Theory Starlike

(* 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

subsectiontag 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 "yS. 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 < d 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 < e by (simp add: field_simps norm_minus_commute)
  then have "z  interior (ball c d)"
    using 0 < e 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 < e by (auto simp: algebra_simps min_def)
          show "dist (x - min (e/2 / norm (x - a)) 1 *R (x - a)) x < e"
            using 0 < e by (auto simp: dist_norm min_def)
          show "x - min (e/2 / norm (x - a)) 1 *R (x - a)  interior S"
            using 0 < e 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


subsectiontag unimportant› ‹Some obvious but surprisingly hard simplex lemmas›

lemma simplex:
  assumes "finite S"
    and "0  S"
  shows "convex hull (insert 0 S) = {y. u. (xS. 0  u x)  sum u S  1  sum (λx. u x *R x) S = y}"
proof -
  { fix x and u :: "'a  real"
    assume "xS. 0  u x" "sum u S  1"
    then have "v. 0  v 0  (xS. 0  v x)  v 0 + sum v S = 1  (xS. v x *R x) = (xS. 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. (iBasis. 0  xi)  (id. xi)  1  (iBasis. i  d  xi = 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: "iBasis. i  d  u i = ?x  i"
      and notind: "(iBasis. i  d  ?x  i = 0)"
      using substdbasis_expansion_unique[OF assms] by blast+
    then have **: "sum u ?D = sum ((∙) ?x) ?D"
      using assms by (auto intro!: 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 "iBasis. 0  x  i" "sum ((∙) x) ?D  1" "(iBasis. 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. (iBasis. 0  xi)  sum (λi. xi) Basis  1}"
  using substd_simplex[of Basis] by auto

lemma interior_std_simplex:
  "interior (convex hull (insert 0 Basis)) =
    {x::'a::euclidean_space. (iBasis. 0 < xi)  sum (λi. xi) 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. (iBasis. 0  x  i)  sum ((∙) x) Basis  1}"
  show "(iBasis. 0 < x  i)  sum ((∙) x) Basis < 1"
  proof safe
    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
    have **: "dist x (x + (e/2) *R (SOME i. iBasis)) < e" using e > 0
      unfolding dist_norm
      by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
    have "i. i  Basis  (x + (e/2) *R (SOME i. iBasis))  i =
      xi + (if i = (SOME i. iBasis) then e/2 else 0)"
      by (auto simp: SOME_Basis inner_Basis inner_simps)
    then have *: "sum ((∙) (x + (e/2) *R (SOME i. iBasis))) Basis =
      sum (λi. xi + (if (SOME i. iBasis) = i then e/2 else 0)) Basis"
      by (auto simp: intro!: sum.cong)
    have "sum ((∙) x) Basis < sum ((∙) (x + (e/2) *R (SOME i. iBasis))) Basis"
      using e > 0 DIM_positive by (auto simp: SOME_Basis sum.distrib *)
    also have "  1"
      using ** as by force
    finally show "sum ((∙) x) Basis < 1" by auto
  qed 
next
  fix x :: 'a
  assume as: "iBasis. 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. (iBasis. 0  x  i)  sum ((∙) x) Basis  1}"
  proof (rule_tac x="min (Min (((∙) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
    fix y
    assume y: "y  ball x (min (Min ((∙) x ` Basis)) ?d)"
    have "sum ((∙) y) Basis  sum (λi. xi + ?d) Basis"
    proof (rule sum_mono)
      fix i :: 'a
      assume i: "i  Basis"
      have "¦yi - xi¦  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 "¦yi - xi¦ < ?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 "(iBasis. 0  y  i)"
    proof safe
      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 "...  xi"
        using i by auto
      finally have "norm (x - y) < xi" .
      then show "0  yi"
        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 (auto intro: sum.cong)
      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. (iD. 0 < xi)  (iD. xi) < 1  (iBasis. i  D  xi = 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. (iBasis. i  D  xi = 0)}"
      using affine_hull_convex_hull affine_hull_substd_basis assms by auto
    have aux: "x::'a. iBasis. (iD. 0  xi)  (iBasis. i  D  xi = 0)  0  xi"
      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. (iBasis. i  D  xai = 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  (iBasis. i  D  yi = 0) 
                            (iD. 0  y  i)  sum ((∙) y) D  1"
        using assms by (force simp: substd_simplex)
      have x0: "(iBasis. i  D  xi = 0)"
        using x rel_interior_subset  substd_simplex[OF assms] by auto
      have "(iD. 0 < x  i)  sum ((∙) x) D < 1  (iBasis. i  D  xi = 0)"
      proof (intro conjI ballI)
        fix i :: 'a
        assume "i  D"
        then have "jD. 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 = xi + (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. xi + (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: "(iBasis. i  D  (x + (e/2) *R a)  i = 0)"
          using x0 D aD 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. iBasis  i  D  xi = 0"
          using x0 by auto
      qed
    }
    moreover
    {
      fix x :: "'a::euclidean_space"
      assume as: "x  ?s"
      have "i. 0 < xi  0 = xi  0  xi"
        by auto
      moreover have "i. i  D  i  D" by auto
      ultimately
      have "i. (iD. 0 < xi)  (i. i  D  xi = 0)  0  xi"
        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. iBasis. 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 < card D by (auto simp: d_def)
        ultimately show "min (Min (((∙) x) ` D)) d > 0"
          by auto
        fix y :: 'a
        assume y2: "iBasis. i  D  yi = 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. xi + d) D"
        proof (rule sum_mono)
          fix i
          assume "i  D"
          with D have i: "i  Basis"
            by auto
          have "¦yi - xi¦  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 "¦yi - xi¦ < d" .
          then show "y  i  x  i + d" by auto
        qed
        also have "  1"
          unfolding sum.distrib sum_constant d_def using 0 < card D
          by auto
        finally show "sum ((∙) y) D  1" .

        fix i :: 'a
        assume i: "i  Basis"
        then show "0  yi"
        proof (cases "iD")
          case True
          have "norm (x - y) < xi"
            using y Min_gr_iff[of "(∙) x ` D" "norm (x - y)"] 0 < card D i  D
            by (simp add: dist_norm card_gt_0_iff)
          then show "0  yi"
            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. (iD. 0 < x  i)  sum ((∙) x) D < 1  (iBasis. 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 = "sum (λb::'a::euclidean_space. inverse (2 * real (card D)) *R b) 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 = sum (λj. if i = j then inverse (2 * real (card D)) else 0) D"
      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) (rule refl, rule **)
      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. xD  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

subsectiontag 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. iBasis. 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. a0  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›

definitiontag 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 "xaffine 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 "xaffine 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 "xS. 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  e1 < 1 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  (xS. 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  (xaffine 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


subsubsectiontag 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  convex (S :: 'n::euclidean_space set)"
    and "(rel_interior ` )  {}"
  shows "(closure ` )  closure ((rel_interior ` ))"
proof -
  obtain x where x: "S. x  rel_interior S"
    using assms by auto
  show ?thesis
  proof
    fix y
    assume y: "y   (closure ` )"
    show "y  closure ((rel_interior ` ))"
    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  "
          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 ` )"
          by auto
        show "x (rel_interior ` ). 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  :: "'n::euclidean_space set set"
  assumes "S. S    convex S" and "(rel_interior ` )  {}"
  shows "closure() = (closure ` )"
proof -
  have "(closure ` )  closure ((rel_interior ` ))"
    by (meson assms convex_closure_rel_interior_Int)
  moreover
  have "closure ((rel_interior ` ))  closure ()"
    using rel_interior_inter_aux closure_mono[of "(rel_interior ` )" ""]
    by auto
  ultimately show ?thesis
    using closure_Int[of ] by blast
qed

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

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

lemma convex_rel_interior_finite_Inter:
  fixes  :: "'n::euclidean_space set set"
  assumes "S. S    convex S"
    and "(rel_interior ` )  {}"
    and "finite "
  shows "rel_interior () = (rel_interior ` )"
proof -
  have "  {}"
    using assms rel_interior_inter_aux[of ] by auto
  have "convex ()"
    using convex_Inter assms by auto
  show ?thesis
  proof (cases " = {}")
    case True
    then show ?thesis
      using Inter_empty rel_interior_UNIV by auto
  next
    case False
    {
      fix z
      assume z: "z  (rel_interior ` )"
      {
        fix x
        assume x: "x  "
        {
          fix S
          assume S: "S  "
          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. 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 ` )"
        then have "e  mS ` " using assms   {} by simp
        then have "e > 1" using mS by auto
        moreover have "S. e  mS S"
          using e_def assms by auto
        ultimately have "e > 1. (1 - e) *R x + e *R z  "
          using mS by auto
      }
      then have "z  rel_interior ()"
        using convex_rel_interior_iff[of "" z]   {} convex () by auto
    }
    then show ?thesis
      using convex_rel_interior_Inter[of ] 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 "(*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 "((*R) c) ` (rel_interior S) = rel_interior (((*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  :: "'n::euclidean_space set set"
  assumes "S. S    convex S  rel_open S"
    and "finite "
  shows "convex ()  rel_open ()"
proof (cases "{rel_interior S |S. S  } = {}")
  case True
  then have " = {}"
    using assms unfolding rel_open_def by auto
  then show ?thesis
    unfolding rel_open_def by auto
next
  case False
  then have "rel_open ()"
    using assms convex_rel_interior_finite_Inter[of ] 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)


subsubsectiontag 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 "iI. convex (S i)  (S i)  {}"
  shows "convex hull ((S ` I)) =
    {sum (λi. c i *R s i) I | c s. (iI. c i  0)  sum c I = 1  (iI. 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"
      "(iI. c i  0)  (iI. s i  S i)" by auto
    then have "iI. 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: "iI. 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 "(iI. 0  c i)  sum c I = 1  (iI. 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  (iI. c i  0)  sum c I = 1  (iI. s i  S i)"
      by auto
    from xy obtain d t where
      yc: "y = sum (λi. d i *R t i) I  (iI. d i  0)  sum d I = 1  (iI. t i  S i)"
      by auto
    define e where "e i = u * c i + v * d i" for i
    have ge0: "iI. 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: "iI. 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 *: "iI. (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) =
    { iI. c i *R sa i | c sa. (iI. 0  c i)  sum c I = 1  (iI. sa i  s i)}"
      using assms s_def I_def
      by (subst convex_hull_finite_union) auto
  moreover have
    "{iI. c i *R sa i | c sa. (iI. 0  c i)  sum c I = 1  (iI. 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 < d" "(a + d *R l)  rel_frontier S"
           "e. 0  e; e < d  (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 < d  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 < e  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: "η. 0  η; η < ε  (a + η *R l)  rel_interior S"
  proof -
    obtain e where "e>0"
            and e: "x'. x'  affine hull S  dist x' a < e  x'  rel_interior S"
      using relin_Ex a by blast
    show thesis
    proof (rule_tac ε = "e / norm l" in that)
      show "0 < e / norm l" by (simp add: 0 < e l  0)
    next
      show "a + η *R l  rel_interior S" if "0  η" "η < e / norm l" for η
      proof (rule e)
        show "a + η *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 + η *R l) a < e"
          using that by (simp add: l  0 dist_norm pos_less_divide_eq)
      qed
    qed
  qed
  have inint: "e. 0  e; e < d  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 < d" 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) < e  x'  rel_interior S"
      using relin_Ex adl by blast
    have "d + e / norm l  x"
      if "0 < x" and nonrel: "a + x *R l  rel_interior S" for x
    proof (cases "x < d")
      case True with inint nonrel 0 < x
      show ?thesis by auto
    next
      case False
      then have dle: "x < d + e / norm l  dist (a + x *R l) (a + d *R l) < e"
        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 < d  a + d *R l  rel_interior S}"
      by (force simp add: intro: cInf_greatest [OF nonMT])
    then show False
      using 0 < e l  0 by (simp add: d_def [symmetric] field_simps)
  qed
  moreover 
  have "yS. dist y (a + d *R l) < η" if "0 < η" for η::real
  proof -
    have 1: "a + (d - min d (η / 2 / norm l)) *R l  S"
    proof (rule subsetD [OF rel_interior_subset inint])
      show "d - min d (η / 2 / norm l) < d"
        using l  0 0 < d 0 < η by auto
    qed auto
    have "norm l * min d (η / (norm l * 2))  norm l * (η / (norm l * 2))"
      by (metis min_def mult_left_mono norm_ge_zero order_refl)
    also have "... < η"
      using l  0 0 < η by (simp add: field_simps)
    finally have 2: "norm l * min d (η / (norm l * 2)) < η" .
    show ?thesis
      using 1 2 0 < d 0 < η 
      by (rule_tac x="a + (d - min d (η / 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 < d 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 < d" "(a + d *R l)  frontier S"
           "e. 0  e; e < d  (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 < d" and df: "(x + d *R (y-x))  rel_frontier S"
                  and di: "e. 0  e; e < d  (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 < 1"
      using xy 0 < d 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 < d 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 < e" "(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 < d" "(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 < e 0 < d 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 < e" "(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 < d" "(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 < e 0 < d 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


subsectiontag 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. iI  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. iI  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  (iI. c i  0)  sum c I = 1  (iI. 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 "iI. 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  (iI. 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 "iI. xs i  S i"
      using x xs_def assms by (simp add: cone_def)
    moreover have "iI. (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 "iI. convex (S i)  S i  {}"
  shows "rel_interior (convex hull ((S ` I))) =
    {sum (λi. c i *R s i) I | c s. (iI. c i > 0)  sum c I = 1 
      (iI. 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 "iI. 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 "iI. K i  {}"
    unfolding K_def using assms
    by (simp add: cone_hull_empty_iff[symmetric])
  have convK: "iI. 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 iI. 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 "iI. 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 iI. 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  (iI. 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 < ci  si  rel_interior (S i)"
        using rel_interior_convex_cone[of "S i"] by auto
    }
    then obtain c s where cs: "iI. k i = (c i, c i *R s i)  0 < c i  s i  rel_interior (S i)"
      by metis
    then have "x = (iI. 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 
        (iI. c i > 0)  sum c I = 1  (iI. 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 < y"
  moreover
  have "open (interior I)" by auto
  from openE[OF this x  interior I]
  obtain e where e: "0 < e" "ball x e  interior I" .
  moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
  ultimately have "x < t" "t < y" "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 < e have "K  ball x e" "K < x"
    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 < t 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 t < y
      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 < x _]]
    show "(f K - f x) / (K - x)  y" by auto
  qed
  then show ?thesis
    using x < y by (simp add: field_simps)
next
  assume "y < x"
  moreover
  have "open (interior I)" by auto
  from openE[OF this x  interior I]
  obtain e where e: "0 < e" "ball x e  interior I" .
  moreover define t where "t = x + e / 2"
  ultimately have "x < t" "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 < x 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 < x]]
    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 < x by (simp add: field_simps)
qed simp

subsectiontag 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  (vS. u v = 0)  (vS. 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: "(vT. u v *R v)  convex hull S"
        and T1: "sum u T = 1"
      then obtain v where v: "xS. 0  v x" "sum v S = 1" "(xS. v x *R x) = (vT. 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]: "(xT. v x *R x) + (xS - T. v x *R x) = (xT. 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 "(xS. if x  T then v x - u x else v x) = 0"
             "(xS. (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 "(xT. 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 "(vT. 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) < card 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)) < DIM('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 < u x  u x < 1)  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 < 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 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 < u 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 "(xS. if x = a then - d else if x = b then d else 0) = 0"
           "(xS. (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: "xS. 0  u x" "sum u S = 1" "¬ 0 < u a" "a  S"
        and yT: "y = (xS. u x *R x)" "y  T" "open T"
        and sb: "T  affine hull S  {w. u. (xS. 0  u x)  sum u S = 1  (xS. u x *R x) = w}"
      for u T a
      proof -
        have ua0: "u a = 0"
          using ua by auto
        obtain b where b: "bS" "a  b"
          using ua False by auto
        obtain e where e: "0 < e" "ball (xS. u x *R x) e  T"
          using yT by (auto elim: openE)
        with b obtain d where d: "0 < d" "norm(d *R (a-b)) < e"
          by (auto intro: that [of "e / 2 / norm(a-b)"])
        have "(xS. u x *R x)  affine hull S"
          using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
        then have "(xS. 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: "xS. 0  v x"
          "sum v S = 1"
          "(xS. v x *R x) = (xS. u x *R x) - d *R (a - b)"
          using subsetD [OF sb] yT
          by auto
        have aff: "u. sum u S = 0  (vS. u v = 0)  (vS. 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 < u x)  sum u S = 1  (xS. 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 < u x  u x < 1)  sum u S = 1  (xS. u x *R x) = y})"
proof -
  { fix u :: "'a  real" and a
    assume "card Basis < card S" and u: "x. xS  0 < u x" "sum u S = 1" and a: "a  S"
    then have cs: "Suc 0 < card 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 < 1"
      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

subsectiontag 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; xS. 0 < v x;
        sum v S = 1; (xS. v x *R x) = (xS. u x *R x)
        u. sum u S = 0  (vS. u v  0)  (vS. 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) < card 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) < card 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 "xS. 0  u x  a  S  u a = 0  sum u S = 1 
            x v. x  S 
                  (xS - {x}. 0  v x) 
                      sum v (S - {x}) = 1  (xS - {x}. v x *R x) = (xS. 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  xS - {a}. 0  u x  sum u (S - {a}) = 1 
            v. (xS. 0  v x) 
                 (xS. v x = 0)  sum v S = 1  (xS. v x *R x) = (xS - {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 < card 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›

definitiontag 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. xS; yS  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 "xS" "yS"
    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 < x  0 < y  0 < z  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


subsectiontag 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 < b} = {}  a = 0  b  0"
proof -
  have "a = 0  b  0" 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_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 < 0"
proof -
  have "a = 0  b < 0" 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

subsectiontag unimportant›‹Use set distance for an easy proof of separation properties›

propositiontag 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 < 0}" in that)
    show "{x. 0 < f x}  {x. f x < 0} = {}"
      by auto
    show "open {x. 0 < f x}"
      by (simp add: open_Collect_less contf)
    show "open {x. f x < 0}"
      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 < f x}" "T  {x. f x < 0}"
      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 -` {..<0})" in that)
    show "(U  f -` {0<..})  (U  f -` {..<0}) = {}"
      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 -` {..<0})"
      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 -` {..<0}"
      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  :: "'a :: euclidean_space set set"
  assumes cc: "S. S    compact S  connected S"
      and linear: "S T. S    T    S  T  T  S"
  shows "connected()"
proof (cases " = {}")
  case True then show ?thesis
    by auto
next
  case False
  then have cf: "compact()"
    by (simp add: cc compact_Inter)
  have False if AB: "closed A" "closed B" "A  B = {}"
                and ABeq: "A  B = " 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  " "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) ` )"
    obtain 𝒟 where "𝒟  ?𝒞" "finite 𝒟" "K  𝒟"
    proof (rule compactE [OF compact K])
      show "K  (insert (U  V) ((-) N ` ))"
        using K  N ABeq A  U B  V by auto
      show "B. B  insert (U  V) ((-) N ` )  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) ` "
      using 𝒟  ?𝒞 by blast
    ultimately obtain 𝒢 where "𝒢  " "finite 𝒢" and Deq: "𝒟 - {U  V} = (λS. N-S) ` 𝒢"
      using finite_subset_image by metis
    obtain J where "J  " and J: "(S𝒢. N - S)  N - J"
    proof (cases "𝒢 = {}")
      case True
      with   {} that show ?thesis
        by auto
    next
      case False
      have "S T. S  𝒢; T  𝒢  S  T  T  S"
        by (meson 𝒢   in_mono local.linear)
      with finite 𝒢 𝒢  {}
      have "J  𝒢. (S𝒢. N - S)  N - J"
      proof induction
        case (insert X )
        show ?case
        proof (cases " = {}")
          case True then show ?thesis by auto
        next
          case False
          then have "S T. S  ; T    S  T  T  S"
            by (simp add: insert.prems)
          with insert.IH False obtain J where "J  " and J: "(Y. N - Y)  N - J"
            by metis
          have "N - J  N - X  N - X  N - J"
            by (meson Diff_mono J   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 ` )  N - J"
              by auto
            with J   show ?thesis
              by blast
          qed
        qed
      qed simp
      with 𝒢   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   K   cc inf.orderE local.linear)
    moreover have "U  (J  K)  {}"
      using ABeq J   K   A  {} A  U by blast
    moreover have "V  (J  K)  {}"
      using ABeq J   K   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  :: "'a :: euclidean_space set set"
  assumes X: "X  " "compact X"
      and cc: "T. T    closed T  connected T"
      and linear: "S T. S    T    S  T  T  S"
  shows "connected()"
proof -
  have " = (T. X  T)"
    using X by blast
  moreover have "connected (T. X  T)"
  proof (rule connected_chain)
    show "T. T  (∩) X `   compact T  connected T"
      using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
    show "S T. S  (∩) X `   T  (∩) X `   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. km. 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. km. 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. (xK. 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: "  {}"
             if "finite "
                and : "t. t    (n. t = Ψ n)"
             for 
    proof -
      obtain m where m: "t. t    km. t = Ψ k"
        by (rule exE [OF finite_indexed_bound [OF finite  ]], force+)
      have "X m  "
        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

subsectiontag 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 < e  d>0. y  affine hull S. ¦f y - f x¦ * norm z < d  ¦f y - f x¦ < e"
        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 "{..<f 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 "{..<f 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 "{..<f z}  f ` S  {}" "{f z<..}  f ` S  {}"
            using x  S y  S by blast+
        qed
        ultimately show False
          using connectedD [OF conn_fS, of "{..<f 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 < e  d>0. y  affine hull S. ¦f y  - f x¦ * norm z < d  ¦f y  - f x¦ < e"
      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

subsubsectiontag unimportant›‹Representing affine hull as a finite intersection of hyperplanes›

propositiontag 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" "k < 1"
      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 < k 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 < r} = (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 < 0 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 < r} = UNIV"
    using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
  moreover have "{x. a  x < r}  {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 < r} =
        (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 < 0 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)

subsectiontag 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" "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 "aff_dim {x. a  x = 0}  0"
    by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
  with assms show "DIM('a) = 1"
    by (simp add: le_Suc_eq)
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

subsectiontag unimportant›‹General case without assuming closure and getting non-strict separation›

propositiontag 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 < b" "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 < 2 * ((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 < b" "x. x  S  a  x > b"
  using separating_hyperplane_closed_point_inset [OF assms] by simp (metis 0  S)


propositiontag 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 < b"
                  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 < b by (metis hull_inc inner_commute less_eq_real_def less_trans)
      then have aa: "a /R (norm a)  (cC. {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

propositiontag 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 < a  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 < a  y" if "y  rel_interior S" for y
  proof -
    obtain e where "0 < e" "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 < e 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 < e 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 < a  y"
using supporting_hyperplane_rel_boundary [of "closure S" x]
by (metis assms convex_closure convex_rel_interior_closure)


subsectiontag 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 
          (vS  T. u v *R v) = (vS. u v *R v)"
      if [simp]:  "sum u S = 1"
                 "sum v T = 1"
         and eq: "(xT. v x *R x) = (xS. 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 "(vS  T. u v *R v) = (vS. u v *R v)"
      by (auto simp: if_smult 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


propositiontag 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)

propositiontag 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)

propositiontag unimportant›
  fixes S :: "'a::euclidean_space set set"
  assumes "¬ affine_dependent (S)"
    shows affine_hull_Inter: "affine hull (S) = (TS. affine hull T)" (is "?A")
      and convex_hull_Inter: "convex hull (S) = (TS. 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

propositiontag 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 = (sS. 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 "(vS. (b v * x) *R v) = x *R (vS. b v *R v)" for x
    by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
  then have *: "(vS. (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: "xT  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 "(xinsert 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: "(xinsert 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: "xT  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 "(xinsert 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: "(xinsert 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

corollarytag 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  where
     "finite "
     "of_nat (card ) + aff_dim S = DIM('a)"
     "affine hull S = "
     "h. h    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 (ac - 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 "... < dim {x + y |x y. x  S  a  y = 0}"
    using span1 span2 by (blast intro: dim_psubset)
  finally have "DIM('a) - 1 < dim {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

propositiontag 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  (xaS. 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 < DIM('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 < e" 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 < e  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 < e 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 < e 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 < dim S" and "subspace S" shows "closure(S - T) = S"
proof -
  have "closure(S - U) = S" if "dim U < dim 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 < e"
        show "yS - U. dist y x < e"
        proof (cases "x  U")
          case True
          let ?y = "x + (e/2 / norm a) *R a"
          show ?thesis
          proof
            show "dist ?y x < e"
              using 0 < e 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 < e 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 < e x  S show ?thesis by force
        qed
      qed
    qed
  qed
  moreover have "S - S  T = S-T"
    by blast
  moreover have "dim (S  T) < dim S"
    by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
  ultimately show ?thesis
    by force
qed

corollarytag unimportant› dense_complement_affine:
  fixes S :: "'a :: euclidean_space set"
  assumes less: "aff_dim T < aff_dim 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)) < int (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

corollarytag unimportant› dense_complement_openin_affine_hull:
  fixes S :: "'a :: euclidean_space set"
  assumes less: "aff_dim T < aff_dim 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

corollarytag unimportant› dense_complement_convex:
  fixes S :: "'a :: euclidean_space set"
  assumes "aff_dim T < aff_dim 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

corollarytag unimportant› dense_complement_convex_closed:
  fixes S :: "'a :: euclidean_space set"
  assumes "aff_dim T < aff_dim S" "convex S" "closed S"
    shows "closure(S - T) = S"
  by (simp add: assms dense_complement_convex)


subsectiontag 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.›

propositiontag 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 η where "η = u + ((b - a  u) / (a  v - a  u)) *R (v - u)"
  have "η  S"
    by (simp add: η_def u  S v  S affine S mem_affine_3_minus)
  moreover have "a  η = b"
    using a  u  b b < a  v
    by (simp add: η_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  where "  G ` S" "countable " " = (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

corollarytag 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)

  
subsectiontag unimportant›‹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

subsectiontag unimportant›‹The union of two collinear segments is another segment›

propositiontag 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: "(iS. 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: "(iS. 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 "... = (xS. v x)"
            by (metis v b = 0 diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
          finally show "(xinsert 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 "... = (xS. 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 "(xinsert 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 "... = (xS. v x)"
            using 0 < u 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 < u b 0 < v b v0 [of i] le_Max [of i] beq that False
            by (auto simp: field_simps split: if_split_asm)
          have "(xinsert 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 + (xS - {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 "... = (xS. v x *R x)"
            using 0 < u b True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
          finally
          show "(xinsert 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. nN. 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. nN. 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  (iI. 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 "(iI. 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 (simp add: interior_diff, rule Diff_mono)
      show "cball a (real n)  ball a (1 + real n)"
        by (simp add: cball_subset_ball_iff)
      have cl: "closed (x- S. ecball 0 (1 / (2 + real n)). {x + e})"
        using assms by (auto intro: closed_compact_sums)
      have "closure (x- S. yball 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. yball 0 (1 / (1 + real n)). {x + y})"
        by (simp add: cball_subset_ball_iff field_split_simps UN_mono)
      finally show "closure (x- S. yball 0 (1 / (2 + real n)). {x + y})
                     (x  -S. yball 0 (1 / (1 + real n)). {x + y})" .
    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"
        using reals_Archimedean2
        by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
      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 < e)
        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›

definitiontag important› orthogonal_comp ("_" [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"
proof
  fix x assume x: "x  B"
  show "x  orthogonal_comp A" using x unfolding orthogonal_comp_def
    by (simp add: orthogonal_def, metis assms in_mono)
qed

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 *: "xB. yB. 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 = "bB. (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"
    proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
      fix y
      assume "y  U"
      with span B = U span_finite [OF finite B]
      obtain u where u: "y = (bB. 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

subsectiontag 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 (subst (asm) orthogonal_comp_self)
      (simp add: adjoint_linear linear_subspace_image)
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