--- a/src/HOL/Analysis/Continuous_Extension.thy Wed Jul 19 22:56:16 2017 +0100
+++ b/src/HOL/Analysis/Continuous_Extension.thy Thu Jul 20 14:05:29 2017 +0100
@@ -5,7 +5,7 @@
section \<open>Continuous extensions of functions: Urysohn's lemma, Dugundji extension theorem, Tietze\<close>
theory Continuous_Extension
-imports Convex_Euclidean_Space
+imports Starlike
begin
subsection\<open>Partitions of unity subordinate to locally finite open coverings\<close>
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Wed Jul 19 22:56:16 2017 +0100
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Thu Jul 20 14:05:29 2017 +0100
@@ -7242,7886 +7242,4 @@
using \<open>d > 0\<close> by auto
qed
-
-subsection \<open>Line segments, Starlike Sets, etc.\<close>
-
-definition midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
- where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
-
-definition 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 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:
- "linear f \<Longrightarrow> closed_segment (f a) (f b) = f ` (closed_segment a b)"
- by (force simp add: in_segment linear_add_cmul)
-
-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 midpoint_idem [simp]: "midpoint x x = x"
- unfolding midpoint_def
- unfolding scaleR_right_distrib
- unfolding scaleR_left_distrib[symmetric]
- by auto
-
-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>Starlike sets\<close>
-
-definition "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
-
-lemma starlike_UNIV [simp]: "starlike UNIV"
- by (simp add: starlike_def)
-
-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_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 convex_imp_starlike:
- "convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
- unfolding convex_contains_segment starlike_def by auto
-
-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 real_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]:
- fixes a :: "'a::euclidean_space"
- shows "closure(open_segment a b) = (if a = b then {} else closed_segment a b)"
-proof -
- have "closure ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1}) = (\<lambda>u. u *\<^sub>R (b - a)) ` closure {0<..<1}" if "a \<noteq> b"
- apply (rule closure_injective_linear_image [symmetric])
- apply (simp add:)
- using that by (simp add: inj_on_def)
- then show ?thesis
- by (simp add: segment_image_interval segment_eq_compose closure_greaterThanLessThan [symmetric]
- closure_translation image_comp [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 show ?thesis
- apply (simp add: open_segment_image_interval segment_eq_compose)
- by (metis image_comp convex_translation)
-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 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 \<in> rel_interior S" "b \<in> closure S"
- shows "open_segment a b \<subseteq> rel_interior S"
-proof
- fix x
- have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
- by (simp add: algebra_simps)
- assume "x \<in> open_segment a b"
- then show "x \<in> 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 \<Union>x \<in> 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 \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
- shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
-proof
- have "T = {} \<Longrightarrow> z \<notin> S"
- using dis z by (auto simp add: disjnt_def)
- then show "?lhs \<subseteq> ?rhs"
- proof (clarsimp simp add: convex_hull_insert_segments)
- fix x y
- assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
- have "y \<in> closure C"
- by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
- moreover have "x \<notin> rel_interior C"
- by (meson \<open>x \<in> S\<close> dis disjnt_iff)
- moreover have "x \<in> open_segment z y \<union> {z, y}"
- using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
- ultimately show "x \<in> convex hull T"
- using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
- using y z by blast
- qed
- show "?rhs \<subseteq> ?lhs"
- by (meson hull_mono inf_mono subset_insertI subset_refl)
-qed
-
-subsection\<open>More results about segments\<close>
-
-lemma dist_half_times2:
- fixes a :: "'a :: real_normed_vector"
- shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a+b) (2 *\<^sub>R x)"
-proof -
- have "norm ((1 / 2) *\<^sub>R (a + b) - x) * 2 = norm (2 *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x))"
- by simp
- also have "... = norm ((a + b) - 2 *\<^sub>R x)"
- by (simp add: real_vector.scale_right_diff_distrib)
- finally show ?thesis
- by (simp only: dist_norm)
-qed
-
-lemma closed_segment_as_ball:
- "closed_segment a b = affine hull {a,b} \<inter> cball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
-proof (cases "b = a")
- case True then show ?thesis by (auto simp: hull_inc)
-next
- case False
- then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
- (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)" for x
- proof -
- have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
- ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a))"
- unfolding eq_diff_eq [symmetric] by simp
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((a+b) - (2 *\<^sub>R x)) \<le> norm (b - a))"
- by (simp add: dist_half_times2) (simp add: dist_norm)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) \<le> norm (b - a))"
- by auto
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((1 - u * 2) *\<^sub>R (b - a)) \<le> norm (b - a))"
- by (simp add: algebra_simps scaleR_2)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- \<bar>1 - u * 2\<bar> * norm (b - a) \<le> norm (b - a))"
- by simp
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> \<le> 1)"
- by (simp add: mult_le_cancel_right2 False)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)"
- by auto
- finally show ?thesis .
- qed
- show ?thesis
- by (simp add: affine_hull_2 Set.set_eq_iff closed_segment_def *)
-qed
-
-lemma open_segment_as_ball:
- "open_segment a b =
- affine hull {a,b} \<inter> ball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
-proof (cases "b = a")
- case True then show ?thesis by (auto simp: hull_inc)
-next
- case False
- then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
- (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)" for x
- proof -
- have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
- ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
- dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a))"
- unfolding eq_diff_eq [symmetric] by simp
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((a+b) - (2 *\<^sub>R x)) < norm (b - a))"
- by (simp add: dist_half_times2) (simp add: dist_norm)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) < norm (b - a))"
- by auto
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- norm ((1 - u * 2) *\<^sub>R (b - a)) < norm (b - a))"
- by (simp add: algebra_simps scaleR_2)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
- \<bar>1 - u * 2\<bar> * norm (b - a) < norm (b - a))"
- by simp
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> < 1)"
- by (simp add: mult_le_cancel_right2 False)
- also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)"
- by auto
- finally show ?thesis .
- qed
- show ?thesis
- using False by (force simp: affine_hull_2 Set.set_eq_iff open_segment_image_interval *)
-qed
-
-lemmas segment_as_ball = closed_segment_as_ball open_segment_as_ball
-
-lemma closed_segment_neq_empty [simp]: "closed_segment a b \<noteq> {}"
- by auto
-
-lemma open_segment_eq_empty [simp]: "open_segment a b = {} \<longleftrightarrow> a = b"
-proof -
- { assume a1: "open_segment a b = {}"
- have "{} \<noteq> {0::real<..<1}"
- by simp
- then have "a = b"
- using a1 open_segment_image_interval by fastforce
- } then show ?thesis by auto
-qed
-
-lemma open_segment_eq_empty' [simp]: "{} = open_segment a b \<longleftrightarrow> a = b"
- using open_segment_eq_empty by blast
-
-lemmas segment_eq_empty = closed_segment_neq_empty open_segment_eq_empty
-
-lemma inj_segment:
- fixes a :: "'a :: real_vector"
- assumes "a \<noteq> b"
- shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
-proof
- fix x y
- assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b"
- then have "x *\<^sub>R (b - a) = y *\<^sub>R (b - a)"
- by (simp add: algebra_simps)
- with assms show "x = y"
- by (simp add: real_vector.scale_right_imp_eq)
-qed
-
-lemma finite_closed_segment [simp]: "finite(closed_segment a b) \<longleftrightarrow> a = b"
- apply auto
- apply (rule ccontr)
- apply (simp add: segment_image_interval)
- using infinite_Icc [OF zero_less_one] finite_imageD [OF _ inj_segment] apply blast
- done
-
-lemma finite_open_segment [simp]: "finite(open_segment a b) \<longleftrightarrow> a = b"
- by (auto simp: open_segment_def)
-
-lemmas finite_segment = finite_closed_segment finite_open_segment
-
-lemma closed_segment_eq_sing: "closed_segment a b = {c} \<longleftrightarrow> a = c \<and> b = c"
- by auto
-
-lemma open_segment_eq_sing: "open_segment a b \<noteq> {c}"
- by (metis finite_insert finite_open_segment insert_not_empty open_segment_image_interval)
-
-lemmas segment_eq_sing = closed_segment_eq_sing open_segment_eq_sing
-
-lemma subset_closed_segment:
- "closed_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
- a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
- by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
-
-lemma subset_co_segment:
- "closed_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
- a \<in> open_segment c d \<and> b \<in> open_segment c d"
-using closed_segment_subset by blast
-
-lemma subset_open_segment:
- fixes a :: "'a::euclidean_space"
- shows "open_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
- a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
- (is "?lhs = ?rhs")
-proof (cases "a = b")
- case True then show ?thesis by simp
-next
- case False show ?thesis
- proof
- assume rhs: ?rhs
- with \<open>a \<noteq> b\<close> have "c \<noteq> d"
- using closed_segment_idem singleton_iff by auto
- have "\<exists>uc. (1 - u) *\<^sub>R ((1 - ua) *\<^sub>R c + ua *\<^sub>R d) + u *\<^sub>R ((1 - ub) *\<^sub>R c + ub *\<^sub>R d) =
- (1 - uc) *\<^sub>R c + uc *\<^sub>R d \<and> 0 < uc \<and> uc < 1"
- if neq: "(1 - ua) *\<^sub>R c + ua *\<^sub>R d \<noteq> (1 - ub) *\<^sub>R c + ub *\<^sub>R d" "c \<noteq> d"
- and "a = (1 - ua) *\<^sub>R c + ua *\<^sub>R d" "b = (1 - ub) *\<^sub>R c + ub *\<^sub>R d"
- and u: "0 < u" "u < 1" and uab: "0 \<le> ua" "ua \<le> 1" "0 \<le> ub" "ub \<le> 1"
- for u ua ub
- proof -
- have "ua \<noteq> ub"
- using neq by auto
- moreover have "(u - 1) * ua \<le> 0" using u uab
- by (simp add: mult_nonpos_nonneg)
- ultimately have lt: "(u - 1) * ua < u * ub" using u uab
- by (metis antisym_conv diff_ge_0_iff_ge le_less_trans mult_eq_0_iff mult_le_0_iff not_less)
- have "p * ua + q * ub < p+q" if p: "0 < p" and q: "0 < q" for p q
- proof -
- have "\<not> p \<le> 0" "\<not> q \<le> 0"
- using p q not_less by blast+
- then show ?thesis
- by (metis \<open>ua \<noteq> ub\<close> add_less_cancel_left add_less_cancel_right add_mono_thms_linordered_field(5)
- less_eq_real_def mult_cancel_left1 mult_less_cancel_left2 uab(2) uab(4))
- qed
- then have "(1 - u) * ua + u * ub < 1" using u \<open>ua \<noteq> ub\<close>
- by (metis diff_add_cancel diff_gt_0_iff_gt)
- with lt show ?thesis
- by (rule_tac x="ua + u*(ub-ua)" in exI) (simp add: algebra_simps)
- qed
- with rhs \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close> show ?lhs
- unfolding open_segment_image_interval closed_segment_def
- by (fastforce simp add:)
- next
- assume lhs: ?lhs
- with \<open>a \<noteq> b\<close> have "c \<noteq> d"
- by (meson finite_open_segment rev_finite_subset)
- have "closure (open_segment a b) \<subseteq> closure (open_segment c d)"
- using lhs closure_mono by blast
- then have "closed_segment a b \<subseteq> closed_segment c d"
- by (simp add: \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close>)
- then show ?rhs
- by (force simp: \<open>a \<noteq> b\<close>)
- qed
-qed
-
-lemma subset_oc_segment:
- fixes a :: "'a::euclidean_space"
- shows "open_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
- a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
-apply (simp add: subset_open_segment [symmetric])
-apply (rule iffI)
- apply (metis closure_closed_segment closure_mono closure_open_segment subset_closed_segment subset_open_segment)
-apply (meson dual_order.trans segment_open_subset_closed)
-done
-
-lemmas subset_segment = subset_closed_segment subset_co_segment subset_oc_segment subset_open_segment
-
-
-subsection\<open>Betweenness\<close>
-
-definition "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
- unfolding between_def split_conv
- by (auto simp add: 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)
- show ?thesis
- unfolding between_def split_conv closed_segment_def mem_Collect_eq
- apply rule
- apply (elim exE conjE)
- apply (subst dist_triangle_eq)
- proof -
- fix u
- assume as: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
- then have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
- unfolding as(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 as(2,3)
- by (auto simp add: field_simps)
- next
- assume as: "dist a b = dist a x + dist x b"
- have "norm (a - x) / norm (a - b) \<le> 1"
- using Fal2 unfolding as[unfolded dist_norm] norm_ge_zero by auto
- then show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
- apply (rule_tac x="dist a x / dist a b" in exI)
- unfolding dist_norm
- apply (subst euclidean_eq_iff)
- apply rule
- defer
- apply rule
- prefer 3
- apply rule
- proof -
- fix i :: 'a
- assume i: "i \<in> Basis"
- have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^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 as[unfolded dist_norm]
- using as[unfolded dist_triangle_eq]
- apply -
- apply (subst (asm) euclidean_eq_iff)
- using i
- apply (erule_tac x=i in ballE)
- apply (auto simp add: field_simps inner_simps)
- done
- finally show "x \<bullet> i =
- ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
- by auto
- qed (insert Fal2, auto)
- qed
-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
- apply(rule_tac[!] *)
- unfolding euclidean_eq_iff[where 'a='a]
- apply (auto simp add: field_simps inner_simps)
- done
-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)
-
-
-subsection \<open>Shrinking towards the interior of a convex set\<close>
-
-lemma mem_interior_convex_shrink:
- fixes s :: "'a::euclidean_space set"
- assumes "convex s"
- and "c \<in> interior s"
- and "x \<in> s"
- and "0 < e"
- and "e \<le> 1"
- shows "x - e *\<^sub>R (x - c) \<in> interior s"
-proof -
- obtain d where "d > 0" and d: "ball c d \<subseteq> s"
- using assms(2) unfolding mem_interior by auto
- show ?thesis
- unfolding mem_interior
- apply (rule_tac x="e*d" in exI)
- apply rule
- defer
- unfolding subset_eq Ball_def mem_ball
- proof (rule, rule)
- fix y
- assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
- have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
- using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
- have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
- unfolding dist_norm
- unfolding norm_scaleR[symmetric]
- apply (rule arg_cong[where f=norm])
- using \<open>e > 0\<close>
- by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
- also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
- by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
- also have "\<dots> < d"
- using as[unfolded dist_norm] and \<open>e > 0\<close>
- by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
- finally show "y \<in> s"
- apply (subst *)
- apply (rule assms(1)[unfolded convex_alt,rule_format])
- apply (rule d[unfolded subset_eq,rule_format])
- unfolding mem_ball
- using assms(3-5)
- apply auto
- done
- qed (insert \<open>e>0\<close> \<open>d>0\<close>, auto)
-qed
-
-lemma mem_interior_closure_convex_shrink:
- fixes s :: "'a::euclidean_space set"
- assumes "convex s"
- and "c \<in> interior s"
- and "x \<in> closure s"
- and "0 < e"
- and "e \<le> 1"
- shows "x - e *\<^sub>R (x - c) \<in> interior s"
-proof -
- obtain d where "d > 0" and d: "ball c d \<subseteq> s"
- using assms(2) unfolding mem_interior by auto
- have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d"
- proof (cases "x \<in> s")
- case True
- then show ?thesis
- using \<open>e > 0\<close> \<open>d > 0\<close>
- apply (rule_tac bexI[where x=x])
- apply (auto)
- done
- 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 \<in> s" "y \<noteq> x" "dist y x < 1"
- using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
- then show ?thesis
- apply (rule_tac x=y in bexI)
- unfolding True
- using \<open>d > 0\<close>
- apply auto
- done
- next
- case False
- then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
- using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
- then obtain y where "y \<in> s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
- using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
- then show ?thesis
- apply (rule_tac x=y in bexI)
- unfolding dist_norm
- using pos_less_divide_eq[OF *]
- apply auto
- done
- qed
- qed
- then obtain y where "y \<in> s" and y: "norm (y - x) * (1 - e) < e * d"
- by auto
- define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
- have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
- unfolding z_def using \<open>e > 0\<close>
- by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
- have "z \<in> interior s"
- apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
- unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
- apply (auto simp add:field_simps norm_minus_commute)
- done
- then show ?thesis
- unfolding *
- apply -
- apply (rule mem_interior_convex_shrink)
- using assms(1,4-5) \<open>y\<in>s\<close>
- apply auto
- done
-qed
-
-lemma in_interior_closure_convex_segment:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
- shows "open_segment a b \<subseteq> interior S"
-proof (clarsimp simp: in_segment)
- fix u::real
- assume u: "0 < u" "u < 1"
- have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
- by (simp add: algebra_simps)
- also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
- by simp
- finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
-qed
-
-lemma closure_open_Int_superset:
- assumes "open S" "S \<subseteq> closure T"
- shows "closure(S \<inter> T) = closure S"
-proof -
- have "closure S \<subseteq> closure(S \<inter> T)"
- by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
- then show ?thesis
- by (simp add: closure_mono dual_order.antisym)
-qed
-
-lemma convex_closure_interior:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" and int: "interior S \<noteq> {}"
- shows "closure(interior S) = closure S"
-proof -
- obtain a where a: "a \<in> interior S"
- using int by auto
- have "closure S \<subseteq> closure(interior S)"
- proof
- fix x
- assume x: "x \<in> closure S"
- show "x \<in> closure (interior S)"
- proof (cases "x=a")
- case True
- then show ?thesis
- using \<open>a \<in> interior S\<close> closure_subset by blast
- next
- case False
- show ?thesis
- proof (clarsimp simp add: closure_def islimpt_approachable)
- fix e::real
- assume xnotS: "x \<notin> interior S" and "0 < e"
- show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
- proof (intro bexI conjI)
- show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
- using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
- show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
- using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
- show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
- apply (clarsimp simp add: min_def a)
- apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
- using \<open>0 < e\<close> False apply (auto simp: divide_simps)
- done
- qed
- qed
- qed
- qed
- then show ?thesis
- by (simp add: closure_mono interior_subset subset_antisym)
-qed
-
-lemma closure_convex_Int_superset:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
- shows "closure(S \<inter> T) = closure S"
-proof -
- have "closure S \<subseteq> closure(interior S)"
- by (simp add: convex_closure_interior assms)
- also have "... \<subseteq> closure (S \<inter> T)"
- using interior_subset [of S] assms
- by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
- finally show ?thesis
- by (simp add: closure_mono dual_order.antisym)
-qed
-
-
-subsection \<open>Some obvious but surprisingly hard simplex lemmas\<close>
-
-lemma simplex:
- assumes "finite s"
- and "0 \<notin> s"
- shows "convex hull (insert 0 s) =
- {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y)}"
- unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
- apply (rule set_eqI, rule)
- unfolding mem_Collect_eq
- apply (erule_tac[!] exE)
- apply (erule_tac[!] conjE)+
- unfolding sum_clauses(2)[OF \<open>finite s\<close>]
- apply (rule_tac x=u in exI)
- defer
- apply (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u s else u x" in exI)
- using assms(2)
- unfolding if_smult and sum_delta_notmem[OF assms(2)]
- apply auto
- done
-
-lemma substd_simplex:
- assumes d: "d \<subseteq> Basis"
- shows "convex hull (insert 0 d) =
- {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
- (is "convex hull (insert 0 ?p) = ?s")
-proof -
- let ?D = d
- have "0 \<notin> ?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 \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
- apply (rule set_eqI)
- unfolding mem_Collect_eq
- apply rule
- apply (elim exE conjE)
- apply (erule_tac[2] conjE)+
- proof -
- fix x :: "'a::euclidean_space"
- fix u
- assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
- have *: "\<forall>i\<in>Basis. i:d \<longrightarrow> u i = x\<bullet>i"
- and "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
- using as(3)
- unfolding substdbasis_expansion_unique[OF assms]
- by auto
- then have **: "sum u ?D = sum (op \<bullet> x) ?D"
- apply -
- apply (rule sum.cong)
- using assms
- apply auto
- done
- have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1"
- proof (rule,rule)
- fix i :: 'a
- assume i: "i \<in> Basis"
- have "i \<in> d \<Longrightarrow> 0 \<le> x\<bullet>i"
- unfolding *[rule_format,OF i,symmetric]
- apply (rule_tac as(1)[rule_format])
- apply auto
- done
- moreover have "i \<notin> d \<Longrightarrow> 0 \<le> x\<bullet>i"
- using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close>[rule_format, OF i] by auto
- ultimately show "0 \<le> x\<bullet>i" by auto
- qed (insert as(2)[unfolded **], auto)
- then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
- using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close> by auto
- next
- fix x :: "'a::euclidean_space"
- assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
- show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
- using as d
- unfolding substdbasis_expansion_unique[OF assms]
- apply (rule_tac x="inner x" in exI)
- apply auto
- done
- qed
-qed
-
-lemma std_simplex:
- "convex hull (insert 0 Basis) =
- {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
- using substd_simplex[of Basis] by auto
-
-lemma interior_std_simplex:
- "interior (convex hull (insert 0 Basis)) =
- {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
- apply (rule set_eqI)
- unfolding mem_interior std_simplex
- unfolding subset_eq mem_Collect_eq Ball_def mem_ball
- unfolding Ball_def[symmetric]
- apply rule
- apply (elim exE conjE)
- defer
- apply (erule conjE)
-proof -
- fix x :: 'a
- fix e
- assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum (op \<bullet> xa) Basis \<le> 1"
- show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum (op \<bullet> x) Basis < 1"
- apply safe
- proof -
- fix i :: 'a
- assume i: "i \<in> Basis"
- then show "0 < x \<bullet> i"
- using as[THEN spec[where x="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close>
- unfolding dist_norm
- by (auto elim!: ballE[where x=i] simp: inner_simps)
- next
- have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
- unfolding dist_norm
- by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
- have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
- x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
- by (auto simp: SOME_Basis inner_Basis inner_simps)
- then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
- sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
- apply (rule_tac sum.cong)
- apply auto
- done
- have "sum (op \<bullet> x) Basis < sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
- unfolding * sum.distrib
- using \<open>e > 0\<close> DIM_positive[where 'a='a]
- apply (subst sum.delta')
- apply (auto simp: SOME_Basis)
- done
- also have "\<dots> \<le> 1"
- using **
- apply (drule_tac as[rule_format])
- apply auto
- done
- finally show "sum (op \<bullet> x) Basis < 1" by auto
- qed
-next
- fix x :: 'a
- assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum (op \<bullet> x) Basis < 1"
- obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
- let ?d = "(1 - sum (op \<bullet> x) Basis) / real (DIM('a))"
- show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
- proof (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI, intro conjI impI allI)
- fix y
- assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
- have "sum (op \<bullet> y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
- proof (rule sum_mono)
- fix i :: 'a
- assume i: "i \<in> Basis"
- then have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
- apply -
- apply (rule le_less_trans)
- using Basis_le_norm[OF i, of "y - x"]
- using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
- apply (auto simp add: norm_minus_commute inner_diff_left)
- done
- then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
- qed
- also have "\<dots> \<le> 1"
- unfolding sum.distrib sum_constant
- by (auto simp add: Suc_le_eq)
- finally show "sum (op \<bullet> y) Basis \<le> 1" .
- show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
- proof safe
- fix i :: 'a
- assume i: "i \<in> Basis"
- have "norm (x - y) < x\<bullet>i"
- apply (rule less_le_trans)
- apply (rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1])
- using i
- apply auto
- done
- then show "0 \<le> y\<bullet>i"
- using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
- by (auto simp: inner_simps)
- qed
- next
- have "Min ((op \<bullet> x) ` Basis) > 0"
- using as by simp
- moreover have "?d > 0"
- using as by (auto simp: Suc_le_eq)
- ultimately show "0 < min (Min (op \<bullet> x ` Basis)) ((1 - sum (op \<bullet> x) Basis) / real DIM('a))"
- by linarith
- qed
-qed
-
-lemma interior_std_simplex_nonempty:
- obtains a :: "'a::euclidean_space" where
- "a \<in> interior(convex hull (insert 0 Basis))"
-proof -
- let ?D = "Basis :: 'a set"
- let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
- {
- fix i :: 'a
- assume i: "i \<in> Basis"
- have "?a \<bullet> i = inverse (2 * real DIM('a))"
- by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
- (simp_all add: sum.If_cases i) }
- note ** = this
- show ?thesis
- apply (rule that[of ?a])
- unfolding interior_std_simplex mem_Collect_eq
- proof safe
- fix i :: 'a
- assume i: "i \<in> Basis"
- show "0 < ?a \<bullet> i"
- unfolding **[OF i] by (auto simp add: Suc_le_eq DIM_positive)
- next
- have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
- apply (rule sum.cong)
- apply rule
- apply auto
- done
- also have "\<dots> < 1"
- unfolding sum_constant divide_inverse[symmetric]
- by (auto simp add: field_simps)
- finally show "sum (op \<bullet> ?a) ?D < 1" by auto
- qed
-qed
-
-lemma rel_interior_substd_simplex:
- assumes d: "d \<subseteq> Basis"
- shows "rel_interior (convex hull (insert 0 d)) =
- {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
- (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
-proof -
- have "finite d"
- apply (rule finite_subset)
- using assms
- apply auto
- done
- 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 ?p)) =
- {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
- using affine_hull_convex_hull affine_hull_substd_basis assms by auto
- have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>d. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
- by auto
- {
- fix x :: "'a::euclidean_space"
- assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
- then obtain e where e0: "e > 0" and
- "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
- using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
- then have as: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
- (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum (op \<bullet> xa) d \<le> 1"
- unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
- have x0: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
- using x rel_interior_subset substd_simplex[OF assms] by auto
- have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
- apply rule
- apply rule
- proof -
- fix i :: 'a
- assume "i \<in> d"
- then have "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> ia"
- apply -
- apply (rule as[rule_format,THEN conjunct1])
- unfolding dist_norm
- using d \<open>e > 0\<close> x0
- apply (auto simp: inner_simps inner_Basis)
- done
- then show "0 < x \<bullet> i"
- apply (erule_tac x=i in ballE)
- using \<open>e > 0\<close> \<open>i \<in> d\<close> d
- apply (auto simp: inner_simps inner_Basis)
- done
- next
- obtain a where a: "a \<in> d"
- using \<open>d \<noteq> {}\<close> by auto
- then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
- using \<open>e > 0\<close> norm_Basis[of a] d
- unfolding dist_norm
- by auto
- have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
- using a d by (auto simp: inner_simps inner_Basis)
- then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
- sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
- using d by (intro sum.cong) auto
- have "a \<in> Basis"
- using \<open>a \<in> d\<close> d by auto
- then have h1: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
- using x0 d \<open>a\<in>d\<close> by (auto simp add: inner_add_left inner_Basis)
- have "sum (op \<bullet> x) d < sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
- unfolding * sum.distrib
- using \<open>e > 0\<close> \<open>a \<in> d\<close>
- using \<open>finite d\<close>
- by (auto simp add: sum.delta')
- also have "\<dots> \<le> 1"
- using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
- by auto
- finally show "sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
- using x0 by auto
- qed
- }
- moreover
- {
- fix x :: "'a::euclidean_space"
- assume as: "x \<in> ?s"
- have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
- by auto
- moreover have "\<forall>i. i \<in> d \<or> i \<notin> d" by auto
- ultimately
- have "\<forall>i. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
- by metis
- then have h2: "x \<in> convex hull (insert 0 ?p)"
- using as assms
- unfolding substd_simplex[OF assms] by fastforce
- obtain a where a: "a \<in> d"
- using \<open>d \<noteq> {}\<close> by auto
- let ?d = "(1 - sum (op \<bullet> x) d) / real (card d)"
- have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
- by (simp add: card_gt_0_iff)
- have "Min ((op \<bullet> x) ` d) > 0"
- using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_gr_iff)
- moreover have "?d > 0" using as using \<open>0 < card d\<close> by auto
- ultimately have h3: "min (Min ((op \<bullet> x) ` d)) ?d > 0"
- by auto
-
- have "x \<in> rel_interior (convex hull (insert 0 ?p))"
- unfolding rel_interior_ball mem_Collect_eq h0
- apply (rule,rule h2)
- unfolding substd_simplex[OF assms]
- apply (rule_tac x="min (Min ((op \<bullet> x) ` d)) ?d" in exI)
- apply (rule, rule h3)
- apply safe
- unfolding mem_ball
- proof -
- fix y :: 'a
- assume y: "dist x y < min (Min (op \<bullet> x ` d)) ?d"
- assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
- have "sum (op \<bullet> y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
- proof (rule sum_mono)
- fix i
- assume "i \<in> d"
- with d have i: "i \<in> Basis"
- by auto
- have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
- apply (rule le_less_trans)
- using Basis_le_norm[OF i, of "y - x"]
- using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
- apply (auto simp add: norm_minus_commute inner_simps)
- done
- then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
- qed
- also have "\<dots> \<le> 1"
- unfolding sum.distrib sum_constant using \<open>0 < card d\<close>
- by auto
- finally show "sum (op \<bullet> y) d \<le> 1" .
-
- fix i :: 'a
- assume i: "i \<in> Basis"
- then show "0 \<le> y\<bullet>i"
- proof (cases "i\<in>d")
- case True
- have "norm (x - y) < x\<bullet>i"
- using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
- using Min_gr_iff[of "op \<bullet> x ` d" "norm (x - y)"] \<open>0 < card d\<close> \<open>i:d\<close>
- by (simp add: card_gt_0_iff)
- then show "0 \<le> y\<bullet>i"
- using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
- by (auto simp: inner_simps)
- qed (insert y2, auto)
- qed
- }
- ultimately have
- "\<And>x. x \<in> rel_interior (convex hull insert 0 d) \<longleftrightarrow>
- x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
- by blast
- then show ?thesis by (rule set_eqI)
- qed
-qed
-
-lemma rel_interior_substd_simplex_nonempty:
- assumes "d \<noteq> {}"
- and "d \<subseteq> Basis"
- obtains a :: "'a::euclidean_space"
- where "a \<in> rel_interior (convex hull (insert 0 d))"
-proof -
- let ?D = d
- let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
- have "finite d"
- apply (rule finite_subset)
- using assms(2)
- apply auto
- done
- then have d1: "0 < real (card d)"
- using \<open>d \<noteq> {}\<close> by auto
- {
- fix i
- assume "i \<in> d"
- have "?a \<bullet> i = inverse (2 * real (card d))"
- apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
- unfolding inner_sum_left
- apply (rule sum.cong)
- using \<open>i \<in> d\<close> \<open>finite d\<close> sum.delta'[of d i "(\<lambda>k. inverse (2 * real (card d)))"]
- d1 assms(2)
- by (auto simp: inner_Basis set_rev_mp[OF _ assms(2)])
- }
- note ** = this
- show ?thesis
- apply (rule that[of ?a])
- unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
- proof safe
- fix i
- assume "i \<in> d"
- have "0 < inverse (2 * real (card d))"
- using d1 by auto
- also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> d\<close>
- by auto
- finally show "0 < ?a \<bullet> i" by auto
- next
- have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
- by (rule sum.cong) (rule refl, rule **)
- also have "\<dots> < 1"
- unfolding sum_constant divide_real_def[symmetric]
- by (auto simp add: field_simps)
- finally show "sum (op \<bullet> ?a) ?D < 1" by auto
- next
- fix i
- assume "i \<in> Basis" and "i \<notin> d"
- have "?a \<in> span d"
- proof (rule span_sum[of d "(\<lambda>b. b /\<^sub>R (2 * real (card d)))" d])
- {
- fix x :: "'a::euclidean_space"
- assume "x \<in> d"
- then have "x \<in> span d"
- using span_superset[of _ "d"] by auto
- then have "x /\<^sub>R (2 * real (card d)) \<in> span d"
- using span_mul[of x "d" "(inverse (real (card d)) / 2)"] by auto
- }
- then show "\<And>x. x\<in>d \<Longrightarrow> x /\<^sub>R (2 * real (card d)) \<in> span d"
- by auto
- qed
- then show "?a \<bullet> i = 0 "
- using \<open>i \<notin> d\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
- qed
-qed
-
-
-subsection \<open>Relative interior of convex set\<close>
-
-lemma rel_interior_convex_nonempty_aux:
- fixes S :: "'n::euclidean_space set"
- assumes "convex S"
- and "0 \<in> S"
- shows "rel_interior S \<noteq> {}"
-proof (cases "S = {0}")
- case True
- then show ?thesis using rel_interior_sing by auto
-next
- case False
- obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
- using basis_exists[of S] by auto
- then have "B \<noteq> {}"
- using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
- have "insert 0 B \<le> span B"
- using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
- then have "span (insert 0 B) \<le> span B"
- using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
- then have "convex hull insert 0 B \<le> span B"
- using convex_hull_subset_span[of "insert 0 B"] by auto
- then have "span (convex hull insert 0 B) \<le> 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 \<in> 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 \<Rightarrow> 'n" where
- fd: "card d = card B" "linear f" "f ` B = d"
- "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
- and d: "d \<subseteq> 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 \<noteq> {}"
- using fd B \<open>B \<noteq> {}\<close> 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)"] \<open>linear f\<close>
- by auto
- moreover have "rel_interior (f ` (convex hull insert 0 B)) =
- f ` rel_interior (convex hull insert 0 B)"
- apply (rule rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
- using \<open>bounded_linear f\<close> fd *
- apply auto
- done
- ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
- using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
- apply auto
- apply blast
- done
- moreover have "convex hull (insert 0 B) \<subseteq> 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 = {} \<longleftrightarrow> S = {}"
-proof -
- {
- assume "S \<noteq> {}"
- then obtain a where "a \<in> S" by auto
- then have "0 \<in> op + (-a) ` S"
- using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
- then have "rel_interior (op + (-a) ` S) \<noteq> {}"
- using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]
- convex_translation[of S "-a"] assms
- by auto
- then have "rel_interior S \<noteq> {}"
- using rel_interior_translation by auto
- }
- then show ?thesis
- using rel_interior_empty 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 \<in> interior (convex hull (insert 0 S))"
-proof -
- have "affine hull (insert 0 S) = UNIV"
- apply (simp add: hull_inc affine_hull_span_0)
- using assms dim_eq_full indep_card_eq_dim_span by fastforce
- moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
- using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
- ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
- 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 \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
- then have "x \<in> S"
- using rel_interior_subset by auto
- have "x - u *\<^sub>R (x-y) \<in> 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 \<open>x \<in> S\<close> by auto
- next
- case True
- then show ?thesis using assm by auto
- qed
- then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> 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) \<le> 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 \<in> rel_interior S"
- using rel_interior_eq_empty assms by auto
- { fix x
- assume x: "x \<in> closure S"
- {
- assume "x = a"
- then have "x \<in> closure (rel_interior S)"
- using a unfolding closure_def by auto
- }
- moreover
- {
- assume "x \<noteq> a"
- {
- fix e :: real
- assume "e > 0"
- define e1 where "e1 = min 1 (e/norm (x - a))"
- then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
- using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
- by simp_all
- then have *: "x - e1 *\<^sub>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 "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
- apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
- using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
- apply simp
- done
- }
- then have "x islimpt rel_interior S"
- unfolding islimpt_approachable_le by auto
- then have "x \<in> closure(rel_interior S)"
- unfolding closure_def by auto
- }
- ultimately have "x \<in> closure(rel_interior S)" by auto
- }
- then show ?thesis using h1 by auto
- next
- case True
- then have "rel_interior S = {}"
- using rel_interior_empty by auto
- then have "closure (rel_interior S) = {}"
- using closure_empty by auto
- with True show ?thesis by auto
- qed
-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 (subtopology euclidean (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) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
- obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
-proof -
- define e where "e = a / (a + b)"
- have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
- apply auto
- using assms
- apply simp
- done
- also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
- using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
- by auto
- also have "\<dots> = y - e *\<^sub>R (y-x)"
- using e_def
- apply (simp add: algebra_simps)
- using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
- apply auto
- done
- finally have "z = y - e *\<^sub>R (y-x)"
- by auto
- moreover have "e > 0" using e_def assms by auto
- moreover have "e \<le> 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) \<supseteq> 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 \<in> rel_interior (closure S)"
- obtain x where x: "x \<in> rel_interior S"
- using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
- have "z \<in> 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 \<inter> affine hull closure S \<le> 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)) *\<^sub>R (z-x)"
- have yball: "y \<in> cball z e"
- using mem_cball y_def dist_norm[of z y] e by auto
- have "x \<in> affine hull closure S"
- using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
- moreover have "z \<in> affine hull closure S"
- using z rel_interior_subset hull_subset[of "closure S"] by blast
- ultimately have "y \<in> 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 \<in> closure S" using e yball by auto
- have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
- using y_def by (simp add: algebra_simps)
- then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>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 \<open>y \<in> closure S\<close>
- by auto
- 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 closure_eq_rel_interior_eq:
- fixes S1 S2 :: "'n::euclidean_space set"
- assumes "convex S1"
- and "convex S2"
- shows "closure S1 = closure S2 \<longleftrightarrow> 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 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
- (is "?A \<longleftrightarrow> ?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 \<subseteq> closure S2"
- by (metis assms(1) convex_closure_rel_interior closure_mono)
- moreover from \<open>?B\<close> have "closure S1 \<supseteq> 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 \<inter> closure S = {} \<longleftrightarrow> A \<inter> 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 show ?thesis
- apply (simp add: rel_interior_eq openin_open)
- apply (rule_tac x="ball (inverse 2 *\<^sub>R (a + b)) (norm(b - a) / 2)" in exI)
- apply (simp add: open_segment_as_ball)
- done
-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
-
-lemma starlike_convex_tweak_boundary_points:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "S \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
- shows "starlike T"
-proof -
- have "rel_interior S \<noteq> {}"
- by (simp add: assms rel_interior_eq_empty)
- then obtain a where a: "a \<in> rel_interior S" by blast
- with ST have "a \<in> T" by blast
- have *: "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
- apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
- using assms by blast
- show ?thesis
- unfolding starlike_def
- apply (rule bexI [OF _ \<open>a \<in> T\<close>])
- apply (simp add: closed_segment_eq_open)
- apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
- apply (simp add: order_trans [OF * ST])
- done
-qed
-
-subsection\<open>The relative frontier of a set\<close>
-
-definition "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 = {} \<longleftrightarrow> affine S"
- apply (simp add: rel_frontier_def)
- apply (simp add: rel_interior_eq_closure [symmetric])
- using rel_interior_subset_closure by blast
-
-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 \<subseteq> 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
- apply simp
- by (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((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
-by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
-
-lemma closed_affine_hull [iff]:
- fixes S :: "'n::euclidean_space set"
- shows "closed (affine hull S)"
- by (metis affine_affine_hull affine_closed)
-
-lemma rel_frontier_nonempty_interior:
- fixes S :: "'n::euclidean_space set"
- shows "interior S \<noteq> {} \<Longrightarrow> 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 \<Longrightarrow> rel_frontier S = frontier S"
-by (simp add: frontier_def rel_frontier_def rel_interior_interior)
-
-lemma closest_point_in_rel_frontier:
- "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
- \<Longrightarrow> closest_point S x \<in> 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 (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
- by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
- show ?thesis
- apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
- unfolding rel_frontier_def
- using * closed_affine_hull
- apply auto
- done
-qed
-
-lemma closed_rel_boundary:
- fixes S :: "'n::euclidean_space set"
- shows "closed S \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
- \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
- \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
- rel_interior S \<subseteq> T \<and> T \<subseteq> 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 \<noteq> {}"
- and "S1 \<le> rel_frontier S2"
- shows "aff_dim S1 < aff_dim S2"
-proof -
- have "S1 \<subseteq> closure S2"
- using assms unfolding rel_frontier_def by auto
- then have *: "affine hull S1 \<subseteq> affine hull S2"
- using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
- then have "aff_dim S1 \<le> 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 \<noteq> {}"
- using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
- have **: "affine hull S1 = affine hull S2"
- apply (rule affine_dim_equal)
- using * affine_affine_hull
- apply auto
- using \<open>S1 \<noteq> {}\<close> hull_subset[of S1]
- apply auto
- using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
- apply auto
- done
- obtain a where a: "a \<in> rel_interior S1"
- using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
- obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
- using mem_rel_interior[of a S1] a by auto
- then have "a \<in> T \<inter> closure S2"
- using a assms unfolding rel_frontier_def by auto
- then obtain b where b: "b \<in> T \<inter> rel_interior S2"
- using open_inter_closure_rel_interior[of S2 T] assms T by auto
- then have "b \<in> affine hull S1"
- using rel_interior_subset hull_subset[of S2] ** by auto
- then have "b \<in> 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 \<in> rel_interior S"
- shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
-proof -
- obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
- using mem_rel_interior_cball[of z S] assms by auto
- {
- fix x
- assume x: "x \<in> affine hull S"
- {
- assume "x \<noteq> z"
- define m where "m = 1 + e1/norm(x-z)"
- hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
- {
- fix e
- assume e: "e > 1 \<and> e \<le> m"
- have "z \<in> affine hull S"
- using assms rel_interior_subset hull_subset[of S] by auto
- then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
- by (simp add: algebra_simps)
- also have "\<dots> = (e - 1) * norm (x-z)"
- using norm_scaleR e by auto
- also have "\<dots> \<le> (m - 1) * norm (x - z)"
- using e mult_right_mono[of _ _ "norm(x-z)"] by auto
- also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
- using m_def by auto
- also have "\<dots> = e1"
- using \<open>x \<noteq> z\<close> e1 by simp
- finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
- by auto
- have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
- using m_def **
- unfolding cball_def dist_norm
- by (auto simp add: algebra_simps)
- then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
- using e * e1 by auto
- }
- then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
- using \<open>m> 1 \<close> 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 \<and> e \<le> m"
- then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
- using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
- then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
- using e by auto
- }
- then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
- using \<open>m > 1\<close> by auto
- }
- ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> rel_interior S"
- shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
- assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
- shows "z \<in> rel_interior S"
-proof -
- obtain x where x: "x \<in> rel_interior S"
- using rel_interior_eq_empty assms by auto
- then have "x \<in> S"
- using rel_interior_subset by auto
- then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
- using assms by auto
- define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
- then have "y \<in> S" using e by auto
- define e1 where "e1 = 1/e"
- then have "0 < e1 \<and> e1 < 1" using e by auto
- then have "z =y - (1 - e1) *\<^sub>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"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
- by auto
-qed
-
-lemma convex_rel_interior_iff:
- fixes S :: "'n::euclidean_space set"
- assumes "convex S"
- and "S \<noteq> {}"
- shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
- shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
-proof (cases "aff_dim S = int DIM('n)")
- case False
- {
- assume "z \<in> interior S"
- then have False
- using False interior_rel_interior_gen[of S] by auto
- }
- moreover
- {
- assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
- {
- fix x
- obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
- using r by auto
- obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
- using r by auto
- define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
- then have x1: "x1 \<in> affine hull S"
- using e1 hull_subset[of S] by auto
- define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
- then have x2: "x2 \<in> 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)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
- using x1_def x2_def
- apply (auto simp add: algebra_simps)
- using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
- apply auto
- done
- then have z: "z \<in> 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) *\<^sub>R (x - z)"
- using x1_def x2_def by (auto simp add: algebra_simps)
- then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
- using e1 e2 by simp
- then have "x \<in> 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 add: aff_dim_UNIV)
- then have False
- using False by auto
- }
- ultimately show ?thesis by auto
-next
- case True
- then have "S \<noteq> {}"
- using aff_dim_empty[of S] by auto
- have *: "affine hull S = UNIV"
- using True affine_hull_UNIV by auto
- {
- assume "z \<in> interior S"
- then have "z \<in> rel_interior S"
- using True interior_rel_interior_gen[of S] by auto
- then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
- using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
- fix x
- obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
- using **[rule_format, of "z-x"] by auto
- define e where [abs_def]: "e = e1 - 1"
- then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
- by (simp add: algebra_simps)
- then have "e > 0" "z + e *\<^sub>R x \<in> S"
- using e1 e_def by auto
- then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
- by auto
- }
- moreover
- {
- assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
- {
- fix x
- obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
- using r[rule_format, of "z-x"] by auto
- define e where "e = e1 + 1"
- then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
- by (simp add: algebra_simps)
- then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
- using e1 e_def by auto
- then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
- }
- then have "z \<in> rel_interior S"
- using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
- then have "z \<in> interior S"
- using True interior_rel_interior_gen[of S] by auto
- }
- ultimately show ?thesis by auto
-qed
-
-
-subsubsection \<open>Relative interior and closure under common operations\<close>
-
-lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S : I} \<subseteq> \<Inter>I"
-proof -
- {
- fix y
- assume "y \<in> \<Inter>{rel_interior S |S. S : I}"
- then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
- by auto
- {
- fix S
- assume "S \<in> I"
- then have "y \<in> S"
- using rel_interior_subset y by auto
- }
- then have "y \<in> \<Inter>I" by auto
- }
- then show ?thesis by auto
-qed
-
-lemma closure_Int: "closure (\<Inter>I) \<le> \<Inter>{closure S |S. S \<in> I}"
-proof -
- {
- fix y
- assume "y \<in> \<Inter>I"
- then have y: "\<forall>S \<in> I. y \<in> S" by auto
- {
- fix S
- assume "S \<in> I"
- then have "y \<in> closure S"
- using closure_subset y by auto
- }
- then have "y \<in> \<Inter>{closure S |S. S \<in> I}"
- by auto
- }
- then have "\<Inter>I \<subseteq> \<Inter>{closure S |S. S \<in> I}"
- by auto
- moreover have "closed (\<Inter>{closure S |S. S \<in> I})"
- unfolding closed_Inter closed_closure by auto
- ultimately show ?thesis using closure_hull[of "\<Inter>I"]
- hull_minimal[of "\<Inter>I" "\<Inter>{closure S |S. S \<in> I}" "closed"] by auto
-qed
-
-lemma convex_closure_rel_interior_inter:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
- and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
- shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
-proof -
- obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
- using assms by auto
- {
- fix y
- assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
- then have y: "\<forall>S \<in> I. y \<in> closure S"
- by auto
- {
- assume "y = x"
- then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
- using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
- }
- moreover
- {
- assume "y \<noteq> x"
- { fix e :: real
- assume e: "e > 0"
- define e1 where "e1 = min 1 (e/norm (y - x))"
- then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
- using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
- by simp_all
- define z where "z = y - e1 *\<^sub>R (y - x)"
- {
- fix S
- assume "S \<in> I"
- then have "z \<in> 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 \<in> \<Inter>{rel_interior S |S. S \<in> I}"
- by auto
- have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
- apply (rule_tac x="z" in exI)
- using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
- apply simp
- done
- }
- then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
- unfolding islimpt_approachable_le by blast
- then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
- unfolding closure_def by auto
- }
- ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
- by auto
- }
- then show ?thesis by auto
-qed
-
-lemma convex_closure_inter:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
- and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
- shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
-proof -
- have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
- using convex_closure_rel_interior_inter assms by auto
- moreover
- have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
- using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
- by auto
- ultimately show ?thesis
- using closure_Int[of I] by auto
-qed
-
-lemma convex_inter_rel_interior_same_closure:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
- and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
- shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
-proof -
- have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
- using convex_closure_rel_interior_inter assms by auto
- moreover
- have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
- using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
- by auto
- ultimately show ?thesis
- using closure_Int[of I] by auto
-qed
-
-lemma convex_rel_interior_inter:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
- and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
- shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
-proof -
- have "convex (\<Inter>I)"
- using assms convex_Inter by auto
- moreover
- have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
- apply (rule convex_Inter)
- using assms convex_rel_interior
- apply auto
- done
- ultimately
- have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
- using convex_inter_rel_interior_same_closure assms
- closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
- by blast
- then show ?thesis
- using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
-qed
-
-lemma convex_rel_interior_finite_inter:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
- and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
- and "finite I"
- shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
-proof -
- have "\<Inter>I \<noteq> {}"
- using assms rel_interior_inter_aux[of I] by auto
- have "convex (\<Inter>I)"
- using convex_Inter assms by auto
- show ?thesis
- proof (cases "I = {}")
- case True
- then show ?thesis
- using Inter_empty rel_interior_UNIV by auto
- next
- case False
- {
- fix z
- assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
- {
- fix x
- assume x: "x \<in> \<Inter>I"
- {
- fix S
- assume S: "S \<in> I"
- then have "z \<in> rel_interior S" "x \<in> S"
- using z x by auto
- then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
- using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
- }
- then obtain mS where
- mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
- define e where "e = Min (mS ` I)"
- then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
- then have "e > 1" using mS by auto
- moreover have "\<forall>S\<in>I. e \<le> mS S"
- using e_def assms by auto
- ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
- using mS by auto
- }
- then have "z \<in> rel_interior (\<Inter>I)"
- using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
- }
- then show ?thesis
- using convex_rel_interior_inter[of I] assms by auto
- qed
-qed
-
-lemma convex_closure_inter_two:
- fixes S T :: "'n::euclidean_space set"
- assumes "convex S"
- and "convex T"
- assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
- shows "closure (S \<inter> T) = closure S \<inter> closure T"
- using convex_closure_inter[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 \<inter> rel_interior T \<noteq> {}"
- shows "rel_interior (S \<inter> T) = rel_interior S \<inter> 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 \<inter> T \<noteq> {}"
- shows "closure (S \<inter> T) = closure S \<inter> T"
-proof -
- have "affine hull T = T"
- using assms by auto
- then have "rel_interior T = T"
- using rel_interior_affine_hull[of T] by metis
- moreover have "closure T = T"
- using assms affine_closed[of T] by auto
- ultimately show ?thesis
- using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
-qed
-
-lemma connected_component_1_gen:
- fixes S :: "'a :: euclidean_space set"
- assumes "DIM('a) = 1"
- shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<inter> T \<noteq> {}"
- shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
-proof -
- have "affine hull T = T"
- using assms by auto
- then have "rel_interior T = T"
- using rel_interior_affine_hull[of T] by metis
- moreover have "closure T = T"
- using assms affine_closed[of T] by auto
- ultimately show ?thesis
- using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
-qed
-
-lemma convex_affine_rel_frontier_Int:
- fixes S T :: "'n::euclidean_space set"
- assumes "convex S"
- and "affine T"
- and "interior S \<inter> T \<noteq> {}"
- shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
-using assms
-apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
-by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
-
-lemma rel_interior_convex_Int_affine:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
- shows "rel_interior(S \<inter> T) = interior S \<inter> T"
-proof -
- obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
- using assms by force
- have "rel_interior S = interior S"
- by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
- then show ?thesis
- by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
-qed
-
-lemma closure_convex_Int_affine:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
- shows "closure(S \<inter> T) = closure S \<inter> T"
-proof
- have "closure (S \<inter> T) \<subseteq> closure T"
- by (simp add: closure_mono)
- also have "... \<subseteq> T"
- by (simp add: affine_closed assms)
- finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
- by (simp add: closure_mono)
-next
- obtain a where "a \<in> rel_interior S" "a \<in> T"
- using assms by auto
- then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
- using affine_diffs_subspace rel_interior_subset assms by blast+
- show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
- proof
- fix x assume "x \<in> closure S \<inter> T"
- show "x \<in> closure (S \<inter> T)"
- proof (cases "x = a")
- case True
- then show ?thesis
- using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
- next
- case False
- then have "x \<in> closure(open_segment a x)"
- by auto
- then show ?thesis
- using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
- qed
- qed
-qed
-
-lemma subset_rel_interior_convex:
- fixes S T :: "'n::euclidean_space set"
- assumes "convex S"
- and "convex T"
- and "S \<le> closure T"
- and "\<not> S \<subseteq> rel_frontier T"
- shows "rel_interior S \<subseteq> rel_interior T"
-proof -
- have *: "S \<inter> closure T = S"
- using assms by auto
- have "\<not> rel_interior S \<subseteq> 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 \<inter> rel_interior (closure T) \<noteq> {}"
- using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
- by auto
- then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> 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 "\<dots> = rel_interior S"
- using * by auto
- finally show ?thesis
- by auto
-qed
-
-lemma rel_interior_convex_linear_image:
- fixes f :: "'m::euclidean_space \<Rightarrow> '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 rel_interior_empty rel_interior_eq_empty by auto
-next
- case False
- have *: "f ` (rel_interior S) \<subseteq> f ` S"
- unfolding image_mono using rel_interior_subset by auto
- have "f ` S \<subseteq> f ` (closure S)"
- unfolding image_mono using closure_subset by auto
- also have "\<dots> = f ` (closure (rel_interior S))"
- using convex_closure_rel_interior assms by auto
- also have "\<dots> \<subseteq> 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) \<subseteq> f ` rel_interior S"
- using rel_interior_subset by auto
- moreover
- {
- fix z
- assume "z \<in> f ` rel_interior S"
- then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
- {
- fix x
- assume "x \<in> f ` S"
- then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
- then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
- using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
- moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
- using x1 z1 \<open>linear f\<close> by (simp add: linear_add_cmul)
- ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
- using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
- then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
- using e by auto
- }
- then have "z \<in> rel_interior (f ` S)"
- using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close>
- \<open>linear f\<close> \<open>S \<noteq> {}\<close> 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 \<Rightarrow> 'n::euclidean_space"
- assumes "linear f"
- and "convex S"
- and "f -` (rel_interior S) \<noteq> {}"
- shows "rel_interior (f -` S) = f -` (rel_interior S)"
-proof -
- have "S \<noteq> {}"
- using assms rel_interior_empty by auto
- have nonemp: "f -` S \<noteq> {}"
- by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
- then have "S \<inter> (range f) \<noteq> {}"
- by auto
- have conv: "convex (f -` S)"
- using convex_linear_vimage assms by auto
- then have "convex (S \<inter> range f)"
- by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
- {
- fix z
- assume "z \<in> f -` (rel_interior S)"
- then have z: "f z : rel_interior S"
- by auto
- {
- fix x
- assume "x \<in> f -` S"
- then have "f x \<in> S" by auto
- then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
- using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
- moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
- using \<open>linear f\<close> by (simp add: linear_iff)
- ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
- using e by auto
- }
- then have "z \<in> rel_interior (f -` S)"
- using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
- }
- moreover
- {
- fix z
- assume z: "z \<in> rel_interior (f -` S)"
- {
- fix x
- assume "x \<in> S \<inter> range f"
- then obtain y where y: "f y = x" "y \<in> f -` S" by auto
- then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
- using convex_rel_interior_iff[of "f -` S" z] z conv by auto
- moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
- using \<open>linear f\<close> y by (simp add: linear_iff)
- ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
- using e by auto
- }
- then have "f z \<in> rel_interior (S \<inter> range f)"
- using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
- convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
- by auto
- moreover have "affine (range f)"
- by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
- ultimately have "f z \<in> rel_interior S"
- using convex_affine_rel_interior_Int[of S "range f"] assms by auto
- then have "z \<in> 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 \<times> T) = rel_interior S \<times> rel_interior T"
-proof -
- { assume "S = {}"
- then have ?thesis
- by auto
- }
- moreover
- { assume "T = {}"
- then have ?thesis
- by auto
- }
- moreover
- {
- assume "S \<noteq> {}" "T \<noteq> {}"
- then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
- using rel_interior_eq_empty assms by auto
- then have "fst -` rel_interior S \<noteq> {}"
- using fst_vimage_eq_Times[of "rel_interior S"] by auto
- then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
- using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
- then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
- by (simp add: fst_vimage_eq_Times)
- from ri have "snd -` rel_interior T \<noteq> {}"
- using snd_vimage_eq_Times[of "rel_interior T"] by auto
- then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
- using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
- then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
- by (simp add: snd_vimage_eq_Times)
- from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
- rel_interior S \<times> rel_interior T" by auto
- have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
- by auto
- then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
- by auto
- also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
- apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> T"])
- using * ri assms convex_Times
- apply auto
- done
- finally have ?thesis using * by auto
- }
- ultimately show ?thesis by blast
-qed
-
-lemma rel_interior_scaleR:
- fixes S :: "'n::euclidean_space set"
- assumes "c \<noteq> 0"
- shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
- using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
- linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms
- by auto
-
-lemma rel_interior_convex_scaleR:
- fixes S :: "'n::euclidean_space set"
- assumes "convex S"
- shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
- by (metis assms linear_scaleR rel_interior_convex_linear_image)
-
-lemma convex_rel_open_scaleR:
- fixes S :: "'n::euclidean_space set"
- assumes "convex S"
- and "rel_open S"
- shows "convex ((op *\<^sub>R c) ` S) \<and> rel_open ((op *\<^sub>R c) ` S)"
- by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
-
-lemma convex_rel_open_finite_inter:
- assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
- and "finite I"
- shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
-proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
- case True
- then have "\<Inter>I = {}"
- using assms unfolding rel_open_def by auto
- then show ?thesis
- unfolding rel_open_def using rel_interior_empty by auto
-next
- case False
- then have "rel_open (\<Inter>I)"
- using assms unfolding rel_open_def
- using convex_rel_interior_finite_inter[of I]
- by auto
- then show ?thesis
- using convex_Inter assms by auto
-qed
-
-lemma convex_rel_open_linear_image:
- fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "linear f"
- and "convex S"
- and "rel_open S"
- shows "convex (f ` S) \<and> 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 \<Rightarrow> 'n::euclidean_space"
- assumes "linear f"
- and "convex S"
- and "rel_open S"
- shows "convex (f -` S) \<and> 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 using rel_interior_empty 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 \<times> 'n::euclidean_space) set"
- and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
- assumes "convex S"
- and "f = (\<lambda>y. {z. (y, z) \<in> S})"
- shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
-proof -
- {
- fix y
- assume "y \<in> {y. f y \<noteq> {}}"
- then obtain z where "(y, z) \<in> S"
- using assms by auto
- then have "\<exists>x. x \<in> S \<and> y = fst x"
- apply (rule_tac x="(y, z)" in exI)
- apply auto
- done
- then obtain x where "x \<in> S" "y = fst x"
- by blast
- then have "y \<in> fst ` S"
- unfolding image_def by auto
- }
- then have "fst ` S = {y. f y \<noteq> {}}"
- unfolding fst_def using assms by auto
- then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
- using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
- {
- fix y
- assume "y \<in> rel_interior {y. f y \<noteq> {}}"
- then have "y \<in> fst ` rel_interior S"
- using h1 by auto
- then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
- by auto
- moreover have aff: "affine (fst -` {y})"
- unfolding affine_alt by (simp add: algebra_simps)
- ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
- using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
- have conv: "convex (S \<inter> fst -` {y})"
- using convex_Int assms aff affine_imp_convex by auto
- {
- fix x
- assume "x \<in> f y"
- then have "(y, x) \<in> S \<inter> (fst -` {y})"
- using assms by auto
- moreover have "x = snd (y, x)" by auto
- ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
- by blast
- }
- then have "snd ` (S \<inter> fst -` {y}) = f y"
- using assms by auto
- then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
- using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
- by auto
- {
- fix z
- assume "z \<in> rel_interior (f y)"
- then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
- using *** by auto
- moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
- using * ** rel_interior_subset by auto
- ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
- by force
- then have "(y,z) \<in> rel_interior S"
- using ** by auto
- }
- moreover
- {
- fix z
- assume "(y, z) \<in> rel_interior S"
- then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
- using ** by auto
- then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
- by (metis Range_iff snd_eq_Range)
- then have "z \<in> rel_interior (f y)"
- using *** by auto
- }
- ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
- by auto
- }
- then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
- (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
- by auto
- {
- fix y z
- assume asm: "(y, z) \<in> rel_interior S"
- then have "y \<in> fst ` rel_interior S"
- by (metis Domain_iff fst_eq_Domain)
- then have "y \<in> rel_interior {t. f t \<noteq> {}}"
- using h1 by auto
- then have "y \<in> rel_interior {t. f t \<noteq> {}}" 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 \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
- by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
-
-
-subsubsection \<open>Relative interior of convex cone\<close>
-
-lemma cone_rel_interior:
- fixes S :: "'m::euclidean_space set"
- assumes "cone S"
- shows "cone ({0} \<union> rel_interior S)"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (simp add: rel_interior_empty cone_0)
-next
- case False
- then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
- using cone_iff[of S] assms by auto
- then have *: "0 \<in> ({0} \<union> rel_interior S)"
- and "\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
- by (auto simp add: rel_interior_scaleR)
- then show ?thesis
- using cone_iff[of "{0} \<union> rel_interior S"] by auto
-qed
-
-lemma rel_interior_convex_cone_aux:
- fixes S :: "'m::euclidean_space set"
- assumes "convex S"
- shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
- c > 0 \<and> x \<in> ((op *\<^sub>R c) ` (rel_interior S))"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (simp add: rel_interior_empty cone_hull_empty)
-next
- case False
- then obtain s where "s \<in> S" by auto
- have conv: "convex ({(1 :: real)} \<times> S)"
- using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
- by auto
- define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
- then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
- (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
- apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
- using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
- apply auto
- done
- {
- fix y :: real
- assume "y \<ge> 0"
- then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
- using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
- then have "f y \<noteq> {}"
- using f_def by auto
- }
- then have "{y. f y \<noteq> {}} = {0..}"
- using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
- then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
- using rel_interior_real_semiline by auto
- {
- fix c :: real
- assume "c > 0"
- then have "f c = (op *\<^sub>R c ` S)"
- using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
- then have "rel_interior (f c) = op *\<^sub>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} \<times> S)) =
- {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
- (is "?lhs = ?rhs")
-proof -
- {
- fix z
- assume "z \<in> ?lhs"
- have *: "z = (fst z, snd z)"
- by auto
- have "z \<in> ?rhs"
- using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close>
- apply auto
- apply (rule_tac x = "fst z" in exI)
- apply (rule_tac x = x in exI)
- using *
- apply auto
- done
- }
- moreover
- {
- fix z
- assume "z \<in> ?rhs"
- then have "z \<in> ?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 "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
- shows "convex hull (\<Union>(S ` I)) =
- {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
- (is "?lhs = ?rhs")
-proof -
- have "?lhs \<supseteq> ?rhs"
- proof
- fix x
- assume "x : ?rhs"
- then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
- "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
- then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
- using hull_subset[of "\<Union>(S ` I)" convex] by auto
- then show "x \<in> ?lhs"
- unfolding *(1)[symmetric]
- apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
- using * assms convex_convex_hull
- apply auto
- done
- qed
-
- {
- fix i
- assume "i \<in> I"
- with assms have "\<exists>p. p \<in> S i" by auto
- }
- then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
-
- {
- fix i
- assume "i \<in> I"
- {
- fix x
- assume "x \<in> S i"
- define c where "c j = (if j = i then 1::real else 0)" for j
- then have *: "sum c I = 1"
- using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
- by auto
- define s where "s j = (if j = i then x else p j)" for j
- then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
- using c_def by (auto simp add: algebra_simps)
- then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
- using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
- by auto
- then have "x \<in> ?rhs"
- apply auto
- apply (rule_tac x = c in exI)
- apply (rule_tac x = s in exI)
- using * c_def s_def p \<open>x \<in> S i\<close>
- apply auto
- done
- }
- then have "?rhs \<supseteq> S i" by auto
- }
- then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
-
- {
- fix u v :: real
- assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
- fix x y
- assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
- from xy obtain c s where
- xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
- by auto
- from xy obtain d t where
- yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
- by auto
- define e where "e i = u * c i + v * d i" for i
- have ge0: "\<forall>i\<in>I. e i \<ge> 0"
- using e_def xc yc uv by simp
- have "sum (\<lambda>i. u * c i) I = u * sum c I"
- by (simp add: sum_distrib_left)
- moreover have "sum (\<lambda>i. 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) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
- for i
- {
- fix i
- assume i: "i \<in> I"
- have "q i \<in> 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: "\<forall>i\<in>I. q i \<in> S i" by auto
- {
- fix i
- assume i: "i \<in> I"
- have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
- proof (cases "e i = 0")
- case True
- have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
- using xc yc uv i by simp
- moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
- using True e_def i by simp
- ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
- with True show ?thesis by auto
- next
- case False
- then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
- using q_def by auto
- then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
- = (e i) *\<^sub>R (q i)" by auto
- with False show ?thesis by (simp add: algebra_simps)
- qed
- }
- then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
- by auto
- have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
- using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
- also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
- using * by auto
- finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
- by auto
- then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
- using ge0 sum1 qs by auto
- }
- then have "convex ?rhs" unfolding convex_def by auto
- then show ?thesis
- using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
- by blast
-qed
-
-lemma convex_hull_union_two:
- fixes S T :: "'m::euclidean_space set"
- assumes "convex S"
- and "S \<noteq> {}"
- and "convex T"
- and "T \<noteq> {}"
- shows "convex hull (S \<union> T) =
- {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> 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 "\<Union>(s ` I) = S \<union> T"
- using s_def I_def by auto
- then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
- by auto
- moreover have "convex hull \<Union>(s ` I) =
- {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
- apply (subst convex_hull_finite_union[of I s])
- using assms s_def I_def
- apply auto
- done
- moreover have
- "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
- using s_def I_def by auto
- ultimately show "?lhs \<subseteq> ?rhs" by auto
- {
- fix x
- assume "x \<in> ?rhs"
- then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
- by auto
- then have "x \<in> convex hull {s, t}"
- using convex_hull_2[of s t] by auto
- then have "x \<in> convex hull (S \<union> T)"
- using * hull_mono[of "{s, t}" "S \<union> T"] by auto
- }
- then show "?lhs \<supseteq> ?rhs" by blast
-qed
-
-
-subsection \<open>Convexity on direct sums\<close>
-
-lemma closure_sum:
- fixes S T :: "'a::real_normed_vector set"
- shows "closure S + closure T \<subseteq> 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 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 = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
- by (simp add: set_plus_image)
- also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
- using rel_interior_Times assms by auto
- also have "\<dots> = rel_interior (S + T)"
- using fst_snd_linear convex_Times assms
- rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
- by (auto simp add: set_plus_image)
- finally show ?thesis ..
-qed
-
-lemma rel_interior_sum_gen:
- fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
- assumes "\<forall>i\<in>I. convex (S i)"
- shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
- apply (subst sum_set_cond_linear[of convex])
- using rel_interior_sum rel_interior_sing[of "0"] assms
- apply (auto simp add: convex_set_plus)
- done
-
-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 \<times> T) \<and> rel_open (S \<times> 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) \<and> 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 \<noteq> {}"
- assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
- shows "convex hull (\<Union>(S ` I)) = sum S I"
- (is "?lhs = ?rhs")
-proof -
- {
- fix x
- assume "x \<in> ?lhs"
- then obtain c xs where
- x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
- using convex_hull_finite_union[of I S] assms by auto
- define s where "s i = c i *\<^sub>R xs i" for i
- {
- fix i
- assume "i \<in> I"
- then have "s i \<in> S i"
- using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
- }
- then have "\<forall>i\<in>I. s i \<in> S i" by auto
- moreover have "x = sum s I" using x s_def by auto
- ultimately have "x \<in> ?rhs"
- using set_sum_alt[of I S] assms by auto
- }
- moreover
- {
- fix x
- assume "x \<in> ?rhs"
- then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
- using set_sum_alt[of I S] assms by auto
- define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
- then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
- using x assms by auto
- moreover have "\<forall>i\<in>I. xs i \<in> S i"
- using x xs_def assms by (simp add: cone_def)
- moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
- by auto
- moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
- using assms by auto
- ultimately have "x \<in> ?lhs"
- apply (subst convex_hull_finite_union[of I S])
- using assms
- apply blast
- using assms
- apply blast
- apply rule
- apply (rule_tac x = "(\<lambda>i. (1 :: real) / of_nat (card I))" in exI)
- apply auto
- done
- }
- 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 \<noteq> {}"
- assumes "convex T"
- and "cone T"
- and "T \<noteq> {}"
- shows "convex hull (S \<union> 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 "\<Union>(A ` I) = S \<union> T"
- using A_def I_def by auto
- then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
- by auto
- moreover have "convex hull \<Union>(A ` I) = sum A I"
- apply (subst convex_hull_finite_union_cones[of I A])
- using assms A_def I_def
- apply auto
- done
- moreover have "sum A I = S + T"
- using A_def I_def
- unfolding set_plus_def
- apply auto
- unfolding set_plus_def
- apply auto
- done
- ultimately show ?thesis by auto
-qed
-
-lemma rel_interior_convex_hull_union:
- fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
- assumes "finite I"
- and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
- shows "rel_interior (convex hull (\<Union>(S ` I))) =
- {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
- (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
- (is "?lhs = ?rhs")
-proof (cases "I = {}")
- case True
- then show ?thesis
- using convex_hull_empty rel_interior_empty by auto
-next
- case False
- define C0 where "C0 = convex hull (\<Union>(S ` I))"
- have "\<forall>i\<in>I. C0 \<ge> S i"
- unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
- define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
- define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
- have "\<forall>i\<in>I. K i \<noteq> {}"
- unfolding K_def using assms
- by (simp add: cone_hull_empty_iff[symmetric])
- {
- fix i
- assume "i \<in> I"
- then have "convex (K i)"
- unfolding K_def
- apply (subst convex_cone_hull)
- apply (subst convex_Times)
- using assms
- apply auto
- done
- }
- then have convK: "\<forall>i\<in>I. convex (K i)"
- by auto
- {
- fix i
- assume "i \<in> I"
- then have "K0 \<supseteq> K i"
- unfolding K0_def K_def
- apply (subst hull_mono)
- using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
- apply auto
- done
- }
- then have "K0 \<supseteq> \<Union>(K ` I)" by auto
- moreover have "convex K0"
- unfolding K0_def
- apply (subst convex_cone_hull)
- apply (subst convex_Times)
- unfolding C0_def
- using convex_convex_hull
- apply auto
- done
- ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
- using hull_minimal[of _ "K0" "convex"] by blast
- have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
- using K_def by (simp add: hull_subset)
- then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
- by auto
- then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
- by (simp add: hull_mono)
- then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
- unfolding C0_def
- using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
- by auto
- moreover have "cone (convex hull (\<Union>(K ` I)))"
- apply (subst cone_convex_hull)
- using cone_Union[of "K ` I"]
- apply auto
- unfolding K_def
- using cone_cone_hull
- apply auto
- done
- ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
- unfolding K0_def
- using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
- by blast
- then have "K0 = convex hull (\<Union>(K ` I))"
- using geq by auto
- also have "\<dots> = sum K I"
- apply (subst convex_hull_finite_union_cones[of I K])
- using assms
- apply blast
- using False
- apply blast
- unfolding K_def
- apply rule
- apply (subst convex_cone_hull)
- apply (subst convex_Times)
- using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
- apply auto
- done
- finally have "K0 = sum K I" by auto
- then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
- using rel_interior_sum_gen[of I K] convK by auto
- {
- fix x
- assume "x \<in> ?lhs"
- then have "(1::real, x) \<in> 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 \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
- using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
- {
- fix i
- assume "i \<in> I"
- then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
- using k K_def assms by auto
- then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
- using rel_interior_convex_cone[of "S i"] by auto
- }
- then obtain c s where
- cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
- by metis
- then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
- using k by (simp add: sum_prod)
- then have "x \<in> ?rhs"
- using k
- apply auto
- apply (rule_tac x = c in exI)
- apply (rule_tac x = s in exI)
- using cs
- apply auto
- done
- }
- moreover
- {
- fix x
- assume "x \<in> ?rhs"
- then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
- (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
- by auto
- define k where "k i = (c i, c i *\<^sub>R s i)" for i
- {
- fix i assume "i:I"
- then have "k i \<in> rel_interior (K i)"
- using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
- by auto
- }
- then have "(1::real, x) \<in> rel_interior K0"
- using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
- apply auto
- apply (rule_tac x = k in exI)
- apply (simp add: sum_prod)
- done
- then have "x \<in> ?lhs"
- using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
- by (auto simp add: convex_convex_hull)
- }
- ultimately show ?thesis by blast
-qed
-
-
-lemma convex_le_Inf_differential:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_on I f"
- and "x \<in> interior I"
- and "y \<in> I"
- shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
- (is "_ \<ge> _ + Inf (?F x) * (y - x)")
-proof (cases rule: linorder_cases)
- assume "x < y"
- moreover
- have "open (interior I)" by auto
- from openE[OF this \<open>x \<in> interior I\<close>]
- obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
- moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
- ultimately have "x < t" "t < y" "t \<in> ball x e"
- by (auto simp: dist_real_def field_simps split: split_min)
- with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "open (interior I)" by auto
- from openE[OF this \<open>x \<in> interior I\<close>]
- obtain e where "0 < e" "ball x e \<subseteq> interior I" .
- moreover define K where "K = x - e / 2"
- with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
- by (auto simp: dist_real_def)
- ultimately have "K \<in> I" "K < x" "x \<in> I"
- using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
-
- have "Inf (?F x) \<le> (f x - f y) / (x - y)"
- proof (intro bdd_belowI cInf_lower2)
- show "(f x - f t) / (x - t) \<in> ?F x"
- using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
- show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
- by (rule convex_on_diff)
- next
- fix y
- assume "y \<in> ?F x"
- with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
- show "(f K - f x) / (K - x) \<le> y" by auto
- qed
- then show ?thesis
- using \<open>x < y\<close> by (simp add: field_simps)
-next
- assume "y < x"
- moreover
- have "open (interior I)" by auto
- from openE[OF this \<open>x \<in> interior I\<close>]
- obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
- moreover define t where "t = x + e / 2"
- ultimately have "x < t" "t \<in> ball x e"
- by (auto simp: dist_real_def field_simps)
- with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
-
- have "(f x - f y) / (x - y) \<le> Inf (?F x)"
- proof (rule cInf_greatest)
- have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
- using \<open>y < x\<close> by (auto simp: field_simps)
- also
- fix z
- assume "z \<in> ?F x"
- with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
- have "(f y - f x) / (y - x) \<le> z"
- by auto
- finally show "(f x - f y) / (x - y) \<le> z" .
- next
- have "open (interior I)" by auto
- from openE[OF this \<open>x \<in> interior I\<close>]
- obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
- then have "x + e / 2 \<in> ball x e"
- by (auto simp: dist_real_def)
- with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
- by auto
- then show "?F x \<noteq> {}"
- by blast
- qed
- then show ?thesis
- using \<open>y < x\<close> by (simp add: field_simps)
-qed simp
-
-subsection\<open>Explicit formulas for interior and relative interior of convex hull\<close>
-
-lemma interior_atLeastAtMost [simp]:
- fixes a::real shows "interior {a..b} = {a<..<b}"
- using interior_cbox [of a b] by auto
-
-lemma interior_atLeastLessThan [simp]:
- fixes a::real shows "interior {a..<b} = {a<..<b}"
- by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost interior_Int interior_interior interior_real_semiline)
-
-lemma interior_lessThanAtMost [simp]:
- fixes a::real shows "interior {a<..b} = {a<..<b}"
- by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost interior_Int
- interior_interior interior_real_semiline)
-
-lemma at_within_closed_interval:
- fixes x::real
- shows "a < x \<Longrightarrow> x < b \<Longrightarrow> (at x within {a..b}) = at x"
- by (metis at_within_interior greaterThanLessThan_iff interior_atLeastAtMost)
-
-lemma affine_independent_convex_affine_hull:
- fixes s :: "'a::euclidean_space set"
- assumes "~affine_dependent s" "t \<subseteq> s"
- shows "convex hull t = affine hull t \<inter> convex hull s"
-proof -
- have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
- { fix u v x
- assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
- "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
- then have s: "s = (s - t) \<union> t" \<comment>\<open>split into separate cases\<close>
- using assms by auto
- have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
- "sum v t + sum v (s - t) = 1"
- using uv fin s
- by (auto simp: sum.union_disjoint [symmetric] Un_commute)
- have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
- "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
- using uv fin
- by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
- } note [simp] = this
- have "convex hull t \<subseteq> affine hull t"
- using convex_hull_subset_affine_hull by blast
- moreover have "convex hull t \<subseteq> convex hull s"
- using assms hull_mono by blast
- moreover have "affine hull t \<inter> convex hull s \<subseteq> convex hull t"
- using assms
- apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
- apply (drule_tac x=s in spec)
- apply (auto simp: fin)
- apply (rule_tac x=u in exI)
- apply (rename_tac v)
- apply (drule_tac x="\<lambda>x. if x \<in> t then v x - u x else v x" in spec)
- apply (force)+
- done
- 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 \<notin> t" "s = insert a t"
- by blast
- then have fin: "finite t" using assms
- by (metis finite_insert aff_independent_finite)
- show ?thesis
- using assms t fin
- apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
- apply (rule subset_antisym)
- apply force
- apply (rule Fun.vimage_subsetD)
- apply (metis add.commute diff_add_cancel surj_def)
- apply (rule card_ge_dim_independent)
- apply (auto simp: card_image inj_on_def dim_subset_UNIV)
- done
-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"
- apply (rule affine_independent_span_eq [OF ind])
- apply (rule antisym)
- using assms
- apply auto
- apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
- done
-
-lemma empty_interior_affine_hull:
- fixes s :: "'a::euclidean_space set"
- assumes "finite s" and dim: "card s \<le> DIM ('a)"
- shows "interior(affine hull s) = {}"
- using assms
- apply (induct s rule: finite_induct)
- apply (simp_all add: affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
- apply (rule empty_interior_lowdim)
- apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
- apply (metis Suc_le_lessD not_less order_trans card_image_le finite_imageI dim_le_card)
- done
-
-lemma empty_interior_convex_hull:
- fixes s :: "'a::euclidean_space set"
- assumes "finite s" and dim: "card s \<le> 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
- \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
- \<subseteq> rel_interior (convex hull s)"
- by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
-
-lemma explicit_subset_rel_interior_convex_hull_minimal:
- fixes s :: "'a::euclidean_space set"
- shows "finite s
- \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
- \<subseteq> rel_interior (convex hull s)"
- by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {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. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
- (is "?lhs = ?rhs")
-proof
- show "?rhs \<le> ?lhs"
- by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
-next
- show "?lhs \<le> ?rhs"
- proof (cases "\<exists>a. s = {a}")
- case True then show "?lhs \<le> ?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 \<in> s" "b \<in> s" "a \<noteq> b"
- then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment>\<open>split into separate cases\<close>
- by auto
- have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
- "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
- using ab fs
- by (subst s, subst sum.union_disjoint, auto)+
- } note [simp] = this
- { fix y
- assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
- { fix u T a
- assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
- and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
- and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
- have ua0: "u a = 0"
- using ua by auto
- obtain b where b: "b\<in>s" "a \<noteq> b"
- using ua False by auto
- obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
- using yT by (auto elim: openE)
- with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
- by (auto intro: that [of "e / 2 / norm(a-b)"])
- have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
- using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
- then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
- using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
- then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
- using d e yT by auto
- then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
- "sum v s = 1"
- "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
- using subsetD [OF sb] yT
- by auto
- then have False
- using assms
- apply (simp add: affine_dependent_explicit_finite fs)
- apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
- using ua b d
- apply (auto simp: algebra_simps sum_subtractf sum.distrib)
- done
- } note * = this
- have "y \<notin> rel_interior (convex hull s)"
- using y
- apply (simp add: mem_rel_interior affine_hull_convex_hull)
- apply (auto simp: convex_hull_finite [OF fs])
- apply (drule_tac x=u in spec)
- apply (auto intro: *)
- done
- } with rel_interior_subset show "?lhs \<le> ?rhs"
- by blast
- qed
-qed
-
-lemma interior_convex_hull_explicit_minimal:
- fixes s :: "'a::euclidean_space set"
- shows
- "~ affine_dependent s
- ==> interior(convex hull s) =
- (if card(s) \<le> DIM('a) then {}
- else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
- apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
- apply (rule trans [of _ "rel_interior(convex hull s)"])
- apply (simp add: affine_hull_convex_hull affine_independent_span_gt rel_interior_interior)
- by (simp add: rel_interior_convex_hull_explicit)
-
-lemma interior_convex_hull_explicit:
- fixes s :: "'a::euclidean_space set"
- assumes "~ affine_dependent s"
- shows
- "interior(convex hull s) =
- (if card(s) \<le> DIM('a) then {}
- else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
-proof -
- { fix u :: "'a \<Rightarrow> real" and a
- assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
- then have cs: "Suc 0 < card s"
- by (metis DIM_positive less_trans_Suc)
- obtain b where b: "b \<in> s" "a \<noteq> b"
- proof (cases "s \<le> {a}")
- case True
- then show thesis
- using cs subset_singletonD by fastforce
- next
- case False
- then show thesis
- by (blast intro: that)
- qed
- have "u a + u b \<le> sum u {a,b}"
- using a b by simp
- also have "... \<le> sum u s"
- apply (rule Groups_Big.sum_mono2)
- using a b u
- apply (auto simp: less_imp_le aff_independent_finite assms)
- done
- finally have "u a < 1"
- using \<open>b \<in> s\<close> u by fastforce
- } note [simp] = this
- show ?thesis
- using assms
- apply (auto simp: interior_convex_hull_explicit_minimal)
- apply (rule_tac x=u in exI)
- apply (auto simp: not_le)
- done
-qed
-
-lemma interior_closed_segment_ge2:
- fixes a :: "'a::euclidean_space"
- assumes "2 \<le> DIM('a)"
- shows "interior(closed_segment a b) = {}"
-using assms unfolding segment_convex_hull
-proof -
- have "card {a, b} \<le> 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 \<le> DIM('a) then {} else open_segment a b)"
-proof (simp add: not_le, intro conjI impI)
- assume "2 \<le> DIM('a)"
- then show "interior (open_segment a b) = {}"
- apply (simp add: segment_convex_hull open_segment_def)
- apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
- done
-next
- assume le2: "DIM('a) < 2"
- show "interior (open_segment a b) = open_segment a b"
- proof (cases "a = b")
- case True then show ?thesis by auto
- next
- case False
- with le2 have "affine hull (open_segment a b) = UNIV"
- apply simp
- apply (rule affine_independent_span_gt)
- apply (simp_all add: affine_dependent_def insert_Diff_if)
- done
- then show "interior (open_segment a b) = open_segment a b"
- using rel_interior_interior rel_interior_open_segment by blast
- qed
-qed
-
-lemma interior_closed_segment:
- fixes a :: "'a::euclidean_space"
- shows "interior(closed_segment a b) =
- (if 2 \<le> 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 \<longleftrightarrow> {a,b} = {c,d}"
-proof
- assume abcd: "closed_segment a b = closed_segment c d"
- show "{a,b} = {c,d}"
- proof (cases "a=b \<or> c=d")
- case True with abcd show ?thesis by force
- next
- case False
- then have neq: "a \<noteq> b \<and> c \<noteq> 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 \<in> {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 \<in> {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 \<noteq> 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 \<noteq> 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 \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
- (is "?lhs = ?rhs")
-proof
- assume abcd: ?lhs
- show ?rhs
- proof (cases "a=b \<or> c=d")
- case True with abcd show ?thesis
- using finite_open_segment by fastforce
- next
- case False
- then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
- with abcd show ?rhs
- unfolding open_segment_def
- by (metis (no_types) abcd closed_segment_eq closure_open_segment)
- qed
-next
- assume ?rhs
- then show ?lhs
- by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
-qed
-
-subsection\<open>Similar results for closure and (relative or absolute) frontier.\<close>
-
-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. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
-proof -
- have fs: "finite s"
- using assms by (simp add: aff_independent_finite)
- show ?thesis
- apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
- apply (auto simp: convex_hull_finite fs)
- apply (drule_tac x=u in spec)
- apply (rule_tac x=u in exI)
- apply force
- apply (rename_tac v)
- apply (rule notE [OF assms])
- apply (simp add: affine_dependent_explicit)
- apply (rule_tac x=s in exI)
- apply (auto simp: fs)
- apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
- apply (force simp: sum_subtractf scaleR_diff_left)
- done
-qed
-
-lemma frontier_convex_hull_explicit:
- fixes s :: "'a::euclidean_space set"
- assumes "~ affine_dependent s"
- shows "frontier(convex hull s) =
- {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
- sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>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 \<le> 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) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
-proof -
- have fs: "finite s"
- using assms by (simp add: aff_independent_finite)
- { fix u a
- have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
- \<exists>x v. x \<in> s \<and>
- (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
- sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>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 \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
- \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
- (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
- apply (rule_tac x="\<lambda>x. 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 (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
- 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 \<le> 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 \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> 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 \<le> Suc (DIM ('a))" "x \<in> s"
- shows "x \<in> frontier(convex hull s)"
-proof (cases "affine_dependent s")
- case True
- with assms show ?thesis
- apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
- by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
-next
- case False
- { assume "card s = Suc (card Basis)"
- then have cs: "Suc 0 < card s"
- by (simp add: DIM_positive)
- with subset_singletonD have "\<exists>y \<in> s. y \<noteq> x"
- by (cases "s \<le> {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 \<le> Suc (DIM ('a))" "x \<in> s"
- shows "x \<notin> 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) = {} \<longleftrightarrow> affine_dependent s"
-proof -
- { fix a b
- assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> 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 have False using ab
- by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
- } then
- show ?thesis
- using assms
- apply auto
- apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
- apply (auto simp: affine_dependent_def)
- done
-qed
-
-
-subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
-
-definition coplanar where
- "coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
-
-lemma collinear_affine_hull:
- "collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
-proof (cases "s={}")
- case True then show ?thesis
- by simp
-next
- case False
- then obtain x where x: "x \<in> s" by auto
- { fix u
- assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
- have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
- apply (rule_tac x=x in exI)
- apply (rule_tac x="x+u" in exI, clarify)
- apply (erule exE [OF * [OF x]])
- apply (rename_tac c)
- apply (rule_tac x="1+c" in exI)
- apply (rule_tac x="-c" in exI)
- apply (simp add: algebra_simps)
- done
- } moreover
- { fix u v x y
- assume *: "s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
- have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R (v-u)"
- apply (drule subsetD [OF *])+
- apply simp
- apply clarify
- apply (rename_tac r1 r2)
- apply (rule_tac x="r1-r2" in exI)
- apply (simp add: algebra_simps)
- apply (metis scaleR_left.add)
- done
- } 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} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>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, hide_lams) 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 \<Rightarrow> real"
- assumes "continuous_on (closed_segment a b) f"
- shows "closed_segment (f a) (f b) \<subseteq> 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 \<Rightarrow> 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) \<subseteq> f ` closed_segment a b"
- by (metis subset_continuous_image_segment_1 contf)
- show "f ` closed_segment a b \<subseteq> 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 \<noteq> f b"
- using inj_onD injf by fastforce
- moreover
- have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
- proof (clarsimp simp add: open_segment_def)
- assume fa: "f a \<in> closed_segment (f c) (f b)"
- moreover have "closed_segment (f c) (f b) \<subseteq> 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 \<in> f ` closed_segment c b"
- by blast
- then have a: "a \<in> closed_segment c b"
- by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
- have cb: "closed_segment c b \<subseteq> 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 \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
- proof (clarsimp simp add: open_segment_def fnot eq_commute)
- assume fb: "f b \<in> closed_segment (f a) (f c)"
- moreover have "closed_segment (f a) (f c) \<subseteq> 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 \<in> f ` closed_segment a c"
- by blast
- then have b: "b \<in> closed_segment a c"
- by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
- have ca: "closed_segment a c \<subseteq> 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 \<Rightarrow> 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, hide_lams) 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 \<le> 2"
- shows "collinear s"
-proof -
- have "card s = 0 \<or> card s = 1 \<or> card s = 2"
- using assms by linarith
- then show ?thesis using assms
- using card_eq_SucD
- by auto (metis collinear_2 numeral_2_eq_2)
-qed
-
-lemma coplanar_small:
- assumes "finite s" "card s \<le> 3"
- shows "coplanar s"
-proof -
- have "card s \<le> 2 \<or> card s = Suc (Suc (Suc 0))"
- using assms by linarith
- then show ?thesis using assms
- apply safe
- apply (simp add: collinear_small collinear_imp_coplanar)
- apply (safe dest!: card_eq_SucD)
- apply (auto simp: coplanar_def)
- apply (metis hull_subset insert_subset)
- done
-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) \<longleftrightarrow> 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) \<longleftrightarrow> 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 \<Rightarrow> 'b::real_normed_vector"
- assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
-proof -
- { fix u v w
- assume "s \<subseteq> affine hull {u, v, w}"
- then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
- by (simp add: image_mono)
- then have "f ` s \<subseteq> 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: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
- unfolding coplanar_def
- apply clarify
- apply (rule_tac x="u+a" in exI)
- apply (rule_tac x="v+a" in exI)
- apply (rule_tac x="w+a" in exI)
- using affine_hull_translation [of a "{u,v,w}" for u v w]
- apply (force simp: add.commute)
- done
-
-lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
- by (metis (no_types) coplanar_translation_imp translation_galois)
-
-lemma coplanar_linear_image_eq:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
-proof
- assume "coplanar s"
- then show "coplanar (f ` s)"
- unfolding coplanar_def
- using affine_hull_linear_image [of f "{u,v,w}" for u v w] assms
- by (meson coplanar_def coplanar_linear_image)
-next
- obtain g where g: "linear g" "g \<circ> f = id"
- using linear_injective_left_inverse [OF assms]
- by blast
- assume "coplanar (f ` s)"
- then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
- by (auto simp: coplanar_def)
- then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
- by blast
- then have "s \<subseteq> g ` (affine hull {u, v, w})"
- using g by (simp add: Fun.image_comp)
- then show "coplanar s"
- unfolding coplanar_def
- using affine_hull_linear_image [of g "{u,v,w}" for u v w] \<open>linear g\<close> linear_conv_bounded_linear
- by fastforce
-qed
-(*The HOL Light proof is simply
- MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
-*)
-
-lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
- by (meson coplanar_def order_trans)
-
-lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
- by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
-
-lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> affine hull {a,b}"
- unfolding collinear_def
- apply clarify
- apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
- apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
- apply (rename_tac y x)
- apply (simp add: affine_hull_2)
- apply (rule_tac x="1 - x/y" in exI)
- apply (simp add: algebra_simps)
- done
-
-lemma collinear_3_affine_hull:
- assumes "a \<noteq> b"
- shows "collinear {a,b,c} \<longleftrightarrow> c \<in> 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} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
-apply (case_tac "a=b", simp)
-apply (case_tac "a=c")
-apply (simp add: insert_commute)
-apply (case_tac "b=c")
-apply (simp add: insert_commute)
-apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
-apply (metis collinear_3_affine_hull insert_commute)+
-done
-
-lemma affine_dependent_imp_collinear_3:
- "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
-by (simp add: collinear_3_eq_affine_dependent)
-
-lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
- by (auto simp add: collinear_def)
-
-lemma collinear_3_expand:
- "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>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 "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
- by (simp add: collinear_lemma)
- also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
- by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
- finally show ?thesis .
-qed
-
-lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
-proof
- assume "collinear S"
- then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
- by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
- then show "aff_dim S \<le> 1"
- using order_trans by fastforce
-next
- assume "aff_dim S \<le> 1"
- then have le1: "aff_dim (affine hull S) \<le> 1"
- by simp
- obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
- using affine_basis_exists [of S] by auto
- then have "finite B" "card B \<le> 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 \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
-qed
-
-lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
- apply (auto simp: collinear_3 collinear_lemma)
- apply (drule_tac x="-1" in spec)
- apply (simp add: algebra_simps)
- done
-
-lemma midpoint_collinear:
- fixes a b c :: "'a::real_normed_vector"
- assumes "a \<noteq> c"
- shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
-proof -
- have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
- "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
- "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
- by (auto simp: algebra_simps)
- have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
- using collinear_midpoint by blast
- moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
- apply (auto simp: collinear_3_expand assms dist_midpoint)
- apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
- apply (simp add: algebra_simps)
- done
- 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 \<or> x = b \<or> a = b")
- case True with assms show ?thesis
- by (auto simp: dist_commute)
-next
- case False with assms show ?thesis
- apply (auto simp: collinear_3 collinear_lemma between_norm)
- apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
- apply (simp add: vector_add_divide_simps eq_vector_fraction_iff real_vector.scale_minus_right [symmetric])
- done
-qed
-
-lemma midpoint_between:
- fixes a b :: "'a::euclidean_space"
- shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
-proof (cases "a = c")
- case True then show ?thesis
- by (auto simp: dist_commute)
-next
- case False
- show ?thesis
- apply (rule iffI)
- apply (simp add: between_midpoint(1) dist_midpoint)
- using False between_imp_collinear midpoint_collinear by blast
-qed
-
-lemma collinear_triples:
- assumes "a \<noteq> b"
- shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
- (is "?lhs = ?rhs")
-proof safe
- fix x
- assume ?lhs and "x \<in> S"
- then show "collinear {a, b, x}"
- using collinear_subset by force
-next
- assume ?rhs
- then have "\<forall>x \<in> S. collinear{a,x,b}"
- by (simp add: insert_commute)
- then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
- using that assms collinear_3_expand by fastforce+
- show ?lhs
- unfolding collinear_def
- apply (rule_tac x="b-a" in exI)
- apply (clarify dest!: *)
- by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
-qed
-
-lemma collinear_4_3:
- assumes "a \<noteq> b"
- shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> 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 \<noteq> 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_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
- using affine_hull_nonempty by blast
-
-lemma affine_hull_2_alt:
- fixes a b :: "'a::real_vector"
- shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
-apply (simp add: affine_hull_2, safe)
-apply (rule_tac x=v in image_eqI)
-apply (simp add: algebra_simps)
-apply (metis scaleR_add_left scaleR_one, simp)
-apply (rule_tac x="1-u" in exI)
-apply (simp add: algebra_simps)
-done
-
-lemma interior_convex_hull_3_minimal:
- fixes a :: "'a::euclidean_space"
- shows "\<lbrakk>~ collinear{a,b,c}; DIM('a) = 2\<rbrakk>
- \<Longrightarrow> interior(convex hull {a,b,c}) =
- {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
- x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
-apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
-apply (rule_tac x="u a" in exI, simp)
-apply (rule_tac x="u b" in exI, simp)
-apply (rule_tac x="u c" in exI, simp)
-apply (rename_tac uu x y z)
-apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
-apply simp
-done
-
-subsection\<open>The infimum of the distance between two sets\<close>
-
-definition setdist :: "'a::metric_space set \<Rightarrow> 'a set \<Rightarrow> real" where
- "setdist s t \<equiv>
- (if s = {} \<or> t = {} then 0
- else Inf {dist x y| x y. x \<in> s \<and> y \<in> t})"
-
-lemma setdist_empty1 [simp]: "setdist {} t = 0"
- by (simp add: setdist_def)
-
-lemma setdist_empty2 [simp]: "setdist t {} = 0"
- by (simp add: setdist_def)
-
-lemma setdist_pos_le [simp]: "0 \<le> setdist s t"
- by (auto simp: setdist_def ex_in_conv [symmetric] intro: cInf_greatest)
-
-lemma le_setdistI:
- assumes "s \<noteq> {}" "t \<noteq> {}" "\<And>x y. \<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> d \<le> dist x y"
- shows "d \<le> setdist s t"
- using assms
- by (auto simp: setdist_def Set.ex_in_conv [symmetric] intro: cInf_greatest)
-
-lemma setdist_le_dist: "\<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> setdist s t \<le> dist x y"
- unfolding setdist_def
- by (auto intro!: bdd_belowI [where m=0] cInf_lower)
-
-lemma le_setdist_iff:
- "d \<le> setdist s t \<longleftrightarrow>
- (\<forall>x \<in> s. \<forall>y \<in> t. d \<le> dist x y) \<and> (s = {} \<or> t = {} \<longrightarrow> d \<le> 0)"
- apply (cases "s = {} \<or> t = {}")
- apply (force simp add: setdist_def)
- apply (intro iffI conjI)
- using setdist_le_dist apply fastforce
- apply (auto simp: intro: le_setdistI)
- done
-
-lemma setdist_ltE:
- assumes "setdist s t < b" "s \<noteq> {}" "t \<noteq> {}"
- obtains x y where "x \<in> s" "y \<in> t" "dist x y < b"
-using assms
-by (auto simp: not_le [symmetric] le_setdist_iff)
-
-lemma setdist_refl: "setdist s s = 0"
- apply (cases "s = {}")
- apply (force simp add: setdist_def)
- apply (rule antisym [OF _ setdist_pos_le])
- apply (metis all_not_in_conv dist_self setdist_le_dist)
- done
-
-lemma setdist_sym: "setdist s t = setdist t s"
- by (force simp: setdist_def dist_commute intro!: arg_cong [where f=Inf])
-
-lemma setdist_triangle: "setdist s t \<le> setdist s {a} + setdist {a} t"
-proof (cases "s = {} \<or> t = {}")
- case True then show ?thesis
- using setdist_pos_le by fastforce
-next
- case False
- have "\<And>x. x \<in> s \<Longrightarrow> setdist s t - dist x a \<le> setdist {a} t"
- apply (rule le_setdistI, blast)
- using False apply (fastforce intro: le_setdistI)
- apply (simp add: algebra_simps)
- apply (metis dist_commute dist_triangle3 order_trans [OF setdist_le_dist])
- done
- then have "setdist s t - setdist {a} t \<le> setdist s {a}"
- using False by (fastforce intro: le_setdistI)
- then show ?thesis
- by (simp add: algebra_simps)
-qed
-
-lemma setdist_singletons [simp]: "setdist {x} {y} = dist x y"
- by (simp add: setdist_def)
-
-lemma setdist_Lipschitz: "\<bar>setdist {x} s - setdist {y} s\<bar> \<le> dist x y"
- apply (subst setdist_singletons [symmetric])
- by (metis abs_diff_le_iff diff_le_eq setdist_triangle setdist_sym)
-
-lemma continuous_at_setdist [continuous_intros]: "continuous (at x) (\<lambda>y. (setdist {y} s))"
- by (force simp: continuous_at_eps_delta dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
-
-lemma continuous_on_setdist [continuous_intros]: "continuous_on t (\<lambda>y. (setdist {y} s))"
- by (metis continuous_at_setdist continuous_at_imp_continuous_on)
-
-lemma uniformly_continuous_on_setdist: "uniformly_continuous_on t (\<lambda>y. (setdist {y} s))"
- by (force simp: uniformly_continuous_on_def dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
-
-lemma setdist_subset_right: "\<lbrakk>t \<noteq> {}; t \<subseteq> u\<rbrakk> \<Longrightarrow> setdist s u \<le> setdist s t"
- apply (cases "s = {} \<or> u = {}", force)
- apply (auto simp: setdist_def intro!: bdd_belowI [where m=0] cInf_superset_mono)
- done
-
-lemma setdist_subset_left: "\<lbrakk>s \<noteq> {}; s \<subseteq> t\<rbrakk> \<Longrightarrow> setdist t u \<le> setdist s u"
- by (metis setdist_subset_right setdist_sym)
-
-lemma setdist_closure_1 [simp]: "setdist (closure s) t = setdist s t"
-proof (cases "s = {} \<or> t = {}")
- case True then show ?thesis by force
-next
- case False
- { fix y
- assume "y \<in> t"
- have "continuous_on (closure s) (\<lambda>a. dist a y)"
- by (auto simp: continuous_intros dist_norm)
- then have *: "\<And>x. x \<in> closure s \<Longrightarrow> setdist s t \<le> dist x y"
- apply (rule continuous_ge_on_closure)
- apply assumption
- apply (blast intro: setdist_le_dist \<open>y \<in> t\<close> )
- done
- } note * = this
- show ?thesis
- apply (rule antisym)
- using False closure_subset apply (blast intro: setdist_subset_left)
- using False *
- apply (force simp add: closure_eq_empty intro!: le_setdistI)
- done
-qed
-
-lemma setdist_closure_2 [simp]: "setdist t (closure s) = setdist t s"
-by (metis setdist_closure_1 setdist_sym)
-
-lemma setdist_compact_closed:
- fixes S :: "'a::euclidean_space set"
- assumes S: "compact S" and T: "closed T"
- and "S \<noteq> {}" "T \<noteq> {}"
- shows "\<exists>x \<in> S. \<exists>y \<in> T. dist x y = setdist S T"
-proof -
- have "(\<Union>x\<in> S. \<Union>y \<in> T. {x - y}) \<noteq> {}"
- using assms by blast
- then have "\<exists>x \<in> S. \<exists>y \<in> T. dist x y \<le> setdist S T"
- apply (rule distance_attains_inf [where a=0, OF compact_closed_differences [OF S T]])
- apply (simp add: dist_norm le_setdist_iff)
- apply blast
- done
- then show ?thesis
- by (blast intro!: antisym [OF _ setdist_le_dist] )
-qed
-
-lemma setdist_closed_compact:
- fixes S :: "'a::euclidean_space set"
- assumes S: "closed S" and T: "compact T"
- and "S \<noteq> {}" "T \<noteq> {}"
- shows "\<exists>x \<in> S. \<exists>y \<in> T. dist x y = setdist S T"
- using setdist_compact_closed [OF T S \<open>T \<noteq> {}\<close> \<open>S \<noteq> {}\<close>]
- by (metis dist_commute setdist_sym)
-
-lemma setdist_eq_0I: "\<lbrakk>x \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> setdist S T = 0"
- by (metis antisym dist_self setdist_le_dist setdist_pos_le)
-
-lemma setdist_eq_0_compact_closed:
- fixes S :: "'a::euclidean_space set"
- assumes S: "compact S" and T: "closed T"
- shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
- apply (cases "S = {} \<or> T = {}", force)
- using setdist_compact_closed [OF S T]
- apply (force intro: setdist_eq_0I )
- done
-
-corollary setdist_gt_0_compact_closed:
- fixes S :: "'a::euclidean_space set"
- assumes S: "compact S" and T: "closed T"
- shows "setdist S T > 0 \<longleftrightarrow> (S \<noteq> {} \<and> T \<noteq> {} \<and> S \<inter> T = {})"
- using setdist_pos_le [of S T] setdist_eq_0_compact_closed [OF assms]
- by linarith
-
-lemma setdist_eq_0_closed_compact:
- fixes S :: "'a::euclidean_space set"
- assumes S: "closed S" and T: "compact T"
- shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
- using setdist_eq_0_compact_closed [OF T S]
- by (metis Int_commute setdist_sym)
-
-lemma setdist_eq_0_bounded:
- fixes S :: "'a::euclidean_space set"
- assumes "bounded S \<or> bounded T"
- shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> closure S \<inter> closure T \<noteq> {}"
- apply (cases "S = {} \<or> T = {}", force)
- using setdist_eq_0_compact_closed [of "closure S" "closure T"]
- setdist_eq_0_closed_compact [of "closure S" "closure T"] assms
- apply (force simp add: bounded_closure compact_eq_bounded_closed)
- done
-
-lemma setdist_unique:
- "\<lbrakk>a \<in> S; b \<in> T; \<And>x y. x \<in> S \<and> y \<in> T ==> dist a b \<le> dist x y\<rbrakk>
- \<Longrightarrow> setdist S T = dist a b"
- by (force simp add: setdist_le_dist le_setdist_iff intro: antisym)
-
-lemma setdist_closest_point:
- "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> setdist {a} S = dist a (closest_point S a)"
- apply (rule setdist_unique)
- using closest_point_le
- apply (auto simp: closest_point_in_set)
- done
-
-lemma setdist_eq_0_sing_1:
- fixes S :: "'a::euclidean_space set"
- shows "setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> closure S"
- by (auto simp: setdist_eq_0_bounded)
-
-lemma setdist_eq_0_sing_2:
- fixes S :: "'a::euclidean_space set"
- shows "setdist S {x} = 0 \<longleftrightarrow> S = {} \<or> x \<in> closure S"
- by (auto simp: setdist_eq_0_bounded)
-
-lemma setdist_neq_0_sing_1:
- fixes S :: "'a::euclidean_space set"
- shows "\<lbrakk>setdist {x} S = a; a \<noteq> 0\<rbrakk> \<Longrightarrow> S \<noteq> {} \<and> x \<notin> closure S"
- by (auto simp: setdist_eq_0_sing_1)
-
-lemma setdist_neq_0_sing_2:
- fixes S :: "'a::euclidean_space set"
- shows "\<lbrakk>setdist S {x} = a; a \<noteq> 0\<rbrakk> \<Longrightarrow> S \<noteq> {} \<and> x \<notin> closure S"
- by (auto simp: setdist_eq_0_sing_2)
-
-lemma setdist_sing_in_set:
- fixes S :: "'a::euclidean_space set"
- shows "x \<in> S \<Longrightarrow> setdist {x} S = 0"
- using closure_subset by (auto simp: setdist_eq_0_sing_1)
-
-lemma setdist_le_sing: "x \<in> S ==> setdist S T \<le> setdist {x} T"
- using setdist_subset_left by auto
-
-lemma setdist_eq_0_closed:
- fixes S :: "'a::euclidean_space set"
- shows "closed S \<Longrightarrow> (setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> S)"
-by (simp add: setdist_eq_0_sing_1)
-
-lemma setdist_eq_0_closedin:
- fixes S :: "'a::euclidean_space set"
- shows "\<lbrakk>closedin (subtopology euclidean u) S; x \<in> u\<rbrakk>
- \<Longrightarrow> (setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> S)"
- by (auto simp: closedin_limpt setdist_eq_0_sing_1 closure_def)
-
-subsection\<open>Basic lemmas about hyperplanes and halfspaces\<close>
-
-lemma hyperplane_eq_Ex:
- assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
- by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
-
-lemma hyperplane_eq_empty:
- "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
- using hyperplane_eq_Ex apply auto[1]
- using inner_zero_right by blast
-
-lemma hyperplane_eq_UNIV:
- "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
-proof -
- have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
- apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
- apply simp_all
- by (metis add_cancel_right_right zero_neq_one)
- then show ?thesis by force
-qed
-
-lemma halfspace_eq_empty_lt:
- "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
-proof -
- have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
- apply (rule ccontr)
- apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
- apply force+
- done
- then show ?thesis by force
-qed
-
-lemma halfspace_eq_empty_gt:
- "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
-using halfspace_eq_empty_lt [of "-a" "-b"]
-by simp
-
-lemma halfspace_eq_empty_le:
- "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
-proof -
- have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
- apply (rule ccontr)
- apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
- apply force+
- done
- then show ?thesis by force
-qed
-
-lemma halfspace_eq_empty_ge:
- "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
-using halfspace_eq_empty_le [of "-a" "-b"]
-by simp
-
-subsection\<open>Use set distance for an easy proof of separation properties\<close>
-
-proposition separation_closures:
- fixes S :: "'a::euclidean_space set"
- assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
- obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
-proof (cases "S = {} \<or> T = {}")
- case True with that show ?thesis by auto
-next
- case False
- define f where "f \<equiv> \<lambda>x. 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} \<inter> {x. f x < 0} = {}"
- by auto
- show "open {x. 0 < f x}"
- by (simp add: open_Collect_less contf continuous_on_const)
- show "open {x. f x < 0}"
- by (simp add: open_Collect_less contf continuous_on_const)
- show "S \<subseteq> {x. 0 < f x}"
- apply (clarsimp simp add: f_def setdist_sing_in_set)
- using assms
- by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
- show "T \<subseteq> {x. f x < 0}"
- apply (clarsimp simp add: f_def setdist_sing_in_set)
- using assms
- by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
- qed
-qed
-
-lemma separation_normal:
- fixes S :: "'a::euclidean_space set"
- assumes "closed S" "closed T" "S \<inter> T = {}"
- obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 (subtopology euclidean U) S"
- and UT: "closedin (subtopology euclidean U) T"
- and "S \<inter> T = {}"
- obtains S' T' where "openin (subtopology euclidean U) S'"
- "openin (subtopology euclidean U) T'"
- "S \<subseteq> S'" "T \<subseteq> T'" "S' \<inter> T' = {}"
-proof (cases "S = {} \<or> T = {}")
- case True with that show ?thesis
- apply safe
- using UT closedin_subset apply blast
- using US closedin_subset apply blast
- done
-next
- case False
- define f where "f \<equiv> \<lambda>x. 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' = "{x\<in>U. f x > 0}" and T' = "{x\<in>U. f x < 0}" in that)
- show "{x \<in> U. 0 < f x} \<inter> {x \<in> U. f x < 0} = {}"
- by auto
- have "openin (subtopology euclidean U) {x \<in> U. f x \<in> {0<..}}"
- by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
- then show "openin (subtopology euclidean U) {x \<in> U. 0 < f x}" by simp
- next
- have "openin (subtopology euclidean U) {x \<in> U. f x \<in> {..<0}}"
- by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
- then show "openin (subtopology euclidean U) {x \<in> U. f x < 0}" by simp
- next
- show "S \<subseteq> {x \<in> U. 0 < f x}"
- apply (clarsimp simp add: f_def setdist_sing_in_set)
- using assms
- by (metis False Int_insert_right closedin_imp_subset empty_not_insert insert_absorb insert_subset linorder_neqE_linordered_idom not_le setdist_eq_0_closedin setdist_pos_le)
- show "T \<subseteq> {x \<in> U. f x < 0}"
- apply (clarsimp simp add: f_def setdist_sing_in_set)
- using assms
- by (metis False closedin_subset disjoint_iff_not_equal insert_absorb insert_subset linorder_neqE_linordered_idom not_less setdist_eq_0_closedin setdist_pos_le topspace_euclidean_subtopology)
- qed
-qed
-
-lemma separation_normal_compact:
- fixes S :: "'a::euclidean_space set"
- assumes "compact S" "closed T" "S \<inter> T = {}"
- obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 \<subseteq> ball 0 r"
- by (auto dest!: bounded_subset_ballD)
- have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
- using assms r by blast+
- then show ?thesis
- apply (rule separation_normal [OF \<open>closed S\<close>])
- apply (rule_tac U=U and V=V in that)
- by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
-qed
-
-subsection\<open>Proper maps, including projections out of compact sets\<close>
-
-lemma finite_indexed_bound:
- assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
- shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
-using A
-proof (induction A)
- case empty then show ?case by force
-next
- case (insert a A)
- then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
- by force
- then show ?case
- apply (rule_tac x="max m n" in exI, safe)
- using max.cobounded2 apply blast
- by (meson le_max_iff_disj)
-qed
-
-proposition proper_map:
- fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
- assumes "closedin (subtopology euclidean S) K"
- and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact {x \<in> S. f x \<in> U}"
- and "f ` S \<subseteq> T"
- shows "closedin (subtopology euclidean T) (f ` K)"
-proof -
- have "K \<subseteq> S"
- using assms closedin_imp_subset by metis
- obtain C where "closed C" and Keq: "K = S \<inter> C"
- using assms by (auto simp: closedin_closed)
- have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
- proof -
- obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
- using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
- then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
- by metis
- then have fX: "\<And>n. f (X n) = h n"
- by metis
- have "compact (C \<inter> {a \<in> S. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))})" for n
- apply (rule closed_Int_compact [OF \<open>closed C\<close>])
- apply (rule com)
- using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> apply blast
- apply (rule compact_sequence_with_limit)
- apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
- done
- then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
- by (simp add: Keq Int_def conj_commute)
- have ne: "\<Inter>\<F> \<noteq> {}"
- if "finite \<F>"
- and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
- (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
- for \<F>
- proof -
- obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
- apply (rule exE)
- apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
- done
- have "X m \<in> \<Inter>\<F>"
- using X le_Suc_ex by (fastforce dest: m)
- then show ?thesis by blast
- qed
- have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
- \<noteq> {}"
- apply (rule compact_fip_heine_borel)
- using comf apply force
- using ne apply (simp add: subset_iff del: insert_iff)
- done
- then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
- by blast
- then show ?thesis
- apply (simp add: image_iff fX)
- by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
- qed
- with assms closedin_subset show ?thesis
- by (force simp: closedin_limpt)
-qed
-
-
-lemma compact_continuous_image_eq:
- fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
- assumes f: "inj_on f S"
- shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> 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: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
- by (metis inv_into_f_f f)
- then have *: "{x \<in> S. f x \<in> U} = g ` U" if "U \<subseteq> f ` S" for U
- using that by fastforce
- have gfim: "g ` f ` S \<subseteq> S" using gf by auto
- have **: "compact {x \<in> f ` S. g x \<in> C}" if C: "C \<subseteq> S" "compact C" for C
- proof -
- obtain h :: "'a set \<Rightarrow> 'a" where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
- by (force simp: C RHS)
- moreover have "f ` C = {b \<in> f ` S. g b \<in> C}"
- using C gf by auto
- ultimately show "compact {b \<in> f ` S. g b \<in> C}"
- using C by auto
- qed
- show ?lhs
- using proper_map [OF _ _ gfim] **
- by (simp add: continuous_on_closed * closedin_imp_subset)
-qed
-
-subsection\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
-
-lemma convex_connected_collinear:
- fixes S :: "'a::euclidean_space set"
- assumes "collinear S"
- shows "convex S \<longleftrightarrow> 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 \<in> S" by auto
- have "collinear (affine hull S)"
- by (simp add: assms collinear_affine_hull_collinear)
- then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
- by (meson \<open>a \<in> S\<close> collinear hull_inc)
- then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
- proof -
- have "f x *\<^sub>R z = x - a"
- by (simp add: f hull_inc x)
- moreover have "f y *\<^sub>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"
- apply (clarsimp simp: dist_norm continuous_on_iff diff)
- by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
- 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 \<in> S" "y \<in> S" "z \<in> closed_segment x y"
- have False if "z \<notin> S"
- proof -
- have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
- apply (rule continuous_injective_image_segment_1)
- apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
- by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
- then have fz: "f z \<in> closed_segment (f x) (f y)"
- using \<open>z \<in> closed_segment x y\<close> by blast
- have "z \<in> affine hull S"
- by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
- then have fz_notin: "f z \<notin> f ` S"
- using hull_subset inj_f inj_onD that by fastforce
- moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
- proof -
- have "{..<f z} \<inter> f ` {x,y} \<noteq> {}" "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
- using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
- apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
- apply (metis image_eqI less_eq_real_def)+
- done
- then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
- using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
- qed
- ultimately show False
- using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
- qed
- then show "z \<in> S" by meson
- qed
- qed
-qed
-
-lemma compact_convex_collinear_segment_alt:
- fixes S :: "'a::euclidean_space set"
- assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
- obtains a b where "S = closed_segment a b"
-proof -
- obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
- have "collinear (affine hull S)"
- by (simp add: assms collinear_affine_hull_collinear)
- then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
- by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
- then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
- by metis
- let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
- have gf: "?g (f x) = x" if "x \<in> 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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
- proof -
- have "f x *\<^sub>R z = x - \<xi>"
- by (simp add: f hull_inc x)
- moreover have "f y *\<^sub>R z = y - \<xi>"
- 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"
- apply (clarsimp simp: dist_norm continuous_on_iff diff)
- by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
- then have "connected (f ` S)"
- by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
- moreover have "compact (f ` S)"
- by (meson \<open>compact S\<close> 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 \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
- obtain a b where "a \<in> S" "f a = x" "b \<in> 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)"
- apply (rule continuous_injective_image_segment_1)
- apply (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
- by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
- then have "f ` (closed_segment a b) = f ` S"
- by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
- then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
- by simp
- moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
- apply safe
- apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
- by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff 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 \<noteq> {}" "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 \<Rightarrow> 'b::euclidean_space"
- assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
- "closedin (subtopology euclidean T) K"
- shows "compact {x. x \<in> S \<and> f x \<in> K}"
-by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
-
-lemma proper_map_fst:
- assumes "compact T" "K \<subseteq> S" "compact K"
- shows "compact {z \<in> S \<times> T. fst z \<in> K}"
-proof -
- have "{z \<in> S \<times> T. fst z \<in> K} = K \<times> 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 (subtopology euclidean (S \<times> T)) c"
- shows "closedin (subtopology euclidean S) (fst ` c)"
-proof -
- have *: "fst ` (S \<times> T) \<subseteq> 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 \<subseteq> T" "compact K"
- shows "compact {z \<in> S \<times> T. snd z \<in> K}"
-proof -
- have "{z \<in> S \<times> T. snd z \<in> K} = S \<times> 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 (subtopology euclidean (S \<times> T)) c"
- shows "closedin (subtopology euclidean T) (snd ` c)"
-proof -
- have *: "snd ` (S \<times> T) \<subseteq> 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 (subtopology euclidean (S \<times> T)) U"
- shows "closedin (subtopology euclidean T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
-proof -
- have "U \<subseteq> S \<times> T"
- by (metis clo closedin_imp_subset)
- then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
- by force
- moreover have "closedin (subtopology euclidean 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. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
-proof -
- have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
- {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
- by auto
- show ?thesis
- apply (subst *)
- apply (rule closedin_closed_trans [OF _ closed_UNIV])
- apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
- by (simp add: clo closedin_closed_Int)
-qed
-
-subsubsection\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
-
-proposition affine_hull_convex_Int_nonempty_interior:
- fixes S :: "'a::real_normed_vector set"
- assumes "convex S" "S \<inter> interior T \<noteq> {}"
- shows "affine hull (S \<inter> T) = affine hull S"
-proof
- show "affine hull (S \<inter> T) \<subseteq> affine hull S"
- by (simp add: hull_mono)
-next
- obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
- using assms interior_subset by blast
- then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
- using mem_interior_cball by blast
- have *: "x \<in> op + a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> 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 \<open>e > 0\<close> False by (auto simp: k_def)
- then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
- by simp
- have "e / norm (x - a) \<ge> k"
- using k_def by linarith
- then have "a + k *\<^sub>R (x - a) \<in> cball a e"
- using \<open>0 < k\<close> False by (simp add: dist_norm field_simps)
- then have T: "a + k *\<^sub>R (x - a) \<in> T"
- using e by blast
- have S: "a + k *\<^sub>R (x - a) \<in> S"
- using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
- by (simp add: algebra_simps)
- have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
- apply (rule span_mul)
- apply (rule span_superset)
- apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
- apply (auto simp: S T)
- done
- with xa image_iff show ?thesis by fastforce
- qed
- show "affine hull S \<subseteq> affine hull (S \<inter> T)"
- apply (simp add: subset_hull)
- apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
- apply (force simp: *)
- done
-qed
-
-corollary affine_hull_convex_Int_open:
- fixes S :: "'a::real_normed_vector set"
- assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
- shows "affine hull (S \<inter> 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 \<inter> interior T \<noteq> {}"
- shows "affine hull (S \<inter> 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 \<inter> T \<noteq> {}"
- shows "affine hull (S \<inter> 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 (subtopology euclidean (affine hull S)) T" "S \<inter> T \<noteq> {}"
- shows "affine hull (S \<inter> 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 (subtopology euclidean (affine hull T)) S" "S \<noteq> {}"
- 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 \<noteq> {}"
- 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 "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 (subtopology euclidean (affine hull S)) S" and "finite F" "F \<subset> S"
- shows "affine hull (S - F) = affine hull S"
-proof -
- have clo: "closedin (subtopology euclidean S) F"
- using assms finite_imp_closedin by auto
- moreover have "S - F \<noteq> {}"
- 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 \<bullet> x < r} = (if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} = (if a = 0 \<and> 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 \<bullet> x < r} = UNIV"
- using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
- moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> 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 \<bullet> x > r} = (if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} = (if a = 0 \<and> 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 \<bullet> x < r} =
- (if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} =
- (if a = 0 \<and> r < 0 then -1 else DIM('a))"
-proof -
- have "int (DIM('a)) = aff_dim (UNIV::'a set)"
- by (simp add: aff_dim_UNIV)
- then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
- using that by (simp add: affine_hull_halfspace_le not_less)
- then show ?thesis
- by (force simp: aff_dim_affine_hull)
-qed
-
-lemma aff_dim_halfspace_gt:
- fixes a :: "'a::euclidean_space"
- shows "aff_dim {x. a \<bullet> x > r} =
- (if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} =
- (if a = 0 \<and> r > 0 then -1 else DIM('a))"
-using aff_dim_halfspace_le [of "-a" "-r"] by simp
-
-subsection\<open>Properties of special hyperplanes\<close>
-
-lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
- by (simp add: subspace_def inner_right_distrib)
-
-lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
- by (simp add: inner_commute inner_right_distrib subspace_def)
-
-lemma special_hyperplane_span:
- fixes S :: "'n::euclidean_space set"
- assumes "k \<in> Basis"
- shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
-proof -
- have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
- proof -
- have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
- by (simp add: euclidean_representation)
- also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
- by (auto simp: sum.remove [of _ k] inner_commute assms that)
- finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
- then show ?thesis
- by (simp add: Linear_Algebra.span_finite) metis
- qed
- show ?thesis
- apply (rule span_subspace [symmetric])
- using assms
- apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
- done
-qed
-
-lemma dim_special_hyperplane:
- fixes k :: "'n::euclidean_space"
- shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
-apply (simp add: special_hyperplane_span)
-apply (rule Linear_Algebra.dim_unique [OF subset_refl])
-apply (auto simp: Diff_subset independent_substdbasis)
-apply (metis member_remove remove_def span_clauses(1))
-done
-
-proposition dim_hyperplane:
- fixes a :: "'a::euclidean_space"
- assumes "a \<noteq> 0"
- shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
-proof -
- have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
- by (rule span_unique) (auto simp: subspace_hyperplane)
- then obtain B where "independent B"
- and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
- and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
- and card0: "(card B = dim {x. a \<bullet> x = 0})"
- and ortho: "pairwise orthogonal B"
- using orthogonal_basis_exists by metis
- with assms have "a \<notin> span B"
- by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0 span_subspace)
- then have ind: "independent (insert a B)"
- by (simp add: \<open>independent B\<close> independent_insert)
- have "finite B"
- using \<open>independent B\<close> independent_bound by blast
- have "UNIV \<subseteq> span (insert a B)"
- proof fix y::'a
- obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
- apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
- using assms
- by (auto simp: algebra_simps)
- show "y \<in> span (insert a B)"
- by (metis (mono_tags, lifting) z Bsub Convex_Euclidean_Space.span_eq
- add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
- qed
- then have dima: "DIM('a) = dim(insert a B)"
- by (metis antisym dim_UNIV dim_subset_UNIV subset_le_dim)
- then show ?thesis
- by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0 card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE subspB)
-qed
-
-lemma lowdim_eq_hyperplane:
- fixes S :: "'a::euclidean_space set"
- assumes "dim S = DIM('a) - 1"
- obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
-proof -
- have [simp]: "dim S < DIM('a)"
- by (simp add: DIM_positive assms)
- then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
- using lowdim_subset_hyperplane [of S] by fastforce
- show ?thesis
- using b that real_vector_class.subspace_span [of S]
- by (simp add: assms dim_hyperplane subspace_dim_equal subspace_hyperplane)
-qed
-
-lemma dim_eq_hyperplane:
- fixes S :: "'n::euclidean_space set"
- shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
-by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
-
-proposition aff_dim_eq_hyperplane:
- fixes S :: "'a::euclidean_space set"
- shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
-proof (cases "S = {}")
- case True then show ?thesis
- by (auto simp: dest: hyperplane_eq_Ex)
-next
- case False
- then obtain c where "c \<in> S" by blast
- show ?thesis
- proof (cases "c = 0")
- case True show ?thesis
- apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
- del: One_nat_def)
- apply (rule ex_cong)
- apply (metis (mono_tags) span_0 \<open>c = 0\<close> image_add_0 inner_zero_right mem_Collect_eq)
- done
- next
- case False
- have xc_im: "x \<in> op + c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
- proof -
- have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
- by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
- then show "x \<in> op + c ` {y. a \<bullet> y = 0}"
- by blast
- qed
- have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
- if "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
- proof -
- have "b = a \<bullet> c"
- using span_0 that by fastforce
- with that have "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
- by simp
- then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
- by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
- also have "... = {x. a \<bullet> x = 0}"
- by (force simp: inner_distrib inner_diff_right
- intro: image_eqI [where x="x+c" for x])
- finally show ?thesis .
- qed
- show ?thesis
- apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
- del: One_nat_def, safe)
- apply (fastforce simp add: inner_distrib intro: xc_im)
- apply (force simp: intro!: 2)
- done
- qed
-qed
-
-corollary aff_dim_hyperplane [simp]:
- fixes a :: "'a::euclidean_space"
- shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
-by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
-
-subsection\<open>Some stepping theorems\<close>
-
-lemma dim_empty [simp]: "dim ({} :: 'a::euclidean_space set) = 0"
- by (force intro!: dim_unique)
-
-lemma dim_insert:
- fixes x :: "'a::euclidean_space"
- shows "dim (insert x S) = (if x \<in> span S then dim S else dim S + 1)"
-proof -
- show ?thesis
- proof (cases "x \<in> span S")
- case True then show ?thesis
- by (metis dim_span span_redundant)
- next
- case False
- obtain B where B: "B \<subseteq> span S" "independent B" "span S \<subseteq> span B" "card B = dim (span S)"
- using basis_exists [of "span S"] by blast
- have 1: "insert x B \<subseteq> span (insert x S)"
- by (meson \<open>B \<subseteq> span S\<close> dual_order.trans insertI1 insert_subsetI span_mono span_superset subset_insertI)
- have 2: "span (insert x S) \<subseteq> span (insert x B)"
- by (metis \<open>B \<subseteq> span S\<close> \<open>span S \<subseteq> span B\<close> span_breakdown_eq span_subspace subsetI subspace_span)
- have 3: "independent (insert x B)"
- by (metis B independent_insert span_subspace subspace_span False)
- have "dim (span (insert x S)) = Suc (dim S)"
- apply (rule dim_unique [OF 1 2 3])
- by (metis B False card_insert_disjoint dim_span independent_imp_finite subsetCE)
- then show ?thesis
- by (simp add: False)
- qed
-qed
-
-lemma dim_singleton [simp]:
- fixes x :: "'a::euclidean_space"
- shows "dim{x} = (if x = 0 then 0 else 1)"
-by (simp add: dim_insert)
-
-lemma dim_eq_0 [simp]:
- fixes S :: "'a::euclidean_space set"
- shows "dim S = 0 \<longleftrightarrow> S \<subseteq> {0}"
-apply safe
-apply (metis DIM_positive DIM_real card_ge_dim_independent contra_subsetD dim_empty dim_insert dim_singleton empty_subsetI independent_empty less_not_refl zero_le)
-by (metis dim_singleton dim_subset le_0_eq)
-
-lemma aff_dim_insert:
- fixes a :: "'a::euclidean_space"
- shows "aff_dim (insert a S) = (if a \<in> 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 \<notin> s'"
- by (meson Set.set_insert all_not_in_conv)
- show ?thesis using S
- apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
- apply (simp add: affine_hull_insert_span_gen hull_inc)
- apply (simp add: insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert span_0)
- apply (metis (no_types, lifting) add_minus_cancel image_iff uminus_add_conv_diff)
- done
-qed
-
-lemma subspace_bounded_eq_trivial:
- fixes S :: "'a::real_normed_vector set"
- assumes "subspace S"
- shows "bounded S \<longleftrightarrow> S = {0}"
-proof -
- have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
- proof -
- obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
- using \<open>bounded S\<close> by (force simp: bounded_pos_less)
- have "(B / norm x) *\<^sub>R x \<in> S"
- using assms subspace_mul \<open>x \<in> S\<close> by auto
- moreover have "norm ((B / norm x) *\<^sub>R x) = B"
- using that B by (simp add: algebra_simps)
- ultimately show False using B by force
- qed
- then have "bounded S \<Longrightarrow> 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 \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
-proof (cases "S = {}")
- case True then show ?thesis
- by simp
-next
- case False
- then obtain b where "b \<in> S" by blast
- with False assms show ?thesis
- apply safe
- using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
- apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
- image_empty image_insert translation_invert)
- apply force
- done
-qed
-
-lemma affine_bounded_eq_lowdim:
- fixes S :: "'a::euclidean_space set"
- assumes "affine S"
- shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
-apply safe
-using affine_bounded_eq_trivial assms apply fastforce
-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)
-
-
-lemma bounded_hyperplane_eq_trivial_0:
- fixes a :: "'a::euclidean_space"
- assumes "a \<noteq> 0"
- shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
-proof
- assume "bounded {x. a \<bullet> x = 0}"
- then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
- by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
- with assms show "DIM('a) = 1"
- by (simp add: le_Suc_eq aff_dim_hyperplane)
-next
- assume "DIM('a) = 1"
- then show "bounded {x. a \<bullet> x = 0}"
- by (simp add: aff_dim_hyperplane affine_bounded_eq_lowdim affine_hyperplane assms)
-qed
-
-lemma bounded_hyperplane_eq_trivial:
- fixes a :: "'a::euclidean_space"
- shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
-proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
- assume "r \<noteq> 0" "a \<noteq> 0"
- have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
- by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
- then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
- by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
-qed
-
-subsection\<open>General case without assuming closure and getting non-strict separation\<close>
-
-proposition separating_hyperplane_closed_point_inset:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
- obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
-proof -
- obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
- using distance_attains_inf [of S z] assms by auto
- then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
- using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
- show ?thesis
- proof (rule that [OF \<open>y \<in> S\<close> *])
- fix x
- assume "x \<in> S"
- have yz: "0 < (y - z) \<bullet> (y - z)"
- using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
- { assume 0: "0 < ((z - y) \<bullet> (x - y))"
- with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
- have False
- using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
- }
- then have "0 \<le> ((y - z) \<bullet> (x - y))"
- by (force simp: not_less inner_diff_left)
- with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
- by (simp add: algebra_simps)
- then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> 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 \<noteq> {}" "0 \<notin> S"
- obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
-using separating_hyperplane_closed_point_inset [OF assms]
-by simp (metis \<open>0 \<notin> S\<close>)
-
-
-proposition separating_hyperplane_set_0_inspan:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
- obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
-proof -
- define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
- have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
- if f': "finite f'" "f' \<subseteq> k ` S" for f'
- proof -
- obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
- using finite_subset_image [OF f'] by blast
- obtain a where "a \<in> S" "a \<noteq> 0"
- using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
- then have "norm (a /\<^sub>R (norm a)) = 1"
- by simp
- moreover have "a /\<^sub>R (norm a) \<in> span S"
- by (simp add: \<open>a \<in> S\<close> span_mul span_superset)
- ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> 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 \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
- moreover have "convex hull C \<noteq> {}"
- by (simp add: False)
- moreover have "0 \<notin> convex hull C"
- by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
- ultimately obtain a b
- where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
- and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
- using separating_hyperplane_closed_0_inset by blast
- then have "a \<in> S"
- by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
- moreover have "norm (a /\<^sub>R (norm a)) = 1"
- using \<open>a \<noteq> 0\<close> by simp
- moreover have "a /\<^sub>R (norm a) \<in> span S"
- by (simp add: \<open>a \<in> S\<close> span_mul span_superset)
- ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
- by simp
- have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
- apply (clarsimp simp add: divide_simps)
- using ab \<open>0 < b\<close>
- by (metis hull_inc inner_commute less_eq_real_def less_trans)
- show ?thesis
- apply (simp add: C k_def)
- using ass aa Int_iff empty_iff by blast
- qed
- qed
- have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
- apply (rule compact_imp_fip)
- apply (blast intro: compact_cball)
- using closed_halfspace_ge k_def apply blast
- apply (metis *)
- done
- 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 \<noteq> {}" and zno: "z \<notin> S"
- obtains a b where "(z + a) \<in> affine hull (insert z S)"
- and "a \<noteq> 0" and "a \<bullet> z \<le> b"
- and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
-proof -
-from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
- have "convex (op + (- z) ` S)"
- by (simp add: \<open>convex S\<close>)
- moreover have "op + (- z) ` S \<noteq> {}"
- by (simp add: \<open>S \<noteq> {}\<close>)
- moreover have "0 \<notin> op + (- z) ` S"
- using zno by auto
- ultimately obtain a where "a \<in> span (op + (- z) ` S)" "a \<noteq> 0"
- and a: "\<And>x. x \<in> (op + (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
- using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
- by blast
- then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> 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)
- show ?thesis
- apply (rule_tac a=a and b = "a \<bullet> z" in that, simp_all)
- using \<open>a \<in> span (op + (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
- apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
- done
-qed
-
-proposition supporting_hyperplane_rel_boundary:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
- obtains a where "a \<noteq> 0"
- and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
- and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
-proof -
- obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
- and "a \<noteq> 0" and "a \<bullet> x \<le> b"
- and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> 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 \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
- proof -
- have con: "continuous_on (closure (rel_interior S)) (op \<bullet> a)"
- by (rule continuous_intros continuous_on_subset | blast)+
- have y: "y \<in> closure (rel_interior S)"
- using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
- by fastforce
- show ?thesis
- using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
- by fastforce
- qed
- have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
- proof -
- obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
- using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
- define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
- have "y' \<in> cball y e"
- unfolding y'_def using \<open>0 < e\<close> by force
- moreover have "y' \<in> affine hull S"
- unfolding y'_def
- by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
- rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
- ultimately have "y' \<in> S"
- using e by auto
- have "a \<bullet> x \<le> a \<bullet> y"
- using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
- moreover have "a \<bullet> x \<noteq> a \<bullet> y"
- using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
- apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
- by (metis \<open>0 < e\<close> add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
- ultimately show ?thesis by force
- qed
- show ?thesis
- by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
-qed
-
-lemma supporting_hyperplane_relative_frontier:
- fixes S :: "'a::euclidean_space set"
- assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
- obtains a where "a \<noteq> 0"
- and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
- and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
-using supporting_hyperplane_rel_boundary [of "closure S" x]
-by (metis assms convex_closure convex_rel_interior_closure)
-
-
-subsection\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
-
-lemma
- fixes s :: "'a::euclidean_space set"
- assumes "~ (affine_dependent(s \<union> t))"
- shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
- and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
-proof -
- have [simp]: "finite s" "finite t"
- using aff_independent_finite assms by blast+
- have "sum u (s \<inter> t) = 1 \<and>
- (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
- if [simp]: "sum u s = 1"
- "sum v t = 1"
- and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
- proof -
- define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
- have "sum f (s \<union> t) = 0"
- apply (simp add: f_def sum_Un sum_subtractf)
- apply (simp add: sum.inter_restrict [symmetric] Int_commute)
- done
- moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
- apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
- apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
- cong del: if_weak_cong)
- done
- ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
- using aff_independent_finite assms unfolding affine_dependent_explicit
- by blast
- then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
- by (simp add: f_def) presburger
- have "sum u (s \<inter> t) = sum u s"
- by (simp add: sum.inter_restrict)
- then have "sum u (s \<inter> t) = 1"
- using that by linarith
- moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>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
-
-
-proposition affine_hull_Int:
- fixes s :: "'a::euclidean_space set"
- assumes "~ (affine_dependent(s \<union> t))"
- shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
-apply (rule subset_antisym)
-apply (simp add: hull_mono)
-by (simp add: affine_hull_Int_subset assms)
-
-proposition convex_hull_Int:
- fixes s :: "'a::euclidean_space set"
- assumes "~ (affine_dependent(s \<union> t))"
- shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
-apply (rule subset_antisym)
-apply (simp add: hull_mono)
-by (simp add: convex_hull_Int_subset assms)
-
-proposition
- fixes s :: "'a::euclidean_space set set"
- assumes "~ (affine_dependent (\<Union>s))"
- shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
- and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
-proof -
- have "finite s"
- using aff_independent_finite assms finite_UnionD by blast
- then have "?A \<and> ?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]: "\<not> affine_dependent (t \<union> \<Inter>F)"
- by (auto intro: affine_dependent_subset)
- have [simp]: "\<not> affine_dependent (\<Union>F)"
- using affine_independent_subset insert.prems by fastforce
- show ?thesis
- by (simp add: affine_hull_Int convex_hull_Int insert.IH)
- qed
- qed
- then show "?A" "?C"
- by auto
-qed
-
-proposition in_convex_hull_exchange_unique:
- fixes S :: "'a::euclidean_space set"
- assumes naff: "~ affine_dependent S" and a: "a \<in> convex hull S"
- and S: "T \<subseteq> S" "T' \<subseteq> S"
- and x: "x \<in> convex hull (insert a T)"
- and x': "x \<in> convex hull (insert a T')"
- shows "x \<in> convex hull (insert a (T \<inter> T'))"
-proof (cases "a \<in> S")
- case True
- then have "\<not> affine_dependent (insert a T \<union> insert a T')"
- using affine_dependent_subset assms by auto
- then have "x \<in> convex hull (insert a T \<inter> insert a T')"
- by (metis IntI convex_hull_Int x x')
- then show ?thesis
- by simp
-next
- case False
- then have anot: "a \<notin> T" "a \<notin> T'"
- using assms by auto
- have [simp]: "finite S"
- by (simp add: aff_independent_finite assms)
- then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
- and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
- using a by (auto simp: convex_hull_finite)
- have fin [simp]: "finite T" "finite T'"
- using assms infinite_super \<open>finite S\<close> by blast+
- then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
- and c1: "sum c (insert a T) = 1"
- and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
- and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
- and c'1: "sum c' (insert a T') = 1"
- and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>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: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
- by simp_all
- have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
- using x'eq fin anot by simp
- define cc where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
- define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
- define dd where "dd \<equiv> \<lambda>x. 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: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>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 "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
- by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
- then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
- using aeq by blast
- have "(\<Sum>v \<in> S. dd v *\<^sub>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 \<in> S" for v
- using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
- apply (simp only: not_ex)
- apply (drule_tac x=S in spec)
- apply (drule_tac x=dd in spec, simp)
- done
- consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
- then show ?thesis
- proof cases
- case 1
- then have "sum cc S \<le> sum cc' S"
- by (simp add: sumSS')
- then have le: "cc x \<le> cc' x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
- using le [OF \<open>x \<in> S\<close>] that c0
- by (force simp: cc_def cc'_def split: if_split_asm)
- show ?thesis
- proof (simp add: convex_hull_finite, intro exI conjI)
- show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
- by (simp add: c0 cc_def)
- show "0 \<le> (cc(a := c a)) a"
- by (simp add: c0)
- have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
- by (simp add: anot)
- also have "... = c a + sum (cc(a := c a)) S"
- apply simp
- apply (rule sum.mono_neutral_left)
- using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
- done
- also have "... = c a + (1 - c a)"
- by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
- finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
- by simp
- have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
- by (simp add: anot)
- also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
- apply simp
- apply (rule sum.mono_neutral_left)
- using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
- done
- also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
- by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
- finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
- by simp
- qed
- next
- case 2
- then have "sum cc' S \<le> sum cc S"
- by (simp add: sumSS')
- then have le: "cc' x \<le> cc x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
- using le [OF \<open>x \<in> S\<close>] that c'0
- by (force simp: cc_def cc'_def split: if_split_asm)
- show ?thesis
- proof (simp add: convex_hull_finite, intro exI conjI)
- show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
- by (simp add: c'0 cc'_def)
- show "0 \<le> (cc'(a := c' a)) a"
- by (simp add: c'0)
- have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
- by (simp add: anot)
- also have "... = c' a + sum (cc'(a := c' a)) S"
- apply simp
- apply (rule sum.mono_neutral_left)
- using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
- done
- also have "... = c' a + (1 - c' a)"
- by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
- finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
- by simp
- have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
- by (simp add: anot)
- also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
- apply simp
- apply (rule sum.mono_neutral_left)
- using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
- done
- also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
- by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
- finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
- by simp
- qed
- qed
-qed
-
-corollary convex_hull_exchange_Int:
- fixes a :: "'a::euclidean_space"
- assumes "~ affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
- shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
- convex hull (insert a (T \<inter> T'))"
-apply (rule subset_antisym)
- using in_convex_hull_exchange_unique assms apply blast
- by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
-
-lemma Int_closed_segment:
- fixes b :: "'a::euclidean_space"
- assumes "b \<in> closed_segment a c \<or> ~collinear{a,b,c}"
- shows "closed_segment a b \<inter> 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 \<in> closed_segment a c"
- moreover have "\<not> affine_dependent {a, c}"
- by (simp add: affine_independent_2)
- 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: "\<not> collinear {a, b, c}"
- have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
- proof -
- have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
- by auto
- with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> 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}"
- apply (rule collinear_3_trans [OF _ _ \<open>b \<noteq> d\<close>])
- using d by (auto simp: insert_commute)
- with ncoll show False ..
- qed
- then show ?thesis
- by blast
- qed
-qed
-
-lemma affine_hull_finite_intersection_hyperplanes:
- fixes s :: "'a::euclidean_space set"
- obtains f where
- "finite f"
- "of_nat (card f) + aff_dim s = DIM('a)"
- "affine hull s = \<Inter>f"
- "\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
-proof -
- obtain b where "b \<subseteq> s"
- and indb: "\<not> affine_dependent b"
- and eq: "affine hull s = affine hull b"
- using affine_basis_exists by blast
- obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> 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 \<le> card c"
- using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
- have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
- by blast
- have card1: "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
- apply (rule card_image [OF inj_onI])
- by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
- have card2: "(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: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent aff_dim_UNIV card_Diff_subset of_nat_diff)
- qed
- show ?thesis
- proof (cases "c = b")
- case True show ?thesis
- apply (rule_tac f="{}" in that)
- using True affc
- apply (simp_all add: eq [symmetric])
- by (metis aff_dim_UNIV aff_dim_affine_hull)
- next
- case False
- have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
- by (rule affine_independent_subset [OF indc]) auto
- have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
- using \<open>b \<subseteq> c\<close> False
- apply (subst affine_hull_Inter [OF ind, symmetric])
- apply (simp add: eq double_diff)
- done
- have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
- if t: "t \<in> 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
- show ?thesis
- apply (rule_tac f = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
- using \<open>finite c\<close> apply blast
- apply (simp add: imeq card1 card2)
- apply (simp add: affeq, clarify)
- apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
- done
- qed
-qed
-
-subsection\<open>Misc results about span\<close>
-
-lemma eq_span_insert_eq:
- assumes "(x - y) \<in> span S"
- shows "span(insert x S) = span(insert y S)"
-proof -
- have *: "span(insert x S) \<subseteq> span(insert y S)" if "(x - y) \<in> span S" for x y
- proof -
- have 1: "(r *\<^sub>R x - r *\<^sub>R y) \<in> span S" for r
- by (metis real_vector.scale_right_diff_distrib span_mul that)
- have 2: "(z - k *\<^sub>R y) - k *\<^sub>R (x - y) = z - k *\<^sub>R x" for z k
- by (simp add: real_vector.scale_right_diff_distrib)
- show ?thesis
- apply (clarsimp simp add: span_breakdown_eq)
- by (metis 1 2 diff_add_cancel real_vector.scale_right_diff_distrib span_add_eq)
- qed
- show ?thesis
- apply (intro subset_antisym * assms)
- using assms subspace_neg subspace_span minus_diff_eq by force
-qed
-
-lemma dim_psubset:
- fixes S :: "'a :: euclidean_space set"
- shows "span S \<subset> span T \<Longrightarrow> dim S < dim T"
-by (metis (no_types, hide_lams) dim_span less_le not_le subspace_dim_equal subspace_span)
-
-
-lemma basis_subspace_exists:
- fixes S :: "'a::euclidean_space set"
- shows
- "subspace S
- \<Longrightarrow> \<exists>b. finite b \<and> b \<subseteq> S \<and>
- independent b \<and> span b = S \<and> card b = dim S"
-by (metis span_subspace basis_exists independent_imp_finite)
-
-lemma affine_hyperplane_sums_eq_UNIV_0:
- fixes S :: "'a :: euclidean_space set"
- assumes "affine S"
- and "0 \<in> S" and "w \<in> S"
- and "a \<bullet> w \<noteq> 0"
- shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
-proof -
- have "subspace S"
- by (simp add: assms subspace_affine)
- have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
- apply (rule span_mono)
- using \<open>0 \<in> S\<close> add.left_neutral by force
- have "w \<notin> span {y. a \<bullet> y = 0}"
- using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
- moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
- using \<open>w \<in> S\<close>
- by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_superset)
- ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
- by blast
- have "a \<noteq> 0" using assms inner_zero_left by blast
- then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
- by (simp add: dim_hyperplane)
- also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
- using span1 span2 by (blast intro: dim_psubset)
- finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
- have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
- using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
- moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
- apply (rule dim_eq_full [THEN iffD1])
- apply (rule antisym [OF dim_subset_UNIV])
- using DIM_lt apply simp
- done
- ultimately show ?thesis
- by (simp add: subs) (metis (lifting) span_eq subs)
-qed
-
-proposition affine_hyperplane_sums_eq_UNIV:
- fixes S :: "'a :: euclidean_space set"
- assumes "affine S"
- and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
- and "S - {v. a \<bullet> v = b} \<noteq> {}"
- shows "{x + y| x y. x \<in> S \<and> a \<bullet> 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 \<in> S" and c: "a \<bullet> c = b"
- using assms by force
- with affine_diffs_subspace [OF \<open>affine S\<close>]
- have "subspace (op + (- c) ` S)" by blast
- then have aff: "affine (op + (- c) ` S)"
- by (simp add: subspace_imp_affine)
- have 0: "0 \<in> op + (- c) ` S"
- by (simp add: \<open>c \<in> S\<close>)
- obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> op + (- c) ` S"
- using assms by auto
- then have adc: "a \<bullet> (d - c) \<noteq> 0"
- by (simp add: c inner_diff_right)
- let ?U = "op + (c+c) ` {x + y |x y. x \<in> op + (- c) ` S \<and> a \<bullet> y = 0}"
- have "u + v \<in> op + (c + c) ` {x + v |x v. x \<in> op + (- c) ` S \<and> a \<bullet> v = 0}"
- if "u \<in> S" "b = a \<bullet> v" for u v
- apply (rule_tac x="u+v-c-c" in image_eqI)
- apply (simp_all add: algebra_simps)
- apply (rule_tac x="u-c" in exI)
- apply (rule_tac x="v-c" in exI)
- apply (simp add: algebra_simps that c)
- done
- moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
- \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
- by (metis add.left_commute c inner_right_distrib pth_d)
- ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
- by (fastforce simp: algebra_simps)
- also have "... = op + (c+c) ` UNIV"
- by (simp add: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
- also have "... = UNIV"
- by (simp add: translation_UNIV)
- finally show ?thesis .
-qed
-
-proposition dim_sums_Int:
- fixes S :: "'a :: euclidean_space set"
- assumes "subspace S" "subspace T"
- shows "dim {x + y |x y. x \<in> S \<and> y \<in> T} + dim(S \<inter> T) = dim S + dim T"
-proof -
- obtain B where B: "B \<subseteq> S \<inter> T" "S \<inter> T \<subseteq> span B"
- and indB: "independent B"
- and cardB: "card B = dim (S \<inter> T)"
- using basis_exists by blast
- then obtain C D where "B \<subseteq> C" "C \<subseteq> S" "independent C" "S \<subseteq> span C"
- and "B \<subseteq> D" "D \<subseteq> T" "independent D" "T \<subseteq> span D"
- using maximal_independent_subset_extend
- by (metis Int_subset_iff \<open>B \<subseteq> S \<inter> T\<close> indB)
- then have "finite B" "finite C" "finite D"
- by (simp_all add: independent_imp_finite indB independent_bound)
- have Beq: "B = C \<inter> D"
- apply (rule sym)
- apply (rule spanning_subset_independent)
- using \<open>B \<subseteq> C\<close> \<open>B \<subseteq> D\<close> apply blast
- apply (meson \<open>independent C\<close> independent_mono inf.cobounded1)
- using B \<open>C \<subseteq> S\<close> \<open>D \<subseteq> T\<close> apply auto
- done
- then have Deq: "D = B \<union> (D - C)"
- by blast
- have CUD: "C \<union> D \<subseteq> {x + y |x y. x \<in> S \<and> y \<in> T}"
- apply safe
- apply (metis add.right_neutral subsetCE \<open>C \<subseteq> S\<close> \<open>subspace T\<close> set_eq_subset span_0 span_minimal)
- apply (metis add.left_neutral subsetCE \<open>D \<subseteq> T\<close> \<open>subspace S\<close> set_eq_subset span_0 span_minimal)
- done
- have "a v = 0" if 0: "(\<Sum>v\<in>C. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v) = 0"
- and v: "v \<in> C \<union> (D-C)" for a v
- proof -
- have eq: "(\<Sum>v\<in>D - C. a v *\<^sub>R v) = - (\<Sum>v\<in>C. a v *\<^sub>R v)"
- using that add_eq_0_iff by blast
- have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> S"
- apply (subst eq)
- apply (rule subspace_neg [OF \<open>subspace S\<close>])
- apply (rule subspace_sum [OF \<open>subspace S\<close>])
- by (meson subsetCE subspace_mul \<open>C \<subseteq> S\<close> \<open>subspace S\<close>)
- moreover have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> T"
- apply (rule subspace_sum [OF \<open>subspace T\<close>])
- by (meson DiffD1 \<open>D \<subseteq> T\<close> \<open>subspace T\<close> subset_eq subspace_def)
- ultimately have "(\<Sum>v \<in> D-C. a v *\<^sub>R v) \<in> span B"
- using B by blast
- then obtain e where e: "(\<Sum>v\<in>B. e v *\<^sub>R v) = (\<Sum>v \<in> D-C. a v *\<^sub>R v)"
- using span_finite [OF \<open>finite B\<close>] by blast
- have "\<And>c v. \<lbrakk>(\<Sum>v\<in>C. c v *\<^sub>R v) = 0; v \<in> C\<rbrakk> \<Longrightarrow> c v = 0"
- using independent_explicit \<open>independent C\<close> by blast
- define cc where "cc x = (if x \<in> B then a x + e x else a x)" for x
- have [simp]: "C \<inter> B = B" "D \<inter> B = B" "C \<inter> - B = C-D" "B \<inter> (D - C) = {}"
- using \<open>B \<subseteq> C\<close> \<open>B \<subseteq> D\<close> Beq by blast+
- have f2: "(\<Sum>v\<in>C \<inter> D. e v *\<^sub>R v) = (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
- using Beq e by presburger
- have f3: "(\<Sum>v\<in>C \<union> D. a v *\<^sub>R v) = (\<Sum>v\<in>C - D. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v) + (\<Sum>v\<in>C \<inter> D. a v *\<^sub>R v)"
- using \<open>finite C\<close> \<open>finite D\<close> sum.union_diff2 by blast
- have f4: "(\<Sum>v\<in>C \<union> (D - C). a v *\<^sub>R v) = (\<Sum>v\<in>C. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
- by (meson Diff_disjoint \<open>finite C\<close> \<open>finite D\<close> finite_Diff sum.union_disjoint)
- have "(\<Sum>v\<in>C. cc v *\<^sub>R v) = 0"
- using 0 f2 f3 f4
- apply (simp add: cc_def Beq if_smult \<open>finite C\<close> sum.If_cases algebra_simps sum.distrib)
- apply (simp add: add.commute add.left_commute diff_eq)
- done
- then have "\<And>v. v \<in> C \<Longrightarrow> cc v = 0"
- using independent_explicit \<open>independent C\<close> by blast
- then have C0: "\<And>v. v \<in> C - B \<Longrightarrow> a v = 0"
- by (simp add: cc_def Beq) meson
- then have [simp]: "(\<Sum>x\<in>C - B. a x *\<^sub>R x) = 0"
- by simp
- have "(\<Sum>x\<in>C. a x *\<^sub>R x) = (\<Sum>x\<in>B. a x *\<^sub>R x)"
- proof -
- have "C - D = C - B"
- using Beq by blast
- then show ?thesis
- using Beq \<open>(\<Sum>x\<in>C - B. a x *\<^sub>R x) = 0\<close> f3 f4 by auto
- qed
- with 0 have Dcc0: "(\<Sum>v\<in>D. a v *\<^sub>R v) = 0"
- apply (subst Deq)
- by (simp add: \<open>finite B\<close> \<open>finite D\<close> sum_Un)
- then have D0: "\<And>v. v \<in> D \<Longrightarrow> a v = 0"
- using independent_explicit \<open>independent D\<close> by blast
- show ?thesis
- using v C0 D0 Beq by blast
- qed
- then have "independent (C \<union> (D - C))"
- by (simp add: independent_explicit \<open>finite C\<close> \<open>finite D\<close> sum_Un del: Un_Diff_cancel)
- then have indCUD: "independent (C \<union> D)" by simp
- have "dim (S \<inter> T) = card B"
- by (rule dim_unique [OF B indB refl])
- moreover have "dim S = card C"
- by (metis \<open>C \<subseteq> S\<close> \<open>independent C\<close> \<open>S \<subseteq> span C\<close> basis_card_eq_dim)
- moreover have "dim T = card D"
- by (metis \<open>D \<subseteq> T\<close> \<open>independent D\<close> \<open>T \<subseteq> span D\<close> basis_card_eq_dim)
- moreover have "dim {x + y |x y. x \<in> S \<and> y \<in> T} = card(C \<union> D)"
- apply (rule dim_unique [OF CUD _ indCUD refl], clarify)
- apply (meson \<open>S \<subseteq> span C\<close> \<open>T \<subseteq> span D\<close> span_add span_inc span_minimal subsetCE subspace_span sup.bounded_iff)
- done
- ultimately show ?thesis
- using \<open>B = C \<inter> D\<close> [symmetric]
- by (simp add: \<open>independent C\<close> \<open>independent D\<close> card_Un_Int independent_finite)
-qed
-
-
-lemma aff_dim_sums_Int_0:
- assumes "affine S"
- and "affine T"
- and "0 \<in> S" "0 \<in> T"
- shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
-proof -
- have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
- using assms by force
- then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> 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 \<inter> T \<noteq> {}"
- shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
-proof -
- obtain a where a: "a \<in> S" "a \<in> T" using assms by force
- have aff: "affine (op+ (-a) ` S)" "affine (op+ (-a) ` T)"
- using assms by (auto simp: affine_translation [symmetric])
- have zero: "0 \<in> (op+ (-a) ` S)" "0 \<in> (op+ (-a) ` T)"
- using a assms by auto
- have [simp]: "{x + y |x y. x \<in> op + (- a) ` S \<and> y \<in> op + (- a) ` T} =
- op + (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
- by (force simp: algebra_simps scaleR_2)
- have [simp]: "op + (- a) ` S \<inter> op + (- a) ` T = op + (- a) ` (S \<inter> T)"
- by auto
- show ?thesis
- using aff_dim_sums_Int_0 [OF aff zero]
- by (auto simp: aff_dim_translation_eq)
-qed
-
-lemma aff_dim_affine_Int_hyperplane:
- fixes a :: "'a::euclidean_space"
- assumes "affine S"
- shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
- (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
- else if S \<subseteq> {v. a \<bullet> 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 \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
- if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
- proof -
- have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> 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) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
-by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
-
-
-lemma dim_Times:
- fixes S :: "'a :: euclidean_space set" and T :: "'a set"
- assumes "subspace S" "subspace T"
- shows "dim(S \<times> T) = dim S + dim T"
-proof -
- have ss: "subspace ((\<lambda>x. (x, 0)) ` S)" "subspace (Pair 0 ` T)"
- by (rule subspace_linear_image, unfold_locales, auto simp: assms)+
- have "dim(S \<times> T) = dim({u + v |u v. u \<in> (\<lambda>x. (x, 0)) ` S \<and> v \<in> Pair 0 ` T})"
- by (simp add: Times_eq_image_sum)
- moreover have "dim ((\<lambda>x. (x, 0::'a)) ` S) = dim S" "dim (Pair (0::'a) ` T) = dim T"
- by (auto simp: additive.intro linear.intro linear_axioms.intro inj_on_def intro: dim_image_eq)
- moreover have "dim ((\<lambda>x. (x, 0)) ` S \<inter> Pair 0 ` T) = 0"
- by (subst dim_eq_0) (force simp: zero_prod_def)
- ultimately show ?thesis
- using dim_sums_Int [OF ss] by linarith
-qed
-
-subsection\<open> Orthogonal bases, Gram-Schmidt process, and related theorems\<close>
-
-lemma span_delete_0 [simp]: "span(S - {0}) = span S"
-proof
- show "span (S - {0}) \<subseteq> span S"
- by (blast intro!: span_mono)
-next
- have "span S \<subseteq> span(insert 0 (S - {0}))"
- by (blast intro!: span_mono)
- also have "... \<subseteq> span(S - {0})"
- using span_insert_0 by blast
- finally show "span S \<subseteq> span (S - {0})" .
-qed
-
-lemma span_image_scale:
- assumes "finite S" and nz: "\<And>x. x \<in> S \<Longrightarrow> c x \<noteq> 0"
- shows "span ((\<lambda>x. c x *\<^sub>R x) ` S) = span S"
-using assms
-proof (induction S arbitrary: c)
- case (empty c) show ?case by simp
-next
- case (insert x F c)
- show ?case
- proof (intro set_eqI iffI)
- fix y
- assume "y \<in> span ((\<lambda>x. c x *\<^sub>R x) ` insert x F)"
- then show "y \<in> span (insert x F)"
- using insert by (force simp: span_breakdown_eq)
- next
- fix y
- assume "y \<in> span (insert x F)"
- then show "y \<in> span ((\<lambda>x. c x *\<^sub>R x) ` insert x F)"
- using insert
- apply (clarsimp simp: span_breakdown_eq)
- apply (rule_tac x="k / c x" in exI)
- by simp
- qed
-qed
-
-lemma pairwise_orthogonal_independent:
- assumes "pairwise orthogonal S" and "0 \<notin> S"
- shows "independent S"
-proof -
- have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- using assms by (simp add: pairwise_def orthogonal_def)
- have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
- proof -
- obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
- using a by (force simp: span_explicit)
- then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
- by simp
- also have "... = 0"
- apply (simp add: inner_sum_right)
- apply (rule comm_monoid_add_class.sum.neutral)
- by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
- finally show ?thesis
- using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
- qed
- then show ?thesis
- by (force simp: dependent_def)
-qed
-
-lemma pairwise_orthogonal_imp_finite:
- fixes S :: "'a::euclidean_space set"
- assumes "pairwise orthogonal S"
- shows "finite S"
-proof -
- have "independent (S - {0})"
- apply (rule pairwise_orthogonal_independent)
- apply (metis Diff_iff assms pairwise_def)
- by blast
- then show ?thesis
- by (meson independent_imp_finite infinite_remove)
-qed
-
-lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
- by (simp add: subspace_def orthogonal_clauses)
-
-lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
- by (simp add: subspace_def orthogonal_clauses)
-
-lemma orthogonal_to_span:
- assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
- shows "orthogonal x a"
-apply (rule span_induct [OF a subspace_orthogonal_to_vector])
-apply (simp add: x)
-done
-
-proposition Gram_Schmidt_step:
- fixes S :: "'a::euclidean_space set"
- assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
- shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
-proof -
- have "finite S"
- by (simp add: S pairwise_orthogonal_imp_finite)
- have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
- if "x \<in> S" for x
- proof -
- have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
- by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
- also have "... = (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
- apply (rule sum.cong [OF refl], simp)
- by (meson S orthogonal_def pairwise_def that)
- finally show ?thesis
- by (simp add: orthogonal_def algebra_simps inner_sum_left)
- qed
- then show ?thesis
- using orthogonal_to_span orthogonal_commute x by blast
-qed
-
-
-lemma orthogonal_extension_aux:
- fixes S :: "'a::euclidean_space set"
- assumes "finite T" "finite S" "pairwise orthogonal S"
- shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
-using assms
-proof (induction arbitrary: S)
- case empty then show ?case
- by simp (metis sup_bot_right)
-next
- case (insert a T)
- have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- using insert by (simp add: pairwise_def orthogonal_def)
- define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
- obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
- and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
- apply (rule exE [OF insert.IH [of "insert a' S"]])
- apply (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute pairwise_orthogonal_insert span_clauses)
- done
- have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
- apply (simp add: a'_def)
- using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
- apply (force simp: orthogonal_def inner_commute span_inc [THEN subsetD])
- done
- have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
- using spanU by simp
- also have "... = span (insert a (S \<union> T))"
- apply (rule eq_span_insert_eq)
- apply (simp add: a'_def span_neg span_sum span_clauses(1) span_mul)
- done
- also have "... = span (S \<union> insert a T)"
- by simp
- finally show ?case
- apply (rule_tac x="insert a' U" in exI)
- using orthU apply auto
- done
-qed
-
-
-proposition orthogonal_extension:
- fixes S :: "'a::euclidean_space set"
- assumes S: "pairwise orthogonal S"
- obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
-proof -
- obtain B where "finite B" "span B = span T"
- using basis_subspace_exists [of "span T"] subspace_span by auto
- with orthogonal_extension_aux [of B S]
- obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
- using assms pairwise_orthogonal_imp_finite by auto
- show ?thesis
- apply (rule_tac U=U in that)
- apply (simp add: \<open>pairwise orthogonal (S \<union> U)\<close>)
- by (metis \<open>span (S \<union> U) = span (S \<union> B)\<close> \<open>span B = span T\<close> span_Un)
-qed
-
-corollary orthogonal_extension_strong:
- fixes S :: "'a::euclidean_space set"
- assumes S: "pairwise orthogonal S"
- obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
- "span (S \<union> U) = span (S \<union> T)"
-proof -
- obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
- using orthogonal_extension assms by blast
- then show ?thesis
- apply (rule_tac U = "U - (insert 0 S)" in that)
- apply blast
- apply (force simp: pairwise_def)
- apply (metis (no_types, lifting) Un_Diff_cancel span_insert_0 span_Un)
- done
-qed
-
-subsection\<open>Decomposing a vector into parts in orthogonal subspaces.\<close>
-
-text\<open>existence of orthonormal basis for a subspace.\<close>
-
-lemma orthogonal_spanningset_subspace:
- fixes S :: "'a :: euclidean_space set"
- assumes "subspace S"
- obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
-proof -
- obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
- using basis_exists by blast
- with orthogonal_extension [of "{}" B]
- show ?thesis
- by (metis Un_empty_left assms pairwise_empty span_inc span_subspace that)
-qed
-
-lemma orthogonal_basis_subspace:
- fixes S :: "'a :: euclidean_space set"
- assumes "subspace S"
- obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
- "card B = dim S" "span B = S"
-proof -
- obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
- using assms orthogonal_spanningset_subspace by blast
- then show ?thesis
- apply (rule_tac B = "B - {0}" in that)
- apply (auto simp: indep_card_eq_dim_span pairwise_subset Diff_subset pairwise_orthogonal_independent elim: pairwise_subset)
- done
-qed
-
-proposition orthonormal_basis_subspace:
- fixes S :: "'a :: euclidean_space set"
- assumes "subspace S"
- obtains B where "B \<subseteq> S" "pairwise orthogonal B"
- and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
- and "independent B" "card B = dim S" "span B = S"
-proof -
- obtain B where "0 \<notin> B" "B \<subseteq> S"
- and orth: "pairwise orthogonal B"
- and "independent B" "card B = dim S" "span B = S"
- by (blast intro: orthogonal_basis_subspace [OF assms])
- have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
- using \<open>span B = S\<close> span_clauses(1) span_mul by fastforce
- have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
- using orth by (force simp: pairwise_def orthogonal_clauses)
- have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
- by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
- have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
- by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
- have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
- proof
- fix x y
- assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
- moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
- using 3 by blast
- ultimately show "x = y"
- by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
- qed
- then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
- by (metis \<open>card B = dim S\<close> card_image)
- have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
- by (metis "1" "4" "5" assms card_eq_dim independent_finite span_subspace)
- show ?thesis
- by (rule that [OF 1 2 3 4 5 6])
-qed
-
-proposition orthogonal_subspace_decomp_exists:
- fixes S :: "'a :: euclidean_space set"
- obtains y z where "y \<in> span S" "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w" "x = y + z"
-proof -
- obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T" "card T = dim (span S)" "span T = span S"
- using orthogonal_basis_subspace subspace_span by blast
- let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
- have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
- by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close> orthogonal_commute that)
- show ?thesis
- apply (rule_tac y = "?a" and z = "x - ?a" in that)
- apply (meson \<open>T \<subseteq> span S\<close> span_mul span_sum subsetCE)
- apply (fact orth, simp)
- done
-qed
-
-lemma orthogonal_subspace_decomp_unique:
- fixes S :: "'a :: euclidean_space set"
- assumes "x + y = x' + y'"
- and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
- and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
- shows "x = x' \<and> y = y'"
-proof -
- have "x + y - y' = x'"
- by (simp add: assms)
- moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
- by (meson orth orthogonal_commute orthogonal_to_span)
- ultimately have "0 = x' - x"
- by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
- with assms show ?thesis by auto
-qed
-
-proposition dim_orthogonal_sum:
- fixes A :: "'a::euclidean_space set"
- assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- shows "dim(A \<union> B) = dim A + dim B"
-proof -
- have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
- have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- apply (erule span_induct [OF _ subspace_hyperplane])
- using 1 by (simp add: )
- then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
- by simp
- have "dim(A \<union> B) = dim (span (A \<union> B))"
- by simp
- also have "... = dim ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
- by (simp add: span_Un)
- also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
- by (auto intro!: arg_cong [where f=dim])
- also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
- by (auto simp: dest: 0)
- also have "... = dim (span A) + dim (span B)"
- by (rule dim_sums_Int) auto
- also have "... = dim A + dim B"
- by simp
- finally show ?thesis .
-qed
-
-lemma dim_subspace_orthogonal_to_vectors:
- fixes A :: "'a::euclidean_space set"
- assumes "subspace A" "subspace B" "A \<subseteq> B"
- shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
-proof -
- have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
- proof (rule arg_cong [where f=dim, OF subset_antisym])
- show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
- by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
- next
- have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
- if "x \<in> B" for x
- proof -
- obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
- using orthogonal_subspace_decomp_exists [of A x] that by auto
- have "y \<in> span B"
- by (metis span_eq \<open>y \<in> span A\<close> assms subset_iff)
- then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
- by simp (metis (no_types) span_eq \<open>x = y + z\<close> \<open>subspace A\<close> \<open>subspace B\<close> orth orthogonal_commute span_add_eq that)
- then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
- by (meson span_inc subset_iff)
- then show ?thesis
- apply (simp add: span_Un image_def)
- apply (rule bexI [OF _ z])
- apply (simp add: \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
- done
- qed
- show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
- by (rule span_minimal) (auto intro: * span_minimal elim: )
- qed
- then show ?thesis
- by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq orthogonal_commute orthogonal_def)
-qed
-
-lemma aff_dim_openin:
- fixes S :: "'a::euclidean_space set"
- assumes ope: "openin (subtopology euclidean T) S" and "affine T" "S \<noteq> {}"
- shows "aff_dim S = aff_dim T"
-proof -
- show ?thesis
- proof (rule order_antisym)
- show "aff_dim S \<le> aff_dim T"
- by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
- next
- obtain a where "a \<in> S"
- using \<open>S \<noteq> {}\<close> by blast
- have "S \<subseteq> T"
- using ope openin_imp_subset by auto
- then have "a \<in> T"
- using \<open>a \<in> S\<close> by auto
- then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
- using affine_diffs_subspace \<open>affine T\<close> by auto
- then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
- and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
- and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
- and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
- by (rule orthonormal_basis_subspace) auto
- obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
- by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
- have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
- by (metis aff_dim_translation_eq)
- also have "... = dim ((\<lambda>x. - a + x) ` T)"
- using aff_dim_subspace subT' by blast
- also have "... = card B"
- by (simp add: cardB)
- also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
- using \<open>0 < e\<close> by (force simp: inj_on_def card_image)
- also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
- proof (simp, rule independent_card_le_dim)
- have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
- using e by (auto simp: dist_norm norm_minus_commute subset_eq)
- have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
- using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
- then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
- using e' by blast
- show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
- using \<open>independent B\<close>
- apply (rule independent_injective_image, simp)
- by (metis \<open>0 < e\<close> injective_scaleR less_irrefl)
- qed
- also have "... = aff_dim S"
- using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by force
- finally show "aff_dim T \<le> aff_dim S" .
- qed
-qed
-
-lemma dim_openin:
- fixes S :: "'a::euclidean_space set"
- assumes ope: "openin (subtopology euclidean T) S" and "subspace T" "S \<noteq> {}"
- shows "dim S = dim T"
-proof (rule order_antisym)
- show "dim S \<le> 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 \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
- also have "... \<le> dim S"
- by (metis aff_dim_subset aff_dim_subspace dim_span span_inc subspace_span)
- finally show "dim T \<le> dim S" by simp
-qed
-
-subsection\<open>Parallel slices, etc.\<close>
-
-text\<open> 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.\<close>
-
-proposition affine_parallel_slice:
- fixes S :: "'a :: euclidean_space set"
- assumes "affine S"
- and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
- and "~ (S \<subseteq> {x. a \<bullet> x \<le> b})"
- obtains a' b' where "a' \<noteq> 0"
- "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
- "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
- "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
-proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
- case True
- then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
- using assms by (auto simp: not_le)
- define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
- have "\<eta> \<in> S"
- by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
- moreover have "a \<bullet> \<eta> = b"
- using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
- by (simp add: \<eta>_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 \<in> S" and z: "a \<bullet> z = b"
- using assms by auto
- with affine_diffs_subspace [OF \<open>affine S\<close>]
- have sub: "subspace (op + (- z) ` S)" by blast
- then have aff: "affine (op + (- z) ` S)" and span: "span (op + (- z) ` S) = (op + (- z) ` S)"
- by (auto simp: subspace_imp_affine)
- obtain a' a'' where a': "a' \<in> span (op + (- z) ` S)" and a: "a = a' + a''"
- and "\<And>w. w \<in> span (op + (- z) ` S) \<Longrightarrow> orthogonal a'' w"
- using orthogonal_subspace_decomp_exists [of "op + (- z) ` S" "a"] by metis
- then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
- by (simp add: imageI orthogonal_def span)
- then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
- by (simp add: a inner_diff_right)
- then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
- by (simp add: inner_diff_left z)
- have "\<And>w. w \<in> op + (- z) ` S \<Longrightarrow> (w + a') \<in> op + (- z) ` S"
- by (metis subspace_add a' span_eq sub)
- then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
- by fastforce
- 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
- apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
- apply (auto simp: a ba'' inner_left_distrib False Sclo)
- done
- qed
-qed
-
-lemma diffs_affine_hull_span:
- assumes "a \<in> S"
- shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
-proof -
- have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
- by (auto simp: algebra_simps)
- show ?thesis
- apply (simp add: affine_hull_span2 [OF assms] *)
- apply (auto simp: algebra_simps)
- done
-qed
-
-lemma aff_dim_dim_affine_diffs:
- fixes S :: "'a :: euclidean_space set"
- assumes "affine S" "a \<in> S"
- shows "aff_dim S = dim {x - a |x. x \<in> S}"
-proof -
- obtain B where aff: "affine hull B = affine hull S"
- and ind: "\<not> affine_dependent B"
- and card: "of_nat (card B) = aff_dim S + 1"
- using aff_dim_basis_exists by blast
- then have "B \<noteq> {}" using assms
- by (metis affine_hull_eq_empty ex_in_conv)
- then obtain c where "c \<in> B" by auto
- then have "c \<in> S"
- by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
- have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
- by (auto simp: algebra_simps)
- have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
- apply safe
- apply (simp_all only: xy)
- using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
- done
- have affS: "affine hull S = S"
- by (simp add: \<open>affine S\<close>)
- have "aff_dim S = of_nat (card B) - 1"
- using card by simp
- also have "... = dim {x - c |x. x \<in> B}"
- by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
- also have "... = dim {x - c | x. x \<in> affine hull B}"
- by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
- also have "... = dim {x - a |x. x \<in> S}"
- by (simp add: affS aff *)
- finally show ?thesis .
-qed
-
-lemma aff_dim_linear_image_le:
- assumes "linear f"
- shows "aff_dim(f ` S) \<le> aff_dim S"
-proof -
- have "aff_dim (f ` T) \<le> 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 \<in> T" by auto
- have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
- by auto
- have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
- by (force simp: linear_diff [OF assms])
- have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
- by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1)
- also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
- by (force simp: linear_diff [OF assms] 2)
- also have "... \<le> int (dim {x - a| x. x \<in> T})"
- by (simp add: dim_image_le [OF assms])
- also have "... \<le> aff_dim T"
- by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
- finally show ?thesis .
- qed
- then
- have "aff_dim (f ` (affine hull S)) \<le> 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) \<le> aff_dim S"
- by (simp add: aff_dim_linear_image_le assms(1))
-next
- obtain g where "linear g" "g \<circ> f = id"
- using linear_injective_left_inverse assms by blast
- then have "aff_dim S \<le> aff_dim(g ` f ` S)"
- by (simp add: image_comp)
- also have "... \<le> aff_dim (f ` S)"
- by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
- finally show "aff_dim S \<le> aff_dim (f ` S)" .
-qed
-
-
-text\<open>Choosing a subspace of a given dimension\<close>
-proposition choose_subspace_of_subspace:
- fixes S :: "'n::euclidean_space set"
- assumes "n \<le> dim S"
- obtains T where "subspace T" "T \<subseteq> span S" "dim T = n"
-proof -
- have "\<exists>T. subspace T \<and> T \<subseteq> span S \<and> dim T = n"
- using assms
- proof (induction n)
- case 0 then show ?case by force
- next
- case (Suc n)
- then obtain T where "subspace T" "T \<subseteq> span S" "dim T = n"
- by force
- then show ?case
- proof (cases "span S \<subseteq> span T")
- case True
- have "dim S = dim T"
- apply (rule span_eq_dim [OF subset_antisym [OF True]])
- by (simp add: \<open>T \<subseteq> span S\<close> span_minimal subspace_span)
- then show ?thesis
- using Suc.prems \<open>dim T = n\<close> by linarith
- next
- case False
- then obtain y where y: "y \<in> S" "y \<notin> T"
- by (meson span_mono subsetI)
- then have "span (insert y T) \<subseteq> span S"
- by (metis (no_types) \<open>T \<subseteq> span S\<close> subsetD insert_subset span_inc span_mono span_span)
- with \<open>dim T = n\<close> \<open>subspace T\<close> y show ?thesis
- apply (rule_tac x="span(insert y T)" in exI)
- apply (auto simp: dim_insert)
- using span_eq by blast
- qed
- qed
- with that show ?thesis by blast
-qed
-
-lemma choose_affine_subset:
- assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
- obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
-proof (cases "d = -1 \<or> 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 \<in> S" "0 \<le> d" by auto
- with assms have ss: "subspace (op + (- a) ` S)"
- by (simp add: affine_diffs_subspace)
- have "nat d \<le> dim (op + (- 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 \<subseteq> span (op + (- a) ` S)"
- and Tdim: "dim T = nat d"
- using choose_subspace_of_subspace [of "nat d" "op + (- a) ` S"] by blast
- then have "affine T"
- using subspace_affine by blast
- then have "affine (op + a ` T)"
- by (metis affine_hull_eq affine_hull_translation)
- moreover have "op + a ` T \<subseteq> S"
- proof -
- have "T \<subseteq> op + (- a) ` S"
- by (metis (no_types) span_eq Tsb ss)
- then show "op + a ` T \<subseteq> S"
- using add_ac by auto
- qed
- moreover have "aff_dim (op + a ` T) = d"
- by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
- ultimately show ?thesis
- by (rule that)
-qed
-
-subsection\<open>Several Variants of Paracompactness\<close>
-
-proposition paracompact:
- fixes S :: "'a :: euclidean_space set"
- assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
- obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
- and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
- and "\<And>x. x \<in> S
- \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and>
- finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
-proof (cases "S = {}")
- case True with that show ?thesis by blast
-next
- case False
- have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
- proof -
- obtain T where "x \<in> T" "T \<in> \<C>" "open T"
- using assms \<open>x \<in> S\<close> by blast
- then obtain e where "e > 0" "cball x e \<subseteq> T"
- by (force simp: open_contains_cball)
- then show ?thesis
- apply (rule_tac x = T in exI)
- apply (rule_tac x = "ball x e" in exI)
- using \<open>T \<in> \<C>\<close>
- apply (simp add: closure_minimal)
- done
- qed
- then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
- and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
- if "x \<in> S" for x
- by metis
- then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = UNION S G"
- using Lindelof [of "G ` S"] by (metis image_iff)
- then obtain K where K: "K \<subseteq> S" "countable K" and eq: "UNION K G = UNION S G"
- by (metis countable_subset_image)
- with False Gin have "K \<noteq> {}" by force
- then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
- by (metis range_from_nat_into \<open>countable K\<close>)
- then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
- using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
- let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
- have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
- proof -
- have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
- then show ?thesis
- using clos K \<open>range a = K\<close> closure_subset by blast
- qed
- have 1: "S \<subseteq> Union ?C"
- proof
- fix x assume "x \<in> S"
- define n where "n \<equiv> LEAST n. x \<in> F(a n)"
- have n: "x \<in> F(a n)"
- using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
- have notn: "x \<notin> F(a m)" if "m < n" for m
- using that not_less_Least by (force simp: n_def)
- then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
- using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
- with n show "x \<in> Union ?C"
- by blast
- qed
- have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
- proof -
- obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
- using \<open>x \<in> S\<close> enum_S by auto
- have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
- proof clarsimp
- fix k assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
- then have "k \<le> n"
- by auto (metis closure_subset not_le subsetCE)
- then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
- \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
- by force
- qed
- moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
- by force
- ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
- using finite_subset by blast
- show ?thesis
- apply (rule_tac x="G (a n)" in exI)
- apply (intro conjI oG n *)
- using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
- done
- qed
- show ?thesis
- apply (rule that [OF 1 _ 3])
- using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply (auto simp: odif)
- done
-qed
-
-corollary paracompact_closedin:
- fixes S :: "'a :: euclidean_space set"
- assumes cin: "closedin (subtopology euclidean U) S"
- and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (subtopology euclidean U) T"
- and "S \<subseteq> \<Union>\<C>"
- obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
- and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
- and "\<And>x. x \<in> U
- \<Longrightarrow> \<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and>
- finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
-proof -
- have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
- using oin [OF that] by (auto simp: openin_open)
- then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
- by metis
- obtain K where K: "closed K" "U \<inter> K = S"
- using cin by (auto simp: closedin_closed)
- have 1: "U \<subseteq> \<Union>insert (- K) (F ` \<C>)"
- by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
- have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
- using \<open>closed K\<close> by (auto simp: opF)
- obtain \<D> where "U \<subseteq> \<Union>\<D>"
- and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
- and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
- using paracompact [OF 1 2] by auto
- let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
- show ?thesis
- proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
- show "S \<subseteq> \<Union>?C"
- using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
- show "\<And>V. V \<in> ?C \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
- using D1 intF by fastforce
- have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
- (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
- by blast
- show "\<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
- if "x \<in> U" for x
- using D2 [OF that]
- apply clarify
- apply (rule_tac x="U \<inter> V" in exI)
- apply (auto intro: that finite_subset [OF *])
- done
- qed
-qed
-
-corollary paracompact_closed:
- fixes S :: "'a :: euclidean_space set"
- assumes "closed S"
- and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
- and "S \<subseteq> \<Union>\<C>"
- obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
- and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
- and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
- finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
-using paracompact_closedin [of UNIV S \<C>] assms
-by (auto simp: open_openin [symmetric] closed_closedin [symmetric])
-
-
-subsection\<open>Closed-graph characterization of continuity\<close>
-
-lemma continuous_closed_graph_gen:
- fixes T :: "'b::real_normed_vector set"
- assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
- shows "closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
-proof -
- have eq: "((\<lambda>x. Pair x (f x)) ` S) = {z. z \<in> S \<times> T \<and> (f \<circ> fst)z - snd z \<in> {0}}"
- using fim by auto
- show ?thesis
- apply (subst eq)
- apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
- by auto
-qed
-
-lemma continuous_closed_graph_eq:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "compact T" and fim: "f ` S \<subseteq> T"
- shows "continuous_on S f \<longleftrightarrow>
- closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. 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 (subtopology euclidean T) U"
- have eq: "{x. x \<in> S \<and> f x \<in> U} = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
- by (force simp: image_iff)
- show "closedin (subtopology euclidean S) {x \<in> S. f x \<in> U}"
- by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
- qed
- with continuous_closed_graph_gen assms show ?thesis by blast
-qed
-
-lemma continuous_closed_graph:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- assumes "closed S" and contf: "continuous_on S f"
- shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
- apply (rule closedin_closed_trans)
- apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
- by (simp add: \<open>closed S\<close> closed_Times)
-
-lemma continuous_from_closed_graph:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
- shows "continuous_on S f"
- using fim clo
- by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
-
-lemma continuous_on_Un_local_open:
- assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
- and opT: "openin (subtopology euclidean (S \<union> T)) T"
- and contf: "continuous_on S f" and contg: "continuous_on T f"
- shows "continuous_on (S \<union> T) f"
-using pasting_lemma [of "{S,T}" "S \<union> T" "\<lambda>i. i" "\<lambda>i. f" f] contf contg opS opT by blast
-
-lemma continuous_on_cases_local_open:
- assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
- and opT: "openin (subtopology euclidean (S \<union> T)) T"
- and contf: "continuous_on S f" and contg: "continuous_on T g"
- and fg: "\<And>x. x \<in> S \<and> ~P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
- shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
-proof -
- have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x" "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
- by (simp_all add: fg)
- then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
- by (simp_all add: contf contg cong: continuous_on_cong)
- then show ?thesis
- by (rule continuous_on_Un_local_open [OF opS opT])
-qed
-
-subsection\<open>The union of two collinear segments is another segment\<close>
-
-proposition in_convex_hull_exchange:
- fixes a :: "'a::euclidean_space"
- assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
- obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
-proof (cases "a \<in> S")
- case True
- with xS insert_Diff that show ?thesis by fastforce
-next
- case False
- show ?thesis
- proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
- case True
- then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
- and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
- using a by (auto simp: convex_hull_finite)
- obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
- and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
- using True xS by (auto simp: convex_hull_finite)
- show ?thesis
- proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
- case True
- then obtain b where b: "b \<in> S" "v b = 0"
- by blast
- show ?thesis
- proof
- have fin: "finite (insert a (S - {b}))"
- using sum.infinite v1 by fastforce
- show "x \<in> convex hull insert a (S - {b})"
- unfolding convex_hull_finite [OF fin] mem_Collect_eq
- proof (intro conjI exI ballI)
- have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
- (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
- apply (rule sum.mono_neutral_right)
- using fin by auto
- also have "... = (\<Sum>x \<in> S - {b}. v x)"
- using b False by (auto intro!: sum.cong split: if_split_asm)
- also have "... = (\<Sum>x\<in>S. v x)"
- by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
- finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
- by (simp add: v1)
- show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
- by (auto simp: v0)
- have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
- (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
- apply (rule sum.mono_neutral_right)
- using fin by auto
- also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
- using b False by (auto intro!: sum.cong split: if_split_asm)
- also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
- by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert real_vector.scale_eq_0_iff sum_diff1)
- finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
- by (simp add: vx)
- qed
- qed (rule \<open>b \<in> S\<close>)
- next
- case False
- have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
- by (simp add: True that)
- have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
- using True v1 by (auto intro: Max_in)
- then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
- by blast
- then have "0 \<noteq> 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 \<open>b \<in> S\<close> u0 v0 by force+
- have fin: "finite (insert a (S - {b}))"
- using sum.infinite v1 by fastforce
- show ?thesis
- proof
- show "x \<in> convex hull insert a (S - {b})"
- unfolding convex_hull_finite [OF fin] mem_Collect_eq
- proof (intro conjI exI ballI)
- have "(\<Sum>x \<in> 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 + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
- using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
- apply (rule sum.cong, auto)
- done
- also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
- by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
- also have "... = (\<Sum>x\<in>S. v x)"
- using \<open>0 < u b\<close> True by (simp add: Groups_Big.sum_diff1 u1 field_simps)
- finally show "sum (\<lambda>x. 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 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
- if "i \<in> insert a (S - {b})" for i
- using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
- by (auto simp: field_simps split: if_split_asm)
- have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
- (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
- using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
- apply (rule sum.cong, auto)
- done
- also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
- by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left real_vector.scale_sum_right)
- also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
- using \<open>0 < u b\<close> True by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
- finally
- show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
- by (simp add: vx)
- qed
- qed (rule \<open>b \<in> S\<close>)
- qed
- next
- case False
- obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
- using xS by (auto simp: caratheodory [of S])
- with False obtain b where b: "b \<in> S" "b \<notin> T"
- by (metis antisym subsetI)
- show ?thesis
- proof
- show "x \<in> convex hull insert a (S - {b})"
- using \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
- qed (rule \<open>b \<in> S\<close>)
- qed
-qed
-
-lemma convex_hull_exchange_Union:
- fixes a :: "'a::euclidean_space"
- assumes "a \<in> convex hull S"
- shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
-proof
- show "?lhs \<subseteq> ?rhs"
- by (blast intro: in_convex_hull_exchange [OF assms])
- show "?rhs \<subseteq> ?lhs"
- proof clarify
- fix x b
- assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
- then show "x \<in> convex hull S" if "b \<in> 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 \<in> closed_segment a c"
- shows "closed_segment a b \<union> 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} \<union> convex hull {b, c} = (\<Union>ba\<in>{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 \<in> open_segment a c"
- shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
-proof -
- have b: "b \<in> closed_segment a c"
- by (simp add: assms open_closed_segment)
- have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
- if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
- proof -
- have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
- 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
- using Un_closed_segment [OF b]
- apply (simp add: closed_segment_eq_open)
- apply (rule equalityI)
- using assms
- apply (simp add: b subset_open_segment)
- using * by (simp add: insert_commute)
-qed
-
-subsection\<open>Covering an open set by a countable chain of compact sets\<close>
-
-proposition open_Union_compact_subsets:
- fixes S :: "'a::euclidean_space set"
- assumes "open S"
- obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
- "\<And>n. C n \<subseteq> interior(C(Suc n))"
- "\<Union>(range C) = S"
- "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (rule_tac C = "\<lambda>n. {}" in that) auto
-next
- case False
- then obtain a where "a \<in> S"
- by auto
- let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
- have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
- if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
- and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
- proof -
- have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
- by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
- have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
- by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
- obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
- proof (rule compactE_image [OF \<open>compact K\<close>])
- show "K \<subseteq> (\<Union>n. interior (f n))"
- using \<open>K \<subseteq> S\<close> \<open>UNION UNIV f = S\<close> * by blast
- qed auto
- { fix n
- assume n: "Max I \<le> n"
- have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
- by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
- then have "K \<subseteq> f n"
- using I by auto
- }
- then show ?thesis
- by blast
- qed
- moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
- ((\<Union>(range f) = S))"
- proof (intro exI conjI allI)
- show "\<And>n. compact (?C n)"
- by (auto simp: compact_diff open_sums)
- show "\<And>n. ?C n \<subseteq> S"
- by auto
- show "?C n \<subseteq> interior (?C (Suc n))" for n
- proof (simp add: interior_diff, rule Diff_mono)
- show "cball a (real n) \<subseteq> ball a (1 + real n)"
- by (simp add: cball_subset_ball_iff)
- have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
- using assms by (auto intro: closed_compact_sums)
- have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
- \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
- by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
- also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
- apply (intro UN_mono order_refl)
- apply (simp add: cball_subset_ball_iff divide_simps)
- done
- finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
- \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
- qed
- have "S \<subseteq> UNION UNIV ?C"
- proof
- fix x
- assume x: "x \<in> S"
- then obtain e where "e > 0" and e: "ball x e \<subseteq> 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) \<le> real N2"
- by (meson real_arch_simple)
- have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
- using \<open>N1 > 0\<close> by (auto simp: divide_simps)
- have "x \<noteq> y + z" if "y \<notin> 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: \<open>0 < e\<close>)
- then have "1 / (1 + (real N1 + real N2)) < e"
- using N1 \<open>e > 0\<close>
- by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
- then have "x - z \<in> ball x e"
- using that by simp
- then have "x - z \<in> S"
- using e by blast
- with that show ?thesis
- by auto
- qed
- with N2 show "x \<in> UNION UNIV ?C"
- by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
- qed
- then show "UNION UNIV ?C = S" by auto
- qed
- ultimately show ?thesis
- using that by metis
-qed
-
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Starlike.thy Thu Jul 20 14:05:29 2017 +0100
@@ -0,0 +1,7896 @@
+(* 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
+*)
+
+section \<open>Line segments, Starlike Sets, etc.\<close>
+
+theory Starlike
+ imports Convex_Euclidean_Space
+
+begin
+
+definition midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
+ where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
+
+definition 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 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:
+ "linear f \<Longrightarrow> closed_segment (f a) (f b) = f ` (closed_segment a b)"
+ by (force simp add: in_segment linear_add_cmul)
+
+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 midpoint_idem [simp]: "midpoint x x = x"
+ unfolding midpoint_def
+ unfolding scaleR_right_distrib
+ unfolding scaleR_left_distrib[symmetric]
+ by auto
+
+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>Starlike sets\<close>
+
+definition "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
+
+lemma starlike_UNIV [simp]: "starlike UNIV"
+ by (simp add: starlike_def)
+
+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_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 convex_imp_starlike:
+ "convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
+ unfolding convex_contains_segment starlike_def by auto
+
+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 real_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]:
+ fixes a :: "'a::euclidean_space"
+ shows "closure(open_segment a b) = (if a = b then {} else closed_segment a b)"
+proof -
+ have "closure ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1}) = (\<lambda>u. u *\<^sub>R (b - a)) ` closure {0<..<1}" if "a \<noteq> b"
+ apply (rule closure_injective_linear_image [symmetric])
+ apply (simp add:)
+ using that by (simp add: inj_on_def)
+ then show ?thesis
+ by (simp add: segment_image_interval segment_eq_compose closure_greaterThanLessThan [symmetric]
+ closure_translation image_comp [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 show ?thesis
+ apply (simp add: open_segment_image_interval segment_eq_compose)
+ by (metis image_comp convex_translation)
+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 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 \<in> rel_interior S" "b \<in> closure S"
+ shows "open_segment a b \<subseteq> rel_interior S"
+proof
+ fix x
+ have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
+ by (simp add: algebra_simps)
+ assume "x \<in> open_segment a b"
+ then show "x \<in> 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 \<Union>x \<in> 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 \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
+ shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
+proof
+ have "T = {} \<Longrightarrow> z \<notin> S"
+ using dis z by (auto simp add: disjnt_def)
+ then show "?lhs \<subseteq> ?rhs"
+ proof (clarsimp simp add: convex_hull_insert_segments)
+ fix x y
+ assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
+ have "y \<in> closure C"
+ by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
+ moreover have "x \<notin> rel_interior C"
+ by (meson \<open>x \<in> S\<close> dis disjnt_iff)
+ moreover have "x \<in> open_segment z y \<union> {z, y}"
+ using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
+ ultimately show "x \<in> convex hull T"
+ using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
+ using y z by blast
+ qed
+ show "?rhs \<subseteq> ?lhs"
+ by (meson hull_mono inf_mono subset_insertI subset_refl)
+qed
+
+subsection\<open>More results about segments\<close>
+
+lemma dist_half_times2:
+ fixes a :: "'a :: real_normed_vector"
+ shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a+b) (2 *\<^sub>R x)"
+proof -
+ have "norm ((1 / 2) *\<^sub>R (a + b) - x) * 2 = norm (2 *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x))"
+ by simp
+ also have "... = norm ((a + b) - 2 *\<^sub>R x)"
+ by (simp add: real_vector.scale_right_diff_distrib)
+ finally show ?thesis
+ by (simp only: dist_norm)
+qed
+
+lemma closed_segment_as_ball:
+ "closed_segment a b = affine hull {a,b} \<inter> cball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
+proof (cases "b = a")
+ case True then show ?thesis by (auto simp: hull_inc)
+next
+ case False
+ then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
+ (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)" for x
+ proof -
+ have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
+ ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a))"
+ unfolding eq_diff_eq [symmetric] by simp
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((a+b) - (2 *\<^sub>R x)) \<le> norm (b - a))"
+ by (simp add: dist_half_times2) (simp add: dist_norm)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) \<le> norm (b - a))"
+ by auto
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((1 - u * 2) *\<^sub>R (b - a)) \<le> norm (b - a))"
+ by (simp add: algebra_simps scaleR_2)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ \<bar>1 - u * 2\<bar> * norm (b - a) \<le> norm (b - a))"
+ by simp
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> \<le> 1)"
+ by (simp add: mult_le_cancel_right2 False)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)"
+ by auto
+ finally show ?thesis .
+ qed
+ show ?thesis
+ by (simp add: affine_hull_2 Set.set_eq_iff closed_segment_def *)
+qed
+
+lemma open_segment_as_ball:
+ "open_segment a b =
+ affine hull {a,b} \<inter> ball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
+proof (cases "b = a")
+ case True then show ?thesis by (auto simp: hull_inc)
+next
+ case False
+ then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
+ (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)" for x
+ proof -
+ have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
+ ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
+ dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a))"
+ unfolding eq_diff_eq [symmetric] by simp
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((a+b) - (2 *\<^sub>R x)) < norm (b - a))"
+ by (simp add: dist_half_times2) (simp add: dist_norm)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) < norm (b - a))"
+ by auto
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ norm ((1 - u * 2) *\<^sub>R (b - a)) < norm (b - a))"
+ by (simp add: algebra_simps scaleR_2)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
+ \<bar>1 - u * 2\<bar> * norm (b - a) < norm (b - a))"
+ by simp
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> < 1)"
+ by (simp add: mult_le_cancel_right2 False)
+ also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)"
+ by auto
+ finally show ?thesis .
+ qed
+ show ?thesis
+ using False by (force simp: affine_hull_2 Set.set_eq_iff open_segment_image_interval *)
+qed
+
+lemmas segment_as_ball = closed_segment_as_ball open_segment_as_ball
+
+lemma closed_segment_neq_empty [simp]: "closed_segment a b \<noteq> {}"
+ by auto
+
+lemma open_segment_eq_empty [simp]: "open_segment a b = {} \<longleftrightarrow> a = b"
+proof -
+ { assume a1: "open_segment a b = {}"
+ have "{} \<noteq> {0::real<..<1}"
+ by simp
+ then have "a = b"
+ using a1 open_segment_image_interval by fastforce
+ } then show ?thesis by auto
+qed
+
+lemma open_segment_eq_empty' [simp]: "{} = open_segment a b \<longleftrightarrow> a = b"
+ using open_segment_eq_empty by blast
+
+lemmas segment_eq_empty = closed_segment_neq_empty open_segment_eq_empty
+
+lemma inj_segment:
+ fixes a :: "'a :: real_vector"
+ assumes "a \<noteq> b"
+ shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
+proof
+ fix x y
+ assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b"
+ then have "x *\<^sub>R (b - a) = y *\<^sub>R (b - a)"
+ by (simp add: algebra_simps)
+ with assms show "x = y"
+ by (simp add: real_vector.scale_right_imp_eq)
+qed
+
+lemma finite_closed_segment [simp]: "finite(closed_segment a b) \<longleftrightarrow> a = b"
+ apply auto
+ apply (rule ccontr)
+ apply (simp add: segment_image_interval)
+ using infinite_Icc [OF zero_less_one] finite_imageD [OF _ inj_segment] apply blast
+ done
+
+lemma finite_open_segment [simp]: "finite(open_segment a b) \<longleftrightarrow> a = b"
+ by (auto simp: open_segment_def)
+
+lemmas finite_segment = finite_closed_segment finite_open_segment
+
+lemma closed_segment_eq_sing: "closed_segment a b = {c} \<longleftrightarrow> a = c \<and> b = c"
+ by auto
+
+lemma open_segment_eq_sing: "open_segment a b \<noteq> {c}"
+ by (metis finite_insert finite_open_segment insert_not_empty open_segment_image_interval)
+
+lemmas segment_eq_sing = closed_segment_eq_sing open_segment_eq_sing
+
+lemma subset_closed_segment:
+ "closed_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
+ a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
+ by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
+
+lemma subset_co_segment:
+ "closed_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
+ a \<in> open_segment c d \<and> b \<in> open_segment c d"
+using closed_segment_subset by blast
+
+lemma subset_open_segment:
+ fixes a :: "'a::euclidean_space"
+ shows "open_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
+ a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
+ (is "?lhs = ?rhs")
+proof (cases "a = b")
+ case True then show ?thesis by simp
+next
+ case False show ?thesis
+ proof
+ assume rhs: ?rhs
+ with \<open>a \<noteq> b\<close> have "c \<noteq> d"
+ using closed_segment_idem singleton_iff by auto
+ have "\<exists>uc. (1 - u) *\<^sub>R ((1 - ua) *\<^sub>R c + ua *\<^sub>R d) + u *\<^sub>R ((1 - ub) *\<^sub>R c + ub *\<^sub>R d) =
+ (1 - uc) *\<^sub>R c + uc *\<^sub>R d \<and> 0 < uc \<and> uc < 1"
+ if neq: "(1 - ua) *\<^sub>R c + ua *\<^sub>R d \<noteq> (1 - ub) *\<^sub>R c + ub *\<^sub>R d" "c \<noteq> d"
+ and "a = (1 - ua) *\<^sub>R c + ua *\<^sub>R d" "b = (1 - ub) *\<^sub>R c + ub *\<^sub>R d"
+ and u: "0 < u" "u < 1" and uab: "0 \<le> ua" "ua \<le> 1" "0 \<le> ub" "ub \<le> 1"
+ for u ua ub
+ proof -
+ have "ua \<noteq> ub"
+ using neq by auto
+ moreover have "(u - 1) * ua \<le> 0" using u uab
+ by (simp add: mult_nonpos_nonneg)
+ ultimately have lt: "(u - 1) * ua < u * ub" using u uab
+ by (metis antisym_conv diff_ge_0_iff_ge le_less_trans mult_eq_0_iff mult_le_0_iff not_less)
+ have "p * ua + q * ub < p+q" if p: "0 < p" and q: "0 < q" for p q
+ proof -
+ have "\<not> p \<le> 0" "\<not> q \<le> 0"
+ using p q not_less by blast+
+ then show ?thesis
+ by (metis \<open>ua \<noteq> ub\<close> add_less_cancel_left add_less_cancel_right add_mono_thms_linordered_field(5)
+ less_eq_real_def mult_cancel_left1 mult_less_cancel_left2 uab(2) uab(4))
+ qed
+ then have "(1 - u) * ua + u * ub < 1" using u \<open>ua \<noteq> ub\<close>
+ by (metis diff_add_cancel diff_gt_0_iff_gt)
+ with lt show ?thesis
+ by (rule_tac x="ua + u*(ub-ua)" in exI) (simp add: algebra_simps)
+ qed
+ with rhs \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close> show ?lhs
+ unfolding open_segment_image_interval closed_segment_def
+ by (fastforce simp add:)
+ next
+ assume lhs: ?lhs
+ with \<open>a \<noteq> b\<close> have "c \<noteq> d"
+ by (meson finite_open_segment rev_finite_subset)
+ have "closure (open_segment a b) \<subseteq> closure (open_segment c d)"
+ using lhs closure_mono by blast
+ then have "closed_segment a b \<subseteq> closed_segment c d"
+ by (simp add: \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close>)
+ then show ?rhs
+ by (force simp: \<open>a \<noteq> b\<close>)
+ qed
+qed
+
+lemma subset_oc_segment:
+ fixes a :: "'a::euclidean_space"
+ shows "open_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
+ a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
+apply (simp add: subset_open_segment [symmetric])
+apply (rule iffI)
+ apply (metis closure_closed_segment closure_mono closure_open_segment subset_closed_segment subset_open_segment)
+apply (meson dual_order.trans segment_open_subset_closed)
+done
+
+lemmas subset_segment = subset_closed_segment subset_co_segment subset_oc_segment subset_open_segment
+
+
+subsection\<open>Betweenness\<close>
+
+definition "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
+ unfolding between_def split_conv
+ by (auto simp add: 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)
+ show ?thesis
+ unfolding between_def split_conv closed_segment_def mem_Collect_eq
+ apply rule
+ apply (elim exE conjE)
+ apply (subst dist_triangle_eq)
+ proof -
+ fix u
+ assume as: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
+ then have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
+ unfolding as(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 as(2,3)
+ by (auto simp add: field_simps)
+ next
+ assume as: "dist a b = dist a x + dist x b"
+ have "norm (a - x) / norm (a - b) \<le> 1"
+ using Fal2 unfolding as[unfolded dist_norm] norm_ge_zero by auto
+ then show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
+ apply (rule_tac x="dist a x / dist a b" in exI)
+ unfolding dist_norm
+ apply (subst euclidean_eq_iff)
+ apply rule
+ defer
+ apply rule
+ prefer 3
+ apply rule
+ proof -
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^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 as[unfolded dist_norm]
+ using as[unfolded dist_triangle_eq]
+ apply -
+ apply (subst (asm) euclidean_eq_iff)
+ using i
+ apply (erule_tac x=i in ballE)
+ apply (auto simp add: field_simps inner_simps)
+ done
+ finally show "x \<bullet> i =
+ ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
+ by auto
+ qed (insert Fal2, auto)
+ qed
+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
+ apply(rule_tac[!] *)
+ unfolding euclidean_eq_iff[where 'a='a]
+ apply (auto simp add: field_simps inner_simps)
+ done
+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)
+
+
+subsection \<open>Shrinking towards the interior of a convex set\<close>
+
+lemma mem_interior_convex_shrink:
+ fixes s :: "'a::euclidean_space set"
+ assumes "convex s"
+ and "c \<in> interior s"
+ and "x \<in> s"
+ and "0 < e"
+ and "e \<le> 1"
+ shows "x - e *\<^sub>R (x - c) \<in> interior s"
+proof -
+ obtain d where "d > 0" and d: "ball c d \<subseteq> s"
+ using assms(2) unfolding mem_interior by auto
+ show ?thesis
+ unfolding mem_interior
+ apply (rule_tac x="e*d" in exI)
+ apply rule
+ defer
+ unfolding subset_eq Ball_def mem_ball
+ proof (rule, rule)
+ fix y
+ assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
+ have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
+ using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
+ have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
+ unfolding dist_norm
+ unfolding norm_scaleR[symmetric]
+ apply (rule arg_cong[where f=norm])
+ using \<open>e > 0\<close>
+ by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
+ also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
+ by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
+ also have "\<dots> < d"
+ using as[unfolded dist_norm] and \<open>e > 0\<close>
+ by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
+ finally show "y \<in> s"
+ apply (subst *)
+ apply (rule assms(1)[unfolded convex_alt,rule_format])
+ apply (rule d[unfolded subset_eq,rule_format])
+ unfolding mem_ball
+ using assms(3-5)
+ apply auto
+ done
+ qed (insert \<open>e>0\<close> \<open>d>0\<close>, auto)
+qed
+
+lemma mem_interior_closure_convex_shrink:
+ fixes s :: "'a::euclidean_space set"
+ assumes "convex s"
+ and "c \<in> interior s"
+ and "x \<in> closure s"
+ and "0 < e"
+ and "e \<le> 1"
+ shows "x - e *\<^sub>R (x - c) \<in> interior s"
+proof -
+ obtain d where "d > 0" and d: "ball c d \<subseteq> s"
+ using assms(2) unfolding mem_interior by auto
+ have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d"
+ proof (cases "x \<in> s")
+ case True
+ then show ?thesis
+ using \<open>e > 0\<close> \<open>d > 0\<close>
+ apply (rule_tac bexI[where x=x])
+ apply (auto)
+ done
+ 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 \<in> s" "y \<noteq> x" "dist y x < 1"
+ using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
+ then show ?thesis
+ apply (rule_tac x=y in bexI)
+ unfolding True
+ using \<open>d > 0\<close>
+ apply auto
+ done
+ next
+ case False
+ then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
+ using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
+ then obtain y where "y \<in> s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
+ using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
+ then show ?thesis
+ apply (rule_tac x=y in bexI)
+ unfolding dist_norm
+ using pos_less_divide_eq[OF *]
+ apply auto
+ done
+ qed
+ qed
+ then obtain y where "y \<in> s" and y: "norm (y - x) * (1 - e) < e * d"
+ by auto
+ define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
+ have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
+ unfolding z_def using \<open>e > 0\<close>
+ by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
+ have "z \<in> interior s"
+ apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
+ unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
+ apply (auto simp add:field_simps norm_minus_commute)
+ done
+ then show ?thesis
+ unfolding *
+ apply -
+ apply (rule mem_interior_convex_shrink)
+ using assms(1,4-5) \<open>y\<in>s\<close>
+ apply auto
+ done
+qed
+
+lemma in_interior_closure_convex_segment:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
+ shows "open_segment a b \<subseteq> interior S"
+proof (clarsimp simp: in_segment)
+ fix u::real
+ assume u: "0 < u" "u < 1"
+ have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
+ by (simp add: algebra_simps)
+ also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
+ by simp
+ finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
+qed
+
+lemma closure_open_Int_superset:
+ assumes "open S" "S \<subseteq> closure T"
+ shows "closure(S \<inter> T) = closure S"
+proof -
+ have "closure S \<subseteq> closure(S \<inter> T)"
+ by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
+ then show ?thesis
+ by (simp add: closure_mono dual_order.antisym)
+qed
+
+lemma convex_closure_interior:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" and int: "interior S \<noteq> {}"
+ shows "closure(interior S) = closure S"
+proof -
+ obtain a where a: "a \<in> interior S"
+ using int by auto
+ have "closure S \<subseteq> closure(interior S)"
+ proof
+ fix x
+ assume x: "x \<in> closure S"
+ show "x \<in> closure (interior S)"
+ proof (cases "x=a")
+ case True
+ then show ?thesis
+ using \<open>a \<in> interior S\<close> closure_subset by blast
+ next
+ case False
+ show ?thesis
+ proof (clarsimp simp add: closure_def islimpt_approachable)
+ fix e::real
+ assume xnotS: "x \<notin> interior S" and "0 < e"
+ show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
+ proof (intro bexI conjI)
+ show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
+ using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
+ show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
+ using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
+ show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
+ apply (clarsimp simp add: min_def a)
+ apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
+ using \<open>0 < e\<close> False apply (auto simp: divide_simps)
+ done
+ qed
+ qed
+ qed
+ qed
+ then show ?thesis
+ by (simp add: closure_mono interior_subset subset_antisym)
+qed
+
+lemma closure_convex_Int_superset:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
+ shows "closure(S \<inter> T) = closure S"
+proof -
+ have "closure S \<subseteq> closure(interior S)"
+ by (simp add: convex_closure_interior assms)
+ also have "... \<subseteq> closure (S \<inter> T)"
+ using interior_subset [of S] assms
+ by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
+ finally show ?thesis
+ by (simp add: closure_mono dual_order.antisym)
+qed
+
+
+subsection \<open>Some obvious but surprisingly hard simplex lemmas\<close>
+
+lemma simplex:
+ assumes "finite s"
+ and "0 \<notin> s"
+ shows "convex hull (insert 0 s) =
+ {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y)}"
+ unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
+ apply (rule set_eqI, rule)
+ unfolding mem_Collect_eq
+ apply (erule_tac[!] exE)
+ apply (erule_tac[!] conjE)+
+ unfolding sum_clauses(2)[OF \<open>finite s\<close>]
+ apply (rule_tac x=u in exI)
+ defer
+ apply (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u s else u x" in exI)
+ using assms(2)
+ unfolding if_smult and sum_delta_notmem[OF assms(2)]
+ apply auto
+ done
+
+lemma substd_simplex:
+ assumes d: "d \<subseteq> Basis"
+ shows "convex hull (insert 0 d) =
+ {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
+ (is "convex hull (insert 0 ?p) = ?s")
+proof -
+ let ?D = d
+ have "0 \<notin> ?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 \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
+ apply (rule set_eqI)
+ unfolding mem_Collect_eq
+ apply rule
+ apply (elim exE conjE)
+ apply (erule_tac[2] conjE)+
+ proof -
+ fix x :: "'a::euclidean_space"
+ fix u
+ assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+ have *: "\<forall>i\<in>Basis. i:d \<longrightarrow> u i = x\<bullet>i"
+ and "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+ using as(3)
+ unfolding substdbasis_expansion_unique[OF assms]
+ by auto
+ then have **: "sum u ?D = sum (op \<bullet> x) ?D"
+ apply -
+ apply (rule sum.cong)
+ using assms
+ apply auto
+ done
+ have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1"
+ proof (rule,rule)
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "i \<in> d \<Longrightarrow> 0 \<le> x\<bullet>i"
+ unfolding *[rule_format,OF i,symmetric]
+ apply (rule_tac as(1)[rule_format])
+ apply auto
+ done
+ moreover have "i \<notin> d \<Longrightarrow> 0 \<le> x\<bullet>i"
+ using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close>[rule_format, OF i] by auto
+ ultimately show "0 \<le> x\<bullet>i" by auto
+ qed (insert as(2)[unfolded **], auto)
+ then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (op \<bullet> x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+ using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close> by auto
+ next
+ fix x :: "'a::euclidean_space"
+ assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum (op \<bullet> x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+ show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+ using as d
+ unfolding substdbasis_expansion_unique[OF assms]
+ apply (rule_tac x="inner x" in exI)
+ apply auto
+ done
+ qed
+qed
+
+lemma std_simplex:
+ "convex hull (insert 0 Basis) =
+ {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
+ using substd_simplex[of Basis] by auto
+
+lemma interior_std_simplex:
+ "interior (convex hull (insert 0 Basis)) =
+ {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
+ apply (rule set_eqI)
+ unfolding mem_interior std_simplex
+ unfolding subset_eq mem_Collect_eq Ball_def mem_ball
+ unfolding Ball_def[symmetric]
+ apply rule
+ apply (elim exE conjE)
+ defer
+ apply (erule conjE)
+proof -
+ fix x :: 'a
+ fix e
+ assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum (op \<bullet> xa) Basis \<le> 1"
+ show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum (op \<bullet> x) Basis < 1"
+ apply safe
+ proof -
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then show "0 < x \<bullet> i"
+ using as[THEN spec[where x="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close>
+ unfolding dist_norm
+ by (auto elim!: ballE[where x=i] simp: inner_simps)
+ next
+ have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
+ unfolding dist_norm
+ by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
+ have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
+ x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
+ by (auto simp: SOME_Basis inner_Basis inner_simps)
+ then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
+ sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
+ apply (rule_tac sum.cong)
+ apply auto
+ done
+ have "sum (op \<bullet> x) Basis < sum (op \<bullet> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
+ unfolding * sum.distrib
+ using \<open>e > 0\<close> DIM_positive[where 'a='a]
+ apply (subst sum.delta')
+ apply (auto simp: SOME_Basis)
+ done
+ also have "\<dots> \<le> 1"
+ using **
+ apply (drule_tac as[rule_format])
+ apply auto
+ done
+ finally show "sum (op \<bullet> x) Basis < 1" by auto
+ qed
+next
+ fix x :: 'a
+ assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum (op \<bullet> x) Basis < 1"
+ obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
+ let ?d = "(1 - sum (op \<bullet> x) Basis) / real (DIM('a))"
+ show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
+ proof (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI, intro conjI impI allI)
+ fix y
+ assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
+ have "sum (op \<bullet> y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
+ proof (rule sum_mono)
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
+ apply -
+ apply (rule le_less_trans)
+ using Basis_le_norm[OF i, of "y - x"]
+ using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
+ apply (auto simp add: norm_minus_commute inner_diff_left)
+ done
+ then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
+ qed
+ also have "\<dots> \<le> 1"
+ unfolding sum.distrib sum_constant
+ by (auto simp add: Suc_le_eq)
+ finally show "sum (op \<bullet> y) Basis \<le> 1" .
+ show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
+ proof safe
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "norm (x - y) < x\<bullet>i"
+ apply (rule less_le_trans)
+ apply (rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1])
+ using i
+ apply auto
+ done
+ then show "0 \<le> y\<bullet>i"
+ using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
+ by (auto simp: inner_simps)
+ qed
+ next
+ have "Min ((op \<bullet> x) ` Basis) > 0"
+ using as by simp
+ moreover have "?d > 0"
+ using as by (auto simp: Suc_le_eq)
+ ultimately show "0 < min (Min (op \<bullet> x ` Basis)) ((1 - sum (op \<bullet> x) Basis) / real DIM('a))"
+ by linarith
+ qed
+qed
+
+lemma interior_std_simplex_nonempty:
+ obtains a :: "'a::euclidean_space" where
+ "a \<in> interior(convex hull (insert 0 Basis))"
+proof -
+ let ?D = "Basis :: 'a set"
+ let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ have "?a \<bullet> i = inverse (2 * real DIM('a))"
+ by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
+ (simp_all add: sum.If_cases i) }
+ note ** = this
+ show ?thesis
+ apply (rule that[of ?a])
+ unfolding interior_std_simplex mem_Collect_eq
+ proof safe
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ show "0 < ?a \<bullet> i"
+ unfolding **[OF i] by (auto simp add: Suc_le_eq DIM_positive)
+ next
+ have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
+ apply (rule sum.cong)
+ apply rule
+ apply auto
+ done
+ also have "\<dots> < 1"
+ unfolding sum_constant divide_inverse[symmetric]
+ by (auto simp add: field_simps)
+ finally show "sum (op \<bullet> ?a) ?D < 1" by auto
+ qed
+qed
+
+lemma rel_interior_substd_simplex:
+ assumes d: "d \<subseteq> Basis"
+ shows "rel_interior (convex hull (insert 0 d)) =
+ {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
+ (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
+proof -
+ have "finite d"
+ apply (rule finite_subset)
+ using assms
+ apply auto
+ done
+ 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 ?p)) =
+ {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
+ using affine_hull_convex_hull affine_hull_substd_basis assms by auto
+ have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>d. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
+ by auto
+ {
+ fix x :: "'a::euclidean_space"
+ assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
+ then obtain e where e0: "e > 0" and
+ "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
+ using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
+ then have as: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
+ (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum (op \<bullet> xa) d \<le> 1"
+ unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
+ have x0: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+ using x rel_interior_subset substd_simplex[OF assms] by auto
+ have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+ apply rule
+ apply rule
+ proof -
+ fix i :: 'a
+ assume "i \<in> d"
+ then have "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> ia"
+ apply -
+ apply (rule as[rule_format,THEN conjunct1])
+ unfolding dist_norm
+ using d \<open>e > 0\<close> x0
+ apply (auto simp: inner_simps inner_Basis)
+ done
+ then show "0 < x \<bullet> i"
+ apply (erule_tac x=i in ballE)
+ using \<open>e > 0\<close> \<open>i \<in> d\<close> d
+ apply (auto simp: inner_simps inner_Basis)
+ done
+ next
+ obtain a where a: "a \<in> d"
+ using \<open>d \<noteq> {}\<close> by auto
+ then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
+ using \<open>e > 0\<close> norm_Basis[of a] d
+ unfolding dist_norm
+ by auto
+ have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
+ using a d by (auto simp: inner_simps inner_Basis)
+ then have *: "sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d =
+ sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
+ using d by (intro sum.cong) auto
+ have "a \<in> Basis"
+ using \<open>a \<in> d\<close> d by auto
+ then have h1: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
+ using x0 d \<open>a\<in>d\<close> by (auto simp add: inner_add_left inner_Basis)
+ have "sum (op \<bullet> x) d < sum (op \<bullet> (x + (e / 2) *\<^sub>R a)) d"
+ unfolding * sum.distrib
+ using \<open>e > 0\<close> \<open>a \<in> d\<close>
+ using \<open>finite d\<close>
+ by (auto simp add: sum.delta')
+ also have "\<dots> \<le> 1"
+ using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
+ by auto
+ finally show "sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+ using x0 by auto
+ qed
+ }
+ moreover
+ {
+ fix x :: "'a::euclidean_space"
+ assume as: "x \<in> ?s"
+ have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
+ by auto
+ moreover have "\<forall>i. i \<in> d \<or> i \<notin> d" by auto
+ ultimately
+ have "\<forall>i. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
+ by metis
+ then have h2: "x \<in> convex hull (insert 0 ?p)"
+ using as assms
+ unfolding substd_simplex[OF assms] by fastforce
+ obtain a where a: "a \<in> d"
+ using \<open>d \<noteq> {}\<close> by auto
+ let ?d = "(1 - sum (op \<bullet> x) d) / real (card d)"
+ have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
+ by (simp add: card_gt_0_iff)
+ have "Min ((op \<bullet> x) ` d) > 0"
+ using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_gr_iff)
+ moreover have "?d > 0" using as using \<open>0 < card d\<close> by auto
+ ultimately have h3: "min (Min ((op \<bullet> x) ` d)) ?d > 0"
+ by auto
+
+ have "x \<in> rel_interior (convex hull (insert 0 ?p))"
+ unfolding rel_interior_ball mem_Collect_eq h0
+ apply (rule,rule h2)
+ unfolding substd_simplex[OF assms]
+ apply (rule_tac x="min (Min ((op \<bullet> x) ` d)) ?d" in exI)
+ apply (rule, rule h3)
+ apply safe
+ unfolding mem_ball
+ proof -
+ fix y :: 'a
+ assume y: "dist x y < min (Min (op \<bullet> x ` d)) ?d"
+ assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
+ have "sum (op \<bullet> y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
+ proof (rule sum_mono)
+ fix i
+ assume "i \<in> d"
+ with d have i: "i \<in> Basis"
+ by auto
+ have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
+ apply (rule le_less_trans)
+ using Basis_le_norm[OF i, of "y - x"]
+ using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
+ apply (auto simp add: norm_minus_commute inner_simps)
+ done
+ then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
+ qed
+ also have "\<dots> \<le> 1"
+ unfolding sum.distrib sum_constant using \<open>0 < card d\<close>
+ by auto
+ finally show "sum (op \<bullet> y) d \<le> 1" .
+
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then show "0 \<le> y\<bullet>i"
+ proof (cases "i\<in>d")
+ case True
+ have "norm (x - y) < x\<bullet>i"
+ using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
+ using Min_gr_iff[of "op \<bullet> x ` d" "norm (x - y)"] \<open>0 < card d\<close> \<open>i:d\<close>
+ by (simp add: card_gt_0_iff)
+ then show "0 \<le> y\<bullet>i"
+ using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
+ by (auto simp: inner_simps)
+ qed (insert y2, auto)
+ qed
+ }
+ ultimately have
+ "\<And>x. x \<in> rel_interior (convex hull insert 0 d) \<longleftrightarrow>
+ x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum (op \<bullet> x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
+ by blast
+ then show ?thesis by (rule set_eqI)
+ qed
+qed
+
+lemma rel_interior_substd_simplex_nonempty:
+ assumes "d \<noteq> {}"
+ and "d \<subseteq> Basis"
+ obtains a :: "'a::euclidean_space"
+ where "a \<in> rel_interior (convex hull (insert 0 d))"
+proof -
+ let ?D = d
+ let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
+ have "finite d"
+ apply (rule finite_subset)
+ using assms(2)
+ apply auto
+ done
+ then have d1: "0 < real (card d)"
+ using \<open>d \<noteq> {}\<close> by auto
+ {
+ fix i
+ assume "i \<in> d"
+ have "?a \<bullet> i = inverse (2 * real (card d))"
+ apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
+ unfolding inner_sum_left
+ apply (rule sum.cong)
+ using \<open>i \<in> d\<close> \<open>finite d\<close> sum.delta'[of d i "(\<lambda>k. inverse (2 * real (card d)))"]
+ d1 assms(2)
+ by (auto simp: inner_Basis set_rev_mp[OF _ assms(2)])
+ }
+ note ** = this
+ show ?thesis
+ apply (rule that[of ?a])
+ unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
+ proof safe
+ fix i
+ assume "i \<in> d"
+ have "0 < inverse (2 * real (card d))"
+ using d1 by auto
+ also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> d\<close>
+ by auto
+ finally show "0 < ?a \<bullet> i" by auto
+ next
+ have "sum (op \<bullet> ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
+ by (rule sum.cong) (rule refl, rule **)
+ also have "\<dots> < 1"
+ unfolding sum_constant divide_real_def[symmetric]
+ by (auto simp add: field_simps)
+ finally show "sum (op \<bullet> ?a) ?D < 1" by auto
+ next
+ fix i
+ assume "i \<in> Basis" and "i \<notin> d"
+ have "?a \<in> span d"
+ proof (rule span_sum[of d "(\<lambda>b. b /\<^sub>R (2 * real (card d)))" d])
+ {
+ fix x :: "'a::euclidean_space"
+ assume "x \<in> d"
+ then have "x \<in> span d"
+ using span_superset[of _ "d"] by auto
+ then have "x /\<^sub>R (2 * real (card d)) \<in> span d"
+ using span_mul[of x "d" "(inverse (real (card d)) / 2)"] by auto
+ }
+ then show "\<And>x. x\<in>d \<Longrightarrow> x /\<^sub>R (2 * real (card d)) \<in> span d"
+ by auto
+ qed
+ then show "?a \<bullet> i = 0 "
+ using \<open>i \<notin> d\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
+ qed
+qed
+
+
+subsection \<open>Relative interior of convex set\<close>
+
+lemma rel_interior_convex_nonempty_aux:
+ fixes S :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "0 \<in> S"
+ shows "rel_interior S \<noteq> {}"
+proof (cases "S = {0}")
+ case True
+ then show ?thesis using rel_interior_sing by auto
+next
+ case False
+ obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
+ using basis_exists[of S] by auto
+ then have "B \<noteq> {}"
+ using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
+ have "insert 0 B \<le> span B"
+ using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
+ then have "span (insert 0 B) \<le> span B"
+ using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
+ then have "convex hull insert 0 B \<le> span B"
+ using convex_hull_subset_span[of "insert 0 B"] by auto
+ then have "span (convex hull insert 0 B) \<le> 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 \<in> 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 \<Rightarrow> 'n" where
+ fd: "card d = card B" "linear f" "f ` B = d"
+ "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
+ and d: "d \<subseteq> 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 \<noteq> {}"
+ using fd B \<open>B \<noteq> {}\<close> 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)"] \<open>linear f\<close>
+ by auto
+ moreover have "rel_interior (f ` (convex hull insert 0 B)) =
+ f ` rel_interior (convex hull insert 0 B)"
+ apply (rule rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
+ using \<open>bounded_linear f\<close> fd *
+ apply auto
+ done
+ ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
+ using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
+ apply auto
+ apply blast
+ done
+ moreover have "convex hull (insert 0 B) \<subseteq> 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 = {} \<longleftrightarrow> S = {}"
+proof -
+ {
+ assume "S \<noteq> {}"
+ then obtain a where "a \<in> S" by auto
+ then have "0 \<in> op + (-a) ` S"
+ using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
+ then have "rel_interior (op + (-a) ` S) \<noteq> {}"
+ using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"]
+ convex_translation[of S "-a"] assms
+ by auto
+ then have "rel_interior S \<noteq> {}"
+ using rel_interior_translation by auto
+ }
+ then show ?thesis
+ using rel_interior_empty 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 \<in> interior (convex hull (insert 0 S))"
+proof -
+ have "affine hull (insert 0 S) = UNIV"
+ apply (simp add: hull_inc affine_hull_span_0)
+ using assms dim_eq_full indep_card_eq_dim_span by fastforce
+ moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
+ using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
+ ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
+ 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 \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
+ then have "x \<in> S"
+ using rel_interior_subset by auto
+ have "x - u *\<^sub>R (x-y) \<in> 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 \<open>x \<in> S\<close> by auto
+ next
+ case True
+ then show ?thesis using assm by auto
+ qed
+ then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> 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) \<le> 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 \<in> rel_interior S"
+ using rel_interior_eq_empty assms by auto
+ { fix x
+ assume x: "x \<in> closure S"
+ {
+ assume "x = a"
+ then have "x \<in> closure (rel_interior S)"
+ using a unfolding closure_def by auto
+ }
+ moreover
+ {
+ assume "x \<noteq> a"
+ {
+ fix e :: real
+ assume "e > 0"
+ define e1 where "e1 = min 1 (e/norm (x - a))"
+ then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
+ using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
+ by simp_all
+ then have *: "x - e1 *\<^sub>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 "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
+ apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
+ using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
+ apply simp
+ done
+ }
+ then have "x islimpt rel_interior S"
+ unfolding islimpt_approachable_le by auto
+ then have "x \<in> closure(rel_interior S)"
+ unfolding closure_def by auto
+ }
+ ultimately have "x \<in> closure(rel_interior S)" by auto
+ }
+ then show ?thesis using h1 by auto
+ next
+ case True
+ then have "rel_interior S = {}"
+ using rel_interior_empty by auto
+ then have "closure (rel_interior S) = {}"
+ using closure_empty by auto
+ with True show ?thesis by auto
+ qed
+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 (subtopology euclidean (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) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
+ obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
+proof -
+ define e where "e = a / (a + b)"
+ have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
+ apply auto
+ using assms
+ apply simp
+ done
+ also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
+ using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
+ by auto
+ also have "\<dots> = y - e *\<^sub>R (y-x)"
+ using e_def
+ apply (simp add: algebra_simps)
+ using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
+ apply auto
+ done
+ finally have "z = y - e *\<^sub>R (y-x)"
+ by auto
+ moreover have "e > 0" using e_def assms by auto
+ moreover have "e \<le> 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) \<supseteq> 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 \<in> rel_interior (closure S)"
+ obtain x where x: "x \<in> rel_interior S"
+ using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
+ have "z \<in> 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 \<inter> affine hull closure S \<le> 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)) *\<^sub>R (z-x)"
+ have yball: "y \<in> cball z e"
+ using mem_cball y_def dist_norm[of z y] e by auto
+ have "x \<in> affine hull closure S"
+ using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
+ moreover have "z \<in> affine hull closure S"
+ using z rel_interior_subset hull_subset[of "closure S"] by blast
+ ultimately have "y \<in> 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 \<in> closure S" using e yball by auto
+ have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
+ using y_def by (simp add: algebra_simps)
+ then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>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 \<open>y \<in> closure S\<close>
+ by auto
+ 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 closure_eq_rel_interior_eq:
+ fixes S1 S2 :: "'n::euclidean_space set"
+ assumes "convex S1"
+ and "convex S2"
+ shows "closure S1 = closure S2 \<longleftrightarrow> 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 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
+ (is "?A \<longleftrightarrow> ?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 \<subseteq> closure S2"
+ by (metis assms(1) convex_closure_rel_interior closure_mono)
+ moreover from \<open>?B\<close> have "closure S1 \<supseteq> 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 \<inter> closure S = {} \<longleftrightarrow> A \<inter> 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 show ?thesis
+ apply (simp add: rel_interior_eq openin_open)
+ apply (rule_tac x="ball (inverse 2 *\<^sub>R (a + b)) (norm(b - a) / 2)" in exI)
+ apply (simp add: open_segment_as_ball)
+ done
+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
+
+lemma starlike_convex_tweak_boundary_points:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "S \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
+ shows "starlike T"
+proof -
+ have "rel_interior S \<noteq> {}"
+ by (simp add: assms rel_interior_eq_empty)
+ then obtain a where a: "a \<in> rel_interior S" by blast
+ with ST have "a \<in> T" by blast
+ have *: "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
+ apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
+ using assms by blast
+ show ?thesis
+ unfolding starlike_def
+ apply (rule bexI [OF _ \<open>a \<in> T\<close>])
+ apply (simp add: closed_segment_eq_open)
+ apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
+ apply (simp add: order_trans [OF * ST])
+ done
+qed
+
+subsection\<open>The relative frontier of a set\<close>
+
+definition "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 = {} \<longleftrightarrow> affine S"
+ apply (simp add: rel_frontier_def)
+ apply (simp add: rel_interior_eq_closure [symmetric])
+ using rel_interior_subset_closure by blast
+
+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 \<subseteq> 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
+ apply simp
+ by (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((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
+by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
+
+lemma closed_affine_hull [iff]:
+ fixes S :: "'n::euclidean_space set"
+ shows "closed (affine hull S)"
+ by (metis affine_affine_hull affine_closed)
+
+lemma rel_frontier_nonempty_interior:
+ fixes S :: "'n::euclidean_space set"
+ shows "interior S \<noteq> {} \<Longrightarrow> 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 \<Longrightarrow> rel_frontier S = frontier S"
+by (simp add: frontier_def rel_frontier_def rel_interior_interior)
+
+lemma closest_point_in_rel_frontier:
+ "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
+ \<Longrightarrow> closest_point S x \<in> 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 (subtopology euclidean (affine hull S)) (closure S - rel_interior S)"
+ by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
+ show ?thesis
+ apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
+ unfolding rel_frontier_def
+ using * closed_affine_hull
+ apply auto
+ done
+qed
+
+lemma closed_rel_boundary:
+ fixes S :: "'n::euclidean_space set"
+ shows "closed S \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
+ \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> 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 "\<lbrakk>convex S; convex T\<rbrakk>
+ \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
+ rel_interior S \<subseteq> T \<and> T \<subseteq> 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 \<noteq> {}"
+ and "S1 \<le> rel_frontier S2"
+ shows "aff_dim S1 < aff_dim S2"
+proof -
+ have "S1 \<subseteq> closure S2"
+ using assms unfolding rel_frontier_def by auto
+ then have *: "affine hull S1 \<subseteq> affine hull S2"
+ using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
+ then have "aff_dim S1 \<le> 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 \<noteq> {}"
+ using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
+ have **: "affine hull S1 = affine hull S2"
+ apply (rule affine_dim_equal)
+ using * affine_affine_hull
+ apply auto
+ using \<open>S1 \<noteq> {}\<close> hull_subset[of S1]
+ apply auto
+ using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
+ apply auto
+ done
+ obtain a where a: "a \<in> rel_interior S1"
+ using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
+ obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
+ using mem_rel_interior[of a S1] a by auto
+ then have "a \<in> T \<inter> closure S2"
+ using a assms unfolding rel_frontier_def by auto
+ then obtain b where b: "b \<in> T \<inter> rel_interior S2"
+ using open_inter_closure_rel_interior[of S2 T] assms T by auto
+ then have "b \<in> affine hull S1"
+ using rel_interior_subset hull_subset[of S2] ** by auto
+ then have "b \<in> 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 \<in> rel_interior S"
+ shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
+proof -
+ obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
+ using mem_rel_interior_cball[of z S] assms by auto
+ {
+ fix x
+ assume x: "x \<in> affine hull S"
+ {
+ assume "x \<noteq> z"
+ define m where "m = 1 + e1/norm(x-z)"
+ hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
+ {
+ fix e
+ assume e: "e > 1 \<and> e \<le> m"
+ have "z \<in> affine hull S"
+ using assms rel_interior_subset hull_subset[of S] by auto
+ then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
+ by (simp add: algebra_simps)
+ also have "\<dots> = (e - 1) * norm (x-z)"
+ using norm_scaleR e by auto
+ also have "\<dots> \<le> (m - 1) * norm (x - z)"
+ using e mult_right_mono[of _ _ "norm(x-z)"] by auto
+ also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
+ using m_def by auto
+ also have "\<dots> = e1"
+ using \<open>x \<noteq> z\<close> e1 by simp
+ finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
+ by auto
+ have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
+ using m_def **
+ unfolding cball_def dist_norm
+ by (auto simp add: algebra_simps)
+ then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
+ using e * e1 by auto
+ }
+ then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
+ using \<open>m> 1 \<close> 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 \<and> e \<le> m"
+ then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
+ using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
+ then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
+ using e by auto
+ }
+ then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
+ using \<open>m > 1\<close> by auto
+ }
+ ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> rel_interior S"
+ shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
+ assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
+ shows "z \<in> rel_interior S"
+proof -
+ obtain x where x: "x \<in> rel_interior S"
+ using rel_interior_eq_empty assms by auto
+ then have "x \<in> S"
+ using rel_interior_subset by auto
+ then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
+ using assms by auto
+ define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
+ then have "y \<in> S" using e by auto
+ define e1 where "e1 = 1/e"
+ then have "0 < e1 \<and> e1 < 1" using e by auto
+ then have "z =y - (1 - e1) *\<^sub>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"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
+ by auto
+qed
+
+lemma convex_rel_interior_iff:
+ fixes S :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "S \<noteq> {}"
+ shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<noteq> {}"
+ shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> 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 \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
+proof (cases "aff_dim S = int DIM('n)")
+ case False
+ {
+ assume "z \<in> interior S"
+ then have False
+ using False interior_rel_interior_gen[of S] by auto
+ }
+ moreover
+ {
+ assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
+ {
+ fix x
+ obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
+ using r by auto
+ obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
+ using r by auto
+ define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
+ then have x1: "x1 \<in> affine hull S"
+ using e1 hull_subset[of S] by auto
+ define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
+ then have x2: "x2 \<in> 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)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
+ using x1_def x2_def
+ apply (auto simp add: algebra_simps)
+ using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
+ apply auto
+ done
+ then have z: "z \<in> 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) *\<^sub>R (x - z)"
+ using x1_def x2_def by (auto simp add: algebra_simps)
+ then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
+ using e1 e2 by simp
+ then have "x \<in> 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 add: aff_dim_UNIV)
+ then have False
+ using False by auto
+ }
+ ultimately show ?thesis by auto
+next
+ case True
+ then have "S \<noteq> {}"
+ using aff_dim_empty[of S] by auto
+ have *: "affine hull S = UNIV"
+ using True affine_hull_UNIV by auto
+ {
+ assume "z \<in> interior S"
+ then have "z \<in> rel_interior S"
+ using True interior_rel_interior_gen[of S] by auto
+ then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
+ using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
+ fix x
+ obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
+ using **[rule_format, of "z-x"] by auto
+ define e where [abs_def]: "e = e1 - 1"
+ then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
+ by (simp add: algebra_simps)
+ then have "e > 0" "z + e *\<^sub>R x \<in> S"
+ using e1 e_def by auto
+ then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
+ by auto
+ }
+ moreover
+ {
+ assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
+ {
+ fix x
+ obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
+ using r[rule_format, of "z-x"] by auto
+ define e where "e = e1 + 1"
+ then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
+ by (simp add: algebra_simps)
+ then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
+ using e1 e_def by auto
+ then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
+ }
+ then have "z \<in> rel_interior S"
+ using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
+ then have "z \<in> interior S"
+ using True interior_rel_interior_gen[of S] by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+
+subsubsection \<open>Relative interior and closure under common operations\<close>
+
+lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S : I} \<subseteq> \<Inter>I"
+proof -
+ {
+ fix y
+ assume "y \<in> \<Inter>{rel_interior S |S. S : I}"
+ then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
+ by auto
+ {
+ fix S
+ assume "S \<in> I"
+ then have "y \<in> S"
+ using rel_interior_subset y by auto
+ }
+ then have "y \<in> \<Inter>I" by auto
+ }
+ then show ?thesis by auto
+qed
+
+lemma closure_Int: "closure (\<Inter>I) \<le> \<Inter>{closure S |S. S \<in> I}"
+proof -
+ {
+ fix y
+ assume "y \<in> \<Inter>I"
+ then have y: "\<forall>S \<in> I. y \<in> S" by auto
+ {
+ fix S
+ assume "S \<in> I"
+ then have "y \<in> closure S"
+ using closure_subset y by auto
+ }
+ then have "y \<in> \<Inter>{closure S |S. S \<in> I}"
+ by auto
+ }
+ then have "\<Inter>I \<subseteq> \<Inter>{closure S |S. S \<in> I}"
+ by auto
+ moreover have "closed (\<Inter>{closure S |S. S \<in> I})"
+ unfolding closed_Inter closed_closure by auto
+ ultimately show ?thesis using closure_hull[of "\<Inter>I"]
+ hull_minimal[of "\<Inter>I" "\<Inter>{closure S |S. S \<in> I}" "closed"] by auto
+qed
+
+lemma convex_closure_rel_interior_inter:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
+ and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
+ shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+proof -
+ obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
+ using assms by auto
+ {
+ fix y
+ assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
+ then have y: "\<forall>S \<in> I. y \<in> closure S"
+ by auto
+ {
+ assume "y = x"
+ then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+ using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
+ }
+ moreover
+ {
+ assume "y \<noteq> x"
+ { fix e :: real
+ assume e: "e > 0"
+ define e1 where "e1 = min 1 (e/norm (y - x))"
+ then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
+ using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
+ by simp_all
+ define z where "z = y - e1 *\<^sub>R (y - x)"
+ {
+ fix S
+ assume "S \<in> I"
+ then have "z \<in> 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 \<in> \<Inter>{rel_interior S |S. S \<in> I}"
+ by auto
+ have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
+ apply (rule_tac x="z" in exI)
+ using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
+ apply simp
+ done
+ }
+ then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
+ unfolding islimpt_approachable_le by blast
+ then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+ unfolding closure_def by auto
+ }
+ ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+ by auto
+ }
+ then show ?thesis by auto
+qed
+
+lemma convex_closure_inter:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
+ and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
+ shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
+proof -
+ have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+ using convex_closure_rel_interior_inter assms by auto
+ moreover
+ have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
+ using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
+ by auto
+ ultimately show ?thesis
+ using closure_Int[of I] by auto
+qed
+
+lemma convex_inter_rel_interior_same_closure:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
+ and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
+ shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
+proof -
+ have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
+ using convex_closure_rel_interior_inter assms by auto
+ moreover
+ have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
+ using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
+ by auto
+ ultimately show ?thesis
+ using closure_Int[of I] by auto
+qed
+
+lemma convex_rel_interior_inter:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
+ and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
+ shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
+proof -
+ have "convex (\<Inter>I)"
+ using assms convex_Inter by auto
+ moreover
+ have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
+ apply (rule convex_Inter)
+ using assms convex_rel_interior
+ apply auto
+ done
+ ultimately
+ have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
+ using convex_inter_rel_interior_same_closure assms
+ closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
+ by blast
+ then show ?thesis
+ using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
+qed
+
+lemma convex_rel_interior_finite_inter:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
+ and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
+ and "finite I"
+ shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
+proof -
+ have "\<Inter>I \<noteq> {}"
+ using assms rel_interior_inter_aux[of I] by auto
+ have "convex (\<Inter>I)"
+ using convex_Inter assms by auto
+ show ?thesis
+ proof (cases "I = {}")
+ case True
+ then show ?thesis
+ using Inter_empty rel_interior_UNIV by auto
+ next
+ case False
+ {
+ fix z
+ assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
+ {
+ fix x
+ assume x: "x \<in> \<Inter>I"
+ {
+ fix S
+ assume S: "S \<in> I"
+ then have "z \<in> rel_interior S" "x \<in> S"
+ using z x by auto
+ then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
+ using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
+ }
+ then obtain mS where
+ mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
+ define e where "e = Min (mS ` I)"
+ then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
+ then have "e > 1" using mS by auto
+ moreover have "\<forall>S\<in>I. e \<le> mS S"
+ using e_def assms by auto
+ ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
+ using mS by auto
+ }
+ then have "z \<in> rel_interior (\<Inter>I)"
+ using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
+ }
+ then show ?thesis
+ using convex_rel_interior_inter[of I] assms by auto
+ qed
+qed
+
+lemma convex_closure_inter_two:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "convex T"
+ assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
+ shows "closure (S \<inter> T) = closure S \<inter> closure T"
+ using convex_closure_inter[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 \<inter> rel_interior T \<noteq> {}"
+ shows "rel_interior (S \<inter> T) = rel_interior S \<inter> 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 \<inter> T \<noteq> {}"
+ shows "closure (S \<inter> T) = closure S \<inter> T"
+proof -
+ have "affine hull T = T"
+ using assms by auto
+ then have "rel_interior T = T"
+ using rel_interior_affine_hull[of T] by metis
+ moreover have "closure T = T"
+ using assms affine_closed[of T] by auto
+ ultimately show ?thesis
+ using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
+qed
+
+lemma connected_component_1_gen:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "DIM('a) = 1"
+ shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<longleftrightarrow> closed_segment a b \<subseteq> 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 \<inter> T \<noteq> {}"
+ shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
+proof -
+ have "affine hull T = T"
+ using assms by auto
+ then have "rel_interior T = T"
+ using rel_interior_affine_hull[of T] by metis
+ moreover have "closure T = T"
+ using assms affine_closed[of T] by auto
+ ultimately show ?thesis
+ using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
+qed
+
+lemma convex_affine_rel_frontier_Int:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "affine T"
+ and "interior S \<inter> T \<noteq> {}"
+ shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
+using assms
+apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
+by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
+
+lemma rel_interior_convex_Int_affine:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
+ shows "rel_interior(S \<inter> T) = interior S \<inter> T"
+proof -
+ obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
+ using assms by force
+ have "rel_interior S = interior S"
+ by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
+ then show ?thesis
+ by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
+qed
+
+lemma closure_convex_Int_affine:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
+ shows "closure(S \<inter> T) = closure S \<inter> T"
+proof
+ have "closure (S \<inter> T) \<subseteq> closure T"
+ by (simp add: closure_mono)
+ also have "... \<subseteq> T"
+ by (simp add: affine_closed assms)
+ finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
+ by (simp add: closure_mono)
+next
+ obtain a where "a \<in> rel_interior S" "a \<in> T"
+ using assms by auto
+ then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
+ using affine_diffs_subspace rel_interior_subset assms by blast+
+ show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
+ proof
+ fix x assume "x \<in> closure S \<inter> T"
+ show "x \<in> closure (S \<inter> T)"
+ proof (cases "x = a")
+ case True
+ then show ?thesis
+ using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
+ next
+ case False
+ then have "x \<in> closure(open_segment a x)"
+ by auto
+ then show ?thesis
+ using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
+ qed
+ qed
+qed
+
+lemma subset_rel_interior_convex:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "convex T"
+ and "S \<le> closure T"
+ and "\<not> S \<subseteq> rel_frontier T"
+ shows "rel_interior S \<subseteq> rel_interior T"
+proof -
+ have *: "S \<inter> closure T = S"
+ using assms by auto
+ have "\<not> rel_interior S \<subseteq> 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 \<inter> rel_interior (closure T) \<noteq> {}"
+ using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
+ by auto
+ then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> 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 "\<dots> = rel_interior S"
+ using * by auto
+ finally show ?thesis
+ by auto
+qed
+
+lemma rel_interior_convex_linear_image:
+ fixes f :: "'m::euclidean_space \<Rightarrow> '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 rel_interior_empty rel_interior_eq_empty by auto
+next
+ case False
+ have *: "f ` (rel_interior S) \<subseteq> f ` S"
+ unfolding image_mono using rel_interior_subset by auto
+ have "f ` S \<subseteq> f ` (closure S)"
+ unfolding image_mono using closure_subset by auto
+ also have "\<dots> = f ` (closure (rel_interior S))"
+ using convex_closure_rel_interior assms by auto
+ also have "\<dots> \<subseteq> 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) \<subseteq> f ` rel_interior S"
+ using rel_interior_subset by auto
+ moreover
+ {
+ fix z
+ assume "z \<in> f ` rel_interior S"
+ then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
+ {
+ fix x
+ assume "x \<in> f ` S"
+ then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
+ then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
+ using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
+ moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
+ using x1 z1 \<open>linear f\<close> by (simp add: linear_add_cmul)
+ ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
+ using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
+ then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
+ using e by auto
+ }
+ then have "z \<in> rel_interior (f ` S)"
+ using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close>
+ \<open>linear f\<close> \<open>S \<noteq> {}\<close> 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 \<Rightarrow> 'n::euclidean_space"
+ assumes "linear f"
+ and "convex S"
+ and "f -` (rel_interior S) \<noteq> {}"
+ shows "rel_interior (f -` S) = f -` (rel_interior S)"
+proof -
+ have "S \<noteq> {}"
+ using assms rel_interior_empty by auto
+ have nonemp: "f -` S \<noteq> {}"
+ by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
+ then have "S \<inter> (range f) \<noteq> {}"
+ by auto
+ have conv: "convex (f -` S)"
+ using convex_linear_vimage assms by auto
+ then have "convex (S \<inter> range f)"
+ by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
+ {
+ fix z
+ assume "z \<in> f -` (rel_interior S)"
+ then have z: "f z : rel_interior S"
+ by auto
+ {
+ fix x
+ assume "x \<in> f -` S"
+ then have "f x \<in> S" by auto
+ then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
+ using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
+ moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
+ using \<open>linear f\<close> by (simp add: linear_iff)
+ ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
+ using e by auto
+ }
+ then have "z \<in> rel_interior (f -` S)"
+ using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
+ }
+ moreover
+ {
+ fix z
+ assume z: "z \<in> rel_interior (f -` S)"
+ {
+ fix x
+ assume "x \<in> S \<inter> range f"
+ then obtain y where y: "f y = x" "y \<in> f -` S" by auto
+ then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
+ using convex_rel_interior_iff[of "f -` S" z] z conv by auto
+ moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
+ using \<open>linear f\<close> y by (simp add: linear_iff)
+ ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
+ using e by auto
+ }
+ then have "f z \<in> rel_interior (S \<inter> range f)"
+ using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
+ convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
+ by auto
+ moreover have "affine (range f)"
+ by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
+ ultimately have "f z \<in> rel_interior S"
+ using convex_affine_rel_interior_Int[of S "range f"] assms by auto
+ then have "z \<in> 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 \<times> T) = rel_interior S \<times> rel_interior T"
+proof -
+ { assume "S = {}"
+ then have ?thesis
+ by auto
+ }
+ moreover
+ { assume "T = {}"
+ then have ?thesis
+ by auto
+ }
+ moreover
+ {
+ assume "S \<noteq> {}" "T \<noteq> {}"
+ then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
+ using rel_interior_eq_empty assms by auto
+ then have "fst -` rel_interior S \<noteq> {}"
+ using fst_vimage_eq_Times[of "rel_interior S"] by auto
+ then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
+ using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
+ then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
+ by (simp add: fst_vimage_eq_Times)
+ from ri have "snd -` rel_interior T \<noteq> {}"
+ using snd_vimage_eq_Times[of "rel_interior T"] by auto
+ then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
+ using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
+ then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
+ by (simp add: snd_vimage_eq_Times)
+ from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
+ rel_interior S \<times> rel_interior T" by auto
+ have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
+ by auto
+ then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
+ by auto
+ also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
+ apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> T"])
+ using * ri assms convex_Times
+ apply auto
+ done
+ finally have ?thesis using * by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma rel_interior_scaleR:
+ fixes S :: "'n::euclidean_space set"
+ assumes "c \<noteq> 0"
+ shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
+ using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
+ linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms
+ by auto
+
+lemma rel_interior_convex_scaleR:
+ fixes S :: "'n::euclidean_space set"
+ assumes "convex S"
+ shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
+ by (metis assms linear_scaleR rel_interior_convex_linear_image)
+
+lemma convex_rel_open_scaleR:
+ fixes S :: "'n::euclidean_space set"
+ assumes "convex S"
+ and "rel_open S"
+ shows "convex ((op *\<^sub>R c) ` S) \<and> rel_open ((op *\<^sub>R c) ` S)"
+ by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
+
+lemma convex_rel_open_finite_inter:
+ assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
+ and "finite I"
+ shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
+proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
+ case True
+ then have "\<Inter>I = {}"
+ using assms unfolding rel_open_def by auto
+ then show ?thesis
+ unfolding rel_open_def using rel_interior_empty by auto
+next
+ case False
+ then have "rel_open (\<Inter>I)"
+ using assms unfolding rel_open_def
+ using convex_rel_interior_finite_inter[of I]
+ by auto
+ then show ?thesis
+ using convex_Inter assms by auto
+qed
+
+lemma convex_rel_open_linear_image:
+ fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "linear f"
+ and "convex S"
+ and "rel_open S"
+ shows "convex (f ` S) \<and> 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 \<Rightarrow> 'n::euclidean_space"
+ assumes "linear f"
+ and "convex S"
+ and "rel_open S"
+ shows "convex (f -` S) \<and> 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 using rel_interior_empty 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 \<times> 'n::euclidean_space) set"
+ and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
+ assumes "convex S"
+ and "f = (\<lambda>y. {z. (y, z) \<in> S})"
+ shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
+proof -
+ {
+ fix y
+ assume "y \<in> {y. f y \<noteq> {}}"
+ then obtain z where "(y, z) \<in> S"
+ using assms by auto
+ then have "\<exists>x. x \<in> S \<and> y = fst x"
+ apply (rule_tac x="(y, z)" in exI)
+ apply auto
+ done
+ then obtain x where "x \<in> S" "y = fst x"
+ by blast
+ then have "y \<in> fst ` S"
+ unfolding image_def by auto
+ }
+ then have "fst ` S = {y. f y \<noteq> {}}"
+ unfolding fst_def using assms by auto
+ then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
+ using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
+ {
+ fix y
+ assume "y \<in> rel_interior {y. f y \<noteq> {}}"
+ then have "y \<in> fst ` rel_interior S"
+ using h1 by auto
+ then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
+ by auto
+ moreover have aff: "affine (fst -` {y})"
+ unfolding affine_alt by (simp add: algebra_simps)
+ ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
+ using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
+ have conv: "convex (S \<inter> fst -` {y})"
+ using convex_Int assms aff affine_imp_convex by auto
+ {
+ fix x
+ assume "x \<in> f y"
+ then have "(y, x) \<in> S \<inter> (fst -` {y})"
+ using assms by auto
+ moreover have "x = snd (y, x)" by auto
+ ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
+ by blast
+ }
+ then have "snd ` (S \<inter> fst -` {y}) = f y"
+ using assms by auto
+ then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
+ using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
+ by auto
+ {
+ fix z
+ assume "z \<in> rel_interior (f y)"
+ then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
+ using *** by auto
+ moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
+ using * ** rel_interior_subset by auto
+ ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
+ by force
+ then have "(y,z) \<in> rel_interior S"
+ using ** by auto
+ }
+ moreover
+ {
+ fix z
+ assume "(y, z) \<in> rel_interior S"
+ then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
+ using ** by auto
+ then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
+ by (metis Range_iff snd_eq_Range)
+ then have "z \<in> rel_interior (f y)"
+ using *** by auto
+ }
+ ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
+ by auto
+ }
+ then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
+ (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
+ by auto
+ {
+ fix y z
+ assume asm: "(y, z) \<in> rel_interior S"
+ then have "y \<in> fst ` rel_interior S"
+ by (metis Domain_iff fst_eq_Domain)
+ then have "y \<in> rel_interior {t. f t \<noteq> {}}"
+ using h1 by auto
+ then have "y \<in> rel_interior {t. f t \<noteq> {}}" 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 \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
+ by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
+
+
+subsubsection \<open>Relative interior of convex cone\<close>
+
+lemma cone_rel_interior:
+ fixes S :: "'m::euclidean_space set"
+ assumes "cone S"
+ shows "cone ({0} \<union> rel_interior S)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (simp add: rel_interior_empty cone_0)
+next
+ case False
+ then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` S = S)"
+ using cone_iff[of S] assms by auto
+ then have *: "0 \<in> ({0} \<union> rel_interior S)"
+ and "\<forall>c. c > 0 \<longrightarrow> op *\<^sub>R c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
+ by (auto simp add: rel_interior_scaleR)
+ then show ?thesis
+ using cone_iff[of "{0} \<union> rel_interior S"] by auto
+qed
+
+lemma rel_interior_convex_cone_aux:
+ fixes S :: "'m::euclidean_space set"
+ assumes "convex S"
+ shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
+ c > 0 \<and> x \<in> ((op *\<^sub>R c) ` (rel_interior S))"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (simp add: rel_interior_empty cone_hull_empty)
+next
+ case False
+ then obtain s where "s \<in> S" by auto
+ have conv: "convex ({(1 :: real)} \<times> S)"
+ using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
+ by auto
+ define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
+ then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
+ (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
+ apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
+ using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
+ apply auto
+ done
+ {
+ fix y :: real
+ assume "y \<ge> 0"
+ then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
+ using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
+ then have "f y \<noteq> {}"
+ using f_def by auto
+ }
+ then have "{y. f y \<noteq> {}} = {0..}"
+ using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
+ then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
+ using rel_interior_real_semiline by auto
+ {
+ fix c :: real
+ assume "c > 0"
+ then have "f c = (op *\<^sub>R c ` S)"
+ using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
+ then have "rel_interior (f c) = op *\<^sub>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} \<times> S)) =
+ {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
+ (is "?lhs = ?rhs")
+proof -
+ {
+ fix z
+ assume "z \<in> ?lhs"
+ have *: "z = (fst z, snd z)"
+ by auto
+ have "z \<in> ?rhs"
+ using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close>
+ apply auto
+ apply (rule_tac x = "fst z" in exI)
+ apply (rule_tac x = x in exI)
+ using *
+ apply auto
+ done
+ }
+ moreover
+ {
+ fix z
+ assume "z \<in> ?rhs"
+ then have "z \<in> ?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 "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
+ shows "convex hull (\<Union>(S ` I)) =
+ {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
+ (is "?lhs = ?rhs")
+proof -
+ have "?lhs \<supseteq> ?rhs"
+ proof
+ fix x
+ assume "x : ?rhs"
+ then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
+ "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
+ then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
+ using hull_subset[of "\<Union>(S ` I)" convex] by auto
+ then show "x \<in> ?lhs"
+ unfolding *(1)[symmetric]
+ apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
+ using * assms convex_convex_hull
+ apply auto
+ done
+ qed
+
+ {
+ fix i
+ assume "i \<in> I"
+ with assms have "\<exists>p. p \<in> S i" by auto
+ }
+ then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
+
+ {
+ fix i
+ assume "i \<in> I"
+ {
+ fix x
+ assume "x \<in> S i"
+ define c where "c j = (if j = i then 1::real else 0)" for j
+ then have *: "sum c I = 1"
+ using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
+ by auto
+ define s where "s j = (if j = i then x else p j)" for j
+ then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
+ using c_def by (auto simp add: algebra_simps)
+ then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
+ using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
+ by auto
+ then have "x \<in> ?rhs"
+ apply auto
+ apply (rule_tac x = c in exI)
+ apply (rule_tac x = s in exI)
+ using * c_def s_def p \<open>x \<in> S i\<close>
+ apply auto
+ done
+ }
+ then have "?rhs \<supseteq> S i" by auto
+ }
+ then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
+
+ {
+ fix u v :: real
+ assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
+ fix x y
+ assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
+ from xy obtain c s where
+ xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
+ by auto
+ from xy obtain d t where
+ yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
+ by auto
+ define e where "e i = u * c i + v * d i" for i
+ have ge0: "\<forall>i\<in>I. e i \<ge> 0"
+ using e_def xc yc uv by simp
+ have "sum (\<lambda>i. u * c i) I = u * sum c I"
+ by (simp add: sum_distrib_left)
+ moreover have "sum (\<lambda>i. 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) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
+ for i
+ {
+ fix i
+ assume i: "i \<in> I"
+ have "q i \<in> 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: "\<forall>i\<in>I. q i \<in> S i" by auto
+ {
+ fix i
+ assume i: "i \<in> I"
+ have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
+ proof (cases "e i = 0")
+ case True
+ have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
+ using xc yc uv i by simp
+ moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
+ using True e_def i by simp
+ ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
+ with True show ?thesis by auto
+ next
+ case False
+ then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
+ using q_def by auto
+ then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
+ = (e i) *\<^sub>R (q i)" by auto
+ with False show ?thesis by (simp add: algebra_simps)
+ qed
+ }
+ then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
+ by auto
+ have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
+ using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
+ also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
+ using * by auto
+ finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
+ by auto
+ then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
+ using ge0 sum1 qs by auto
+ }
+ then have "convex ?rhs" unfolding convex_def by auto
+ then show ?thesis
+ using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
+ by blast
+qed
+
+lemma convex_hull_union_two:
+ fixes S T :: "'m::euclidean_space set"
+ assumes "convex S"
+ and "S \<noteq> {}"
+ and "convex T"
+ and "T \<noteq> {}"
+ shows "convex hull (S \<union> T) =
+ {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> 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 "\<Union>(s ` I) = S \<union> T"
+ using s_def I_def by auto
+ then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
+ by auto
+ moreover have "convex hull \<Union>(s ` I) =
+ {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
+ apply (subst convex_hull_finite_union[of I s])
+ using assms s_def I_def
+ apply auto
+ done
+ moreover have
+ "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
+ using s_def I_def by auto
+ ultimately show "?lhs \<subseteq> ?rhs" by auto
+ {
+ fix x
+ assume "x \<in> ?rhs"
+ then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
+ by auto
+ then have "x \<in> convex hull {s, t}"
+ using convex_hull_2[of s t] by auto
+ then have "x \<in> convex hull (S \<union> T)"
+ using * hull_mono[of "{s, t}" "S \<union> T"] by auto
+ }
+ then show "?lhs \<supseteq> ?rhs" by blast
+qed
+
+
+subsection \<open>Convexity on direct sums\<close>
+
+lemma closure_sum:
+ fixes S T :: "'a::real_normed_vector set"
+ shows "closure S + closure T \<subseteq> 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 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 = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
+ by (simp add: set_plus_image)
+ also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
+ using rel_interior_Times assms by auto
+ also have "\<dots> = rel_interior (S + T)"
+ using fst_snd_linear convex_Times assms
+ rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
+ by (auto simp add: set_plus_image)
+ finally show ?thesis ..
+qed
+
+lemma rel_interior_sum_gen:
+ fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
+ assumes "\<forall>i\<in>I. convex (S i)"
+ shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
+ apply (subst sum_set_cond_linear[of convex])
+ using rel_interior_sum rel_interior_sing[of "0"] assms
+ apply (auto simp add: convex_set_plus)
+ done
+
+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 \<times> T) \<and> rel_open (S \<times> 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) \<and> 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 \<noteq> {}"
+ assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
+ shows "convex hull (\<Union>(S ` I)) = sum S I"
+ (is "?lhs = ?rhs")
+proof -
+ {
+ fix x
+ assume "x \<in> ?lhs"
+ then obtain c xs where
+ x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
+ using convex_hull_finite_union[of I S] assms by auto
+ define s where "s i = c i *\<^sub>R xs i" for i
+ {
+ fix i
+ assume "i \<in> I"
+ then have "s i \<in> S i"
+ using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
+ }
+ then have "\<forall>i\<in>I. s i \<in> S i" by auto
+ moreover have "x = sum s I" using x s_def by auto
+ ultimately have "x \<in> ?rhs"
+ using set_sum_alt[of I S] assms by auto
+ }
+ moreover
+ {
+ fix x
+ assume "x \<in> ?rhs"
+ then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
+ using set_sum_alt[of I S] assms by auto
+ define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
+ then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
+ using x assms by auto
+ moreover have "\<forall>i\<in>I. xs i \<in> S i"
+ using x xs_def assms by (simp add: cone_def)
+ moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
+ by auto
+ moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
+ using assms by auto
+ ultimately have "x \<in> ?lhs"
+ apply (subst convex_hull_finite_union[of I S])
+ using assms
+ apply blast
+ using assms
+ apply blast
+ apply rule
+ apply (rule_tac x = "(\<lambda>i. (1 :: real) / of_nat (card I))" in exI)
+ apply auto
+ done
+ }
+ 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 \<noteq> {}"
+ assumes "convex T"
+ and "cone T"
+ and "T \<noteq> {}"
+ shows "convex hull (S \<union> 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 "\<Union>(A ` I) = S \<union> T"
+ using A_def I_def by auto
+ then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
+ by auto
+ moreover have "convex hull \<Union>(A ` I) = sum A I"
+ apply (subst convex_hull_finite_union_cones[of I A])
+ using assms A_def I_def
+ apply auto
+ done
+ moreover have "sum A I = S + T"
+ using A_def I_def
+ unfolding set_plus_def
+ apply auto
+ unfolding set_plus_def
+ apply auto
+ done
+ ultimately show ?thesis by auto
+qed
+
+lemma rel_interior_convex_hull_union:
+ fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
+ assumes "finite I"
+ and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
+ shows "rel_interior (convex hull (\<Union>(S ` I))) =
+ {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
+ (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
+ (is "?lhs = ?rhs")
+proof (cases "I = {}")
+ case True
+ then show ?thesis
+ using convex_hull_empty rel_interior_empty by auto
+next
+ case False
+ define C0 where "C0 = convex hull (\<Union>(S ` I))"
+ have "\<forall>i\<in>I. C0 \<ge> S i"
+ unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
+ define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
+ define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
+ have "\<forall>i\<in>I. K i \<noteq> {}"
+ unfolding K_def using assms
+ by (simp add: cone_hull_empty_iff[symmetric])
+ {
+ fix i
+ assume "i \<in> I"
+ then have "convex (K i)"
+ unfolding K_def
+ apply (subst convex_cone_hull)
+ apply (subst convex_Times)
+ using assms
+ apply auto
+ done
+ }
+ then have convK: "\<forall>i\<in>I. convex (K i)"
+ by auto
+ {
+ fix i
+ assume "i \<in> I"
+ then have "K0 \<supseteq> K i"
+ unfolding K0_def K_def
+ apply (subst hull_mono)
+ using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
+ apply auto
+ done
+ }
+ then have "K0 \<supseteq> \<Union>(K ` I)" by auto
+ moreover have "convex K0"
+ unfolding K0_def
+ apply (subst convex_cone_hull)
+ apply (subst convex_Times)
+ unfolding C0_def
+ using convex_convex_hull
+ apply auto
+ done
+ ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
+ using hull_minimal[of _ "K0" "convex"] by blast
+ have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
+ using K_def by (simp add: hull_subset)
+ then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
+ by auto
+ then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
+ by (simp add: hull_mono)
+ then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
+ unfolding C0_def
+ using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
+ by auto
+ moreover have "cone (convex hull (\<Union>(K ` I)))"
+ apply (subst cone_convex_hull)
+ using cone_Union[of "K ` I"]
+ apply auto
+ unfolding K_def
+ using cone_cone_hull
+ apply auto
+ done
+ ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
+ unfolding K0_def
+ using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
+ by blast
+ then have "K0 = convex hull (\<Union>(K ` I))"
+ using geq by auto
+ also have "\<dots> = sum K I"
+ apply (subst convex_hull_finite_union_cones[of I K])
+ using assms
+ apply blast
+ using False
+ apply blast
+ unfolding K_def
+ apply rule
+ apply (subst convex_cone_hull)
+ apply (subst convex_Times)
+ using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
+ apply auto
+ done
+ finally have "K0 = sum K I" by auto
+ then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
+ using rel_interior_sum_gen[of I K] convK by auto
+ {
+ fix x
+ assume "x \<in> ?lhs"
+ then have "(1::real, x) \<in> 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 \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
+ using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
+ {
+ fix i
+ assume "i \<in> I"
+ then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
+ using k K_def assms by auto
+ then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
+ using rel_interior_convex_cone[of "S i"] by auto
+ }
+ then obtain c s where
+ cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
+ by metis
+ then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
+ using k by (simp add: sum_prod)
+ then have "x \<in> ?rhs"
+ using k
+ apply auto
+ apply (rule_tac x = c in exI)
+ apply (rule_tac x = s in exI)
+ using cs
+ apply auto
+ done
+ }
+ moreover
+ {
+ fix x
+ assume "x \<in> ?rhs"
+ then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
+ (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
+ by auto
+ define k where "k i = (c i, c i *\<^sub>R s i)" for i
+ {
+ fix i assume "i:I"
+ then have "k i \<in> rel_interior (K i)"
+ using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
+ by auto
+ }
+ then have "(1::real, x) \<in> rel_interior K0"
+ using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
+ apply auto
+ apply (rule_tac x = k in exI)
+ apply (simp add: sum_prod)
+ done
+ then have "x \<in> ?lhs"
+ using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
+ by (auto simp add: convex_convex_hull)
+ }
+ ultimately show ?thesis by blast
+qed
+
+
+lemma convex_le_Inf_differential:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex_on I f"
+ and "x \<in> interior I"
+ and "y \<in> I"
+ shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
+ (is "_ \<ge> _ + Inf (?F x) * (y - x)")
+proof (cases rule: linorder_cases)
+ assume "x < y"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this \<open>x \<in> interior I\<close>]
+ obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
+ moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
+ ultimately have "x < t" "t < y" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps split: split_min)
+ with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "open (interior I)" by auto
+ from openE[OF this \<open>x \<in> interior I\<close>]
+ obtain e where "0 < e" "ball x e \<subseteq> interior I" .
+ moreover define K where "K = x - e / 2"
+ with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
+ by (auto simp: dist_real_def)
+ ultimately have "K \<in> I" "K < x" "x \<in> I"
+ using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
+
+ have "Inf (?F x) \<le> (f x - f y) / (x - y)"
+ proof (intro bdd_belowI cInf_lower2)
+ show "(f x - f t) / (x - t) \<in> ?F x"
+ using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
+ show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+ using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
+ by (rule convex_on_diff)
+ next
+ fix y
+ assume "y \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
+ show "(f K - f x) / (K - x) \<le> y" by auto
+ qed
+ then show ?thesis
+ using \<open>x < y\<close> by (simp add: field_simps)
+next
+ assume "y < x"
+ moreover
+ have "open (interior I)" by auto
+ from openE[OF this \<open>x \<in> interior I\<close>]
+ obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
+ moreover define t where "t = x + e / 2"
+ ultimately have "x < t" "t \<in> ball x e"
+ by (auto simp: dist_real_def field_simps)
+ with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
+
+ have "(f x - f y) / (x - y) \<le> Inf (?F x)"
+ proof (rule cInf_greatest)
+ have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
+ using \<open>y < x\<close> by (auto simp: field_simps)
+ also
+ fix z
+ assume "z \<in> ?F x"
+ with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
+ have "(f y - f x) / (y - x) \<le> z"
+ by auto
+ finally show "(f x - f y) / (x - y) \<le> z" .
+ next
+ have "open (interior I)" by auto
+ from openE[OF this \<open>x \<in> interior I\<close>]
+ obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
+ then have "x + e / 2 \<in> ball x e"
+ by (auto simp: dist_real_def)
+ with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
+ by auto
+ then show "?F x \<noteq> {}"
+ by blast
+ qed
+ then show ?thesis
+ using \<open>y < x\<close> by (simp add: field_simps)
+qed simp
+
+subsection\<open>Explicit formulas for interior and relative interior of convex hull\<close>
+
+lemma interior_atLeastAtMost [simp]:
+ fixes a::real shows "interior {a..b} = {a<..<b}"
+ using interior_cbox [of a b] by auto
+
+lemma interior_atLeastLessThan [simp]:
+ fixes a::real shows "interior {a..<b} = {a<..<b}"
+ by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost interior_Int interior_interior interior_real_semiline)
+
+lemma interior_lessThanAtMost [simp]:
+ fixes a::real shows "interior {a<..b} = {a<..<b}"
+ by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost interior_Int
+ interior_interior interior_real_semiline)
+
+lemma at_within_closed_interval:
+ fixes x::real
+ shows "a < x \<Longrightarrow> x < b \<Longrightarrow> (at x within {a..b}) = at x"
+ by (metis at_within_interior greaterThanLessThan_iff interior_atLeastAtMost)
+
+lemma affine_independent_convex_affine_hull:
+ fixes s :: "'a::euclidean_space set"
+ assumes "~affine_dependent s" "t \<subseteq> s"
+ shows "convex hull t = affine hull t \<inter> convex hull s"
+proof -
+ have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
+ { fix u v x
+ assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
+ "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
+ then have s: "s = (s - t) \<union> t" \<comment>\<open>split into separate cases\<close>
+ using assms by auto
+ have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
+ "sum v t + sum v (s - t) = 1"
+ using uv fin s
+ by (auto simp: sum.union_disjoint [symmetric] Un_commute)
+ have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
+ "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
+ using uv fin
+ by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
+ } note [simp] = this
+ have "convex hull t \<subseteq> affine hull t"
+ using convex_hull_subset_affine_hull by blast
+ moreover have "convex hull t \<subseteq> convex hull s"
+ using assms hull_mono by blast
+ moreover have "affine hull t \<inter> convex hull s \<subseteq> convex hull t"
+ using assms
+ apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
+ apply (drule_tac x=s in spec)
+ apply (auto simp: fin)
+ apply (rule_tac x=u in exI)
+ apply (rename_tac v)
+ apply (drule_tac x="\<lambda>x. if x \<in> t then v x - u x else v x" in spec)
+ apply (force)+
+ done
+ 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 \<notin> t" "s = insert a t"
+ by blast
+ then have fin: "finite t" using assms
+ by (metis finite_insert aff_independent_finite)
+ show ?thesis
+ using assms t fin
+ apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
+ apply (rule subset_antisym)
+ apply force
+ apply (rule Fun.vimage_subsetD)
+ apply (metis add.commute diff_add_cancel surj_def)
+ apply (rule card_ge_dim_independent)
+ apply (auto simp: card_image inj_on_def dim_subset_UNIV)
+ done
+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"
+ apply (rule affine_independent_span_eq [OF ind])
+ apply (rule antisym)
+ using assms
+ apply auto
+ apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
+ done
+
+lemma empty_interior_affine_hull:
+ fixes s :: "'a::euclidean_space set"
+ assumes "finite s" and dim: "card s \<le> DIM ('a)"
+ shows "interior(affine hull s) = {}"
+ using assms
+ apply (induct s rule: finite_induct)
+ apply (simp_all add: affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
+ apply (rule empty_interior_lowdim)
+ apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
+ apply (metis Suc_le_lessD not_less order_trans card_image_le finite_imageI dim_le_card)
+ done
+
+lemma empty_interior_convex_hull:
+ fixes s :: "'a::euclidean_space set"
+ assumes "finite s" and dim: "card s \<le> 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
+ \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
+ \<subseteq> rel_interior (convex hull s)"
+ by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
+
+lemma explicit_subset_rel_interior_convex_hull_minimal:
+ fixes s :: "'a::euclidean_space set"
+ shows "finite s
+ \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
+ \<subseteq> rel_interior (convex hull s)"
+ by (force simp add: rel_interior_convex_hull_union [where S="\<lambda>x. {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. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
+ (is "?lhs = ?rhs")
+proof
+ show "?rhs \<le> ?lhs"
+ by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
+next
+ show "?lhs \<le> ?rhs"
+ proof (cases "\<exists>a. s = {a}")
+ case True then show "?lhs \<le> ?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 \<in> s" "b \<in> s" "a \<noteq> b"
+ then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment>\<open>split into separate cases\<close>
+ by auto
+ have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
+ "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
+ using ab fs
+ by (subst s, subst sum.union_disjoint, auto)+
+ } note [simp] = this
+ { fix y
+ assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
+ { fix u T a
+ assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
+ and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
+ and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
+ have ua0: "u a = 0"
+ using ua by auto
+ obtain b where b: "b\<in>s" "a \<noteq> b"
+ using ua False by auto
+ obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
+ using yT by (auto elim: openE)
+ with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
+ by (auto intro: that [of "e / 2 / norm(a-b)"])
+ have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
+ using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
+ then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
+ using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
+ then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
+ using d e yT by auto
+ then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
+ "sum v s = 1"
+ "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
+ using subsetD [OF sb] yT
+ by auto
+ then have False
+ using assms
+ apply (simp add: affine_dependent_explicit_finite fs)
+ apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
+ using ua b d
+ apply (auto simp: algebra_simps sum_subtractf sum.distrib)
+ done
+ } note * = this
+ have "y \<notin> rel_interior (convex hull s)"
+ using y
+ apply (simp add: mem_rel_interior affine_hull_convex_hull)
+ apply (auto simp: convex_hull_finite [OF fs])
+ apply (drule_tac x=u in spec)
+ apply (auto intro: *)
+ done
+ } with rel_interior_subset show "?lhs \<le> ?rhs"
+ by blast
+ qed
+qed
+
+lemma interior_convex_hull_explicit_minimal:
+ fixes s :: "'a::euclidean_space set"
+ shows
+ "~ affine_dependent s
+ ==> interior(convex hull s) =
+ (if card(s) \<le> DIM('a) then {}
+ else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
+ apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
+ apply (rule trans [of _ "rel_interior(convex hull s)"])
+ apply (simp add: affine_hull_convex_hull affine_independent_span_gt rel_interior_interior)
+ by (simp add: rel_interior_convex_hull_explicit)
+
+lemma interior_convex_hull_explicit:
+ fixes s :: "'a::euclidean_space set"
+ assumes "~ affine_dependent s"
+ shows
+ "interior(convex hull s) =
+ (if card(s) \<le> DIM('a) then {}
+ else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
+proof -
+ { fix u :: "'a \<Rightarrow> real" and a
+ assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
+ then have cs: "Suc 0 < card s"
+ by (metis DIM_positive less_trans_Suc)
+ obtain b where b: "b \<in> s" "a \<noteq> b"
+ proof (cases "s \<le> {a}")
+ case True
+ then show thesis
+ using cs subset_singletonD by fastforce
+ next
+ case False
+ then show thesis
+ by (blast intro: that)
+ qed
+ have "u a + u b \<le> sum u {a,b}"
+ using a b by simp
+ also have "... \<le> sum u s"
+ apply (rule Groups_Big.sum_mono2)
+ using a b u
+ apply (auto simp: less_imp_le aff_independent_finite assms)
+ done
+ finally have "u a < 1"
+ using \<open>b \<in> s\<close> u by fastforce
+ } note [simp] = this
+ show ?thesis
+ using assms
+ apply (auto simp: interior_convex_hull_explicit_minimal)
+ apply (rule_tac x=u in exI)
+ apply (auto simp: not_le)
+ done
+qed
+
+lemma interior_closed_segment_ge2:
+ fixes a :: "'a::euclidean_space"
+ assumes "2 \<le> DIM('a)"
+ shows "interior(closed_segment a b) = {}"
+using assms unfolding segment_convex_hull
+proof -
+ have "card {a, b} \<le> 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 \<le> DIM('a) then {} else open_segment a b)"
+proof (simp add: not_le, intro conjI impI)
+ assume "2 \<le> DIM('a)"
+ then show "interior (open_segment a b) = {}"
+ apply (simp add: segment_convex_hull open_segment_def)
+ apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
+ done
+next
+ assume le2: "DIM('a) < 2"
+ show "interior (open_segment a b) = open_segment a b"
+ proof (cases "a = b")
+ case True then show ?thesis by auto
+ next
+ case False
+ with le2 have "affine hull (open_segment a b) = UNIV"
+ apply simp
+ apply (rule affine_independent_span_gt)
+ apply (simp_all add: affine_dependent_def insert_Diff_if)
+ done
+ then show "interior (open_segment a b) = open_segment a b"
+ using rel_interior_interior rel_interior_open_segment by blast
+ qed
+qed
+
+lemma interior_closed_segment:
+ fixes a :: "'a::euclidean_space"
+ shows "interior(closed_segment a b) =
+ (if 2 \<le> 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 \<longleftrightarrow> {a,b} = {c,d}"
+proof
+ assume abcd: "closed_segment a b = closed_segment c d"
+ show "{a,b} = {c,d}"
+ proof (cases "a=b \<or> c=d")
+ case True with abcd show ?thesis by force
+ next
+ case False
+ then have neq: "a \<noteq> b \<and> c \<noteq> 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 \<in> {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 \<in> {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 \<noteq> 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 \<noteq> 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 \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
+ (is "?lhs = ?rhs")
+proof
+ assume abcd: ?lhs
+ show ?rhs
+ proof (cases "a=b \<or> c=d")
+ case True with abcd show ?thesis
+ using finite_open_segment by fastforce
+ next
+ case False
+ then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
+ with abcd show ?rhs
+ unfolding open_segment_def
+ by (metis (no_types) abcd closed_segment_eq closure_open_segment)
+ qed
+next
+ assume ?rhs
+ then show ?lhs
+ by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
+qed
+
+subsection\<open>Similar results for closure and (relative or absolute) frontier.\<close>
+
+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. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
+proof -
+ have fs: "finite s"
+ using assms by (simp add: aff_independent_finite)
+ show ?thesis
+ apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
+ apply (auto simp: convex_hull_finite fs)
+ apply (drule_tac x=u in spec)
+ apply (rule_tac x=u in exI)
+ apply force
+ apply (rename_tac v)
+ apply (rule notE [OF assms])
+ apply (simp add: affine_dependent_explicit)
+ apply (rule_tac x=s in exI)
+ apply (auto simp: fs)
+ apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
+ apply (force simp: sum_subtractf scaleR_diff_left)
+ done
+qed
+
+lemma frontier_convex_hull_explicit:
+ fixes s :: "'a::euclidean_space set"
+ assumes "~ affine_dependent s"
+ shows "frontier(convex hull s) =
+ {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
+ sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>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 \<le> 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) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
+proof -
+ have fs: "finite s"
+ using assms by (simp add: aff_independent_finite)
+ { fix u a
+ have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
+ \<exists>x v. x \<in> s \<and>
+ (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
+ sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>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 \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
+ \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
+ (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
+ apply (rule_tac x="\<lambda>x. 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 (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
+ 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 \<le> 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 \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> 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 \<le> Suc (DIM ('a))" "x \<in> s"
+ shows "x \<in> frontier(convex hull s)"
+proof (cases "affine_dependent s")
+ case True
+ with assms show ?thesis
+ apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
+ by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
+next
+ case False
+ { assume "card s = Suc (card Basis)"
+ then have cs: "Suc 0 < card s"
+ by (simp add: DIM_positive)
+ with subset_singletonD have "\<exists>y \<in> s. y \<noteq> x"
+ by (cases "s \<le> {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 \<le> Suc (DIM ('a))" "x \<in> s"
+ shows "x \<notin> 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) = {} \<longleftrightarrow> affine_dependent s"
+proof -
+ { fix a b
+ assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> 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 have False using ab
+ by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
+ } then
+ show ?thesis
+ using assms
+ apply auto
+ apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
+ apply (auto simp: affine_dependent_def)
+ done
+qed
+
+
+subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
+
+definition coplanar where
+ "coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
+
+lemma collinear_affine_hull:
+ "collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
+proof (cases "s={}")
+ case True then show ?thesis
+ by simp
+next
+ case False
+ then obtain x where x: "x \<in> s" by auto
+ { fix u
+ assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
+ have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
+ apply (rule_tac x=x in exI)
+ apply (rule_tac x="x+u" in exI, clarify)
+ apply (erule exE [OF * [OF x]])
+ apply (rename_tac c)
+ apply (rule_tac x="1+c" in exI)
+ apply (rule_tac x="-c" in exI)
+ apply (simp add: algebra_simps)
+ done
+ } moreover
+ { fix u v x y
+ assume *: "s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
+ have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R (v-u)"
+ apply (drule subsetD [OF *])+
+ apply simp
+ apply clarify
+ apply (rename_tac r1 r2)
+ apply (rule_tac x="r1-r2" in exI)
+ apply (simp add: algebra_simps)
+ apply (metis scaleR_left.add)
+ done
+ } 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} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>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, hide_lams) 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 \<Rightarrow> real"
+ assumes "continuous_on (closed_segment a b) f"
+ shows "closed_segment (f a) (f b) \<subseteq> 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 \<Rightarrow> 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) \<subseteq> f ` closed_segment a b"
+ by (metis subset_continuous_image_segment_1 contf)
+ show "f ` closed_segment a b \<subseteq> 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 \<noteq> f b"
+ using inj_onD injf by fastforce
+ moreover
+ have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
+ proof (clarsimp simp add: open_segment_def)
+ assume fa: "f a \<in> closed_segment (f c) (f b)"
+ moreover have "closed_segment (f c) (f b) \<subseteq> 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 \<in> f ` closed_segment c b"
+ by blast
+ then have a: "a \<in> closed_segment c b"
+ by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+ have cb: "closed_segment c b \<subseteq> 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 \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
+ proof (clarsimp simp add: open_segment_def fnot eq_commute)
+ assume fb: "f b \<in> closed_segment (f a) (f c)"
+ moreover have "closed_segment (f a) (f c) \<subseteq> 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 \<in> f ` closed_segment a c"
+ by blast
+ then have b: "b \<in> closed_segment a c"
+ by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
+ have ca: "closed_segment a c \<subseteq> 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 \<Rightarrow> 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, hide_lams) 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 \<le> 2"
+ shows "collinear s"
+proof -
+ have "card s = 0 \<or> card s = 1 \<or> card s = 2"
+ using assms by linarith
+ then show ?thesis using assms
+ using card_eq_SucD
+ by auto (metis collinear_2 numeral_2_eq_2)
+qed
+
+lemma coplanar_small:
+ assumes "finite s" "card s \<le> 3"
+ shows "coplanar s"
+proof -
+ have "card s \<le> 2 \<or> card s = Suc (Suc (Suc 0))"
+ using assms by linarith
+ then show ?thesis using assms
+ apply safe
+ apply (simp add: collinear_small collinear_imp_coplanar)
+ apply (safe dest!: card_eq_SucD)
+ apply (auto simp: coplanar_def)
+ apply (metis hull_subset insert_subset)
+ done
+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) \<longleftrightarrow> 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) \<longleftrightarrow> 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 \<Rightarrow> 'b::real_normed_vector"
+ assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
+proof -
+ { fix u v w
+ assume "s \<subseteq> affine hull {u, v, w}"
+ then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
+ by (simp add: image_mono)
+ then have "f ` s \<subseteq> 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: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
+ unfolding coplanar_def
+ apply clarify
+ apply (rule_tac x="u+a" in exI)
+ apply (rule_tac x="v+a" in exI)
+ apply (rule_tac x="w+a" in exI)
+ using affine_hull_translation [of a "{u,v,w}" for u v w]
+ apply (force simp: add.commute)
+ done
+
+lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
+ by (metis (no_types) coplanar_translation_imp translation_galois)
+
+lemma coplanar_linear_image_eq:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
+proof
+ assume "coplanar s"
+ then show "coplanar (f ` s)"
+ unfolding coplanar_def
+ using affine_hull_linear_image [of f "{u,v,w}" for u v w] assms
+ by (meson coplanar_def coplanar_linear_image)
+next
+ obtain g where g: "linear g" "g \<circ> f = id"
+ using linear_injective_left_inverse [OF assms]
+ by blast
+ assume "coplanar (f ` s)"
+ then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
+ by (auto simp: coplanar_def)
+ then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
+ by blast
+ then have "s \<subseteq> g ` (affine hull {u, v, w})"
+ using g by (simp add: Fun.image_comp)
+ then show "coplanar s"
+ unfolding coplanar_def
+ using affine_hull_linear_image [of g "{u,v,w}" for u v w] \<open>linear g\<close> linear_conv_bounded_linear
+ by fastforce
+qed
+(*The HOL Light proof is simply
+ MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
+*)
+
+lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
+ by (meson coplanar_def order_trans)
+
+lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
+ by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
+
+lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> affine hull {a,b}"
+ unfolding collinear_def
+ apply clarify
+ apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
+ apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
+ apply (rename_tac y x)
+ apply (simp add: affine_hull_2)
+ apply (rule_tac x="1 - x/y" in exI)
+ apply (simp add: algebra_simps)
+ done
+
+lemma collinear_3_affine_hull:
+ assumes "a \<noteq> b"
+ shows "collinear {a,b,c} \<longleftrightarrow> c \<in> 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} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
+apply (case_tac "a=b", simp)
+apply (case_tac "a=c")
+apply (simp add: insert_commute)
+apply (case_tac "b=c")
+apply (simp add: insert_commute)
+apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
+apply (metis collinear_3_affine_hull insert_commute)+
+done
+
+lemma affine_dependent_imp_collinear_3:
+ "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
+by (simp add: collinear_3_eq_affine_dependent)
+
+lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
+ by (auto simp add: collinear_def)
+
+lemma collinear_3_expand:
+ "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>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 "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
+ by (simp add: collinear_lemma)
+ also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
+ by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
+ finally show ?thesis .
+qed
+
+lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
+proof
+ assume "collinear S"
+ then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
+ by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
+ then show "aff_dim S \<le> 1"
+ using order_trans by fastforce
+next
+ assume "aff_dim S \<le> 1"
+ then have le1: "aff_dim (affine hull S) \<le> 1"
+ by simp
+ obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
+ using affine_basis_exists [of S] by auto
+ then have "finite B" "card B \<le> 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 \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
+qed
+
+lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
+ apply (auto simp: collinear_3 collinear_lemma)
+ apply (drule_tac x="-1" in spec)
+ apply (simp add: algebra_simps)
+ done
+
+lemma midpoint_collinear:
+ fixes a b c :: "'a::real_normed_vector"
+ assumes "a \<noteq> c"
+ shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
+proof -
+ have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
+ "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
+ "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
+ by (auto simp: algebra_simps)
+ have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
+ using collinear_midpoint by blast
+ moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
+ apply (auto simp: collinear_3_expand assms dist_midpoint)
+ apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
+ apply (simp add: algebra_simps)
+ done
+ 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 \<or> x = b \<or> a = b")
+ case True with assms show ?thesis
+ by (auto simp: dist_commute)
+next
+ case False with assms show ?thesis
+ apply (auto simp: collinear_3 collinear_lemma between_norm)
+ apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
+ apply (simp add: vector_add_divide_simps eq_vector_fraction_iff real_vector.scale_minus_right [symmetric])
+ done
+qed
+
+lemma midpoint_between:
+ fixes a b :: "'a::euclidean_space"
+ shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
+proof (cases "a = c")
+ case True then show ?thesis
+ by (auto simp: dist_commute)
+next
+ case False
+ show ?thesis
+ apply (rule iffI)
+ apply (simp add: between_midpoint(1) dist_midpoint)
+ using False between_imp_collinear midpoint_collinear by blast
+qed
+
+lemma collinear_triples:
+ assumes "a \<noteq> b"
+ shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
+ (is "?lhs = ?rhs")
+proof safe
+ fix x
+ assume ?lhs and "x \<in> S"
+ then show "collinear {a, b, x}"
+ using collinear_subset by force
+next
+ assume ?rhs
+ then have "\<forall>x \<in> S. collinear{a,x,b}"
+ by (simp add: insert_commute)
+ then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
+ using that assms collinear_3_expand by fastforce+
+ show ?lhs
+ unfolding collinear_def
+ apply (rule_tac x="b-a" in exI)
+ apply (clarify dest!: *)
+ by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
+qed
+
+lemma collinear_4_3:
+ assumes "a \<noteq> b"
+ shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> 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 \<noteq> 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_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
+ using affine_hull_nonempty by blast
+
+lemma affine_hull_2_alt:
+ fixes a b :: "'a::real_vector"
+ shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
+apply (simp add: affine_hull_2, safe)
+apply (rule_tac x=v in image_eqI)
+apply (simp add: algebra_simps)
+apply (metis scaleR_add_left scaleR_one, simp)
+apply (rule_tac x="1-u" in exI)
+apply (simp add: algebra_simps)
+done
+
+lemma interior_convex_hull_3_minimal:
+ fixes a :: "'a::euclidean_space"
+ shows "\<lbrakk>~ collinear{a,b,c}; DIM('a) = 2\<rbrakk>
+ \<Longrightarrow> interior(convex hull {a,b,c}) =
+ {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
+ x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
+apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
+apply (rule_tac x="u a" in exI, simp)
+apply (rule_tac x="u b" in exI, simp)
+apply (rule_tac x="u c" in exI, simp)
+apply (rename_tac uu x y z)
+apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
+apply simp
+done
+
+subsection\<open>The infimum of the distance between two sets\<close>
+
+definition setdist :: "'a::metric_space set \<Rightarrow> 'a set \<Rightarrow> real" where
+ "setdist s t \<equiv>
+ (if s = {} \<or> t = {} then 0
+ else Inf {dist x y| x y. x \<in> s \<and> y \<in> t})"
+
+lemma setdist_empty1 [simp]: "setdist {} t = 0"
+ by (simp add: setdist_def)
+
+lemma setdist_empty2 [simp]: "setdist t {} = 0"
+ by (simp add: setdist_def)
+
+lemma setdist_pos_le [simp]: "0 \<le> setdist s t"
+ by (auto simp: setdist_def ex_in_conv [symmetric] intro: cInf_greatest)
+
+lemma le_setdistI:
+ assumes "s \<noteq> {}" "t \<noteq> {}" "\<And>x y. \<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> d \<le> dist x y"
+ shows "d \<le> setdist s t"
+ using assms
+ by (auto simp: setdist_def Set.ex_in_conv [symmetric] intro: cInf_greatest)
+
+lemma setdist_le_dist: "\<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> setdist s t \<le> dist x y"
+ unfolding setdist_def
+ by (auto intro!: bdd_belowI [where m=0] cInf_lower)
+
+lemma le_setdist_iff:
+ "d \<le> setdist s t \<longleftrightarrow>
+ (\<forall>x \<in> s. \<forall>y \<in> t. d \<le> dist x y) \<and> (s = {} \<or> t = {} \<longrightarrow> d \<le> 0)"
+ apply (cases "s = {} \<or> t = {}")
+ apply (force simp add: setdist_def)
+ apply (intro iffI conjI)
+ using setdist_le_dist apply fastforce
+ apply (auto simp: intro: le_setdistI)
+ done
+
+lemma setdist_ltE:
+ assumes "setdist s t < b" "s \<noteq> {}" "t \<noteq> {}"
+ obtains x y where "x \<in> s" "y \<in> t" "dist x y < b"
+using assms
+by (auto simp: not_le [symmetric] le_setdist_iff)
+
+lemma setdist_refl: "setdist s s = 0"
+ apply (cases "s = {}")
+ apply (force simp add: setdist_def)
+ apply (rule antisym [OF _ setdist_pos_le])
+ apply (metis all_not_in_conv dist_self setdist_le_dist)
+ done
+
+lemma setdist_sym: "setdist s t = setdist t s"
+ by (force simp: setdist_def dist_commute intro!: arg_cong [where f=Inf])
+
+lemma setdist_triangle: "setdist s t \<le> setdist s {a} + setdist {a} t"
+proof (cases "s = {} \<or> t = {}")
+ case True then show ?thesis
+ using setdist_pos_le by fastforce
+next
+ case False
+ have "\<And>x. x \<in> s \<Longrightarrow> setdist s t - dist x a \<le> setdist {a} t"
+ apply (rule le_setdistI, blast)
+ using False apply (fastforce intro: le_setdistI)
+ apply (simp add: algebra_simps)
+ apply (metis dist_commute dist_triangle3 order_trans [OF setdist_le_dist])
+ done
+ then have "setdist s t - setdist {a} t \<le> setdist s {a}"
+ using False by (fastforce intro: le_setdistI)
+ then show ?thesis
+ by (simp add: algebra_simps)
+qed
+
+lemma setdist_singletons [simp]: "setdist {x} {y} = dist x y"
+ by (simp add: setdist_def)
+
+lemma setdist_Lipschitz: "\<bar>setdist {x} s - setdist {y} s\<bar> \<le> dist x y"
+ apply (subst setdist_singletons [symmetric])
+ by (metis abs_diff_le_iff diff_le_eq setdist_triangle setdist_sym)
+
+lemma continuous_at_setdist [continuous_intros]: "continuous (at x) (\<lambda>y. (setdist {y} s))"
+ by (force simp: continuous_at_eps_delta dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
+
+lemma continuous_on_setdist [continuous_intros]: "continuous_on t (\<lambda>y. (setdist {y} s))"
+ by (metis continuous_at_setdist continuous_at_imp_continuous_on)
+
+lemma uniformly_continuous_on_setdist: "uniformly_continuous_on t (\<lambda>y. (setdist {y} s))"
+ by (force simp: uniformly_continuous_on_def dist_real_def intro: le_less_trans [OF setdist_Lipschitz])
+
+lemma setdist_subset_right: "\<lbrakk>t \<noteq> {}; t \<subseteq> u\<rbrakk> \<Longrightarrow> setdist s u \<le> setdist s t"
+ apply (cases "s = {} \<or> u = {}", force)
+ apply (auto simp: setdist_def intro!: bdd_belowI [where m=0] cInf_superset_mono)
+ done
+
+lemma setdist_subset_left: "\<lbrakk>s \<noteq> {}; s \<subseteq> t\<rbrakk> \<Longrightarrow> setdist t u \<le> setdist s u"
+ by (metis setdist_subset_right setdist_sym)
+
+lemma setdist_closure_1 [simp]: "setdist (closure s) t = setdist s t"
+proof (cases "s = {} \<or> t = {}")
+ case True then show ?thesis by force
+next
+ case False
+ { fix y
+ assume "y \<in> t"
+ have "continuous_on (closure s) (\<lambda>a. dist a y)"
+ by (auto simp: continuous_intros dist_norm)
+ then have *: "\<And>x. x \<in> closure s \<Longrightarrow> setdist s t \<le> dist x y"
+ apply (rule continuous_ge_on_closure)
+ apply assumption
+ apply (blast intro: setdist_le_dist \<open>y \<in> t\<close> )
+ done
+ } note * = this
+ show ?thesis
+ apply (rule antisym)
+ using False closure_subset apply (blast intro: setdist_subset_left)
+ using False *
+ apply (force simp add: closure_eq_empty intro!: le_setdistI)
+ done
+qed
+
+lemma setdist_closure_2 [simp]: "setdist t (closure s) = setdist t s"
+by (metis setdist_closure_1 setdist_sym)
+
+lemma setdist_compact_closed:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "compact S" and T: "closed T"
+ and "S \<noteq> {}" "T \<noteq> {}"
+ shows "\<exists>x \<in> S. \<exists>y \<in> T. dist x y = setdist S T"
+proof -
+ have "(\<Union>x\<in> S. \<Union>y \<in> T. {x - y}) \<noteq> {}"
+ using assms by blast
+ then have "\<exists>x \<in> S. \<exists>y \<in> T. dist x y \<le> setdist S T"
+ apply (rule distance_attains_inf [where a=0, OF compact_closed_differences [OF S T]])
+ apply (simp add: dist_norm le_setdist_iff)
+ apply blast
+ done
+ then show ?thesis
+ by (blast intro!: antisym [OF _ setdist_le_dist] )
+qed
+
+lemma setdist_closed_compact:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "closed S" and T: "compact T"
+ and "S \<noteq> {}" "T \<noteq> {}"
+ shows "\<exists>x \<in> S. \<exists>y \<in> T. dist x y = setdist S T"
+ using setdist_compact_closed [OF T S \<open>T \<noteq> {}\<close> \<open>S \<noteq> {}\<close>]
+ by (metis dist_commute setdist_sym)
+
+lemma setdist_eq_0I: "\<lbrakk>x \<in> S; x \<in> T\<rbrakk> \<Longrightarrow> setdist S T = 0"
+ by (metis antisym dist_self setdist_le_dist setdist_pos_le)
+
+lemma setdist_eq_0_compact_closed:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "compact S" and T: "closed T"
+ shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
+ apply (cases "S = {} \<or> T = {}", force)
+ using setdist_compact_closed [OF S T]
+ apply (force intro: setdist_eq_0I )
+ done
+
+corollary setdist_gt_0_compact_closed:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "compact S" and T: "closed T"
+ shows "setdist S T > 0 \<longleftrightarrow> (S \<noteq> {} \<and> T \<noteq> {} \<and> S \<inter> T = {})"
+ using setdist_pos_le [of S T] setdist_eq_0_compact_closed [OF assms]
+ by linarith
+
+lemma setdist_eq_0_closed_compact:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "closed S" and T: "compact T"
+ shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {}"
+ using setdist_eq_0_compact_closed [OF T S]
+ by (metis Int_commute setdist_sym)
+
+lemma setdist_eq_0_bounded:
+ fixes S :: "'a::euclidean_space set"
+ assumes "bounded S \<or> bounded T"
+ shows "setdist S T = 0 \<longleftrightarrow> S = {} \<or> T = {} \<or> closure S \<inter> closure T \<noteq> {}"
+ apply (cases "S = {} \<or> T = {}", force)
+ using setdist_eq_0_compact_closed [of "closure S" "closure T"]
+ setdist_eq_0_closed_compact [of "closure S" "closure T"] assms
+ apply (force simp add: bounded_closure compact_eq_bounded_closed)
+ done
+
+lemma setdist_unique:
+ "\<lbrakk>a \<in> S; b \<in> T; \<And>x y. x \<in> S \<and> y \<in> T ==> dist a b \<le> dist x y\<rbrakk>
+ \<Longrightarrow> setdist S T = dist a b"
+ by (force simp add: setdist_le_dist le_setdist_iff intro: antisym)
+
+lemma setdist_closest_point:
+ "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> setdist {a} S = dist a (closest_point S a)"
+ apply (rule setdist_unique)
+ using closest_point_le
+ apply (auto simp: closest_point_in_set)
+ done
+
+lemma setdist_eq_0_sing_1:
+ fixes S :: "'a::euclidean_space set"
+ shows "setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> closure S"
+ by (auto simp: setdist_eq_0_bounded)
+
+lemma setdist_eq_0_sing_2:
+ fixes S :: "'a::euclidean_space set"
+ shows "setdist S {x} = 0 \<longleftrightarrow> S = {} \<or> x \<in> closure S"
+ by (auto simp: setdist_eq_0_bounded)
+
+lemma setdist_neq_0_sing_1:
+ fixes S :: "'a::euclidean_space set"
+ shows "\<lbrakk>setdist {x} S = a; a \<noteq> 0\<rbrakk> \<Longrightarrow> S \<noteq> {} \<and> x \<notin> closure S"
+ by (auto simp: setdist_eq_0_sing_1)
+
+lemma setdist_neq_0_sing_2:
+ fixes S :: "'a::euclidean_space set"
+ shows "\<lbrakk>setdist S {x} = a; a \<noteq> 0\<rbrakk> \<Longrightarrow> S \<noteq> {} \<and> x \<notin> closure S"
+ by (auto simp: setdist_eq_0_sing_2)
+
+lemma setdist_sing_in_set:
+ fixes S :: "'a::euclidean_space set"
+ shows "x \<in> S \<Longrightarrow> setdist {x} S = 0"
+ using closure_subset by (auto simp: setdist_eq_0_sing_1)
+
+lemma setdist_le_sing: "x \<in> S ==> setdist S T \<le> setdist {x} T"
+ using setdist_subset_left by auto
+
+lemma setdist_eq_0_closed:
+ fixes S :: "'a::euclidean_space set"
+ shows "closed S \<Longrightarrow> (setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> S)"
+by (simp add: setdist_eq_0_sing_1)
+
+lemma setdist_eq_0_closedin:
+ fixes S :: "'a::euclidean_space set"
+ shows "\<lbrakk>closedin (subtopology euclidean u) S; x \<in> u\<rbrakk>
+ \<Longrightarrow> (setdist {x} S = 0 \<longleftrightarrow> S = {} \<or> x \<in> S)"
+ by (auto simp: closedin_limpt setdist_eq_0_sing_1 closure_def)
+
+subsection\<open>Basic lemmas about hyperplanes and halfspaces\<close>
+
+lemma hyperplane_eq_Ex:
+ assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
+ by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
+
+lemma hyperplane_eq_empty:
+ "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
+ using hyperplane_eq_Ex apply auto[1]
+ using inner_zero_right by blast
+
+lemma hyperplane_eq_UNIV:
+ "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
+proof -
+ have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
+ apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
+ apply simp_all
+ by (metis add_cancel_right_right zero_neq_one)
+ then show ?thesis by force
+qed
+
+lemma halfspace_eq_empty_lt:
+ "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
+proof -
+ have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
+ apply (rule ccontr)
+ apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
+ apply force+
+ done
+ then show ?thesis by force
+qed
+
+lemma halfspace_eq_empty_gt:
+ "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
+using halfspace_eq_empty_lt [of "-a" "-b"]
+by simp
+
+lemma halfspace_eq_empty_le:
+ "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
+proof -
+ have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
+ apply (rule ccontr)
+ apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
+ apply force+
+ done
+ then show ?thesis by force
+qed
+
+lemma halfspace_eq_empty_ge:
+ "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
+using halfspace_eq_empty_le [of "-a" "-b"]
+by simp
+
+subsection\<open>Use set distance for an easy proof of separation properties\<close>
+
+proposition separation_closures:
+ fixes S :: "'a::euclidean_space set"
+ assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
+ obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
+proof (cases "S = {} \<or> T = {}")
+ case True with that show ?thesis by auto
+next
+ case False
+ define f where "f \<equiv> \<lambda>x. 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} \<inter> {x. f x < 0} = {}"
+ by auto
+ show "open {x. 0 < f x}"
+ by (simp add: open_Collect_less contf continuous_on_const)
+ show "open {x. f x < 0}"
+ by (simp add: open_Collect_less contf continuous_on_const)
+ show "S \<subseteq> {x. 0 < f x}"
+ apply (clarsimp simp add: f_def setdist_sing_in_set)
+ using assms
+ by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
+ show "T \<subseteq> {x. f x < 0}"
+ apply (clarsimp simp add: f_def setdist_sing_in_set)
+ using assms
+ by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
+ qed
+qed
+
+lemma separation_normal:
+ fixes S :: "'a::euclidean_space set"
+ assumes "closed S" "closed T" "S \<inter> T = {}"
+ obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 (subtopology euclidean U) S"
+ and UT: "closedin (subtopology euclidean U) T"
+ and "S \<inter> T = {}"
+ obtains S' T' where "openin (subtopology euclidean U) S'"
+ "openin (subtopology euclidean U) T'"
+ "S \<subseteq> S'" "T \<subseteq> T'" "S' \<inter> T' = {}"
+proof (cases "S = {} \<or> T = {}")
+ case True with that show ?thesis
+ apply safe
+ using UT closedin_subset apply blast
+ using US closedin_subset apply blast
+ done
+next
+ case False
+ define f where "f \<equiv> \<lambda>x. 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' = "{x\<in>U. f x > 0}" and T' = "{x\<in>U. f x < 0}" in that)
+ show "{x \<in> U. 0 < f x} \<inter> {x \<in> U. f x < 0} = {}"
+ by auto
+ have "openin (subtopology euclidean U) {x \<in> U. f x \<in> {0<..}}"
+ by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
+ then show "openin (subtopology euclidean U) {x \<in> U. 0 < f x}" by simp
+ next
+ have "openin (subtopology euclidean U) {x \<in> U. f x \<in> {..<0}}"
+ by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
+ then show "openin (subtopology euclidean U) {x \<in> U. f x < 0}" by simp
+ next
+ show "S \<subseteq> {x \<in> U. 0 < f x}"
+ apply (clarsimp simp add: f_def setdist_sing_in_set)
+ using assms
+ by (metis False Int_insert_right closedin_imp_subset empty_not_insert insert_absorb insert_subset linorder_neqE_linordered_idom not_le setdist_eq_0_closedin setdist_pos_le)
+ show "T \<subseteq> {x \<in> U. f x < 0}"
+ apply (clarsimp simp add: f_def setdist_sing_in_set)
+ using assms
+ by (metis False closedin_subset disjoint_iff_not_equal insert_absorb insert_subset linorder_neqE_linordered_idom not_less setdist_eq_0_closedin setdist_pos_le topspace_euclidean_subtopology)
+ qed
+qed
+
+lemma separation_normal_compact:
+ fixes S :: "'a::euclidean_space set"
+ assumes "compact S" "closed T" "S \<inter> T = {}"
+ obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> 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 \<subseteq> ball 0 r"
+ by (auto dest!: bounded_subset_ballD)
+ have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
+ using assms r by blast+
+ then show ?thesis
+ apply (rule separation_normal [OF \<open>closed S\<close>])
+ apply (rule_tac U=U and V=V in that)
+ by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
+qed
+
+subsection\<open>Proper maps, including projections out of compact sets\<close>
+
+lemma finite_indexed_bound:
+ assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
+ shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
+using A
+proof (induction A)
+ case empty then show ?case by force
+next
+ case (insert a A)
+ then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
+ by force
+ then show ?case
+ apply (rule_tac x="max m n" in exI, safe)
+ using max.cobounded2 apply blast
+ by (meson le_max_iff_disj)
+qed
+
+proposition proper_map:
+ fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
+ assumes "closedin (subtopology euclidean S) K"
+ and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact {x \<in> S. f x \<in> U}"
+ and "f ` S \<subseteq> T"
+ shows "closedin (subtopology euclidean T) (f ` K)"
+proof -
+ have "K \<subseteq> S"
+ using assms closedin_imp_subset by metis
+ obtain C where "closed C" and Keq: "K = S \<inter> C"
+ using assms by (auto simp: closedin_closed)
+ have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
+ proof -
+ obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
+ using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
+ then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
+ by metis
+ then have fX: "\<And>n. f (X n) = h n"
+ by metis
+ have "compact (C \<inter> {a \<in> S. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))})" for n
+ apply (rule closed_Int_compact [OF \<open>closed C\<close>])
+ apply (rule com)
+ using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> apply blast
+ apply (rule compact_sequence_with_limit)
+ apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
+ done
+ then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
+ by (simp add: Keq Int_def conj_commute)
+ have ne: "\<Inter>\<F> \<noteq> {}"
+ if "finite \<F>"
+ and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
+ (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
+ for \<F>
+ proof -
+ obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
+ apply (rule exE)
+ apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
+ done
+ have "X m \<in> \<Inter>\<F>"
+ using X le_Suc_ex by (fastforce dest: m)
+ then show ?thesis by blast
+ qed
+ have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
+ \<noteq> {}"
+ apply (rule compact_fip_heine_borel)
+ using comf apply force
+ using ne apply (simp add: subset_iff del: insert_iff)
+ done
+ then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
+ by blast
+ then show ?thesis
+ apply (simp add: image_iff fX)
+ by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
+ qed
+ with assms closedin_subset show ?thesis
+ by (force simp: closedin_limpt)
+qed
+
+
+lemma compact_continuous_image_eq:
+ fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
+ assumes f: "inj_on f S"
+ shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> 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: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
+ by (metis inv_into_f_f f)
+ then have *: "{x \<in> S. f x \<in> U} = g ` U" if "U \<subseteq> f ` S" for U
+ using that by fastforce
+ have gfim: "g ` f ` S \<subseteq> S" using gf by auto
+ have **: "compact {x \<in> f ` S. g x \<in> C}" if C: "C \<subseteq> S" "compact C" for C
+ proof -
+ obtain h :: "'a set \<Rightarrow> 'a" where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
+ by (force simp: C RHS)
+ moreover have "f ` C = {b \<in> f ` S. g b \<in> C}"
+ using C gf by auto
+ ultimately show "compact {b \<in> f ` S. g b \<in> C}"
+ using C by auto
+ qed
+ show ?lhs
+ using proper_map [OF _ _ gfim] **
+ by (simp add: continuous_on_closed * closedin_imp_subset)
+qed
+
+subsection\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
+
+lemma convex_connected_collinear:
+ fixes S :: "'a::euclidean_space set"
+ assumes "collinear S"
+ shows "convex S \<longleftrightarrow> 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 \<in> S" by auto
+ have "collinear (affine hull S)"
+ by (simp add: assms collinear_affine_hull_collinear)
+ then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
+ by (meson \<open>a \<in> S\<close> collinear hull_inc)
+ then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
+ proof -
+ have "f x *\<^sub>R z = x - a"
+ by (simp add: f hull_inc x)
+ moreover have "f y *\<^sub>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"
+ apply (clarsimp simp: dist_norm continuous_on_iff diff)
+ by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
+ 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 \<in> S" "y \<in> S" "z \<in> closed_segment x y"
+ have False if "z \<notin> S"
+ proof -
+ have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
+ apply (rule continuous_injective_image_segment_1)
+ apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
+ by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
+ then have fz: "f z \<in> closed_segment (f x) (f y)"
+ using \<open>z \<in> closed_segment x y\<close> by blast
+ have "z \<in> affine hull S"
+ by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
+ then have fz_notin: "f z \<notin> f ` S"
+ using hull_subset inj_f inj_onD that by fastforce
+ moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
+ proof -
+ have "{..<f z} \<inter> f ` {x,y} \<noteq> {}" "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
+ using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
+ apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
+ apply (metis image_eqI less_eq_real_def)+
+ done
+ then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
+ using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
+ qed
+ ultimately show False
+ using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
+ qed
+ then show "z \<in> S" by meson
+ qed
+ qed
+qed
+
+lemma compact_convex_collinear_segment_alt:
+ fixes S :: "'a::euclidean_space set"
+ assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
+ obtains a b where "S = closed_segment a b"
+proof -
+ obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
+ have "collinear (affine hull S)"
+ by (simp add: assms collinear_affine_hull_collinear)
+ then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
+ by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
+ then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
+ by metis
+ let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
+ have gf: "?g (f x) = x" if "x \<in> 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) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
+ proof -
+ have "f x *\<^sub>R z = x - \<xi>"
+ by (simp add: f hull_inc x)
+ moreover have "f y *\<^sub>R z = y - \<xi>"
+ 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"
+ apply (clarsimp simp: dist_norm continuous_on_iff diff)
+ by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
+ then have "connected (f ` S)"
+ by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
+ moreover have "compact (f ` S)"
+ by (meson \<open>compact S\<close> 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 \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
+ obtain a b where "a \<in> S" "f a = x" "b \<in> 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)"
+ apply (rule continuous_injective_image_segment_1)
+ apply (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
+ by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
+ then have "f ` (closed_segment a b) = f ` S"
+ by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
+ then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
+ by simp
+ moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
+ apply safe
+ apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
+ by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff 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 \<noteq> {}" "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 \<Rightarrow> 'b::euclidean_space"
+ assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
+ "closedin (subtopology euclidean T) K"
+ shows "compact {x. x \<in> S \<and> f x \<in> K}"
+by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
+
+lemma proper_map_fst:
+ assumes "compact T" "K \<subseteq> S" "compact K"
+ shows "compact {z \<in> S \<times> T. fst z \<in> K}"
+proof -
+ have "{z \<in> S \<times> T. fst z \<in> K} = K \<times> 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 (subtopology euclidean (S \<times> T)) c"
+ shows "closedin (subtopology euclidean S) (fst ` c)"
+proof -
+ have *: "fst ` (S \<times> T) \<subseteq> 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 \<subseteq> T" "compact K"
+ shows "compact {z \<in> S \<times> T. snd z \<in> K}"
+proof -
+ have "{z \<in> S \<times> T. snd z \<in> K} = S \<times> 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 (subtopology euclidean (S \<times> T)) c"
+ shows "closedin (subtopology euclidean T) (snd ` c)"
+proof -
+ have *: "snd ` (S \<times> T) \<subseteq> 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 (subtopology euclidean (S \<times> T)) U"
+ shows "closedin (subtopology euclidean T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
+proof -
+ have "U \<subseteq> S \<times> T"
+ by (metis clo closedin_imp_subset)
+ then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
+ by force
+ moreover have "closedin (subtopology euclidean 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. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
+proof -
+ have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
+ {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
+ by auto
+ show ?thesis
+ apply (subst *)
+ apply (rule closedin_closed_trans [OF _ closed_UNIV])
+ apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
+ by (simp add: clo closedin_closed_Int)
+qed
+
+subsubsection\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
+
+proposition affine_hull_convex_Int_nonempty_interior:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "convex S" "S \<inter> interior T \<noteq> {}"
+ shows "affine hull (S \<inter> T) = affine hull S"
+proof
+ show "affine hull (S \<inter> T) \<subseteq> affine hull S"
+ by (simp add: hull_mono)
+next
+ obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
+ using assms interior_subset by blast
+ then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
+ using mem_interior_cball by blast
+ have *: "x \<in> op + a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> 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 \<open>e > 0\<close> False by (auto simp: k_def)
+ then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
+ by simp
+ have "e / norm (x - a) \<ge> k"
+ using k_def by linarith
+ then have "a + k *\<^sub>R (x - a) \<in> cball a e"
+ using \<open>0 < k\<close> False by (simp add: dist_norm field_simps)
+ then have T: "a + k *\<^sub>R (x - a) \<in> T"
+ using e by blast
+ have S: "a + k *\<^sub>R (x - a) \<in> S"
+ using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
+ by (simp add: algebra_simps)
+ have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
+ apply (rule span_mul)
+ apply (rule span_superset)
+ apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
+ apply (auto simp: S T)
+ done
+ with xa image_iff show ?thesis by fastforce
+ qed
+ show "affine hull S \<subseteq> affine hull (S \<inter> T)"
+ apply (simp add: subset_hull)
+ apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
+ apply (force simp: *)
+ done
+qed
+
+corollary affine_hull_convex_Int_open:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
+ shows "affine hull (S \<inter> 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 \<inter> interior T \<noteq> {}"
+ shows "affine hull (S \<inter> 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 \<inter> T \<noteq> {}"
+ shows "affine hull (S \<inter> 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 (subtopology euclidean (affine hull S)) T" "S \<inter> T \<noteq> {}"
+ shows "affine hull (S \<inter> 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 (subtopology euclidean (affine hull T)) S" "S \<noteq> {}"
+ 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 \<noteq> {}"
+ 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 "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> 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 (subtopology euclidean (affine hull S)) S" and "finite F" "F \<subset> S"
+ shows "affine hull (S - F) = affine hull S"
+proof -
+ have clo: "closedin (subtopology euclidean S) F"
+ using assms finite_imp_closedin by auto
+ moreover have "S - F \<noteq> {}"
+ 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 \<bullet> x < r} = (if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} = (if a = 0 \<and> 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 \<bullet> x < r} = UNIV"
+ using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
+ moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> 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 \<bullet> x > r} = (if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} = (if a = 0 \<and> 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 \<bullet> x < r} =
+ (if a = 0 \<and> r \<le> 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 \<bullet> x \<le> r} =
+ (if a = 0 \<and> r < 0 then -1 else DIM('a))"
+proof -
+ have "int (DIM('a)) = aff_dim (UNIV::'a set)"
+ by (simp add: aff_dim_UNIV)
+ then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
+ using that by (simp add: affine_hull_halfspace_le not_less)
+ then show ?thesis
+ by (force simp: aff_dim_affine_hull)
+qed
+
+lemma aff_dim_halfspace_gt:
+ fixes a :: "'a::euclidean_space"
+ shows "aff_dim {x. a \<bullet> x > r} =
+ (if a = 0 \<and> r \<ge> 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 \<bullet> x \<ge> r} =
+ (if a = 0 \<and> r > 0 then -1 else DIM('a))"
+using aff_dim_halfspace_le [of "-a" "-r"] by simp
+
+subsection\<open>Properties of special hyperplanes\<close>
+
+lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
+ by (simp add: subspace_def inner_right_distrib)
+
+lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
+ by (simp add: inner_commute inner_right_distrib subspace_def)
+
+lemma special_hyperplane_span:
+ fixes S :: "'n::euclidean_space set"
+ assumes "k \<in> Basis"
+ shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
+proof -
+ have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
+ proof -
+ have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
+ by (simp add: euclidean_representation)
+ also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
+ by (auto simp: sum.remove [of _ k] inner_commute assms that)
+ finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
+ then show ?thesis
+ by (simp add: Linear_Algebra.span_finite) metis
+ qed
+ show ?thesis
+ apply (rule span_subspace [symmetric])
+ using assms
+ apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
+ done
+qed
+
+lemma dim_special_hyperplane:
+ fixes k :: "'n::euclidean_space"
+ shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
+apply (simp add: special_hyperplane_span)
+apply (rule Linear_Algebra.dim_unique [OF subset_refl])
+apply (auto simp: Diff_subset independent_substdbasis)
+apply (metis member_remove remove_def span_clauses(1))
+done
+
+proposition dim_hyperplane:
+ fixes a :: "'a::euclidean_space"
+ assumes "a \<noteq> 0"
+ shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
+proof -
+ have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
+ by (rule span_unique) (auto simp: subspace_hyperplane)
+ then obtain B where "independent B"
+ and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
+ and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
+ and card0: "(card B = dim {x. a \<bullet> x = 0})"
+ and ortho: "pairwise orthogonal B"
+ using orthogonal_basis_exists by metis
+ with assms have "a \<notin> span B"
+ by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0 span_subspace)
+ then have ind: "independent (insert a B)"
+ by (simp add: \<open>independent B\<close> independent_insert)
+ have "finite B"
+ using \<open>independent B\<close> independent_bound by blast
+ have "UNIV \<subseteq> span (insert a B)"
+ proof fix y::'a
+ obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
+ apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
+ using assms
+ by (auto simp: algebra_simps)
+ show "y \<in> span (insert a B)"
+ by (metis (mono_tags, lifting) z Bsub Convex_Euclidean_Space.span_eq
+ add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
+ qed
+ then have dima: "DIM('a) = dim(insert a B)"
+ by (metis antisym dim_UNIV dim_subset_UNIV subset_le_dim)
+ then show ?thesis
+ by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0 card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE subspB)
+qed
+
+lemma lowdim_eq_hyperplane:
+ fixes S :: "'a::euclidean_space set"
+ assumes "dim S = DIM('a) - 1"
+ obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
+proof -
+ have [simp]: "dim S < DIM('a)"
+ by (simp add: DIM_positive assms)
+ then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
+ using lowdim_subset_hyperplane [of S] by fastforce
+ show ?thesis
+ using b that real_vector_class.subspace_span [of S]
+ by (simp add: assms dim_hyperplane subspace_dim_equal subspace_hyperplane)
+qed
+
+lemma dim_eq_hyperplane:
+ fixes S :: "'n::euclidean_space set"
+ shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
+by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
+
+proposition aff_dim_eq_hyperplane:
+ fixes S :: "'a::euclidean_space set"
+ shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
+proof (cases "S = {}")
+ case True then show ?thesis
+ by (auto simp: dest: hyperplane_eq_Ex)
+next
+ case False
+ then obtain c where "c \<in> S" by blast
+ show ?thesis
+ proof (cases "c = 0")
+ case True show ?thesis
+ apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
+ del: One_nat_def)
+ apply (rule ex_cong)
+ apply (metis (mono_tags) span_0 \<open>c = 0\<close> image_add_0 inner_zero_right mem_Collect_eq)
+ done
+ next
+ case False
+ have xc_im: "x \<in> op + c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
+ proof -
+ have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
+ by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
+ then show "x \<in> op + c ` {y. a \<bullet> y = 0}"
+ by blast
+ qed
+ have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
+ if "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
+ proof -
+ have "b = a \<bullet> c"
+ using span_0 that by fastforce
+ with that have "op + c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
+ by simp
+ then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
+ by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
+ also have "... = {x. a \<bullet> x = 0}"
+ by (force simp: inner_distrib inner_diff_right
+ intro: image_eqI [where x="x+c" for x])
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
+ del: One_nat_def, safe)
+ apply (fastforce simp add: inner_distrib intro: xc_im)
+ apply (force simp: intro!: 2)
+ done
+ qed
+qed
+
+corollary aff_dim_hyperplane [simp]:
+ fixes a :: "'a::euclidean_space"
+ shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
+by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
+
+subsection\<open>Some stepping theorems\<close>
+
+lemma dim_empty [simp]: "dim ({} :: 'a::euclidean_space set) = 0"
+ by (force intro!: dim_unique)
+
+lemma dim_insert:
+ fixes x :: "'a::euclidean_space"
+ shows "dim (insert x S) = (if x \<in> span S then dim S else dim S + 1)"
+proof -
+ show ?thesis
+ proof (cases "x \<in> span S")
+ case True then show ?thesis
+ by (metis dim_span span_redundant)
+ next
+ case False
+ obtain B where B: "B \<subseteq> span S" "independent B" "span S \<subseteq> span B" "card B = dim (span S)"
+ using basis_exists [of "span S"] by blast
+ have 1: "insert x B \<subseteq> span (insert x S)"
+ by (meson \<open>B \<subseteq> span S\<close> dual_order.trans insertI1 insert_subsetI span_mono span_superset subset_insertI)
+ have 2: "span (insert x S) \<subseteq> span (insert x B)"
+ by (metis \<open>B \<subseteq> span S\<close> \<open>span S \<subseteq> span B\<close> span_breakdown_eq span_subspace subsetI subspace_span)
+ have 3: "independent (insert x B)"
+ by (metis B independent_insert span_subspace subspace_span False)
+ have "dim (span (insert x S)) = Suc (dim S)"
+ apply (rule dim_unique [OF 1 2 3])
+ by (metis B False card_insert_disjoint dim_span independent_imp_finite subsetCE)
+ then show ?thesis
+ by (simp add: False)
+ qed
+qed
+
+lemma dim_singleton [simp]:
+ fixes x :: "'a::euclidean_space"
+ shows "dim{x} = (if x = 0 then 0 else 1)"
+by (simp add: dim_insert)
+
+lemma dim_eq_0 [simp]:
+ fixes S :: "'a::euclidean_space set"
+ shows "dim S = 0 \<longleftrightarrow> S \<subseteq> {0}"
+apply safe
+apply (metis DIM_positive DIM_real card_ge_dim_independent contra_subsetD dim_empty dim_insert dim_singleton empty_subsetI independent_empty less_not_refl zero_le)
+by (metis dim_singleton dim_subset le_0_eq)
+
+lemma aff_dim_insert:
+ fixes a :: "'a::euclidean_space"
+ shows "aff_dim (insert a S) = (if a \<in> 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 \<notin> s'"
+ by (meson Set.set_insert all_not_in_conv)
+ show ?thesis using S
+ apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
+ apply (simp add: affine_hull_insert_span_gen hull_inc)
+ apply (simp add: insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert span_0)
+ apply (metis (no_types, lifting) add_minus_cancel image_iff uminus_add_conv_diff)
+ done
+qed
+
+lemma subspace_bounded_eq_trivial:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "subspace S"
+ shows "bounded S \<longleftrightarrow> S = {0}"
+proof -
+ have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
+ proof -
+ obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
+ using \<open>bounded S\<close> by (force simp: bounded_pos_less)
+ have "(B / norm x) *\<^sub>R x \<in> S"
+ using assms subspace_mul \<open>x \<in> S\<close> by auto
+ moreover have "norm ((B / norm x) *\<^sub>R x) = B"
+ using that B by (simp add: algebra_simps)
+ ultimately show False using B by force
+ qed
+ then have "bounded S \<Longrightarrow> 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 \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
+proof (cases "S = {}")
+ case True then show ?thesis
+ by simp
+next
+ case False
+ then obtain b where "b \<in> S" by blast
+ with False assms show ?thesis
+ apply safe
+ using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
+ apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
+ image_empty image_insert translation_invert)
+ apply force
+ done
+qed
+
+lemma affine_bounded_eq_lowdim:
+ fixes S :: "'a::euclidean_space set"
+ assumes "affine S"
+ shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
+apply safe
+using affine_bounded_eq_trivial assms apply fastforce
+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)
+
+
+lemma bounded_hyperplane_eq_trivial_0:
+ fixes a :: "'a::euclidean_space"
+ assumes "a \<noteq> 0"
+ shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
+proof
+ assume "bounded {x. a \<bullet> x = 0}"
+ then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
+ by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
+ with assms show "DIM('a) = 1"
+ by (simp add: le_Suc_eq aff_dim_hyperplane)
+next
+ assume "DIM('a) = 1"
+ then show "bounded {x. a \<bullet> x = 0}"
+ by (simp add: aff_dim_hyperplane affine_bounded_eq_lowdim affine_hyperplane assms)
+qed
+
+lemma bounded_hyperplane_eq_trivial:
+ fixes a :: "'a::euclidean_space"
+ shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
+proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
+ assume "r \<noteq> 0" "a \<noteq> 0"
+ have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
+ by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
+ then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
+ by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
+qed
+
+subsection\<open>General case without assuming closure and getting non-strict separation\<close>
+
+proposition separating_hyperplane_closed_point_inset:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
+ obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
+proof -
+ obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
+ using distance_attains_inf [of S z] assms by auto
+ then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
+ using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
+ show ?thesis
+ proof (rule that [OF \<open>y \<in> S\<close> *])
+ fix x
+ assume "x \<in> S"
+ have yz: "0 < (y - z) \<bullet> (y - z)"
+ using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
+ { assume 0: "0 < ((z - y) \<bullet> (x - y))"
+ with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
+ have False
+ using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
+ }
+ then have "0 \<le> ((y - z) \<bullet> (x - y))"
+ by (force simp: not_less inner_diff_left)
+ with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
+ by (simp add: algebra_simps)
+ then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> 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 \<noteq> {}" "0 \<notin> S"
+ obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
+using separating_hyperplane_closed_point_inset [OF assms]
+by simp (metis \<open>0 \<notin> S\<close>)
+
+
+proposition separating_hyperplane_set_0_inspan:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
+ obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
+proof -
+ define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
+ have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
+ if f': "finite f'" "f' \<subseteq> k ` S" for f'
+ proof -
+ obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
+ using finite_subset_image [OF f'] by blast
+ obtain a where "a \<in> S" "a \<noteq> 0"
+ using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
+ then have "norm (a /\<^sub>R (norm a)) = 1"
+ by simp
+ moreover have "a /\<^sub>R (norm a) \<in> span S"
+ by (simp add: \<open>a \<in> S\<close> span_mul span_superset)
+ ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> 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 \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
+ moreover have "convex hull C \<noteq> {}"
+ by (simp add: False)
+ moreover have "0 \<notin> convex hull C"
+ by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
+ ultimately obtain a b
+ where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
+ and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
+ using separating_hyperplane_closed_0_inset by blast
+ then have "a \<in> S"
+ by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
+ moreover have "norm (a /\<^sub>R (norm a)) = 1"
+ using \<open>a \<noteq> 0\<close> by simp
+ moreover have "a /\<^sub>R (norm a) \<in> span S"
+ by (simp add: \<open>a \<in> S\<close> span_mul span_superset)
+ ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
+ by simp
+ have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
+ apply (clarsimp simp add: divide_simps)
+ using ab \<open>0 < b\<close>
+ by (metis hull_inc inner_commute less_eq_real_def less_trans)
+ show ?thesis
+ apply (simp add: C k_def)
+ using ass aa Int_iff empty_iff by blast
+ qed
+ qed
+ have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
+ apply (rule compact_imp_fip)
+ apply (blast intro: compact_cball)
+ using closed_halfspace_ge k_def apply blast
+ apply (metis *)
+ done
+ 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 \<noteq> {}" and zno: "z \<notin> S"
+ obtains a b where "(z + a) \<in> affine hull (insert z S)"
+ and "a \<noteq> 0" and "a \<bullet> z \<le> b"
+ and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
+proof -
+from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
+ have "convex (op + (- z) ` S)"
+ by (simp add: \<open>convex S\<close>)
+ moreover have "op + (- z) ` S \<noteq> {}"
+ by (simp add: \<open>S \<noteq> {}\<close>)
+ moreover have "0 \<notin> op + (- z) ` S"
+ using zno by auto
+ ultimately obtain a where "a \<in> span (op + (- z) ` S)" "a \<noteq> 0"
+ and a: "\<And>x. x \<in> (op + (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
+ using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
+ by blast
+ then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> 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)
+ show ?thesis
+ apply (rule_tac a=a and b = "a \<bullet> z" in that, simp_all)
+ using \<open>a \<in> span (op + (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
+ apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
+ done
+qed
+
+proposition supporting_hyperplane_rel_boundary:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
+ obtains a where "a \<noteq> 0"
+ and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
+ and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
+proof -
+ obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
+ and "a \<noteq> 0" and "a \<bullet> x \<le> b"
+ and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> 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 \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
+ proof -
+ have con: "continuous_on (closure (rel_interior S)) (op \<bullet> a)"
+ by (rule continuous_intros continuous_on_subset | blast)+
+ have y: "y \<in> closure (rel_interior S)"
+ using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
+ by fastforce
+ show ?thesis
+ using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
+ by fastforce
+ qed
+ have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
+ proof -
+ obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
+ using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
+ define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
+ have "y' \<in> cball y e"
+ unfolding y'_def using \<open>0 < e\<close> by force
+ moreover have "y' \<in> affine hull S"
+ unfolding y'_def
+ by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
+ rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
+ ultimately have "y' \<in> S"
+ using e by auto
+ have "a \<bullet> x \<le> a \<bullet> y"
+ using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
+ moreover have "a \<bullet> x \<noteq> a \<bullet> y"
+ using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
+ apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
+ by (metis \<open>0 < e\<close> add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
+ ultimately show ?thesis by force
+ qed
+ show ?thesis
+ by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
+qed
+
+lemma supporting_hyperplane_relative_frontier:
+ fixes S :: "'a::euclidean_space set"
+ assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
+ obtains a where "a \<noteq> 0"
+ and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
+ and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
+using supporting_hyperplane_rel_boundary [of "closure S" x]
+by (metis assms convex_closure convex_rel_interior_closure)
+
+
+subsection\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
+
+lemma
+ fixes s :: "'a::euclidean_space set"
+ assumes "~ (affine_dependent(s \<union> t))"
+ shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
+ and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
+proof -
+ have [simp]: "finite s" "finite t"
+ using aff_independent_finite assms by blast+
+ have "sum u (s \<inter> t) = 1 \<and>
+ (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
+ if [simp]: "sum u s = 1"
+ "sum v t = 1"
+ and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
+ proof -
+ define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
+ have "sum f (s \<union> t) = 0"
+ apply (simp add: f_def sum_Un sum_subtractf)
+ apply (simp add: sum.inter_restrict [symmetric] Int_commute)
+ done
+ moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
+ apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
+ apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
+ cong del: if_weak_cong)
+ done
+ ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
+ using aff_independent_finite assms unfolding affine_dependent_explicit
+ by blast
+ then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
+ by (simp add: f_def) presburger
+ have "sum u (s \<inter> t) = sum u s"
+ by (simp add: sum.inter_restrict)
+ then have "sum u (s \<inter> t) = 1"
+ using that by linarith
+ moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>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
+
+
+proposition affine_hull_Int:
+ fixes s :: "'a::euclidean_space set"
+ assumes "~ (affine_dependent(s \<union> t))"
+ shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
+apply (rule subset_antisym)
+apply (simp add: hull_mono)
+by (simp add: affine_hull_Int_subset assms)
+
+proposition convex_hull_Int:
+ fixes s :: "'a::euclidean_space set"
+ assumes "~ (affine_dependent(s \<union> t))"
+ shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
+apply (rule subset_antisym)
+apply (simp add: hull_mono)
+by (simp add: convex_hull_Int_subset assms)
+
+proposition
+ fixes s :: "'a::euclidean_space set set"
+ assumes "~ (affine_dependent (\<Union>s))"
+ shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
+ and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
+proof -
+ have "finite s"
+ using aff_independent_finite assms finite_UnionD by blast
+ then have "?A \<and> ?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]: "\<not> affine_dependent (t \<union> \<Inter>F)"
+ by (auto intro: affine_dependent_subset)
+ have [simp]: "\<not> affine_dependent (\<Union>F)"
+ using affine_independent_subset insert.prems by fastforce
+ show ?thesis
+ by (simp add: affine_hull_Int convex_hull_Int insert.IH)
+ qed
+ qed
+ then show "?A" "?C"
+ by auto
+qed
+
+proposition in_convex_hull_exchange_unique:
+ fixes S :: "'a::euclidean_space set"
+ assumes naff: "~ affine_dependent S" and a: "a \<in> convex hull S"
+ and S: "T \<subseteq> S" "T' \<subseteq> S"
+ and x: "x \<in> convex hull (insert a T)"
+ and x': "x \<in> convex hull (insert a T')"
+ shows "x \<in> convex hull (insert a (T \<inter> T'))"
+proof (cases "a \<in> S")
+ case True
+ then have "\<not> affine_dependent (insert a T \<union> insert a T')"
+ using affine_dependent_subset assms by auto
+ then have "x \<in> convex hull (insert a T \<inter> insert a T')"
+ by (metis IntI convex_hull_Int x x')
+ then show ?thesis
+ by simp
+next
+ case False
+ then have anot: "a \<notin> T" "a \<notin> T'"
+ using assms by auto
+ have [simp]: "finite S"
+ by (simp add: aff_independent_finite assms)
+ then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
+ and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
+ using a by (auto simp: convex_hull_finite)
+ have fin [simp]: "finite T" "finite T'"
+ using assms infinite_super \<open>finite S\<close> by blast+
+ then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
+ and c1: "sum c (insert a T) = 1"
+ and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
+ and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
+ and c'1: "sum c' (insert a T') = 1"
+ and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>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: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
+ by simp_all
+ have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
+ using x'eq fin anot by simp
+ define cc where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
+ define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
+ define dd where "dd \<equiv> \<lambda>x. 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: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>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 "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
+ by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
+ then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
+ using aeq by blast
+ have "(\<Sum>v \<in> S. dd v *\<^sub>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 \<in> S" for v
+ using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
+ apply (simp only: not_ex)
+ apply (drule_tac x=S in spec)
+ apply (drule_tac x=dd in spec, simp)
+ done
+ consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
+ then show ?thesis
+ proof cases
+ case 1
+ then have "sum cc S \<le> sum cc' S"
+ by (simp add: sumSS')
+ then have le: "cc x \<le> cc' x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
+ using le [OF \<open>x \<in> S\<close>] that c0
+ by (force simp: cc_def cc'_def split: if_split_asm)
+ show ?thesis
+ proof (simp add: convex_hull_finite, intro exI conjI)
+ show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
+ by (simp add: c0 cc_def)
+ show "0 \<le> (cc(a := c a)) a"
+ by (simp add: c0)
+ have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
+ by (simp add: anot)
+ also have "... = c a + sum (cc(a := c a)) S"
+ apply simp
+ apply (rule sum.mono_neutral_left)
+ using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
+ done
+ also have "... = c a + (1 - c a)"
+ by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
+ finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
+ by simp
+ have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
+ by (simp add: anot)
+ also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
+ apply simp
+ apply (rule sum.mono_neutral_left)
+ using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
+ done
+ also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
+ by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
+ finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
+ by simp
+ qed
+ next
+ case 2
+ then have "sum cc' S \<le> sum cc S"
+ by (simp add: sumSS')
+ then have le: "cc' x \<le> cc x" if "x \<in> 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 \<in> S" "x \<notin> T \<inter> T'" for x
+ using le [OF \<open>x \<in> S\<close>] that c'0
+ by (force simp: cc_def cc'_def split: if_split_asm)
+ show ?thesis
+ proof (simp add: convex_hull_finite, intro exI conjI)
+ show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
+ by (simp add: c'0 cc'_def)
+ show "0 \<le> (cc'(a := c' a)) a"
+ by (simp add: c'0)
+ have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
+ by (simp add: anot)
+ also have "... = c' a + sum (cc'(a := c' a)) S"
+ apply simp
+ apply (rule sum.mono_neutral_left)
+ using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
+ done
+ also have "... = c' a + (1 - c' a)"
+ by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
+ finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
+ by simp
+ have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
+ by (simp add: anot)
+ also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
+ apply simp
+ apply (rule sum.mono_neutral_left)
+ using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
+ done
+ also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
+ by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
+ finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
+ by simp
+ qed
+ qed
+qed
+
+corollary convex_hull_exchange_Int:
+ fixes a :: "'a::euclidean_space"
+ assumes "~ affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
+ shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
+ convex hull (insert a (T \<inter> T'))"
+apply (rule subset_antisym)
+ using in_convex_hull_exchange_unique assms apply blast
+ by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
+
+lemma Int_closed_segment:
+ fixes b :: "'a::euclidean_space"
+ assumes "b \<in> closed_segment a c \<or> ~collinear{a,b,c}"
+ shows "closed_segment a b \<inter> 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 \<in> closed_segment a c"
+ moreover have "\<not> affine_dependent {a, c}"
+ by (simp add: affine_independent_2)
+ 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: "\<not> collinear {a, b, c}"
+ have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
+ proof -
+ have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
+ by auto
+ with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> 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}"
+ apply (rule collinear_3_trans [OF _ _ \<open>b \<noteq> d\<close>])
+ using d by (auto simp: insert_commute)
+ with ncoll show False ..
+ qed
+ then show ?thesis
+ by blast
+ qed
+qed
+
+lemma affine_hull_finite_intersection_hyperplanes:
+ fixes s :: "'a::euclidean_space set"
+ obtains f where
+ "finite f"
+ "of_nat (card f) + aff_dim s = DIM('a)"
+ "affine hull s = \<Inter>f"
+ "\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
+proof -
+ obtain b where "b \<subseteq> s"
+ and indb: "\<not> affine_dependent b"
+ and eq: "affine hull s = affine hull b"
+ using affine_basis_exists by blast
+ obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> 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 \<le> card c"
+ using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
+ have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
+ by blast
+ have card1: "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
+ apply (rule card_image [OF inj_onI])
+ by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
+ have card2: "(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: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent aff_dim_UNIV card_Diff_subset of_nat_diff)
+ qed
+ show ?thesis
+ proof (cases "c = b")
+ case True show ?thesis
+ apply (rule_tac f="{}" in that)
+ using True affc
+ apply (simp_all add: eq [symmetric])
+ by (metis aff_dim_UNIV aff_dim_affine_hull)
+ next
+ case False
+ have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
+ by (rule affine_independent_subset [OF indc]) auto
+ have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
+ using \<open>b \<subseteq> c\<close> False
+ apply (subst affine_hull_Inter [OF ind, symmetric])
+ apply (simp add: eq double_diff)
+ done
+ have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
+ if t: "t \<in> 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
+ show ?thesis
+ apply (rule_tac f = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
+ using \<open>finite c\<close> apply blast
+ apply (simp add: imeq card1 card2)
+ apply (simp add: affeq, clarify)
+ apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
+ done
+ qed
+qed
+
+subsection\<open>Misc results about span\<close>
+
+lemma eq_span_insert_eq:
+ assumes "(x - y) \<in> span S"
+ shows "span(insert x S) = span(insert y S)"
+proof -
+ have *: "span(insert x S) \<subseteq> span(insert y S)" if "(x - y) \<in> span S" for x y
+ proof -
+ have 1: "(r *\<^sub>R x - r *\<^sub>R y) \<in> span S" for r
+ by (metis real_vector.scale_right_diff_distrib span_mul that)
+ have 2: "(z - k *\<^sub>R y) - k *\<^sub>R (x - y) = z - k *\<^sub>R x" for z k
+ by (simp add: real_vector.scale_right_diff_distrib)
+ show ?thesis
+ apply (clarsimp simp add: span_breakdown_eq)
+ by (metis 1 2 diff_add_cancel real_vector.scale_right_diff_distrib span_add_eq)
+ qed
+ show ?thesis
+ apply (intro subset_antisym * assms)
+ using assms subspace_neg subspace_span minus_diff_eq by force
+qed
+
+lemma dim_psubset:
+ fixes S :: "'a :: euclidean_space set"
+ shows "span S \<subset> span T \<Longrightarrow> dim S < dim T"
+by (metis (no_types, hide_lams) dim_span less_le not_le subspace_dim_equal subspace_span)
+
+
+lemma basis_subspace_exists:
+ fixes S :: "'a::euclidean_space set"
+ shows
+ "subspace S
+ \<Longrightarrow> \<exists>b. finite b \<and> b \<subseteq> S \<and>
+ independent b \<and> span b = S \<and> card b = dim S"
+by (metis span_subspace basis_exists independent_imp_finite)
+
+lemma affine_hyperplane_sums_eq_UNIV_0:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "affine S"
+ and "0 \<in> S" and "w \<in> S"
+ and "a \<bullet> w \<noteq> 0"
+ shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
+proof -
+ have "subspace S"
+ by (simp add: assms subspace_affine)
+ have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
+ apply (rule span_mono)
+ using \<open>0 \<in> S\<close> add.left_neutral by force
+ have "w \<notin> span {y. a \<bullet> y = 0}"
+ using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
+ moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
+ using \<open>w \<in> S\<close>
+ by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_superset)
+ ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
+ by blast
+ have "a \<noteq> 0" using assms inner_zero_left by blast
+ then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
+ by (simp add: dim_hyperplane)
+ also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
+ using span1 span2 by (blast intro: dim_psubset)
+ finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
+ have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
+ using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
+ moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
+ apply (rule dim_eq_full [THEN iffD1])
+ apply (rule antisym [OF dim_subset_UNIV])
+ using DIM_lt apply simp
+ done
+ ultimately show ?thesis
+ by (simp add: subs) (metis (lifting) span_eq subs)
+qed
+
+proposition affine_hyperplane_sums_eq_UNIV:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "affine S"
+ and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
+ and "S - {v. a \<bullet> v = b} \<noteq> {}"
+ shows "{x + y| x y. x \<in> S \<and> a \<bullet> 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 \<in> S" and c: "a \<bullet> c = b"
+ using assms by force
+ with affine_diffs_subspace [OF \<open>affine S\<close>]
+ have "subspace (op + (- c) ` S)" by blast
+ then have aff: "affine (op + (- c) ` S)"
+ by (simp add: subspace_imp_affine)
+ have 0: "0 \<in> op + (- c) ` S"
+ by (simp add: \<open>c \<in> S\<close>)
+ obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> op + (- c) ` S"
+ using assms by auto
+ then have adc: "a \<bullet> (d - c) \<noteq> 0"
+ by (simp add: c inner_diff_right)
+ let ?U = "op + (c+c) ` {x + y |x y. x \<in> op + (- c) ` S \<and> a \<bullet> y = 0}"
+ have "u + v \<in> op + (c + c) ` {x + v |x v. x \<in> op + (- c) ` S \<and> a \<bullet> v = 0}"
+ if "u \<in> S" "b = a \<bullet> v" for u v
+ apply (rule_tac x="u+v-c-c" in image_eqI)
+ apply (simp_all add: algebra_simps)
+ apply (rule_tac x="u-c" in exI)
+ apply (rule_tac x="v-c" in exI)
+ apply (simp add: algebra_simps that c)
+ done
+ moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
+ \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
+ by (metis add.left_commute c inner_right_distrib pth_d)
+ ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
+ by (fastforce simp: algebra_simps)
+ also have "... = op + (c+c) ` UNIV"
+ by (simp add: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
+ also have "... = UNIV"
+ by (simp add: translation_UNIV)
+ finally show ?thesis .
+qed
+
+proposition dim_sums_Int:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "subspace S" "subspace T"
+ shows "dim {x + y |x y. x \<in> S \<and> y \<in> T} + dim(S \<inter> T) = dim S + dim T"
+proof -
+ obtain B where B: "B \<subseteq> S \<inter> T" "S \<inter> T \<subseteq> span B"
+ and indB: "independent B"
+ and cardB: "card B = dim (S \<inter> T)"
+ using basis_exists by blast
+ then obtain C D where "B \<subseteq> C" "C \<subseteq> S" "independent C" "S \<subseteq> span C"
+ and "B \<subseteq> D" "D \<subseteq> T" "independent D" "T \<subseteq> span D"
+ using maximal_independent_subset_extend
+ by (metis Int_subset_iff \<open>B \<subseteq> S \<inter> T\<close> indB)
+ then have "finite B" "finite C" "finite D"
+ by (simp_all add: independent_imp_finite indB independent_bound)
+ have Beq: "B = C \<inter> D"
+ apply (rule sym)
+ apply (rule spanning_subset_independent)
+ using \<open>B \<subseteq> C\<close> \<open>B \<subseteq> D\<close> apply blast
+ apply (meson \<open>independent C\<close> independent_mono inf.cobounded1)
+ using B \<open>C \<subseteq> S\<close> \<open>D \<subseteq> T\<close> apply auto
+ done
+ then have Deq: "D = B \<union> (D - C)"
+ by blast
+ have CUD: "C \<union> D \<subseteq> {x + y |x y. x \<in> S \<and> y \<in> T}"
+ apply safe
+ apply (metis add.right_neutral subsetCE \<open>C \<subseteq> S\<close> \<open>subspace T\<close> set_eq_subset span_0 span_minimal)
+ apply (metis add.left_neutral subsetCE \<open>D \<subseteq> T\<close> \<open>subspace S\<close> set_eq_subset span_0 span_minimal)
+ done
+ have "a v = 0" if 0: "(\<Sum>v\<in>C. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v) = 0"
+ and v: "v \<in> C \<union> (D-C)" for a v
+ proof -
+ have eq: "(\<Sum>v\<in>D - C. a v *\<^sub>R v) = - (\<Sum>v\<in>C. a v *\<^sub>R v)"
+ using that add_eq_0_iff by blast
+ have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> S"
+ apply (subst eq)
+ apply (rule subspace_neg [OF \<open>subspace S\<close>])
+ apply (rule subspace_sum [OF \<open>subspace S\<close>])
+ by (meson subsetCE subspace_mul \<open>C \<subseteq> S\<close> \<open>subspace S\<close>)
+ moreover have "(\<Sum>v\<in>D - C. a v *\<^sub>R v) \<in> T"
+ apply (rule subspace_sum [OF \<open>subspace T\<close>])
+ by (meson DiffD1 \<open>D \<subseteq> T\<close> \<open>subspace T\<close> subset_eq subspace_def)
+ ultimately have "(\<Sum>v \<in> D-C. a v *\<^sub>R v) \<in> span B"
+ using B by blast
+ then obtain e where e: "(\<Sum>v\<in>B. e v *\<^sub>R v) = (\<Sum>v \<in> D-C. a v *\<^sub>R v)"
+ using span_finite [OF \<open>finite B\<close>] by blast
+ have "\<And>c v. \<lbrakk>(\<Sum>v\<in>C. c v *\<^sub>R v) = 0; v \<in> C\<rbrakk> \<Longrightarrow> c v = 0"
+ using independent_explicit \<open>independent C\<close> by blast
+ define cc where "cc x = (if x \<in> B then a x + e x else a x)" for x
+ have [simp]: "C \<inter> B = B" "D \<inter> B = B" "C \<inter> - B = C-D" "B \<inter> (D - C) = {}"
+ using \<open>B \<subseteq> C\<close> \<open>B \<subseteq> D\<close> Beq by blast+
+ have f2: "(\<Sum>v\<in>C \<inter> D. e v *\<^sub>R v) = (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
+ using Beq e by presburger
+ have f3: "(\<Sum>v\<in>C \<union> D. a v *\<^sub>R v) = (\<Sum>v\<in>C - D. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v) + (\<Sum>v\<in>C \<inter> D. a v *\<^sub>R v)"
+ using \<open>finite C\<close> \<open>finite D\<close> sum.union_diff2 by blast
+ have f4: "(\<Sum>v\<in>C \<union> (D - C). a v *\<^sub>R v) = (\<Sum>v\<in>C. a v *\<^sub>R v) + (\<Sum>v\<in>D - C. a v *\<^sub>R v)"
+ by (meson Diff_disjoint \<open>finite C\<close> \<open>finite D\<close> finite_Diff sum.union_disjoint)
+ have "(\<Sum>v\<in>C. cc v *\<^sub>R v) = 0"
+ using 0 f2 f3 f4
+ apply (simp add: cc_def Beq if_smult \<open>finite C\<close> sum.If_cases algebra_simps sum.distrib)
+ apply (simp add: add.commute add.left_commute diff_eq)
+ done
+ then have "\<And>v. v \<in> C \<Longrightarrow> cc v = 0"
+ using independent_explicit \<open>independent C\<close> by blast
+ then have C0: "\<And>v. v \<in> C - B \<Longrightarrow> a v = 0"
+ by (simp add: cc_def Beq) meson
+ then have [simp]: "(\<Sum>x\<in>C - B. a x *\<^sub>R x) = 0"
+ by simp
+ have "(\<Sum>x\<in>C. a x *\<^sub>R x) = (\<Sum>x\<in>B. a x *\<^sub>R x)"
+ proof -
+ have "C - D = C - B"
+ using Beq by blast
+ then show ?thesis
+ using Beq \<open>(\<Sum>x\<in>C - B. a x *\<^sub>R x) = 0\<close> f3 f4 by auto
+ qed
+ with 0 have Dcc0: "(\<Sum>v\<in>D. a v *\<^sub>R v) = 0"
+ apply (subst Deq)
+ by (simp add: \<open>finite B\<close> \<open>finite D\<close> sum_Un)
+ then have D0: "\<And>v. v \<in> D \<Longrightarrow> a v = 0"
+ using independent_explicit \<open>independent D\<close> by blast
+ show ?thesis
+ using v C0 D0 Beq by blast
+ qed
+ then have "independent (C \<union> (D - C))"
+ by (simp add: independent_explicit \<open>finite C\<close> \<open>finite D\<close> sum_Un del: Un_Diff_cancel)
+ then have indCUD: "independent (C \<union> D)" by simp
+ have "dim (S \<inter> T) = card B"
+ by (rule dim_unique [OF B indB refl])
+ moreover have "dim S = card C"
+ by (metis \<open>C \<subseteq> S\<close> \<open>independent C\<close> \<open>S \<subseteq> span C\<close> basis_card_eq_dim)
+ moreover have "dim T = card D"
+ by (metis \<open>D \<subseteq> T\<close> \<open>independent D\<close> \<open>T \<subseteq> span D\<close> basis_card_eq_dim)
+ moreover have "dim {x + y |x y. x \<in> S \<and> y \<in> T} = card(C \<union> D)"
+ apply (rule dim_unique [OF CUD _ indCUD refl], clarify)
+ apply (meson \<open>S \<subseteq> span C\<close> \<open>T \<subseteq> span D\<close> span_add span_inc span_minimal subsetCE subspace_span sup.bounded_iff)
+ done
+ ultimately show ?thesis
+ using \<open>B = C \<inter> D\<close> [symmetric]
+ by (simp add: \<open>independent C\<close> \<open>independent D\<close> card_Un_Int independent_finite)
+qed
+
+
+lemma aff_dim_sums_Int_0:
+ assumes "affine S"
+ and "affine T"
+ and "0 \<in> S" "0 \<in> T"
+ shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
+proof -
+ have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
+ using assms by force
+ then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> 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 \<inter> T \<noteq> {}"
+ shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
+proof -
+ obtain a where a: "a \<in> S" "a \<in> T" using assms by force
+ have aff: "affine (op+ (-a) ` S)" "affine (op+ (-a) ` T)"
+ using assms by (auto simp: affine_translation [symmetric])
+ have zero: "0 \<in> (op+ (-a) ` S)" "0 \<in> (op+ (-a) ` T)"
+ using a assms by auto
+ have [simp]: "{x + y |x y. x \<in> op + (- a) ` S \<and> y \<in> op + (- a) ` T} =
+ op + (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
+ by (force simp: algebra_simps scaleR_2)
+ have [simp]: "op + (- a) ` S \<inter> op + (- a) ` T = op + (- a) ` (S \<inter> T)"
+ by auto
+ show ?thesis
+ using aff_dim_sums_Int_0 [OF aff zero]
+ by (auto simp: aff_dim_translation_eq)
+qed
+
+lemma aff_dim_affine_Int_hyperplane:
+ fixes a :: "'a::euclidean_space"
+ assumes "affine S"
+ shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
+ (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
+ else if S \<subseteq> {v. a \<bullet> 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 \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
+ if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
+ proof -
+ have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> 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) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
+by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
+
+
+lemma dim_Times:
+ fixes S :: "'a :: euclidean_space set" and T :: "'a set"
+ assumes "subspace S" "subspace T"
+ shows "dim(S \<times> T) = dim S + dim T"
+proof -
+ have ss: "subspace ((\<lambda>x. (x, 0)) ` S)" "subspace (Pair 0 ` T)"
+ by (rule subspace_linear_image, unfold_locales, auto simp: assms)+
+ have "dim(S \<times> T) = dim({u + v |u v. u \<in> (\<lambda>x. (x, 0)) ` S \<and> v \<in> Pair 0 ` T})"
+ by (simp add: Times_eq_image_sum)
+ moreover have "dim ((\<lambda>x. (x, 0::'a)) ` S) = dim S" "dim (Pair (0::'a) ` T) = dim T"
+ by (auto simp: additive.intro linear.intro linear_axioms.intro inj_on_def intro: dim_image_eq)
+ moreover have "dim ((\<lambda>x. (x, 0)) ` S \<inter> Pair 0 ` T) = 0"
+ by (subst dim_eq_0) (force simp: zero_prod_def)
+ ultimately show ?thesis
+ using dim_sums_Int [OF ss] by linarith
+qed
+
+subsection\<open> Orthogonal bases, Gram-Schmidt process, and related theorems\<close>
+
+lemma span_delete_0 [simp]: "span(S - {0}) = span S"
+proof
+ show "span (S - {0}) \<subseteq> span S"
+ by (blast intro!: span_mono)
+next
+ have "span S \<subseteq> span(insert 0 (S - {0}))"
+ by (blast intro!: span_mono)
+ also have "... \<subseteq> span(S - {0})"
+ using span_insert_0 by blast
+ finally show "span S \<subseteq> span (S - {0})" .
+qed
+
+lemma span_image_scale:
+ assumes "finite S" and nz: "\<And>x. x \<in> S \<Longrightarrow> c x \<noteq> 0"
+ shows "span ((\<lambda>x. c x *\<^sub>R x) ` S) = span S"
+using assms
+proof (induction S arbitrary: c)
+ case (empty c) show ?case by simp
+next
+ case (insert x F c)
+ show ?case
+ proof (intro set_eqI iffI)
+ fix y
+ assume "y \<in> span ((\<lambda>x. c x *\<^sub>R x) ` insert x F)"
+ then show "y \<in> span (insert x F)"
+ using insert by (force simp: span_breakdown_eq)
+ next
+ fix y
+ assume "y \<in> span (insert x F)"
+ then show "y \<in> span ((\<lambda>x. c x *\<^sub>R x) ` insert x F)"
+ using insert
+ apply (clarsimp simp: span_breakdown_eq)
+ apply (rule_tac x="k / c x" in exI)
+ by simp
+ qed
+qed
+
+lemma pairwise_orthogonal_independent:
+ assumes "pairwise orthogonal S" and "0 \<notin> S"
+ shows "independent S"
+proof -
+ have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ using assms by (simp add: pairwise_def orthogonal_def)
+ have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
+ proof -
+ obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
+ using a by (force simp: span_explicit)
+ then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
+ by simp
+ also have "... = 0"
+ apply (simp add: inner_sum_right)
+ apply (rule comm_monoid_add_class.sum.neutral)
+ by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
+ finally show ?thesis
+ using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
+ qed
+ then show ?thesis
+ by (force simp: dependent_def)
+qed
+
+lemma pairwise_orthogonal_imp_finite:
+ fixes S :: "'a::euclidean_space set"
+ assumes "pairwise orthogonal S"
+ shows "finite S"
+proof -
+ have "independent (S - {0})"
+ apply (rule pairwise_orthogonal_independent)
+ apply (metis Diff_iff assms pairwise_def)
+ by blast
+ then show ?thesis
+ by (meson independent_imp_finite infinite_remove)
+qed
+
+lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
+ by (simp add: subspace_def orthogonal_clauses)
+
+lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
+ by (simp add: subspace_def orthogonal_clauses)
+
+lemma orthogonal_to_span:
+ assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
+ shows "orthogonal x a"
+apply (rule span_induct [OF a subspace_orthogonal_to_vector])
+apply (simp add: x)
+done
+
+proposition Gram_Schmidt_step:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
+ shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
+proof -
+ have "finite S"
+ by (simp add: S pairwise_orthogonal_imp_finite)
+ have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
+ if "x \<in> S" for x
+ proof -
+ have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
+ by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
+ also have "... = (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
+ apply (rule sum.cong [OF refl], simp)
+ by (meson S orthogonal_def pairwise_def that)
+ finally show ?thesis
+ by (simp add: orthogonal_def algebra_simps inner_sum_left)
+ qed
+ then show ?thesis
+ using orthogonal_to_span orthogonal_commute x by blast
+qed
+
+
+lemma orthogonal_extension_aux:
+ fixes S :: "'a::euclidean_space set"
+ assumes "finite T" "finite S" "pairwise orthogonal S"
+ shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
+using assms
+proof (induction arbitrary: S)
+ case empty then show ?case
+ by simp (metis sup_bot_right)
+next
+ case (insert a T)
+ have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ using insert by (simp add: pairwise_def orthogonal_def)
+ define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
+ obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
+ and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
+ apply (rule exE [OF insert.IH [of "insert a' S"]])
+ apply (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute pairwise_orthogonal_insert span_clauses)
+ done
+ have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
+ apply (simp add: a'_def)
+ using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
+ apply (force simp: orthogonal_def inner_commute span_inc [THEN subsetD])
+ done
+ have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
+ using spanU by simp
+ also have "... = span (insert a (S \<union> T))"
+ apply (rule eq_span_insert_eq)
+ apply (simp add: a'_def span_neg span_sum span_clauses(1) span_mul)
+ done
+ also have "... = span (S \<union> insert a T)"
+ by simp
+ finally show ?case
+ apply (rule_tac x="insert a' U" in exI)
+ using orthU apply auto
+ done
+qed
+
+
+proposition orthogonal_extension:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "pairwise orthogonal S"
+ obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
+proof -
+ obtain B where "finite B" "span B = span T"
+ using basis_subspace_exists [of "span T"] subspace_span by auto
+ with orthogonal_extension_aux [of B S]
+ obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
+ using assms pairwise_orthogonal_imp_finite by auto
+ show ?thesis
+ apply (rule_tac U=U in that)
+ apply (simp add: \<open>pairwise orthogonal (S \<union> U)\<close>)
+ by (metis \<open>span (S \<union> U) = span (S \<union> B)\<close> \<open>span B = span T\<close> span_Un)
+qed
+
+corollary orthogonal_extension_strong:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "pairwise orthogonal S"
+ obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
+ "span (S \<union> U) = span (S \<union> T)"
+proof -
+ obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
+ using orthogonal_extension assms by blast
+ then show ?thesis
+ apply (rule_tac U = "U - (insert 0 S)" in that)
+ apply blast
+ apply (force simp: pairwise_def)
+ apply (metis (no_types, lifting) Un_Diff_cancel span_insert_0 span_Un)
+ done
+qed
+
+subsection\<open>Decomposing a vector into parts in orthogonal subspaces.\<close>
+
+text\<open>existence of orthonormal basis for a subspace.\<close>
+
+lemma orthogonal_spanningset_subspace:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "subspace S"
+ obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
+proof -
+ obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
+ using basis_exists by blast
+ with orthogonal_extension [of "{}" B]
+ show ?thesis
+ by (metis Un_empty_left assms pairwise_empty span_inc span_subspace that)
+qed
+
+lemma orthogonal_basis_subspace:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "subspace S"
+ obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
+ "card B = dim S" "span B = S"
+proof -
+ obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
+ using assms orthogonal_spanningset_subspace by blast
+ then show ?thesis
+ apply (rule_tac B = "B - {0}" in that)
+ apply (auto simp: indep_card_eq_dim_span pairwise_subset Diff_subset pairwise_orthogonal_independent elim: pairwise_subset)
+ done
+qed
+
+proposition orthonormal_basis_subspace:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "subspace S"
+ obtains B where "B \<subseteq> S" "pairwise orthogonal B"
+ and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+ and "independent B" "card B = dim S" "span B = S"
+proof -
+ obtain B where "0 \<notin> B" "B \<subseteq> S"
+ and orth: "pairwise orthogonal B"
+ and "independent B" "card B = dim S" "span B = S"
+ by (blast intro: orthogonal_basis_subspace [OF assms])
+ have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
+ using \<open>span B = S\<close> span_clauses(1) span_mul by fastforce
+ have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
+ using orth by (force simp: pairwise_def orthogonal_clauses)
+ have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
+ by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
+ have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
+ by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
+ have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
+ proof
+ fix x y
+ assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
+ moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
+ using 3 by blast
+ ultimately show "x = y"
+ by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
+ qed
+ then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
+ by (metis \<open>card B = dim S\<close> card_image)
+ have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
+ by (metis "1" "4" "5" assms card_eq_dim independent_finite span_subspace)
+ show ?thesis
+ by (rule that [OF 1 2 3 4 5 6])
+qed
+
+proposition orthogonal_subspace_decomp_exists:
+ fixes S :: "'a :: euclidean_space set"
+ obtains y z where "y \<in> span S" "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w" "x = y + z"
+proof -
+ obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T" "card T = dim (span S)" "span T = span S"
+ using orthogonal_basis_subspace subspace_span by blast
+ let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
+ have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
+ by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close> orthogonal_commute that)
+ show ?thesis
+ apply (rule_tac y = "?a" and z = "x - ?a" in that)
+ apply (meson \<open>T \<subseteq> span S\<close> span_mul span_sum subsetCE)
+ apply (fact orth, simp)
+ done
+qed
+
+lemma orthogonal_subspace_decomp_unique:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "x + y = x' + y'"
+ and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
+ and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
+ shows "x = x' \<and> y = y'"
+proof -
+ have "x + y - y' = x'"
+ by (simp add: assms)
+ moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
+ by (meson orth orthogonal_commute orthogonal_to_span)
+ ultimately have "0 = x' - x"
+ by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
+ with assms show ?thesis by auto
+qed
+
+proposition dim_orthogonal_sum:
+ fixes A :: "'a::euclidean_space set"
+ assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ shows "dim(A \<union> B) = dim A + dim B"
+proof -
+ have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
+ have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ apply (erule span_induct [OF _ subspace_hyperplane])
+ using 1 by (simp add: )
+ then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+ by simp
+ have "dim(A \<union> B) = dim (span (A \<union> B))"
+ by simp
+ also have "... = dim ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
+ by (simp add: span_Un)
+ also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
+ by (auto intro!: arg_cong [where f=dim])
+ also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
+ by (auto simp: dest: 0)
+ also have "... = dim (span A) + dim (span B)"
+ by (rule dim_sums_Int) auto
+ also have "... = dim A + dim B"
+ by simp
+ finally show ?thesis .
+qed
+
+lemma dim_subspace_orthogonal_to_vectors:
+ fixes A :: "'a::euclidean_space set"
+ assumes "subspace A" "subspace B" "A \<subseteq> B"
+ shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
+proof -
+ have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
+ proof (rule arg_cong [where f=dim, OF subset_antisym])
+ show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
+ by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
+ next
+ have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
+ if "x \<in> B" for x
+ proof -
+ obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
+ using orthogonal_subspace_decomp_exists [of A x] that by auto
+ have "y \<in> span B"
+ by (metis span_eq \<open>y \<in> span A\<close> assms subset_iff)
+ then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
+ by simp (metis (no_types) span_eq \<open>x = y + z\<close> \<open>subspace A\<close> \<open>subspace B\<close> orth orthogonal_commute span_add_eq that)
+ then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
+ by (meson span_inc subset_iff)
+ then show ?thesis
+ apply (simp add: span_Un image_def)
+ apply (rule bexI [OF _ z])
+ apply (simp add: \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
+ done
+ qed
+ show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
+ by (rule span_minimal) (auto intro: * span_minimal elim: )
+ qed
+ then show ?thesis
+ by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq orthogonal_commute orthogonal_def)
+qed
+
+lemma aff_dim_openin:
+ fixes S :: "'a::euclidean_space set"
+ assumes ope: "openin (subtopology euclidean T) S" and "affine T" "S \<noteq> {}"
+ shows "aff_dim S = aff_dim T"
+proof -
+ show ?thesis
+ proof (rule order_antisym)
+ show "aff_dim S \<le> aff_dim T"
+ by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
+ next
+ obtain a where "a \<in> S"
+ using \<open>S \<noteq> {}\<close> by blast
+ have "S \<subseteq> T"
+ using ope openin_imp_subset by auto
+ then have "a \<in> T"
+ using \<open>a \<in> S\<close> by auto
+ then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
+ using affine_diffs_subspace \<open>affine T\<close> by auto
+ then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
+ and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
+ and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
+ and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
+ by (rule orthonormal_basis_subspace) auto
+ obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
+ by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
+ have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
+ by (metis aff_dim_translation_eq)
+ also have "... = dim ((\<lambda>x. - a + x) ` T)"
+ using aff_dim_subspace subT' by blast
+ also have "... = card B"
+ by (simp add: cardB)
+ also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
+ using \<open>0 < e\<close> by (force simp: inj_on_def card_image)
+ also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
+ proof (simp, rule independent_card_le_dim)
+ have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
+ using e by (auto simp: dist_norm norm_minus_commute subset_eq)
+ have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
+ using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
+ then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
+ using e' by blast
+ show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
+ using \<open>independent B\<close>
+ apply (rule independent_injective_image, simp)
+ by (metis \<open>0 < e\<close> injective_scaleR less_irrefl)
+ qed
+ also have "... = aff_dim S"
+ using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by force
+ finally show "aff_dim T \<le> aff_dim S" .
+ qed
+qed
+
+lemma dim_openin:
+ fixes S :: "'a::euclidean_space set"
+ assumes ope: "openin (subtopology euclidean T) S" and "subspace T" "S \<noteq> {}"
+ shows "dim S = dim T"
+proof (rule order_antisym)
+ show "dim S \<le> 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 \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
+ also have "... \<le> dim S"
+ by (metis aff_dim_subset aff_dim_subspace dim_span span_inc subspace_span)
+ finally show "dim T \<le> dim S" by simp
+qed
+
+subsection\<open>Parallel slices, etc.\<close>
+
+text\<open> 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.\<close>
+
+proposition affine_parallel_slice:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "affine S"
+ and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
+ and "~ (S \<subseteq> {x. a \<bullet> x \<le> b})"
+ obtains a' b' where "a' \<noteq> 0"
+ "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
+ "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
+ "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
+proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
+ case True
+ then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
+ using assms by (auto simp: not_le)
+ define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
+ have "\<eta> \<in> S"
+ by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
+ moreover have "a \<bullet> \<eta> = b"
+ using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
+ by (simp add: \<eta>_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 \<in> S" and z: "a \<bullet> z = b"
+ using assms by auto
+ with affine_diffs_subspace [OF \<open>affine S\<close>]
+ have sub: "subspace (op + (- z) ` S)" by blast
+ then have aff: "affine (op + (- z) ` S)" and span: "span (op + (- z) ` S) = (op + (- z) ` S)"
+ by (auto simp: subspace_imp_affine)
+ obtain a' a'' where a': "a' \<in> span (op + (- z) ` S)" and a: "a = a' + a''"
+ and "\<And>w. w \<in> span (op + (- z) ` S) \<Longrightarrow> orthogonal a'' w"
+ using orthogonal_subspace_decomp_exists [of "op + (- z) ` S" "a"] by metis
+ then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
+ by (simp add: imageI orthogonal_def span)
+ then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
+ by (simp add: a inner_diff_right)
+ then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
+ by (simp add: inner_diff_left z)
+ have "\<And>w. w \<in> op + (- z) ` S \<Longrightarrow> (w + a') \<in> op + (- z) ` S"
+ by (metis subspace_add a' span_eq sub)
+ then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
+ by fastforce
+ 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
+ apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
+ apply (auto simp: a ba'' inner_left_distrib False Sclo)
+ done
+ qed
+qed
+
+lemma diffs_affine_hull_span:
+ assumes "a \<in> S"
+ shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
+proof -
+ have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
+ by (auto simp: algebra_simps)
+ show ?thesis
+ apply (simp add: affine_hull_span2 [OF assms] *)
+ apply (auto simp: algebra_simps)
+ done
+qed
+
+lemma aff_dim_dim_affine_diffs:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "affine S" "a \<in> S"
+ shows "aff_dim S = dim {x - a |x. x \<in> S}"
+proof -
+ obtain B where aff: "affine hull B = affine hull S"
+ and ind: "\<not> affine_dependent B"
+ and card: "of_nat (card B) = aff_dim S + 1"
+ using aff_dim_basis_exists by blast
+ then have "B \<noteq> {}" using assms
+ by (metis affine_hull_eq_empty ex_in_conv)
+ then obtain c where "c \<in> B" by auto
+ then have "c \<in> S"
+ by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
+ have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
+ by (auto simp: algebra_simps)
+ have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
+ apply safe
+ apply (simp_all only: xy)
+ using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
+ done
+ have affS: "affine hull S = S"
+ by (simp add: \<open>affine S\<close>)
+ have "aff_dim S = of_nat (card B) - 1"
+ using card by simp
+ also have "... = dim {x - c |x. x \<in> B}"
+ by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
+ also have "... = dim {x - c | x. x \<in> affine hull B}"
+ by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
+ also have "... = dim {x - a |x. x \<in> S}"
+ by (simp add: affS aff *)
+ finally show ?thesis .
+qed
+
+lemma aff_dim_linear_image_le:
+ assumes "linear f"
+ shows "aff_dim(f ` S) \<le> aff_dim S"
+proof -
+ have "aff_dim (f ` T) \<le> 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 \<in> T" by auto
+ have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
+ by auto
+ have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
+ by (force simp: linear_diff [OF assms])
+ have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
+ by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1)
+ also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
+ by (force simp: linear_diff [OF assms] 2)
+ also have "... \<le> int (dim {x - a| x. x \<in> T})"
+ by (simp add: dim_image_le [OF assms])
+ also have "... \<le> aff_dim T"
+ by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
+ finally show ?thesis .
+ qed
+ then
+ have "aff_dim (f ` (affine hull S)) \<le> 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) \<le> aff_dim S"
+ by (simp add: aff_dim_linear_image_le assms(1))
+next
+ obtain g where "linear g" "g \<circ> f = id"
+ using linear_injective_left_inverse assms by blast
+ then have "aff_dim S \<le> aff_dim(g ` f ` S)"
+ by (simp add: image_comp)
+ also have "... \<le> aff_dim (f ` S)"
+ by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
+ finally show "aff_dim S \<le> aff_dim (f ` S)" .
+qed
+
+
+text\<open>Choosing a subspace of a given dimension\<close>
+proposition choose_subspace_of_subspace:
+ fixes S :: "'n::euclidean_space set"
+ assumes "n \<le> dim S"
+ obtains T where "subspace T" "T \<subseteq> span S" "dim T = n"
+proof -
+ have "\<exists>T. subspace T \<and> T \<subseteq> span S \<and> dim T = n"
+ using assms
+ proof (induction n)
+ case 0 then show ?case by force
+ next
+ case (Suc n)
+ then obtain T where "subspace T" "T \<subseteq> span S" "dim T = n"
+ by force
+ then show ?case
+ proof (cases "span S \<subseteq> span T")
+ case True
+ have "dim S = dim T"
+ apply (rule span_eq_dim [OF subset_antisym [OF True]])
+ by (simp add: \<open>T \<subseteq> span S\<close> span_minimal subspace_span)
+ then show ?thesis
+ using Suc.prems \<open>dim T = n\<close> by linarith
+ next
+ case False
+ then obtain y where y: "y \<in> S" "y \<notin> T"
+ by (meson span_mono subsetI)
+ then have "span (insert y T) \<subseteq> span S"
+ by (metis (no_types) \<open>T \<subseteq> span S\<close> subsetD insert_subset span_inc span_mono span_span)
+ with \<open>dim T = n\<close> \<open>subspace T\<close> y show ?thesis
+ apply (rule_tac x="span(insert y T)" in exI)
+ apply (auto simp: dim_insert)
+ using span_eq by blast
+ qed
+ qed
+ with that show ?thesis by blast
+qed
+
+lemma choose_affine_subset:
+ assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
+ obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
+proof (cases "d = -1 \<or> 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 \<in> S" "0 \<le> d" by auto
+ with assms have ss: "subspace (op + (- a) ` S)"
+ by (simp add: affine_diffs_subspace)
+ have "nat d \<le> dim (op + (- 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 \<subseteq> span (op + (- a) ` S)"
+ and Tdim: "dim T = nat d"
+ using choose_subspace_of_subspace [of "nat d" "op + (- a) ` S"] by blast
+ then have "affine T"
+ using subspace_affine by blast
+ then have "affine (op + a ` T)"
+ by (metis affine_hull_eq affine_hull_translation)
+ moreover have "op + a ` T \<subseteq> S"
+ proof -
+ have "T \<subseteq> op + (- a) ` S"
+ by (metis (no_types) span_eq Tsb ss)
+ then show "op + a ` T \<subseteq> S"
+ using add_ac by auto
+ qed
+ moreover have "aff_dim (op + a ` T) = d"
+ by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
+ ultimately show ?thesis
+ by (rule that)
+qed
+
+subsection\<open>Several Variants of Paracompactness\<close>
+
+proposition paracompact:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
+ obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
+ and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
+ and "\<And>x. x \<in> S
+ \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and>
+ finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
+proof (cases "S = {}")
+ case True with that show ?thesis by blast
+next
+ case False
+ have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
+ proof -
+ obtain T where "x \<in> T" "T \<in> \<C>" "open T"
+ using assms \<open>x \<in> S\<close> by blast
+ then obtain e where "e > 0" "cball x e \<subseteq> T"
+ by (force simp: open_contains_cball)
+ then show ?thesis
+ apply (rule_tac x = T in exI)
+ apply (rule_tac x = "ball x e" in exI)
+ using \<open>T \<in> \<C>\<close>
+ apply (simp add: closure_minimal)
+ done
+ qed
+ then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
+ and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
+ if "x \<in> S" for x
+ by metis
+ then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = UNION S G"
+ using Lindelof [of "G ` S"] by (metis image_iff)
+ then obtain K where K: "K \<subseteq> S" "countable K" and eq: "UNION K G = UNION S G"
+ by (metis countable_subset_image)
+ with False Gin have "K \<noteq> {}" by force
+ then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
+ by (metis range_from_nat_into \<open>countable K\<close>)
+ then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
+ using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
+ let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
+ have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
+ proof -
+ have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
+ then show ?thesis
+ using clos K \<open>range a = K\<close> closure_subset by blast
+ qed
+ have 1: "S \<subseteq> Union ?C"
+ proof
+ fix x assume "x \<in> S"
+ define n where "n \<equiv> LEAST n. x \<in> F(a n)"
+ have n: "x \<in> F(a n)"
+ using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
+ have notn: "x \<notin> F(a m)" if "m < n" for m
+ using that not_less_Least by (force simp: n_def)
+ then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
+ using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
+ with n show "x \<in> Union ?C"
+ by blast
+ qed
+ have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
+ proof -
+ obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
+ using \<open>x \<in> S\<close> enum_S by auto
+ have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
+ proof clarsimp
+ fix k assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
+ then have "k \<le> n"
+ by auto (metis closure_subset not_le subsetCE)
+ then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
+ \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
+ by force
+ qed
+ moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
+ by force
+ ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
+ using finite_subset by blast
+ show ?thesis
+ apply (rule_tac x="G (a n)" in exI)
+ apply (intro conjI oG n *)
+ using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
+ done
+ qed
+ show ?thesis
+ apply (rule that [OF 1 _ 3])
+ using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply (auto simp: odif)
+ done
+qed
+
+corollary paracompact_closedin:
+ fixes S :: "'a :: euclidean_space set"
+ assumes cin: "closedin (subtopology euclidean U) S"
+ and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (subtopology euclidean U) T"
+ and "S \<subseteq> \<Union>\<C>"
+ obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
+ and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
+ and "\<And>x. x \<in> U
+ \<Longrightarrow> \<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and>
+ finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
+proof -
+ have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
+ using oin [OF that] by (auto simp: openin_open)
+ then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
+ by metis
+ obtain K where K: "closed K" "U \<inter> K = S"
+ using cin by (auto simp: closedin_closed)
+ have 1: "U \<subseteq> \<Union>insert (- K) (F ` \<C>)"
+ by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
+ have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
+ using \<open>closed K\<close> by (auto simp: opF)
+ obtain \<D> where "U \<subseteq> \<Union>\<D>"
+ and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
+ and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
+ using paracompact [OF 1 2] by auto
+ let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
+ show ?thesis
+ proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
+ show "S \<subseteq> \<Union>?C"
+ using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
+ show "\<And>V. V \<in> ?C \<Longrightarrow> openin (subtopology euclidean U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
+ using D1 intF by fastforce
+ have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
+ (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
+ by blast
+ show "\<exists>V. openin (subtopology euclidean U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
+ if "x \<in> U" for x
+ using D2 [OF that]
+ apply clarify
+ apply (rule_tac x="U \<inter> V" in exI)
+ apply (auto intro: that finite_subset [OF *])
+ done
+ qed
+qed
+
+corollary paracompact_closed:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "closed S"
+ and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
+ and "S \<subseteq> \<Union>\<C>"
+ obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
+ and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
+ and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
+ finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
+using paracompact_closedin [of UNIV S \<C>] assms
+by (auto simp: open_openin [symmetric] closed_closedin [symmetric])
+
+
+subsection\<open>Closed-graph characterization of continuity\<close>
+
+lemma continuous_closed_graph_gen:
+ fixes T :: "'b::real_normed_vector set"
+ assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
+ shows "closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
+proof -
+ have eq: "((\<lambda>x. Pair x (f x)) ` S) = {z. z \<in> S \<times> T \<and> (f \<circ> fst)z - snd z \<in> {0}}"
+ using fim by auto
+ show ?thesis
+ apply (subst eq)
+ apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
+ by auto
+qed
+
+lemma continuous_closed_graph_eq:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact T" and fim: "f ` S \<subseteq> T"
+ shows "continuous_on S f \<longleftrightarrow>
+ closedin (subtopology euclidean (S \<times> T)) ((\<lambda>x. 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 (subtopology euclidean T) U"
+ have eq: "{x. x \<in> S \<and> f x \<in> U} = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
+ by (force simp: image_iff)
+ show "closedin (subtopology euclidean S) {x \<in> S. f x \<in> U}"
+ by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
+ qed
+ with continuous_closed_graph_gen assms show ?thesis by blast
+qed
+
+lemma continuous_closed_graph:
+ fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "closed S" and contf: "continuous_on S f"
+ shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
+ apply (rule closedin_closed_trans)
+ apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
+ by (simp add: \<open>closed S\<close> closed_Times)
+
+lemma continuous_from_closed_graph:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
+ shows "continuous_on S f"
+ using fim clo
+ by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
+
+lemma continuous_on_Un_local_open:
+ assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
+ and opT: "openin (subtopology euclidean (S \<union> T)) T"
+ and contf: "continuous_on S f" and contg: "continuous_on T f"
+ shows "continuous_on (S \<union> T) f"
+using pasting_lemma [of "{S,T}" "S \<union> T" "\<lambda>i. i" "\<lambda>i. f" f] contf contg opS opT by blast
+
+lemma continuous_on_cases_local_open:
+ assumes opS: "openin (subtopology euclidean (S \<union> T)) S"
+ and opT: "openin (subtopology euclidean (S \<union> T)) T"
+ and contf: "continuous_on S f" and contg: "continuous_on T g"
+ and fg: "\<And>x. x \<in> S \<and> ~P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
+ shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
+proof -
+ have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x" "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
+ by (simp_all add: fg)
+ then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
+ by (simp_all add: contf contg cong: continuous_on_cong)
+ then show ?thesis
+ by (rule continuous_on_Un_local_open [OF opS opT])
+qed
+
+subsection\<open>The union of two collinear segments is another segment\<close>
+
+proposition in_convex_hull_exchange:
+ fixes a :: "'a::euclidean_space"
+ assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
+ obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
+proof (cases "a \<in> S")
+ case True
+ with xS insert_Diff that show ?thesis by fastforce
+next
+ case False
+ show ?thesis
+ proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
+ case True
+ then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
+ and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
+ using a by (auto simp: convex_hull_finite)
+ obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
+ and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
+ using True xS by (auto simp: convex_hull_finite)
+ show ?thesis
+ proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
+ case True
+ then obtain b where b: "b \<in> S" "v b = 0"
+ by blast
+ show ?thesis
+ proof
+ have fin: "finite (insert a (S - {b}))"
+ using sum.infinite v1 by fastforce
+ show "x \<in> convex hull insert a (S - {b})"
+ unfolding convex_hull_finite [OF fin] mem_Collect_eq
+ proof (intro conjI exI ballI)
+ have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
+ (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
+ apply (rule sum.mono_neutral_right)
+ using fin by auto
+ also have "... = (\<Sum>x \<in> S - {b}. v x)"
+ using b False by (auto intro!: sum.cong split: if_split_asm)
+ also have "... = (\<Sum>x\<in>S. v x)"
+ by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
+ finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
+ by (simp add: v1)
+ show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
+ by (auto simp: v0)
+ have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
+ (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
+ apply (rule sum.mono_neutral_right)
+ using fin by auto
+ also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
+ using b False by (auto intro!: sum.cong split: if_split_asm)
+ also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
+ by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert real_vector.scale_eq_0_iff sum_diff1)
+ finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
+ by (simp add: vx)
+ qed
+ qed (rule \<open>b \<in> S\<close>)
+ next
+ case False
+ have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
+ by (simp add: True that)
+ have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
+ using True v1 by (auto intro: Max_in)
+ then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
+ by blast
+ then have "0 \<noteq> 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 \<open>b \<in> S\<close> u0 v0 by force+
+ have fin: "finite (insert a (S - {b}))"
+ using sum.infinite v1 by fastforce
+ show ?thesis
+ proof
+ show "x \<in> convex hull insert a (S - {b})"
+ unfolding convex_hull_finite [OF fin] mem_Collect_eq
+ proof (intro conjI exI ballI)
+ have "(\<Sum>x \<in> 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 + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
+ using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
+ apply (rule sum.cong, auto)
+ done
+ also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
+ by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
+ also have "... = (\<Sum>x\<in>S. v x)"
+ using \<open>0 < u b\<close> True by (simp add: Groups_Big.sum_diff1 u1 field_simps)
+ finally show "sum (\<lambda>x. 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 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
+ if "i \<in> insert a (S - {b})" for i
+ using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
+ by (auto simp: field_simps split: if_split_asm)
+ have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
+ (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
+ using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True apply simp
+ apply (rule sum.cong, auto)
+ done
+ also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
+ by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left real_vector.scale_sum_right)
+ also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
+ using \<open>0 < u b\<close> True by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
+ finally
+ show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
+ by (simp add: vx)
+ qed
+ qed (rule \<open>b \<in> S\<close>)
+ qed
+ next
+ case False
+ obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
+ using xS by (auto simp: caratheodory [of S])
+ with False obtain b where b: "b \<in> S" "b \<notin> T"
+ by (metis antisym subsetI)
+ show ?thesis
+ proof
+ show "x \<in> convex hull insert a (S - {b})"
+ using \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
+ qed (rule \<open>b \<in> S\<close>)
+ qed
+qed
+
+lemma convex_hull_exchange_Union:
+ fixes a :: "'a::euclidean_space"
+ assumes "a \<in> convex hull S"
+ shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
+proof
+ show "?lhs \<subseteq> ?rhs"
+ by (blast intro: in_convex_hull_exchange [OF assms])
+ show "?rhs \<subseteq> ?lhs"
+ proof clarify
+ fix x b
+ assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
+ then show "x \<in> convex hull S" if "b \<in> 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 \<in> closed_segment a c"
+ shows "closed_segment a b \<union> 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} \<union> convex hull {b, c} = (\<Union>ba\<in>{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 \<in> open_segment a c"
+ shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
+proof -
+ have b: "b \<in> closed_segment a c"
+ by (simp add: assms open_closed_segment)
+ have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
+ if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
+ proof -
+ have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
+ 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
+ using Un_closed_segment [OF b]
+ apply (simp add: closed_segment_eq_open)
+ apply (rule equalityI)
+ using assms
+ apply (simp add: b subset_open_segment)
+ using * by (simp add: insert_commute)
+qed
+
+subsection\<open>Covering an open set by a countable chain of compact sets\<close>
+
+proposition open_Union_compact_subsets:
+ fixes S :: "'a::euclidean_space set"
+ assumes "open S"
+ obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
+ "\<And>n. C n \<subseteq> interior(C(Suc n))"
+ "\<Union>(range C) = S"
+ "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by (rule_tac C = "\<lambda>n. {}" in that) auto
+next
+ case False
+ then obtain a where "a \<in> S"
+ by auto
+ let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
+ have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
+ if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
+ and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
+ proof -
+ have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
+ by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
+ have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
+ by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
+ obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
+ proof (rule compactE_image [OF \<open>compact K\<close>])
+ show "K \<subseteq> (\<Union>n. interior (f n))"
+ using \<open>K \<subseteq> S\<close> \<open>UNION UNIV f = S\<close> * by blast
+ qed auto
+ { fix n
+ assume n: "Max I \<le> n"
+ have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
+ by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
+ then have "K \<subseteq> f n"
+ using I by auto
+ }
+ then show ?thesis
+ by blast
+ qed
+ moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
+ ((\<Union>(range f) = S))"
+ proof (intro exI conjI allI)
+ show "\<And>n. compact (?C n)"
+ by (auto simp: compact_diff open_sums)
+ show "\<And>n. ?C n \<subseteq> S"
+ by auto
+ show "?C n \<subseteq> interior (?C (Suc n))" for n
+ proof (simp add: interior_diff, rule Diff_mono)
+ show "cball a (real n) \<subseteq> ball a (1 + real n)"
+ by (simp add: cball_subset_ball_iff)
+ have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
+ using assms by (auto intro: closed_compact_sums)
+ have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
+ \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
+ by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
+ also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
+ apply (intro UN_mono order_refl)
+ apply (simp add: cball_subset_ball_iff divide_simps)
+ done
+ finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
+ \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
+ qed
+ have "S \<subseteq> UNION UNIV ?C"
+ proof
+ fix x
+ assume x: "x \<in> S"
+ then obtain e where "e > 0" and e: "ball x e \<subseteq> 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) \<le> real N2"
+ by (meson real_arch_simple)
+ have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
+ using \<open>N1 > 0\<close> by (auto simp: divide_simps)
+ have "x \<noteq> y + z" if "y \<notin> 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: \<open>0 < e\<close>)
+ then have "1 / (1 + (real N1 + real N2)) < e"
+ using N1 \<open>e > 0\<close>
+ by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
+ then have "x - z \<in> ball x e"
+ using that by simp
+ then have "x - z \<in> S"
+ using e by blast
+ with that show ?thesis
+ by auto
+ qed
+ with N2 show "x \<in> UNION UNIV ?C"
+ by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
+ qed
+ then show "UNION UNIV ?C = S" by auto
+ qed
+ ultimately show ?thesis
+ using that by metis
+qed
+
+end
+
\ No newline at end of file