(*  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

   assume alt[rule_format]: ?alt

   {

     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[OF mem] by auto

   }

   then show "convex s"

     unfolding convex_def by auto

 qed (auto simp: convex_def)

 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: "(\<forall>s\<in>f. 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: "\<forall>i\<in>A. 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)

     assume z: "setsum a s = 0"

     with \<open>a i + setsum a s = 1\<close> have "a i = 1"

       by simp

     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"

       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

     assume nz: "setsum a s \<noteq> 0"

     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 nz)

   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

 proof safe

   fix t u

   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"

     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"

   have *: "s \<inter> t = t"

     using as(2) by auto

   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

   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"

    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *

    by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)

 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)

 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_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 *: "\<And>u c fx v fy :: real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"

     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"

     using assms 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 setsum_nonneg_0[where 'b=real] insert by fastforce

     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_right_distrib[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 "\<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"

     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 -

   def 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 unfolding add_divide_distrib by (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

     unfolding atLeastAtMost_iff le_less by auto

 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 \<in> C" "y \<in> C"

   shows "f' x * (y - x) \<le> f y - f x"

   using assms

 proof -

   {

     fix x y :: real

     assume *: "x \<in> C" "y \<in> C" "y > x"

     then 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 have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"

       using res by auto

   } note less_imp = this

   {

     fix x y :: real

     assume "x \<in> C" "y \<in> C" "x \<noteq> y"

     then have"f y - f x \<ge> f' x * (y - x)"

     unfolding neq_iff using less_imp by auto

   }

   moreover

   {

     fix x y :: real

     assume "x \<in> C" "y \<in> C" "x = y"

     then have "f y - f x \<ge> f' x * (y - x)" by auto

   }

   ultimately show ?thesis using assms by blast

 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

 end