src/HOL/Library/Convex.thy
author paulson <lp15@cam.ac.uk>
Tue Oct 13 12:42:08 2015 +0100 (2015-10-13)
changeset 61426 d53db136e8fd
parent 61070 b72a990adfe2
child 61518 ff12606337e9
permissions -rw-r--r--
new material on path_component_sets, inside, outside, etc. And more default simprules
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(*  Title:      HOL/Library/Convex.thy
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    Author:     Armin Heller, TU Muenchen
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    Author:     Johannes Hoelzl, TU Muenchen
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*)
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section \<open>Convexity in real vector spaces\<close>
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theory Convex
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imports Product_Vector
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begin
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subsection \<open>Convexity\<close>
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definition convex :: "'a::real_vector set \<Rightarrow> bool"
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  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)"
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lemma convexI:
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  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"
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  shows "convex s"
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  using assms unfolding convex_def by fast
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lemma convexD:
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  assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
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  shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
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  using assms unfolding convex_def by fast
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lemma convex_alt:
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  "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)"
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  (is "_ \<longleftrightarrow> ?alt")
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proof
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  assume alt[rule_format]: ?alt
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  {
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    fix x y and u v :: real
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    assume mem: "x \<in> s" "y \<in> s"
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    assume "0 \<le> u" "0 \<le> v"
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    moreover
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    assume "u + v = 1"
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    then have "u = 1 - v" by auto
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    ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
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      using alt[OF mem] by auto
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  }
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  then show "convex s"
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    unfolding convex_def by auto
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qed (auto simp: convex_def)
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lemma convexD_alt:
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  assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
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  shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
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  using assms unfolding convex_alt by auto
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lemma convex_empty[intro,simp]: "convex {}"
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  unfolding convex_def by simp
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lemma convex_singleton[intro,simp]: "convex {a}"
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  unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
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lemma convex_UNIV[intro,simp]: "convex UNIV"
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  unfolding convex_def by auto
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lemma convex_Inter: "(\<forall>s\<in>f. convex s) \<Longrightarrow> convex(\<Inter>f)"
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  unfolding convex_def by auto
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lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
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  unfolding convex_def by auto
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lemma convex_INT: "\<forall>i\<in>A. convex (B i) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
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  unfolding convex_def by auto
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lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
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  unfolding convex_def by auto
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lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
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  unfolding convex_def
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  by (auto simp: inner_add intro!: convex_bound_le)
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lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
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proof -
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  have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
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    by auto
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  show ?thesis
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    unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
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qed
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lemma convex_hyperplane: "convex {x. inner a x = b}"
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proof -
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  have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
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    by auto
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  show ?thesis using convex_halfspace_le convex_halfspace_ge
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    by (auto intro!: convex_Int simp: *)
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qed
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lemma convex_halfspace_lt: "convex {x. inner a x < b}"
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  unfolding convex_def
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  by (auto simp: convex_bound_lt inner_add)
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lemma convex_halfspace_gt: "convex {x. inner a x > b}"
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   using convex_halfspace_lt[of "-a" "-b"] by auto
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lemma convex_real_interval:
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  fixes a b :: "real"
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  shows "convex {a..}" and "convex {..b}"
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    and "convex {a<..}" and "convex {..<b}"
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    and "convex {a..b}" and "convex {a<..b}"
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    and "convex {a..<b}" and "convex {a<..<b}"
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proof -
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  have "{a..} = {x. a \<le> inner 1 x}"
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    by auto
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  then show 1: "convex {a..}"
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    by (simp only: convex_halfspace_ge)
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  have "{..b} = {x. inner 1 x \<le> b}"
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    by auto
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  then show 2: "convex {..b}"
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    by (simp only: convex_halfspace_le)
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  have "{a<..} = {x. a < inner 1 x}"
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    by auto
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  then show 3: "convex {a<..}"
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    by (simp only: convex_halfspace_gt)
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  have "{..<b} = {x. inner 1 x < b}"
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    by auto
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  then show 4: "convex {..<b}"
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    by (simp only: convex_halfspace_lt)
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  have "{a..b} = {a..} \<inter> {..b}"
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    by auto
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  then show "convex {a..b}"
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    by (simp only: convex_Int 1 2)
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  have "{a<..b} = {a<..} \<inter> {..b}"
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    by auto
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  then show "convex {a<..b}"
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    by (simp only: convex_Int 3 2)
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  have "{a..<b} = {a..} \<inter> {..<b}"
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    by auto
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  then show "convex {a..<b}"
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    by (simp only: convex_Int 1 4)
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  have "{a<..<b} = {a<..} \<inter> {..<b}"
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    by auto
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  then show "convex {a<..<b}"
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    by (simp only: convex_Int 3 4)
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qed
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lemma convex_Reals: "convex \<real>"
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  by (simp add: convex_def scaleR_conv_of_real)
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subsection \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
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lemma convex_setsum:
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  fixes C :: "'a::real_vector set"
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  assumes "finite s"
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    and "convex C"
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    and "(\<Sum> i \<in> s. a i) = 1"
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  assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
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    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
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  shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
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  using assms(1,3,4,5)
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proof (induct arbitrary: a set: finite)
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  case empty
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  then show ?case by simp
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next
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  case (insert i s) note IH = this(3)
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  have "a i + setsum a s = 1"
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    and "0 \<le> a i"
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    and "\<forall>j\<in>s. 0 \<le> a j"
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    and "y i \<in> C"
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    and "\<forall>j\<in>s. y j \<in> C"
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    using insert.hyps(1,2) insert.prems by simp_all
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  then have "0 \<le> setsum a s"
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    by (simp add: setsum_nonneg)
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  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
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  proof (cases)
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    assume z: "setsum a s = 0"
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    with \<open>a i + setsum a s = 1\<close> have "a i = 1"
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      by simp
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    from setsum_nonneg_0 [OF \<open>finite s\<close> _ z] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
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      by simp
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    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>
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      by simp
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  next
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    assume nz: "setsum a s \<noteq> 0"
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    with \<open>0 \<le> setsum a s\<close> have "0 < setsum a s"
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      by simp
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    then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
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      using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
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      by (simp add: IH setsum_divide_distrib [symmetric])
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    from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
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      and \<open>0 \<le> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
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    have "a i *\<^sub>R y i + setsum a s *\<^sub>R (\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
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      by (rule convexD)
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    then show ?thesis
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      by (simp add: scaleR_setsum_right nz)
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  qed
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  then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
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    by simp
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qed
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lemma convex:
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  "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> (setsum u {1..k} = 1)
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      \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
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proof safe
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  fix k :: nat
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  fix u :: "nat \<Rightarrow> real"
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  fix x
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  assume "convex s"
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    "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
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    "setsum u {1..k} = 1"
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  with convex_setsum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
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    by auto
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next
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  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> setsum u {1..k} = 1
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    \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
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  {
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    fix \<mu> :: real
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    fix x y :: 'a
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    assume xy: "x \<in> s" "y \<in> s"
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    assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
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    let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
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    let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
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    have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
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      by auto
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    then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
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      by simp
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    then have "setsum ?u {1 .. 2} = 1"
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      using setsum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
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      by auto
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    with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
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      using mu xy by auto
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    have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
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      using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
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    from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
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    have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
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      by auto
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    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
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      using s by (auto simp: add.commute)
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  }
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  then show "convex s"
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    unfolding convex_alt by auto
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qed
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lemma convex_explicit:
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  fixes s :: "'a::real_vector set"
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  shows "convex s \<longleftrightarrow>
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    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
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proof safe
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  fix t
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  fix u :: "'a \<Rightarrow> real"
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  assume "convex s"
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    and "finite t"
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    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
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  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
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    using convex_setsum[of t s u "\<lambda> x. x"] by auto
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next
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  assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
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    setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
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  show "convex s"
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    unfolding convex_alt
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  proof safe
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    fix x y
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    fix \<mu> :: real
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    assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
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    show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
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    proof (cases "x = y")
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      case False
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      then show ?thesis
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        using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
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          by auto
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    next
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      case True
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      then show ?thesis
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        using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
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          by (auto simp: field_simps real_vector.scale_left_diff_distrib)
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    qed
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  qed
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qed
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lemma convex_finite:
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  assumes "finite s"
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  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
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  unfolding convex_explicit
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proof safe
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  fix t u
wenzelm@49609
   281
  assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
hoelzl@36623
   282
    and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
wenzelm@56796
   283
  have *: "s \<inter> t = t"
wenzelm@56796
   284
    using as(2) by auto
wenzelm@49609
   285
  have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
wenzelm@49609
   286
    by simp
hoelzl@36623
   287
  show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
hoelzl@36623
   288
   using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
haftmann@57418
   289
   by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
hoelzl@36623
   290
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
hoelzl@36623
   291
wenzelm@56796
   292
wenzelm@60423
   293
subsection \<open>Functions that are convex on a set\<close>
huffman@55909
   294
wenzelm@49609
   295
definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
wenzelm@49609
   296
  where "convex_on s f \<longleftrightarrow>
wenzelm@49609
   297
    (\<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)"
hoelzl@36623
   298
hoelzl@36623
   299
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
hoelzl@36623
   300
  unfolding convex_on_def by auto
hoelzl@36623
   301
huffman@53620
   302
lemma convex_on_add [intro]:
wenzelm@56796
   303
  assumes "convex_on s f"
wenzelm@56796
   304
    and "convex_on s g"
hoelzl@36623
   305
  shows "convex_on s (\<lambda>x. f x + g x)"
wenzelm@49609
   306
proof -
wenzelm@56796
   307
  {
wenzelm@56796
   308
    fix x y
wenzelm@56796
   309
    assume "x \<in> s" "y \<in> s"
wenzelm@49609
   310
    moreover
wenzelm@49609
   311
    fix u v :: real
wenzelm@49609
   312
    assume "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@49609
   313
    ultimately
wenzelm@49609
   314
    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)"
wenzelm@60423
   315
      using assms unfolding convex_on_def by (auto simp: add_mono)
wenzelm@49609
   316
    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)"
wenzelm@49609
   317
      by (simp add: field_simps)
wenzelm@49609
   318
  }
wenzelm@56796
   319
  then show ?thesis
wenzelm@56796
   320
    unfolding convex_on_def by auto
hoelzl@36623
   321
qed
hoelzl@36623
   322
huffman@53620
   323
lemma convex_on_cmul [intro]:
wenzelm@56796
   324
  fixes c :: real
wenzelm@56796
   325
  assumes "0 \<le> c"
wenzelm@56796
   326
    and "convex_on s f"
hoelzl@36623
   327
  shows "convex_on s (\<lambda>x. c * f x)"
wenzelm@56796
   328
proof -
wenzelm@60423
   329
  have *: "\<And>u c fx v fy :: real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
wenzelm@49609
   330
    by (simp add: field_simps)
wenzelm@49609
   331
  show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
wenzelm@49609
   332
    unfolding convex_on_def and * by auto
hoelzl@36623
   333
qed
hoelzl@36623
   334
hoelzl@36623
   335
lemma convex_lower:
wenzelm@56796
   336
  assumes "convex_on s f"
wenzelm@56796
   337
    and "x \<in> s"
wenzelm@56796
   338
    and "y \<in> s"
wenzelm@56796
   339
    and "0 \<le> u"
wenzelm@56796
   340
    and "0 \<le> v"
wenzelm@56796
   341
    and "u + v = 1"
hoelzl@36623
   342
  shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
wenzelm@56796
   343
proof -
hoelzl@36623
   344
  let ?m = "max (f x) (f y)"
hoelzl@36623
   345
  have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
wenzelm@60423
   346
    using assms(4,5) by (auto simp: mult_left_mono add_mono)
wenzelm@56796
   347
  also have "\<dots> = max (f x) (f y)"
wenzelm@60423
   348
    using assms(6) by (simp add: distrib_right [symmetric])
hoelzl@36623
   349
  finally show ?thesis
nipkow@44890
   350
    using assms unfolding convex_on_def by fastforce
hoelzl@36623
   351
qed
hoelzl@36623
   352
huffman@53620
   353
lemma convex_on_dist [intro]:
hoelzl@36623
   354
  fixes s :: "'a::real_normed_vector set"
hoelzl@36623
   355
  shows "convex_on s (\<lambda>x. dist a x)"
wenzelm@60423
   356
proof (auto simp: convex_on_def dist_norm)
wenzelm@49609
   357
  fix x y
wenzelm@56796
   358
  assume "x \<in> s" "y \<in> s"
wenzelm@49609
   359
  fix u v :: real
wenzelm@56796
   360
  assume "0 \<le> u"
wenzelm@56796
   361
  assume "0 \<le> v"
wenzelm@56796
   362
  assume "u + v = 1"
wenzelm@49609
   363
  have "a = u *\<^sub>R a + v *\<^sub>R a"
wenzelm@60423
   364
    unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
wenzelm@49609
   365
  then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
wenzelm@60423
   366
    by (auto simp: algebra_simps)
hoelzl@36623
   367
  show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
hoelzl@36623
   368
    unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
wenzelm@60423
   369
    using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
hoelzl@36623
   370
qed
hoelzl@36623
   371
wenzelm@49609
   372
wenzelm@60423
   373
subsection \<open>Arithmetic operations on sets preserve convexity\<close>
wenzelm@49609
   374
huffman@53620
   375
lemma convex_linear_image:
wenzelm@56796
   376
  assumes "linear f"
wenzelm@56796
   377
    and "convex s"
wenzelm@56796
   378
  shows "convex (f ` s)"
huffman@53620
   379
proof -
huffman@53620
   380
  interpret f: linear f by fact
wenzelm@60423
   381
  from \<open>convex s\<close> show "convex (f ` s)"
huffman@53620
   382
    by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
hoelzl@36623
   383
qed
hoelzl@36623
   384
huffman@53620
   385
lemma convex_linear_vimage:
wenzelm@56796
   386
  assumes "linear f"
wenzelm@56796
   387
    and "convex s"
wenzelm@56796
   388
  shows "convex (f -` s)"
huffman@53620
   389
proof -
huffman@53620
   390
  interpret f: linear f by fact
wenzelm@60423
   391
  from \<open>convex s\<close> show "convex (f -` s)"
huffman@53620
   392
    by (simp add: convex_def f.add f.scaleR)
huffman@53620
   393
qed
huffman@53620
   394
huffman@53620
   395
lemma convex_scaling:
wenzelm@56796
   396
  assumes "convex s"
wenzelm@56796
   397
  shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
huffman@53620
   398
proof -
wenzelm@56796
   399
  have "linear (\<lambda>x. c *\<^sub>R x)"
wenzelm@56796
   400
    by (simp add: linearI scaleR_add_right)
wenzelm@56796
   401
  then show ?thesis
wenzelm@60423
   402
    using \<open>convex s\<close> by (rule convex_linear_image)
huffman@53620
   403
qed
huffman@53620
   404
immler@60178
   405
lemma convex_scaled:
immler@60178
   406
  assumes "convex s"
immler@60178
   407
  shows "convex ((\<lambda>x. x *\<^sub>R c) ` s)"
immler@60178
   408
proof -
immler@60178
   409
  have "linear (\<lambda>x. x *\<^sub>R c)"
immler@60178
   410
    by (simp add: linearI scaleR_add_left)
immler@60178
   411
  then show ?thesis
wenzelm@60423
   412
    using \<open>convex s\<close> by (rule convex_linear_image)
immler@60178
   413
qed
immler@60178
   414
huffman@53620
   415
lemma convex_negations:
wenzelm@56796
   416
  assumes "convex s"
wenzelm@56796
   417
  shows "convex ((\<lambda>x. - x) ` s)"
huffman@53620
   418
proof -
wenzelm@56796
   419
  have "linear (\<lambda>x. - x)"
wenzelm@56796
   420
    by (simp add: linearI)
wenzelm@56796
   421
  then show ?thesis
wenzelm@60423
   422
    using \<open>convex s\<close> by (rule convex_linear_image)
hoelzl@36623
   423
qed
hoelzl@36623
   424
hoelzl@36623
   425
lemma convex_sums:
wenzelm@56796
   426
  assumes "convex s"
wenzelm@56796
   427
    and "convex t"
hoelzl@36623
   428
  shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
huffman@53620
   429
proof -
huffman@53620
   430
  have "linear (\<lambda>(x, y). x + y)"
wenzelm@60423
   431
    by (auto intro: linearI simp: scaleR_add_right)
huffman@53620
   432
  with assms have "convex ((\<lambda>(x, y). x + y) ` (s \<times> t))"
huffman@53620
   433
    by (intro convex_linear_image convex_Times)
huffman@53620
   434
  also have "((\<lambda>(x, y). x + y) ` (s \<times> t)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
huffman@53620
   435
    by auto
huffman@53620
   436
  finally show ?thesis .
hoelzl@36623
   437
qed
hoelzl@36623
   438
hoelzl@36623
   439
lemma convex_differences:
hoelzl@36623
   440
  assumes "convex s" "convex t"
hoelzl@36623
   441
  shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
hoelzl@36623
   442
proof -
hoelzl@36623
   443
  have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}"
wenzelm@60423
   444
    by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
wenzelm@49609
   445
  then show ?thesis
wenzelm@49609
   446
    using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
hoelzl@36623
   447
qed
hoelzl@36623
   448
wenzelm@49609
   449
lemma convex_translation:
wenzelm@49609
   450
  assumes "convex s"
wenzelm@49609
   451
  shows "convex ((\<lambda>x. a + x) ` s)"
wenzelm@49609
   452
proof -
wenzelm@56796
   453
  have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s"
wenzelm@56796
   454
    by auto
wenzelm@49609
   455
  then show ?thesis
wenzelm@49609
   456
    using convex_sums[OF convex_singleton[of a] assms] by auto
wenzelm@49609
   457
qed
hoelzl@36623
   458
wenzelm@49609
   459
lemma convex_affinity:
wenzelm@49609
   460
  assumes "convex s"
wenzelm@49609
   461
  shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
wenzelm@49609
   462
proof -
wenzelm@56796
   463
  have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s"
wenzelm@56796
   464
    by auto
wenzelm@49609
   465
  then show ?thesis
wenzelm@49609
   466
    using convex_translation[OF convex_scaling[OF assms], of a c] by auto
wenzelm@49609
   467
qed
hoelzl@36623
   468
wenzelm@49609
   469
lemma pos_is_convex: "convex {0 :: real <..}"
wenzelm@49609
   470
  unfolding convex_alt
hoelzl@36623
   471
proof safe
hoelzl@36623
   472
  fix y x \<mu> :: real
wenzelm@60423
   473
  assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
wenzelm@56796
   474
  {
wenzelm@56796
   475
    assume "\<mu> = 0"
wenzelm@49609
   476
    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
wenzelm@60423
   477
    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
wenzelm@56796
   478
  }
hoelzl@36623
   479
  moreover
wenzelm@56796
   480
  {
wenzelm@56796
   481
    assume "\<mu> = 1"
wenzelm@60423
   482
    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
wenzelm@56796
   483
  }
hoelzl@36623
   484
  moreover
wenzelm@56796
   485
  {
wenzelm@56796
   486
    assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
wenzelm@60423
   487
    then have "\<mu> > 0" "(1 - \<mu>) > 0" using * by auto
wenzelm@60423
   488
    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using *
wenzelm@60423
   489
      by (auto simp: add_pos_pos)
wenzelm@56796
   490
  }
wenzelm@56796
   491
  ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
wenzelm@56796
   492
    using assms by fastforce
hoelzl@36623
   493
qed
hoelzl@36623
   494
hoelzl@36623
   495
lemma convex_on_setsum:
hoelzl@36623
   496
  fixes a :: "'a \<Rightarrow> real"
wenzelm@49609
   497
    and y :: "'a \<Rightarrow> 'b::real_vector"
wenzelm@49609
   498
    and f :: "'b \<Rightarrow> real"
hoelzl@36623
   499
  assumes "finite s" "s \<noteq> {}"
wenzelm@49609
   500
    and "convex_on C f"
wenzelm@49609
   501
    and "convex C"
wenzelm@49609
   502
    and "(\<Sum> i \<in> s. a i) = 1"
wenzelm@49609
   503
    and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
wenzelm@49609
   504
    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
hoelzl@36623
   505
  shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
wenzelm@49609
   506
  using assms
wenzelm@49609
   507
proof (induct s arbitrary: a rule: finite_ne_induct)
hoelzl@36623
   508
  case (singleton i)
wenzelm@49609
   509
  then have ai: "a i = 1" by auto
wenzelm@49609
   510
  then show ?case by auto
hoelzl@36623
   511
next
wenzelm@60423
   512
  case (insert i s)
wenzelm@49609
   513
  then have "convex_on C f" by simp
hoelzl@36623
   514
  from this[unfolded convex_on_def, rule_format]
wenzelm@56796
   515
  have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
wenzelm@56796
   516
      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
hoelzl@36623
   517
    by simp
wenzelm@60423
   518
  show ?case
wenzelm@60423
   519
  proof (cases "a i = 1")
wenzelm@60423
   520
    case True
wenzelm@49609
   521
    then have "(\<Sum> j \<in> s. a j) = 0"
wenzelm@60423
   522
      using insert by auto
wenzelm@49609
   523
    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
wenzelm@60423
   524
      using setsum_nonneg_0[where 'b=real] insert by fastforce
wenzelm@60423
   525
    then show ?thesis
wenzelm@60423
   526
      using insert by auto
wenzelm@60423
   527
  next
wenzelm@60423
   528
    case False
wenzelm@60423
   529
    from insert have yai: "y i \<in> C" "a i \<ge> 0"
wenzelm@60423
   530
      by auto
wenzelm@60423
   531
    have fis: "finite (insert i s)"
wenzelm@60423
   532
      using insert by auto
wenzelm@60423
   533
    then have ai1: "a i \<le> 1"
wenzelm@60423
   534
      using setsum_nonneg_leq_bound[of "insert i s" a] insert by simp
wenzelm@60423
   535
    then have "a i < 1"
wenzelm@60423
   536
      using False by auto
wenzelm@60423
   537
    then have i0: "1 - a i > 0"
wenzelm@60423
   538
      by auto
wenzelm@49609
   539
    let ?a = "\<lambda>j. a j / (1 - a i)"
wenzelm@60423
   540
    have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
wenzelm@60449
   541
      using i0 insert that by fastforce
wenzelm@60423
   542
    have "(\<Sum> j \<in> insert i s. a j) = 1"
wenzelm@60423
   543
      using insert by auto
wenzelm@60423
   544
    then have "(\<Sum> j \<in> s. a j) = 1 - a i"
wenzelm@60423
   545
      using setsum.insert insert by fastforce
wenzelm@60423
   546
    then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
wenzelm@60423
   547
      using i0 by auto
wenzelm@60423
   548
    then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
wenzelm@60423
   549
      unfolding setsum_divide_distrib by simp
wenzelm@60423
   550
    have "convex C" using insert by auto
wenzelm@49609
   551
    then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
wenzelm@60423
   552
      using insert convex_setsum[OF \<open>finite s\<close>
wenzelm@60423
   553
        \<open>convex C\<close> a1 a_nonneg] by auto
hoelzl@36623
   554
    have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
wenzelm@60423
   555
      using a_nonneg a1 insert by blast
hoelzl@36623
   556
    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)"
wenzelm@60423
   557
      using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
wenzelm@60423
   558
      by (auto simp only: add.commute)
hoelzl@36623
   559
    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)"
hoelzl@36623
   560
      using i0 by auto
hoelzl@36623
   561
    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)"
wenzelm@49609
   562
      using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
wenzelm@60423
   563
      by (auto simp: algebra_simps)
hoelzl@36623
   564
    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)"
huffman@36778
   565
      by (auto simp: divide_inverse)
hoelzl@36623
   566
    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)"
hoelzl@36623
   567
      using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
wenzelm@60423
   568
      by (auto simp: add.commute)
hoelzl@36623
   569
    also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
hoelzl@36623
   570
      using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
hoelzl@36623
   571
        OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
hoelzl@36623
   572
    also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
huffman@44282
   573
      unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
wenzelm@60423
   574
    also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
wenzelm@60423
   575
      using i0 by auto
wenzelm@60423
   576
    also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
wenzelm@60423
   577
      using insert by auto
wenzelm@60423
   578
    finally show ?thesis
wenzelm@56796
   579
      by simp
wenzelm@60423
   580
  qed
hoelzl@36623
   581
qed
hoelzl@36623
   582
hoelzl@36623
   583
lemma convex_on_alt:
hoelzl@36623
   584
  fixes C :: "'a::real_vector set"
hoelzl@36623
   585
  assumes "convex C"
wenzelm@56796
   586
  shows "convex_on C f \<longleftrightarrow>
wenzelm@56796
   587
    (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
wenzelm@56796
   588
      f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
hoelzl@36623
   589
proof safe
wenzelm@49609
   590
  fix x y
wenzelm@49609
   591
  fix \<mu> :: real
wenzelm@60423
   592
  assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
hoelzl@36623
   593
  from this[unfolded convex_on_def, rule_format]
wenzelm@56796
   594
  have "\<And>u v. 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"
wenzelm@56796
   595
    by auto
wenzelm@60423
   596
  from this[of "\<mu>" "1 - \<mu>", simplified] *
wenzelm@56796
   597
  show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
wenzelm@56796
   598
    by auto
hoelzl@36623
   599
next
wenzelm@60423
   600
  assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
wenzelm@56796
   601
    f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
wenzelm@56796
   602
  {
wenzelm@56796
   603
    fix x y
wenzelm@49609
   604
    fix u v :: real
wenzelm@60423
   605
    assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
wenzelm@49609
   606
    then have[simp]: "1 - u = v" by auto
wenzelm@60423
   607
    from *[rule_format, of x y u]
wenzelm@56796
   608
    have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
wenzelm@60423
   609
      using ** by auto
wenzelm@49609
   610
  }
wenzelm@56796
   611
  then show "convex_on C f"
wenzelm@56796
   612
    unfolding convex_on_def by auto
hoelzl@36623
   613
qed
hoelzl@36623
   614
hoelzl@43337
   615
lemma convex_on_diff:
hoelzl@43337
   616
  fixes f :: "real \<Rightarrow> real"
wenzelm@56796
   617
  assumes f: "convex_on I f"
wenzelm@56796
   618
    and I: "x \<in> I" "y \<in> I"
wenzelm@56796
   619
    and t: "x < t" "t < y"
wenzelm@49609
   620
  shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
wenzelm@56796
   621
    and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
hoelzl@43337
   622
proof -
hoelzl@43337
   623
  def a \<equiv> "(t - y) / (x - y)"
wenzelm@56796
   624
  with t have "0 \<le> a" "0 \<le> 1 - a"
wenzelm@56796
   625
    by (auto simp: field_simps)
wenzelm@60423
   626
  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"
hoelzl@43337
   627
    by (auto simp: convex_on_def)
wenzelm@56796
   628
  have "a * x + (1 - a) * y = a * (x - y) + y"
wenzelm@56796
   629
    by (simp add: field_simps)
wenzelm@56796
   630
  also have "\<dots> = t"
wenzelm@60423
   631
    unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
wenzelm@56796
   632
  finally have "f t \<le> a * f x + (1 - a) * f y"
wenzelm@56796
   633
    using cvx by simp
wenzelm@56796
   634
  also have "\<dots> = a * (f x - f y) + f y"
wenzelm@56796
   635
    by (simp add: field_simps)
wenzelm@56796
   636
  finally have "f t - f y \<le> a * (f x - f y)"
wenzelm@56796
   637
    by simp
hoelzl@43337
   638
  with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
huffman@44142
   639
    by (simp add: le_divide_eq divide_le_eq field_simps a_def)
hoelzl@43337
   640
  with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
huffman@44142
   641
    by (simp add: le_divide_eq divide_le_eq field_simps)
hoelzl@43337
   642
qed
hoelzl@36623
   643
hoelzl@36623
   644
lemma pos_convex_function:
hoelzl@36623
   645
  fixes f :: "real \<Rightarrow> real"
hoelzl@36623
   646
  assumes "convex C"
wenzelm@56796
   647
    and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
hoelzl@36623
   648
  shows "convex_on C f"
wenzelm@49609
   649
  unfolding convex_on_alt[OF assms(1)]
wenzelm@49609
   650
  using assms
hoelzl@36623
   651
proof safe
hoelzl@36623
   652
  fix x y \<mu> :: real
hoelzl@36623
   653
  let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
wenzelm@60423
   654
  assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
wenzelm@49609
   655
  then have "1 - \<mu> \<ge> 0" by auto
wenzelm@56796
   656
  then have xpos: "?x \<in> C"
wenzelm@60423
   657
    using * unfolding convex_alt by fastforce
wenzelm@56796
   658
  have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
wenzelm@56796
   659
      \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
wenzelm@60423
   660
    using add_mono[OF mult_left_mono[OF leq[OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
wenzelm@60423
   661
      mult_left_mono[OF leq[OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
wenzelm@56796
   662
    by auto
wenzelm@49609
   663
  then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
wenzelm@60423
   664
    by (auto simp: field_simps)
wenzelm@49609
   665
  then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
hoelzl@36623
   666
    using convex_on_alt by auto
hoelzl@36623
   667
qed
hoelzl@36623
   668
hoelzl@36623
   669
lemma atMostAtLeast_subset_convex:
hoelzl@36623
   670
  fixes C :: "real set"
hoelzl@36623
   671
  assumes "convex C"
wenzelm@49609
   672
    and "x \<in> C" "y \<in> C" "x < y"
hoelzl@36623
   673
  shows "{x .. y} \<subseteq> C"
hoelzl@36623
   674
proof safe
wenzelm@60423
   675
  fix z assume z: "z \<in> {x .. y}"
wenzelm@60423
   676
  have less: "z \<in> C" if *: "x < z" "z < y"
wenzelm@60423
   677
  proof -
wenzelm@49609
   678
    let ?\<mu> = "(y - z) / (y - x)"
wenzelm@56796
   679
    have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
wenzelm@60423
   680
      using assms * by (auto simp: field_simps)
wenzelm@49609
   681
    then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
wenzelm@49609
   682
      using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
wenzelm@49609
   683
      by (simp add: algebra_simps)
hoelzl@36623
   684
    have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
wenzelm@60423
   685
      by (auto simp: field_simps)
hoelzl@36623
   686
    also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
wenzelm@49609
   687
      using assms unfolding add_divide_distrib by (auto simp: field_simps)
hoelzl@36623
   688
    also have "\<dots> = z"
wenzelm@49609
   689
      using assms by (auto simp: field_simps)
wenzelm@60423
   690
    finally show ?thesis
wenzelm@56796
   691
      using comb by auto
wenzelm@60423
   692
  qed
wenzelm@60423
   693
  show "z \<in> C" using z less assms
hoelzl@36623
   694
    unfolding atLeastAtMost_iff le_less by auto
hoelzl@36623
   695
qed
hoelzl@36623
   696
hoelzl@36623
   697
lemma f''_imp_f':
hoelzl@36623
   698
  fixes f :: "real \<Rightarrow> real"
hoelzl@36623
   699
  assumes "convex C"
wenzelm@49609
   700
    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
wenzelm@49609
   701
    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
wenzelm@49609
   702
    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
wenzelm@49609
   703
    and "x \<in> C" "y \<in> C"
hoelzl@36623
   704
  shows "f' x * (y - x) \<le> f y - f x"
wenzelm@49609
   705
  using assms
hoelzl@36623
   706
proof -
wenzelm@56796
   707
  {
wenzelm@56796
   708
    fix x y :: real
wenzelm@60423
   709
    assume *: "x \<in> C" "y \<in> C" "y > x"
wenzelm@60423
   710
    then have ge: "y - x > 0" "y - x \<ge> 0"
wenzelm@60423
   711
      by auto
wenzelm@60423
   712
    from * have le: "x - y < 0" "x - y \<le> 0"
wenzelm@60423
   713
      by auto
hoelzl@36623
   714
    then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
wenzelm@60423
   715
      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>],
wenzelm@60423
   716
        THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
hoelzl@36623
   717
      by auto
wenzelm@60423
   718
    then have "z1 \<in> C"
wenzelm@60423
   719
      using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
wenzelm@60423
   720
      by fastforce
hoelzl@36623
   721
    from z1 have z1': "f x - f y = (x - y) * f' z1"
wenzelm@60423
   722
      by (simp add: field_simps)
hoelzl@36623
   723
    obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
wenzelm@60423
   724
      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>],
wenzelm@60423
   725
        THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
hoelzl@36623
   726
      by auto
hoelzl@36623
   727
    obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
wenzelm@60423
   728
      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>],
wenzelm@60423
   729
        THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
hoelzl@36623
   730
      by auto
hoelzl@36623
   731
    have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
wenzelm@60423
   732
      using * z1' by auto
wenzelm@60423
   733
    also have "\<dots> = (y - z1) * f'' z3"
wenzelm@60423
   734
      using z3 by auto
wenzelm@60423
   735
    finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
wenzelm@60423
   736
      by simp
wenzelm@60423
   737
    have A': "y - z1 \<ge> 0"
wenzelm@60423
   738
      using z1 by auto
wenzelm@60423
   739
    have "z3 \<in> C"
wenzelm@60423
   740
      using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
wenzelm@60423
   741
      by fastforce
wenzelm@60423
   742
    then have B': "f'' z3 \<ge> 0"
wenzelm@60423
   743
      using assms by auto
wenzelm@60423
   744
    from A' B' have "(y - z1) * f'' z3 \<ge> 0"
wenzelm@60423
   745
      by auto
wenzelm@60423
   746
    from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
wenzelm@60423
   747
      by auto
hoelzl@36623
   748
    from mult_right_mono_neg[OF this le(2)]
hoelzl@36623
   749
    have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
huffman@36778
   750
      by (simp add: algebra_simps)
wenzelm@60423
   751
    then have "f' y * (x - y) - (f x - f y) \<le> 0"
wenzelm@60423
   752
      using le by auto
wenzelm@60423
   753
    then have res: "f' y * (x - y) \<le> f x - f y"
wenzelm@60423
   754
      by auto
hoelzl@36623
   755
    have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
wenzelm@60423
   756
      using * z1 by auto
wenzelm@60423
   757
    also have "\<dots> = (z1 - x) * f'' z2"
wenzelm@60423
   758
      using z2 by auto
wenzelm@60423
   759
    finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
wenzelm@60423
   760
      by simp
wenzelm@60423
   761
    have A: "z1 - x \<ge> 0"
wenzelm@60423
   762
      using z1 by auto
wenzelm@60423
   763
    have "z2 \<in> C"
wenzelm@60423
   764
      using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
wenzelm@60423
   765
      by fastforce
wenzelm@60423
   766
    then have B: "f'' z2 \<ge> 0"
wenzelm@60423
   767
      using assms by auto
wenzelm@60423
   768
    from A B have "(z1 - x) * f'' z2 \<ge> 0"
wenzelm@60423
   769
      by auto
wenzelm@60423
   770
    with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
wenzelm@60423
   771
      by auto
hoelzl@36623
   772
    from mult_right_mono[OF this ge(2)]
hoelzl@36623
   773
    have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
huffman@36778
   774
      by (simp add: algebra_simps)
wenzelm@60423
   775
    then have "f y - f x - f' x * (y - x) \<ge> 0"
wenzelm@60423
   776
      using ge by auto
wenzelm@49609
   777
    then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
wenzelm@60423
   778
      using res by auto
wenzelm@60423
   779
  } note less_imp = this
wenzelm@56796
   780
  {
wenzelm@56796
   781
    fix x y :: real
wenzelm@49609
   782
    assume "x \<in> C" "y \<in> C" "x \<noteq> y"
wenzelm@49609
   783
    then have"f y - f x \<ge> f' x * (y - x)"
wenzelm@56796
   784
    unfolding neq_iff using less_imp by auto
wenzelm@56796
   785
  }
hoelzl@36623
   786
  moreover
wenzelm@56796
   787
  {
wenzelm@56796
   788
    fix x y :: real
wenzelm@60423
   789
    assume "x \<in> C" "y \<in> C" "x = y"
wenzelm@56796
   790
    then have "f y - f x \<ge> f' x * (y - x)" by auto
wenzelm@56796
   791
  }
hoelzl@36623
   792
  ultimately show ?thesis using assms by blast
hoelzl@36623
   793
qed
hoelzl@36623
   794
hoelzl@36623
   795
lemma f''_ge0_imp_convex:
hoelzl@36623
   796
  fixes f :: "real \<Rightarrow> real"
hoelzl@36623
   797
  assumes conv: "convex C"
wenzelm@49609
   798
    and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
wenzelm@49609
   799
    and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
wenzelm@49609
   800
    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
hoelzl@36623
   801
  shows "convex_on C f"
wenzelm@56796
   802
  using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
wenzelm@56796
   803
  by fastforce
hoelzl@36623
   804
hoelzl@36623
   805
lemma minus_log_convex:
hoelzl@36623
   806
  fixes b :: real
hoelzl@36623
   807
  assumes "b > 1"
hoelzl@36623
   808
  shows "convex_on {0 <..} (\<lambda> x. - log b x)"
hoelzl@36623
   809
proof -
wenzelm@56796
   810
  have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
wenzelm@56796
   811
    using DERIV_log by auto
wenzelm@49609
   812
  then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
hoelzl@56479
   813
    by (auto simp: DERIV_minus)
wenzelm@49609
   814
  have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
hoelzl@36623
   815
    using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
hoelzl@36623
   816
  from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
wenzelm@49609
   817
  have "\<And>z :: real. z > 0 \<Longrightarrow>
wenzelm@49609
   818
    DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
hoelzl@36623
   819
    by auto
wenzelm@56796
   820
  then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
wenzelm@56796
   821
    DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
wenzelm@60423
   822
    unfolding inverse_eq_divide by (auto simp: mult.assoc)
wenzelm@56796
   823
  have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
wenzelm@60423
   824
    using \<open>b > 1\<close> by (auto intro!: less_imp_le)
hoelzl@36623
   825
  from f''_ge0_imp_convex[OF pos_is_convex,
hoelzl@36623
   826
    unfolded greaterThan_iff, OF f' f''0 f''_ge0]
hoelzl@36623
   827
  show ?thesis by auto
hoelzl@36623
   828
qed
hoelzl@36623
   829
hoelzl@36623
   830
end