Theory Convex

theory Convex
imports Product_Vector
(*  Title:      HOL/Library/Convex.thy
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
*)

section ‹Convexity in real vector spaces›

theory Convex
imports Product_Vector
begin

subsection ‹Convexity›

definition convex :: "'a::real_vector set ⇒ bool"
  where "convex s ⟷ (∀x∈s. ∀y∈s. ∀u≥0. ∀v≥0. u + v = 1 ⟶ u *R x + v *R y ∈ s)"

lemma convexI:
  assumes "⋀x y u v. x ∈ s ⟹ y ∈ s ⟹ 0 ≤ u ⟹ 0 ≤ v ⟹ u + v = 1 ⟹ u *R x + v *R y ∈ s"
  shows "convex s"
  using assms unfolding convex_def by fast

lemma convexD:
  assumes "convex s" and "x ∈ s" and "y ∈ s" and "0 ≤ u" and "0 ≤ v" and "u + v = 1"
  shows "u *R x + v *R y ∈ s"
  using assms unfolding convex_def by fast

lemma convex_alt:
  "convex s ⟷ (∀x∈s. ∀y∈s. ∀u. 0 ≤ u ∧ u ≤ 1 ⟶ ((1 - u) *R x + u *R y) ∈ s)"
  (is "_ ⟷ ?alt")
proof
  assume alt[rule_format]: ?alt
  {
    fix x y and u v :: real
    assume mem: "x ∈ s" "y ∈ s"
    assume "0 ≤ u" "0 ≤ v"
    moreover
    assume "u + v = 1"
    then have "u = 1 - v" by auto
    ultimately have "u *R x + v *R y ∈ 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 ∈ s" "b ∈ s" "0 ≤ u" "u ≤ 1"
  shows "((1 - u) *R a + u *R b) ∈ s"
  using assms unfolding convex_alt by auto

lemma mem_convex_alt:
  assumes "convex S" "x ∈ S" "y ∈ S" "u ≥ 0" "v ≥ 0" "u + v > 0"
  shows "((u/(u+v)) *R x + (v/(u+v)) *R y) ∈ 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: "(∀s∈f. convex s) ⟹ convex(⋂f)"
  unfolding convex_def by auto

lemma convex_Int: "convex s ⟹ convex t ⟹ convex (s ∩ t)"
  unfolding convex_def by auto

lemma convex_INT: "∀i∈A. convex (B i) ⟹ convex (⋂i∈A. B i)"
  unfolding convex_def by auto

lemma convex_Times: "convex s ⟹ convex t ⟹ convex (s × t)"
  unfolding convex_def by auto

lemma convex_halfspace_le: "convex {x. inner a x ≤ b}"
  unfolding convex_def
  by (auto simp: inner_add intro!: convex_bound_le)

lemma convex_halfspace_ge: "convex {x. inner a x ≥ b}"
proof -
  have *: "{x. inner a x ≥ b} = {x. inner (-a) x ≤ -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 ≤ b} ∩ {x. inner a x ≥ 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 ≤ inner 1 x}"
    by auto
  then show 1: "convex {a..}"
    by (simp only: convex_halfspace_ge)
  have "{..b} = {x. inner 1 x ≤ 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..} ∩ {..b}"
    by auto
  then show "convex {a..b}"
    by (simp only: convex_Int 1 2)
  have "{a<..b} = {a<..} ∩ {..b}"
    by auto
  then show "convex {a<..b}"
    by (simp only: convex_Int 3 2)
  have "{a..<b} = {a..} ∩ {..<b}"
    by auto
  then show "convex {a..<b}"
    by (simp only: convex_Int 1 4)
  have "{a<..<b} = {a<..} ∩ {..<b}"
    by auto
  then show "convex {a<..<b}"
    by (simp only: convex_Int 3 4)
qed

lemma convex_Reals: "convex ℝ"
  by (simp add: convex_def scaleR_conv_of_real)


subsection ‹Explicit expressions for convexity in terms of arbitrary sums›

lemma convex_setsum:
  fixes C :: "'a::real_vector set"
  assumes "finite s"
    and "convex C"
    and "(∑ i ∈ s. a i) = 1"
  assumes "⋀i. i ∈ s ⟹ a i ≥ 0"
    and "⋀i. i ∈ s ⟹ y i ∈ C"
  shows "(∑ j ∈ s. a j *R y j) ∈ 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 ≤ a i"
    and "∀j∈s. 0 ≤ a j"
    and "y i ∈ C"
    and "∀j∈s. y j ∈ C"
    using insert.hyps(1,2) insert.prems by simp_all
  then have "0 ≤ setsum a s"
    by (simp add: setsum_nonneg)
  have "a i *R y i + (∑j∈s. a j *R y j) ∈ C"
  proof (cases)
    assume z: "setsum a s = 0"
    with ‹a i + setsum a s = 1› have "a i = 1"
      by simp
    from setsum_nonneg_0 [OF ‹finite s› _ z] ‹∀j∈s. 0 ≤ a j› have "∀j∈s. a j = 0"
      by simp
    show ?thesis using ‹a i = 1› and ‹∀j∈s. a j = 0› and ‹y i ∈ C›
      by simp
  next
    assume nz: "setsum a s ≠ 0"
    with ‹0 ≤ setsum a s› have "0 < setsum a s"
      by simp
    then have "(∑j∈s. (a j / setsum a s) *R y j) ∈ C"
      using ‹∀j∈s. 0 ≤ a j› and ‹∀j∈s. y j ∈ C›
      by (simp add: IH setsum_divide_distrib [symmetric])
    from ‹convex C› and ‹y i ∈ C› and this and ‹0 ≤ a i›
      and ‹0 ≤ setsum a s› and ‹a i + setsum a s = 1›
    have "a i *R y i + setsum a s *R (∑j∈s. (a j / setsum a s) *R y j) ∈ C"
      by (rule convexD)
    then show ?thesis
      by (simp add: scaleR_setsum_right nz)
  qed
  then show ?case using ‹finite s› and ‹i ∉ s›
    by simp
qed

lemma convex:
  "convex s ⟷ (∀(k::nat) u x. (∀i. 1≤i ∧ i≤k ⟶ 0 ≤ u i ∧ x i ∈s) ∧ (setsum u {1..k} = 1)
      ⟶ setsum (λi. u i *R x i) {1..k} ∈ s)"
proof safe
  fix k :: nat
  fix u :: "nat ⇒ real"
  fix x
  assume "convex s"
    "∀i. 1 ≤ i ∧ i ≤ k ⟶ 0 ≤ u i ∧ x i ∈ s"
    "setsum u {1..k} = 1"
  with convex_setsum[of "{1 .. k}" s] show "(∑j∈{1 .. k}. u j *R x j) ∈ s"
    by auto
next
  assume *: "∀k u x. (∀ i :: nat. 1 ≤ i ∧ i ≤ k ⟶ 0 ≤ u i ∧ x i ∈ s) ∧ setsum u {1..k} = 1
    ⟶ (∑i = 1..k. u i *R (x i :: 'a)) ∈ s"
  {
    fix μ :: real
    fix x y :: 'a
    assume xy: "x ∈ s" "y ∈ s"
    assume mu: "μ ≥ 0" "μ ≤ 1"
    let ?u = "λi. if (i :: nat) = 1 then μ else 1 - μ"
    let ?x = "λi. if (i :: nat) = 1 then x else y"
    have "{1 :: nat .. 2} ∩ - {x. x = 1} = {2}"
      by auto
    then have card: "card ({1 :: nat .. 2} ∩ - {x. x = 1}) = 1"
      by simp
    then have "setsum ?u {1 .. 2} = 1"
      using setsum.If_cases[of "{(1 :: nat) .. 2}" "λ x. x = 1" "λ x. μ" "λ x. 1 - μ"]
      by auto
    with *[rule_format, of "2" ?u ?x] have s: "(∑j ∈ {1..2}. ?u j *R ?x j) ∈ s"
      using mu xy by auto
    have grarr: "(∑j ∈ {Suc (Suc 0)..2}. ?u j *R ?x j) = (1 - μ) *R y"
      using setsum_head_Suc[of "Suc (Suc 0)" 2 "λ j. (1 - μ) *R y"] by auto
    from setsum_head_Suc[of "Suc 0" 2 "λ j. ?u j *R ?x j", simplified this]
    have "(∑j ∈ {1..2}. ?u j *R ?x j) = μ *R x + (1 - μ) *R y"
      by auto
    then have "(1 - μ) *R y + μ *R x ∈ 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 ⟷
    (∀t u. finite t ∧ t ⊆ s ∧ (∀x∈t. 0 ≤ u x) ∧ setsum u t = 1 ⟶ setsum (λx. u x *R x) t ∈ s)"
proof safe
  fix t
  fix u :: "'a ⇒ real"
  assume "convex s"
    and "finite t"
    and "t ⊆ s" "∀x∈t. 0 ≤ u x" "setsum u t = 1"
  then show "(∑x∈t. u x *R x) ∈ s"
    using convex_setsum[of t s u "λ x. x"] by auto
next
  assume *: "∀t. ∀ u. finite t ∧ t ⊆ s ∧ (∀x∈t. 0 ≤ u x) ∧
    setsum u t = 1 ⟶ (∑x∈t. u x *R x) ∈ s"
  show "convex s"
    unfolding convex_alt
  proof safe
    fix x y
    fix μ :: real
    assume **: "x ∈ s" "y ∈ s" "0 ≤ μ" "μ ≤ 1"
    show "(1 - μ) *R x + μ *R y ∈ s"
    proof (cases "x = y")
      case False
      then show ?thesis
        using *[rule_format, of "{x, y}" "λ z. if z = x then 1 - μ else μ"] **
          by auto
    next
      case True
      then show ?thesis
        using *[rule_format, of "{x, y}" "λ 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 ⟷ (∀u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ⟶ setsum (λx. u x *R x) s ∈ s)"
  unfolding convex_explicit
proof safe
  fix t u
  assume sum: "∀u. (∀x∈s. 0 ≤ u x) ∧ setsum u s = 1 ⟶ (∑x∈s. u x *R x) ∈ s"
    and as: "finite t" "t ⊆ s" "∀x∈t. 0 ≤ u x" "setsum u t = (1::real)"
  have *: "s ∩ t = t"
    using as(2) by auto
  have if_distrib_arg: "⋀P f g x. (if P then f else g) x = (if P then f x else g x)"
    by simp
  show "(∑x∈t. u x *R x) ∈ s"
   using sum[THEN spec[where x="λx. if x∈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 ‹Functions that are convex on a set›

definition convex_on :: "'a::real_vector set ⇒ ('a ⇒ real) ⇒ bool"
  where "convex_on s f ⟷
    (∀x∈s. ∀y∈s. ∀u≥0. ∀v≥0. u + v = 1 ⟶ f (u *R x + v *R y) ≤ u * f x + v * f y)"

lemma convex_onI [intro?]:
  assumes "⋀t x y. t > 0 ⟹ t < 1 ⟹ x ∈ A ⟹ y ∈ A ⟹
             f ((1 - t) *R x + t *R y) ≤ (1 - t) * f x + t * f y"
  shows   "convex_on A f"
  unfolding convex_on_def
proof clarify
  fix x y u v assume A: "x ∈ A" "y ∈ A" "(u::real) ≥ 0" "v ≥ 0" "u + v = 1"
  from A(5) have [simp]: "v = 1 - u" by (simp add: algebra_simps)
  from A(1-4) show "f (u *R x + v *R y) ≤ u * f x + v * f y" using assms[of u y x]
    by (cases "u = 0 ∨ u = 1") (auto simp: algebra_simps)
qed

lemma convex_on_linorderI [intro?]:
  fixes A :: "('a::{linorder,real_vector}) set"
  assumes "⋀t x y. t > 0 ⟹ t < 1 ⟹ x ∈ A ⟹ y ∈ A ⟹ x < y ⟹
             f ((1 - t) *R x + t *R y) ≤ (1 - t) * f x + t * f y"
  shows   "convex_on A f"
proof
  fix t x y assume A: "x ∈ A" "y ∈ A" "(t::real) > 0" "t < 1"
  case (goal1 t x y)
  with goal1 assms[of t x y] assms[of "1 - t" y x]
    show ?case by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
qed

lemma convex_onD:
  assumes "convex_on A f"
  shows   "⋀t x y. t ≥ 0 ⟹ t ≤ 1 ⟹ x ∈ A ⟹ y ∈ A ⟹
             f ((1 - t) *R x + t *R y) ≤ (1 - t) * f x + t * f y"
  using assms unfolding convex_on_def by auto

lemma convex_onD_Icc:
  assumes "convex_on {x..y} f" "x ≤ (y :: _ :: {real_vector,preorder})"
  shows   "⋀t. t ≥ 0 ⟹ t ≤ 1 ⟹
             f ((1 - t) *R x + t *R y) ≤ (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 ⟹ s ⊆ t ⟹ 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 (λx. f x + g x)"
proof -
  {
    fix x y
    assume "x ∈ s" "y ∈ s"
    moreover
    fix u v :: real
    assume "0 ≤ u" "0 ≤ v" "u + v = 1"
    ultimately
    have "f (u *R x + v *R y) + g (u *R x + v *R y) ≤ (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 *R x + v *R y) + g (u *R x + v *R y) ≤ 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 ≤ c"
    and "convex_on s f"
  shows "convex_on s (λx. c * f x)"
proof -
  have *: "⋀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 ∈ s"
    and "y ∈ s"
    and "0 ≤ u"
    and "0 ≤ v"
    and "u + v = 1"
  shows "f (u *R x + v *R y) ≤ max (f x) (f y)"
proof -
  let ?m = "max (f x) (f y)"
  have "u * f x + v * f y ≤ 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 "… = 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 (λx. dist a x)"
proof (auto simp: convex_on_def dist_norm)
  fix x y
  assume "x ∈ s" "y ∈ s"
  fix u v :: real
  assume "0 ≤ u"
  assume "0 ≤ v"
  assume "u + v = 1"
  have "a = u *R a + v *R a"
    unfolding scaleR_left_distrib[symmetric] and ‹u + v = 1› by simp
  then have *: "a - (u *R x + v *R y) = (u *R (a - x)) + (v *R (a - y))"
    by (auto simp: algebra_simps)
  show "norm (a - (u *R x + v *R y)) ≤ u * norm (a - x) + v * norm (a - y)"
    unfolding * using norm_triangle_ineq[of "u *R (a - x)" "v *R (a - y)"]
    using ‹0 ≤ u› ‹0 ≤ v› by auto
qed


subsection ‹Arithmetic operations on sets preserve convexity›

lemma convex_linear_image:
  assumes "linear f"
    and "convex s"
  shows "convex (f ` s)"
proof -
  interpret f: linear f by fact
  from ‹convex s› 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 ‹convex s› show "convex (f -` s)"
    by (simp add: convex_def f.add f.scaleR)
qed

lemma convex_scaling:
  assumes "convex s"
  shows "convex ((λx. c *R x) ` s)"
proof -
  have "linear (λx. c *R x)"
    by (simp add: linearI scaleR_add_right)
  then show ?thesis
    using ‹convex s› by (rule convex_linear_image)
qed

lemma convex_scaled:
  assumes "convex s"
  shows "convex ((λx. x *R c) ` s)"
proof -
  have "linear (λx. x *R c)"
    by (simp add: linearI scaleR_add_left)
  then show ?thesis
    using ‹convex s› by (rule convex_linear_image)
qed

lemma convex_negations:
  assumes "convex s"
  shows "convex ((λx. - x) ` s)"
proof -
  have "linear (λx. - x)"
    by (simp add: linearI)
  then show ?thesis
    using ‹convex s› by (rule convex_linear_image)
qed

lemma convex_sums:
  assumes "convex s"
    and "convex t"
  shows "convex {x + y| x y. x ∈ s ∧ y ∈ t}"
proof -
  have "linear (λ(x, y). x + y)"
    by (auto intro: linearI simp: scaleR_add_right)
  with assms have "convex ((λ(x, y). x + y) ` (s × t))"
    by (intro convex_linear_image convex_Times)
  also have "((λ(x, y). x + y) ` (s × t)) = {x + y| x y. x ∈ s ∧ y ∈ t}"
    by auto
  finally show ?thesis .
qed

lemma convex_differences:
  assumes "convex s" "convex t"
  shows "convex {x - y| x y. x ∈ s ∧ y ∈ t}"
proof -
  have "{x - y| x y. x ∈ s ∧ y ∈ t} = {x + y |x y. x ∈ s ∧ y ∈ 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 ((λx. a + x) ` s)"
proof -
  have "{a + y |y. y ∈ s} = (λx. 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 ((λx. a + c *R x) ` s)"
proof -
  have "(λx. a + c *R x) ` s = op + a ` op *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 μ :: real
  assume *: "y > 0" "x > 0" "μ ≥ 0" "μ ≤ 1"
  {
    assume "μ = 0"
    then have "μ *R x + (1 - μ) *R y = y" by simp
    then have "μ *R x + (1 - μ) *R y > 0" using * by simp
  }
  moreover
  {
    assume "μ = 1"
    then have "μ *R x + (1 - μ) *R y > 0" using * by simp
  }
  moreover
  {
    assume "μ ≠ 1" "μ ≠ 0"
    then have "μ > 0" "(1 - μ) > 0" using * by auto
    then have "μ *R x + (1 - μ) *R y > 0" using *
      by (auto simp: add_pos_pos)
  }
  ultimately show "(1 - μ) *R y + μ *R x > 0"
    using assms by fastforce
qed

lemma convex_on_setsum:
  fixes a :: "'a ⇒ real"
    and y :: "'a ⇒ 'b::real_vector"
    and f :: "'b ⇒ real"
  assumes "finite s" "s ≠ {}"
    and "convex_on C f"
    and "convex C"
    and "(∑ i ∈ s. a i) = 1"
    and "⋀i. i ∈ s ⟹ a i ≥ 0"
    and "⋀i. i ∈ s ⟹ y i ∈ C"
  shows "f (∑ i ∈ s. a i *R y i) ≤ (∑ i ∈ 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: "⋀x y μ. x ∈ C ⟹ y ∈ C ⟹ 0 ≤ μ ⟹ μ ≤ 1 ⟹
      f (μ *R x + (1 - μ) *R y) ≤ μ * f x + (1 - μ) * f y"
    by simp
  show ?case
  proof (cases "a i = 1")
    case True
    then have "(∑ j ∈ s. a j) = 0"
      using insert by auto
    then have "⋀j. j ∈ s ⟹ 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 ∈ C" "a i ≥ 0"
      by auto
    have fis: "finite (insert i s)"
      using insert by auto
    then have ai1: "a i ≤ 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 = "λj. a j / (1 - a i)"
    have a_nonneg: "?a j ≥ 0" if "j ∈ s" for j
      using i0 insert that by fastforce
    have "(∑ j ∈ insert i s. a j) = 1"
      using insert by auto
    then have "(∑ j ∈ s. a j) = 1 - a i"
      using setsum.insert insert by fastforce
    then have "(∑ j ∈ s. a j) / (1 - a i) = 1"
      using i0 by auto
    then have a1: "(∑ j ∈ s. ?a j) = 1"
      unfolding setsum_divide_distrib by simp
    have "convex C" using insert by auto
    then have asum: "(∑ j ∈ s. ?a j *R y j) ∈ C"
      using insert convex_setsum[OF ‹finite s›
        ‹convex C› a1 a_nonneg] by auto
    have asum_le: "f (∑ j ∈ s. ?a j *R y j) ≤ (∑ j ∈ s. ?a j * f (y j))"
      using a_nonneg a1 insert by blast
    have "f (∑ j ∈ insert i s. a j *R y j) = f ((∑ j ∈ s. a j *R y j) + a i *R y i)"
      using setsum.insert[of s i "λ j. a j *R y j", OF ‹finite s› ‹i ∉ s›] insert
      by (auto simp only: add.commute)
    also have "… = f (((1 - a i) * inverse (1 - a i)) *R (∑ j ∈ s. a j *R y j) + a i *R y i)"
      using i0 by auto
    also have "… = f ((1 - a i) *R (∑ j ∈ s. (a j * inverse (1 - a i)) *R y j) + a i *R y i)"
      using scaleR_right.setsum[of "inverse (1 - a i)" "λ j. a j *R y j" s, symmetric]
      by (auto simp: algebra_simps)
    also have "… = f ((1 - a i) *R (∑ j ∈ s. ?a j *R y j) + a i *R y i)"
      by (auto simp: divide_inverse)
    also have "… ≤ (1 - a i) *R f ((∑ j ∈ s. ?a j *R y j)) + a i * f (y i)"
      using conv[of "y i" "(∑ j ∈ s. ?a j *R y j)" "a i", OF yai(1) asum yai(2) ai1]
      by (auto simp: add.commute)
    also have "… ≤ (1 - a i) * (∑ j ∈ 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 "… = (∑ j ∈ s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
      unfolding setsum_right_distrib[of "1 - a i" "λ j. ?a j * f (y j)"] using i0 by auto
    also have "… = (∑ j ∈ s. a j * f (y j)) + a i * f (y i)"
      using i0 by auto
    also have "… = (∑ j ∈ 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 ⟷
    (∀x ∈ C. ∀ y ∈ C. ∀ μ :: real. μ ≥ 0 ∧ μ ≤ 1 ⟶
      f (μ *R x + (1 - μ) *R y) ≤ μ * f x + (1 - μ) * f y)"
proof safe
  fix x y
  fix μ :: real
  assume *: "convex_on C f" "x ∈ C" "y ∈ C" "0 ≤ μ" "μ ≤ 1"
  from this[unfolded convex_on_def, rule_format]
  have "⋀u v. 0 ≤ u ⟹ 0 ≤ v ⟹ u + v = 1 ⟹ f (u *R x + v *R y) ≤ u * f x + v * f y"
    by auto
  from this[of "μ" "1 - μ", simplified] *
  show "f (μ *R x + (1 - μ) *R y) ≤ μ * f x + (1 - μ) * f y"
    by auto
next
  assume *: "∀x∈C. ∀y∈C. ∀μ. 0 ≤ μ ∧ μ ≤ 1 ⟶
    f (μ *R x + (1 - μ) *R y) ≤ μ * f x + (1 - μ) * f y"
  {
    fix x y
    fix u v :: real
    assume **: "x ∈ C" "y ∈ C" "u ≥ 0" "v ≥ 0" "u + v = 1"
    then have[simp]: "1 - u = v" by auto
    from *[rule_format, of x y u]
    have "f (u *R x + v *R y) ≤ 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 ⇒ real"
  assumes f: "convex_on I f"
    and I: "x ∈ I" "y ∈ I"
    and t: "x < t" "t < y"
  shows "(f x - f t) / (x - t) ≤ (f x - f y) / (x - y)"
    and "(f x - f y) / (x - y) ≤ (f t - f y) / (t - y)"
proof -
  def a  "(t - y) / (x - y)"
  with t have "0 ≤ a" "0 ≤ 1 - a"
    by (auto simp: field_simps)
  with f ‹x ∈ I› ‹y ∈ I› have cvx: "f (a * x + (1 - a) * y) ≤ 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 "… = t"
    unfolding a_def using ‹x < t› ‹t < y› by simp
  finally have "f t ≤ a * f x + (1 - a) * f y"
    using cvx by simp
  also have "… = a * (f x - f y) + f y"
    by (simp add: field_simps)
  finally have "f t - f y ≤ a * (f x - f y)"
    by simp
  with t show "(f x - f t) / (x - t) ≤ (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) ≤ (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 ⇒ real"
  assumes "convex C"
    and leq: "⋀x y. x ∈ C ⟹ y ∈ C ⟹ f' x * (y - x) ≤ f y - f x"
  shows "convex_on C f"
  unfolding convex_on_alt[OF assms(1)]
  using assms
proof safe
  fix x y μ :: real
  let ?x = "μ *R x + (1 - μ) *R y"
  assume *: "convex C" "x ∈ C" "y ∈ C" "μ ≥ 0" "μ ≤ 1"
  then have "1 - μ ≥ 0" by auto
  then have xpos: "?x ∈ C"
    using * unfolding convex_alt by fastforce
  have geq: "μ * (f x - f ?x) + (1 - μ) * (f y - f ?x) ≥
      μ * f' ?x * (x - ?x) + (1 - μ) * f' ?x * (y - ?x)"
    using add_mono[OF mult_left_mono[OF leq[OF xpos *(2)] ‹μ ≥ 0›]
      mult_left_mono[OF leq[OF xpos *(3)] ‹1 - μ ≥ 0›]]
    by auto
  then have "μ * f x + (1 - μ) * f y - f ?x ≥ 0"
    by (auto simp: field_simps)
  then show "f (μ *R x + (1 - μ) *R y) ≤ μ * f x + (1 - μ) * f y"
    using convex_on_alt by auto
qed

lemma atMostAtLeast_subset_convex:
  fixes C :: "real set"
  assumes "convex C"
    and "x ∈ C" "y ∈ C" "x < y"
  shows "{x .. y} ⊆ C"
proof safe
  fix z assume z: "z ∈ {x .. y}"
  have less: "z ∈ C" if *: "x < z" "z < y"
  proof -
    let  = "(y - z) / (y - x)"
    have "0 ≤ ?μ" "?μ ≤ 1"
      using assms * by (auto simp: field_simps)
    then have comb: "?μ * x + (1 - ?μ) * y ∈ C"
      using assms iffD1[OF convex_alt, rule_format, of C y x ]
      by (simp add: algebra_simps)
    have "?μ * x + (1 - ?μ) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
      by (auto simp: field_simps)
    also have "… = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
      using assms unfolding add_divide_distrib by (auto simp: field_simps)
    also have "… = z"
      using assms by (auto simp: field_simps)
    finally show ?thesis
      using comb by auto
  qed
  show "z ∈ C" using z less assms
    unfolding atLeastAtMost_iff le_less by auto
qed

lemma f''_imp_f':
  fixes f :: "real ⇒ real"
  assumes "convex C"
    and f': "⋀x. x ∈ C ⟹ DERIV f x :> (f' x)"
    and f'': "⋀x. x ∈ C ⟹ DERIV f' x :> (f'' x)"
    and pos: "⋀x. x ∈ C ⟹ f'' x ≥ 0"
    and "x ∈ C" "y ∈ C"
  shows "f' x * (y - x) ≤ f y - f x"
  using assms
proof -
  {
    fix x y :: real
    assume *: "x ∈ C" "y ∈ C" "y > x"
    then have ge: "y - x > 0" "y - x ≥ 0"
      by auto
    from * have le: "x - y < 0" "x - y ≤ 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 ‹convex C› ‹x ∈ C› ‹y ∈ C› ‹x < y›],
        THEN f', THEN MVT2[OF ‹x < y›, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
      by auto
    then have "z1 ∈ C"
      using atMostAtLeast_subset_convex ‹convex C› ‹x ∈ C› ‹y ∈ C› ‹x < y›
      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 ‹convex C› ‹x ∈ C› ‹z1 ∈ C› ‹x < z1›],
        THEN f'', THEN MVT2[OF ‹x < z1›, 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 ‹convex C› ‹z1 ∈ C› ‹y ∈ C› ‹z1 < y›],
        THEN f'', THEN MVT2[OF ‹z1 < y›, 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 "… = (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 ≥ 0"
      using z1 by auto
    have "z3 ∈ C"
      using z3 * atMostAtLeast_subset_convex ‹convex C› ‹x ∈ C› ‹z1 ∈ C› ‹x < z1›
      by fastforce
    then have B': "f'' z3 ≥ 0"
      using assms by auto
    from A' B' have "(y - z1) * f'' z3 ≥ 0"
      by auto
    from cool' this have "f' y - (f x - f y) / (x - y) ≥ 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) ≤ 0 * (x - y)"
      by (simp add: algebra_simps)
    then have "f' y * (x - y) - (f x - f y) ≤ 0"
      using le by auto
    then have res: "f' y * (x - y) ≤ 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 "… = (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 ≥ 0"
      using z1 by auto
    have "z2 ∈ C"
      using z2 z1 * atMostAtLeast_subset_convex ‹convex C› ‹z1 ∈ C› ‹y ∈ C› ‹z1 < y›
      by fastforce
    then have B: "f'' z2 ≥ 0"
      using assms by auto
    from A B have "(z1 - x) * f'' z2 ≥ 0"
      by auto
    with cool have "(f y - f x) / (y - x) - f' x ≥ 0"
      by auto
    from mult_right_mono[OF this ge(2)]
    have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) ≥ 0 * (y - x)"
      by (simp add: algebra_simps)
    then have "f y - f x - f' x * (y - x) ≥ 0"
      using ge by auto
    then have "f y - f x ≥ f' x * (y - x)" "f' y * (x - y) ≤ f x - f y"
      using res by auto
  } note less_imp = this
  {
    fix x y :: real
    assume "x ∈ C" "y ∈ C" "x ≠ y"
    then have"f y - f x ≥ f' x * (y - x)"
    unfolding neq_iff using less_imp by auto
  }
  moreover
  {
    fix x y :: real
    assume "x ∈ C" "y ∈ C" "x = y"
    then have "f y - f x ≥ f' x * (y - x)" by auto
  }
  ultimately show ?thesis using assms by blast
qed

lemma f''_ge0_imp_convex:
  fixes f :: "real ⇒ real"
  assumes conv: "convex C"
    and f': "⋀x. x ∈ C ⟹ DERIV f x :> (f' x)"
    and f'': "⋀x. x ∈ C ⟹ DERIV f' x :> (f'' x)"
    and pos: "⋀x. x ∈ C ⟹ f'' x ≥ 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 <..} (λ x. - log b x)"
proof -
  have "⋀z. z > 0 ⟹ DERIV (log b) z :> 1 / (ln b * z)"
    using DERIV_log by auto
  then have f': "⋀z. z > 0 ⟹ DERIV (λ z. - log b z) z :> - 1 / (ln b * z)"
    by (auto simp: DERIV_minus)
  have "⋀z :: real. z > 0 ⟹ 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 "⋀z :: real. z > 0 ⟹
    DERIV (λ z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
    by auto
  then have f''0: "⋀z::real. z > 0 ⟹
    DERIV (λ z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
    unfolding inverse_eq_divide by (auto simp: mult.assoc)
  have f''_ge0: "⋀z::real. z > 0 ⟹ 1 / (ln b * z * z) ≥ 0"
    using ‹b > 1› 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 ‹Convexity of real functions›

lemma convex_on_realI:
  assumes "connected A"
  assumes "⋀x. x ∈ A ⟹ (f has_real_derivative f' x) (at x)"
  assumes "⋀x y. x ∈ A ⟹ y ∈ A ⟹ x ≤ y ⟹ f' x ≤ f' y"
  shows   "convex_on A f"
proof (rule convex_on_linorderI)
  fix t x y :: real
  assume t: "t > 0" "t < 1" and xy: "x ∈ A" "y ∈ A" "x < y"
  def z  "(1 - t) * x + t * y"
  with ‹connected A› and xy have ivl: "{x..y} ⊆ 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 "... > 0" by (intro mult_pos_pos) simp_all
  finally have yz: "z < y" by simp

  from assms xz yz ivl t have "∃ξ. ξ > x ∧ ξ < z ∧ f z - f x = (z - x) * f' ξ"
    by (intro MVT2) (auto intro!: assms(2))
  then obtain ξ where ξ: "ξ > x" "ξ < z" "f' ξ = (f z - f x) / (z - x)" by auto
  from assms xz yz ivl t have "∃η. η > z ∧ η < y ∧ f y - f z = (y - z) * f' η"
    by (intro MVT2) (auto intro!: assms(2))
  then obtain η where η: "η > z" "η < y" "f' η = (f y - f z) / (y - z)" by auto

  from η(3) have "(f y - f z) / (y - z) = f' η" ..
  also from ξ η ivl have "ξ ∈ A" "η ∈ A" by auto
  with ξ η have "f' η ≥ f' ξ" by (intro assms(3)) auto
  also from ξ(3) have "f' ξ = (f z - f x) / (z - x)" .
  finally have "(f y - f z) * (z - x) ≥ (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 ≥ (f z - f x) * (1 - t)" using xy by simp
  thus "(1 - t) * f x + t * f y ≥ f ((1 - t) *R x + t *R y)"
    by (simp add: z_def algebra_simps)
qed

lemma convex_on_inverse:
  assumes "A ⊆ {0<..}"
  shows   "convex_on A (inverse :: real ⇒ real)"
proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "λx. -inverse (x^2)"])
  fix u v :: real assume "u ∈ {0<..}" "v ∈ {0<..}" "u ≤ v"
  with assms show "-inverse (u^2) ≤ -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 ∈ {x..y}"
  defines "d ≡ y - x"
  shows   "f c ≤ (f y - f x) / d * (c - x) + f x"
proof (cases y x rule: linorder_cases)
  assume less: "x < y"
  hence d: "d > 0" by (simp add: d_def)
  from assms(2) less have A: "0 ≤ (c - x) / d" "(c - x) / d ≤ 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) * … = (1 - (c - x) / d) *R x + ((c - x) / d) *R y"
    by (simp add: field_simps)
  also have "f … ≤ (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 "… = (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 ∈ {x..y}"
  defines "d ≡ y - x"
  shows   "f c ≤ (f x - f y) / d * (y - c) + f y"
proof (cases y x rule: linorder_cases)
  assume less: "x < y"
  hence d: "d > 0" by (simp add: d_def)
  from assms(2) less have A: "0 ≤ (y - c) / d" "(y - c) / d ≤ 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) * … = (1 - (1 - (y - c) / d)) *R x + (1 - (y - c) / d) *R y"
    by (simp add: field_simps)
  also have "f … ≤ (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 "… = (f x - f y) / d * (y - c) + f y" by (simp add: field_simps)
  finally show ?thesis .
qed (insert assms(2), simp_all)


end