HOL-Analysis: move Library/Convex to Convex_Euclidean_Space

--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Fri Sep 30 11:35:39 2016 +0200
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Fri Sep 30 12:00:17 2016 +0200
@@ -137,7 +137,7 @@
         moreover have "open ({a<..<c} - s)"
           by (blast intro: open_greaterThanLessThan \<open>finite s\<close> finite_imp_closed)
         ultimately show "f differentiable at x within {a..b}"
-          using x le by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI) 
+          using x le by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI)
       qed (use x le st dist_real_def in auto)
     next
       case ge show ?diff_fg
@@ -149,7 +149,7 @@
         moreover have "open ({c<..<b} - t)"
           by (blast intro: open_greaterThanLessThan \<open>finite t\<close> finite_imp_closed)
         ultimately show "g differentiable at x within {a..b}"
-          using x ge by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI) 
+          using x ge by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI)
       qed (use x ge st dist_real_def in auto)
     qed
   }
@@ -1753,7 +1753,7 @@
     apply (simp add: segment_convex_hull)
     apply (rule convex_hull_subset)
     using assms
-    apply (auto simp: hull_inc c' Convex.convexD_alt)
+    apply (auto simp: hull_inc c' convexD_alt)
     done
 qed
 
@@ -1770,7 +1770,7 @@
     apply (simp_all add: segment_convex_hull)
     apply (rule_tac [!] convex_hull_subset)
     using assms
-    apply (auto simp: hull_inc c' Convex.convexD_alt)
+    apply (auto simp: hull_inc c' convexD_alt)
     done
   show ?thesis
     apply (rule contour_integral_unique)

--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Fri Sep 30 11:35:39 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Fri Sep 30 12:00:17 2016 +0200
@@ -1,5 +1,9 @@
-(*  Author:     L C Paulson, University of Cambridge
-    Authors: Robert Himmelmann, TU Muenchen; Bogdan Grechuk, University of Edinburgh
+(* 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, functions and related things\<close>
@@ -7,28 +11,10 @@
 theory Convex_Euclidean_Space
 imports
   Topology_Euclidean_Space
-  "~~/src/HOL/Library/Convex"
+  "~~/src/HOL/Library/Product_Vector"
   "~~/src/HOL/Library/Set_Algebras"
 begin
 
-(*FIXME move up e.g. to Library/Convex.thy, but requires the "support" primitive*)
-lemma convex_supp_setsum:
-  assumes "convex S" and 1: "supp_setsum u I = 1"
-      and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
-    shows "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
-proof -
-  have fin: "finite {i \<in> I. u i \<noteq> 0}"
-    using 1 setsum.infinite by (force simp: supp_setsum_def support_on_def)
-  then have eq: "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I = setsum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
-    by (force intro: setsum.mono_neutral_left simp: supp_setsum_def support_on_def)
-  show ?thesis
-    apply (simp add: eq)
-    apply (rule convex_setsum [OF fin \<open>convex S\<close>])
-    using 1 assms apply (auto simp: supp_setsum_def support_on_def)
-    done
-qed
-
-
 lemma dim_image_eq:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
@@ -158,9 +144,6 @@
   apply auto
   done
 
-lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
-  by (metis convex_translation translation_galois)
-
 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)"
@@ -179,6 +162,1012 @@
   then show ?thesis by auto
 qed
 
+subsection \<open>Convexity\<close>
+
+definition 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_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_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 \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
+
+lemma convex_setsum:
+  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 + setsum 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> setsum a s"
+    by (simp add: setsum_nonneg)
+  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
+  proof (cases "setsum a s = 0")
+    case True
+    with \<open>a i + setsum a s = 1\<close> have "a i = 1"
+      by simp
+    from setsum_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> setsum a s\<close> have "0 < setsum a s"
+      by simp
+    then have "(\<Sum>j\<in>s. (a j / setsum 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 setsum_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> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
+    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"
+      by (rule convexD)
+    then show ?thesis
+      by (simp add: scaleR_setsum_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> (setsum u {1..k} = 1)
+      \<longrightarrow> setsum (\<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"
+    "setsum u {1..k} = 1"
+  with convex_setsum[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> setsum 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 "setsum ?u {1 .. 2} = 1"
+      using setsum.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 setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
+    from setsum_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> setsum u t = 1 \<longrightarrow> setsum (\<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" "setsum u t = 1"
+  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+    using convex_setsum[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>
+    setsum 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> setsum u s = 1 \<longrightarrow> setsum (\<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> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
+    assume *: "\<forall>x\<in>t. 0 \<le> u x" "setsum 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 setsum.If_cases if_distrib if_distrib_arg)
+  qed
+  done
+
+
+subsection \<open>Functions that are convex on a set\<close>
+
+definition 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 \<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 {x + y| x y. x \<in> s \<and> y \<in> t}"
+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)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
+    by auto
+  finally show ?thesis .
+qed
+
+lemma convex_differences:
+  assumes "convex s" "convex t"
+  shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
+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 "{a + y |y. y \<in> s} = (\<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 = op + a ` op *\<^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_setsum:
+  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: setsum_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 setsum_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 setsum.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 setsum_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_setsum [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 setsum.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.setsum[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 setsum_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 \<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_supp_setsum:
+  assumes "convex S" and 1: "supp_setsum u I = 1"
+      and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
+    shows "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
+proof -
+  have fin: "finite {i \<in> I. u i \<noteq> 0}"
+    using 1 setsum.infinite by (force simp: supp_setsum_def support_on_def)
+  then have eq: "supp_setsum (\<lambda>i. u i *\<^sub>R f i) I = setsum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
+    by (force intro: setsum.mono_neutral_left simp: supp_setsum_def support_on_def)
+  show ?thesis
+    apply (simp add: eq)
+    apply (rule convex_setsum [OF fin \<open>convex S\<close>])
+    using 1 assms apply (auto simp: supp_setsum_def support_on_def)
+    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"

--- a/src/HOL/Library/Convex.thy	Fri Sep 30 11:35:39 2016 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,999 +0,0 @@
-(*  Title:      HOL/Library/Convex.thy
-    Author:     Armin Heller, TU Muenchen
-    Author:     Johannes Hoelzl, TU Muenchen
-*)
-
-section \<open>Convexity in real vector spaces\<close>
-
-theory Convex
-  imports Product_Vector
-begin
-
-subsection \<open>Convexity\<close>
-
-definition 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_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_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 \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
-
-lemma convex_setsum:
-  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 + setsum 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> setsum a s"
-    by (simp add: setsum_nonneg)
-  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
-  proof (cases "setsum a s = 0")
-    case True
-    with \<open>a i + setsum a s = 1\<close> have "a i = 1"
-      by simp
-    from setsum_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> setsum a s\<close> have "0 < setsum a s"
-      by simp
-    then have "(\<Sum>j\<in>s. (a j / setsum 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 setsum_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> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
-    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"
-      by (rule convexD)
-    then show ?thesis
-      by (simp add: scaleR_setsum_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> (setsum u {1..k} = 1)
-      \<longrightarrow> setsum (\<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"
-    "setsum u {1..k} = 1"
-  with convex_setsum[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> setsum 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 "setsum ?u {1 .. 2} = 1"
-      using setsum.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 setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
-    from setsum_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> setsum u t = 1 \<longrightarrow> setsum (\<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" "setsum u t = 1"
-  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-    using convex_setsum[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>
-    setsum 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> setsum u s = 1 \<longrightarrow> setsum (\<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> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
-    assume *: "\<forall>x\<in>t. 0 \<le> u x" "setsum 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 setsum.If_cases if_distrib if_distrib_arg)
-  qed
-  done 
-
-
-subsection \<open>Functions that are convex on a set\<close>
-
-definition 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 \<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 {x + y| x y. x \<in> s \<and> y \<in> t}"
-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)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
-    by auto
-  finally show ?thesis .
-qed
-
-lemma convex_differences:
-  assumes "convex s" "convex t"
-  shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
-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 "{a + y |y. y \<in> s} = (\<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 = op + a ` op *\<^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_setsum:
-  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: setsum_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 setsum_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 setsum.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 setsum_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_setsum [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 setsum.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.setsum[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 setsum_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 \<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)
-
-end

--- a/src/HOL/Library/Library.thy	Fri Sep 30 11:35:39 2016 +0200
+++ b/src/HOL/Library/Library.thy	Fri Sep 30 12:00:17 2016 +0200
@@ -11,7 +11,6 @@
   Char_ord
   Code_Test
   Continuum_Not_Denumerable
-  Convex
   Combine_PER
   Complete_Partial_Order2
   Countable