--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Convex.thy Mon Jan 07 14:06:54 2019 +0100
@@ -0,0 +1,4195 @@
+(* Title: HOL/Analysis/Convex_Euclidean_Space.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>Convex Sets and Functions\<close>
+
+theory Convex
+imports
+ Linear_Algebra
+ "HOL-Library.Set_Algebras"
+begin
+
+lemma substdbasis_expansion_unique:
+ assumes d: "d \<subseteq> Basis"
+ shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
+ (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
+proof -
+ have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
+ by auto
+ have **: "finite d"
+ by (auto intro: finite_subset[OF assms])
+ have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
+ using d
+ by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
+ show ?thesis
+ unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
+qed
+
+lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
+ by (rule independent_mono[OF independent_Basis])
+
+lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
+ by (rule ccontr) auto
+
+lemma subset_translation_eq [simp]:
+ fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
+ by auto
+
+lemma translate_inj_on:
+ fixes A :: "'a::ab_group_add set"
+ shows "inj_on (\<lambda>x. a + x) A"
+ unfolding inj_on_def by auto
+
+lemma translation_assoc:
+ fixes a b :: "'a::ab_group_add"
+ shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
+ by auto
+
+lemma translation_invert:
+ fixes a :: "'a::ab_group_add"
+ assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
+ shows "A = B"
+proof -
+ have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
+ using assms by auto
+ then show ?thesis
+ using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
+qed
+
+lemma translation_galois:
+ fixes a :: "'a::ab_group_add"
+ shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
+ using translation_assoc[of "-a" a S]
+ apply auto
+ using translation_assoc[of a "-a" T]
+ apply auto
+ done
+
+lemma translation_inverse_subset:
+ assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
+ shows "V \<le> ((\<lambda>x. a + x) ` S)"
+proof -
+ {
+ fix x
+ assume "x \<in> V"
+ then have "x-a \<in> S" using assms by auto
+ then have "x \<in> {a + v |v. v \<in> S}"
+ apply auto
+ apply (rule exI[of _ "x-a"], simp)
+ done
+ then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
+ }
+ then show ?thesis by auto
+qed
+
+subsection \<open>Convexity\<close>
+
+definition%important convex :: "'a::real_vector set \<Rightarrow> bool"
+ where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
+
+lemma convexI:
+ assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ shows "convex s"
+ using assms unfolding convex_def by fast
+
+lemma convexD:
+ assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
+ shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ using assms unfolding convex_def by fast
+
+lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
+ (is "_ \<longleftrightarrow> ?alt")
+proof
+ show "convex s" if alt: ?alt
+ proof -
+ {
+ fix x y and u v :: real
+ assume mem: "x \<in> s" "y \<in> s"
+ assume "0 \<le> u" "0 \<le> v"
+ moreover
+ assume "u + v = 1"
+ then have "u = 1 - v" by auto
+ ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ using alt [rule_format, OF mem] by auto
+ }
+ then show ?thesis
+ unfolding convex_def by auto
+ qed
+ show ?alt if "convex s"
+ using that by (auto simp: convex_def)
+qed
+
+lemma convexD_alt:
+ assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
+ shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
+ using assms unfolding convex_alt by auto
+
+lemma mem_convex_alt:
+ assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
+ shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
+ apply (rule convexD)
+ using assms
+ apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
+ done
+
+lemma convex_empty[intro,simp]: "convex {}"
+ unfolding convex_def by simp
+
+lemma convex_singleton[intro,simp]: "convex {a}"
+ unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
+
+lemma convex_UNIV[intro,simp]: "convex UNIV"
+ unfolding convex_def by auto
+
+lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
+ unfolding convex_def by auto
+
+lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
+ unfolding convex_def by auto
+
+lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
+ unfolding convex_def by auto
+
+lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
+ unfolding convex_def by auto
+
+lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
+ unfolding convex_def
+ by (auto simp: inner_add intro!: convex_bound_le)
+
+lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
+proof -
+ have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
+ by auto
+ show ?thesis
+ unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
+qed
+
+lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
+proof -
+ have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
+ by auto
+ show ?thesis
+ unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
+qed
+
+lemma convex_hyperplane: "convex {x. inner a x = b}"
+proof -
+ have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
+ by auto
+ show ?thesis using convex_halfspace_le convex_halfspace_ge
+ by (auto intro!: convex_Int simp: *)
+qed
+
+lemma convex_halfspace_lt: "convex {x. inner a x < b}"
+ unfolding convex_def
+ by (auto simp: convex_bound_lt inner_add)
+
+lemma convex_halfspace_gt: "convex {x. inner a x > b}"
+ using convex_halfspace_lt[of "-a" "-b"] by auto
+
+lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
+ using convex_halfspace_ge[of b "1::complex"] by simp
+
+lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
+ using convex_halfspace_le[of "1::complex" b] by simp
+
+lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
+ using convex_halfspace_ge[of b \<i>] by simp
+
+lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
+ using convex_halfspace_le[of \<i> b] by simp
+
+lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
+ using convex_halfspace_gt[of b "1::complex"] by simp
+
+lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
+ using convex_halfspace_lt[of "1::complex" b] by simp
+
+lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
+ using convex_halfspace_gt[of b \<i>] by simp
+
+lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
+ using convex_halfspace_lt[of \<i> b] by simp
+
+lemma convex_real_interval [iff]:
+ fixes a b :: "real"
+ shows "convex {a..}" and "convex {..b}"
+ and "convex {a<..}" and "convex {..<b}"
+ and "convex {a..b}" and "convex {a<..b}"
+ and "convex {a..<b}" and "convex {a<..<b}"
+proof -
+ have "{a..} = {x. a \<le> inner 1 x}"
+ by auto
+ then show 1: "convex {a..}"
+ by (simp only: convex_halfspace_ge)
+ have "{..b} = {x. inner 1 x \<le> b}"
+ by auto
+ then show 2: "convex {..b}"
+ by (simp only: convex_halfspace_le)
+ have "{a<..} = {x. a < inner 1 x}"
+ by auto
+ then show 3: "convex {a<..}"
+ by (simp only: convex_halfspace_gt)
+ have "{..<b} = {x. inner 1 x < b}"
+ by auto
+ then show 4: "convex {..<b}"
+ by (simp only: convex_halfspace_lt)
+ have "{a..b} = {a..} \<inter> {..b}"
+ by auto
+ then show "convex {a..b}"
+ by (simp only: convex_Int 1 2)
+ have "{a<..b} = {a<..} \<inter> {..b}"
+ by auto
+ then show "convex {a<..b}"
+ by (simp only: convex_Int 3 2)
+ have "{a..<b} = {a..} \<inter> {..<b}"
+ by auto
+ then show "convex {a..<b}"
+ by (simp only: convex_Int 1 4)
+ have "{a<..<b} = {a<..} \<inter> {..<b}"
+ by auto
+ then show "convex {a<..<b}"
+ by (simp only: convex_Int 3 4)
+qed
+
+lemma convex_Reals: "convex \<real>"
+ by (simp add: convex_def scaleR_conv_of_real)
+
+
+subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
+
+lemma convex_sum:
+ fixes C :: "'a::real_vector set"
+ assumes "finite s"
+ and "convex C"
+ and "(\<Sum> i \<in> s. a i) = 1"
+ assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
+ and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
+ shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
+ using assms(1,3,4,5)
+proof (induct arbitrary: a set: finite)
+ case empty
+ then show ?case by simp
+next
+ case (insert i s) note IH = this(3)
+ have "a i + sum a s = 1"
+ and "0 \<le> a i"
+ and "\<forall>j\<in>s. 0 \<le> a j"
+ and "y i \<in> C"
+ and "\<forall>j\<in>s. y j \<in> C"
+ using insert.hyps(1,2) insert.prems by simp_all
+ then have "0 \<le> sum a s"
+ by (simp add: sum_nonneg)
+ have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
+ proof (cases "sum a s = 0")
+ case True
+ with \<open>a i + sum a s = 1\<close> have "a i = 1"
+ by simp
+ from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
+ by simp
+ show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
+ by simp
+ next
+ case False
+ with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
+ by simp
+ then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
+ using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
+ by (simp add: IH sum_divide_distrib [symmetric])
+ from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
+ and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
+ have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
+ by (rule convexD)
+ then show ?thesis
+ by (simp add: scaleR_sum_right False)
+ qed
+ then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
+ by simp
+qed
+
+lemma convex:
+ "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (sum u {1..k} = 1)
+ \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
+proof safe
+ fix k :: nat
+ fix u :: "nat \<Rightarrow> real"
+ fix x
+ assume "convex s"
+ "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
+ "sum u {1..k} = 1"
+ with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
+ by auto
+next
+ assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
+ \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
+ {
+ fix \<mu> :: real
+ fix x y :: 'a
+ assume xy: "x \<in> s" "y \<in> s"
+ assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
+ let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
+ let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
+ have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
+ by auto
+ then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
+ by simp
+ then have "sum ?u {1 .. 2} = 1"
+ using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
+ by auto
+ with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
+ using mu xy by auto
+ have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
+ using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
+ from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
+ have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
+ by auto
+ then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
+ using s by (auto simp: add.commute)
+ }
+ then show "convex s"
+ unfolding convex_alt by auto
+qed
+
+
+lemma convex_explicit:
+ fixes s :: "'a::real_vector set"
+ shows "convex s \<longleftrightarrow>
+ (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
+proof safe
+ fix t
+ fix u :: "'a \<Rightarrow> real"
+ assume "convex s"
+ and "finite t"
+ and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
+ then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ using convex_sum[of t s u "\<lambda> x. x"] by auto
+next
+ assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
+ sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ show "convex s"
+ unfolding convex_alt
+ proof safe
+ fix x y
+ fix \<mu> :: real
+ assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
+ show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+ proof (cases "x = y")
+ case False
+ then show ?thesis
+ using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
+ by auto
+ next
+ case True
+ then show ?thesis
+ using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
+ by (auto simp: field_simps real_vector.scale_left_diff_distrib)
+ qed
+ qed
+qed
+
+lemma convex_finite:
+ assumes "finite s"
+ shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
+ unfolding convex_explicit
+ apply safe
+ subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
+ subgoal for t u
+ proof -
+ have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
+ by simp
+ assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
+ assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
+ assume "t \<subseteq> s"
+ then have "s \<inter> t = t" by auto
+ with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
+ qed
+ done
+
+
+subsection \<open>Functions that are convex on a set\<close>
+
+definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
+ where "convex_on s f \<longleftrightarrow>
+ (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
+
+lemma convex_onI [intro?]:
+ assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ shows "convex_on A f"
+ unfolding convex_on_def
+proof clarify
+ fix x y
+ fix u v :: real
+ assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+ from A(5) have [simp]: "v = 1 - u"
+ by (simp add: algebra_simps)
+ from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+ using assms[of u y x]
+ by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
+qed
+
+lemma convex_on_linorderI [intro?]:
+ fixes A :: "('a::{linorder,real_vector}) set"
+ assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ shows "convex_on A f"
+proof
+ fix x y
+ fix t :: real
+ assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
+ with assms [of t x y] assms [of "1 - t" y x]
+ show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
+qed
+
+lemma convex_onD:
+ assumes "convex_on A f"
+ shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ using assms by (auto simp: convex_on_def)
+
+lemma convex_onD_Icc:
+ assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
+ shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
+
+lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
+ unfolding convex_on_def by auto
+
+lemma convex_on_add [intro]:
+ assumes "convex_on s f"
+ and "convex_on s g"
+ shows "convex_on s (\<lambda>x. f x + g x)"
+proof -
+ {
+ fix x y
+ assume "x \<in> s" "y \<in> s"
+ moreover
+ fix u v :: real
+ assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+ ultimately
+ have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
+ using assms unfolding convex_on_def by (auto simp: add_mono)
+ then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
+ by (simp add: field_simps)
+ }
+ then show ?thesis
+ unfolding convex_on_def by auto
+qed
+
+lemma convex_on_cmul [intro]:
+ fixes c :: real
+ assumes "0 \<le> c"
+ and "convex_on s f"
+ shows "convex_on s (\<lambda>x. c * f x)"
+proof -
+ have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+ for u c fx v fy :: real
+ by (simp add: field_simps)
+ show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
+ unfolding convex_on_def and * by auto
+qed
+
+lemma convex_lower:
+ assumes "convex_on s f"
+ and "x \<in> s"
+ and "y \<in> s"
+ and "0 \<le> u"
+ and "0 \<le> v"
+ and "u + v = 1"
+ shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
+proof -
+ let ?m = "max (f x) (f y)"
+ have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
+ using assms(4,5) by (auto simp: mult_left_mono add_mono)
+ also have "\<dots> = max (f x) (f y)"
+ using assms(6) by (simp add: distrib_right [symmetric])
+ finally show ?thesis
+ using assms unfolding convex_on_def by fastforce
+qed
+
+lemma convex_on_dist [intro]:
+ fixes s :: "'a::real_normed_vector set"
+ shows "convex_on s (\<lambda>x. dist a x)"
+proof (auto simp: convex_on_def dist_norm)
+ fix x y
+ assume "x \<in> s" "y \<in> s"
+ fix u v :: real
+ assume "0 \<le> u"
+ assume "0 \<le> v"
+ assume "u + v = 1"
+ have "a = u *\<^sub>R a + v *\<^sub>R a"
+ unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
+ then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
+ by (auto simp: algebra_simps)
+ show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
+ unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
+ using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
+qed
+
+
+subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close>
+
+lemma convex_linear_image:
+ assumes "linear f"
+ and "convex s"
+ shows "convex (f ` s)"
+proof -
+ interpret f: linear f by fact
+ from \<open>convex s\<close> show "convex (f ` s)"
+ by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
+qed
+
+lemma convex_linear_vimage:
+ assumes "linear f"
+ and "convex s"
+ shows "convex (f -` s)"
+proof -
+ interpret f: linear f by fact
+ from \<open>convex s\<close> show "convex (f -` s)"
+ by (simp add: convex_def f.add f.scaleR)
+qed
+
+lemma convex_scaling:
+ assumes "convex s"
+ shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
+proof -
+ have "linear (\<lambda>x. c *\<^sub>R x)"
+ by (simp add: linearI scaleR_add_right)
+ then show ?thesis
+ using \<open>convex s\<close> by (rule convex_linear_image)
+qed
+
+lemma convex_scaled:
+ assumes "convex S"
+ shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
+proof -
+ have "linear (\<lambda>x. x *\<^sub>R c)"
+ by (simp add: linearI scaleR_add_left)
+ then show ?thesis
+ using \<open>convex S\<close> by (rule convex_linear_image)
+qed
+
+lemma convex_negations:
+ assumes "convex S"
+ shows "convex ((\<lambda>x. - x) ` S)"
+proof -
+ have "linear (\<lambda>x. - x)"
+ by (simp add: linearI)
+ then show ?thesis
+ using \<open>convex S\<close> by (rule convex_linear_image)
+qed
+
+lemma convex_sums:
+ assumes "convex S"
+ and "convex T"
+ shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+proof -
+ have "linear (\<lambda>(x, y). x + y)"
+ by (auto intro: linearI simp: scaleR_add_right)
+ with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
+ by (intro convex_linear_image convex_Times)
+ also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma convex_differences:
+ assumes "convex S" "convex T"
+ shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+proof -
+ have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
+ by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
+ then show ?thesis
+ using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
+qed
+
+lemma convex_translation:
+ assumes "convex S"
+ shows "convex ((\<lambda>x. a + x) ` S)"
+proof -
+ have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (\<lambda>x. a + x) ` S"
+ by auto
+ then show ?thesis
+ using convex_sums[OF convex_singleton[of a] assms] by auto
+qed
+
+lemma convex_affinity:
+ assumes "convex S"
+ shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
+proof -
+ have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
+ by auto
+ then show ?thesis
+ using convex_translation[OF convex_scaling[OF assms], of a c] by auto
+qed
+
+lemma pos_is_convex: "convex {0 :: real <..}"
+ unfolding convex_alt
+proof safe
+ fix y x \<mu> :: real
+ assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
+ {
+ assume "\<mu> = 0"
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
+ by simp
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by simp
+ }
+ moreover
+ {
+ assume "\<mu> = 1"
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by simp
+ }
+ moreover
+ {
+ assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
+ then have "\<mu> > 0" "(1 - \<mu>) > 0"
+ using * by auto
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by (auto simp: add_pos_pos)
+ }
+ ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
+ by fastforce
+qed
+
+lemma convex_on_sum:
+ fixes a :: "'a \<Rightarrow> real"
+ and y :: "'a \<Rightarrow> 'b::real_vector"
+ and f :: "'b \<Rightarrow> real"
+ assumes "finite s" "s \<noteq> {}"
+ and "convex_on C f"
+ and "convex C"
+ and "(\<Sum> i \<in> s. a i) = 1"
+ and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
+ and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
+ shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
+ using assms
+proof (induct s arbitrary: a rule: finite_ne_induct)
+ case (singleton i)
+ then have ai: "a i = 1"
+ by auto
+ then show ?case
+ by auto
+next
+ case (insert i s)
+ then have "convex_on C f"
+ by simp
+ from this[unfolded convex_on_def, rule_format]
+ have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
+ f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ by simp
+ show ?case
+ proof (cases "a i = 1")
+ case True
+ then have "(\<Sum> j \<in> s. a j) = 0"
+ using insert by auto
+ then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
+ using insert by (fastforce simp: sum_nonneg_eq_0_iff)
+ then show ?thesis
+ using insert by auto
+ next
+ case False
+ from insert have yai: "y i \<in> C" "a i \<ge> 0"
+ by auto
+ have fis: "finite (insert i s)"
+ using insert by auto
+ then have ai1: "a i \<le> 1"
+ using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
+ then have "a i < 1"
+ using False by auto
+ then have i0: "1 - a i > 0"
+ by auto
+ let ?a = "\<lambda>j. a j / (1 - a i)"
+ have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
+ using i0 insert that by fastforce
+ have "(\<Sum> j \<in> insert i s. a j) = 1"
+ using insert by auto
+ then have "(\<Sum> j \<in> s. a j) = 1 - a i"
+ using sum.insert insert by fastforce
+ then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
+ using i0 by auto
+ then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
+ unfolding sum_divide_distrib by simp
+ have "convex C" using insert by auto
+ then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
+ using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
+ have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
+ using a_nonneg a1 insert by blast
+ have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
+ using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
+ by (auto simp only: add.commute)
+ also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
+ using i0 by auto
+ also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
+ using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
+ by (auto simp: algebra_simps)
+ also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
+ by (auto simp: divide_inverse)
+ also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
+ using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
+ by (auto simp: add.commute)
+ also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
+ using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
+ OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
+ by simp
+ also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
+ unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
+ using i0 by auto
+ also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
+ using i0 by auto
+ also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
+ using insert by auto
+ finally show ?thesis
+ by simp
+ qed
+qed
+
+lemma convex_on_alt:
+ fixes C :: "'a::real_vector set"
+ assumes "convex C"
+ shows "convex_on C f \<longleftrightarrow>
+ (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
+ f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
+proof safe
+ fix x y
+ fix \<mu> :: real
+ assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
+ from this[unfolded convex_on_def, rule_format]
+ have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
+ by auto
+ from this [of "\<mu>" "1 - \<mu>", simplified] *
+ show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ by auto
+next
+ assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
+ f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ {
+ fix x y
+ fix u v :: real
+ assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+ then have[simp]: "1 - u = v" by auto
+ from *[rule_format, of x y u]
+ have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+ using ** by auto
+ }
+ then show "convex_on C f"
+ unfolding convex_on_def by auto
+qed
+
+lemma convex_on_diff:
+ fixes f :: "real \<Rightarrow> real"
+ assumes f: "convex_on I f"
+ and I: "x \<in> I" "y \<in> I"
+ and t: "x < t" "t < y"
+ shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+ and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
+proof -
+ define a where "a \<equiv> (t - y) / (x - y)"
+ with t have "0 \<le> a" "0 \<le> 1 - a"
+ by (auto simp: field_simps)
+ with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
+ by (auto simp: convex_on_def)
+ have "a * x + (1 - a) * y = a * (x - y) + y"
+ by (simp add: field_simps)
+ also have "\<dots> = t"
+ unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
+ finally have "f t \<le> a * f x + (1 - a) * f y"
+ using cvx by simp
+ also have "\<dots> = a * (f x - f y) + f y"
+ by (simp add: field_simps)
+ finally have "f t - f y \<le> a * (f x - f y)"
+ by simp
+ with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+ by (simp add: le_divide_eq divide_le_eq field_simps a_def)
+ with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
+ by (simp add: le_divide_eq divide_le_eq field_simps)
+qed
+
+lemma pos_convex_function:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex C"
+ and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
+ shows "convex_on C f"
+ unfolding convex_on_alt[OF assms(1)]
+ using assms
+proof safe
+ fix x y \<mu> :: real
+ let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
+ assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
+ then have "1 - \<mu> \<ge> 0" by auto
+ then have xpos: "?x \<in> C"
+ using * unfolding convex_alt by fastforce
+ have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
+ \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
+ using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
+ mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
+ by auto
+ then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
+ by (auto simp: field_simps)
+ then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ using convex_on_alt by auto
+qed
+
+lemma atMostAtLeast_subset_convex:
+ fixes C :: "real set"
+ assumes "convex C"
+ and "x \<in> C" "y \<in> C" "x < y"
+ shows "{x .. y} \<subseteq> C"
+proof safe
+ fix z assume z: "z \<in> {x .. y}"
+ have less: "z \<in> C" if *: "x < z" "z < y"
+ proof -
+ let ?\<mu> = "(y - z) / (y - x)"
+ have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
+ using assms * by (auto simp: field_simps)
+ then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
+ using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
+ by (simp add: algebra_simps)
+ have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
+ by (auto simp: field_simps)
+ also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
+ using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
+ also have "\<dots> = z"
+ using assms by (auto simp: field_simps)
+ finally show ?thesis
+ using comb by auto
+ qed
+ show "z \<in> C"
+ using z less assms by (auto simp: le_less)
+qed
+
+lemma f''_imp_f':
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex C"
+ and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
+ and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
+ and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
+ and x: "x \<in> C"
+ and y: "y \<in> C"
+ shows "f' x * (y - x) \<le> f y - f x"
+ using assms
+proof -
+ have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+ if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
+ proof -
+ from * have ge: "y - x > 0" "y - x \<ge> 0"
+ by auto
+ from * have le: "x - y < 0" "x - y \<le> 0"
+ by auto
+ then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
+ using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
+ THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
+ by auto
+ then have "z1 \<in> C"
+ using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
+ by fastforce
+ from z1 have z1': "f x - f y = (x - y) * f' z1"
+ by (simp add: field_simps)
+ obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
+ using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
+ THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ by auto
+ obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
+ using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
+ THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ by auto
+ have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
+ using * z1' by auto
+ also have "\<dots> = (y - z1) * f'' z3"
+ using z3 by auto
+ finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
+ by simp
+ have A': "y - z1 \<ge> 0"
+ using z1 by auto
+ have "z3 \<in> C"
+ using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
+ by fastforce
+ then have B': "f'' z3 \<ge> 0"
+ using assms by auto
+ from A' B' have "(y - z1) * f'' z3 \<ge> 0"
+ by auto
+ from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
+ by auto
+ from mult_right_mono_neg[OF this le(2)]
+ have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
+ by (simp add: algebra_simps)
+ then have "f' y * (x - y) - (f x - f y) \<le> 0"
+ using le by auto
+ then have res: "f' y * (x - y) \<le> f x - f y"
+ by auto
+ have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
+ using * z1 by auto
+ also have "\<dots> = (z1 - x) * f'' z2"
+ using z2 by auto
+ finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
+ by simp
+ have A: "z1 - x \<ge> 0"
+ using z1 by auto
+ have "z2 \<in> C"
+ using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
+ by fastforce
+ then have B: "f'' z2 \<ge> 0"
+ using assms by auto
+ from A B have "(z1 - x) * f'' z2 \<ge> 0"
+ by auto
+ with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
+ by auto
+ from mult_right_mono[OF this ge(2)]
+ have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
+ by (simp add: algebra_simps)
+ then have "f y - f x - f' x * (y - x) \<ge> 0"
+ using ge by auto
+ then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+ using res by auto
+ qed
+ show ?thesis
+ proof (cases "x = y")
+ case True
+ with x y show ?thesis by auto
+ next
+ case False
+ with less_imp x y show ?thesis
+ by (auto simp: neq_iff)
+ qed
+qed
+
+lemma f''_ge0_imp_convex:
+ fixes f :: "real \<Rightarrow> real"
+ assumes conv: "convex C"
+ and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
+ and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
+ and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
+ shows "convex_on C f"
+ using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
+ by fastforce
+
+lemma minus_log_convex:
+ fixes b :: real
+ assumes "b > 1"
+ shows "convex_on {0 <..} (\<lambda> x. - log b x)"
+proof -
+ have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
+ using DERIV_log by auto
+ then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
+ by (auto simp: DERIV_minus)
+ have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
+ using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
+ from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
+ have "\<And>z::real. z > 0 \<Longrightarrow>
+ DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
+ by auto
+ then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
+ DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
+ unfolding inverse_eq_divide by (auto simp: mult.assoc)
+ have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
+ using \<open>b > 1\<close> by (auto intro!: less_imp_le)
+ from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
+ show ?thesis
+ by auto
+qed
+
+
+subsection%unimportant \<open>Convexity of real functions\<close>
+
+lemma convex_on_realI:
+ assumes "connected A"
+ and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
+ and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
+ shows "convex_on A f"
+proof (rule convex_on_linorderI)
+ fix t x y :: real
+ assume t: "t > 0" "t < 1"
+ assume xy: "x \<in> A" "y \<in> A" "x < y"
+ define z where "z = (1 - t) * x + t * y"
+ with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
+ using connected_contains_Icc by blast
+
+ from xy t have xz: "z > x"
+ by (simp add: z_def algebra_simps)
+ have "y - z = (1 - t) * (y - x)"
+ by (simp add: z_def algebra_simps)
+ also from xy t have "\<dots> > 0"
+ by (intro mult_pos_pos) simp_all
+ finally have yz: "z < y"
+ by simp
+
+ from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
+ by (intro MVT2) (auto intro!: assms(2))
+ then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
+ by auto
+ from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
+ by (intro MVT2) (auto intro!: assms(2))
+ then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
+ by auto
+
+ from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
+ also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
+ by auto
+ with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
+ by (intro assms(3)) auto
+ also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
+ finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
+ using xz yz by (simp add: field_simps)
+ also have "z - x = t * (y - x)"
+ by (simp add: z_def algebra_simps)
+ also have "y - z = (1 - t) * (y - x)"
+ by (simp add: z_def algebra_simps)
+ finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
+ using xy by simp
+ then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
+ by (simp add: z_def algebra_simps)
+qed
+
+lemma convex_on_inverse:
+ assumes "A \<subseteq> {0<..}"
+ shows "convex_on A (inverse :: real \<Rightarrow> real)"
+proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
+ fix u v :: real
+ assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
+ with assms show "-inverse (u^2) \<le> -inverse (v^2)"
+ by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
+qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
+
+lemma convex_onD_Icc':
+ assumes "convex_on {x..y} f" "c \<in> {x..y}"
+ defines "d \<equiv> y - x"
+ shows "f c \<le> (f y - f x) / d * (c - x) + f x"
+proof (cases x y rule: linorder_cases)
+ case less
+ then have d: "d > 0"
+ by (simp add: d_def)
+ from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
+ by (simp_all add: d_def divide_simps)
+ have "f c = f (x + (c - x) * 1)"
+ by simp
+ also from less have "1 = ((y - x) / d)"
+ by (simp add: d_def)
+ also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
+ by (simp add: field_simps)
+ also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
+ using assms less by (intro convex_onD_Icc) simp_all
+ also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
+ by (simp add: field_simps)
+ finally show ?thesis .
+qed (insert assms(2), simp_all)
+
+lemma convex_onD_Icc'':
+ assumes "convex_on {x..y} f" "c \<in> {x..y}"
+ defines "d \<equiv> y - x"
+ shows "f c \<le> (f x - f y) / d * (y - c) + f y"
+proof (cases x y rule: linorder_cases)
+ case less
+ then have d: "d > 0"
+ by (simp add: d_def)
+ from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
+ by (simp_all add: d_def divide_simps)
+ have "f c = f (y - (y - c) * 1)"
+ by simp
+ also from less have "1 = ((y - x) / d)"
+ by (simp add: d_def)
+ also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
+ by (simp add: field_simps)
+ also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
+ using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
+ also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
+ by (simp add: field_simps)
+ finally show ?thesis .
+qed (insert assms(2), simp_all)
+
+lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
+ by (metis convex_translation translation_galois)
+
+lemma convex_linear_image_eq [simp]:
+ fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
+ shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
+ by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
+
+lemma fst_linear: "linear fst"
+ unfolding linear_iff by (simp add: algebra_simps)
+
+lemma snd_linear: "linear snd"
+ unfolding linear_iff by (simp add: algebra_simps)
+
+lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
+ unfolding linear_iff by (simp add: algebra_simps)
+
+lemma vector_choose_size:
+ assumes "0 \<le> c"
+ obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
+proof -
+ obtain a::'a where "a \<noteq> 0"
+ using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
+ then show ?thesis
+ by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
+qed
+
+lemma vector_choose_dist:
+ assumes "0 \<le> c"
+ obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
+by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
+
+lemma sum_delta_notmem:
+ assumes "x \<notin> s"
+ shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
+ and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
+ apply (rule_tac [!] sum.cong)
+ using assms
+ apply auto
+ done
+
+lemma sum_delta'':
+ fixes s::"'a::real_vector set"
+ assumes "finite s"
+ shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
+proof -
+ have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
+ by auto
+ show ?thesis
+ unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
+qed
+
+lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
+ by (fact if_distrib)
+
+lemma dist_triangle_eq:
+ fixes x y z :: "'a::real_inner"
+ shows "dist x z = dist x y + dist y z \<longleftrightarrow>
+ norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
+proof -
+ have *: "x - y + (y - z) = x - z" by auto
+ show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
+ by (auto simp:norm_minus_commute)
+qed
+
+
+subsection \<open>Affine set and affine hull\<close>
+
+definition%important affine :: "'a::real_vector set \<Rightarrow> bool"
+ where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
+
+lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
+ unfolding affine_def by (metis eq_diff_eq')
+
+lemma affine_empty [iff]: "affine {}"
+ unfolding affine_def by auto
+
+lemma affine_sing [iff]: "affine {x}"
+ unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
+
+lemma affine_UNIV [iff]: "affine UNIV"
+ unfolding affine_def by auto
+
+lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
+ unfolding affine_def by auto
+
+lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
+ unfolding affine_def by auto
+
+lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
+ apply (clarsimp simp add: affine_def)
+ apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
+ apply (auto simp: algebra_simps)
+ done
+
+lemma affine_affine_hull [simp]: "affine(affine hull s)"
+ unfolding hull_def
+ using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
+
+lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
+ by (metis affine_affine_hull hull_same)
+
+lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
+ by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
+
+
+subsubsection%unimportant \<open>Some explicit formulations\<close>
+
+text "Formalized by Lars Schewe."
+
+lemma affine:
+ fixes V::"'a::real_vector set"
+ shows "affine V \<longleftrightarrow>
+ (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
+proof -
+ have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
+ and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
+ proof (cases "x = y")
+ case True
+ then show ?thesis
+ using that by (metis scaleR_add_left scaleR_one)
+ next
+ case False
+ then show ?thesis
+ using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
+ qed
+ moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
+ and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
+ proof -
+ define n where "n = card S"
+ consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
+ then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ proof cases
+ assume "card S = 1"
+ then obtain a where "S={a}"
+ by (auto simp: card_Suc_eq)
+ then show ?thesis
+ using that by simp
+ next
+ assume "card S = 2"
+ then obtain a b where "S = {a, b}"
+ by (metis Suc_1 card_1_singletonE card_Suc_eq)
+ then show ?thesis
+ using *[of a b] that
+ by (auto simp: sum_clauses(2))
+ next
+ assume "card S > 2"
+ then show ?thesis using that n_def
+ proof (induct n arbitrary: u S)
+ case 0
+ then show ?case by auto
+ next
+ case (Suc n u S)
+ have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
+ using that unfolding card_eq_sum by auto
+ with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
+ have c: "card (S - {x}) = card S - 1"
+ by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
+ have "sum u (S - {x}) = 1 - u x"
+ by (simp add: Suc.prems sum_diff1_ring \<open>x \<in> S\<close>)
+ with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
+ by auto
+ have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
+ proof (cases "card (S - {x}) > 2")
+ case True
+ then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
+ using Suc.prems c by force+
+ show ?thesis
+ proof (rule Suc.hyps)
+ show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
+ by (auto simp: eq1 sum_distrib_left[symmetric])
+ qed (use S Suc.prems True in auto)
+ next
+ case False
+ then have "card (S - {x}) = Suc (Suc 0)"
+ using Suc.prems c by auto
+ then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
+ unfolding card_Suc_eq by auto
+ then show ?thesis
+ using eq1 \<open>S \<subseteq> V\<close>
+ by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
+ qed
+ have "u x + (1 - u x) = 1 \<Longrightarrow>
+ u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
+ by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
+ moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
+ by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
+ ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
+ by (simp add: x)
+ qed
+ qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
+ qed
+ ultimately show ?thesis
+ unfolding affine_def by meson
+qed
+
+
+lemma affine_hull_explicit:
+ "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
+ (is "_ = ?rhs")
+proof (rule hull_unique)
+ show "p \<subseteq> ?rhs"
+ proof (intro subsetI CollectI exI conjI)
+ show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
+ by auto
+ qed auto
+ show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
+ using that unfolding affine by blast
+ show "affine ?rhs"
+ unfolding affine_def
+ proof clarify
+ fix u v :: real and sx ux sy uy
+ assume uv: "u + v = 1"
+ and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
+ and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
+ have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
+ by auto
+ show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
+ sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
+ proof (intro exI conjI)
+ show "finite (sx \<union> sy)"
+ using x y by auto
+ show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
+ using x y uv
+ by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
+ have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
+ = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
+ using x y
+ unfolding scaleR_left_distrib scaleR_zero_left if_smult
+ by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **)
+ also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
+ unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
+ finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
+ = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
+ qed (use x y in auto)
+ qed
+qed
+
+lemma affine_hull_finite:
+ assumes "finite S"
+ shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
+proof -
+ have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
+ if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
+ proof -
+ have "S \<inter> F = F"
+ using that by auto
+ show ?thesis
+ proof (intro exI conjI)
+ show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
+ by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
+ show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
+ by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
+ qed
+ qed
+ show ?thesis
+ unfolding affine_hull_explicit using assms
+ by (fastforce dest: *)
+qed
+
+
+subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close>
+
+lemma affine_hull_empty[simp]: "affine hull {} = {}"
+ by simp
+
+lemma affine_hull_finite_step:
+ fixes y :: "'a::real_vector"
+ shows "finite S \<Longrightarrow>
+ (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
+ (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
+proof -
+ assume fin: "finite S"
+ show "?lhs = ?rhs"
+ proof
+ assume ?lhs
+ then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
+ by auto
+ show ?rhs
+ proof (cases "a \<in> S")
+ case True
+ then show ?thesis
+ using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
+ next
+ case False
+ show ?thesis
+ by (rule exI [where x="u a"]) (use u fin False in auto)
+ qed
+ next
+ assume ?rhs
+ then obtain v u where vu: "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+ by auto
+ have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
+ by auto
+ show ?lhs
+ proof (cases "a \<in> S")
+ case True
+ show ?thesis
+ by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
+ (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
+ next
+ case False
+ then show ?thesis
+ apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
+ apply (simp add: vu sum_clauses(2)[OF fin] *)
+ by (simp add: sum_delta_notmem(3) vu)
+ qed
+ qed
+qed
+
+lemma affine_hull_2:
+ fixes a b :: "'a::real_vector"
+ shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
+ (is "?lhs = ?rhs")
+proof -
+ have *:
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
+ have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
+ using affine_hull_finite[of "{a,b}"] by auto
+ also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
+ by (simp add: affine_hull_finite_step[of "{b}" a])
+ also have "\<dots> = ?rhs" unfolding * by auto
+ finally show ?thesis by auto
+qed
+
+lemma affine_hull_3:
+ fixes a b c :: "'a::real_vector"
+ shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
+proof -
+ have *:
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
+ "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
+ show ?thesis
+ apply (simp add: affine_hull_finite affine_hull_finite_step)
+ unfolding *
+ apply safe
+ apply (metis add.assoc)
+ apply (rule_tac x=u in exI, force)
+ done
+qed
+
+lemma mem_affine:
+ assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
+ shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
+ using assms affine_def[of S] by auto
+
+lemma mem_affine_3:
+ assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
+ shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
+proof -
+ have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
+ using affine_hull_3[of x y z] assms by auto
+ moreover
+ have "affine hull {x, y, z} \<subseteq> affine hull S"
+ using hull_mono[of "{x, y, z}" "S"] assms by auto
+ moreover
+ have "affine hull S = S"
+ using assms affine_hull_eq[of S] by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma mem_affine_3_minus:
+ assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
+ shows "x + v *\<^sub>R (y-z) \<in> S"
+ using mem_affine_3[of S x y z 1 v "-v"] assms
+ by (simp add: algebra_simps)
+
+corollary mem_affine_3_minus2:
+ "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
+ by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
+
+
+subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close>
+
+lemma affine_hull_insert_subset_span:
+ "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
+proof -
+ have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
+ if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
+ for x F u
+ proof -
+ have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
+ using that by auto
+ show ?thesis
+ proof (intro exI conjI)
+ show "finite ((\<lambda>x. x - a) ` (F - {a}))"
+ by (simp add: that(1))
+ show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
+ by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
+ sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
+ qed (use \<open>F \<subseteq> insert a S\<close> in auto)
+ qed
+ then show ?thesis
+ unfolding affine_hull_explicit span_explicit by blast
+qed
+
+lemma affine_hull_insert_span:
+ assumes "a \<notin> S"
+ shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}"
+proof -
+ have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
+ if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
+ proof -
+ from that
+ obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
+ unfolding span_explicit by auto
+ define F where "F = (\<lambda>x. x + a) ` T"
+ have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
+ unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
+ have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
+ using F assms by auto
+ show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
+ apply (rule_tac x = "insert a F" in exI)
+ apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
+ using assms F
+ apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
+ done
+ qed
+ show ?thesis
+ by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
+qed
+
+lemma affine_hull_span:
+ assumes "a \<in> S"
+ shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
+ using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
+
+
+subsubsection%unimportant \<open>Parallel affine sets\<close>
+
+definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
+ where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
+
+lemma affine_parallel_expl_aux:
+ fixes S T :: "'a::real_vector set"
+ assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
+ shows "T = (\<lambda>x. a + x) ` S"
+proof -
+ have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
+ using that
+ by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
+ moreover have "T \<ge> (\<lambda>x. a + x) ` S"
+ using assms by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
+ unfolding affine_parallel_def
+ using affine_parallel_expl_aux[of S _ T] by auto
+
+lemma affine_parallel_reflex: "affine_parallel S S"
+ unfolding affine_parallel_def
+ using image_add_0 by blast
+
+lemma affine_parallel_commut:
+ assumes "affine_parallel A B"
+ shows "affine_parallel B A"
+proof -
+ from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
+ unfolding affine_parallel_def by auto
+ have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
+ from B show ?thesis
+ using translation_galois [of B a A]
+ unfolding affine_parallel_def by auto
+qed
+
+lemma affine_parallel_assoc:
+ assumes "affine_parallel A B"
+ and "affine_parallel B C"
+ shows "affine_parallel A C"
+proof -
+ from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
+ unfolding affine_parallel_def by auto
+ moreover
+ from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
+ unfolding affine_parallel_def by auto
+ ultimately show ?thesis
+ using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
+qed
+
+lemma affine_translation_aux:
+ fixes a :: "'a::real_vector"
+ assumes "affine ((\<lambda>x. a + x) ` S)"
+ shows "affine S"
+proof -
+ {
+ fix x y u v
+ assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
+ then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
+ by auto
+ then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
+ using xy assms unfolding affine_def by auto
+ have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
+ by (simp add: algebra_simps)
+ also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
+ using \<open>u + v = 1\<close> by auto
+ ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
+ using h1 by auto
+ then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
+ }
+ then show ?thesis unfolding affine_def by auto
+qed
+
+lemma affine_translation:
+ fixes a :: "'a::real_vector"
+ shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
+proof -
+ have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
+ using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
+ using translation_assoc[of "-a" a S] by auto
+ then show ?thesis using affine_translation_aux by auto
+qed
+
+lemma parallel_is_affine:
+ fixes S T :: "'a::real_vector set"
+ assumes "affine S" "affine_parallel S T"
+ shows "affine T"
+proof -
+ from assms obtain a where "T = (\<lambda>x. a + x) ` S"
+ unfolding affine_parallel_def by auto
+ then show ?thesis
+ using affine_translation assms by auto
+qed
+
+lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
+ unfolding subspace_def affine_def by auto
+
+
+subsubsection%unimportant \<open>Subspace parallel to an affine set\<close>
+
+lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
+proof -
+ have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
+ using subspace_imp_affine[of S] subspace_0 by auto
+ {
+ assume assm: "affine S \<and> 0 \<in> S"
+ {
+ fix c :: real
+ fix x
+ assume x: "x \<in> S"
+ have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
+ moreover
+ have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
+ using affine_alt[of S] assm x by auto
+ ultimately have "c *\<^sub>R x \<in> S" by auto
+ }
+ then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
+
+ {
+ fix x y
+ assume xy: "x \<in> S" "y \<in> S"
+ define u where "u = (1 :: real)/2"
+ have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
+ by auto
+ moreover
+ have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
+ by (simp add: algebra_simps)
+ moreover
+ have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
+ using affine_alt[of S] assm xy by auto
+ ultimately
+ have "(1/2) *\<^sub>R (x+y) \<in> S"
+ using u_def by auto
+ moreover
+ have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
+ by auto
+ ultimately
+ have "x + y \<in> S"
+ using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
+ }
+ then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
+ by auto
+ then have "subspace S"
+ using h1 assm unfolding subspace_def by auto
+ }
+ then show ?thesis using h0 by metis
+qed
+
+lemma affine_diffs_subspace:
+ assumes "affine S" "a \<in> S"
+ shows "subspace ((\<lambda>x. (-a)+x) ` S)"
+proof -
+ have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
+ have "affine ((\<lambda>x. (-a)+x) ` S)"
+ using affine_translation assms by auto
+ moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
+ using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
+ ultimately show ?thesis using subspace_affine by auto
+qed
+
+lemma parallel_subspace_explicit:
+ assumes "affine S"
+ and "a \<in> S"
+ assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
+ shows "subspace L \<and> affine_parallel S L"
+proof -
+ from assms have "L = plus (- a) ` S" by auto
+ then have par: "affine_parallel S L"
+ unfolding affine_parallel_def ..
+ then have "affine L" using assms parallel_is_affine by auto
+ moreover have "0 \<in> L"
+ using assms by auto
+ ultimately show ?thesis
+ using subspace_affine par by auto
+qed
+
+lemma parallel_subspace_aux:
+ assumes "subspace A"
+ and "subspace B"
+ and "affine_parallel A B"
+ shows "A \<supseteq> B"
+proof -
+ from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
+ using affine_parallel_expl[of A B] by auto
+ then have "-a \<in> A"
+ using assms subspace_0[of B] by auto
+ then have "a \<in> A"
+ using assms subspace_neg[of A "-a"] by auto
+ then show ?thesis
+ using assms a unfolding subspace_def by auto
+qed
+
+lemma parallel_subspace:
+ assumes "subspace A"
+ and "subspace B"
+ and "affine_parallel A B"
+ shows "A = B"
+proof
+ show "A \<supseteq> B"
+ using assms parallel_subspace_aux by auto
+ show "A \<subseteq> B"
+ using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
+qed
+
+lemma affine_parallel_subspace:
+ assumes "affine S" "S \<noteq> {}"
+ shows "\<exists>!L. subspace L \<and> affine_parallel S L"
+proof -
+ have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
+ using assms parallel_subspace_explicit by auto
+ {
+ fix L1 L2
+ assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
+ then have "affine_parallel L1 L2"
+ using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
+ then have "L1 = L2"
+ using ass parallel_subspace by auto
+ }
+ then show ?thesis using ex by auto
+qed
+
+
+subsection \<open>Cones\<close>
+
+definition%important cone :: "'a::real_vector set \<Rightarrow> bool"
+ where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
+
+lemma cone_empty[intro, simp]: "cone {}"
+ unfolding cone_def by auto
+
+lemma cone_univ[intro, simp]: "cone UNIV"
+ unfolding cone_def by auto
+
+lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
+ unfolding cone_def by auto
+
+lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
+ by (simp add: cone_def subspace_scale)
+
+
+subsubsection \<open>Conic hull\<close>
+
+lemma cone_cone_hull: "cone (cone hull s)"
+ unfolding hull_def by auto
+
+lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
+ apply (rule hull_eq)
+ using cone_Inter
+ unfolding subset_eq
+ apply auto
+ done
+
+lemma mem_cone:
+ assumes "cone S" "x \<in> S" "c \<ge> 0"
+ shows "c *\<^sub>R x \<in> S"
+ using assms cone_def[of S] by auto
+
+lemma cone_contains_0:
+ assumes "cone S"
+ shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
+proof -
+ {
+ assume "S \<noteq> {}"
+ then obtain a where "a \<in> S" by auto
+ then have "0 \<in> S"
+ using assms mem_cone[of S a 0] by auto
+ }
+ then show ?thesis by auto
+qed
+
+lemma cone_0: "cone {0}"
+ unfolding cone_def by auto
+
+lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
+ unfolding cone_def by blast
+
+lemma cone_iff:
+ assumes "S \<noteq> {}"
+ shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
+proof -
+ {
+ assume "cone S"
+ {
+ fix c :: real
+ assume "c > 0"
+ {
+ fix x
+ assume "x \<in> S"
+ then have "x \<in> ((*\<^sub>R) c) ` S"
+ unfolding image_def
+ using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
+ exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
+ by auto
+ }
+ moreover
+ {
+ fix x
+ assume "x \<in> ((*\<^sub>R) c) ` S"
+ then have "x \<in> S"
+ using \<open>cone S\<close> \<open>c > 0\<close>
+ unfolding cone_def image_def \<open>c > 0\<close> by auto
+ }
+ ultimately have "((*\<^sub>R) c) ` S = S" by auto
+ }
+ then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
+ using \<open>cone S\<close> cone_contains_0[of S] assms by auto
+ }
+ moreover
+ {
+ assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
+ {
+ fix x
+ assume "x \<in> S"
+ fix c1 :: real
+ assume "c1 \<ge> 0"
+ then have "c1 = 0 \<or> c1 > 0" by auto
+ then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
+ }
+ then have "cone S" unfolding cone_def by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma cone_hull_empty: "cone hull {} = {}"
+ by (metis cone_empty cone_hull_eq)
+
+lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
+ by (metis bot_least cone_hull_empty hull_subset xtrans(5))
+
+lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
+ using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
+ by auto
+
+lemma mem_cone_hull:
+ assumes "x \<in> S" "c \<ge> 0"
+ shows "c *\<^sub>R x \<in> cone hull S"
+ by (metis assms cone_cone_hull hull_inc mem_cone)
+
+proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
+ (is "?lhs = ?rhs")
+proof -
+ {
+ fix x
+ assume "x \<in> ?rhs"
+ then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
+ by auto
+ fix c :: real
+ assume c: "c \<ge> 0"
+ then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
+ using x by (simp add: algebra_simps)
+ moreover
+ have "c * cx \<ge> 0" using c x by auto
+ ultimately
+ have "c *\<^sub>R x \<in> ?rhs" using x by auto
+ }
+ then have "cone ?rhs"
+ unfolding cone_def by auto
+ then have "?rhs \<in> Collect cone"
+ unfolding mem_Collect_eq by auto
+ {
+ fix x
+ assume "x \<in> S"
+ then have "1 *\<^sub>R x \<in> ?rhs"
+ apply auto
+ apply (rule_tac x = 1 in exI, auto)
+ done
+ then have "x \<in> ?rhs" by auto
+ }
+ then have "S \<subseteq> ?rhs" by auto
+ then have "?lhs \<subseteq> ?rhs"
+ using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
+ moreover
+ {
+ fix x
+ assume "x \<in> ?rhs"
+ then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
+ by auto
+ then have "xx \<in> cone hull S"
+ using hull_subset[of S] by auto
+ then have "x \<in> ?lhs"
+ using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+
+subsection \<open>Affine dependence and consequential theorems\<close>
+
+text "Formalized by Lars Schewe."
+
+definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
+ where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
+
+lemma affine_dependent_subset:
+ "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
+apply (simp add: affine_dependent_def Bex_def)
+apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
+done
+
+lemma affine_independent_subset:
+ shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
+by (metis affine_dependent_subset)
+
+lemma affine_independent_Diff:
+ "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
+by (meson Diff_subset affine_dependent_subset)
+
+proposition affine_dependent_explicit:
+ "affine_dependent p \<longleftrightarrow>
+ (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
+proof -
+ have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
+ if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
+ proof (intro exI conjI)
+ have "x \<notin> S"
+ using that by auto
+ then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
+ using that by (simp add: sum_delta_notmem)
+ show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
+ using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
+ qed (use that in auto)
+ moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
+ if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
+ proof (intro bexI exI conjI)
+ have "S \<noteq> {v}"
+ using that by auto
+ then show "S - {v} \<noteq> {}"
+ using that by auto
+ show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
+ unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
+ show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
+ unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
+ scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
+ using that by auto
+ show "S - {v} \<subseteq> p - {v}"
+ using that by auto
+ qed (use that in auto)
+ ultimately show ?thesis
+ unfolding affine_dependent_def affine_hull_explicit by auto
+qed
+
+lemma affine_dependent_explicit_finite:
+ fixes S :: "'a::real_vector set"
+ assumes "finite S"
+ shows "affine_dependent S \<longleftrightarrow>
+ (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
+ (is "?lhs = ?rhs")
+proof
+ have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
+ by auto
+ assume ?lhs
+ then obtain t u v where
+ "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
+ unfolding affine_dependent_explicit by auto
+ then show ?rhs
+ apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
+ apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
+ done
+next
+ assume ?rhs
+ then obtain u v where "sum u S = 0" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
+ by auto
+ then show ?lhs unfolding affine_dependent_explicit
+ using assms by auto
+qed
+
+
+subsection%unimportant \<open>Connectedness of convex sets\<close>
+
+lemma connectedD:
+ "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
+ by (rule Topological_Spaces.topological_space_class.connectedD)
+
+lemma convex_connected:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "convex S"
+ shows "connected S"
+proof (rule connectedI)
+ fix A B
+ assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
+ moreover
+ assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
+ then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
+ define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
+ then have "continuous_on {0 .. 1} f"
+ by (auto intro!: continuous_intros)
+ then have "connected (f ` {0 .. 1})"
+ by (auto intro!: connected_continuous_image)
+ note connectedD[OF this, of A B]
+ moreover have "a \<in> A \<inter> f ` {0 .. 1}"
+ using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
+ moreover have "b \<in> B \<inter> f ` {0 .. 1}"
+ using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
+ moreover have "f ` {0 .. 1} \<subseteq> S"
+ using \<open>convex S\<close> a b unfolding convex_def f_def by auto
+ ultimately show False by auto
+qed
+
+corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
+ by (simp add: convex_connected)
+
+lemma convex_prod:
+ assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
+ shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
+ using assms unfolding convex_def
+ by (auto simp: inner_add_left)
+
+lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
+ by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
+
+subsection \<open>Convex hull\<close>
+
+lemma convex_convex_hull [iff]: "convex (convex hull s)"
+ unfolding hull_def
+ using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
+ by auto
+
+lemma convex_hull_subset:
+ "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
+ by (simp add: convex_convex_hull subset_hull)
+
+lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
+ by (metis convex_convex_hull hull_same)
+
+subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close>
+
+lemma convex_hull_linear_image:
+ assumes f: "linear f"
+ shows "f ` (convex hull s) = convex hull (f ` s)"
+proof
+ show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
+ by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
+ show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
+ proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
+ show "s \<subseteq> f -` (convex hull (f ` s))"
+ by (fast intro: hull_inc)
+ show "convex (f -` (convex hull (f ` s)))"
+ by (intro convex_linear_vimage [OF f] convex_convex_hull)
+ qed
+qed
+
+lemma in_convex_hull_linear_image:
+ assumes "linear f"
+ and "x \<in> convex hull s"
+ shows "f x \<in> convex hull (f ` s)"
+ using convex_hull_linear_image[OF assms(1)] assms(2) by auto
+
+lemma convex_hull_Times:
+ "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
+proof
+ show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
+ by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
+ have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
+ proof (rule hull_induct [OF x], rule hull_induct [OF y])
+ fix x y assume "x \<in> s" and "y \<in> t"
+ then show "(x, y) \<in> convex hull (s \<times> t)"
+ by (simp add: hull_inc)
+ next
+ fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
+ have "convex ?S"
+ by (intro convex_linear_vimage convex_translation convex_convex_hull,
+ simp add: linear_iff)
+ also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
+ by (auto simp: image_def Bex_def)
+ finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
+ next
+ show "convex {x. (x, y) \<in> convex hull s \<times> t}"
+ proof -
+ fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
+ have "convex ?S"
+ by (intro convex_linear_vimage convex_translation convex_convex_hull,
+ simp add: linear_iff)
+ also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
+ by (auto simp: image_def Bex_def)
+ finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
+ qed
+ qed
+ then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
+ unfolding subset_eq split_paired_Ball_Sigma by blast
+qed
+
+
+subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close>
+
+lemma convex_hull_empty[simp]: "convex hull {} = {}"
+ by (rule hull_unique) auto
+
+lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
+ by (rule hull_unique) auto
+
+lemma convex_hull_insert:
+ fixes S :: "'a::real_vector set"
+ assumes "S \<noteq> {}"
+ shows "convex hull (insert a S) =
+ {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
+ (is "_ = ?hull")
+proof (intro equalityI hull_minimal subsetI)
+ fix x
+ assume "x \<in> insert a S"
+ then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
+ unfolding insert_iff
+ proof
+ assume "x = a"
+ then show ?thesis
+ by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
+ next
+ assume "x \<in> S"
+ with hull_subset[of S convex] show ?thesis
+ by force
+ qed
+ then show "x \<in> ?hull"
+ by simp
+next
+ fix x
+ assume "x \<in> ?hull"
+ then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
+ by auto
+ have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
+ using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
+ by auto
+ then show "x \<in> convex hull insert a S"
+ unfolding obt(5) using obt(1-3)
+ by (rule convexD [OF convex_convex_hull])
+next
+ show "convex ?hull"
+ proof (rule convexI)
+ fix x y u v
+ assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
+ from x obtain u1 v1 b1 where
+ obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
+ by auto
+ from y obtain u2 v2 b2 where
+ obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
+ by auto
+ have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
+ by (auto simp: algebra_simps)
+ have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
+ (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
+ proof (cases "u * v1 + v * v2 = 0")
+ case True
+ have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
+ by (auto simp: algebra_simps)
+ have eq0: "u * v1 = 0" "v * v2 = 0"
+ using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
+ by arith+
+ then have "u * u1 + v * u2 = 1"
+ using as(3) obt1(3) obt2(3) by auto
+ then show ?thesis
+ using "*" eq0 as obt1(4) xeq yeq by auto
+ next
+ case False
+ have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
+ using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
+ also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
+ using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
+ also have "\<dots> = u * v1 + v * v2"
+ by simp
+ finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
+ let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
+ have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
+ using as(1,2) obt1(1,2) obt2(1,2) by auto
+ show ?thesis
+ proof
+ show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
+ unfolding xeq yeq * **
+ using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
+ show "?b \<in> convex hull S"
+ using False zeroes obt1(4) obt2(4)
+ by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib add_divide_distrib[symmetric] zero_le_divide_iff)
+ qed
+ qed
+ then obtain b where b: "b \<in> convex hull S"
+ "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
+
+ have u1: "u1 \<le> 1"
+ unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
+ have u2: "u2 \<le> 1"
+ unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
+ have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
+ proof (rule add_mono)
+ show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
+ by (simp_all add: as mult_right_mono)
+ qed
+ also have "\<dots> \<le> 1"
+ unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
+ finally have le1: "u1 * u + u2 * v \<le> 1" .
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
+ proof (intro CollectI exI conjI)
+ show "0 \<le> u * u1 + v * u2"
+ by (simp add: as(1) as(2) obt1(1) obt2(1))
+ show "0 \<le> 1 - u * u1 - v * u2"
+ by (simp add: le1 diff_diff_add mult.commute)
+ qed (use b in \<open>auto simp: algebra_simps\<close>)
+ qed
+qed
+
+lemma convex_hull_insert_alt:
+ "convex hull (insert a S) =
+ (if S = {} then {a}
+ else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
+ apply (auto simp: convex_hull_insert)
+ using diff_eq_eq apply fastforce
+ by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
+
+subsubsection%unimportant \<open>Explicit expression for convex hull\<close>
+
+proposition convex_hull_indexed:
+ fixes S :: "'a::real_vector set"
+ shows "convex hull S =
+ {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
+ (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
+ (is "?xyz = ?hull")
+proof (rule hull_unique [OF _ convexI])
+ show "S \<subseteq> ?hull"
+ by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
+next
+ fix T
+ assume "S \<subseteq> T" "convex T"
+ then show "?hull \<subseteq> T"
+ by (blast intro: convex_sum)
+next
+ fix x y u v
+ assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
+ assume xy: "x \<in> ?hull" "y \<in> ?hull"
+ from xy obtain k1 u1 x1 where
+ x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
+ "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
+ by auto
+ from xy obtain k2 u2 x2 where
+ y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
+ "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
+ by auto
+ have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
+ "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
+ by auto
+ have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
+ unfolding inj_on_def by auto
+ let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
+ let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
+ show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
+ proof (intro CollectI exI conjI ballI)
+ show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
+ using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
+ show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1" "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
+ unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
+ sum.reindex[OF inj] Collect_mem_eq o_def
+ unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
+ by (simp_all add: sum_distrib_left[symmetric] x(2,3) y(2,3) uv(3))
+ qed
+qed
+
+lemma convex_hull_finite:
+ fixes S :: "'a::real_vector set"
+ assumes "finite S"
+ shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
+ (is "?HULL = _")
+proof (rule hull_unique [OF _ convexI]; clarify)
+ fix x
+ assume "x \<in> S"
+ then show "\<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) = x"
+ by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
+next
+ fix u v :: real
+ assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
+ fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
+ fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
+ have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
+ by (simp add: that uv ux(1) uy(1))
+ moreover
+ have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
+ unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
+ using uv(3) by auto
+ moreover
+ have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
+ unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
+ by auto
+ ultimately
+ show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
+ (\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
+ by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
+qed (use assms in \<open>auto simp: convex_explicit\<close>)
+
+
+subsubsection%unimportant \<open>Another formulation\<close>
+
+text "Formalized by Lars Schewe."
+
+lemma convex_hull_explicit:
+ fixes p :: "'a::real_vector set"
+ shows "convex hull p =
+ {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
+ (is "?lhs = ?rhs")
+proof -
+ {
+ fix x
+ assume "x\<in>?lhs"
+ then obtain k u y where
+ obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
+ unfolding convex_hull_indexed by auto
+
+ have fin: "finite {1..k}" by auto
+ have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
+ {
+ fix j
+ assume "j\<in>{1..k}"
+ then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
+ using obt(1)[THEN bspec[where x=j]] and obt(2)
+ apply simp
+ apply (rule sum_nonneg)
+ using obt(1)
+ apply auto
+ done
+ }
+ moreover
+ have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
+ unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
+ moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
+ using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
+ unfolding scaleR_left.sum using obt(3) by auto
+ ultimately
+ have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
+ apply (rule_tac x="y ` {1..k}" in exI)
+ apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
+ done
+ then have "x\<in>?rhs" by auto
+ }
+ moreover
+ {
+ fix y
+ assume "y\<in>?rhs"
+ then obtain S u where
+ obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
+ by auto
+
+ obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
+ using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
+
+ {
+ fix i :: nat
+ assume "i\<in>{1..card S}"
+ then have "f i \<in> S"
+ using f(2) by blast
+ then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
+ }
+ moreover have *: "finite {1..card S}" by auto
+ {
+ fix y
+ assume "y\<in>S"
+ then obtain i where "i\<in>{1..card S}" "f i = y"
+ using f using image_iff[of y f "{1..card S}"]
+ by auto
+ then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
+ apply auto
+ using f(1)[unfolded inj_on_def]
+ by (metis One_nat_def atLeastAtMost_iff)
+ then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
+ then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
+ "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
+ by (auto simp: sum_constant_scaleR)
+ }
+ then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
+ unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
+ and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
+ unfolding f
+ using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
+ using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
+ unfolding obt(4,5)
+ by auto
+ ultimately
+ have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
+ (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
+ apply (rule_tac x="card S" in exI)
+ apply (rule_tac x="u \<circ> f" in exI)
+ apply (rule_tac x=f in exI, fastforce)
+ done
+ then have "y \<in> ?lhs"
+ unfolding convex_hull_indexed by auto
+ }
+ ultimately show ?thesis
+ unfolding set_eq_iff by blast
+qed
+
+
+subsubsection%unimportant \<open>A stepping theorem for that expansion\<close>
+
+lemma convex_hull_finite_step:
+ fixes S :: "'a::real_vector set"
+ assumes "finite S"
+ shows
+ "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
+ \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
+ (is "?lhs = ?rhs")
+proof (rule, case_tac[!] "a\<in>S")
+ assume "a \<in> S"
+ then have *: "insert a S = S" by auto
+ assume ?lhs
+ then show ?rhs
+ unfolding * by (rule_tac x=0 in exI, auto)
+next
+ assume ?lhs
+ then obtain u where
+ u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
+ by auto
+ assume "a \<notin> S"
+ then show ?rhs
+ apply (rule_tac x="u a" in exI)
+ using u(1)[THEN bspec[where x=a]]
+ apply simp
+ apply (rule_tac x=u in exI)
+ using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
+ apply auto
+ done
+next
+ assume "a \<in> S"
+ then have *: "insert a S = S" by auto
+ have fin: "finite (insert a S)" using assms by auto
+ assume ?rhs
+ then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+ by auto
+ show ?lhs
+ apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
+ unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
+ unfolding sum_clauses(2)[OF assms]
+ using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
+ apply auto
+ done
+next
+ assume ?rhs
+ then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+ by auto
+ moreover assume "a \<notin> S"
+ moreover
+ have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S" "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
+ using \<open>a \<notin> S\<close>
+ by (auto simp: intro!: sum.cong)
+ ultimately show ?lhs
+ by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
+qed
+
+
+subsubsection%unimportant \<open>Hence some special cases\<close>
+
+lemma convex_hull_2:
+ "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
+proof -
+ have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
+ by auto
+ have **: "finite {b}" by auto
+ show ?thesis
+ apply (simp add: convex_hull_finite)
+ unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
+ apply auto
+ apply (rule_tac x=v in exI)
+ apply (rule_tac x="1 - v" in exI, simp)
+ apply (rule_tac x=u in exI, simp)
+ apply (rule_tac x="\<lambda>x. v" in exI, simp)
+ done
+qed
+
+lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
+ unfolding convex_hull_2
+proof (rule Collect_cong)
+ have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
+ by auto
+ fix x
+ show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
+ (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
+ unfolding *
+ apply auto
+ apply (rule_tac[!] x=u in exI)
+ apply (auto simp: algebra_simps)
+ done
+qed
+
+lemma convex_hull_3:
+ "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
+proof -
+ have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
+ by auto
+ have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
+ by (auto simp: field_simps)
+ show ?thesis
+ unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
+ unfolding convex_hull_finite_step[OF fin(3)]
+ apply (rule Collect_cong, simp)
+ apply auto
+ apply (rule_tac x=va in exI)
+ apply (rule_tac x="u c" in exI, simp)
+ apply (rule_tac x="1 - v - w" in exI, simp)
+ apply (rule_tac x=v in exI, simp)
+ apply (rule_tac x="\<lambda>x. w" in exI, simp)
+ done
+qed
+
+lemma convex_hull_3_alt:
+ "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
+proof -
+ have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
+ by auto
+ show ?thesis
+ unfolding convex_hull_3
+ apply (auto simp: *)
+ apply (rule_tac x=v in exI)
+ apply (rule_tac x=w in exI)
+ apply (simp add: algebra_simps)
+ apply (rule_tac x=u in exI)
+ apply (rule_tac x=v in exI)
+ apply (simp add: algebra_simps)
+ done
+qed
+
+
+subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close>
+
+lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
+ unfolding affine_def convex_def by auto
+
+lemma convex_affine_hull [simp]: "convex (affine hull S)"
+ by (simp add: affine_imp_convex)
+
+lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
+ using subspace_imp_affine affine_imp_convex by auto
+
+lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
+ by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
+
+lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
+ by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
+
+lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
+ by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
+
+lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
+ unfolding affine_dependent_def dependent_def
+ using affine_hull_subset_span by auto
+
+lemma dependent_imp_affine_dependent:
+ assumes "dependent {x - a| x . x \<in> s}"
+ and "a \<notin> s"
+ shows "affine_dependent (insert a s)"
+proof -
+ from assms(1)[unfolded dependent_explicit] obtain S u v
+ where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
+ by auto
+ define t where "t = (\<lambda>x. x + a) ` S"
+
+ have inj: "inj_on (\<lambda>x. x + a) S"
+ unfolding inj_on_def by auto
+ have "0 \<notin> S"
+ using obt(2) assms(2) unfolding subset_eq by auto
+ have fin: "finite t" and "t \<subseteq> s"
+ unfolding t_def using obt(1,2) by auto
+ then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
+ by auto
+ moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
+ apply (rule sum.cong)
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
+ apply auto
+ done
+ have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
+ unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
+ moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
+ using obt(3,4) \<open>0\<notin>S\<close>
+ by (rule_tac x="v + a" in bexI) (auto simp: t_def)
+ moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
+ have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
+ unfolding scaleR_left.sum
+ unfolding t_def and sum.reindex[OF inj] and o_def
+ using obt(5)
+ by (auto simp: sum.distrib scaleR_right_distrib)
+ then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
+ unfolding sum_clauses(2)[OF fin]
+ using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
+ by (auto simp: *)
+ ultimately show ?thesis
+ unfolding affine_dependent_explicit
+ apply (rule_tac x="insert a t" in exI, auto)
+ done
+qed
+
+lemma convex_cone:
+ "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
+ (is "?lhs = ?rhs")
+proof -
+ {
+ fix x y
+ assume "x\<in>s" "y\<in>s" and ?lhs
+ then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
+ unfolding cone_def by auto
+ then have "x + y \<in> s"
+ using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
+ apply (erule_tac x="2*\<^sub>R x" in ballE)
+ apply (erule_tac x="2*\<^sub>R y" in ballE)
+ apply (erule_tac x="1/2" in allE, simp)
+ apply (erule_tac x="1/2" in allE, auto)
+ done
+ }
+ then show ?thesis
+ unfolding convex_def cone_def by blast
+qed
+
+lemma affine_dependent_biggerset:
+ fixes s :: "'a::euclidean_space set"
+ assumes "finite s" "card s \<ge> DIM('a) + 2"
+ shows "affine_dependent s"
+proof -
+ have "s \<noteq> {}" using assms by auto
+ then obtain a where "a\<in>s" by auto
+ have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
+ by auto
+ have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
+ unfolding * by (simp add: card_image inj_on_def)
+ also have "\<dots> > DIM('a)" using assms(2)
+ unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
+ finally show ?thesis
+ apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
+ apply (rule dependent_imp_affine_dependent)
+ apply (rule dependent_biggerset, auto)
+ done
+qed
+
+lemma affine_dependent_biggerset_general:
+ assumes "finite (S :: 'a::euclidean_space set)"
+ and "card S \<ge> dim S + 2"
+ shows "affine_dependent S"
+proof -
+ from assms(2) have "S \<noteq> {}" by auto
+ then obtain a where "a\<in>S" by auto
+ have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
+ by auto
+ have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
+ by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
+ have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
+ using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
+ also have "\<dots> < dim S + 1" by auto
+ also have "\<dots> \<le> card (S - {a})"
+ using assms
+ using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
+ by auto
+ finally show ?thesis
+ apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
+ apply (rule dependent_imp_affine_dependent)
+ apply (rule dependent_biggerset_general)
+ unfolding **
+ apply auto
+ done
+qed
+
+
+subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close>
+
+lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
+ by (simp add: affine_dependent_def)
+
+lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
+ by (simp add: affine_dependent_def)
+
+lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
+ by (simp add: affine_dependent_def insert_Diff_if hull_same)
+
+lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)"
+proof -
+ have "affine ((\<lambda>x. a + x) ` (affine hull S))"
+ using affine_translation affine_affine_hull by blast
+ moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
+ using hull_subset[of S] by auto
+ ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
+ by (metis hull_minimal)
+ have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))"
+ using affine_translation affine_affine_hull by blast
+ moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))"
+ using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto
+ moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S"
+ using translation_assoc[of "-a" a] by auto
+ ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)"
+ by (metis hull_minimal)
+ then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
+ by auto
+ then show ?thesis using h1 by auto
+qed
+
+lemma affine_dependent_translation:
+ assumes "affine_dependent S"
+ shows "affine_dependent ((\<lambda>x. a + x) ` S)"
+proof -
+ obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
+ using assms affine_dependent_def by auto
+ have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
+ by auto
+ then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
+ using affine_hull_translation[of a "S - {x}"] x by auto
+ moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
+ using x by auto
+ ultimately show ?thesis
+ unfolding affine_dependent_def by auto
+qed
+
+lemma affine_dependent_translation_eq:
+ "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
+proof -
+ {
+ assume "affine_dependent ((\<lambda>x. a + x) ` S)"
+ then have "affine_dependent S"
+ using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
+ by auto
+ }
+ then show ?thesis
+ using affine_dependent_translation by auto
+qed
+
+lemma affine_hull_0_dependent:
+ assumes "0 \<in> affine hull S"
+ shows "dependent S"
+proof -
+ obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+ using assms affine_hull_explicit[of S] by auto
+ then have "\<exists>v\<in>s. u v \<noteq> 0"
+ using sum_not_0[of "u" "s"] by auto
+ then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
+ using s_u by auto
+ then show ?thesis
+ unfolding dependent_explicit[of S] by auto
+qed
+
+lemma affine_dependent_imp_dependent2:
+ assumes "affine_dependent (insert 0 S)"
+ shows "dependent S"
+proof -
+ obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
+ using affine_dependent_def[of "(insert 0 S)"] assms by blast
+ then have "x \<in> span (insert 0 S - {x})"
+ using affine_hull_subset_span by auto
+ moreover have "span (insert 0 S - {x}) = span (S - {x})"
+ using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
+ ultimately have "x \<in> span (S - {x})" by auto
+ then have "x \<noteq> 0 \<Longrightarrow> dependent S"
+ using x dependent_def by auto
+ moreover
+ {
+ assume "x = 0"
+ then have "0 \<in> affine hull S"
+ using x hull_mono[of "S - {0}" S] by auto
+ then have "dependent S"
+ using affine_hull_0_dependent by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_dependent_iff_dependent:
+ assumes "a \<notin> S"
+ shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
+proof -
+ have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
+ then show ?thesis
+ using affine_dependent_translation_eq[of "(insert a S)" "-a"]
+ affine_dependent_imp_dependent2 assms
+ dependent_imp_affine_dependent[of a S]
+ by (auto simp del: uminus_add_conv_diff)
+qed
+
+lemma affine_dependent_iff_dependent2:
+ assumes "a \<in> S"
+ shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
+proof -
+ have "insert a (S - {a}) = S"
+ using assms by auto
+ then show ?thesis
+ using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
+qed
+
+lemma affine_hull_insert_span_gen:
+ "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
+proof -
+ have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
+ by auto
+ {
+ assume "a \<notin> s"
+ then have ?thesis
+ using affine_hull_insert_span[of a s] h1 by auto
+ }
+ moreover
+ {
+ assume a1: "a \<in> s"
+ have "\<exists>x. x \<in> s \<and> -a+x=0"
+ apply (rule exI[of _ a])
+ using a1
+ apply auto
+ done
+ then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
+ by auto
+ then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
+ using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
+ moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
+ by auto
+ moreover have "insert a (s - {a}) = insert a s"
+ by auto
+ ultimately have ?thesis
+ using affine_hull_insert_span[of "a" "s-{a}"] by auto
+ }
+ ultimately show ?thesis by auto
+qed
+
+lemma affine_hull_span2:
+ assumes "a \<in> s"
+ shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
+ using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
+ by auto
+
+lemma affine_hull_span_gen:
+ assumes "a \<in> affine hull s"
+ shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
+proof -
+ have "affine hull (insert a s) = affine hull s"
+ using hull_redundant[of a affine s] assms by auto
+ then show ?thesis
+ using affine_hull_insert_span_gen[of a "s"] by auto
+qed
+
+lemma affine_hull_span_0:
+ assumes "0 \<in> affine hull S"
+ shows "affine hull S = span S"
+ using affine_hull_span_gen[of "0" S] assms by auto
+
+lemma extend_to_affine_basis_nonempty:
+ fixes S V :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
+ shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
+proof -
+ obtain a where a: "a \<in> S"
+ using assms by auto
+ then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))"
+ using affine_dependent_iff_dependent2 assms by auto
+ obtain B where B:
+ "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
+ using assms
+ by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
+ define T where "T = (\<lambda>x. a+x) ` insert 0 B"
+ then have "T = insert a ((\<lambda>x. a+x) ` B)"
+ by auto
+ then have "affine hull T = (\<lambda>x. a+x) ` span B"
+ using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
+ by auto
+ then have "V \<subseteq> affine hull T"
+ using B assms translation_inverse_subset[of a V "span B"]
+ by auto
+ moreover have "T \<subseteq> V"
+ using T_def B a assms by auto
+ ultimately have "affine hull T = affine hull V"
+ by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
+ moreover have "S \<subseteq> T"
+ using T_def B translation_inverse_subset[of a "S-{a}" B]
+ by auto
+ moreover have "\<not> affine_dependent T"
+ using T_def affine_dependent_translation_eq[of "insert 0 B"]
+ affine_dependent_imp_dependent2 B
+ by auto
+ ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
+qed
+
+lemma affine_basis_exists:
+ fixes V :: "'n::euclidean_space set"
+ shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
+proof (cases "V = {}")
+ case True
+ then show ?thesis
+ using affine_independent_0 by auto
+next
+ case False
+ then obtain x where "x \<in> V" by auto
+ then show ?thesis
+ using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
+ by auto
+qed
+
+proposition extend_to_affine_basis:
+ fixes S V :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent S" "S \<subseteq> V"
+ obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
+proof (cases "S = {}")
+ case True then show ?thesis
+ using affine_basis_exists by (metis empty_subsetI that)
+next
+ case False
+ then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
+qed
+
+subsection \<open>Affine Dimension of a Set\<close>
+
+definition%important aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
+ where "aff_dim V =
+ (SOME d :: int.
+ \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
+
+lemma aff_dim_basis_exists:
+ fixes V :: "('n::euclidean_space) set"
+ shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
+proof -
+ obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
+ using affine_basis_exists[of V] by auto
+ then show ?thesis
+ unfolding aff_dim_def
+ some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
+ apply auto
+ apply (rule exI[of _ "int (card B) - (1 :: int)"])
+ apply (rule exI[of _ "B"], auto)
+ done
+qed
+
+lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
+proof -
+ have "S = {} \<Longrightarrow> affine hull S = {}"
+ using affine_hull_empty by auto
+ moreover have "affine hull S = {} \<Longrightarrow> S = {}"
+ unfolding hull_def by auto
+ ultimately show ?thesis by blast
+qed
+
+lemma aff_dim_parallel_subspace_aux:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B" "a \<in> B"
+ shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
+proof -
+ have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
+ using affine_dependent_iff_dependent2 assms by auto
+ then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
+ "finite ((\<lambda>x. -a + x) ` (B - {a}))"
+ using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
+ show ?thesis
+ proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
+ case True
+ have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
+ using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
+ then have "B = {a}" using True by auto
+ then show ?thesis using assms fin by auto
+ next
+ case False
+ then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
+ using fin by auto
+ moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
+ by (rule card_image) (use translate_inj_on in blast)
+ ultimately have "card (B-{a}) > 0" by auto
+ then have *: "finite (B - {a})"
+ using card_gt_0_iff[of "(B - {a})"] by auto
+ then have "card (B - {a}) = card B - 1"
+ using card_Diff_singleton assms by auto
+ with * show ?thesis using fin h1 by auto
+ qed
+qed
+
+lemma aff_dim_parallel_subspace:
+ fixes V L :: "'n::euclidean_space set"
+ assumes "V \<noteq> {}"
+ and "subspace L"
+ and "affine_parallel (affine hull V) L"
+ shows "aff_dim V = int (dim L)"
+proof -
+ obtain B where
+ B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
+ using aff_dim_basis_exists by auto
+ then have "B \<noteq> {}"
+ using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
+ by auto
+ then obtain a where a: "a \<in> B" by auto
+ define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
+ moreover have "affine_parallel (affine hull B) Lb"
+ using Lb_def B assms affine_hull_span2[of a B] a
+ affine_parallel_commut[of "Lb" "(affine hull B)"]
+ unfolding affine_parallel_def
+ by auto
+ moreover have "subspace Lb"
+ using Lb_def subspace_span by auto
+ moreover have "affine hull B \<noteq> {}"
+ using assms B affine_hull_nonempty[of V] by auto
+ ultimately have "L = Lb"
+ using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
+ by auto
+ then have "dim L = dim Lb"
+ by auto
+ moreover have "card B - 1 = dim Lb" and "finite B"
+ using Lb_def aff_dim_parallel_subspace_aux a B by auto
+ ultimately show ?thesis
+ using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
+qed
+
+lemma aff_independent_finite:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B"
+ shows "finite B"
+proof -
+ {
+ assume "B \<noteq> {}"
+ then obtain a where "a \<in> B" by auto
+ then have ?thesis
+ using aff_dim_parallel_subspace_aux assms by auto
+ }
+ then show ?thesis by auto
+qed
+
+lemmas independent_finite = independent_imp_finite
+
+lemma span_substd_basis:
+ assumes d: "d \<subseteq> Basis"
+ shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+ (is "_ = ?B")
+proof -
+ have "d \<subseteq> ?B"
+ using d by (auto simp: inner_Basis)
+ moreover have s: "subspace ?B"
+ using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
+ ultimately have "span d \<subseteq> ?B"
+ using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
+ moreover have *: "card d \<le> dim (span d)"
+ using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
+ span_superset[of d]
+ by auto
+ moreover from * have "dim ?B \<le> dim (span d)"
+ using dim_substandard[OF assms] by auto
+ ultimately show ?thesis
+ using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
+qed
+
+lemma basis_to_substdbasis_subspace_isomorphism:
+ fixes B :: "'a::euclidean_space set"
+ assumes "independent B"
+ shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
+ f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
+proof -
+ have B: "card B = dim B"
+ using dim_unique[of B B "card B"] assms span_superset[of B] by auto
+ have "dim B \<le> card (Basis :: 'a set)"
+ using dim_subset_UNIV[of B] by simp
+ from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
+ by auto
+ let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+ have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
+ proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
+ show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
+ using d inner_not_same_Basis by blast
+ qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
+ with t \<open>card B = dim B\<close> d show ?thesis by auto
+qed
+
+lemma aff_dim_empty:
+ fixes S :: "'n::euclidean_space set"
+ shows "S = {} \<longleftrightarrow> aff_dim S = -1"
+proof -
+ obtain B where *: "affine hull B = affine hull S"
+ and "\<not> affine_dependent B"
+ and "int (card B) = aff_dim S + 1"
+ using aff_dim_basis_exists by auto
+ moreover
+ from * have "S = {} \<longleftrightarrow> B = {}"
+ using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
+ ultimately show ?thesis
+ using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
+qed
+
+lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
+ by (simp add: aff_dim_empty [symmetric])
+
+lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
+ unfolding aff_dim_def using hull_hull[of _ S] by auto
+
+lemma aff_dim_affine_hull2:
+ assumes "affine hull S = affine hull T"
+ shows "aff_dim S = aff_dim T"
+ unfolding aff_dim_def using assms by auto
+
+lemma aff_dim_unique:
+ fixes B V :: "'n::euclidean_space set"
+ assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
+ shows "of_nat (card B) = aff_dim V + 1"
+proof (cases "B = {}")
+ case True
+ then have "V = {}"
+ using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
+ by auto
+ then have "aff_dim V = (-1::int)"
+ using aff_dim_empty by auto
+ then show ?thesis
+ using \<open>B = {}\<close> by auto
+next
+ case False
+ then obtain a where a: "a \<in> B" by auto
+ define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
+ have "affine_parallel (affine hull B) Lb"
+ using Lb_def affine_hull_span2[of a B] a
+ affine_parallel_commut[of "Lb" "(affine hull B)"]
+ unfolding affine_parallel_def by auto
+ moreover have "subspace Lb"
+ using Lb_def subspace_span by auto
+ ultimately have "aff_dim B = int(dim Lb)"
+ using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
+ moreover have "(card B) - 1 = dim Lb" "finite B"
+ using Lb_def aff_dim_parallel_subspace_aux a assms by auto
+ ultimately have "of_nat (card B) = aff_dim B + 1"
+ using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
+ then show ?thesis
+ using aff_dim_affine_hull2 assms by auto
+qed
+
+lemma aff_dim_affine_independent:
+ fixes B :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent B"
+ shows "of_nat (card B) = aff_dim B + 1"
+ using aff_dim_unique[of B B] assms by auto
+
+lemma affine_independent_iff_card:
+ fixes s :: "'a::euclidean_space set"
+ shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
+ apply (rule iffI)
+ apply (simp add: aff_dim_affine_independent aff_independent_finite)
+ by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
+
+lemma aff_dim_sing [simp]:
+ fixes a :: "'n::euclidean_space"
+ shows "aff_dim {a} = 0"
+ using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
+
+lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
+proof (clarsimp)
+ assume "a \<noteq> b"
+ then have "aff_dim{a,b} = card{a,b} - 1"
+ using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
+ also have "\<dots> = 1"
+ using \<open>a \<noteq> b\<close> by simp
+ finally show "aff_dim {a, b} = 1" .
+qed
+
+lemma aff_dim_inner_basis_exists:
+ fixes V :: "('n::euclidean_space) set"
+ shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
+ \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
+proof -
+ obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
+ using affine_basis_exists[of V] by auto
+ then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_le_card:
+ fixes V :: "'n::euclidean_space set"
+ assumes "finite V"
+ shows "aff_dim V \<le> of_nat (card V) - 1"
+proof -
+ obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
+ using aff_dim_inner_basis_exists[of V] by auto
+ then have "card B \<le> card V"
+ using assms card_mono by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_parallel_eq:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "affine_parallel (affine hull S) (affine hull T)"
+ shows "aff_dim S = aff_dim T"
+proof -
+ {
+ assume "T \<noteq> {}" "S \<noteq> {}"
+ then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
+ using affine_parallel_subspace[of "affine hull T"]
+ affine_affine_hull[of T] affine_hull_nonempty
+ by auto
+ then have "aff_dim T = int (dim L)"
+ using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
+ moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
+ using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
+ moreover from * have "aff_dim S = int (dim L)"
+ using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
+ ultimately have ?thesis by auto
+ }
+ moreover
+ {
+ assume "S = {}"
+ then have "S = {}" and "T = {}"
+ using assms affine_hull_nonempty
+ unfolding affine_parallel_def
+ by auto
+ then have ?thesis using aff_dim_empty by auto
+ }
+ moreover
+ {
+ assume "T = {}"
+ then have "S = {}" and "T = {}"
+ using assms affine_hull_nonempty
+ unfolding affine_parallel_def
+ by auto
+ then have ?thesis
+ using aff_dim_empty by auto
+ }
+ ultimately show ?thesis by blast
+qed
+
+lemma aff_dim_translation_eq:
+ fixes a :: "'n::euclidean_space"
+ shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
+proof -
+ have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
+ unfolding affine_parallel_def
+ apply (rule exI[of _ "a"])
+ using affine_hull_translation[of a S]
+ apply auto
+ done
+ then show ?thesis
+ using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
+qed
+
+lemma aff_dim_affine:
+ fixes S L :: "'n::euclidean_space set"
+ assumes "S \<noteq> {}"
+ and "affine S"
+ and "subspace L"
+ and "affine_parallel S L"
+ shows "aff_dim S = int (dim L)"
+proof -
+ have *: "affine hull S = S"
+ using assms affine_hull_eq[of S] by auto
+ then have "affine_parallel (affine hull S) L"
+ using assms by (simp add: *)
+ then show ?thesis
+ using assms aff_dim_parallel_subspace[of S L] by blast
+qed
+
+lemma dim_affine_hull:
+ fixes S :: "'n::euclidean_space set"
+ shows "dim (affine hull S) = dim S"
+proof -
+ have "dim (affine hull S) \<ge> dim S"
+ using dim_subset by auto
+ moreover have "dim (span S) \<ge> dim (affine hull S)"
+ using dim_subset affine_hull_subset_span by blast
+ moreover have "dim (span S) = dim S"
+ using dim_span by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma aff_dim_subspace:
+ fixes S :: "'n::euclidean_space set"
+ assumes "subspace S"
+ shows "aff_dim S = int (dim S)"
+proof (cases "S={}")
+ case True with assms show ?thesis
+ by (simp add: subspace_affine)
+next
+ case False
+ with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
+ show ?thesis by auto
+qed
+
+lemma aff_dim_zero:
+ fixes S :: "'n::euclidean_space set"
+ assumes "0 \<in> affine hull S"
+ shows "aff_dim S = int (dim S)"
+proof -
+ have "subspace (affine hull S)"
+ using subspace_affine[of "affine hull S"] affine_affine_hull assms
+ by auto
+ then have "aff_dim (affine hull S) = int (dim (affine hull S))"
+ using assms aff_dim_subspace[of "affine hull S"] by auto
+ then show ?thesis
+ using aff_dim_affine_hull[of S] dim_affine_hull[of S]
+ by auto
+qed
+
+lemma aff_dim_eq_dim:
+ fixes S :: "'n::euclidean_space set"
+ assumes "a \<in> affine hull S"
+ shows "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
+proof -
+ have "0 \<in> affine hull ((\<lambda>x. -a+x) ` S)"
+ unfolding affine_hull_translation
+ using assms by (simp add: ab_group_add_class.ab_left_minus image_iff)
+ with aff_dim_zero show ?thesis
+ by (metis aff_dim_translation_eq)
+qed
+
+lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
+ using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
+ dim_UNIV[where 'a="'n::euclidean_space"]
+ by auto
+
+lemma aff_dim_geq:
+ fixes V :: "'n::euclidean_space set"
+ shows "aff_dim V \<ge> -1"
+proof -
+ obtain B where "affine hull B = affine hull V"
+ and "\<not> affine_dependent B"
+ and "int (card B) = aff_dim V + 1"
+ using aff_dim_basis_exists by auto
+ then show ?thesis by auto
+qed
+
+lemma aff_dim_negative_iff [simp]:
+ fixes S :: "'n::euclidean_space set"
+ shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
+by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
+
+lemma aff_lowdim_subset_hyperplane:
+ fixes S :: "'a::euclidean_space set"
+ assumes "aff_dim S < DIM('a)"
+ obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
+proof (cases "S={}")
+ case True
+ moreover
+ have "(SOME b. b \<in> Basis) \<noteq> 0"
+ by (metis norm_some_Basis norm_zero zero_neq_one)
+ ultimately show ?thesis
+ using that by blast
+next
+ case False
+ then obtain c S' where "c \<notin> S'" "S = insert c S'"
+ by (meson equals0I mk_disjoint_insert)
+ have "dim ((+) (-c) ` S) < DIM('a)"
+ by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
+ then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
+ using lowdim_subset_hyperplane by blast
+ moreover
+ have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
+ proof -
+ have "w-c \<in> span ((+) (- c) ` S)"
+ by (simp add: span_base \<open>w \<in> S\<close>)
+ with that have "w-c \<in> {x. a \<bullet> x = 0}"
+ by blast
+ then show ?thesis
+ by (auto simp: algebra_simps)
+ qed
+ ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
+ by blast
+ then show ?thesis
+ by (rule that[OF \<open>a \<noteq> 0\<close>])
+qed
+
+lemma affine_independent_card_dim_diffs:
+ fixes S :: "'a :: euclidean_space set"
+ assumes "\<not> affine_dependent S" "a \<in> S"
+ shows "card S = dim {x - a|x. x \<in> S} + 1"
+proof -
+ have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
+ have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
+ proof (cases "x = a")
+ case True then show ?thesis by (simp add: span_clauses)
+ next
+ case False then show ?thesis
+ using assms by (blast intro: span_base that)
+ qed
+ have "\<not> affine_dependent (insert a S)"
+ by (simp add: assms insert_absorb)
+ then have 3: "independent {b - a |b. b \<in> S - {a}}"
+ using dependent_imp_affine_dependent by fastforce
+ have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
+ by blast
+ then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
+ by simp
+ also have "\<dots> = card (S - {a})"
+ by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
+ also have "\<dots> = card S - 1"
+ by (simp add: aff_independent_finite assms)
+ finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
+ have "finite S"
+ by (meson assms aff_independent_finite)
+ with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
+ moreover have "dim {x - a |x. x \<in> S} = card S - 1"
+ using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
+ ultimately show ?thesis
+ by auto
+qed
+
+lemma independent_card_le_aff_dim:
+ fixes B :: "'n::euclidean_space set"
+ assumes "B \<subseteq> V"
+ assumes "\<not> affine_dependent B"
+ shows "int (card B) \<le> aff_dim V + 1"
+proof -
+ obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
+ by (metis assms extend_to_affine_basis[of B V])
+ then have "of_nat (card T) = aff_dim V + 1"
+ using aff_dim_unique by auto
+ then show ?thesis
+ using T card_mono[of T B] aff_independent_finite[of T] by auto
+qed
+
+lemma aff_dim_subset:
+ fixes S T :: "'n::euclidean_space set"
+ assumes "S \<subseteq> T"
+ shows "aff_dim S \<le> aff_dim T"
+proof -
+ obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
+ "of_nat (card B) = aff_dim S + 1"
+ using aff_dim_inner_basis_exists[of S] by auto
+ then have "int (card B) \<le> aff_dim T + 1"
+ using assms independent_card_le_aff_dim[of B T] by auto
+ with B show ?thesis by auto
+qed
+
+lemma aff_dim_le_DIM:
+ fixes S :: "'n::euclidean_space set"
+ shows "aff_dim S \<le> int (DIM('n))"
+proof -
+ have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
+ using aff_dim_UNIV by auto
+ then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
+ using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
+qed
+
+lemma affine_dim_equal:
+ fixes S :: "'n::euclidean_space set"
+ assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
+ shows "S = T"
+proof -
+ obtain a where "a \<in> S" using assms by auto
+ then have "a \<in> T" using assms by auto
+ define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
+ then have ls: "subspace LS" "affine_parallel S LS"
+ using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
+ then have h1: "int(dim LS) = aff_dim S"
+ using assms aff_dim_affine[of S LS] by auto
+ have "T \<noteq> {}" using assms by auto
+ define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
+ then have lt: "subspace LT \<and> affine_parallel T LT"
+ using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
+ then have "int(dim LT) = aff_dim T"
+ using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
+ then have "dim LS = dim LT"
+ using h1 assms by auto
+ moreover have "LS \<le> LT"
+ using LS_def LT_def assms by auto
+ ultimately have "LS = LT"
+ using subspace_dim_equal[of LS LT] ls lt by auto
+ moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
+ using LS_def by auto
+ moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
+ using LT_def by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma aff_dim_eq_0:
+ fixes S :: "'a::euclidean_space set"
+ shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ by auto
+next
+ case False
+ then obtain a where "a \<in> S" by auto
+ show ?thesis
+ proof safe
+ assume 0: "aff_dim S = 0"
+ have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
+ by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
+ then show "\<exists>a. S = {a}"
+ using \<open>a \<in> S\<close> by blast
+ qed auto
+qed
+
+lemma affine_hull_UNIV:
+ fixes S :: "'n::euclidean_space set"
+ assumes "aff_dim S = int(DIM('n))"
+ shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
+proof -
+ have "S \<noteq> {}"
+ using assms aff_dim_empty[of S] by auto
+ have h0: "S \<subseteq> affine hull S"
+ using hull_subset[of S _] by auto
+ have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
+ using aff_dim_UNIV assms by auto
+ then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
+ using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
+ have h3: "aff_dim S \<le> aff_dim (affine hull S)"
+ using h0 aff_dim_subset[of S "affine hull S"] assms by auto
+ then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
+ using h0 h1 h2 by auto
+ then show ?thesis
+ using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
+ affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
+ by auto
+qed
+
+lemma disjoint_affine_hull:
+ fixes s :: "'n::euclidean_space set"
+ assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
+ shows "(affine hull t) \<inter> (affine hull u) = {}"
+proof -
+ have "finite s" using assms by (simp add: aff_independent_finite)
+ then have "finite t" "finite u" using assms finite_subset by blast+
+ { fix y
+ assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
+ then obtain a b
+ where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
+ and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
+ by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
+ define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
+ have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
+ have "sum c s = 0"
+ by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
+ moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
+ by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
+ moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
+ by (simp add: c_def if_smult sum_negf
+ comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
+ ultimately have False
+ using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
+ }
+ then show ?thesis by blast
+qed
+
+lemma aff_dim_convex_hull:
+ fixes S :: "'n::euclidean_space set"
+ shows "aff_dim (convex hull S) = aff_dim S"
+ using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
+ hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
+ aff_dim_subset[of "convex hull S" "affine hull S"]
+ by auto
+
+subsection \<open>Caratheodory's theorem\<close>
+
+lemma convex_hull_caratheodory_aff_dim:
+ fixes p :: "('a::euclidean_space) set"
+ shows "convex hull p =
+ {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
+ (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
+ unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
+proof (intro allI iffI)
+ fix y
+ let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
+ sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+ assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+ then obtain N where "?P N" by auto
+ then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
+ apply (rule_tac ex_least_nat_le, auto)
+ done
+ then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
+ by blast
+ then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
+ "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
+
+ have "card s \<le> aff_dim p + 1"
+ proof (rule ccontr, simp only: not_le)
+ assume "aff_dim p + 1 < card s"
+ then have "affine_dependent s"
+ using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
+ by blast
+ then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
+ using affine_dependent_explicit_finite[OF obt(1)] by auto
+ define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
+ define t where "t = Min i"
+ have "\<exists>x\<in>s. w x < 0"
+ proof (rule ccontr, simp add: not_less)
+ assume as:"\<forall>x\<in>s. 0 \<le> w x"
+ then have "sum w (s - {v}) \<ge> 0"
+ apply (rule_tac sum_nonneg, auto)
+ done
+ then have "sum w s > 0"
+ unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
+ using as[THEN bspec[where x=v]] \<open>v\<in>s\<close> \<open>w v \<noteq> 0\<close> by auto
+ then show False using wv(1) by auto
+ qed
+ then have "i \<noteq> {}" unfolding i_def by auto
+ then have "t \<ge> 0"
+ using Min_ge_iff[of i 0 ] and obt(1)
+ unfolding t_def i_def
+ using obt(4)[unfolded le_less]
+ by (auto simp: divide_le_0_iff)
+ have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
+ proof
+ fix v
+ assume "v \<in> s"
+ then have v: "0 \<le> u v"
+ using obt(4)[THEN bspec[where x=v]] by auto
+ show "0 \<le> u v + t * w v"
+ proof (cases "w v < 0")
+ case False
+ thus ?thesis using v \<open>t\<ge>0\<close> by auto
+ next
+ case True
+ then have "t \<le> u v / (- w v)"
+ using \<open>v\<in>s\<close> unfolding t_def i_def
+ apply (rule_tac Min_le)
+ using obt(1) apply auto
+ done
+ then show ?thesis
+ unfolding real_0_le_add_iff
+ using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
+ by auto
+ qed
+ qed
+ obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
+ using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
+ then have a: "a \<in> s" "u a + t * w a = 0" by auto
+ have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
+ unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
+ have "(\<Sum>v\<in>s. u v + t * w v) = 1"
+ unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
+ moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
+ unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
+ using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
+ ultimately have "?P (n - 1)"
+ apply (rule_tac x="(s - {a})" in exI)
+ apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
+ using obt(1-3) and t and a
+ apply (auto simp: * scaleR_left_distrib)
+ done
+ then show False
+ using smallest[THEN spec[where x="n - 1"]] by auto
+ qed
+ then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
+ (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+ using obt by auto
+qed auto
+
+lemma caratheodory_aff_dim:
+ fixes p :: "('a::euclidean_space) set"
+ shows "convex hull p = {x. \<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> x \<in> convex hull s}"
+ (is "?lhs = ?rhs")
+proof
+ show "?lhs \<subseteq> ?rhs"
+ apply (subst convex_hull_caratheodory_aff_dim, clarify)
+ apply (rule_tac x=s in exI)
+ apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
+ done
+next
+ show "?rhs \<subseteq> ?lhs"
+ using hull_mono by blast
+qed
+
+lemma convex_hull_caratheodory:
+ fixes p :: "('a::euclidean_space) set"
+ shows "convex hull p =
+ {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
+ (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
+ (is "?lhs = ?rhs")
+proof (intro set_eqI iffI)
+ fix x
+ assume "x \<in> ?lhs" then show "x \<in> ?rhs"
+ apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
+ apply (erule ex_forward)+
+ using aff_dim_le_DIM [of p]
+ apply simp
+ done
+next
+ fix x
+ assume "x \<in> ?rhs" then show "x \<in> ?lhs"
+ by (auto simp: convex_hull_explicit)
+qed
+
+theorem caratheodory:
+ "convex hull p =
+ {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
+ card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
+proof safe
+ fix x
+ assume "x \<in> convex hull p"
+ then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
+ "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+ unfolding convex_hull_caratheodory by auto
+ then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
+ apply (rule_tac x=s in exI)
+ using hull_subset[of s convex]
+ using convex_convex_hull[simplified convex_explicit, of s,
+ THEN spec[where x=s], THEN spec[where x=u]]
+ apply auto
+ done
+next
+ fix x s
+ assume "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
+ then show "x \<in> convex hull p"
+ using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
+qed
+
+subsection%unimportant\<open>Some Properties of subset of standard basis\<close>
+
+lemma affine_hull_substd_basis:
+ assumes "d \<subseteq> Basis"
+ shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
+ (is "affine hull (insert 0 ?A) = ?B")
+proof -
+ have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
+ by auto
+ show ?thesis
+ unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
+qed
+
+lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
+ by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
+
+
+subsection%unimportant \<open>Moving and scaling convex hulls\<close>
+
+lemma convex_hull_set_plus:
+ "convex hull (S + T) = convex hull S + convex hull T"
+ unfolding set_plus_image
+ apply (subst convex_hull_linear_image [symmetric])
+ apply (simp add: linear_iff scaleR_right_distrib)
+ apply (simp add: convex_hull_Times)
+ done
+
+lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
+ unfolding set_plus_def by auto
+
+lemma convex_hull_translation:
+ "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
+ unfolding translation_eq_singleton_plus
+ by (simp only: convex_hull_set_plus convex_hull_singleton)
+
+lemma convex_hull_scaling:
+ "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
+ using linear_scaleR by (rule convex_hull_linear_image [symmetric])
+
+lemma convex_hull_affinity:
+ "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
+ by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
+
+
+subsection%unimportant \<open>Convexity of cone hulls\<close>
+
+lemma convex_cone_hull:
+ assumes "convex S"
+ shows "convex (cone hull S)"
+proof (rule convexI)
+ fix x y
+ assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
+ then have "S \<noteq> {}"
+ using cone_hull_empty_iff[of S] by auto
+ fix u v :: real
+ assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+ then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
+ using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
+ from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
+ using cone_hull_expl[of S] by auto
+ from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
+ using cone_hull_expl[of S] by auto
+ {
+ assume "cx + cy \<le> 0"
+ then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
+ using x y by auto
+ then have "u *\<^sub>R x + v *\<^sub>R y = 0"
+ by auto
+ then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
+ using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
+ }
+ moreover
+ {
+ assume "cx + cy > 0"
+ then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
+ using assms mem_convex_alt[of S xx yy cx cy] x y by auto
+ then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
+ using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
+ by (auto simp: scaleR_right_distrib)
+ then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
+ using x y by auto
+ }
+ moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
+ ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
+qed
+
+lemma cone_convex_hull:
+ assumes "cone S"
+ shows "cone (convex hull S)"
+proof (cases "S = {}")
+ case True
+ then show ?thesis by auto
+next
+ case False
+ then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
+ using cone_iff[of S] assms by auto
+ {
+ fix c :: real
+ assume "c > 0"
+ then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
+ using convex_hull_scaling[of _ S] by auto
+ also have "\<dots> = convex hull S"
+ using * \<open>c > 0\<close> by auto
+ finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
+ by auto
+ }
+ then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
+ using * hull_subset[of S convex] by auto
+ then show ?thesis
+ using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
+qed
+
+subsection \<open>Radon's theorem\<close>
+
+text "Formalized by Lars Schewe."
+
+lemma Radon_ex_lemma:
+ assumes "finite c" "affine_dependent c"
+ shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
+proof -
+ from assms(2)[unfolded affine_dependent_explicit]
+ obtain s u where
+ "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+ by blast
+ then show ?thesis
+ apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
+ unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
+ apply (auto simp: Int_absorb1)
+ done
+qed
+
+lemma Radon_s_lemma:
+ assumes "finite s"
+ and "sum f s = (0::real)"
+ shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
+proof -
+ have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
+ by auto
+ show ?thesis
+ unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
+ and sum.distrib[symmetric] and *
+ using assms(2)
+ by assumption
+qed
+
+lemma Radon_v_lemma:
+ assumes "finite s"
+ and "sum f s = 0"
+ and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
+ shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
+proof -
+ have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
+ using assms(3) by auto
+ show ?thesis
+ unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
+ and sum.distrib[symmetric] and *
+ using assms(2)
+ apply assumption
+ done
+qed
+
+lemma Radon_partition:
+ assumes "finite c" "affine_dependent c"
+ shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
+proof -
+ obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
+ using Radon_ex_lemma[OF assms] by auto
+ have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
+ using assms(1) by auto
+ define z where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
+ have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
+ proof (cases "u v \<ge> 0")
+ case False
+ then have "u v < 0" by auto
+ then show ?thesis
+ proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
+ case True
+ then show ?thesis
+ using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
+ next
+ case False
+ then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
+ apply (rule_tac sum_mono, auto)
+ done
+ then show ?thesis
+ unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
+ qed
+ qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
+
+ then have *: "sum u {x\<in>c. u x > 0} > 0"
+ unfolding less_le
+ apply (rule_tac conjI)
+ apply (rule_tac sum_nonneg, auto)
+ done
+ moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
+ "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
+ using assms(1)
+ apply (rule_tac[!] sum.mono_neutral_left, auto)
+ done
+ then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
+ "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
+ unfolding eq_neg_iff_add_eq_0
+ using uv(1,4)
+ by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
+ moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
+ apply rule
+ apply (rule mult_nonneg_nonneg)
+ using *
+ apply auto
+ done
+ ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
+ unfolding convex_hull_explicit mem_Collect_eq
+ apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
+ apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
+ using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
+ apply (auto simp: sum_negf sum_distrib_left[symmetric])
+ done
+ moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
+ apply rule
+ apply (rule mult_nonneg_nonneg)
+ using *
+ apply auto
+ done
+ then have "z \<in> convex hull {v \<in> c. u v > 0}"
+ unfolding convex_hull_explicit mem_Collect_eq
+ apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
+ apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
+ using assms(1)
+ unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
+ using *
+ apply (auto simp: sum_negf sum_distrib_left[symmetric])
+ done
+ ultimately show ?thesis
+ apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
+ apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto)
+ done
+qed
+
+theorem Radon:
+ assumes "affine_dependent c"
+ obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
+proof -
+ from assms[unfolded affine_dependent_explicit]
+ obtain s u where
+ "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+ by blast
+ then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
+ unfolding affine_dependent_explicit by auto
+ from Radon_partition[OF *]
+ obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
+ by blast
+ then show ?thesis
+ apply (rule_tac that[of p m])
+ using s
+ apply auto
+ done
+qed
+
+
+subsection \<open>Helly's theorem\<close>
+
+lemma Helly_induct:
+ fixes f :: "'a::euclidean_space set set"
+ assumes "card f = n"
+ and "n \<ge> DIM('a) + 1"
+ and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
+ shows "\<Inter>f \<noteq> {}"
+ using assms
+proof (induction n arbitrary: f)
+ case 0
+ then show ?case by auto
+next
+ case (Suc n)
+ have "finite f"
+ using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
+ show "\<Inter>f \<noteq> {}"
+ proof (cases "n = DIM('a)")
+ case True
+ then show ?thesis
+ by (simp add: Suc.prems(1) Suc.prems(4))
+ next
+ case False
+ have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
+ proof (rule Suc.IH[rule_format])
+ show "card (f - {s}) = n"
+ by (simp add: Suc.prems(1) \<open>finite f\<close> that)
+ show "DIM('a) + 1 \<le> n"
+ using False Suc.prems(2) by linarith
+ show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
+ by (simp add: Suc.prems(4) subset_Diff_insert)
+ qed (use Suc in auto)
+ then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
+ by blast
+ then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
+ by metis
+ show ?thesis
+ proof (cases "inj_on X f")
+ case False
+ then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
+ unfolding inj_on_def by auto
+ then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
+ show ?thesis
+ by (metis "*" X disjoint_iff_not_equal st)
+ next
+ case True
+ then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
+ using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
+ unfolding card_image[OF True] and \<open>card f = Suc n\<close>
+ using Suc(3) \<open>finite f\<close> and False
+ by auto
+ have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
+ using mp(2) by auto
+ then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
+ unfolding subset_image_iff by auto
+ then have "f \<union> (g \<union> h) = f" by auto
+ then have f: "f = g \<union> h"
+ using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
+ unfolding mp(2)[unfolded image_Un[symmetric] gh]
+ by auto
+ have *: "g \<inter> h = {}"
+ using mp(1)
+ unfolding gh
+ using inj_on_image_Int[OF True gh(3,4)]
+ by auto
+ have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
+ by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
+ then show ?thesis
+ unfolding f using mp(3)[unfolded gh] by blast
+ qed
+ qed
+qed
+
+theorem Helly:
+ fixes f :: "'a::euclidean_space set set"
+ assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
+ and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
+ shows "\<Inter>f \<noteq> {}"
+ apply (rule Helly_induct)
+ using assms
+ apply auto
+ done
+
+subsection \<open>Epigraphs of convex functions\<close>
+
+definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
+
+lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
+ unfolding epigraph_def by auto
+
+lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
+proof safe
+ assume L: "convex (epigraph S f)"
+ then show "convex_on S f"
+ by (auto simp: convex_def convex_on_def epigraph_def)
+ show "convex S"
+ using L
+ apply (clarsimp simp: convex_def convex_on_def epigraph_def)
+ apply (erule_tac x=x in allE)
+ apply (erule_tac x="f x" in allE, safe)
+ apply (erule_tac x=y in allE)
+ apply (erule_tac x="f y" in allE)
+ apply (auto simp: )
+ done
+next
+ assume "convex_on S f" "convex S"
+ then show "convex (epigraph S f)"
+ unfolding convex_def convex_on_def epigraph_def
+ apply safe
+ apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
+ apply (auto intro!:mult_left_mono add_mono)
+ done
+qed
+
+lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
+ unfolding convex_epigraph by auto
+
+lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
+ by (simp add: convex_epigraph)
+
+
+subsubsection%unimportant \<open>Use this to derive general bound property of convex function\<close>
+
+lemma convex_on:
+ assumes "convex S"
+ shows "convex_on S f \<longleftrightarrow>
+ (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
+ f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
+
+ unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
+ unfolding fst_sum snd_sum fst_scaleR snd_scaleR
+ apply safe
+ apply (drule_tac x=k in spec)
+ apply (drule_tac x=u in spec)
+ apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
+ apply simp
+ using assms[unfolded convex] apply simp
+ apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
+ apply (rule sum_mono)
+ apply (erule_tac x=i in allE)
+ unfolding real_scaleR_def
+ apply (rule mult_left_mono)
+ using assms[unfolded convex] apply auto
+ done
+
+subsection%unimportant \<open>A bound within a convex hull\<close>
+
+lemma convex_on_convex_hull_bound:
+ assumes "convex_on (convex hull s) f"
+ and "\<forall>x\<in>s. f x \<le> b"
+ shows "\<forall>x\<in> convex hull s. f x \<le> b"
+proof
+ fix x
+ assume "x \<in> convex hull s"
+ then obtain k u v where
+ obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
+ unfolding convex_hull_indexed mem_Collect_eq by auto
+ have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
+ using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
+ unfolding sum_distrib_right[symmetric] obt(2) mult_1
+ apply (drule_tac meta_mp)
+ apply (rule mult_left_mono)
+ using assms(2) obt(1)
+ apply auto
+ done
+ then show "f x \<le> b"
+ using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
+ unfolding obt(2-3)
+ using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
+ by auto
+qed
+
+lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
+ by (simp add: inner_sum_left sum.If_cases inner_Basis)
+
+lemma convex_set_plus:
+ assumes "convex S" and "convex T" shows "convex (S + T)"
+proof -
+ have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+ using assms by (rule convex_sums)
+ moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
+ unfolding set_plus_def by auto
+ finally show "convex (S + T)" .
+qed
+
+lemma convex_set_sum:
+ assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
+ shows "convex (\<Sum>i\<in>A. B i)"
+proof (cases "finite A")
+ case True then show ?thesis using assms
+ by induct (auto simp: convex_set_plus)
+qed auto
+
+lemma finite_set_sum:
+ assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
+ using assms by (induct set: finite, simp, simp add: finite_set_plus)
+
+lemma box_eq_set_sum_Basis:
+ shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
+ apply (subst set_sum_alt [OF finite_Basis], safe)
+ apply (fast intro: euclidean_representation [symmetric])
+ apply (subst inner_sum_left)
+apply (rename_tac f)
+ apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
+ apply (drule (1) bspec)
+ apply clarsimp
+ apply (frule sum.remove [OF finite_Basis])
+ apply (erule trans, simp)
+ apply (rule sum.neutral, clarsimp)
+ apply (frule_tac x=i in bspec, assumption)
+ apply (drule_tac x=x in bspec, assumption, clarsimp)
+ apply (cut_tac u=x and v=i in inner_Basis, assumption+)
+ apply (rule ccontr, simp)
+ done
+
+lemma convex_hull_set_sum:
+ "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
+proof (cases "finite A")
+ assume "finite A" then show ?thesis
+ by (induct set: finite, simp, simp add: convex_hull_set_plus)
+qed simp
+
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Mon Jan 07 13:33:29 2019 +0100
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Mon Jan 07 14:06:54 2019 +0100
@@ -6,1143 +6,15 @@
Author: Johannes Hoelzl, TU Muenchen
*)
-section \<open>Convex Sets and Functions\<close>
+section \<open>Convex Sets and Functions on (Normed) Euclidean Spaces\<close>
theory Convex_Euclidean_Space
imports
+ Convex
Topology_Euclidean_Space
- "HOL-Library.Set_Algebras"
begin
-lemma swap_continuous: (*move to Topological_Spaces?*)
- assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
- shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
-proof -
- have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
- by auto
- then show ?thesis
- apply (rule ssubst)
- apply (rule continuous_on_compose)
- apply (simp add: split_def)
- apply (rule continuous_intros | simp add: assms)+
- done
-qed
-
-lemma substdbasis_expansion_unique:
- assumes d: "d \<subseteq> Basis"
- shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
- (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
-proof -
- have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
- by auto
- have **: "finite d"
- by (auto intro: finite_subset[OF assms])
- have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
- using d
- by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
- show ?thesis
- unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
-qed
-
-lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
- by (rule independent_mono[OF independent_Basis])
-
-lemma dim_cball:
- assumes "e > 0"
- shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
-proof -
- {
- fix x :: "'n::euclidean_space"
- define y where "y = (e / norm x) *\<^sub>R x"
- then have "y \<in> cball 0 e"
- using assms by auto
- moreover have *: "x = (norm x / e) *\<^sub>R y"
- using y_def assms by simp
- moreover from * have "x = (norm x/e) *\<^sub>R y"
- by auto
- ultimately have "x \<in> span (cball 0 e)"
- using span_scale[of y "cball 0 e" "norm x/e"]
- span_superset[of "cball 0 e"]
- by (simp add: span_base)
- }
- then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
- by auto
- then show ?thesis
- using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
-qed
-
-lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
- by (rule ccontr) auto
-
-lemma subset_translation_eq [simp]:
- fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
- by auto
-
-lemma translate_inj_on:
- fixes A :: "'a::ab_group_add set"
- shows "inj_on (\<lambda>x. a + x) A"
- unfolding inj_on_def by auto
-
-lemma translation_assoc:
- fixes a b :: "'a::ab_group_add"
- shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
- by auto
-
-lemma translation_invert:
- fixes a :: "'a::ab_group_add"
- assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
- shows "A = B"
-proof -
- have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
- using assms by auto
- then show ?thesis
- using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
-qed
-
-lemma translation_galois:
- fixes a :: "'a::ab_group_add"
- shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
- using translation_assoc[of "-a" a S]
- apply auto
- using translation_assoc[of a "-a" T]
- apply auto
- done
-
-lemma translation_inverse_subset:
- assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
- shows "V \<le> ((\<lambda>x. a + x) ` S)"
-proof -
- {
- fix x
- assume "x \<in> V"
- then have "x-a \<in> S" using assms by auto
- then have "x \<in> {a + v |v. v \<in> S}"
- apply auto
- apply (rule exI[of _ "x-a"], simp)
- done
- then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
- }
- then show ?thesis by auto
-qed
-
-subsection \<open>Convexity\<close>
-
-definition%important convex :: "'a::real_vector set \<Rightarrow> bool"
- where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
-
-lemma convexI:
- assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
- shows "convex s"
- using assms unfolding convex_def by fast
-
-lemma convexD:
- assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
- using assms unfolding convex_def by fast
-
-lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
- (is "_ \<longleftrightarrow> ?alt")
-proof
- show "convex s" if alt: ?alt
- proof -
- {
- fix x y and u v :: real
- assume mem: "x \<in> s" "y \<in> s"
- assume "0 \<le> u" "0 \<le> v"
- moreover
- assume "u + v = 1"
- then have "u = 1 - v" by auto
- ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
- using alt [rule_format, OF mem] by auto
- }
- then show ?thesis
- unfolding convex_def by auto
- qed
- show ?alt if "convex s"
- using that by (auto simp: convex_def)
-qed
-
-lemma convexD_alt:
- assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
- shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
- using assms unfolding convex_alt by auto
-
-lemma mem_convex_alt:
- assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
- shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
- apply (rule convexD)
- using assms
- apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
- done
-
-lemma convex_empty[intro,simp]: "convex {}"
- unfolding convex_def by simp
-
-lemma convex_singleton[intro,simp]: "convex {a}"
- unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
-
-lemma convex_UNIV[intro,simp]: "convex UNIV"
- unfolding convex_def by auto
-
-lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
- unfolding convex_def by auto
-
-lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
- unfolding convex_def by auto
-
-lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
- unfolding convex_def by auto
-
-lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
- unfolding convex_def by auto
-
-lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
- unfolding convex_def
- by (auto simp: inner_add intro!: convex_bound_le)
-
-lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
-proof -
- have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
- by auto
- show ?thesis
- unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
-qed
-
-lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
-proof -
- have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
- by auto
- show ?thesis
- unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
-qed
-
-lemma convex_hyperplane: "convex {x. inner a x = b}"
-proof -
- have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
- by auto
- show ?thesis using convex_halfspace_le convex_halfspace_ge
- by (auto intro!: convex_Int simp: *)
-qed
-
-lemma convex_halfspace_lt: "convex {x. inner a x < b}"
- unfolding convex_def
- by (auto simp: convex_bound_lt inner_add)
-
-lemma convex_halfspace_gt: "convex {x. inner a x > b}"
- using convex_halfspace_lt[of "-a" "-b"] by auto
-
-lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
- using convex_halfspace_ge[of b "1::complex"] by simp
-
-lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
- using convex_halfspace_le[of "1::complex" b] by simp
-
-lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
- using convex_halfspace_ge[of b \<i>] by simp
-
-lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
- using convex_halfspace_le[of \<i> b] by simp
-
-lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
- using convex_halfspace_gt[of b "1::complex"] by simp
-
-lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
- using convex_halfspace_lt[of "1::complex" b] by simp
-
-lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
- using convex_halfspace_gt[of b \<i>] by simp
-
-lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
- using convex_halfspace_lt[of \<i> b] by simp
-
-lemma convex_real_interval [iff]:
- fixes a b :: "real"
- shows "convex {a..}" and "convex {..b}"
- and "convex {a<..}" and "convex {..<b}"
- and "convex {a..b}" and "convex {a<..b}"
- and "convex {a..<b}" and "convex {a<..<b}"
-proof -
- have "{a..} = {x. a \<le> inner 1 x}"
- by auto
- then show 1: "convex {a..}"
- by (simp only: convex_halfspace_ge)
- have "{..b} = {x. inner 1 x \<le> b}"
- by auto
- then show 2: "convex {..b}"
- by (simp only: convex_halfspace_le)
- have "{a<..} = {x. a < inner 1 x}"
- by auto
- then show 3: "convex {a<..}"
- by (simp only: convex_halfspace_gt)
- have "{..<b} = {x. inner 1 x < b}"
- by auto
- then show 4: "convex {..<b}"
- by (simp only: convex_halfspace_lt)
- have "{a..b} = {a..} \<inter> {..b}"
- by auto
- then show "convex {a..b}"
- by (simp only: convex_Int 1 2)
- have "{a<..b} = {a<..} \<inter> {..b}"
- by auto
- then show "convex {a<..b}"
- by (simp only: convex_Int 3 2)
- have "{a..<b} = {a..} \<inter> {..<b}"
- by auto
- then show "convex {a..<b}"
- by (simp only: convex_Int 1 4)
- have "{a<..<b} = {a<..} \<inter> {..<b}"
- by auto
- then show "convex {a<..<b}"
- by (simp only: convex_Int 3 4)
-qed
-
-lemma convex_Reals: "convex \<real>"
- by (simp add: convex_def scaleR_conv_of_real)
-
-
-subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
-
-lemma convex_sum:
- fixes C :: "'a::real_vector set"
- assumes "finite s"
- and "convex C"
- and "(\<Sum> i \<in> s. a i) = 1"
- assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
- and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
- shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
- using assms(1,3,4,5)
-proof (induct arbitrary: a set: finite)
- case empty
- then show ?case by simp
-next
- case (insert i s) note IH = this(3)
- have "a i + sum a s = 1"
- and "0 \<le> a i"
- and "\<forall>j\<in>s. 0 \<le> a j"
- and "y i \<in> C"
- and "\<forall>j\<in>s. y j \<in> C"
- using insert.hyps(1,2) insert.prems by simp_all
- then have "0 \<le> sum a s"
- by (simp add: sum_nonneg)
- have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
- proof (cases "sum a s = 0")
- case True
- with \<open>a i + sum a s = 1\<close> have "a i = 1"
- by simp
- from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
- by simp
- show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
- by simp
- next
- case False
- with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
- by simp
- then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
- using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
- by (simp add: IH sum_divide_distrib [symmetric])
- from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
- and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
- have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
- by (rule convexD)
- then show ?thesis
- by (simp add: scaleR_sum_right False)
- qed
- then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
- by simp
-qed
-
-lemma convex:
- "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (sum u {1..k} = 1)
- \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
-proof safe
- fix k :: nat
- fix u :: "nat \<Rightarrow> real"
- fix x
- assume "convex s"
- "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
- "sum u {1..k} = 1"
- with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
- by auto
-next
- assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
- \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
- {
- fix \<mu> :: real
- fix x y :: 'a
- assume xy: "x \<in> s" "y \<in> s"
- assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
- let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
- let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
- have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
- by auto
- then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
- by simp
- then have "sum ?u {1 .. 2} = 1"
- using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
- by auto
- with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
- using mu xy by auto
- have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
- using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
- from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
- have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
- by auto
- then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
- using s by (auto simp: add.commute)
- }
- then show "convex s"
- unfolding convex_alt by auto
-qed
-
-
-lemma convex_explicit:
- fixes s :: "'a::real_vector set"
- shows "convex s \<longleftrightarrow>
- (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
-proof safe
- fix t
- fix u :: "'a \<Rightarrow> real"
- assume "convex s"
- and "finite t"
- and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
- then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- using convex_sum[of t s u "\<lambda> x. x"] by auto
-next
- assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
- sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- show "convex s"
- unfolding convex_alt
- proof safe
- fix x y
- fix \<mu> :: real
- assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
- show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
- proof (cases "x = y")
- case False
- then show ?thesis
- using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
- by auto
- next
- case True
- then show ?thesis
- using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
- by (auto simp: field_simps real_vector.scale_left_diff_distrib)
- qed
- qed
-qed
-
-lemma convex_finite:
- assumes "finite s"
- shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
- unfolding convex_explicit
- apply safe
- subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
- subgoal for t u
- proof -
- have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
- by simp
- assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
- assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
- assume "t \<subseteq> s"
- then have "s \<inter> t = t" by auto
- with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
- qed
- done
-
-
-subsection \<open>Functions that are convex on a set\<close>
-
-definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- where "convex_on s f \<longleftrightarrow>
- (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
-
-lemma convex_onI [intro?]:
- assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
- unfolding convex_on_def
-proof clarify
- fix x y
- fix u v :: real
- assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
- from A(5) have [simp]: "v = 1 - u"
- by (simp add: algebra_simps)
- from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
- using assms[of u y x]
- by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
-qed
-
-lemma convex_on_linorderI [intro?]:
- fixes A :: "('a::{linorder,real_vector}) set"
- assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
-proof
- fix x y
- fix t :: real
- assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
- with assms [of t x y] assms [of "1 - t" y x]
- show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
-qed
-
-lemma convex_onD:
- assumes "convex_on A f"
- shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms by (auto simp: convex_on_def)
-
-lemma convex_onD_Icc:
- assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
- shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
-
-lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
- unfolding convex_on_def by auto
-
-lemma convex_on_add [intro]:
- assumes "convex_on s f"
- and "convex_on s g"
- shows "convex_on s (\<lambda>x. f x + g x)"
-proof -
- {
- fix x y
- assume "x \<in> s" "y \<in> s"
- moreover
- fix u v :: real
- assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- ultimately
- have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
- using assms unfolding convex_on_def by (auto simp: add_mono)
- then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
- by (simp add: field_simps)
- }
- then show ?thesis
- unfolding convex_on_def by auto
-qed
-
-lemma convex_on_cmul [intro]:
- fixes c :: real
- assumes "0 \<le> c"
- and "convex_on s f"
- shows "convex_on s (\<lambda>x. c * f x)"
-proof -
- have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
- for u c fx v fy :: real
- by (simp add: field_simps)
- show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
- unfolding convex_on_def and * by auto
-qed
-
-lemma convex_lower:
- assumes "convex_on s f"
- and "x \<in> s"
- and "y \<in> s"
- and "0 \<le> u"
- and "0 \<le> v"
- and "u + v = 1"
- shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
-proof -
- let ?m = "max (f x) (f y)"
- have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
- using assms(4,5) by (auto simp: mult_left_mono add_mono)
- also have "\<dots> = max (f x) (f y)"
- using assms(6) by (simp add: distrib_right [symmetric])
- finally show ?thesis
- using assms unfolding convex_on_def by fastforce
-qed
-
-lemma convex_on_dist [intro]:
- fixes s :: "'a::real_normed_vector set"
- shows "convex_on s (\<lambda>x. dist a x)"
-proof (auto simp: convex_on_def dist_norm)
- fix x y
- assume "x \<in> s" "y \<in> s"
- fix u v :: real
- assume "0 \<le> u"
- assume "0 \<le> v"
- assume "u + v = 1"
- have "a = u *\<^sub>R a + v *\<^sub>R a"
- unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
- then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
- by (auto simp: algebra_simps)
- show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
- unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
- using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
-qed
-
-
-subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close>
-
-lemma convex_linear_image:
- assumes "linear f"
- and "convex s"
- shows "convex (f ` s)"
-proof -
- interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f ` s)"
- by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
-qed
-
-lemma convex_linear_vimage:
- assumes "linear f"
- and "convex s"
- shows "convex (f -` s)"
-proof -
- interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f -` s)"
- by (simp add: convex_def f.add f.scaleR)
-qed
-
-lemma convex_scaling:
- assumes "convex s"
- shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
- have "linear (\<lambda>x. c *\<^sub>R x)"
- by (simp add: linearI scaleR_add_right)
- then show ?thesis
- using \<open>convex s\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_scaled:
- assumes "convex S"
- shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
-proof -
- have "linear (\<lambda>x. x *\<^sub>R c)"
- by (simp add: linearI scaleR_add_left)
- then show ?thesis
- using \<open>convex S\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_negations:
- assumes "convex S"
- shows "convex ((\<lambda>x. - x) ` S)"
-proof -
- have "linear (\<lambda>x. - x)"
- by (simp add: linearI)
- then show ?thesis
- using \<open>convex S\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_sums:
- assumes "convex S"
- and "convex T"
- shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-proof -
- have "linear (\<lambda>(x, y). x + y)"
- by (auto intro: linearI simp: scaleR_add_right)
- with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
- by (intro convex_linear_image convex_Times)
- also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- by auto
- finally show ?thesis .
-qed
-
-lemma convex_differences:
- assumes "convex S" "convex T"
- shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-proof -
- have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
- by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
- then show ?thesis
- using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
-qed
-
-lemma convex_translation:
- assumes "convex S"
- shows "convex ((\<lambda>x. a + x) ` S)"
-proof -
- have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (\<lambda>x. a + x) ` S"
- by auto
- then show ?thesis
- using convex_sums[OF convex_singleton[of a] assms] by auto
-qed
-
-lemma convex_affinity:
- assumes "convex S"
- shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
-proof -
- have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
- by auto
- then show ?thesis
- using convex_translation[OF convex_scaling[OF assms], of a c] by auto
-qed
-
-lemma pos_is_convex: "convex {0 :: real <..}"
- unfolding convex_alt
-proof safe
- fix y x \<mu> :: real
- assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- {
- assume "\<mu> = 0"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
- by simp
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
- using * by simp
- }
- moreover
- {
- assume "\<mu> = 1"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
- using * by simp
- }
- moreover
- {
- assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
- then have "\<mu> > 0" "(1 - \<mu>) > 0"
- using * by auto
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
- using * by (auto simp: add_pos_pos)
- }
- ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
- by fastforce
-qed
-
-lemma convex_on_sum:
- fixes a :: "'a \<Rightarrow> real"
- and y :: "'a \<Rightarrow> 'b::real_vector"
- and f :: "'b \<Rightarrow> real"
- assumes "finite s" "s \<noteq> {}"
- and "convex_on C f"
- and "convex C"
- and "(\<Sum> i \<in> s. a i) = 1"
- and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
- and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
- shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
- using assms
-proof (induct s arbitrary: a rule: finite_ne_induct)
- case (singleton i)
- then have ai: "a i = 1"
- by auto
- then show ?case
- by auto
-next
- case (insert i s)
- then have "convex_on C f"
- by simp
- from this[unfolded convex_on_def, rule_format]
- have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by simp
- show ?case
- proof (cases "a i = 1")
- case True
- then have "(\<Sum> j \<in> s. a j) = 0"
- using insert by auto
- then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
- using insert by (fastforce simp: sum_nonneg_eq_0_iff)
- then show ?thesis
- using insert by auto
- next
- case False
- from insert have yai: "y i \<in> C" "a i \<ge> 0"
- by auto
- have fis: "finite (insert i s)"
- using insert by auto
- then have ai1: "a i \<le> 1"
- using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
- then have "a i < 1"
- using False by auto
- then have i0: "1 - a i > 0"
- by auto
- let ?a = "\<lambda>j. a j / (1 - a i)"
- have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
- using i0 insert that by fastforce
- have "(\<Sum> j \<in> insert i s. a j) = 1"
- using insert by auto
- then have "(\<Sum> j \<in> s. a j) = 1 - a i"
- using sum.insert insert by fastforce
- then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
- using i0 by auto
- then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
- unfolding sum_divide_distrib by simp
- have "convex C" using insert by auto
- then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
- using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
- have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
- using a_nonneg a1 insert by blast
- have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
- using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
- by (auto simp only: add.commute)
- also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
- using i0 by auto
- also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
- using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
- by (auto simp: algebra_simps)
- also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
- by (auto simp: divide_inverse)
- also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
- using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
- by (auto simp: add.commute)
- also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
- using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
- OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
- by simp
- also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
- unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
- using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
- using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
- using insert by auto
- finally show ?thesis
- by simp
- qed
-qed
-
-lemma convex_on_alt:
- fixes C :: "'a::real_vector set"
- assumes "convex C"
- shows "convex_on C f \<longleftrightarrow>
- (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
-proof safe
- fix x y
- fix \<mu> :: real
- assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
- from this[unfolded convex_on_def, rule_format]
- have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
- by auto
- from this [of "\<mu>" "1 - \<mu>", simplified] *
- show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by auto
-next
- assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- {
- fix x y
- fix u v :: real
- assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
- then have[simp]: "1 - u = v" by auto
- from *[rule_format, of x y u]
- have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
- using ** by auto
- }
- then show "convex_on C f"
- unfolding convex_on_def by auto
-qed
-
-lemma convex_on_diff:
- fixes f :: "real \<Rightarrow> real"
- assumes f: "convex_on I f"
- and I: "x \<in> I" "y \<in> I"
- and t: "x < t" "t < y"
- shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
-proof -
- define a where "a \<equiv> (t - y) / (x - y)"
- with t have "0 \<le> a" "0 \<le> 1 - a"
- by (auto simp: field_simps)
- with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
- by (auto simp: convex_on_def)
- have "a * x + (1 - a) * y = a * (x - y) + y"
- by (simp add: field_simps)
- also have "\<dots> = t"
- unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
- finally have "f t \<le> a * f x + (1 - a) * f y"
- using cvx by simp
- also have "\<dots> = a * (f x - f y) + f y"
- by (simp add: field_simps)
- finally have "f t - f y \<le> a * (f x - f y)"
- by simp
- with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- by (simp add: le_divide_eq divide_le_eq field_simps a_def)
- with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
- by (simp add: le_divide_eq divide_le_eq field_simps)
-qed
-
-lemma pos_convex_function:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex C"
- and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
- shows "convex_on C f"
- unfolding convex_on_alt[OF assms(1)]
- using assms
-proof safe
- fix x y \<mu> :: real
- let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
- assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- then have "1 - \<mu> \<ge> 0" by auto
- then have xpos: "?x \<in> C"
- using * unfolding convex_alt by fastforce
- have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
- \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
- using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
- mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
- by auto
- then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
- by (auto simp: field_simps)
- then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- using convex_on_alt by auto
-qed
-
-lemma atMostAtLeast_subset_convex:
- fixes C :: "real set"
- assumes "convex C"
- and "x \<in> C" "y \<in> C" "x < y"
- shows "{x .. y} \<subseteq> C"
-proof safe
- fix z assume z: "z \<in> {x .. y}"
- have less: "z \<in> C" if *: "x < z" "z < y"
- proof -
- let ?\<mu> = "(y - z) / (y - x)"
- have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
- using assms * by (auto simp: field_simps)
- then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
- using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
- by (simp add: algebra_simps)
- have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
- by (auto simp: field_simps)
- also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
- using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
- also have "\<dots> = z"
- using assms by (auto simp: field_simps)
- finally show ?thesis
- using comb by auto
- qed
- show "z \<in> C"
- using z less assms by (auto simp: le_less)
-qed
-
-lemma f''_imp_f':
- fixes f :: "real \<Rightarrow> real"
- assumes "convex C"
- and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- and x: "x \<in> C"
- and y: "y \<in> C"
- shows "f' x * (y - x) \<le> f y - f x"
- using assms
-proof -
- have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
- if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
- proof -
- from * have ge: "y - x > 0" "y - x \<ge> 0"
- by auto
- from * have le: "x - y < 0" "x - y \<le> 0"
- by auto
- then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
- THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
- by auto
- then have "z1 \<in> C"
- using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
- by fastforce
- from z1 have z1': "f x - f y = (x - y) * f' z1"
- by (simp add: field_simps)
- obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
- THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
- THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
- using * z1' by auto
- also have "\<dots> = (y - z1) * f'' z3"
- using z3 by auto
- finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
- by simp
- have A': "y - z1 \<ge> 0"
- using z1 by auto
- have "z3 \<in> C"
- using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
- by fastforce
- then have B': "f'' z3 \<ge> 0"
- using assms by auto
- from A' B' have "(y - z1) * f'' z3 \<ge> 0"
- by auto
- from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
- by auto
- from mult_right_mono_neg[OF this le(2)]
- have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
- by (simp add: algebra_simps)
- then have "f' y * (x - y) - (f x - f y) \<le> 0"
- using le by auto
- then have res: "f' y * (x - y) \<le> f x - f y"
- by auto
- have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
- using * z1 by auto
- also have "\<dots> = (z1 - x) * f'' z2"
- using z2 by auto
- finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
- by simp
- have A: "z1 - x \<ge> 0"
- using z1 by auto
- have "z2 \<in> C"
- using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
- by fastforce
- then have B: "f'' z2 \<ge> 0"
- using assms by auto
- from A B have "(z1 - x) * f'' z2 \<ge> 0"
- by auto
- with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
- by auto
- from mult_right_mono[OF this ge(2)]
- have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
- by (simp add: algebra_simps)
- then have "f y - f x - f' x * (y - x) \<ge> 0"
- using ge by auto
- then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
- using res by auto
- qed
- show ?thesis
- proof (cases "x = y")
- case True
- with x y show ?thesis by auto
- next
- case False
- with less_imp x y show ?thesis
- by (auto simp: neq_iff)
- qed
-qed
-
-lemma f''_ge0_imp_convex:
- fixes f :: "real \<Rightarrow> real"
- assumes conv: "convex C"
- and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- shows "convex_on C f"
- using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
- by fastforce
-
-lemma minus_log_convex:
- fixes b :: real
- assumes "b > 1"
- shows "convex_on {0 <..} (\<lambda> x. - log b x)"
-proof -
- have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
- using DERIV_log by auto
- then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
- by (auto simp: DERIV_minus)
- have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
- using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
- from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
- have "\<And>z::real. z > 0 \<Longrightarrow>
- DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
- by auto
- then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
- DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
- unfolding inverse_eq_divide by (auto simp: mult.assoc)
- have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
- using \<open>b > 1\<close> by (auto intro!: less_imp_le)
- from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
- show ?thesis
- by auto
-qed
-
-
-subsection%unimportant \<open>Convexity of real functions\<close>
-
-lemma convex_on_realI:
- assumes "connected A"
- and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
- and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
- shows "convex_on A f"
-proof (rule convex_on_linorderI)
- fix t x y :: real
- assume t: "t > 0" "t < 1"
- assume xy: "x \<in> A" "y \<in> A" "x < y"
- define z where "z = (1 - t) * x + t * y"
- with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
- using connected_contains_Icc by blast
-
- from xy t have xz: "z > x"
- by (simp add: z_def algebra_simps)
- have "y - z = (1 - t) * (y - x)"
- by (simp add: z_def algebra_simps)
- also from xy t have "\<dots> > 0"
- by (intro mult_pos_pos) simp_all
- finally have yz: "z < y"
- by simp
-
- from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
- by (intro MVT2) (auto intro!: assms(2))
- then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
- by auto
- from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
- by (intro MVT2) (auto intro!: assms(2))
- then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
- by auto
-
- from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
- also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
- by auto
- with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
- by (intro assms(3)) auto
- also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
- finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
- using xz yz by (simp add: field_simps)
- also have "z - x = t * (y - x)"
- by (simp add: z_def algebra_simps)
- also have "y - z = (1 - t) * (y - x)"
- by (simp add: z_def algebra_simps)
- finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
- using xy by simp
- then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
- by (simp add: z_def algebra_simps)
-qed
-
-lemma convex_on_inverse:
- assumes "A \<subseteq> {0<..}"
- shows "convex_on A (inverse :: real \<Rightarrow> real)"
-proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
- fix u v :: real
- assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
- with assms show "-inverse (u^2) \<le> -inverse (v^2)"
- by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
-qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
-
-lemma convex_onD_Icc':
- assumes "convex_on {x..y} f" "c \<in> {x..y}"
- defines "d \<equiv> y - x"
- shows "f c \<le> (f y - f x) / d * (c - x) + f x"
-proof (cases x y rule: linorder_cases)
- case less
- then have d: "d > 0"
- by (simp add: d_def)
- from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
- have "f c = f (x + (c - x) * 1)"
- by simp
- also from less have "1 = ((y - x) / d)"
- by (simp add: d_def)
- also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
- by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
- using assms less by (intro convex_onD_Icc) simp_all
- also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
- by (simp add: field_simps)
- finally show ?thesis .
-qed (insert assms(2), simp_all)
-
-lemma convex_onD_Icc'':
- assumes "convex_on {x..y} f" "c \<in> {x..y}"
- defines "d \<equiv> y - x"
- shows "f c \<le> (f x - f y) / d * (y - c) + f y"
-proof (cases x y rule: linorder_cases)
- case less
- then have d: "d > 0"
- by (simp add: d_def)
- from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
- have "f c = f (y - (y - c) * 1)"
- by simp
- also from less have "1 = ((y - x) / d)"
- by (simp add: d_def)
- also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
- by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
- using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
- also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
- by (simp add: field_simps)
- finally show ?thesis .
-qed (insert assms(2), simp_all)
+subsection%unimportant \<open>Topological Properties of Convex Sets and Functions\<close>
lemma convex_supp_sum:
assumes "convex S" and 1: "supp_sum u I = 1"
@@ -1160,14 +32,6 @@
done
qed
-lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
- by (metis convex_translation translation_galois)
-
-lemma convex_linear_image_eq [simp]:
- fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
- shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
- by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
-
lemma closure_bounded_linear_image_subset:
assumes f: "bounded_linear f"
shows "f ` closure S \<subseteq> closure (f ` S)"
@@ -1238,822 +102,11 @@
by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
qed
-lemma fst_linear: "linear fst"
- unfolding linear_iff by (simp add: algebra_simps)
-
-lemma snd_linear: "linear snd"
- unfolding linear_iff by (simp add: algebra_simps)
-
-lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
- unfolding linear_iff by (simp add: algebra_simps)
-
-lemma vector_choose_size:
- assumes "0 \<le> c"
- obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
-proof -
- obtain a::'a where "a \<noteq> 0"
- using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
- then show ?thesis
- by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
-qed
-
-lemma vector_choose_dist:
- assumes "0 \<le> c"
- obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
-by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
-
lemma sphere_eq_empty [simp]:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "sphere a r = {} \<longleftrightarrow> r < 0"
by (auto simp: sphere_def dist_norm) (metis dist_norm le_less_linear vector_choose_dist)
-lemma sum_delta_notmem:
- assumes "x \<notin> s"
- shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
- and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
- apply (rule_tac [!] sum.cong)
- using assms
- apply auto
- done
-
-lemma sum_delta'':
- fixes s::"'a::real_vector set"
- assumes "finite s"
- shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
-proof -
- have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
- by auto
- show ?thesis
- unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
-qed
-
-lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
- by (fact if_distrib)
-
-lemma dist_triangle_eq:
- fixes x y z :: "'a::real_inner"
- shows "dist x z = dist x y + dist y z \<longleftrightarrow>
- norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
-proof -
- have *: "x - y + (y - z) = x - z" by auto
- show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
- by (auto simp:norm_minus_commute)
-qed
-
-
-subsection \<open>Affine set and affine hull\<close>
-
-definition%important affine :: "'a::real_vector set \<Rightarrow> bool"
- where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
-
-lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
- unfolding affine_def by (metis eq_diff_eq')
-
-lemma affine_empty [iff]: "affine {}"
- unfolding affine_def by auto
-
-lemma affine_sing [iff]: "affine {x}"
- unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
-
-lemma affine_UNIV [iff]: "affine UNIV"
- unfolding affine_def by auto
-
-lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
- unfolding affine_def by auto
-
-lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
- unfolding affine_def by auto
-
-lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
- apply (clarsimp simp add: affine_def)
- apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
- apply (auto simp: algebra_simps)
- done
-
-lemma affine_affine_hull [simp]: "affine(affine hull s)"
- unfolding hull_def
- using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
-
-lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
- by (metis affine_affine_hull hull_same)
-
-lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
- by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
-
-
-subsubsection%unimportant \<open>Some explicit formulations\<close>
-
-text "Formalized by Lars Schewe."
-
-lemma affine:
- fixes V::"'a::real_vector set"
- shows "affine V \<longleftrightarrow>
- (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
-proof -
- have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
- and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
- proof (cases "x = y")
- case True
- then show ?thesis
- using that by (metis scaleR_add_left scaleR_one)
- next
- case False
- then show ?thesis
- using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
- qed
- moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
- and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
- proof -
- define n where "n = card S"
- consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
- then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- proof cases
- assume "card S = 1"
- then obtain a where "S={a}"
- by (auto simp: card_Suc_eq)
- then show ?thesis
- using that by simp
- next
- assume "card S = 2"
- then obtain a b where "S = {a, b}"
- by (metis Suc_1 card_1_singletonE card_Suc_eq)
- then show ?thesis
- using *[of a b] that
- by (auto simp: sum_clauses(2))
- next
- assume "card S > 2"
- then show ?thesis using that n_def
- proof (induct n arbitrary: u S)
- case 0
- then show ?case by auto
- next
- case (Suc n u S)
- have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
- using that unfolding card_eq_sum by auto
- with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
- have c: "card (S - {x}) = card S - 1"
- by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
- have "sum u (S - {x}) = 1 - u x"
- by (simp add: Suc.prems sum_diff1_ring \<open>x \<in> S\<close>)
- with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
- by auto
- have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
- proof (cases "card (S - {x}) > 2")
- case True
- then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
- using Suc.prems c by force+
- show ?thesis
- proof (rule Suc.hyps)
- show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
- by (auto simp: eq1 sum_distrib_left[symmetric])
- qed (use S Suc.prems True in auto)
- next
- case False
- then have "card (S - {x}) = Suc (Suc 0)"
- using Suc.prems c by auto
- then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
- unfolding card_Suc_eq by auto
- then show ?thesis
- using eq1 \<open>S \<subseteq> V\<close>
- by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
- qed
- have "u x + (1 - u x) = 1 \<Longrightarrow>
- u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
- by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
- moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
- by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
- ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
- by (simp add: x)
- qed
- qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
- qed
- ultimately show ?thesis
- unfolding affine_def by meson
-qed
-
-
-lemma affine_hull_explicit:
- "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
- (is "_ = ?rhs")
-proof (rule hull_unique)
- show "p \<subseteq> ?rhs"
- proof (intro subsetI CollectI exI conjI)
- show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
- by auto
- qed auto
- show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
- using that unfolding affine by blast
- show "affine ?rhs"
- unfolding affine_def
- proof clarify
- fix u v :: real and sx ux sy uy
- assume uv: "u + v = 1"
- and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
- and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)"
- have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
- by auto
- show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
- sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
- proof (intro exI conjI)
- show "finite (sx \<union> sy)"
- using x y by auto
- show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
- using x y uv
- by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
- have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
- = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
- using x y
- unfolding scaleR_left_distrib scaleR_zero_left if_smult
- by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **)
- also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
- unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
- finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
- = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
- qed (use x y in auto)
- qed
-qed
-
-lemma affine_hull_finite:
- assumes "finite S"
- shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
-proof -
- have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x"
- if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
- proof -
- have "S \<inter> F = F"
- using that by auto
- show ?thesis
- proof (intro exI conjI)
- show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
- by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
- show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
- by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
- qed
- qed
- show ?thesis
- unfolding affine_hull_explicit using assms
- by (fastforce dest: *)
-qed
-
-
-subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close>
-
-lemma affine_hull_empty[simp]: "affine hull {} = {}"
- by simp
-
-lemma affine_hull_finite_step:
- fixes y :: "'a::real_vector"
- shows "finite S \<Longrightarrow>
- (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
- (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
-proof -
- assume fin: "finite S"
- show "?lhs = ?rhs"
- proof
- assume ?lhs
- then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
- by auto
- show ?rhs
- proof (cases "a \<in> S")
- case True
- then show ?thesis
- using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
- next
- case False
- show ?thesis
- by (rule exI [where x="u a"]) (use u fin False in auto)
- qed
- next
- assume ?rhs
- then obtain v u where vu: "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
- by auto
- have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
- by auto
- show ?lhs
- proof (cases "a \<in> S")
- case True
- show ?thesis
- by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
- (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
- next
- case False
- then show ?thesis
- apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
- apply (simp add: vu sum_clauses(2)[OF fin] *)
- by (simp add: sum_delta_notmem(3) vu)
- qed
- qed
-qed
-
-lemma affine_hull_2:
- fixes a b :: "'a::real_vector"
- shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
- (is "?lhs = ?rhs")
-proof -
- have *:
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
- have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
- using affine_hull_finite[of "{a,b}"] by auto
- also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
- by (simp add: affine_hull_finite_step[of "{b}" a])
- also have "\<dots> = ?rhs" unfolding * by auto
- finally show ?thesis by auto
-qed
-
-lemma affine_hull_3:
- fixes a b c :: "'a::real_vector"
- shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
-proof -
- have *:
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
- "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
- show ?thesis
- apply (simp add: affine_hull_finite affine_hull_finite_step)
- unfolding *
- apply safe
- apply (metis add.assoc)
- apply (rule_tac x=u in exI, force)
- done
-qed
-
-lemma mem_affine:
- assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
- using assms affine_def[of S] by auto
-
-lemma mem_affine_3:
- assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
-proof -
- have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
- using affine_hull_3[of x y z] assms by auto
- moreover
- have "affine hull {x, y, z} \<subseteq> affine hull S"
- using hull_mono[of "{x, y, z}" "S"] assms by auto
- moreover
- have "affine hull S = S"
- using assms affine_hull_eq[of S] by auto
- ultimately show ?thesis by auto
-qed
-
-lemma mem_affine_3_minus:
- assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
- shows "x + v *\<^sub>R (y-z) \<in> S"
- using mem_affine_3[of S x y z 1 v "-v"] assms
- by (simp add: algebra_simps)
-
-corollary mem_affine_3_minus2:
- "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
- by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
-
-
-subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close>
-
-lemma affine_hull_insert_subset_span:
- "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
-proof -
- have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
- if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
- for x F u
- proof -
- have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
- using that by auto
- show ?thesis
- proof (intro exI conjI)
- show "finite ((\<lambda>x. x - a) ` (F - {a}))"
- by (simp add: that(1))
- show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
- by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
- sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
- qed (use \<open>F \<subseteq> insert a S\<close> in auto)
- qed
- then show ?thesis
- unfolding affine_hull_explicit span_explicit by blast
-qed
-
-lemma affine_hull_insert_span:
- assumes "a \<notin> S"
- shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}"
-proof -
- have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
- if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
- proof -
- from that
- obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
- unfolding span_explicit by auto
- define F where "F = (\<lambda>x. x + a) ` T"
- have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
- unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
- have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
- using F assms by auto
- show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
- apply (rule_tac x = "insert a F" in exI)
- apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
- using assms F
- apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
- done
- qed
- show ?thesis
- by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
-qed
-
-lemma affine_hull_span:
- assumes "a \<in> S"
- shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
- using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
-
-
-subsubsection%unimportant \<open>Parallel affine sets\<close>
-
-definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
- where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
-
-lemma affine_parallel_expl_aux:
- fixes S T :: "'a::real_vector set"
- assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
- shows "T = (\<lambda>x. a + x) ` S"
-proof -
- have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
- using that
- by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
- moreover have "T \<ge> (\<lambda>x. a + x) ` S"
- using assms by auto
- ultimately show ?thesis by auto
-qed
-
-lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
- unfolding affine_parallel_def
- using affine_parallel_expl_aux[of S _ T] by auto
-
-lemma affine_parallel_reflex: "affine_parallel S S"
- unfolding affine_parallel_def
- using image_add_0 by blast
-
-lemma affine_parallel_commut:
- assumes "affine_parallel A B"
- shows "affine_parallel B A"
-proof -
- from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
- unfolding affine_parallel_def by auto
- have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
- from B show ?thesis
- using translation_galois [of B a A]
- unfolding affine_parallel_def by auto
-qed
-
-lemma affine_parallel_assoc:
- assumes "affine_parallel A B"
- and "affine_parallel B C"
- shows "affine_parallel A C"
-proof -
- from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
- unfolding affine_parallel_def by auto
- moreover
- from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
- unfolding affine_parallel_def by auto
- ultimately show ?thesis
- using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
-qed
-
-lemma affine_translation_aux:
- fixes a :: "'a::real_vector"
- assumes "affine ((\<lambda>x. a + x) ` S)"
- shows "affine S"
-proof -
- {
- fix x y u v
- assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
- then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
- by auto
- then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
- using xy assms unfolding affine_def by auto
- have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
- by (simp add: algebra_simps)
- also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
- using \<open>u + v = 1\<close> by auto
- ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
- using h1 by auto
- then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
- }
- then show ?thesis unfolding affine_def by auto
-qed
-
-lemma affine_translation:
- fixes a :: "'a::real_vector"
- shows "affine S \<longleftrightarrow> affine ((\<lambda>x. a + x) ` S)"
-proof -
- have "affine S \<Longrightarrow> affine ((\<lambda>x. a + x) ` S)"
- using affine_translation_aux[of "-a" "((\<lambda>x. a + x) ` S)"]
- using translation_assoc[of "-a" a S] by auto
- then show ?thesis using affine_translation_aux by auto
-qed
-
-lemma parallel_is_affine:
- fixes S T :: "'a::real_vector set"
- assumes "affine S" "affine_parallel S T"
- shows "affine T"
-proof -
- from assms obtain a where "T = (\<lambda>x. a + x) ` S"
- unfolding affine_parallel_def by auto
- then show ?thesis
- using affine_translation assms by auto
-qed
-
-lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
- unfolding subspace_def affine_def by auto
-
-
-subsubsection%unimportant \<open>Subspace parallel to an affine set\<close>
-
-lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
-proof -
- have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
- using subspace_imp_affine[of S] subspace_0 by auto
- {
- assume assm: "affine S \<and> 0 \<in> S"
- {
- fix c :: real
- fix x
- assume x: "x \<in> S"
- have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
- moreover
- have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
- using affine_alt[of S] assm x by auto
- ultimately have "c *\<^sub>R x \<in> S" by auto
- }
- then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
-
- {
- fix x y
- assume xy: "x \<in> S" "y \<in> S"
- define u where "u = (1 :: real)/2"
- have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
- by auto
- moreover
- have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
- by (simp add: algebra_simps)
- moreover
- have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
- using affine_alt[of S] assm xy by auto
- ultimately
- have "(1/2) *\<^sub>R (x+y) \<in> S"
- using u_def by auto
- moreover
- have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
- by auto
- ultimately
- have "x + y \<in> S"
- using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
- }
- then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
- by auto
- then have "subspace S"
- using h1 assm unfolding subspace_def by auto
- }
- then show ?thesis using h0 by metis
-qed
-
-lemma affine_diffs_subspace:
- assumes "affine S" "a \<in> S"
- shows "subspace ((\<lambda>x. (-a)+x) ` S)"
-proof -
- have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
- have "affine ((\<lambda>x. (-a)+x) ` S)"
- using affine_translation assms by auto
- moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
- using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
- ultimately show ?thesis using subspace_affine by auto
-qed
-
-lemma parallel_subspace_explicit:
- assumes "affine S"
- and "a \<in> S"
- assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
- shows "subspace L \<and> affine_parallel S L"
-proof -
- from assms have "L = plus (- a) ` S" by auto
- then have par: "affine_parallel S L"
- unfolding affine_parallel_def ..
- then have "affine L" using assms parallel_is_affine by auto
- moreover have "0 \<in> L"
- using assms by auto
- ultimately show ?thesis
- using subspace_affine par by auto
-qed
-
-lemma parallel_subspace_aux:
- assumes "subspace A"
- and "subspace B"
- and "affine_parallel A B"
- shows "A \<supseteq> B"
-proof -
- from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
- using affine_parallel_expl[of A B] by auto
- then have "-a \<in> A"
- using assms subspace_0[of B] by auto
- then have "a \<in> A"
- using assms subspace_neg[of A "-a"] by auto
- then show ?thesis
- using assms a unfolding subspace_def by auto
-qed
-
-lemma parallel_subspace:
- assumes "subspace A"
- and "subspace B"
- and "affine_parallel A B"
- shows "A = B"
-proof
- show "A \<supseteq> B"
- using assms parallel_subspace_aux by auto
- show "A \<subseteq> B"
- using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
-qed
-
-lemma affine_parallel_subspace:
- assumes "affine S" "S \<noteq> {}"
- shows "\<exists>!L. subspace L \<and> affine_parallel S L"
-proof -
- have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
- using assms parallel_subspace_explicit by auto
- {
- fix L1 L2
- assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
- then have "affine_parallel L1 L2"
- using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
- then have "L1 = L2"
- using ass parallel_subspace by auto
- }
- then show ?thesis using ex by auto
-qed
-
-
-subsection \<open>Cones\<close>
-
-definition%important cone :: "'a::real_vector set \<Rightarrow> bool"
- where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
-
-lemma cone_empty[intro, simp]: "cone {}"
- unfolding cone_def by auto
-
-lemma cone_univ[intro, simp]: "cone UNIV"
- unfolding cone_def by auto
-
-lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
- unfolding cone_def by auto
-
-lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
- by (simp add: cone_def subspace_scale)
-
-
-subsubsection \<open>Conic hull\<close>
-
-lemma cone_cone_hull: "cone (cone hull s)"
- unfolding hull_def by auto
-
-lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
- apply (rule hull_eq)
- using cone_Inter
- unfolding subset_eq
- apply auto
- done
-
-lemma mem_cone:
- assumes "cone S" "x \<in> S" "c \<ge> 0"
- shows "c *\<^sub>R x \<in> S"
- using assms cone_def[of S] by auto
-
-lemma cone_contains_0:
- assumes "cone S"
- shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
-proof -
- {
- assume "S \<noteq> {}"
- then obtain a where "a \<in> S" by auto
- then have "0 \<in> S"
- using assms mem_cone[of S a 0] by auto
- }
- then show ?thesis by auto
-qed
-
-lemma cone_0: "cone {0}"
- unfolding cone_def by auto
-
-lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
- unfolding cone_def by blast
-
-lemma cone_iff:
- assumes "S \<noteq> {}"
- shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
-proof -
- {
- assume "cone S"
- {
- fix c :: real
- assume "c > 0"
- {
- fix x
- assume "x \<in> S"
- then have "x \<in> ((*\<^sub>R) c) ` S"
- unfolding image_def
- using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
- exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
- by auto
- }
- moreover
- {
- fix x
- assume "x \<in> ((*\<^sub>R) c) ` S"
- then have "x \<in> S"
- using \<open>cone S\<close> \<open>c > 0\<close>
- unfolding cone_def image_def \<open>c > 0\<close> by auto
- }
- ultimately have "((*\<^sub>R) c) ` S = S" by auto
- }
- then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
- using \<open>cone S\<close> cone_contains_0[of S] assms by auto
- }
- moreover
- {
- assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
- {
- fix x
- assume "x \<in> S"
- fix c1 :: real
- assume "c1 \<ge> 0"
- then have "c1 = 0 \<or> c1 > 0" by auto
- then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
- }
- then have "cone S" unfolding cone_def by auto
- }
- ultimately show ?thesis by blast
-qed
-
-lemma cone_hull_empty: "cone hull {} = {}"
- by (metis cone_empty cone_hull_eq)
-
-lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
- by (metis bot_least cone_hull_empty hull_subset xtrans(5))
-
-lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
- using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
- by auto
-
-lemma mem_cone_hull:
- assumes "x \<in> S" "c \<ge> 0"
- shows "c *\<^sub>R x \<in> cone hull S"
- by (metis assms cone_cone_hull hull_inc mem_cone)
-
-proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
- (is "?lhs = ?rhs")
-proof -
- {
- fix x
- assume "x \<in> ?rhs"
- then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
- by auto
- fix c :: real
- assume c: "c \<ge> 0"
- then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
- using x by (simp add: algebra_simps)
- moreover
- have "c * cx \<ge> 0" using c x by auto
- ultimately
- have "c *\<^sub>R x \<in> ?rhs" using x by auto
- }
- then have "cone ?rhs"
- unfolding cone_def by auto
- then have "?rhs \<in> Collect cone"
- unfolding mem_Collect_eq by auto
- {
- fix x
- assume "x \<in> S"
- then have "1 *\<^sub>R x \<in> ?rhs"
- apply auto
- apply (rule_tac x = 1 in exI, auto)
- done
- then have "x \<in> ?rhs" by auto
- }
- then have "S \<subseteq> ?rhs" by auto
- then have "?lhs \<subseteq> ?rhs"
- using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
- moreover
- {
- fix x
- assume "x \<in> ?rhs"
- then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
- by auto
- then have "xx \<in> cone hull S"
- using hull_subset[of S] by auto
- then have "x \<in> ?lhs"
- using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
- }
- ultimately show ?thesis by auto
-qed
-
lemma cone_closure:
fixes S :: "'a::real_normed_vector set"
assumes "cone S"
@@ -2071,122 +124,6 @@
using False cone_iff[of "closure S"] by auto
qed
-
-subsection \<open>Affine dependence and consequential theorems\<close>
-
-text "Formalized by Lars Schewe."
-
-definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
- where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
-
-lemma affine_dependent_subset:
- "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
-apply (simp add: affine_dependent_def Bex_def)
-apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
-done
-
-lemma affine_independent_subset:
- shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
-by (metis affine_dependent_subset)
-
-lemma affine_independent_Diff:
- "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
-by (meson Diff_subset affine_dependent_subset)
-
-proposition affine_dependent_explicit:
- "affine_dependent p \<longleftrightarrow>
- (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
-proof -
- have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
- if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
- proof (intro exI conjI)
- have "x \<notin> S"
- using that by auto
- then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
- using that by (simp add: sum_delta_notmem)
- show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
- using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
- qed (use that in auto)
- moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
- if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
- proof (intro bexI exI conjI)
- have "S \<noteq> {v}"
- using that by auto
- then show "S - {v} \<noteq> {}"
- using that by auto
- show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
- unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
- show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
- unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
- scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>]
- using that by auto
- show "S - {v} \<subseteq> p - {v}"
- using that by auto
- qed (use that in auto)
- ultimately show ?thesis
- unfolding affine_dependent_def affine_hull_explicit by auto
-qed
-
-lemma affine_dependent_explicit_finite:
- fixes S :: "'a::real_vector set"
- assumes "finite S"
- shows "affine_dependent S \<longleftrightarrow>
- (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
- (is "?lhs = ?rhs")
-proof
- have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
- by auto
- assume ?lhs
- then obtain t u v where
- "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
- unfolding affine_dependent_explicit by auto
- then show ?rhs
- apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
- apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
- done
-next
- assume ?rhs
- then obtain u v where "sum u S = 0" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
- by auto
- then show ?lhs unfolding affine_dependent_explicit
- using assms by auto
-qed
-
-
-subsection%unimportant \<open>Connectedness of convex sets\<close>
-
-lemma connectedD:
- "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
- by (rule Topological_Spaces.topological_space_class.connectedD)
-
-lemma convex_connected:
- fixes S :: "'a::real_normed_vector set"
- assumes "convex S"
- shows "connected S"
-proof (rule connectedI)
- fix A B
- assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
- moreover
- assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
- then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
- define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
- then have "continuous_on {0 .. 1} f"
- by (auto intro!: continuous_intros)
- then have "connected (f ` {0 .. 1})"
- by (auto intro!: connected_continuous_image)
- note connectedD[OF this, of A B]
- moreover have "a \<in> A \<inter> f ` {0 .. 1}"
- using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
- moreover have "b \<in> B \<inter> f ` {0 .. 1}"
- using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
- moreover have "f ` {0 .. 1} \<subseteq> S"
- using \<open>convex S\<close> a b unfolding convex_def f_def by auto
- ultimately show False by auto
-qed
-
-corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
- by (simp add: convex_connected)
-
corollary component_complement_connected:
fixes S :: "'a::real_normed_vector set"
assumes "connected S" "C \<in> components (-S)"
@@ -2216,15 +153,6 @@
text \<open>Balls, being convex, are connected.\<close>
-lemma convex_prod:
- assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
- shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
- using assms unfolding convex_def
- by (auto simp: inner_add_left)
-
-lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
- by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval)
-
lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
assumes "e > 0"
@@ -2310,20 +238,6 @@
using convex_connected convex_cball by auto
-subsection \<open>Convex hull\<close>
-
-lemma convex_convex_hull [iff]: "convex (convex hull s)"
- unfolding hull_def
- using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
- by auto
-
-lemma convex_hull_subset:
- "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
- by (simp add: convex_convex_hull subset_hull)
-
-lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
- by (metis convex_convex_hull hull_same)
-
lemma bounded_convex_hull:
fixes s :: "'a::real_normed_vector set"
assumes "bounded s"
@@ -2345,1499 +259,6 @@
using bounded_convex_hull finite_imp_bounded
by auto
-
-subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close>
-
-lemma convex_hull_linear_image:
- assumes f: "linear f"
- shows "f ` (convex hull s) = convex hull (f ` s)"
-proof
- show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
- by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
- show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
- proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
- show "s \<subseteq> f -` (convex hull (f ` s))"
- by (fast intro: hull_inc)
- show "convex (f -` (convex hull (f ` s)))"
- by (intro convex_linear_vimage [OF f] convex_convex_hull)
- qed
-qed
-
-lemma in_convex_hull_linear_image:
- assumes "linear f"
- and "x \<in> convex hull s"
- shows "f x \<in> convex hull (f ` s)"
- using convex_hull_linear_image[OF assms(1)] assms(2) by auto
-
-lemma convex_hull_Times:
- "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
-proof
- show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
- by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
- have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
- proof (rule hull_induct [OF x], rule hull_induct [OF y])
- fix x y assume "x \<in> s" and "y \<in> t"
- then show "(x, y) \<in> convex hull (s \<times> t)"
- by (simp add: hull_inc)
- next
- fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
- have "convex ?S"
- by (intro convex_linear_vimage convex_translation convex_convex_hull,
- simp add: linear_iff)
- also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
- by (auto simp: image_def Bex_def)
- finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
- next
- show "convex {x. (x, y) \<in> convex hull s \<times> t}"
- proof -
- fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
- have "convex ?S"
- by (intro convex_linear_vimage convex_translation convex_convex_hull,
- simp add: linear_iff)
- also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
- by (auto simp: image_def Bex_def)
- finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
- qed
- qed
- then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
- unfolding subset_eq split_paired_Ball_Sigma by blast
-qed
-
-
-subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close>
-
-lemma convex_hull_empty[simp]: "convex hull {} = {}"
- by (rule hull_unique) auto
-
-lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
- by (rule hull_unique) auto
-
-lemma convex_hull_insert:
- fixes S :: "'a::real_vector set"
- assumes "S \<noteq> {}"
- shows "convex hull (insert a S) =
- {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
- (is "_ = ?hull")
-proof (intro equalityI hull_minimal subsetI)
- fix x
- assume "x \<in> insert a S"
- then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
- unfolding insert_iff
- proof
- assume "x = a"
- then show ?thesis
- by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
- next
- assume "x \<in> S"
- with hull_subset[of S convex] show ?thesis
- by force
- qed
- then show "x \<in> ?hull"
- by simp
-next
- fix x
- assume "x \<in> ?hull"
- then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
- by auto
- have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
- using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
- by auto
- then show "x \<in> convex hull insert a S"
- unfolding obt(5) using obt(1-3)
- by (rule convexD [OF convex_convex_hull])
-next
- show "convex ?hull"
- proof (rule convexI)
- fix x y u v
- assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
- from x obtain u1 v1 b1 where
- obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
- by auto
- from y obtain u2 v2 b2 where
- obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
- by auto
- have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
- by (auto simp: algebra_simps)
- have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
- (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
- proof (cases "u * v1 + v * v2 = 0")
- case True
- have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
- by (auto simp: algebra_simps)
- have eq0: "u * v1 = 0" "v * v2 = 0"
- using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
- by arith+
- then have "u * u1 + v * u2 = 1"
- using as(3) obt1(3) obt2(3) by auto
- then show ?thesis
- using "*" eq0 as obt1(4) xeq yeq by auto
- next
- case False
- have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
- using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
- also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
- using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
- also have "\<dots> = u * v1 + v * v2"
- by simp
- finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
- let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
- have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
- using as(1,2) obt1(1,2) obt2(1,2) by auto
- show ?thesis
- proof
- show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
- unfolding xeq yeq * **
- using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
- show "?b \<in> convex hull S"
- using False zeroes obt1(4) obt2(4)
- by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib add_divide_distrib[symmetric] zero_le_divide_iff)
- qed
- qed
- then obtain b where b: "b \<in> convex hull S"
- "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
-
- have u1: "u1 \<le> 1"
- unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
- have u2: "u2 \<le> 1"
- unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
- have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
- proof (rule add_mono)
- show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
- by (simp_all add: as mult_right_mono)
- qed
- also have "\<dots> \<le> 1"
- unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
- finally have le1: "u1 * u + u2 * v \<le> 1" .
- show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
- proof (intro CollectI exI conjI)
- show "0 \<le> u * u1 + v * u2"
- by (simp add: as(1) as(2) obt1(1) obt2(1))
- show "0 \<le> 1 - u * u1 - v * u2"
- by (simp add: le1 diff_diff_add mult.commute)
- qed (use b in \<open>auto simp: algebra_simps\<close>)
- qed
-qed
-
-lemma convex_hull_insert_alt:
- "convex hull (insert a S) =
- (if S = {} then {a}
- else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
- apply (auto simp: convex_hull_insert)
- using diff_eq_eq apply fastforce
- by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
-
-subsubsection%unimportant \<open>Explicit expression for convex hull\<close>
-
-proposition convex_hull_indexed:
- fixes S :: "'a::real_vector set"
- shows "convex hull S =
- {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
- (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
- (is "?xyz = ?hull")
-proof (rule hull_unique [OF _ convexI])
- show "S \<subseteq> ?hull"
- by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
-next
- fix T
- assume "S \<subseteq> T" "convex T"
- then show "?hull \<subseteq> T"
- by (blast intro: convex_sum)
-next
- fix x y u v
- assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
- assume xy: "x \<in> ?hull" "y \<in> ?hull"
- from xy obtain k1 u1 x1 where
- x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
- "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
- by auto
- from xy obtain k2 u2 x2 where
- y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
- "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
- by auto
- have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
- "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
- by auto
- have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
- unfolding inj_on_def by auto
- let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
- let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
- show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
- proof (intro CollectI exI conjI ballI)
- show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
- using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
- show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1" "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
- unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
- sum.reindex[OF inj] Collect_mem_eq o_def
- unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
- by (simp_all add: sum_distrib_left[symmetric] x(2,3) y(2,3) uv(3))
- qed
-qed
-
-lemma convex_hull_finite:
- fixes S :: "'a::real_vector set"
- assumes "finite S"
- shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
- (is "?HULL = _")
-proof (rule hull_unique [OF _ convexI]; clarify)
- fix x
- assume "x \<in> S"
- then show "\<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) = x"
- by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
-next
- fix u v :: real
- assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
- fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
- fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
- have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
- by (simp add: that uv ux(1) uy(1))
- moreover
- have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
- unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
- using uv(3) by auto
- moreover
- have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
- unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
- by auto
- ultimately
- show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
- (\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
- by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
-qed (use assms in \<open>auto simp: convex_explicit\<close>)
-
-
-subsubsection%unimportant \<open>Another formulation\<close>
-
-text "Formalized by Lars Schewe."
-
-lemma convex_hull_explicit:
- fixes p :: "'a::real_vector set"
- shows "convex hull p =
- {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
- (is "?lhs = ?rhs")
-proof -
- {
- fix x
- assume "x\<in>?lhs"
- then obtain k u y where
- obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
- unfolding convex_hull_indexed by auto
-
- have fin: "finite {1..k}" by auto
- have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
- {
- fix j
- assume "j\<in>{1..k}"
- then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
- using obt(1)[THEN bspec[where x=j]] and obt(2)
- apply simp
- apply (rule sum_nonneg)
- using obt(1)
- apply auto
- done
- }
- moreover
- have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
- unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
- moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
- using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
- unfolding scaleR_left.sum using obt(3) by auto
- ultimately
- have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
- apply (rule_tac x="y ` {1..k}" in exI)
- apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
- done
- then have "x\<in>?rhs" by auto
- }
- moreover
- {
- fix y
- assume "y\<in>?rhs"
- then obtain S u where
- obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
- by auto
-
- obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
- using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
-
- {
- fix i :: nat
- assume "i\<in>{1..card S}"
- then have "f i \<in> S"
- using f(2) by blast
- then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
- }
- moreover have *: "finite {1..card S}" by auto
- {
- fix y
- assume "y\<in>S"
- then obtain i where "i\<in>{1..card S}" "f i = y"
- using f using image_iff[of y f "{1..card S}"]
- by auto
- then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
- apply auto
- using f(1)[unfolded inj_on_def]
- by (metis One_nat_def atLeastAtMost_iff)
- then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
- then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
- "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
- by (auto simp: sum_constant_scaleR)
- }
- then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
- unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
- and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
- unfolding f
- using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
- using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
- unfolding obt(4,5)
- by auto
- ultimately
- have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
- (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
- apply (rule_tac x="card S" in exI)
- apply (rule_tac x="u \<circ> f" in exI)
- apply (rule_tac x=f in exI, fastforce)
- done
- then have "y \<in> ?lhs"
- unfolding convex_hull_indexed by auto
- }
- ultimately show ?thesis
- unfolding set_eq_iff by blast
-qed
-
-
-subsubsection%unimportant \<open>A stepping theorem for that expansion\<close>
-
-lemma convex_hull_finite_step:
- fixes S :: "'a::real_vector set"
- assumes "finite S"
- shows
- "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
- \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
- (is "?lhs = ?rhs")
-proof (rule, case_tac[!] "a\<in>S")
- assume "a \<in> S"
- then have *: "insert a S = S" by auto
- assume ?lhs
- then show ?rhs
- unfolding * by (rule_tac x=0 in exI, auto)
-next
- assume ?lhs
- then obtain u where
- u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
- by auto
- assume "a \<notin> S"
- then show ?rhs
- apply (rule_tac x="u a" in exI)
- using u(1)[THEN bspec[where x=a]]
- apply simp
- apply (rule_tac x=u in exI)
- using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
- apply auto
- done
-next
- assume "a \<in> S"
- then have *: "insert a S = S" by auto
- have fin: "finite (insert a S)" using assms by auto
- assume ?rhs
- then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
- by auto
- show ?lhs
- apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
- unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
- unfolding sum_clauses(2)[OF assms]
- using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
- apply auto
- done
-next
- assume ?rhs
- then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
- by auto
- moreover assume "a \<notin> S"
- moreover
- have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S" "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
- using \<open>a \<notin> S\<close>
- by (auto simp: intro!: sum.cong)
- ultimately show ?lhs
- by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
-qed
-
-
-subsubsection%unimportant \<open>Hence some special cases\<close>
-
-lemma convex_hull_2:
- "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
-proof -
- have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
- by auto
- have **: "finite {b}" by auto
- show ?thesis
- apply (simp add: convex_hull_finite)
- unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
- apply auto
- apply (rule_tac x=v in exI)
- apply (rule_tac x="1 - v" in exI, simp)
- apply (rule_tac x=u in exI, simp)
- apply (rule_tac x="\<lambda>x. v" in exI, simp)
- done
-qed
-
-lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
- unfolding convex_hull_2
-proof (rule Collect_cong)
- have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
- by auto
- fix x
- show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
- (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
- unfolding *
- apply auto
- apply (rule_tac[!] x=u in exI)
- apply (auto simp: algebra_simps)
- done
-qed
-
-lemma convex_hull_3:
- "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
-proof -
- have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
- by auto
- have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
- by (auto simp: field_simps)
- show ?thesis
- unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
- unfolding convex_hull_finite_step[OF fin(3)]
- apply (rule Collect_cong, simp)
- apply auto
- apply (rule_tac x=va in exI)
- apply (rule_tac x="u c" in exI, simp)
- apply (rule_tac x="1 - v - w" in exI, simp)
- apply (rule_tac x=v in exI, simp)
- apply (rule_tac x="\<lambda>x. w" in exI, simp)
- done
-qed
-
-lemma convex_hull_3_alt:
- "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
-proof -
- have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
- by auto
- show ?thesis
- unfolding convex_hull_3
- apply (auto simp: *)
- apply (rule_tac x=v in exI)
- apply (rule_tac x=w in exI)
- apply (simp add: algebra_simps)
- apply (rule_tac x=u in exI)
- apply (rule_tac x=v in exI)
- apply (simp add: algebra_simps)
- done
-qed
-
-
-subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close>
-
-lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
- unfolding affine_def convex_def by auto
-
-lemma convex_affine_hull [simp]: "convex (affine hull S)"
- by (simp add: affine_imp_convex)
-
-lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
- using subspace_imp_affine affine_imp_convex by auto
-
-lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
- by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
-
-lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
- by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
-
-lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
- by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
-
-lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
- unfolding affine_dependent_def dependent_def
- using affine_hull_subset_span by auto
-
-lemma dependent_imp_affine_dependent:
- assumes "dependent {x - a| x . x \<in> s}"
- and "a \<notin> s"
- shows "affine_dependent (insert a s)"
-proof -
- from assms(1)[unfolded dependent_explicit] obtain S u v
- where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
- by auto
- define t where "t = (\<lambda>x. x + a) ` S"
-
- have inj: "inj_on (\<lambda>x. x + a) S"
- unfolding inj_on_def by auto
- have "0 \<notin> S"
- using obt(2) assms(2) unfolding subset_eq by auto
- have fin: "finite t" and "t \<subseteq> s"
- unfolding t_def using obt(1,2) by auto
- then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
- by auto
- moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
- apply (rule sum.cong)
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
- apply auto
- done
- have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
- unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
- moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
- using obt(3,4) \<open>0\<notin>S\<close>
- by (rule_tac x="v + a" in bexI) (auto simp: t_def)
- moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
- have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
- unfolding scaleR_left.sum
- unfolding t_def and sum.reindex[OF inj] and o_def
- using obt(5)
- by (auto simp: sum.distrib scaleR_right_distrib)
- then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
- unfolding sum_clauses(2)[OF fin]
- using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
- by (auto simp: *)
- ultimately show ?thesis
- unfolding affine_dependent_explicit
- apply (rule_tac x="insert a t" in exI, auto)
- done
-qed
-
-lemma convex_cone:
- "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
- (is "?lhs = ?rhs")
-proof -
- {
- fix x y
- assume "x\<in>s" "y\<in>s" and ?lhs
- then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
- unfolding cone_def by auto
- then have "x + y \<in> s"
- using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
- apply (erule_tac x="2*\<^sub>R x" in ballE)
- apply (erule_tac x="2*\<^sub>R y" in ballE)
- apply (erule_tac x="1/2" in allE, simp)
- apply (erule_tac x="1/2" in allE, auto)
- done
- }
- then show ?thesis
- unfolding convex_def cone_def by blast
-qed
-
-lemma affine_dependent_biggerset:
- fixes s :: "'a::euclidean_space set"
- assumes "finite s" "card s \<ge> DIM('a) + 2"
- shows "affine_dependent s"
-proof -
- have "s \<noteq> {}" using assms by auto
- then obtain a where "a\<in>s" by auto
- have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
- by auto
- have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
- unfolding * by (simp add: card_image inj_on_def)
- also have "\<dots> > DIM('a)" using assms(2)
- unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
- finally show ?thesis
- apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
- apply (rule dependent_imp_affine_dependent)
- apply (rule dependent_biggerset, auto)
- done
-qed
-
-lemma affine_dependent_biggerset_general:
- assumes "finite (S :: 'a::euclidean_space set)"
- and "card S \<ge> dim S + 2"
- shows "affine_dependent S"
-proof -
- from assms(2) have "S \<noteq> {}" by auto
- then obtain a where "a\<in>S" by auto
- have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
- by auto
- have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
- by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
- have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
- using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
- also have "\<dots> < dim S + 1" by auto
- also have "\<dots> \<le> card (S - {a})"
- using assms
- using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
- by auto
- finally show ?thesis
- apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
- apply (rule dependent_imp_affine_dependent)
- apply (rule dependent_biggerset_general)
- unfolding **
- apply auto
- done
-qed
-
-
-subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close>
-
-lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
- by (simp add: affine_dependent_def)
-
-lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
- by (simp add: affine_dependent_def)
-
-lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
- by (simp add: affine_dependent_def insert_Diff_if hull_same)
-
-lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)"
-proof -
- have "affine ((\<lambda>x. a + x) ` (affine hull S))"
- using affine_translation affine_affine_hull by blast
- moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
- using hull_subset[of S] by auto
- ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
- by (metis hull_minimal)
- have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))"
- using affine_translation affine_affine_hull by blast
- moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))"
- using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto
- moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S"
- using translation_assoc[of "-a" a] by auto
- ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)"
- by (metis hull_minimal)
- then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
- by auto
- then show ?thesis using h1 by auto
-qed
-
-lemma affine_dependent_translation:
- assumes "affine_dependent S"
- shows "affine_dependent ((\<lambda>x. a + x) ` S)"
-proof -
- obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
- using assms affine_dependent_def by auto
- have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
- by auto
- then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
- using affine_hull_translation[of a "S - {x}"] x by auto
- moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
- using x by auto
- ultimately show ?thesis
- unfolding affine_dependent_def by auto
-qed
-
-lemma affine_dependent_translation_eq:
- "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
-proof -
- {
- assume "affine_dependent ((\<lambda>x. a + x) ` S)"
- then have "affine_dependent S"
- using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
- by auto
- }
- then show ?thesis
- using affine_dependent_translation by auto
-qed
-
-lemma affine_hull_0_dependent:
- assumes "0 \<in> affine hull S"
- shows "dependent S"
-proof -
- obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
- using assms affine_hull_explicit[of S] by auto
- then have "\<exists>v\<in>s. u v \<noteq> 0"
- using sum_not_0[of "u" "s"] by auto
- then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
- using s_u by auto
- then show ?thesis
- unfolding dependent_explicit[of S] by auto
-qed
-
-lemma affine_dependent_imp_dependent2:
- assumes "affine_dependent (insert 0 S)"
- shows "dependent S"
-proof -
- obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
- using affine_dependent_def[of "(insert 0 S)"] assms by blast
- then have "x \<in> span (insert 0 S - {x})"
- using affine_hull_subset_span by auto
- moreover have "span (insert 0 S - {x}) = span (S - {x})"
- using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
- ultimately have "x \<in> span (S - {x})" by auto
- then have "x \<noteq> 0 \<Longrightarrow> dependent S"
- using x dependent_def by auto
- moreover
- {
- assume "x = 0"
- then have "0 \<in> affine hull S"
- using x hull_mono[of "S - {0}" S] by auto
- then have "dependent S"
- using affine_hull_0_dependent by auto
- }
- ultimately show ?thesis by auto
-qed
-
-lemma affine_dependent_iff_dependent:
- assumes "a \<notin> S"
- shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
-proof -
- have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
- then show ?thesis
- using affine_dependent_translation_eq[of "(insert a S)" "-a"]
- affine_dependent_imp_dependent2 assms
- dependent_imp_affine_dependent[of a S]
- by (auto simp del: uminus_add_conv_diff)
-qed
-
-lemma affine_dependent_iff_dependent2:
- assumes "a \<in> S"
- shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
-proof -
- have "insert a (S - {a}) = S"
- using assms by auto
- then show ?thesis
- using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
-qed
-
-lemma affine_hull_insert_span_gen:
- "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
-proof -
- have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
- by auto
- {
- assume "a \<notin> s"
- then have ?thesis
- using affine_hull_insert_span[of a s] h1 by auto
- }
- moreover
- {
- assume a1: "a \<in> s"
- have "\<exists>x. x \<in> s \<and> -a+x=0"
- apply (rule exI[of _ a])
- using a1
- apply auto
- done
- then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
- by auto
- then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
- using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
- moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
- by auto
- moreover have "insert a (s - {a}) = insert a s"
- by auto
- ultimately have ?thesis
- using affine_hull_insert_span[of "a" "s-{a}"] by auto
- }
- ultimately show ?thesis by auto
-qed
-
-lemma affine_hull_span2:
- assumes "a \<in> s"
- shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
- using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
- by auto
-
-lemma affine_hull_span_gen:
- assumes "a \<in> affine hull s"
- shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
-proof -
- have "affine hull (insert a s) = affine hull s"
- using hull_redundant[of a affine s] assms by auto
- then show ?thesis
- using affine_hull_insert_span_gen[of a "s"] by auto
-qed
-
-lemma affine_hull_span_0:
- assumes "0 \<in> affine hull S"
- shows "affine hull S = span S"
- using affine_hull_span_gen[of "0" S] assms by auto
-
-lemma extend_to_affine_basis_nonempty:
- fixes S V :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
- shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
-proof -
- obtain a where a: "a \<in> S"
- using assms by auto
- then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))"
- using affine_dependent_iff_dependent2 assms by auto
- obtain B where B:
- "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
- using assms
- by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
- define T where "T = (\<lambda>x. a+x) ` insert 0 B"
- then have "T = insert a ((\<lambda>x. a+x) ` B)"
- by auto
- then have "affine hull T = (\<lambda>x. a+x) ` span B"
- using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
- by auto
- then have "V \<subseteq> affine hull T"
- using B assms translation_inverse_subset[of a V "span B"]
- by auto
- moreover have "T \<subseteq> V"
- using T_def B a assms by auto
- ultimately have "affine hull T = affine hull V"
- by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
- moreover have "S \<subseteq> T"
- using T_def B translation_inverse_subset[of a "S-{a}" B]
- by auto
- moreover have "\<not> affine_dependent T"
- using T_def affine_dependent_translation_eq[of "insert 0 B"]
- affine_dependent_imp_dependent2 B
- by auto
- ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
-qed
-
-lemma affine_basis_exists:
- fixes V :: "'n::euclidean_space set"
- shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
-proof (cases "V = {}")
- case True
- then show ?thesis
- using affine_independent_0 by auto
-next
- case False
- then obtain x where "x \<in> V" by auto
- then show ?thesis
- using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
- by auto
-qed
-
-proposition extend_to_affine_basis:
- fixes S V :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent S" "S \<subseteq> V"
- obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
-proof (cases "S = {}")
- case True then show ?thesis
- using affine_basis_exists by (metis empty_subsetI that)
-next
- case False
- then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
-qed
-
-
-subsection \<open>Affine Dimension of a Set\<close>
-
-definition%important aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
- where "aff_dim V =
- (SOME d :: int.
- \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
-
-lemma aff_dim_basis_exists:
- fixes V :: "('n::euclidean_space) set"
- shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
-proof -
- obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
- using affine_basis_exists[of V] by auto
- then show ?thesis
- unfolding aff_dim_def
- some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
- apply auto
- apply (rule exI[of _ "int (card B) - (1 :: int)"])
- apply (rule exI[of _ "B"], auto)
- done
-qed
-
-lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
-proof -
- have "S = {} \<Longrightarrow> affine hull S = {}"
- using affine_hull_empty by auto
- moreover have "affine hull S = {} \<Longrightarrow> S = {}"
- unfolding hull_def by auto
- ultimately show ?thesis by blast
-qed
-
-lemma aff_dim_parallel_subspace_aux:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B" "a \<in> B"
- shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
-proof -
- have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
- using affine_dependent_iff_dependent2 assms by auto
- then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
- "finite ((\<lambda>x. -a + x) ` (B - {a}))"
- using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
- show ?thesis
- proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
- case True
- have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
- using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
- then have "B = {a}" using True by auto
- then show ?thesis using assms fin by auto
- next
- case False
- then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
- using fin by auto
- moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
- by (rule card_image) (use translate_inj_on in blast)
- ultimately have "card (B-{a}) > 0" by auto
- then have *: "finite (B - {a})"
- using card_gt_0_iff[of "(B - {a})"] by auto
- then have "card (B - {a}) = card B - 1"
- using card_Diff_singleton assms by auto
- with * show ?thesis using fin h1 by auto
- qed
-qed
-
-lemma aff_dim_parallel_subspace:
- fixes V L :: "'n::euclidean_space set"
- assumes "V \<noteq> {}"
- and "subspace L"
- and "affine_parallel (affine hull V) L"
- shows "aff_dim V = int (dim L)"
-proof -
- obtain B where
- B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
- using aff_dim_basis_exists by auto
- then have "B \<noteq> {}"
- using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
- by auto
- then obtain a where a: "a \<in> B" by auto
- define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
- moreover have "affine_parallel (affine hull B) Lb"
- using Lb_def B assms affine_hull_span2[of a B] a
- affine_parallel_commut[of "Lb" "(affine hull B)"]
- unfolding affine_parallel_def
- by auto
- moreover have "subspace Lb"
- using Lb_def subspace_span by auto
- moreover have "affine hull B \<noteq> {}"
- using assms B affine_hull_nonempty[of V] by auto
- ultimately have "L = Lb"
- using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
- by auto
- then have "dim L = dim Lb"
- by auto
- moreover have "card B - 1 = dim Lb" and "finite B"
- using Lb_def aff_dim_parallel_subspace_aux a B by auto
- ultimately show ?thesis
- using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
-qed
-
-lemma aff_independent_finite:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B"
- shows "finite B"
-proof -
- {
- assume "B \<noteq> {}"
- then obtain a where "a \<in> B" by auto
- then have ?thesis
- using aff_dim_parallel_subspace_aux assms by auto
- }
- then show ?thesis by auto
-qed
-
-lemmas independent_finite = independent_imp_finite
-
-lemma span_substd_basis:
- assumes d: "d \<subseteq> Basis"
- shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- (is "_ = ?B")
-proof -
- have "d \<subseteq> ?B"
- using d by (auto simp: inner_Basis)
- moreover have s: "subspace ?B"
- using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
- ultimately have "span d \<subseteq> ?B"
- using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
- moreover have *: "card d \<le> dim (span d)"
- using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
- span_superset[of d]
- by auto
- moreover from * have "dim ?B \<le> dim (span d)"
- using dim_substandard[OF assms] by auto
- ultimately show ?thesis
- using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
-qed
-
-lemma basis_to_substdbasis_subspace_isomorphism:
- fixes B :: "'a::euclidean_space set"
- assumes "independent B"
- shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
- f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
-proof -
- have B: "card B = dim B"
- using dim_unique[of B B "card B"] assms span_superset[of B] by auto
- have "dim B \<le> card (Basis :: 'a set)"
- using dim_subset_UNIV[of B] by simp
- from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
- by auto
- let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
- proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
- show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
- using d inner_not_same_Basis by blast
- qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
- with t \<open>card B = dim B\<close> d show ?thesis by auto
-qed
-
-lemma aff_dim_empty:
- fixes S :: "'n::euclidean_space set"
- shows "S = {} \<longleftrightarrow> aff_dim S = -1"
-proof -
- obtain B where *: "affine hull B = affine hull S"
- and "\<not> affine_dependent B"
- and "int (card B) = aff_dim S + 1"
- using aff_dim_basis_exists by auto
- moreover
- from * have "S = {} \<longleftrightarrow> B = {}"
- using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
- ultimately show ?thesis
- using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
-qed
-
-lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
- by (simp add: aff_dim_empty [symmetric])
-
-lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
- unfolding aff_dim_def using hull_hull[of _ S] by auto
-
-lemma aff_dim_affine_hull2:
- assumes "affine hull S = affine hull T"
- shows "aff_dim S = aff_dim T"
- unfolding aff_dim_def using assms by auto
-
-lemma aff_dim_unique:
- fixes B V :: "'n::euclidean_space set"
- assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
- shows "of_nat (card B) = aff_dim V + 1"
-proof (cases "B = {}")
- case True
- then have "V = {}"
- using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
- by auto
- then have "aff_dim V = (-1::int)"
- using aff_dim_empty by auto
- then show ?thesis
- using \<open>B = {}\<close> by auto
-next
- case False
- then obtain a where a: "a \<in> B" by auto
- define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
- have "affine_parallel (affine hull B) Lb"
- using Lb_def affine_hull_span2[of a B] a
- affine_parallel_commut[of "Lb" "(affine hull B)"]
- unfolding affine_parallel_def by auto
- moreover have "subspace Lb"
- using Lb_def subspace_span by auto
- ultimately have "aff_dim B = int(dim Lb)"
- using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
- moreover have "(card B) - 1 = dim Lb" "finite B"
- using Lb_def aff_dim_parallel_subspace_aux a assms by auto
- ultimately have "of_nat (card B) = aff_dim B + 1"
- using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
- then show ?thesis
- using aff_dim_affine_hull2 assms by auto
-qed
-
-lemma aff_dim_affine_independent:
- fixes B :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent B"
- shows "of_nat (card B) = aff_dim B + 1"
- using aff_dim_unique[of B B] assms by auto
-
-lemma affine_independent_iff_card:
- fixes s :: "'a::euclidean_space set"
- shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
- apply (rule iffI)
- apply (simp add: aff_dim_affine_independent aff_independent_finite)
- by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
-
-lemma aff_dim_sing [simp]:
- fixes a :: "'n::euclidean_space"
- shows "aff_dim {a} = 0"
- using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
-
-lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
-proof (clarsimp)
- assume "a \<noteq> b"
- then have "aff_dim{a,b} = card{a,b} - 1"
- using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
- also have "\<dots> = 1"
- using \<open>a \<noteq> b\<close> by simp
- finally show "aff_dim {a, b} = 1" .
-qed
-
-lemma aff_dim_inner_basis_exists:
- fixes V :: "('n::euclidean_space) set"
- shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
- \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
-proof -
- obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
- using affine_basis_exists[of V] by auto
- then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_le_card:
- fixes V :: "'n::euclidean_space set"
- assumes "finite V"
- shows "aff_dim V \<le> of_nat (card V) - 1"
-proof -
- obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
- using aff_dim_inner_basis_exists[of V] by auto
- then have "card B \<le> card V"
- using assms card_mono by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_parallel_eq:
- fixes S T :: "'n::euclidean_space set"
- assumes "affine_parallel (affine hull S) (affine hull T)"
- shows "aff_dim S = aff_dim T"
-proof -
- {
- assume "T \<noteq> {}" "S \<noteq> {}"
- then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
- using affine_parallel_subspace[of "affine hull T"]
- affine_affine_hull[of T] affine_hull_nonempty
- by auto
- then have "aff_dim T = int (dim L)"
- using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
- moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
- using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
- moreover from * have "aff_dim S = int (dim L)"
- using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
- ultimately have ?thesis by auto
- }
- moreover
- {
- assume "S = {}"
- then have "S = {}" and "T = {}"
- using assms affine_hull_nonempty
- unfolding affine_parallel_def
- by auto
- then have ?thesis using aff_dim_empty by auto
- }
- moreover
- {
- assume "T = {}"
- then have "S = {}" and "T = {}"
- using assms affine_hull_nonempty
- unfolding affine_parallel_def
- by auto
- then have ?thesis
- using aff_dim_empty by auto
- }
- ultimately show ?thesis by blast
-qed
-
-lemma aff_dim_translation_eq:
- fixes a :: "'n::euclidean_space"
- shows "aff_dim ((\<lambda>x. a + x) ` S) = aff_dim S"
-proof -
- have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
- unfolding affine_parallel_def
- apply (rule exI[of _ "a"])
- using affine_hull_translation[of a S]
- apply auto
- done
- then show ?thesis
- using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
-qed
-
-lemma aff_dim_affine:
- fixes S L :: "'n::euclidean_space set"
- assumes "S \<noteq> {}"
- and "affine S"
- and "subspace L"
- and "affine_parallel S L"
- shows "aff_dim S = int (dim L)"
-proof -
- have *: "affine hull S = S"
- using assms affine_hull_eq[of S] by auto
- then have "affine_parallel (affine hull S) L"
- using assms by (simp add: *)
- then show ?thesis
- using assms aff_dim_parallel_subspace[of S L] by blast
-qed
-
-lemma dim_affine_hull:
- fixes S :: "'n::euclidean_space set"
- shows "dim (affine hull S) = dim S"
-proof -
- have "dim (affine hull S) \<ge> dim S"
- using dim_subset by auto
- moreover have "dim (span S) \<ge> dim (affine hull S)"
- using dim_subset affine_hull_subset_span by blast
- moreover have "dim (span S) = dim S"
- using dim_span by auto
- ultimately show ?thesis by auto
-qed
-
-lemma aff_dim_subspace:
- fixes S :: "'n::euclidean_space set"
- assumes "subspace S"
- shows "aff_dim S = int (dim S)"
-proof (cases "S={}")
- case True with assms show ?thesis
- by (simp add: subspace_affine)
-next
- case False
- with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
- show ?thesis by auto
-qed
-
-lemma aff_dim_zero:
- fixes S :: "'n::euclidean_space set"
- assumes "0 \<in> affine hull S"
- shows "aff_dim S = int (dim S)"
-proof -
- have "subspace (affine hull S)"
- using subspace_affine[of "affine hull S"] affine_affine_hull assms
- by auto
- then have "aff_dim (affine hull S) = int (dim (affine hull S))"
- using assms aff_dim_subspace[of "affine hull S"] by auto
- then show ?thesis
- using aff_dim_affine_hull[of S] dim_affine_hull[of S]
- by auto
-qed
-
-lemma aff_dim_eq_dim:
- fixes S :: "'n::euclidean_space set"
- assumes "a \<in> affine hull S"
- shows "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
-proof -
- have "0 \<in> affine hull ((\<lambda>x. -a+x) ` S)"
- unfolding Convex_Euclidean_Space.affine_hull_translation
- using assms by (simp add: ab_group_add_class.ab_left_minus image_iff)
- with aff_dim_zero show ?thesis
- by (metis aff_dim_translation_eq)
-qed
-
-lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
- using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
- dim_UNIV[where 'a="'n::euclidean_space"]
- by auto
-
-lemma aff_dim_geq:
- fixes V :: "'n::euclidean_space set"
- shows "aff_dim V \<ge> -1"
-proof -
- obtain B where "affine hull B = affine hull V"
- and "\<not> affine_dependent B"
- and "int (card B) = aff_dim V + 1"
- using aff_dim_basis_exists by auto
- then show ?thesis by auto
-qed
-
-lemma aff_dim_negative_iff [simp]:
- fixes S :: "'n::euclidean_space set"
- shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
-by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
-
-lemma aff_lowdim_subset_hyperplane:
- fixes S :: "'a::euclidean_space set"
- assumes "aff_dim S < DIM('a)"
- obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
-proof (cases "S={}")
- case True
- moreover
- have "(SOME b. b \<in> Basis) \<noteq> 0"
- by (metis norm_some_Basis norm_zero zero_neq_one)
- ultimately show ?thesis
- using that by blast
-next
- case False
- then obtain c S' where "c \<notin> S'" "S = insert c S'"
- by (meson equals0I mk_disjoint_insert)
- have "dim ((+) (-c) ` S) < DIM('a)"
- by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
- then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
- using lowdim_subset_hyperplane by blast
- moreover
- have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
- proof -
- have "w-c \<in> span ((+) (- c) ` S)"
- by (simp add: span_base \<open>w \<in> S\<close>)
- with that have "w-c \<in> {x. a \<bullet> x = 0}"
- by blast
- then show ?thesis
- by (auto simp: algebra_simps)
- qed
- ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
- by blast
- then show ?thesis
- by (rule that[OF \<open>a \<noteq> 0\<close>])
-qed
-
-lemma affine_independent_card_dim_diffs:
- fixes S :: "'a :: euclidean_space set"
- assumes "\<not> affine_dependent S" "a \<in> S"
- shows "card S = dim {x - a|x. x \<in> S} + 1"
-proof -
- have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
- have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
- proof (cases "x = a")
- case True then show ?thesis by (simp add: span_clauses)
- next
- case False then show ?thesis
- using assms by (blast intro: span_base that)
- qed
- have "\<not> affine_dependent (insert a S)"
- by (simp add: assms insert_absorb)
- then have 3: "independent {b - a |b. b \<in> S - {a}}"
- using dependent_imp_affine_dependent by fastforce
- have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
- by blast
- then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
- by simp
- also have "\<dots> = card (S - {a})"
- by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
- also have "\<dots> = card S - 1"
- by (simp add: aff_independent_finite assms)
- finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
- have "finite S"
- by (meson assms aff_independent_finite)
- with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
- moreover have "dim {x - a |x. x \<in> S} = card S - 1"
- using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
- ultimately show ?thesis
- by auto
-qed
-
-lemma independent_card_le_aff_dim:
- fixes B :: "'n::euclidean_space set"
- assumes "B \<subseteq> V"
- assumes "\<not> affine_dependent B"
- shows "int (card B) \<le> aff_dim V + 1"
-proof -
- obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
- by (metis assms extend_to_affine_basis[of B V])
- then have "of_nat (card T) = aff_dim V + 1"
- using aff_dim_unique by auto
- then show ?thesis
- using T card_mono[of T B] aff_independent_finite[of T] by auto
-qed
-
-lemma aff_dim_subset:
- fixes S T :: "'n::euclidean_space set"
- assumes "S \<subseteq> T"
- shows "aff_dim S \<le> aff_dim T"
-proof -
- obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
- "of_nat (card B) = aff_dim S + 1"
- using aff_dim_inner_basis_exists[of S] by auto
- then have "int (card B) \<le> aff_dim T + 1"
- using assms independent_card_le_aff_dim[of B T] by auto
- with B show ?thesis by auto
-qed
-
-lemma aff_dim_le_DIM:
- fixes S :: "'n::euclidean_space set"
- shows "aff_dim S \<le> int (DIM('n))"
-proof -
- have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
- using aff_dim_UNIV by auto
- then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
- using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
-qed
-
-lemma affine_dim_equal:
- fixes S :: "'n::euclidean_space set"
- assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
- shows "S = T"
-proof -
- obtain a where "a \<in> S" using assms by auto
- then have "a \<in> T" using assms by auto
- define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
- then have ls: "subspace LS" "affine_parallel S LS"
- using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
- then have h1: "int(dim LS) = aff_dim S"
- using assms aff_dim_affine[of S LS] by auto
- have "T \<noteq> {}" using assms by auto
- define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
- then have lt: "subspace LT \<and> affine_parallel T LT"
- using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
- then have "int(dim LT) = aff_dim T"
- using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
- then have "dim LS = dim LT"
- using h1 assms by auto
- moreover have "LS \<le> LT"
- using LS_def LT_def assms by auto
- ultimately have "LS = LT"
- using subspace_dim_equal[of LS LT] ls lt by auto
- moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
- using LS_def by auto
- moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
- using LT_def by auto
- ultimately show ?thesis by auto
-qed
-
-lemma aff_dim_eq_0:
- fixes S :: "'a::euclidean_space set"
- shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by auto
-next
- case False
- then obtain a where "a \<in> S" by auto
- show ?thesis
- proof safe
- assume 0: "aff_dim S = 0"
- have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
- by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
- then show "\<exists>a. S = {a}"
- using \<open>a \<in> S\<close> by blast
- qed auto
-qed
-
-lemma affine_hull_UNIV:
- fixes S :: "'n::euclidean_space set"
- assumes "aff_dim S = int(DIM('n))"
- shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
-proof -
- have "S \<noteq> {}"
- using assms aff_dim_empty[of S] by auto
- have h0: "S \<subseteq> affine hull S"
- using hull_subset[of S _] by auto
- have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
- using aff_dim_UNIV assms by auto
- then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
- using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
- have h3: "aff_dim S \<le> aff_dim (affine hull S)"
- using h0 aff_dim_subset[of S "affine hull S"] assms by auto
- then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
- using h0 h1 h2 by auto
- then show ?thesis
- using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
- affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
- by auto
-qed
-
-lemma disjoint_affine_hull:
- fixes s :: "'n::euclidean_space set"
- assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
- shows "(affine hull t) \<inter> (affine hull u) = {}"
-proof -
- have "finite s" using assms by (simp add: aff_independent_finite)
- then have "finite t" "finite u" using assms finite_subset by blast+
- { fix y
- assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
- then obtain a b
- where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
- and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
- by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
- define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
- have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
- have "sum c s = 0"
- by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
- moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
- by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
- moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
- by (simp add: c_def if_smult sum_negf
- comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
- ultimately have False
- using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
- }
- then show ?thesis by blast
-qed
-
-lemma aff_dim_convex_hull:
- fixes S :: "'n::euclidean_space set"
- shows "aff_dim (convex hull S) = aff_dim S"
- using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
- hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
- aff_dim_subset[of "convex hull S" "affine hull S"]
- by auto
-
lemma aff_dim_cball:
fixes a :: "'n::euclidean_space"
assumes "e > 0"
@@ -3895,162 +316,6 @@
by (metis low_dim_interior)
-subsection \<open>Caratheodory's theorem\<close>
-
-lemma convex_hull_caratheodory_aff_dim:
- fixes p :: "('a::euclidean_space) set"
- shows "convex hull p =
- {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
- (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
- unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
-proof (intro allI iffI)
- fix y
- let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
- sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
- assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
- then obtain N where "?P N" by auto
- then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
- apply (rule_tac ex_least_nat_le, auto)
- done
- then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
- by blast
- then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
- "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
-
- have "card s \<le> aff_dim p + 1"
- proof (rule ccontr, simp only: not_le)
- assume "aff_dim p + 1 < card s"
- then have "affine_dependent s"
- using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
- by blast
- then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
- using affine_dependent_explicit_finite[OF obt(1)] by auto
- define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
- define t where "t = Min i"
- have "\<exists>x\<in>s. w x < 0"
- proof (rule ccontr, simp add: not_less)
- assume as:"\<forall>x\<in>s. 0 \<le> w x"
- then have "sum w (s - {v}) \<ge> 0"
- apply (rule_tac sum_nonneg, auto)
- done
- then have "sum w s > 0"
- unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
- using as[THEN bspec[where x=v]] \<open>v\<in>s\<close> \<open>w v \<noteq> 0\<close> by auto
- then show False using wv(1) by auto
- qed
- then have "i \<noteq> {}" unfolding i_def by auto
- then have "t \<ge> 0"
- using Min_ge_iff[of i 0 ] and obt(1)
- unfolding t_def i_def
- using obt(4)[unfolded le_less]
- by (auto simp: divide_le_0_iff)
- have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
- proof
- fix v
- assume "v \<in> s"
- then have v: "0 \<le> u v"
- using obt(4)[THEN bspec[where x=v]] by auto
- show "0 \<le> u v + t * w v"
- proof (cases "w v < 0")
- case False
- thus ?thesis using v \<open>t\<ge>0\<close> by auto
- next
- case True
- then have "t \<le> u v / (- w v)"
- using \<open>v\<in>s\<close> unfolding t_def i_def
- apply (rule_tac Min_le)
- using obt(1) apply auto
- done
- then show ?thesis
- unfolding real_0_le_add_iff
- using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
- by auto
- qed
- qed
- obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
- using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
- then have a: "a \<in> s" "u a + t * w a = 0" by auto
- have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
- unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
- have "(\<Sum>v\<in>s. u v + t * w v) = 1"
- unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
- moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
- unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
- using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
- ultimately have "?P (n - 1)"
- apply (rule_tac x="(s - {a})" in exI)
- apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
- using obt(1-3) and t and a
- apply (auto simp: * scaleR_left_distrib)
- done
- then show False
- using smallest[THEN spec[where x="n - 1"]] by auto
- qed
- then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
- (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
- using obt by auto
-qed auto
-
-lemma caratheodory_aff_dim:
- fixes p :: "('a::euclidean_space) set"
- shows "convex hull p = {x. \<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> x \<in> convex hull s}"
- (is "?lhs = ?rhs")
-proof
- show "?lhs \<subseteq> ?rhs"
- apply (subst convex_hull_caratheodory_aff_dim, clarify)
- apply (rule_tac x=s in exI)
- apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
- done
-next
- show "?rhs \<subseteq> ?lhs"
- using hull_mono by blast
-qed
-
-lemma convex_hull_caratheodory:
- fixes p :: "('a::euclidean_space) set"
- shows "convex hull p =
- {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
- (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
- (is "?lhs = ?rhs")
-proof (intro set_eqI iffI)
- fix x
- assume "x \<in> ?lhs" then show "x \<in> ?rhs"
- apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
- apply (erule ex_forward)+
- using aff_dim_le_DIM [of p]
- apply simp
- done
-next
- fix x
- assume "x \<in> ?rhs" then show "x \<in> ?lhs"
- by (auto simp: convex_hull_explicit)
-qed
-
-theorem caratheodory:
- "convex hull p =
- {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
- card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
-proof safe
- fix x
- assume "x \<in> convex hull p"
- then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
- "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
- unfolding convex_hull_caratheodory by auto
- then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
- apply (rule_tac x=s in exI)
- using hull_subset[of s convex]
- using convex_convex_hull[simplified convex_explicit, of s,
- THEN spec[where x=s], THEN spec[where x=u]]
- apply auto
- done
-next
- fix x s
- assume "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
- then show "x \<in> convex hull p"
- using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
-qed
-
-
subsection \<open>Relative interior of a set\<close>
definition%important "rel_interior S =
@@ -4731,23 +996,6 @@
by auto
-subsection%unimportant\<open>Some Properties of subset of standard basis\<close>
-
-lemma affine_hull_substd_basis:
- assumes "d \<subseteq> Basis"
- shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- (is "affine hull (insert 0 ?A) = ?B")
-proof -
- have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
- by auto
- show ?thesis
- unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
-qed
-
-lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
- by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
-
-
subsection%unimportant \<open>Openness and compactness are preserved by convex hull operation\<close>
lemma open_convex_hull[intro]:
@@ -5641,101 +1889,6 @@
using hull_subset[of S convex] convex_hull_empty by auto
-subsection%unimportant \<open>Moving and scaling convex hulls\<close>
-
-lemma convex_hull_set_plus:
- "convex hull (S + T) = convex hull S + convex hull T"
- unfolding set_plus_image
- apply (subst convex_hull_linear_image [symmetric])
- apply (simp add: linear_iff scaleR_right_distrib)
- apply (simp add: convex_hull_Times)
- done
-
-lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
- unfolding set_plus_def by auto
-
-lemma convex_hull_translation:
- "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
- unfolding translation_eq_singleton_plus
- by (simp only: convex_hull_set_plus convex_hull_singleton)
-
-lemma convex_hull_scaling:
- "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
- using linear_scaleR by (rule convex_hull_linear_image [symmetric])
-
-lemma convex_hull_affinity:
- "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
- by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
-
-
-subsection%unimportant \<open>Convexity of cone hulls\<close>
-
-lemma convex_cone_hull:
- assumes "convex S"
- shows "convex (cone hull S)"
-proof (rule convexI)
- fix x y
- assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
- then have "S \<noteq> {}"
- using cone_hull_empty_iff[of S] by auto
- fix u v :: real
- assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
- then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
- using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
- from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
- using cone_hull_expl[of S] by auto
- from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
- using cone_hull_expl[of S] by auto
- {
- assume "cx + cy \<le> 0"
- then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
- using x y by auto
- then have "u *\<^sub>R x + v *\<^sub>R y = 0"
- by auto
- then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
- using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
- }
- moreover
- {
- assume "cx + cy > 0"
- then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
- using assms mem_convex_alt[of S xx yy cx cy] x y by auto
- then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
- using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
- by (auto simp: scaleR_right_distrib)
- then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
- using x y by auto
- }
- moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
- ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
-qed
-
-lemma cone_convex_hull:
- assumes "cone S"
- shows "cone (convex hull S)"
-proof (cases "S = {}")
- case True
- then show ?thesis by auto
-next
- case False
- then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
- using cone_iff[of S] assms by auto
- {
- fix c :: real
- assume "c > 0"
- then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
- using convex_hull_scaling[of _ S] by auto
- also have "\<dots> = convex hull S"
- using * \<open>c > 0\<close> by auto
- finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
- by auto
- }
- then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
- using * hull_subset[of S convex] by auto
- then show ?thesis
- using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
-qed
-
subsection%unimportant \<open>Convex set as intersection of halfspaces\<close>
lemma convex_halfspace_intersection:
@@ -5760,300 +1913,6 @@
qed auto
-subsection \<open>Radon's theorem\<close>
-
-text "Formalized by Lars Schewe."
-
-lemma Radon_ex_lemma:
- assumes "finite c" "affine_dependent c"
- shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
-proof -
- from assms(2)[unfolded affine_dependent_explicit]
- obtain s u where
- "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
- by blast
- then show ?thesis
- apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
- unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
- apply (auto simp: Int_absorb1)
- done
-qed
-
-lemma Radon_s_lemma:
- assumes "finite s"
- and "sum f s = (0::real)"
- shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
-proof -
- have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
- by auto
- show ?thesis
- unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
- and sum.distrib[symmetric] and *
- using assms(2)
- by assumption
-qed
-
-lemma Radon_v_lemma:
- assumes "finite s"
- and "sum f s = 0"
- and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
- shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
-proof -
- have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
- using assms(3) by auto
- show ?thesis
- unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
- and sum.distrib[symmetric] and *
- using assms(2)
- apply assumption
- done
-qed
-
-lemma Radon_partition:
- assumes "finite c" "affine_dependent c"
- shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
-proof -
- obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
- using Radon_ex_lemma[OF assms] by auto
- have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
- using assms(1) by auto
- define z where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
- have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
- proof (cases "u v \<ge> 0")
- case False
- then have "u v < 0" by auto
- then show ?thesis
- proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
- case True
- then show ?thesis
- using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
- next
- case False
- then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
- apply (rule_tac sum_mono, auto)
- done
- then show ?thesis
- unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
- qed
- qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
-
- then have *: "sum u {x\<in>c. u x > 0} > 0"
- unfolding less_le
- apply (rule_tac conjI)
- apply (rule_tac sum_nonneg, auto)
- done
- moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
- "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
- using assms(1)
- apply (rule_tac[!] sum.mono_neutral_left, auto)
- done
- then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
- "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
- unfolding eq_neg_iff_add_eq_0
- using uv(1,4)
- by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
- moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
- apply rule
- apply (rule mult_nonneg_nonneg)
- using *
- apply auto
- done
- ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
- unfolding convex_hull_explicit mem_Collect_eq
- apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
- apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
- using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
- apply (auto simp: sum_negf sum_distrib_left[symmetric])
- done
- moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
- apply rule
- apply (rule mult_nonneg_nonneg)
- using *
- apply auto
- done
- then have "z \<in> convex hull {v \<in> c. u v > 0}"
- unfolding convex_hull_explicit mem_Collect_eq
- apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
- apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
- using assms(1)
- unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
- using *
- apply (auto simp: sum_negf sum_distrib_left[symmetric])
- done
- ultimately show ?thesis
- apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
- apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto)
- done
-qed
-
-theorem Radon:
- assumes "affine_dependent c"
- obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
-proof -
- from assms[unfolded affine_dependent_explicit]
- obtain s u where
- "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
- by blast
- then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
- unfolding affine_dependent_explicit by auto
- from Radon_partition[OF *]
- obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
- by blast
- then show ?thesis
- apply (rule_tac that[of p m])
- using s
- apply auto
- done
-qed
-
-
-subsection \<open>Helly's theorem\<close>
-
-lemma Helly_induct:
- fixes f :: "'a::euclidean_space set set"
- assumes "card f = n"
- and "n \<ge> DIM('a) + 1"
- and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
- shows "\<Inter>f \<noteq> {}"
- using assms
-proof (induction n arbitrary: f)
- case 0
- then show ?case by auto
-next
- case (Suc n)
- have "finite f"
- using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
- show "\<Inter>f \<noteq> {}"
- proof (cases "n = DIM('a)")
- case True
- then show ?thesis
- by (simp add: Suc.prems(1) Suc.prems(4))
- next
- case False
- have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
- proof (rule Suc.IH[rule_format])
- show "card (f - {s}) = n"
- by (simp add: Suc.prems(1) \<open>finite f\<close> that)
- show "DIM('a) + 1 \<le> n"
- using False Suc.prems(2) by linarith
- show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
- by (simp add: Suc.prems(4) subset_Diff_insert)
- qed (use Suc in auto)
- then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
- by blast
- then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
- by metis
- show ?thesis
- proof (cases "inj_on X f")
- case False
- then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
- unfolding inj_on_def by auto
- then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
- show ?thesis
- by (metis "*" X disjoint_iff_not_equal st)
- next
- case True
- then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
- using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
- unfolding card_image[OF True] and \<open>card f = Suc n\<close>
- using Suc(3) \<open>finite f\<close> and False
- by auto
- have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
- using mp(2) by auto
- then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
- unfolding subset_image_iff by auto
- then have "f \<union> (g \<union> h) = f" by auto
- then have f: "f = g \<union> h"
- using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
- unfolding mp(2)[unfolded image_Un[symmetric] gh]
- by auto
- have *: "g \<inter> h = {}"
- using mp(1)
- unfolding gh
- using inj_on_image_Int[OF True gh(3,4)]
- by auto
- have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
- by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
- then show ?thesis
- unfolding f using mp(3)[unfolded gh] by blast
- qed
- qed
-qed
-
-theorem Helly:
- fixes f :: "'a::euclidean_space set set"
- assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
- and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
- shows "\<Inter>f \<noteq> {}"
- apply (rule Helly_induct)
- using assms
- apply auto
- done
-
-
-subsection \<open>Epigraphs of convex functions\<close>
-
-definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
-
-lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
- unfolding epigraph_def by auto
-
-lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
-proof safe
- assume L: "convex (epigraph S f)"
- then show "convex_on S f"
- by (auto simp: convex_def convex_on_def epigraph_def)
- show "convex S"
- using L
- apply (clarsimp simp: convex_def convex_on_def epigraph_def)
- apply (erule_tac x=x in allE)
- apply (erule_tac x="f x" in allE, safe)
- apply (erule_tac x=y in allE)
- apply (erule_tac x="f y" in allE)
- apply (auto simp: )
- done
-next
- assume "convex_on S f" "convex S"
- then show "convex (epigraph S f)"
- unfolding convex_def convex_on_def epigraph_def
- apply safe
- apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
- apply (auto intro!:mult_left_mono add_mono)
- done
-qed
-
-lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
- unfolding convex_epigraph by auto
-
-lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
- by (simp add: convex_epigraph)
-
-
-subsubsection%unimportant \<open>Use this to derive general bound property of convex function\<close>
-
-lemma convex_on:
- assumes "convex S"
- shows "convex_on S f \<longleftrightarrow>
- (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
- f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
-
- unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
- unfolding fst_sum snd_sum fst_scaleR snd_scaleR
- apply safe
- apply (drule_tac x=k in spec)
- apply (drule_tac x=u in spec)
- apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
- apply simp
- using assms[unfolded convex] apply simp
- apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
- apply (rule sum_mono)
- apply (erule_tac x=i in allE)
- unfolding real_scaleR_def
- apply (rule mult_left_mono)
- using assms[unfolded convex] apply auto
- done
-
-
subsection%unimportant \<open>Convexity of general and special intervals\<close>
lemma is_interval_convex:
@@ -6268,82 +2127,7 @@
by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
-subsection%unimportant \<open>A bound within a convex hull, and so an interval\<close>
-
-lemma convex_on_convex_hull_bound:
- assumes "convex_on (convex hull s) f"
- and "\<forall>x\<in>s. f x \<le> b"
- shows "\<forall>x\<in> convex hull s. f x \<le> b"
-proof
- fix x
- assume "x \<in> convex hull s"
- then obtain k u v where
- obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
- unfolding convex_hull_indexed mem_Collect_eq by auto
- have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
- using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
- unfolding sum_distrib_right[symmetric] obt(2) mult_1
- apply (drule_tac meta_mp)
- apply (rule mult_left_mono)
- using assms(2) obt(1)
- apply auto
- done
- then show "f x \<le> b"
- using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
- unfolding obt(2-3)
- using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
- by auto
-qed
-
-lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
- by (simp add: inner_sum_left sum.If_cases inner_Basis)
-
-lemma convex_set_plus:
- assumes "convex S" and "convex T" shows "convex (S + T)"
-proof -
- have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- using assms by (rule convex_sums)
- moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
- unfolding set_plus_def by auto
- finally show "convex (S + T)" .
-qed
-
-lemma convex_set_sum:
- assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
- shows "convex (\<Sum>i\<in>A. B i)"
-proof (cases "finite A")
- case True then show ?thesis using assms
- by induct (auto simp: convex_set_plus)
-qed auto
-
-lemma finite_set_sum:
- assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
- using assms by (induct set: finite, simp, simp add: finite_set_plus)
-
-lemma box_eq_set_sum_Basis:
- shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
- apply (subst set_sum_alt [OF finite_Basis], safe)
- apply (fast intro: euclidean_representation [symmetric])
- apply (subst inner_sum_left)
-apply (rename_tac f)
- apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
- apply (drule (1) bspec)
- apply clarsimp
- apply (frule sum.remove [OF finite_Basis])
- apply (erule trans, simp)
- apply (rule sum.neutral, clarsimp)
- apply (frule_tac x=i in bspec, assumption)
- apply (drule_tac x=x in bspec, assumption, clarsimp)
- apply (cut_tac u=x and v=i in inner_Basis, assumption+)
- apply (rule ccontr, simp)
- done
-
-lemma convex_hull_set_sum:
- "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
-proof (cases "finite A")
- assume "finite A" then show ?thesis
- by (induct set: finite, simp, simp add: convex_hull_set_plus)
-qed simp
+subsection%unimportant \<open>A bound within an interval\<close>
lemma convex_hull_eq_real_cbox:
fixes x y :: real assumes "x \<le> y"
--- a/src/HOL/Analysis/Further_Topology.thy Mon Jan 07 13:33:29 2019 +0100
+++ b/src/HOL/Analysis/Further_Topology.thy Mon Jan 07 14:06:54 2019 +0100
@@ -5272,7 +5272,7 @@
using \<open>S \<noteq> {}\<close> \<open>T \<noteq> {}\<close> by blast+
qed
then show False
- by (metis Compl_disjoint Convex_Euclidean_Space.connected_UNIV compl_bot_eq compl_unique connected_closedD inf_sup_absorb sup_compl_top_left1 top.extremum_uniqueI)
+ by (metis Compl_disjoint connected_UNIV compl_bot_eq compl_unique connected_closedD inf_sup_absorb sup_compl_top_left1 top.extremum_uniqueI)
qed
--- a/src/HOL/Analysis/Linear_Algebra.thy Mon Jan 07 13:33:29 2019 +0100
+++ b/src/HOL/Analysis/Linear_Algebra.thy Mon Jan 07 14:06:54 2019 +0100
@@ -111,6 +111,49 @@
then show "x = y" by simp
qed simp
+subsection \<open>Substandard Basis\<close>
+
+lemma ex_card:
+ assumes "n \<le> card A"
+ shows "\<exists>S\<subseteq>A. card S = n"
+proof (cases "finite A")
+ case True
+ from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
+ moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
+ by (auto simp: bij_betw_def intro: subset_inj_on)
+ ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
+ by (auto simp: bij_betw_def card_image)
+ then show ?thesis by blast
+next
+ case False
+ with \<open>n \<le> card A\<close> show ?thesis by force
+qed
+
+lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
+ by (auto simp: subspace_def inner_add_left)
+
+lemma dim_substandard:
+ assumes d: "d \<subseteq> Basis"
+ shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
+proof (rule dim_unique)
+ from d show "d \<subseteq> ?A"
+ by (auto simp: inner_Basis)
+ from d show "independent d"
+ by (rule independent_mono [OF independent_Basis])
+ have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
+ proof -
+ have "finite d"
+ by (rule finite_subset [OF d finite_Basis])
+ then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
+ by (simp add: span_sum span_clauses)
+ also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
+ by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
+ finally show "x \<in> span d"
+ by (simp only: euclidean_representation)
+ qed
+ then show "?A \<subseteq> span d" by auto
+qed simp
+
subsection \<open>Orthogonality\<close>
@@ -858,6 +901,7 @@
shows "f = g"
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
+
subsection \<open>Infinity norm\<close>
definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Mon Jan 07 13:33:29 2019 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Mon Jan 07 14:06:54 2019 +0100
@@ -263,6 +263,30 @@
then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto
qed
+lemma dim_cball:
+ assumes "e > 0"
+ shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
+proof -
+ {
+ fix x :: "'n::euclidean_space"
+ define y where "y = (e / norm x) *\<^sub>R x"
+ then have "y \<in> cball 0 e"
+ using assms by auto
+ moreover have *: "x = (norm x / e) *\<^sub>R y"
+ using y_def assms by simp
+ moreover from * have "x = (norm x/e) *\<^sub>R y"
+ by auto
+ ultimately have "x \<in> span (cball 0 e)"
+ using span_scale[of y "cball 0 e" "norm x/e"]
+ span_superset[of "cball 0 e"]
+ by (simp add: span_base)
+ }
+ then have "span (cball 0 e) = (UNIV :: 'n::euclidean_space set)"
+ by auto
+ then show ?thesis
+ using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp: dim_UNIV)
+qed
+
subsection \<open>Boxes\<close>
@@ -829,6 +853,20 @@
(if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
using image_affinity_cbox[of m 0 a b] by auto
+lemma swap_continuous:
+ assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
+ shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
+proof -
+ have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
+ by auto
+ then show ?thesis
+ apply (rule ssubst)
+ apply (rule continuous_on_compose)
+ apply (simp add: split_def)
+ apply (rule continuous_intros | simp add: assms)+
+ done
+qed
+
subsection \<open>General Intervals\<close>
@@ -2134,9 +2172,6 @@
subsection%unimportant \<open>Some properties of a canonical subspace\<close>
-lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
- by (auto simp: subspace_def inner_add_left)
-
lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"
(is "closed ?A")
proof -
@@ -2149,45 +2184,6 @@
finally show "closed ?A" .
qed
-lemma dim_substandard:
- assumes d: "d \<subseteq> Basis"
- shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
-proof (rule dim_unique)
- from d show "d \<subseteq> ?A"
- by (auto simp: inner_Basis)
- from d show "independent d"
- by (rule independent_mono [OF independent_Basis])
- have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
- proof -
- have "finite d"
- by (rule finite_subset [OF d finite_Basis])
- then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
- by (simp add: span_sum span_clauses)
- also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
- by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
- finally show "x \<in> span d"
- by (simp only: euclidean_representation)
- qed
- then show "?A \<subseteq> span d" by auto
-qed simp
-
-text \<open>Hence closure and completeness of all subspaces.\<close>
-lemma ex_card:
- assumes "n \<le> card A"
- shows "\<exists>S\<subseteq>A. card S = n"
-proof (cases "finite A")
- case True
- from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
- moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
- by (auto simp: bij_betw_def intro: subset_inj_on)
- ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
- by (auto simp: bij_betw_def card_image)
- then show ?thesis by blast
-next
- case False
- with \<open>n \<le> card A\<close> show ?thesis by force
-qed
-
lemma closed_subspace:
fixes s :: "'a::euclidean_space set"
assumes "subspace s"