src/HOL/Analysis/Convex_Euclidean_Space.thy

   Convex
   Topology_Euclidean_Space
 begin
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Topological Properties of Convex Sets and Functions\<close>
-
-lemma convex_supp_sum:
-  assumes "convex S" and 1: "supp_sum u I = 1"
-      and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
-    shows "supp_sum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
-proof -
-  have fin: "finite {i \<in> I. u i \<noteq> 0}"
-    using 1 sum.infinite by (force simp: supp_sum_def support_on_def)
-  then have eq: "supp_sum (\<lambda>i. u i *\<^sub>R f i) I = sum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
-    by (force intro: sum.mono_neutral_left simp: supp_sum_def support_on_def)
-  show ?thesis
-    apply (simp add: eq)
-    apply (rule convex_sum [OF fin \<open>convex S\<close>])
-    using 1 assms apply (auto simp: supp_sum_def support_on_def)
-    done
-qed
-
-lemma closure_bounded_linear_image_subset:
-  assumes f: "bounded_linear f"
-  shows "f ` closure S \<subseteq> closure (f ` S)"
-  using linear_continuous_on [OF f] closed_closure closure_subset
-  by (rule image_closure_subset)
-
-lemma closure_linear_image_subset:
-  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
-  assumes "linear f"
-  shows "f ` (closure S) \<subseteq> closure (f ` S)"
-  using assms unfolding linear_conv_bounded_linear
-  by (rule closure_bounded_linear_image_subset)
-
-lemma closed_injective_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    assumes S: "closed S" and f: "linear f" "inj f"
-    shows "closed (f ` S)"
-proof -
-  obtain g where g: "linear g" "g \<circ> f = id"
-    using linear_injective_left_inverse [OF f] by blast
-  then have confg: "continuous_on (range f) g"
-    using linear_continuous_on linear_conv_bounded_linear by blast
-  have [simp]: "g ` f ` S = S"
-    using g by (simp add: image_comp)
-  have cgf: "closed (g ` f ` S)"
-    by (simp add: \<open>g \<circ> f = id\<close> S image_comp)
-  have [simp]: "(range f \<inter> g -` S) = f ` S"
-    using g unfolding o_def id_def image_def by auto metis+
-  show ?thesis
-  proof (rule closedin_closed_trans [of "range f"])
-    show "closedin (top_of_set (range f)) (f ` S)"
-      using continuous_closedin_preimage [OF confg cgf] by simp
-    show "closed (range f)"
-      apply (rule closed_injective_image_subspace)
-      using f apply (auto simp: linear_linear linear_injective_0)
-      done
-  qed
-qed
-
-lemma closed_injective_linear_image_eq:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    assumes f: "linear f" "inj f"
-      shows "(closed(image f s) \<longleftrightarrow> closed s)"
-  by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)
-
-lemma closure_injective_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
-  apply (rule subset_antisym)
-  apply (simp add: closure_linear_image_subset)
-  by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
-
-lemma closure_bounded_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    shows "\<lbrakk>linear f; bounded S\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
-  apply (rule subset_antisym, simp add: closure_linear_image_subset)
-  apply (rule closure_minimal, simp add: closure_subset image_mono)
-  by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
-
-lemma closure_scaleR:
-  fixes S :: "'a::real_normed_vector set"
-  shows "((*\<^sub>R) c) ` (closure S) = closure (((*\<^sub>R) c) ` S)"
-proof
-  show "((*\<^sub>R) c) ` (closure S) \<subseteq> closure (((*\<^sub>R) c) ` S)"
-    using bounded_linear_scaleR_right
-    by (rule closure_bounded_linear_image_subset)
-  show "closure (((*\<^sub>R) c) ` S) \<subseteq> ((*\<^sub>R) c) ` (closure S)"
-    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
-qed
-
-lemma sphere_eq_empty [simp]:
-  fixes a :: "'a::{real_normed_vector, perfect_space}"
-  shows "sphere a r = {} \<longleftrightarrow> r < 0"
-by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
-
-lemma cone_closure:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "cone S"
-  shows "cone (closure S)"
-proof (cases "S = {}")
-  case True
-  then show ?thesis by auto
-next
-  case False
-  then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
-    using cone_iff[of S] assms by auto
-  then have "0 \<in> closure S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` closure S = closure S)"
-    using closure_subset by (auto simp: closure_scaleR)
-  then show ?thesis
-    using False cone_iff[of "closure S"] by auto
-qed
-
-corollary component_complement_connected:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "connected S" "C \<in> components (-S)"
-  shows "connected(-C)"
-  using component_diff_connected [of S UNIV] assms
-  by (auto simp: Compl_eq_Diff_UNIV)
-
-proposition clopen:
-  fixes S :: "'a :: real_normed_vector set"
-  shows "closed S \<and> open S \<longleftrightarrow> S = {} \<or> S = UNIV"
-    by (force intro!: connected_UNIV [unfolded connected_clopen, rule_format])
-
-corollary compact_open:
-  fixes S :: "'a :: euclidean_space set"
-  shows "compact S \<and> open S \<longleftrightarrow> S = {}"
-  by (auto simp: compact_eq_bounded_closed clopen)
-
-corollary finite_imp_not_open:
-    fixes S :: "'a::{real_normed_vector, perfect_space} set"
-    shows "\<lbrakk>finite S; open S\<rbrakk> \<Longrightarrow> S={}"
-  using clopen [of S] finite_imp_closed not_bounded_UNIV by blast
-
-corollary empty_interior_finite:
-    fixes S :: "'a::{real_normed_vector, perfect_space} set"
-    shows "finite S \<Longrightarrow> interior S = {}"
-  by (metis interior_subset finite_subset open_interior [of S] finite_imp_not_open)
-
-text \<open>Balls, being convex, are connected.\<close>
-
-lemma convex_local_global_minimum:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "e > 0"
-    and "convex_on s f"
-    and "ball x e \<subseteq> s"
-    and "\<forall>y\<in>ball x e. f x \<le> f y"
-  shows "\<forall>y\<in>s. f x \<le> f y"
-proof (rule ccontr)
-  have "x \<in> s" using assms(1,3) by auto
-  assume "\<not> ?thesis"
-  then obtain y where "y\<in>s" and y: "f x > f y" by auto
-  then have xy: "0 < dist x y"  by auto
-  then obtain u where "0 < u" "u \<le> 1" and u: "u < e / dist x y"
-    using field_lbound_gt_zero[of 1 "e / dist x y"] xy \<open>e>0\<close> by auto
-  then have "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y"
-    using \<open>x\<in>s\<close> \<open>y\<in>s\<close>
-    using assms(2)[unfolded convex_on_def,
-      THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]]
-    by auto
-  moreover
-  have *: "x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)"
-    by (simp add: algebra_simps)
-  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e"
-    unfolding mem_ball dist_norm
-    unfolding * and norm_scaleR and abs_of_pos[OF \<open>0<u\<close>]
-    unfolding dist_norm[symmetric]
-    using u
-    unfolding pos_less_divide_eq[OF xy]
-    by auto
-  then have "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)"
-    using assms(4) by auto
-  ultimately show False
-    using mult_strict_left_mono[OF y \<open>u>0\<close>]
-    unfolding left_diff_distrib
-    by auto
-qed
-
-lemma convex_ball [iff]:
-  fixes x :: "'a::real_normed_vector"
-  shows "convex (ball x e)"
-proof (auto simp: convex_def)
-  fix y z
-  assume yz: "dist x y < e" "dist x z < e"
-  fix u v :: real
-  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
-  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
-    using uv yz
-    using convex_on_dist [of "ball x e" x, unfolded convex_on_def,
-      THEN bspec[where x=y], THEN bspec[where x=z]]
-    by auto
-  then show "dist x (u *\<^sub>R y + v *\<^sub>R z) < e"
-    using convex_bound_lt[OF yz uv] by auto
-qed
-
-lemma convex_cball [iff]:
-  fixes x :: "'a::real_normed_vector"
-  shows "convex (cball x e)"
-proof -
-  {
-    fix y z
-    assume yz: "dist x y \<le> e" "dist x z \<le> e"
-    fix u v :: real
-    assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
-    have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z"
-      using uv yz
-      using convex_on_dist [of "cball x e" x, unfolded convex_on_def,
-        THEN bspec[where x=y], THEN bspec[where x=z]]
-      by auto
-    then have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e"
-      using convex_bound_le[OF yz uv] by auto
-  }
-  then show ?thesis by (auto simp: convex_def Ball_def)
-qed
-
-lemma connected_ball [iff]:
-  fixes x :: "'a::real_normed_vector"
-  shows "connected (ball x e)"
-  using convex_connected convex_ball by auto
-
-lemma connected_cball [iff]:
-  fixes x :: "'a::real_normed_vector"
-  shows "connected (cball x e)"
-  using convex_connected convex_cball by auto
-
-
-lemma bounded_convex_hull:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "bounded s"
-  shows "bounded (convex hull s)"
-proof -
-  from assms obtain B where B: "\<forall>x\<in>s. norm x \<le> B"
-    unfolding bounded_iff by auto
-  show ?thesis
-    apply (rule bounded_subset[OF bounded_cball, of _ 0 B])
-    unfolding subset_hull[of convex, OF convex_cball]
-    unfolding subset_eq mem_cball dist_norm using B
-    apply auto
-    done
-qed
-
-lemma finite_imp_bounded_convex_hull:
-  fixes s :: "'a::real_normed_vector set"
-  shows "finite s \<Longrightarrow> bounded (convex hull s)"
-  using bounded_convex_hull finite_imp_bounded
-  by auto
 
 lemma aff_dim_cball:
   fixes a :: "'n::euclidean_space"
   assumes "e > 0"
   shows "aff_dim (cball a e) = int (DIM('n))"

 lemma is_interval_convex_1:
   fixes s :: "real set"
   shows "is_interval s \<longleftrightarrow> convex s"
   by (metis is_interval_convex convex_connected is_interval_connected_1)
 
-lemma is_interval_ball_real: "is_interval (ball a b)" for a b::real
-  by (metis connected_ball is_interval_connected_1)
-
 lemma connected_compact_interval_1:
      "connected S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = {a..b::real})"
   by (auto simp: is_interval_connected_1 [symmetric] is_interval_compact)
 
 lemma connected_convex_1:

   then have "f -` (f ` s) = s"
     by (simp add: inj_vimage_image_eq)
   then show ?thesis
     by (metis connected_convex_1 convex_linear_vimage linf convex_connected connected_linear_image)
 qed
-
-lemma is_interval_cball_1[intro, simp]: "is_interval (cball a b)" for a b::real
-  by (simp add: is_interval_convex_1)
 
 lemma [simp]:
   fixes r s::real
   shows is_interval_io: "is_interval {..<r}"
     and is_interval_oi: "is_interval {r<..}"

   then show "continuous (at x) f"
     unfolding continuous_on_eq_continuous_at[OF open_ball]
     using \<open>d > 0\<close> by auto
 qed
 
-
-section \<open>Line Segments\<close>
-
-subsection \<open>Midpoint\<close>
-
-definition\<^marker>\<open>tag important\<close> midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
-  where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
-
-lemma midpoint_idem [simp]: "midpoint x x = x"
-  unfolding midpoint_def  by simp
-
-lemma midpoint_sym: "midpoint a b = midpoint b a"
-  unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
-
-lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
-proof -
-  have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
-    by simp
-  then show ?thesis
-    unfolding midpoint_def scaleR_2 [symmetric] by simp
-qed
-
-lemma
-  fixes a::real
-  assumes "a \<le> b" shows ge_midpoint_1: "a \<le> midpoint a b"
-                    and le_midpoint_1: "midpoint a b \<le> b"
-  by (simp_all add: midpoint_def assms)
-
-lemma dist_midpoint:
-  fixes a b :: "'a::real_normed_vector" shows
-  "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
-  "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
-  "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
-  "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
-proof -
-  have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2"
-    unfolding equation_minus_iff by auto
-  have **: "\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2"
-    by auto
-  note scaleR_right_distrib [simp]
-  show ?t1
-    unfolding midpoint_def dist_norm
-    apply (rule **)
-    apply (simp add: scaleR_right_diff_distrib)
-    apply (simp add: scaleR_2)
-    done
-  show ?t2
-    unfolding midpoint_def dist_norm
-    apply (rule *)
-    apply (simp add: scaleR_right_diff_distrib)
-    apply (simp add: scaleR_2)
-    done
-  show ?t3
-    unfolding midpoint_def dist_norm
-    apply (rule *)
-    apply (simp add: scaleR_right_diff_distrib)
-    apply (simp add: scaleR_2)
-    done
-  show ?t4
-    unfolding midpoint_def dist_norm
-    apply (rule **)
-    apply (simp add: scaleR_right_diff_distrib)
-    apply (simp add: scaleR_2)
-    done
-qed
-
-lemma midpoint_eq_endpoint [simp]:
-  "midpoint a b = a \<longleftrightarrow> a = b"
-  "midpoint a b = b \<longleftrightarrow> a = b"
-  unfolding midpoint_eq_iff by auto
-
-lemma midpoint_plus_self [simp]: "midpoint a b + midpoint a b = a + b"
-  using midpoint_eq_iff by metis
-
-lemma midpoint_linear_image:
-   "linear f \<Longrightarrow> midpoint(f a)(f b) = f(midpoint a b)"
-by (simp add: linear_iff midpoint_def)
-
-
-subsection \<open>Line segments\<close>
-
-definition\<^marker>\<open>tag important\<close> closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set"
-  where "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
-
-definition\<^marker>\<open>tag important\<close> open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
-  "open_segment a b \<equiv> closed_segment a b - {a,b}"
-
-lemmas segment = open_segment_def closed_segment_def
-
-lemma in_segment:
-    "x \<in> closed_segment a b \<longleftrightarrow> (\<exists>u. 0 \<le> u \<and> u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
-    "x \<in> open_segment a b \<longleftrightarrow> a \<noteq> b \<and> (\<exists>u. 0 < u \<and> u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
-  using less_eq_real_def by (auto simp: segment algebra_simps)
-
-lemma closed_segment_linear_image:
-  "closed_segment (f a) (f b) = f ` (closed_segment a b)" if "linear f"
-proof -
-  interpret linear f by fact
-  show ?thesis
-    by (force simp add: in_segment add scale)
-qed
-
-lemma open_segment_linear_image:
-    "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> open_segment (f a) (f b) = f ` (open_segment a b)"
-  by (force simp: open_segment_def closed_segment_linear_image inj_on_def)
-
-lemma closed_segment_translation:
-    "closed_segment (c + a) (c + b) = image (\<lambda>x. c + x) (closed_segment a b)"
-apply safe
-apply (rule_tac x="x-c" in image_eqI)
-apply (auto simp: in_segment algebra_simps)
-done
-
-lemma open_segment_translation:
-    "open_segment (c + a) (c + b) = image (\<lambda>x. c + x) (open_segment a b)"
-by (simp add: open_segment_def closed_segment_translation translation_diff)
-
-lemma closed_segment_of_real:
-    "closed_segment (of_real x) (of_real y) = of_real ` closed_segment x y"
-  apply (auto simp: image_iff in_segment scaleR_conv_of_real)
-    apply (rule_tac x="(1-u)*x + u*y" in bexI)
-  apply (auto simp: in_segment)
-  done
-
-lemma open_segment_of_real:
-    "open_segment (of_real x) (of_real y) = of_real ` open_segment x y"
-  apply (auto simp: image_iff in_segment scaleR_conv_of_real)
-    apply (rule_tac x="(1-u)*x + u*y" in bexI)
-  apply (auto simp: in_segment)
-  done
-
-lemma closed_segment_Reals:
-    "\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> closed_segment x y = of_real ` closed_segment (Re x) (Re y)"
-  by (metis closed_segment_of_real of_real_Re)
-
-lemma open_segment_Reals:
-    "\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> open_segment x y = of_real ` open_segment (Re x) (Re y)"
-  by (metis open_segment_of_real of_real_Re)
-
-lemma open_segment_PairD:
-    "(x, x') \<in> open_segment (a, a') (b, b')
-     \<Longrightarrow> (x \<in> open_segment a b \<or> a = b) \<and> (x' \<in> open_segment a' b' \<or> a' = b')"
-  by (auto simp: in_segment)
-
-lemma closed_segment_PairD:
-  "(x, x') \<in> closed_segment (a, a') (b, b') \<Longrightarrow> x \<in> closed_segment a b \<and> x' \<in> closed_segment a' b'"
-  by (auto simp: closed_segment_def)
-
-lemma closed_segment_translation_eq [simp]:
-    "d + x \<in> closed_segment (d + a) (d + b) \<longleftrightarrow> x \<in> closed_segment a b"
-proof -
-  have *: "\<And>d x a b. x \<in> closed_segment a b \<Longrightarrow> d + x \<in> closed_segment (d + a) (d + b)"
-    apply (simp add: closed_segment_def)
-    apply (erule ex_forward)
-    apply (simp add: algebra_simps)
-    done
-  show ?thesis
-  using * [where d = "-d"] *
-  by (fastforce simp add:)
-qed
-
-lemma open_segment_translation_eq [simp]:
-    "d + x \<in> open_segment (d + a) (d + b) \<longleftrightarrow> x \<in> open_segment a b"
-  by (simp add: open_segment_def)
-
-lemma of_real_closed_segment [simp]:
-  "of_real x \<in> closed_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> closed_segment a b"
-  apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward)
-  using of_real_eq_iff by fastforce
-
-lemma of_real_open_segment [simp]:
-  "of_real x \<in> open_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> open_segment a b"
-  apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward del: exE)
-  using of_real_eq_iff by fastforce
-
-lemma convex_contains_segment:
-  "convex S \<longleftrightarrow> (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
-  unfolding convex_alt closed_segment_def by auto
-
-lemma closed_segment_in_Reals:
-   "\<lbrakk>x \<in> closed_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
-  by (meson subsetD convex_Reals convex_contains_segment)
-
-lemma open_segment_in_Reals:
-   "\<lbrakk>x \<in> open_segment a b; a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> x \<in> Reals"
-  by (metis Diff_iff closed_segment_in_Reals open_segment_def)
-
-lemma closed_segment_subset: "\<lbrakk>x \<in> S; y \<in> S; convex S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> S"
-  by (simp add: convex_contains_segment)
-
-lemma closed_segment_subset_convex_hull:
-    "\<lbrakk>x \<in> convex hull S; y \<in> convex hull S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> convex hull S"
-  using convex_contains_segment by blast
-
-lemma segment_convex_hull:
-  "closed_segment a b = convex hull {a,b}"
-proof -
-  have *: "\<And>x. {x} \<noteq> {}" by auto
-  show ?thesis
-    unfolding segment convex_hull_insert[OF *] convex_hull_singleton
-    by (safe; rule_tac x="1 - u" in exI; force)
-qed
-
-lemma open_closed_segment: "u \<in> open_segment w z \<Longrightarrow> u \<in> closed_segment w z"
-  by (auto simp add: closed_segment_def open_segment_def)
-
-lemma segment_open_subset_closed:
-   "open_segment a b \<subseteq> closed_segment a b"
-  by (auto simp: closed_segment_def open_segment_def)
-
-lemma bounded_closed_segment:
-    fixes a :: "'a::euclidean_space" shows "bounded (closed_segment a b)"
-  by (simp add: segment_convex_hull compact_convex_hull compact_imp_bounded)
-
-lemma bounded_open_segment:
-    fixes a :: "'a::euclidean_space" shows "bounded (open_segment a b)"
-  by (rule bounded_subset [OF bounded_closed_segment segment_open_subset_closed])
-
-lemmas bounded_segment = bounded_closed_segment open_closed_segment
-
-lemma ends_in_segment [iff]: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
-  unfolding segment_convex_hull
-  by (auto intro!: hull_subset[unfolded subset_eq, rule_format])
-
-lemma eventually_closed_segment:
-  fixes x0::"'a::real_normed_vector"
-  assumes "open X0" "x0 \<in> X0"
-  shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
-proof -
-  from openE[OF assms]
-  obtain e where e: "0 < e" "ball x0 e \<subseteq> X0" .
-  then have "\<forall>\<^sub>F x in at x0 within U. x \<in> ball x0 e"
-    by (auto simp: dist_commute eventually_at)
-  then show ?thesis
-  proof eventually_elim
-    case (elim x)
-    have "x0 \<in> ball x0 e" using \<open>e > 0\<close> by simp
-    from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
-    have "closed_segment x0 x \<subseteq> ball x0 e" .
-    also note \<open>\<dots> \<subseteq> X0\<close>
-    finally show ?case .
-  qed
-qed
-
-lemma segment_furthest_le:
-  fixes a b x y :: "'a::euclidean_space"
-  assumes "x \<in> closed_segment a b"
-  shows "norm (y - x) \<le> norm (y - a) \<or>  norm (y - x) \<le> norm (y - b)"
-proof -
-  obtain z where "z \<in> {a, b}" "norm (x - y) \<le> norm (z - y)"
-    using simplex_furthest_le[of "{a, b}" y]
-    using assms[unfolded segment_convex_hull]
-    by auto
-  then show ?thesis
-    by (auto simp add:norm_minus_commute)
-qed
-
-lemma closed_segment_commute: "closed_segment a b = closed_segment b a"
-proof -
-  have "{a, b} = {b, a}" by auto
-  thus ?thesis
-    by (simp add: segment_convex_hull)
-qed
-
-lemma segment_bound1:
-  assumes "x \<in> closed_segment a b"
-  shows "norm (x - a) \<le> norm (b - a)"
-proof -
-  obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
-    using assms by (auto simp add: closed_segment_def)
-  then show "norm (x - a) \<le> norm (b - a)"
-    apply clarify
-    apply (auto simp: algebra_simps)
-    apply (simp add: scaleR_diff_right [symmetric] mult_left_le_one_le)
-    done
-qed
-
-lemma segment_bound:
-  assumes "x \<in> closed_segment a b"
-  shows "norm (x - a) \<le> norm (b - a)" "norm (x - b) \<le> norm (b - a)"
-apply (simp add: assms segment_bound1)
-by (metis assms closed_segment_commute dist_commute dist_norm segment_bound1)
-
-lemma open_segment_commute: "open_segment a b = open_segment b a"
-proof -
-  have "{a, b} = {b, a}" by auto
-  thus ?thesis
-    by (simp add: closed_segment_commute open_segment_def)
-qed
-
-lemma closed_segment_idem [simp]: "closed_segment a a = {a}"
-  unfolding segment by (auto simp add: algebra_simps)
-
-lemma open_segment_idem [simp]: "open_segment a a = {}"
-  by (simp add: open_segment_def)
-
-lemma closed_segment_eq_open: "closed_segment a b = open_segment a b \<union> {a,b}"
-  using open_segment_def by auto
-
-lemma convex_contains_open_segment:
-  "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. open_segment a b \<subseteq> s)"
-  by (simp add: convex_contains_segment closed_segment_eq_open)
-
-lemma closed_segment_eq_real_ivl:
-  fixes a b::real
-  shows "closed_segment a b = (if a \<le> b then {a .. b} else {b .. a})"
-proof -
-  have "b \<le> a \<Longrightarrow> closed_segment b a = {b .. a}"
-    and "a \<le> b \<Longrightarrow> closed_segment a b = {a .. b}"
-    by (auto simp: convex_hull_eq_real_cbox segment_convex_hull)
-  thus ?thesis
-    by (auto simp: closed_segment_commute)
-qed
-
-lemma open_segment_eq_real_ivl:
-  fixes a b::real
-  shows "open_segment a b = (if a \<le> b then {a<..<b} else {b<..<a})"
-by (auto simp: closed_segment_eq_real_ivl open_segment_def split: if_split_asm)
-
-lemma closed_segment_real_eq:
-  fixes u::real shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
-  by (simp add: add.commute [of u] image_affinity_atLeastAtMost [where c=u] closed_segment_eq_real_ivl)
-
-lemma dist_in_closed_segment:
-  fixes a :: "'a :: euclidean_space"
-  assumes "x \<in> closed_segment a b"
-    shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
-proof (intro conjI)
-  obtain u where u: "0 \<le> u" "u \<le> 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
-    using assms by (force simp: in_segment algebra_simps)
-  have "dist x a = u * dist a b"
-    apply (simp add: dist_norm algebra_simps x)
-    by (metis \<open>0 \<le> u\<close> abs_of_nonneg norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib)
-  also have "...  \<le> dist a b"
-    by (simp add: mult_left_le_one_le u)
-  finally show "dist x a \<le> dist a b" .
-  have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
-    by (simp add: dist_norm algebra_simps x)
-  also have "... = (1-u) * dist a b"
-  proof -
-    have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
-      using \<open>u \<le> 1\<close> by force
-    then show ?thesis
-      by (simp add: dist_norm real_vector.scale_right_diff_distrib)
-  qed
-  also have "... \<le> dist a b"
-    by (simp add: mult_left_le_one_le u)
-  finally show "dist x b \<le> dist a b" .
-qed
-
-lemma dist_in_open_segment:
-  fixes a :: "'a :: euclidean_space"
-  assumes "x \<in> open_segment a b"
-    shows "dist x a < dist a b \<and> dist x b < dist a b"
-proof (intro conjI)
-  obtain u where u: "0 < u" "u < 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
-    using assms by (force simp: in_segment algebra_simps)
-  have "dist x a = u * dist a b"
-    apply (simp add: dist_norm algebra_simps x)
-    by (metis abs_of_nonneg less_eq_real_def norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib \<open>0 < u\<close>)
-  also have *: "...  < dist a b"
-    by (metis (no_types) assms dist_eq_0_iff dist_not_less_zero in_segment(2) linorder_neqE_linordered_idom mult.left_neutral real_mult_less_iff1 \<open>u < 1\<close>)
-  finally show "dist x a < dist a b" .
-  have ab_ne0: "dist a b \<noteq> 0"
-    using * by fastforce
-  have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
-    by (simp add: dist_norm algebra_simps x)
-  also have "... = (1-u) * dist a b"
-  proof -
-    have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
-      using \<open>u < 1\<close> by force
-    then show ?thesis
-      by (simp add: dist_norm real_vector.scale_right_diff_distrib)
-  qed
-  also have "... < dist a b"
-    using ab_ne0 \<open>0 < u\<close> by simp
-  finally show "dist x b < dist a b" .
-qed
-
-lemma dist_decreases_open_segment_0:
-  fixes x :: "'a :: euclidean_space"
-  assumes "x \<in> open_segment 0 b"
-    shows "dist c x < dist c 0 \<or> dist c x < dist c b"
-proof (rule ccontr, clarsimp simp: not_less)
-  obtain u where u: "0 \<noteq> b" "0 < u" "u < 1" and x: "x = u *\<^sub>R b"
-    using assms by (auto simp: in_segment)
-  have xb: "x \<bullet> b < b \<bullet> b"
-    using u x by auto
-  assume "norm c \<le> dist c x"
-  then have "c \<bullet> c \<le> (c - x) \<bullet> (c - x)"
-    by (simp add: dist_norm norm_le)
-  moreover have "0 < x \<bullet> b"
-    using u x by auto
-  ultimately have less: "c \<bullet> b < x \<bullet> b"
-    by (simp add: x algebra_simps inner_commute u)
-  assume "dist c b \<le> dist c x"
-  then have "(c - b) \<bullet> (c - b) \<le> (c - x) \<bullet> (c - x)"
-    by (simp add: dist_norm norm_le)
-  then have "(b \<bullet> b) * (1 - u*u) \<le> 2 * (b \<bullet> c) * (1-u)"
-    by (simp add: x algebra_simps inner_commute)
-  then have "(1+u) * (b \<bullet> b) * (1-u) \<le> 2 * (b \<bullet> c) * (1-u)"
-    by (simp add: algebra_simps)
-  then have "(1+u) * (b \<bullet> b) \<le> 2 * (b \<bullet> c)"
-    using \<open>u < 1\<close> by auto
-  with xb have "c \<bullet> b \<ge> x \<bullet> b"
-    by (auto simp: x algebra_simps inner_commute)
-  with less show False by auto
-qed
-
-proposition dist_decreases_open_segment:
-  fixes a :: "'a :: euclidean_space"
-  assumes "x \<in> open_segment a b"
-    shows "dist c x < dist c a \<or> dist c x < dist c b"
-proof -
-  have *: "x - a \<in> open_segment 0 (b - a)" using assms
-    by (metis diff_self open_segment_translation_eq uminus_add_conv_diff)
-  show ?thesis
-    using dist_decreases_open_segment_0 [OF *, of "c-a"] assms
-    by (simp add: dist_norm)
-qed
-
-corollary open_segment_furthest_le:
-  fixes a b x y :: "'a::euclidean_space"
-  assumes "x \<in> open_segment a b"
-  shows "norm (y - x) < norm (y - a) \<or>  norm (y - x) < norm (y - b)"
-  by (metis assms dist_decreases_open_segment dist_norm)
-
-corollary dist_decreases_closed_segment:
-  fixes a :: "'a :: euclidean_space"
-  assumes "x \<in> closed_segment a b"
-    shows "dist c x \<le> dist c a \<or> dist c x \<le> dist c b"
-apply (cases "x \<in> open_segment a b")
- using dist_decreases_open_segment less_eq_real_def apply blast
-by (metis DiffI assms empty_iff insertE open_segment_def order_refl)
-
-lemma convex_intermediate_ball:
-  fixes a :: "'a :: euclidean_space"
-  shows "\<lbrakk>ball a r \<subseteq> T; T \<subseteq> cball a r\<rbrakk> \<Longrightarrow> convex T"
-apply (simp add: convex_contains_open_segment, clarify)
-by (metis (no_types, hide_lams) less_le_trans mem_ball mem_cball subsetCE dist_decreases_open_segment)
-
-lemma csegment_midpoint_subset: "closed_segment (midpoint a b) b \<subseteq> closed_segment a b"
-  apply (clarsimp simp: midpoint_def in_segment)
-  apply (rule_tac x="(1 + u) / 2" in exI)
-  apply (auto simp: algebra_simps add_divide_distrib diff_divide_distrib)
-  by (metis field_sum_of_halves scaleR_left.add)
-
-lemma notin_segment_midpoint:
-  fixes a :: "'a :: euclidean_space"
-  shows "a \<noteq> b \<Longrightarrow> a \<notin> closed_segment (midpoint a b) b"
-by (auto simp: dist_midpoint dest!: dist_in_closed_segment)
-
-lemma segment_to_closest_point:
-  fixes S :: "'a :: euclidean_space set"
-  shows "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> open_segment a (closest_point S a) \<inter> S = {}"
-  apply (subst disjoint_iff_not_equal)
-  apply (clarify dest!: dist_in_open_segment)
-  by (metis closest_point_le dist_commute le_less_trans less_irrefl)
-
-lemma segment_to_point_exists:
-  fixes S :: "'a :: euclidean_space set"
-    assumes "closed S" "S \<noteq> {}"
-    obtains b where "b \<in> S" "open_segment a b \<inter> S = {}"
-  by (metis assms segment_to_closest_point closest_point_exists that)
-
-subsubsection\<open>More lemmas, especially for working with the underlying formula\<close>
-
-lemma segment_eq_compose:
-  fixes a :: "'a :: real_vector"
-  shows "(\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) = (\<lambda>x. a + x) o (\<lambda>u. u *\<^sub>R (b - a))"
-    by (simp add: o_def algebra_simps)
-
-lemma segment_degen_1:
-  fixes a :: "'a :: real_vector"
-  shows "(1 - u) *\<^sub>R a + u *\<^sub>R b = b \<longleftrightarrow> a=b \<or> u=1"
-proof -
-  { assume "(1 - u) *\<^sub>R a + u *\<^sub>R b = b"
-    then have "(1 - u) *\<^sub>R a = (1 - u) *\<^sub>R b"
-      by (simp add: algebra_simps)
-    then have "a=b \<or> u=1"
-      by simp
-  } then show ?thesis
-      by (auto simp: algebra_simps)
-qed
-
-lemma segment_degen_0:
-    fixes a :: "'a :: real_vector"
-    shows "(1 - u) *\<^sub>R a + u *\<^sub>R b = a \<longleftrightarrow> a=b \<or> u=0"
-  using segment_degen_1 [of "1-u" b a]
-  by (auto simp: algebra_simps)
-
-lemma add_scaleR_degen:
-  fixes a b ::"'a::real_vector"
-  assumes  "(u *\<^sub>R b + v *\<^sub>R a) = (u *\<^sub>R a + v *\<^sub>R b)"  "u \<noteq> v"
-  shows "a=b"
-  by (metis (no_types, hide_lams) add.commute add_diff_eq diff_add_cancel real_vector.scale_cancel_left real_vector.scale_left_diff_distrib assms)
-  
-lemma closed_segment_image_interval:
-     "closed_segment a b = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0..1}"
-  by (auto simp: set_eq_iff image_iff closed_segment_def)
-
-lemma open_segment_image_interval:
-     "open_segment a b = (if a=b then {} else (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1})"
-  by (auto simp:  open_segment_def closed_segment_def segment_degen_0 segment_degen_1)
-
-lemmas segment_image_interval = closed_segment_image_interval open_segment_image_interval
-
-lemma open_segment_bound1:
-  assumes "x \<in> open_segment a b"
-  shows "norm (x - a) < norm (b - a)"
-proof -
-  obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 < u" "u < 1" "a \<noteq> b"
-    using assms by (auto simp add: open_segment_image_interval split: if_split_asm)
-  then show "norm (x - a) < norm (b - a)"
-    apply clarify
-    apply (auto simp: algebra_simps)
-    apply (simp add: scaleR_diff_right [symmetric])
-    done
-qed
-
-lemma compact_segment [simp]:
-  fixes a :: "'a::real_normed_vector"
-  shows "compact (closed_segment a b)"
-  by (auto simp: segment_image_interval intro!: compact_continuous_image continuous_intros)
-
-lemma closed_segment [simp]:
-  fixes a :: "'a::real_normed_vector"
-  shows "closed (closed_segment a b)"
-  by (simp add: compact_imp_closed)
-
-lemma closure_closed_segment [simp]:
-  fixes a :: "'a::real_normed_vector"
-  shows "closure(closed_segment a b) = closed_segment a b"
-  by simp
-
-lemma open_segment_bound:
-  assumes "x \<in> open_segment a b"
-  shows "norm (x - a) < norm (b - a)" "norm (x - b) < norm (b - a)"
-apply (simp add: assms open_segment_bound1)
-by (metis assms norm_minus_commute open_segment_bound1 open_segment_commute)
-
-lemma closure_open_segment [simp]:
-  "closure (open_segment a b) = (if a = b then {} else closed_segment a b)"
-    for a :: "'a::euclidean_space"
-proof (cases "a = b")
-  case True
-  then show ?thesis
-    by simp
-next
-  case False
-  have "closure ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1}) = (\<lambda>u. u *\<^sub>R (b - a)) ` closure {0<..<1}"
-    apply (rule closure_injective_linear_image [symmetric])
-     apply (use False in \<open>auto intro!: injI\<close>)
-    done
-  then have "closure
-     ((\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1}) =
-    (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b) ` closure {0<..<1}"
-    using closure_translation [of a "((\<lambda>x. x *\<^sub>R b - x *\<^sub>R a) ` {0<..<1})"]
-    by (simp add: segment_eq_compose field_simps scaleR_diff_left scaleR_diff_right image_image)
-  then show ?thesis
-    by (simp add: segment_image_interval closure_greaterThanLessThan [symmetric] del: closure_greaterThanLessThan)
-qed
-
-lemma closed_open_segment_iff [simp]:
-    fixes a :: "'a::euclidean_space"  shows "closed(open_segment a b) \<longleftrightarrow> a = b"
-  by (metis open_segment_def DiffE closure_eq closure_open_segment ends_in_segment(1) insert_iff segment_image_interval(2))
-
-lemma compact_open_segment_iff [simp]:
-    fixes a :: "'a::euclidean_space"  shows "compact(open_segment a b) \<longleftrightarrow> a = b"
-  by (simp add: bounded_open_segment compact_eq_bounded_closed)
-
-lemma convex_closed_segment [iff]: "convex (closed_segment a b)"
-  unfolding segment_convex_hull by(rule convex_convex_hull)
-
-lemma convex_open_segment [iff]: "convex (open_segment a b)"
-proof -
-  have "convex ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1})"
-    by (rule convex_linear_image) auto
-  then have "convex ((+) a ` (\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1})"
-    by (rule convex_translation)
-  then show ?thesis
-    by (simp add: image_image open_segment_image_interval segment_eq_compose field_simps scaleR_diff_left scaleR_diff_right)
-qed
-
-lemmas convex_segment = convex_closed_segment convex_open_segment
-
-lemma connected_segment [iff]:
-  fixes x :: "'a :: real_normed_vector"
-  shows "connected (closed_segment x y)"
-  by (simp add: convex_connected)
-
-lemma is_interval_closed_segment_1[intro, simp]: "is_interval (closed_segment a b)" for a b::real
-  by (auto simp: is_interval_convex_1)
-
-lemma IVT'_closed_segment_real:
-  fixes f :: "real \<Rightarrow> real"
-  assumes "y \<in> closed_segment (f a) (f b)"
-  assumes "continuous_on (closed_segment a b) f"
-  shows "\<exists>x \<in> closed_segment a b. f x = y"
-  using IVT'[of f a y b]
-    IVT'[of "-f" a "-y" b]
-    IVT'[of f b y a]
-    IVT'[of "-f" b "-y" a] assms
-  by (cases "a \<le> b"; cases "f b \<ge> f a") (auto simp: closed_segment_eq_real_ivl continuous_on_minus)
-
-subsection \<open>Betweenness\<close>
-
-definition\<^marker>\<open>tag important\<close> "between = (\<lambda>(a,b) x. x \<in> closed_segment a b)"
-
-lemma betweenI:
-  assumes "0 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
-  shows "between (a, b) x"
-using assms unfolding between_def closed_segment_def by auto
-
-lemma betweenE:
-  assumes "between (a, b) x"
-  obtains u where "0 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
-using assms unfolding between_def closed_segment_def by auto
-
-lemma between_implies_scaled_diff:
-  assumes "between (S, T) X" "between (S, T) Y" "S \<noteq> Y"
-  obtains c where "(X - Y) = c *\<^sub>R (S - Y)"
-proof -
-  from \<open>between (S, T) X\<close> obtain u\<^sub>X where X: "X = u\<^sub>X *\<^sub>R S + (1 - u\<^sub>X) *\<^sub>R T"
-    by (metis add.commute betweenE eq_diff_eq)
-  from \<open>between (S, T) Y\<close> obtain u\<^sub>Y where Y: "Y = u\<^sub>Y *\<^sub>R S + (1 - u\<^sub>Y) *\<^sub>R T"
-    by (metis add.commute betweenE eq_diff_eq)
-  have "X - Y = (u\<^sub>X - u\<^sub>Y) *\<^sub>R (S - T)"
-  proof -
-    from X Y have "X - Y =  u\<^sub>X *\<^sub>R S - u\<^sub>Y *\<^sub>R S + ((1 - u\<^sub>X) *\<^sub>R T - (1 - u\<^sub>Y) *\<^sub>R T)" by simp
-    also have "\<dots> = (u\<^sub>X - u\<^sub>Y) *\<^sub>R S - (u\<^sub>X - u\<^sub>Y) *\<^sub>R T" by (simp add: scaleR_left.diff)
-    finally show ?thesis by (simp add: real_vector.scale_right_diff_distrib)
-  qed
-  moreover from Y have "S - Y = (1 - u\<^sub>Y) *\<^sub>R (S - T)"
-    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
-  moreover note \<open>S \<noteq> Y\<close>
-  ultimately have "(X - Y) = ((u\<^sub>X - u\<^sub>Y) / (1 - u\<^sub>Y)) *\<^sub>R (S - Y)" by auto
-  from this that show thesis by blast
-qed
-
-lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
-  unfolding between_def by auto
-
-lemma between: "between (a, b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
-proof (cases "a = b")
-  case True
-  then show ?thesis
-    by (auto simp add: between_def dist_commute)
-next
-  case False
-  then have Fal: "norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0"
-    by auto
-  have *: "\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)"
-    by (auto simp add: algebra_simps)
-  have "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)" if "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" for u
-  proof -
-    have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
-      unfolding that(1) by (auto simp add:algebra_simps)
-    show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
-      unfolding norm_minus_commute[of x a] * using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>
-      by simp
-  qed
-  moreover have "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" if "dist a b = dist a x + dist x b" 
-  proof -
-    let ?\<beta> = "norm (a - x) / norm (a - b)"
-    show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
-    proof (intro exI conjI)
-      show "?\<beta> \<le> 1"
-        using Fal2 unfolding that[unfolded dist_norm] norm_ge_zero by auto
-      show "x = (1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b"
-      proof (subst euclidean_eq_iff; intro ballI)
-        fix i :: 'a
-        assume i: "i \<in> Basis"
-        have "((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> i 
-              = ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
-          using Fal by (auto simp add: field_simps inner_simps)
-        also have "\<dots> = x\<bullet>i"
-          apply (rule divide_eq_imp[OF Fal])
-          unfolding that[unfolded dist_norm]
-          using that[unfolded dist_triangle_eq] i
-          apply (subst (asm) euclidean_eq_iff)
-           apply (auto simp add: field_simps inner_simps)
-          done
-        finally show "x \<bullet> i = ((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> i"
-          by auto
-      qed
-    qed (use Fal2 in auto)
-  qed
-  ultimately show ?thesis
-    by (force simp add: between_def closed_segment_def dist_triangle_eq)
-qed
-
-lemma between_midpoint:
-  fixes a :: "'a::euclidean_space"
-  shows "between (a,b) (midpoint a b)" (is ?t1)
-    and "between (b,a) (midpoint a b)" (is ?t2)
-proof -
-  have *: "\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y"
-    by auto
-  show ?t1 ?t2
-    unfolding between midpoint_def dist_norm
-    by (auto simp add: field_simps inner_simps euclidean_eq_iff[where 'a='a] intro!: *)
-qed
-
-lemma between_mem_convex_hull:
-  "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
-  unfolding between_mem_segment segment_convex_hull ..
-
-lemma between_triv_iff [simp]: "between (a,a) b \<longleftrightarrow> a=b"
-  by (auto simp: between_def)
-
-lemma between_triv1 [simp]: "between (a,b) a"
-  by (auto simp: between_def)
-
-lemma between_triv2 [simp]: "between (a,b) b"
-  by (auto simp: between_def)
-
-lemma between_commute:
-   "between (a,b) = between (b,a)"
-by (auto simp: between_def closed_segment_commute)
-
-lemma between_antisym:
-  fixes a :: "'a :: euclidean_space"
-  shows "\<lbrakk>between (b,c) a; between (a,c) b\<rbrakk> \<Longrightarrow> a = b"
-by (auto simp: between dist_commute)
-
-lemma between_trans:
-    fixes a :: "'a :: euclidean_space"
-    shows "\<lbrakk>between (b,c) a; between (a,c) d\<rbrakk> \<Longrightarrow> between (b,c) d"
-  using dist_triangle2 [of b c d] dist_triangle3 [of b d a]
-  by (auto simp: between dist_commute)
-
-lemma between_norm:
-    fixes a :: "'a :: euclidean_space"
-    shows "between (a,b) x \<longleftrightarrow> norm(x - a) *\<^sub>R (b - x) = norm(b - x) *\<^sub>R (x - a)"
-  by (auto simp: between dist_triangle_eq norm_minus_commute algebra_simps)
-
-lemma between_swap:
-  fixes A B X Y :: "'a::euclidean_space"
-  assumes "between (A, B) X"
-  assumes "between (A, B) Y"
-  shows "between (X, B) Y \<longleftrightarrow> between (A, Y) X"
-using assms by (auto simp add: between)
-
-lemma between_translation [simp]: "between (a + y,a + z) (a + x) \<longleftrightarrow> between (y,z) x"
-  by (auto simp: between_def)
-
-lemma between_trans_2:
-  fixes a :: "'a :: euclidean_space"
-  shows "\<lbrakk>between (b,c) a; between (a,b) d\<rbrakk> \<Longrightarrow> between (c,d) a"
-  by (metis between_commute between_swap between_trans)
-
-lemma between_scaleR_lift [simp]:
-  fixes v :: "'a::euclidean_space"
-  shows "between (a *\<^sub>R v, b *\<^sub>R v) (c *\<^sub>R v) \<longleftrightarrow> v = 0 \<or> between (a, b) c"
-  by (simp add: between dist_norm scaleR_left_diff_distrib [symmetric] distrib_right [symmetric])
-
-lemma between_1:
-  fixes x::real
-  shows "between (a,b) x \<longleftrightarrow> (a \<le> x \<and> x \<le> b) \<or> (b \<le> x \<and> x \<le> a)"
-  by (auto simp: between_mem_segment closed_segment_eq_real_ivl)
-
-
 end