. x - a) ` S)"
 proof -
  have *: "((λx. x - a) ` (S - {a})) = ((λx. x - a) ` S) - {0}"
  by (auto simp: algebra_simps)
  show ?thesis
    by (auto simp add: algebra_simps affine_hull_span2 [OF assms] *)
qed

lemma aff_dim_dim_affine_diffs:
  fixes S :: "'a :: euclidean_space set"
  assumes "affine S" "a ∈ S"
    shows "aff_dim S = dim ((λx. x - a) ` S)"
proof -
  obtain B where aff: "affine hull B = affine hull S"
             and ind: "¬ affine_dependent B"
             and card: "of_nat (card B) = aff_dim S + 1"
    using aff_dim_basis_exists by blast
  then have "B ≠ {}" using assms
    by (metis affine_hull_eq_empty ex_in_conv)
  then obtain c where "c ∈ B" by auto
  then have "c ∈ S"
    by (metis aff affine_hull_eq ‹affine S› hull_inc)
  have xy: "x - c = y - a ⟷ y = x + 1 *🪙R (a - c)" for x y c and a::'a
    by (auto simp: algebra_simps)
  have *: "(λx. x - c) ` S = (λx. x - a) ` S"
    using assms ‹c ∈ S›
    by (auto simp: image_iff xy; metis mem_affine_3_minus pth_1)
  have affS: "affine hull S = S"
    by (simp add: ‹affine S›)
  have "aff_dim S = of_nat (card B) - 1"
    using card by simp
  also have "... = dim ((λx. x - c) ` B)"
    using affine_independent_card_dim_diffs [OF ind ‹c ∈ B›]
    by (simp add: affine_independent_card_dim_diffs [OF ind ‹c ∈ B›])
  also have "... = dim ((λx. x - c) ` (affine hull B))"
    by (simp add: diffs_affine_hull_span ‹c ∈ B›)
  also have "... = dim ((λx. x - a) ` S)"
    by (simp add: affS aff *)
  finally show ?thesis .
qed

lemma aff_dim_linear_image_le:
  assumes "linear f"
    shows "aff_dim(f ` S) ≤ aff_dim S"
proof -
  have "aff_dim (f ` T) ≤ aff_dim T" if "affine T" for T
  proof (cases "T = {}")
    case True then show ?thesis by (simp add: aff_dim_geq)
  next
    case False
    then obtain a where "a ∈ T" by auto
    have 1: "((λx. x - f a) ` f ` T) = {x - f a |x. x ∈ f ` T}"
      by auto
    have 2: "{x - f a| x. x ∈ f ` T} = f ` ((λx. x - a) ` T)"
      by (force simp: linear_diff [OF assms])
    have "aff_dim (f ` T) = int (dim {x - f a |x. x ∈ f ` T})"
      by (simp add: ‹a ∈ T› hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
    also have "... = int (dim (f ` ((λx. x - a) ` T)))"
      by (force simp: linear_diff [OF assms] 2)
    also have "... ≤ int (dim ((λx. x - a) ` T))"
      by (simp add: dim_image_le [OF assms])
    also have "... ≤ aff_dim T"
      by (simp add: aff_dim_dim_affine_diffs [symmetric] ‹a ∈ T› ‹affine T›)
    finally show ?thesis .
  qed
  then
  have "aff_dim (f ` (affine hull S)) ≤ aff_dim (affine hull S)"
    using affine_affine_hull [of S] by blast
  then show ?thesis
    using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
qed

lemma aff_dim_injective_linear_image [simp]:
  assumes "linear f" "inj f"
    shows "aff_dim (f ` S) = aff_dim S"
proof (rule antisym)
  show "aff_dim (f ` S) ≤ aff_dim S"
    by (simp add: aff_dim_linear_image_le assms(1))
next
  obtain g where "linear g" "g ∘ f = id"
    using assms(1) assms(2) linear_injective_left_inverse by blast
  then have "aff_dim S ≤ aff_dim(g ` f ` S)"
    by (simp add: image_comp)
  also have "... ≤ aff_dim (f ` S)"
    by (simp add: ‹linear g› aff_dim_linear_image_le)
  finally show "aff_dim S ≤ aff_dim (f ` S)" .
qed


lemma choose_affine_subset:
  assumes "affine S" "-1 ≤ d" and dle: "d ≤ aff_dim S"
  obtains T where "affine T" "T ⊆ S" "aff_dim T = d"
proof (cases "d = -1 ∨ S={}")
  case True with assms show ?thesis
    by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
next
  case False
  with assms obtain a where "a ∈ S" "0 ≤ d" by auto
  with assms have ss: "subspace ((+) (- a) ` S)"
    by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
  have "nat d ≤ dim ((+) (- a) ` S)"
    by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
  then obtain T where "subspace T" and Tsb: "T ⊆ span ((+) (- a) ` S)"
                  and Tdim: "dim T = nat d"
    using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
  then have "affine T"
    using subspace_affine by blast
  then have "affine ((+) a ` T)"
    by (metis affine_hull_eq affine_hull_translation)
  moreover have "(+) a ` T ⊆ S"
  proof -
    have "T ⊆ (+) (- a) ` S"
      by (metis (no_types) span_eq_iff Tsb ss)
    then show "(+) a ` T ⊆ S"
      using add_ac by auto
  qed
  moreover have "aff_dim ((+) a ` T) = d"
    by (simp add: aff_dim_subspace Tdim ‹0 ≤ d› ‹subspace T› aff_dim_translation_eq)
  ultimately show ?thesis
    by (rule that)
qed

subsection‹Paracompactness›

proposition paracompact:
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
  assumes "S ⊆ ∪C" and opC: "∧T. T ∈ C ==> open T"
  obtains C' where "S ⊆ ∪ C'"
               and "∧U. U ∈ C' ==> open U ∧ (∃T. T ∈ C ∧ U ⊆ T)"
               and "∧x. x ∈ S
                       ==> ∃V. open V ∧ x ∈ V ∧ finite {U. U ∈ C' ∧ (U ∩ V ≠ {})}"
proof (cases "S = {}")
  case True with that show ?thesis by blast
next
  case False
  have "∃T U. x ∈ U ∧ open U ∧ closure U ⊆ T ∧ T ∈ C" if "x ∈ S" for x
  proof -
    obtain T where "x ∈ T" "T ∈ C" "open T"
      using assms ‹x ∈ S› by blast
    then obtain e where "e > 0" "cball x e ⊆ T"
      by (force simp: open_contains_cball)
    then show ?thesis
      by (meson open_ball ‹T ∈ C› ball_subset_cball centre_in_ball closed_cball closure_minimal dual_order.trans)
  qed
  then obtain F G where Gin: "x ∈ G x" and oG: "open (G x)"
    and clos: "closure (G x) ⊆ F x" and Fin: "F x ∈ C"
  if "x ∈ S" for x
    by metis
  then obtain F where "F ⊆ G ` S" "countable F" "∪F = ∪(G ` S)"
    using Lindelof [of "G ` S"] by (metis image_iff)
  then obtain K where K: "K ⊆ S" "countable K" and eq: "∪(G ` K) = ∪(G ` S)"
    by (metis countable_subset_image)
  with False Gin have "K ≠ {}" by force
  then obtain a :: "nat ==> 'a" where "range a = K"
    by (metis range_from_nat_into ‹countable K›)
  then have odif: "∧n. open (F (a n) - ∪{closure (G (a m)) |m. m < n})"
    using ‹K ⊆ S› Fin opC by (fastforce simp add:)
  let ?C = "range (λn. F(a n) - ∪{closure(G(a m)) |m. m < n})"
  have enum_S: "∃n. x ∈ F(a n) ∧ x ∈ G(a n)" if "x ∈ S" for x
  proof -
    have "∃y ∈ K. x ∈ G y" using eq that Gin by fastforce
    then show ?thesis
      using clos K ‹range a = K› closure_subset by blast
  qed
  show ?thesis
  proof
    show "S ⊆ Union ?C"
    proof
      fix x assume "x ∈ S"
      define n where "n ≡ LEAST n. x ∈ F(a n)"
      have n: "x ∈ F(a n)"
        using enum_S [OF ‹x ∈ S›] by (force simp: n_def intro: LeastI)
      have notn: "x ∉ F(a m)" if "m < n" for m
        using that not_less_Least by (force simp: n_def)
      then have "x ∉ ∪{closure (G (a m)) |m. m < n}"
        using n ‹K ⊆ S› ‹range a = K› clos notn by fastforce
      with n show "x ∈ Union ?C"
        by blast
    qed
    show "∧U. U ∈ ?C ==> open U ∧ (∃T. T ∈ C ∧ U ⊆ T)"
      using Fin ‹K ⊆ S› ‹range a = K› by (auto simp: odif)
    show "∃V. open V ∧ x ∈ V ∧ finite {U. U ∈ ?C ∧ (U ∩ V ≠ {})}" if "x ∈ S" for x
    proof -
      obtain n where n: "x ∈ F(a n)" "x ∈ G(a n)"
        using ‹x ∈ S› enum_S by auto
      have "{U ∈ ?C. U ∩ G (a n) ≠ {}} ⊆ (λn. F(a n) - ∪{closure(G(a m)) |m. m < n}) ` atMost n"
      proof clarsimp
        fix k  assume "(F (a k) - ∪{closure (G (a m)) |m. m < k}) ∩ G (a n) ≠ {}"
        then have "k ≤ n"
          by auto (metis closure_subset not_le subsetCE)
        then show "F (a k) - ∪{closure (G (a m)) |m. m < k}
                 ∈ (λn. F (a n) - ∪{closure (G (a m)) |m. m < n}) ` {..n}"
          by force
      qed
      moreover have "finite ((λn. F(a n) - ∪{closure(G(a m)) |m. m < n}) ` atMost n)"
        by force
      ultimately have *: "finite {U ∈ ?C. U ∩ G (a n) ≠ {}}"
        using finite_subset by blast
      have "a n ∈ S"
        using ‹K ⊆ S› ‹range a = K› by blast
      then show ?thesis
        by (blast intro: oG n *)
    qed
  qed
qed

corollary paracompact_closedin:
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
  assumes cin: "closedin (top_of_set U) S"
      and oin: "∧T. T ∈ C ==> openin (top_of_set U) T"
      and "S ⊆ ∪C"
  obtains C' where "S ⊆ ∪ C'"
               and "∧V. V ∈ C' ==> openin (top_of_set U) V ∧ (∃T. T ∈ C ∧ V ⊆ T)"
               and "∧x. x ∈ U
                       ==> ∃V. openin (top_of_set U) V ∧ x ∈ V ∧
                               finite {X. X ∈ C' ∧ (X ∩ V ≠ {})}"
proof -
  have "∃Z. open Z ∧ (T = U ∩ Z)" if "T ∈ C" for T
    using oin [OF that] by (auto simp: openin_open)
  then obtain F where opF: "open (F T)" and intF: "U ∩ F T = T" if "T ∈ C" for T
    by metis
  obtain K where K: "closed K" "U ∩ K = S"
    using cin by (auto simp: closedin_closed)
  have 1: "U ⊆ ∪(insert (- K) (F ` C))"
    by clarsimp (metis Int_iff Union_iff ‹U ∩ K = S› ‹S ⊆ ∪C› subsetD intF)
  have 2: "∧T. T ∈ insert (- K) (F ` C) ==> open T"
    using ‹closed K› by (auto simp: opF)
  obtain D where "U ⊆ ∪D"
             and D1: "∧U. U ∈ D ==> open U ∧ (∃T. T ∈ insert (- K) (F ` C) ∧ U ⊆ T)"
             and D2: "∧x. x ∈ U ==> ∃V. open V ∧ x ∈ V ∧ finite {U ∈ D. U ∩ V ≠ {}}"
    by (blast intro: paracompact [OF 1 2])
  let ?C = "{U ∩ V |V. V ∈ D ∧ (V ∩ K ≠ {})}"
  show ?thesis
  proof (rule_tac C' = "{U ∩ V |V. V ∈ D ∧ (V ∩ K ≠ {})}" in that)
    show "S ⊆ ∪?C"
      using ‹U ∩ K = S› ‹U ⊆ ∪D› K by (blast dest!: subsetD)
    show "∧V. V ∈ ?C ==> openin (top_of_set U) V ∧ (∃T. T ∈ C ∧ V ⊆ T)"
      using D1 intF by fastforce
    have *: "{X. (∃V. X = U ∩ V ∧ V ∈ D ∧ V ∩ K ≠ {}) ∧ X ∩ (U ∩ V) ≠ {}} ⊆
             (λx. U ∩ x) ` {U ∈ D. U ∩ V ≠ {}}" for V
      by blast
    show "∃V. openin (top_of_set U) V ∧ x ∈ V ∧ finite {X ∈ ?C. X ∩ V ≠ {}}"
      if "x ∈ U" for x
    proof -
      from D2 [OF that] obtain V where "open V" "x ∈ V" "finite {U ∈ D. U ∩ V ≠ {}}"
        by auto
      with * show ?thesis
        by (rule_tac x="U ∩ V" in exI) (auto intro: that finite_subset [OF *])
    qed
  qed
qed

corollary🍋‹tag unimportant› paracompact_closed:
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
  assumes "closed S"
      and opC: "∧T. T ∈ C ==> open T"
      and "S ⊆ ∪C"
  obtains C' where "S ⊆ ∪C'"
               and "∧U. U ∈ C' ==> open U ∧ (∃T. T ∈ C ∧ U ⊆ T)"
               and "∧x. ∃V. open V ∧ x ∈ V ∧
                               finite {U. U ∈ C' ∧ (U ∩ V ≠ {})}"
  by (rule paracompact_closedin [of UNIV S C]) (auto simp: assms)

  
subsection🍋‹tag unimportant›\<open>Closed-graph characterization of continuity›

lemma continuous_closed_graph_gen:
  fixes T :: "'b::real_normed_vector set"
  assumes contf: "continuous_on S f" and fim: "f ` S ⊆ T"
    shows "closedin (top_of_set (S × T)) ((λx. Pair x (f x)) ` S)"
proof -
  have eq: "((λx. Pair x (f x)) ` S) = (S × T ∩ (λz. (f ∘ fst)z - snd z) -` {0})"
    using fim by auto
  show ?thesis
    unfolding eq
    by (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf]) auto
qed

lemma continuous_closed_graph_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "compact T" and fim: "f ∈ S → T"
  shows "continuous_on S f ⟷
         closedin (top_of_set (S × T)) ((λx. Pair x (f x)) ` S)"
         (is "?lhs = ?rhs")
proof -
  have "?lhs" if ?rhs
  proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
    fix U
    assume U: "closedin (top_of_set T) U"
    have eq: "(S ∩ f -` U) = fst ` (((λx. Pair x (f x)) ` S) ∩ (S × U))"
      by (force simp: image_iff)
    show "closedin (top_of_set S) (S ∩ f -` U)"
      by (simp add: U closedin_Int closedin_Times closed_map_fst [OF ‹compact T›] that eq)
  qed
  with continuous_closed_graph_gen assms show ?thesis by blast
qed

lemma continuous_closed_graph:
  fixes f :: "'a::topological_space ==> 'b::real_normed_vector"
  assumes "closed S" and contf: "continuous_on S f"
  shows "closed ((λx. Pair x (f x)) ` S)"
proof (rule closedin_closed_trans)
  show "closedin (top_of_set (S × UNIV)) ((λx. (x, f x)) ` S)"
    by (rule continuous_closed_graph_gen [OF contf subset_UNIV])
qed (simp add: ‹closed S› closed_Times)

lemma continuous_from_closed_graph:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "compact T" and fim: "f ∈ S → T" and clo: "closed ((λx. Pair x (f x)) ` S)"
  shows "continuous_on S f"
    using fim clo
    by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF ‹compact T› fim])

lemma continuous_on_Un_local_open:
  assumes opS: "openin (top_of_set (S ∪ T)) S"
      and opT: "openin (top_of_set (S ∪ T)) T"
      and contf: "continuous_on S f" and contg: "continuous_on T f"
    shows "continuous_on (S ∪ T) f"
  using pasting_lemma [of "{S,T}" "top_of_set (S ∪ T)" id euclidean "λi. f" f] contf contg opS opT
  by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  

lemma continuous_on_cases_local_open:
  assumes opS: "openin (top_of_set (S ∪ T)) S"
      and opT: "openin (top_of_set (S ∪ T)) T"
      and contf: "continuous_on S f" and contg: "continuous_on T g"
      and fg: "∧x. x ∈ S ∧ ¬P x ∨ x ∈ T ∧ P x ==> f x = g x"
    shows "continuous_on (S ∪ T) (λx. if P x then f x else g x)"
proof -
  have "∧x. x ∈ S ==> (if P x then f x else g x) = f x"  "∧x. x ∈ T ==> (if P x then f x else g x) = g x"
    by (simp_all add: fg)
  then have "continuous_on S (λx. if P x then f x else g x)" "continuous_on T (λx. if P x then f x else g x)"
    by (simp_all add: contf contg cong: continuous_on_cong)
  then show ?thesis
    by (rule continuous_on_Un_local_open [OF opS opT])
qed

subsection🍋‹tag unimportant›\<open>The union of two collinear segments is another segment›

proposition🍋‹tag unimportant› in_convex_hull_exchange:
  fixes a :: "'a::euclidean_space"
  assumes a: "a ∈ convex hull S" and xS: "x ∈ convex hull S"
  obtains b where "b ∈ S" "x ∈ convex hull (insert a (S - {b}))"
proof (cases "a ∈ S")
  case True
  with xS insert_Diff that  show ?thesis by fastforce
next
  case False
  show ?thesis
  proof (cases "finite S ∧ card S ≤ Suc (DIM('a))")
    case True
    then obtain u where u0: "∧i. i ∈ S ==> 0 ≤ u i" and u1: "sum u S = 1"
                    and ua: "(∑i∈S. u i *🪙R i) = a"
        using a by (auto simp: convex_hull_finite)
    obtain v where v0: "∧i. i ∈ S ==> 0 ≤ v i" and v1: "sum v S = 1"
               and vx: "(∑i∈S. v i *🪙R i) = x"
      using True xS by (auto simp: convex_hull_finite)
    show ?thesis
    proof (cases "∃b. b ∈ S ∧ v b = 0")
      case True
      then obtain b where b: "b ∈ S" "v b = 0"
        by blast
      show ?thesis
      proof
        have fin: "finite (insert a (S - {b}))"
          using sum.infinite v1 by fastforce
        show "x ∈ convex hull insert a (S - {b})"
          unfolding convex_hull_finite [OF fin] mem_Collect_eq
        proof (intro conjI exI ballI)
          have "(∑x ∈ insert a (S - {b}). if x = a then 0 else v x) =
                (∑x ∈ S - {b}. if x = a then 0 else v x)"
            using fin by (force intro: sum.mono_neutral_right)
          also have "... = (∑x ∈ S - {b}. v x)"
            using b False by (auto intro!: sum.cong split: if_split_asm)
          also have "... = (∑x∈S. v x)"
            by (metis ‹v b = 0› diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
          finally show "(∑x∈insert a (S - {b}). if x = a then 0 else v x) = 1"
            by (simp add: v1)
          show "∧x. x ∈ insert a (S - {b}) ==> 0 ≤ (if x = a then 0 else v x)"
            by (auto simp: v0)
          have "(∑x ∈ insert a (S - {b}). (if x = a then 0 else v x) *🪙R x) =
                (∑x ∈ S - {b}. (if x = a then 0 else v x) *🪙R x)"
            using fin by (force intro: sum.mono_neutral_right)
          also have "... = (∑x ∈ S - {b}. v x *🪙R x)"
            using b False by (auto intro!: sum.cong split: if_split_asm)
          also have "... = (∑x∈S. v x *🪙R x)"
            by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
          finally show "(∑x∈insert a (S - {b}). (if x = a then 0 else v x) *🪙R x) = x"
            by (simp add: vx)
        qed
      qed (rule ‹b ∈ S›)
    next
      case False
      have le_Max: "u i / v i ≤ Max ((λi. u i / v i) ` S)" if "i ∈ S" for i
        by (simp add: True that)
      have "Max ((λi. u i / v i) ` S) ∈ (λi. u i / v i) ` S"
        using True v1 by (auto intro: Max_in)
      then obtain b where "b ∈ S" and beq: "Max ((λb. u b / v b) ` S) = u b / v b"
        by blast
      then have "0 ≠ u b / v b"
        using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
        by (metis False eq_iff v0)
      then have  "0 < u b" "0 < v b"
        using False ‹b ∈ S› u0 v0 by force+
      have fin: "finite (insert a (S - {b}))"
        using sum.infinite v1 by fastforce
      show ?thesis
      proof
        show "x ∈ convex hull insert a (S - {b})"
          unfolding convex_hull_finite [OF fin] mem_Collect_eq
        proof (intro conjI exI ballI)
          have "(∑x ∈ insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
                v b / u b + (∑x ∈ S - {b}. v x - (v b / u b) * u x)"
            using ‹a ∉ S› ‹b ∈ S› True  
            by (auto intro!: sum.cong split: if_split_asm)
          also have "... = v b / u b + (∑x ∈ S - {b}. v x) - (v b / u b) * (∑x ∈ S - {b}. u x)"
            by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
          also have "... = (∑x∈S. v x)"
            using ‹0 🚫 b› True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
          finally show "sum (λx. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
            by (simp add: v1)
          show "0 ≤ (if i = a then v b / u b else v i - v b / u b * u i)"
            if "i ∈ insert a (S - {b})" for i
            using ‹0 🚫 b› ‹0 🚫 b› v0 [of i] le_Max [of i] beq that False
            by (auto simp: field_simps split: if_split_asm)
          have "(∑x∈insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *🪙R x) =
                (v b / u b) *🪙R a + (∑x∈S - {b}. (v x - v b / u b * u x) *🪙R x)"
            using ‹a ∉ S› ‹b ∈ S› True  by (auto intro!: sum.cong split: if_split_asm)
          also have "... = (v b / u b) *🪙R a + (∑x ∈ S - {b}. v x *🪙R x) - (v b / u b) *🪙R (∑x ∈ S - {b}. u x *🪙R x)"
            by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
          also have "... = (∑x∈S. v x *🪙R x)"
            using ‹0 🚫 b› True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
          finally
          show "(∑x∈insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *🪙R x) = x"
            by (simp add: vx)
        qed
      qed (rule ‹b ∈ S›)
    qed
  next
    case False
    obtain T where "finite T" "T ⊆ S" and caT: "card T ≤ Suc (DIM('a))" and xT: "x ∈ convex hull T"
      using xS by (auto simp: caratheodory [of S])
    with False obtain b where b: "b ∈ S" "b ∉ T"
      by (metis antisym subsetI)
    show ?thesis
    proof
      show "x ∈ convex hull insert a (S - {b})"
        using  ‹T ⊆ S› b by (blast intro: subsetD [OF hull_mono xT])
    qed (rule ‹b ∈ S›)
  qed
qed

lemma convex_hull_exchange_Union:
  fixes a :: "'a::euclidean_space"
  assumes "a ∈ convex hull S"
  shows "convex hull S = (∪b ∈ S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
proof
  show "?lhs ⊆ ?rhs"
    by (blast intro: in_convex_hull_exchange [OF assms])
  show "?rhs ⊆ ?lhs"
  proof clarify
    fix x b
    assume"b ∈ S" "x ∈ convex hull insert a (S - {b})"
    then show "x ∈ convex hull S" if "b ∈ S"
      by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
  qed
qed

lemma Un_closed_segment:
  fixes a :: "'a::euclidean_space"
  assumes "b ∈ closed_segment a c"
    shows "closed_segment a b ∪ closed_segment b c = closed_segment a c"
proof (cases "c = a")
  case True
  with assms show ?thesis by simp
next
  case False
  with assms have "convex hull {a, b} ∪ convex hull {b, c} = (∪ba∈{a, c}. convex hull insert b ({a, c} - {ba}))"
    by (auto simp: insert_Diff_if insert_commute)
  then show ?thesis
    using convex_hull_exchange_Union
    by (metis assms segment_convex_hull)
qed

lemma Un_open_segment:
  fixes a :: "'a::euclidean_space"
  assumes "b ∈ open_segment a c"
  shows "open_segment a b ∪ {b} ∪ open_segment b c = open_segment a c" (is "?lhs = ?rhs")
proof -
  have b: "b ∈ closed_segment a c"
    by (simp add: assms open_closed_segment)
  have *: "?rhs ⊆ insert b (open_segment a b ∪ open_segment b c)"
          if "{b,c,a} ∪ open_segment a b ∪ open_segment b c = {c,a} ∪ ?rhs"
  proof -
    have "insert a (insert c (insert b (open_segment a b ∪ open_segment b c))) = insert a (insert c (?rhs))"
      using that by (simp add: insert_commute)
    then show ?thesis
      by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
  qed
  show ?thesis
  proof
    show "?lhs ⊆ ?rhs"
      by (simp add: assms b subset_open_segment)
    show "?rhs ⊆ ?lhs"
      using Un_closed_segment [OF b] *
      by (simp add: closed_segment_eq_open insert_commute)
  qed
qed

subsection‹Covering an open set by a countable chain of compact sets›
  
proposition open_Union_compact_subsets:
  fixes S :: "'a::euclidean_space set"
  assumes "open S"
  obtains C where "∧n. compact(C n)" "∧n. C n ⊆ S"
                  "∧n. C n ⊆ interior(C(Suc n))"
                  "∪(range C) = S"
                  "∧K. [compact K; K ⊆ S] ==> ∃N. ∀n≥N. K ⊆ (C n)"
proof (cases "S = {}")
  case True
  then show ?thesis
    by (rule_tac C = "λn. {}" in that) auto
next
  case False
  then obtain a where "a ∈ S"
    by auto
  let ?C = "λn. cball a (real n) - (∪x ∈ -S. ∪e ∈ ball 0 (1 / real(Suc n)). {x + e})"
  have "∃N. ∀n≥N. K ⊆ (f n)"
        if "∧n. compact(f n)" and sub_int: "∧n. f n ⊆ interior (f(Suc n))"
            and eq: "∪(range f) = S" and "compact K" "K ⊆ S" for f K
  proof -
    have *: "∀n. f n ⊆ (∪n. interior (f n))"
      by (meson Sup_upper2 UNIV_I ‹∧n. f n ⊆ interior (f (Suc n))› image_iff)
    have mono: "∧m n. m ≤ n ==>f m ⊆ f n"
      by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
    obtain I where "finite I" and I: "K ⊆ (∪i∈I. interior (f i))"
    proof (rule compactE_image [OF ‹compact K›])
      show "K ⊆ (∪n. interior (f n))"
        using ‹K ⊆ S› ‹∪(f ` UNIV) = S› * by blast
    qed auto
    { fix n
      assume n: "Max I ≤ n"
      have "(∪i∈I. interior (f i)) ⊆ f n"
        by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF ‹finite I›] n)
      then have "K ⊆ f n"
        using I by auto
    }
    then show ?thesis
      by blast
  qed
  moreover have "∃f. (∀n. compact(f n)) ∧ (∀n. (f n) ⊆ S) ∧ (∀n. (f n) ⊆ interior(f(Suc n))) ∧
                     ((∪(range f) = S))"
  proof (intro exI conjI allI)
    show "∧n. compact (?C n)"
      by (auto simp: compact_diff open_sums)
    show "∧n. ?C n ⊆ S"
      by auto
    show "?C n ⊆ interior (?C (Suc n))" for n
    proof -
      have 🍋: "cball a (real n) ⊆ ball a (1 + real n)"
        by (simp add: cball_subset_ball_iff)
      have cl: "closed (∪x∈- S. ∪e∈cball 0 (1 / (2 + real n)). {x + e})"
        using assms by (auto intro: closed_compact_sums)
      have "closure (∪x∈- S. ∪y∈ball 0 (1 / (2 + real n)). {x + y})
            ⊆ (∪x ∈ -S. ∪e ∈ cball 0 (1 / (2 + real n)). {x + e})"
        by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
      also have "... ⊆ (∪x ∈ -S. ∪y∈ball 0 (1 / (1 + real n)). {x + y})"
        by (simp add: cball_subset_ball_iff field_split_simps UN_mono)
      finally have "closure (∪x∈- S. ∪y∈ball 0 (1 / (2 + real n)). {x + y})
                    ⊆ (∪x ∈ -S. ∪y∈ball 0 (1 / (1 + real n)). {x + y})" .
      with 🍋 show ?thesis
        by (auto simp: interior_diff)
    qed
    have "S ⊆ ∪ (range ?C)"
    proof
      fix x
      assume x: "x ∈ S"
      then obtain e where "e > 0" and e: "ball x e ⊆ S"
        using assms open_contains_ball by blast
      then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
        by (metis divide_less_0_1_iff gr0I of_nat_0 order_less_imp_triv reals_Archimedean2)
      obtain N2 where N2: "norm(x - a) ≤ real N2"
        by (meson real_arch_simple)
      have N12: "inverse((N1 + N2) + 1) ≤ inverse(N1)"
        using ‹N1 > 0› by (auto simp: field_split_simps)
      have "x ≠ y + z" if "y ∉ S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
      proof -
        have "e * real N1 < e * (1 + (real N1 + real N2))"
          by (simp add: ‹0 🚫›)
        then have "1 / (1 + (real N1 + real N2)) < e"
          using N1 ‹e > 0›
          by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
        then have "x - z ∈ ball x e"
          using that by simp
        then have "x - z ∈ S"
          using e by blast
        with that show ?thesis
          by auto
      qed
      with N2 show "x ∈ ∪ (range ?C)"
        by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
    qed
    then show "∪ (range ?C) = S" by auto
  qed
  ultimately show ?thesis
    using that by metis
qed


subsection‹Orthogonal complement›

definition🍋‹tag important› orthogonal_comp (‹(‹open_block notation=‹postfix ⊥›\??⊥)› [80] 80)
  where "orthogonal_comp W ≡ {x. ∀y ∈ W. orthogonal y x}"

proposition subspace_orthogonal_comp: "subspace (W🪙⊥)"
  unfolding subspace_def orthogonal_comp_def orthogonal_def
  by (auto simp: inner_right_distrib)

lemma orthogonal_comp_anti_mono:
  assumes "A ⊆ B"
  shows "B🪙⊥ ⊆ A🪙⊥"
  using assms by (force simp add: orthogonal_comp_def orthogonal_def)

lemma orthogonal_comp_null [simp]: "{0}🪙⊥ = UNIV"
  by (auto simp: orthogonal_comp_def orthogonal_def)

lemma orthogonal_comp_UNIV [simp]: "UNIV🪙⊥ = {0}"
  unfolding orthogonal_comp_def orthogonal_def
  by auto (use inner_eq_zero_iff in blast)

lemma orthogonal_comp_subset: "U ⊆ U🪙⊥🪙⊥"
  by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)

lemma subspace_sum_minimal:
  assumes "S ⊆ U" "T ⊆ U" "subspace U"
  shows "S + T ⊆ U"
proof
  fix x
  assume "x ∈ S + T"
  then obtain xs xt where "xs ∈ S" "xt ∈ T" "x = xs+xt"
    by (meson set_plus_elim)
  then show "x ∈ U"
    by (meson assms subsetCE subspace_add)
qed

proposition subspace_sum_orthogonal_comp:
  fixes U :: "'a :: euclidean_space set"
  assumes "subspace U"
  shows "U + U🪙⊥ = UNIV"
proof -
  obtain B where "B ⊆ U"
    and ortho: "pairwise orthogonal B" "∧x. x ∈ B ==> norm x = 1"
    and "independent B" "card B = dim U" "span B = U"
    using orthonormal_basis_subspace [OF assms] by metis
  then have "finite B"
    by (simp add: indep_card_eq_dim_span)
  have *: "∀x∈B. ∀y∈B. x ∙ y = (if x=y then 1 else 0)"
    using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
  { fix v
    let ?u = "∑b∈B. (v ∙ b) *🪙R b"
    have "v = ?u + (v - ?u)"
      by simp
    moreover have "?u ∈ U"
      by (metis (no_types, lifting) ‹span B = U› assms subspace_sum span_base span_mul)
    moreover have "(v - ?u) ∈ U🪙⊥"
      unfolding orthogonal_comp_def orthogonal_def mem_Collect_eq
    proof 
      fix y
      assume "y ∈ U"
      with ‹span B = U› span_finite [OF ‹finite B›]
      obtain u where u: "y = (∑b∈B. u b *🪙R b)"
        by auto
      have "b ∙ (v - ?u) = 0" if "b ∈ B" for b
        using that ‹finite B›
        by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
      then show "y ∙ (v - ?u) = 0"
        by (simp add: u inner_sum_left)
    qed
    ultimately have "v ∈ U + U🪙⊥"
      using set_plus_intro by fastforce
  } then show ?thesis
    by auto
qed

lemma orthogonal_Int_0:
  assumes "subspace U"
  shows "U ∩ U🪙⊥ = {0}"
  using orthogonal_comp_def orthogonal_self
  by (force simp: assms subspace_0 subspace_orthogonal_comp)

lemma orthogonal_comp_self:
  fixes U :: "'a :: euclidean_space set"
  assumes "subspace U"
  shows "U🪙⊥🪙⊥ = U"
proof
  have ssU': "subspace (U🪙⊥)"
    by (simp add: subspace_orthogonal_comp)
  have "u ∈ U" if "u ∈ U🪙⊥🪙⊥" for u
  proof -
    obtain v w where "u = v+w" "v ∈ U" "w ∈ U🪙⊥"
      using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
    then have "u-v ∈ U🪙⊥"
      by simp
    moreover have "v ∈ U🪙⊥🪙⊥"
      using ‹v ∈ U› orthogonal_comp_subset by blast
    then have "u-v ∈ U🪙⊥🪙⊥"
      by (simp add: subspace_diff subspace_orthogonal_comp that)
    ultimately have "u-v = 0"
      using orthogonal_Int_0 ssU' by blast
    with ‹v ∈ U› show ?thesis
      by auto
  qed
  then show "U🪙⊥🪙⊥ ⊆ U"
    by auto
qed (use orthogonal_comp_subset in auto)

lemma ker_orthogonal_comp_adjoint:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
  shows "f -` {0} = (range (adjoint f))🪙⊥"
proof -
  have "∧x. [∀y. y ∙ f x = 0] ==> f x = 0"
    using assms inner_commute all_zero_iff by metis
  then show ?thesis
    using assms
    by (auto simp: orthogonal_comp_def orthogonal_def adjoint_works inner_commute)
qed

subsection🍋‹tag unimportant› ‹A non-injective linear function maps into a hyperplane.›

lemma linear_surj_adj_imp_inj:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f" "surj (adjoint f)"
  shows "inj f"
proof -
  have "∃x. y = adjoint f x" for y
    using assms by (simp add: surjD)
  then show "inj f"
    using assms unfolding inj_on_def image_def
    by (metis (no_types) adjoint_works euclidean_eqI)
qed

\ ‹🪙‹https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map›\
lemma surj_adjoint_iff_inj [simp]:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
  shows "surj (adjoint f) ⟷ inj f"
proof
  assume "surj (adjoint f)"
  then show "inj f"
    by (simp add: assms linear_surj_adj_imp_inj)
next
  assume "inj f"
  have "f -` {0} = {0}"
    using assms ‹inj f› linear_0 linear_injective_0 by fastforce
  moreover have "f -` {0} = range (adjoint f)🪙⊥"
    by (intro ker_orthogonal_comp_adjoint assms)
  ultimately have "range (adjoint f)🪙⊥🪙⊥ = UNIV"
    by (metis orthogonal_comp_null)
  then show "surj (adjoint f)"
    using adjoint_linear ‹linear f›
    by (metis linear_subspace_image orthogonal_comp_self subspace_UNIV)
qed

lemma inj_adjoint_iff_surj [simp]:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes "linear f"
  shows "inj (adjoint f) ⟷ surj f"
proof
  assume "inj (adjoint f)"
  have "(adjoint f) -` {0} = {0}"
    by (metis ‹inj (adjoint f)› adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
  then have "(range(f))🪙⊥ = {0}"
    by (metis (no_types, opaque_lifting) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
  then show "surj f"
    by (metis ‹inj (adjoint f)› adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
next
  assume "surj f"
  then have "range f = (adjoint f -` {0})🪙⊥"
    by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
  then have "{0} = adjoint f -` {0}"
    using ‹surj f› adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
  then show "inj (adjoint f)"
    by (simp add: ‹surj f› adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
qed

lemma linear_singular_into_hyperplane:
  fixes f :: "'n::euclidean_space ==> 'n"
  assumes "linear f"
  shows "¬ inj f ⟷ (∃a. a ≠ 0 ∧ (∀x. a ∙ f x = 0))" (is "_ = ?rhs")
proof
  assume "¬inj f"
  then show ?rhs
    using all_zero_iff
    by (metis (no_types, opaque_lifting) adjoint_clauses(2) adjoint_linear assms
        linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
next
  assume ?rhs
  then show "¬inj f"
    by (metis assms linear_injective_isomorphism all_zero_iff)
qed

lemma linear_singular_image_hyperplane:
  fixes f :: "'n::euclidean_space ==> 'n"
  assumes "linear f" "¬inj f"
  obtains a where "a ≠ 0" "∧S. f ` S ⊆ {x. a ∙ x = 0}"
  using assms by (fastforce simp add: linear_singular_into_hyperplane)

end