Theory Convex

(* Title:      HOL/Analysis/Convex.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 ‹Convex Sets and Functions›

theory Convex
imports
  Affine  "HOL-Library.Set_Algebras"  "HOL-Library.FuncSet"
begin

subsection ‹Convex Sets›

definitiontag important› convex :: "'a::real_vector set  bool"
  where "convex s  (xs. ys. u0. v0. 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"
  by (simp add: assms convex_def)

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  (xs. ys. u. 0  u  u  1  ((1 - u) *R x + u *R y)  s)"
  (is "_  ?alt")
  by (smt (verit) convexD convexI)

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"
  using assms
  by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric])

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. sf  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. i  A  convex (B i))  convex (iA. 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_halfspace_abs_le: "convex {x. ¦inner a x¦  b}"
proof -
  have *: "{x. ¦inner a x¦  b} = {x. inner a x  b}  {x. -b  inner a x}"
    by auto
  show ?thesis
    unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
qed

lemma convex_hyperplane: "convex {x. inner a x = b}"
proof -
  have *: "{x. inner a x = b} = {x. inner a x  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_halfspace_Re_ge: "convex {x. Re x  b}"
  using convex_halfspace_ge[of b "1::complex"] by simp

lemma convex_halfspace_Re_le: "convex {x. Re x  b}"
  using convex_halfspace_le[of "1::complex" b] by simp

lemma convex_halfspace_Im_ge: "convex {x. Im x  b}"
  using convex_halfspace_ge[of b 𝗂] by simp

lemma convex_halfspace_Im_le: "convex {x. Im x  b}"
  using convex_halfspace_le[of 𝗂 b] by simp

lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
  using convex_halfspace_gt[of b "1::complex"] by simp

lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
  using convex_halfspace_lt[of "1::complex" b] by simp

lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
  using convex_halfspace_gt[of b 𝗂] by simp

lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
  using convex_halfspace_lt[of 𝗂 b] by simp

lemma convex_real_interval [iff]:
  fixes a b :: "real"
  shows "convex {a..}" and "convex {..b}"
    and "convex {a<..}" and "convex {..<b}"
    and "convex {a..b}" and "convex {a<..b}"
    and "convex {a..<b}" and "convex {a<..<b}"
proof -
  have "{a..} = {x. a  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 "
  by (simp add: convex_def scaleR_conv_of_real)


subsectiontag unimportant› ‹Explicit expressions for convexity in terms of arbitrary sums›

lemma convex_sum:
  fixes C :: "'a::real_vector set"
  assumes "finite S"
    and "convex C"
    and a: "( i  S. a i) = 1" "i. i  S  a i  0"
    and C: "i. i  S  y i  C"
  shows "( j  S. a j *R y j)  C"
  using finite S a C
proof (induction arbitrary: a set: finite)
  case empty
  then show ?case by simp
next
  case (insert i S) 
  then have "0  sum a S"
    by (simp add: sum_nonneg)
  have "a i *R y i + (jS. a j *R y j)  C"
  proof (cases "sum a S = 0")
    case True with insert show ?thesis
      by (simp add: sum_nonneg_eq_0_iff)
  next
    case False
    with 0  sum a S have "0 < sum a S"
      by simp
    then have "(jS. (a j / sum a S) *R y j)  C"
      using insert
      by (simp add: insert.IH flip: sum_divide_distrib)
    with convex C insert 0  sum a S 
    have "a i *R y i + sum a S *R (jS. (a j / sum a S) *R y j)  C"
      by (simp add: convex_def)
    then show ?thesis
      by (simp add: scaleR_sum_right False)
  qed
  then show ?case using finite S and i  S
    by simp
qed

lemma convex:
  "convex S  
    ((k::nat) u x. (i. 1i  ik  0  u i  x i S)  (sum u {1..k} = 1)
       sum (λi. u i *R x i) {1..k}  S)"  
  (is "?lhs = ?rhs")
proof
  show "?lhs  ?rhs"
    by (metis (full_types) atLeastAtMost_iff convex_sum finite_atLeastAtMost)
  assume *: "k u x. ( i :: nat. 1  i  i  k  0  u i  x i  S)  sum 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 S: "(j  {1..2}. ?u j *R ?x j)  S"
      using sum.If_cases[of "{(1 :: nat) .. 2}" "λx. x = 1" "λx. μ" "λx. 1 - μ"]
      using mu xy "*" by auto
    have grarr: "(j  {Suc (Suc 0)..2}. ?u j *R ?x j) = (1 - μ) *R y"
      using sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "λ j. (1 - μ) *R y"] by auto
    with sum.atLeast_Suc_atMost
    have "(j  {1..2}. ?u j *R ?x j) = μ *R x + (1 - μ) *R y"
      by (smt (verit, best) Suc_1 Suc_eq_plus1 add_0 le_add1)
    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  (xt. 0  u x)  sum u t = 1  sum (λx. u x *R x) t  S)"
proof safe
  fix t
  fix u :: "'a  real"
  assume "convex S"
    and "finite t"
    and "t  S" "xt. 0  u x" "sum u t = 1"
  then show "(xt. u x *R x)  S"
    by (simp add: convex_sum subsetD)
next
  assume *: "t.  u. finite t  t  S  (xt. 0  u x) 
    sum u t = 1  (xt. 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. (xS. 0  u x)  sum u S = 1  sum (λx. u x *R x) S  S)"
       (is "?lhs = ?rhs")
proof 
  { 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
    fix T :: "'a set" and u :: "'a  real"
    assume sum: "u. (xS. 0  u x)  sum u S = 1  (xS. u x *R x)  S"
    assume *: "xT. 0  u x" "sum u T = 1"
    assume "T  S"
    then have "S  T = T" by auto
    with sum[THEN spec[where x="λx. if xT then u x else 0"]] *
    have "(xT. u x *R x)  S"
      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) }
  moreover assume ?rhs
  ultimately show ?lhs
    unfolding convex_explicit by auto
qed (auto simp: convex_explicit assms)


subsection ‹Convex Functions on a Set›

definitiontag important› convex_on :: "'a::real_vector set  ('a  real)  bool"
  where "convex_on S f  convex S 
    (xS. yS. u0. v0. u + v = 1  f (u *R x + v *R y)  u * f x + v * f y)"

definitiontag important› concave_on :: "'a::real_vector set  ('a  real)  bool"
  where "concave_on S f  convex_on S (λx. - f x)"

lemma convex_on_iff_concave: "convex_on S f = concave_on S (λx. - f x)"
  by (simp add: concave_on_def)

lemma concave_on_iff:
  "concave_on S f  convex S 
    (xS. yS. u0. v0. u + v = 1  f (u *R x + v *R y)  u * f x + v * f y)"
  by (auto simp: concave_on_def convex_on_def algebra_simps)

lemma concave_onD:
  assumes "concave_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 by (auto simp: concave_on_iff)

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"
    and "convex A"
  shows "convex_on A f"
  unfolding convex_on_def
  by (smt (verit, del_insts) assms mult_cancel_right1 mult_eq_0_iff scaleR_collapse scaleR_eq_0_iff)

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 by (auto simp: convex_on_def)

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"
    and "convex A"
  shows "convex_on A f"
  by (smt (verit, best) add.commute assms convex_onI distrib_left linorder_cases mult.commute mult_cancel_left2 scaleR_collapse)

lemma concave_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"
    and "convex A"
  shows "concave_on A f"
  by (smt (verit) assms concave_on_def convex_on_linorderI mult_minus_right)

lemma convex_on_imp_convex: "convex_on A f  convex A"
  by (auto simp: convex_on_def)

lemma concave_on_imp_convex: "concave_on A f  convex A"
  by (simp add: concave_on_def convex_on_imp_convex)

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 S  convex_on S f"
  by (simp add: convex_on_def subset_iff)

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)
  }
  with assms show ?thesis
    unfolding convex_on_def by auto
qed

lemma convex_on_ident: "convex_on S (λx. x)  convex S"
  by (simp add: convex_on_def)

lemma concave_on_ident: "concave_on S (λx. x)  convex S"
  by (simp add: concave_on_iff)

lemma convex_on_const: "convex_on S (λx. a)  convex S"
  by (simp add: convex_on_def flip: distrib_right)

lemma concave_on_const: "concave_on S (λx. a)  convex S"
  by (simp add: concave_on_iff flip: distrib_right)

lemma convex_on_diff:
  assumes "convex_on S f" and "concave_on S g"
  shows "convex_on S (λx. f x - g x)"
  using assms concave_on_def convex_on_add by fastforce

lemma concave_on_diff:
  assumes "concave_on S f"
    and "convex_on S g"
  shows "concave_on S (λx. f x - g x)"
  using convex_on_diff assms concave_on_def by fastforce

lemma concave_on_add:
  assumes "concave_on S f"
    and "concave_on S g"
  shows "concave_on S (λx. f x + g x)"
  using assms convex_on_iff_concave concave_on_diff concave_on_def by fastforce

lemma convex_on_mul:
  fixes S::"real set"
  assumes "convex_on S f" "convex_on S g"
  assumes "mono_on S f" "mono_on S g"
  assumes fty: "f  S  {0..}" and gty: "g  S  {0..}"
  shows "convex_on S (λx. f x*g x)"
proof (intro convex_on_linorderI)
  show "convex S"
    using assms convex_on_imp_convex by auto
  fix t::real and x y
  assume t: "0 < t" "t < 1" and xy: "x  S" "y  S" "x<y"
  have *: "t*(1-t) * f x * g y + t*(1-t) * f y * g x  t*(1-t) * f x * g x + t*(1-t) * f y * g y"
    using t mono_on S f mono_on S g xy
    by (smt (verit, ccfv_SIG) left_diff_distrib mono_onD mult_left_less_imp_less zero_le_mult_iff)
  have inS: "(1-t)*x + t*y  S"
    using t xy convex S by (simp add: convex_alt)
  then have "f ((1-t)*x + t*y) * g ((1-t)*x + t*y)  ((1-t) * f x + t * f y)*g ((1-t)*x + t*y)"
    using convex_onD [OF convex_on S f, of t x y] t xy fty gty
    by (intro mult_mono add_nonneg_nonneg) (auto simp: Pi_iff zero_le_mult_iff)
  also have "  ((1-t) * f x + t * f y) * ((1-t)*g x + t*g y)"
    using convex_onD [OF convex_on S g, of t x y] t xy fty gty inS
    by (intro mult_mono add_nonneg_nonneg) (auto simp: Pi_iff zero_le_mult_iff)
  also have "  (1-t) * (f x*g x) + t * (f y*g y)"
    using * by (simp add: algebra_simps)
  finally show "f ((1-t) *R x + t *R y) * g ((1-t) *R x + t *R y)  (1-t)*(f x*g x) + t*(f y*g y)" 
    by simp
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 * (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_on_cdiv [intro]:
  fixes c :: real
  assumes "0  c" and "convex_on S f"
  shows "convex_on S (λx. f x / c)"
  unfolding divide_inverse
  using convex_on_cmul [of "inverse c" S f] by (simp add: mult.commute assms)

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"
  assumes "convex S"
  shows "convex_on S (λx. dist a x)"
unfolding convex_on_def dist_norm
proof (intro conjI strip)
  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"
    by (metis u + v = 1 scaleR_left.add scaleR_one)
  then have "a - (u *R x + v *R y) = (u *R (a - x)) + (v *R (a - y))"
    by (auto simp: algebra_simps)
  then show "norm (a - (u *R x + v *R y))  u * norm (a - x) + v * norm (a - y)"
    by (smt (verit, best) 0  u 0  v norm_scaleR norm_triangle_ineq)
qed (use assms in auto)

lemma concave_on_mul:
  fixes S::"real set"
  assumes f: "concave_on S f" and g: "concave_on S g"
  assumes "mono_on S f" "antimono_on S g"
  assumes fty: "f  S  {0..}" and gty: "g  S  {0..}"
  shows "concave_on S (λx. f x * g x)"
proof (intro concave_on_linorderI)
  show "convex S"
    using concave_on_imp_convex f by blast
  fix t::real and x y
  assume t: "0 < t" "t < 1" and xy: "x  S" "y  S" "x<y"
  have inS: "(1-t)*x + t*y  S"
    using t xy convex S by (simp add: convex_alt)
  have "f x * g y + f y * g x  f x * g x + f y * g y"
    using mono_on S f antimono_on S g
    unfolding monotone_on_def by (smt (verit, best) left_diff_distrib mult_left_mono xy)
  with t have *: "t*(1-t) * f x * g y + t*(1-t) * f y * g x  t*(1-t) * f x * g x + t*(1-t) * f y * g y"
    by (smt (verit, ccfv_SIG) distrib_left mult_left_mono diff_ge_0_iff_ge mult.assoc)
  have "(1 - t) * (f x * g x) + t * (f y * g y)  ((1-t) * f x + t * f y) * ((1-t) * g x + t * g y)"
    using * by (simp add: algebra_simps)
  also have "  ((1-t) * f x + t * f y)*g ((1-t)*x + t*y)"
    using concave_onD [OF concave_on S g, of t x y] t xy fty gty inS
    by (intro mult_mono add_nonneg_nonneg) (auto simp: Pi_iff zero_le_mult_iff)
  also have "  f ((1-t)*x + t*y) * g ((1-t)*x + t*y)"
    using concave_onD [OF concave_on S f, of t x y] t xy fty gty inS
    by (intro mult_mono add_nonneg_nonneg) (auto simp: Pi_iff zero_le_mult_iff)
  finally show "(1 - t) * (f x * g x) + t * (f y * g y)
            f ((1 - t) *R x + t *R y) * g ((1 - t) *R x + t *R y)" 
    by simp
qed

lemma concave_on_cmul [intro]:
  fixes c :: real
  assumes "0  c" and "concave_on S f"
  shows "concave_on S (λx. c * f x)"
  using assms convex_on_cmul [of c S "λx. - f x"]
  by (auto simp: concave_on_def)

lemma concave_on_cdiv [intro]:
  fixes c :: real
  assumes "0  c" and "concave_on S f"
  shows "concave_on S (λx. f x / c)"
  unfolding divide_inverse
  using concave_on_cmul [of "inverse c" S f] by (simp add: mult.commute assms)

subsectiontag unimportant› ‹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)"
  by (simp add: assms convex_linear_image)

lemma convex_scaled:
  assumes "convex S"
  shows "convex ((λx. x *R c) ` S)"
  by (simp add: assms convex_linear_image)

lemma convex_negations:
  assumes "convex S"
  shows "convex ((λx. - x) ` S)"
  by (simp add: assms convex_linear_image module_hom_uminus)

lemma convex_sums:
  assumes "convex S"
    and "convex T"
  shows "convex (x S. y  T. {x + y})"
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 S. y  T. {x + y})"
    by auto
  finally show ?thesis .
qed

lemma convex_differences:
  assumes "convex S" "convex T"
  shows "convex (x S. y  T. {x - y})"
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:
  "convex ((+) a ` S)" if "convex S"
proof -
  have "( x {a}. y  S. {x + y}) = (+) a ` S"
    by auto
  then show ?thesis
    using convex_sums [OF convex_singleton [of a] that] by auto
qed

lemma convex_translation_subtract:
  "convex ((λb. b - a) ` S)" if "convex S"
  using convex_translation [of S "- a"] that by (simp cong: image_cong_simp)

lemma convex_affinity:
  assumes "convex S"
  shows "convex ((λx. a + c *R x) ` S)"
proof -
  have "(λx. a + c *R x) ` S = (+) a ` (*R) c ` S"
    by auto
  then show ?thesis
    using convex_translation[OF convex_scaling[OF assms], of a c] by auto
qed

lemma convex_on_sum:
  fixes a :: "'a  real"
    and y :: "'a  'b::real_vector"
    and f :: "'b  real"
  assumes "finite S" "S  {}"
    and "convex_on C f"
    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 show ?case
    by auto
next
  case (insert i S)
  then have yai: "y i  C" "a i  0"
    by auto
  with insert 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 add: convex_on_def)
  show ?case
  proof (cases "a i = 1")
    case True
    with insert have "( j  S. a j) = 0"
      by auto
    with insert show ?thesis
      by (simp add: sum_nonneg_eq_0_iff)
  next
    case False
    then have ai1: "a i < 1"
      using sum_nonneg_leq_bound[of "insert i S" a] insert by force
    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 sum.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 sum_divide_distrib by simp
    have "convex C"
      using convex_on C f by (simp add: convex_on_def)
    have asum: "( j  S. ?a j *R y j)  C"
      using insert convex_sum [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)"
      by (simp add: add.commute insert.hyps)
    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.sum[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 ai1 by (smt (verit) asum conv real_scaleR_def yai)
    also have "  (1 - a i) * ( j  S. ?a j * f (y j)) + a i * f (y i)"
      using asum_le i0 by fastforce
    also have " = ( j  S. a j * f (y j)) + a i * f (y i)"
      using i0 by (auto simp: sum_distrib_left)
    finally show ?thesis
      using insert by auto
  qed
qed

lemma concave_on_sum:
  fixes a :: "'a  real"
    and y :: "'a  'b::real_vector"
    and f :: "'b  real"
  assumes "finite S" "S  {}"
    and "concave_on C f" 
    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))"
proof -
  have "(uminus  f) (iS. a i *R y i)  (iS. a i * (uminus  f) (y i))"
  proof (intro convex_on_sum)
    show "convex_on C (uminus  f)"
      by (simp add: assms convex_on_iff_concave)
  qed (use assms in auto)
  then show ?thesis
    by (simp add: sum_negf o_def)
qed

lemma convex_on_alt:
  fixes C :: "'a::real_vector set"
  shows "convex_on C f  convex C 
         (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)"
  by (smt (verit) convex_on_def)

lemma convex_on_slope_le:
  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 -
  define a where "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
  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"
    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  " "  1"
      using assms * by (auto simp: field_simps)
    then have comb: " * x + (1 - ) * y  C"
      using assms iffD1[OF convex_alt, rule_format, of C y x ]
      by (simp add: algebra_simps)
    have " * x + (1 - ) * 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 by (simp only: add_divide_distrib) (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 by (auto simp: le_less)
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: "x  C"
    and y: "y  C"
  shows "f' x * (y - x)  f y - f x"
  using assms
proof -
  have "f y - f x  f' x * (y - x)" "f' y * (x - y)  f x - f y"
    if *: "x  C" "y  C" "y > x" for x y :: real
  proof -
    from * have ge: "y - x > 0" "y - x  0" and 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
    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
    from z1 have "f x - f y = (x - y) * f' z1"
      by (simp add: field_simps)
    then have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
      using le(1) z3(3) 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
    with cool' have "f' y - (f x - f y) / (x - y)  0"
      using z1 by auto
    then have res: "f' y * (x - y)  f x - f y"
      by (meson diff_ge_0_iff_ge le(1) neg_divide_le_eq)
    have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
      using le(1) z1(3) z2(3) by auto
    have "z2  C"
      using z2 z1 * atMostAtLeast_subset_convex convex C z1  C y  C z1 < y
      by fastforce
    with z1 assms have "(z1 - x) * f'' z2  0"
      by auto
    then show "f y - f x  f' x * (y - x)" "f' y * (x - y)  f x - f y"
      using that(3) z1(3) res cool by auto
  qed
  then show ?thesis
    using x y by fastforce
qed

lemma f''_ge0_imp_convex:
  fixes f :: "real  real"
  assumes "convex C"
    and "x. x  C  DERIV f x :> (f' x)"
    and "x. x  C  DERIV f' x :> (f'' x)"
    and "x. x  C  f'' x  0"
  shows "convex_on C f"
  by (metis assms f''_imp_f' pos_convex_function)

lemma f''_le0_imp_concave:
  fixes f :: "real  real"
  assumes "convex C"
    and "x. x  C  DERIV f x :> (f' x)"
    and "x. x  C  DERIV f' x :> (f'' x)"
    and "x. x  C  f'' x  0"
  shows "concave_on C f"
  unfolding concave_on_def
  by (rule assms f''_ge0_imp_convex derivative_eq_intros | simp)+

lemma convex_power_even:
  assumes "even n"
  shows "convex_on (UNIV::real set) (λx. x^n)"
proof (intro f''_ge0_imp_convex)
  show "((λx. x ^ n) has_real_derivative of_nat n * x^(n-1)) (at x)" for x
    by (rule derivative_eq_intros | simp)+
  show "((λx. of_nat n * x^(n-1)) has_real_derivative of_nat n * of_nat (n-1) * x^(n-2)) (at x)" for x
    by (rule derivative_eq_intros | simp add: eval_nat_numeral)+
  show "x. 0  real n * real (n - 1) * x ^ (n - 2)"
    using assms by (auto simp: zero_le_mult_iff zero_le_even_power)
qed auto

lemma convex_power_odd:
  assumes "odd n"
  shows "convex_on {0::real..} (λx. x^n)"
proof (intro f''_ge0_imp_convex)
  show "((λx. x ^ n) has_real_derivative of_nat n * x^(n-1)) (at x)" for x
    by (rule derivative_eq_intros | simp)+
  show "((λx. of_nat n * x^(n-1)) has_real_derivative of_nat n * of_nat (n-1) * x^(n-2)) (at x)" for x
    by (rule derivative_eq_intros | simp add: eval_nat_numeral)+
  show "x. x  {0::real..}  0  real n * real (n - 1) * x ^ (n - 2)"
    using assms by (auto simp: zero_le_mult_iff zero_le_even_power)
qed auto

lemma convex_power2: "convex_on (UNIV::real set) power2"
  by (simp add: convex_power_even)

lemma log_concave:
  fixes b :: real
  assumes "b > 1"
  shows "concave_on {0<..} (λ x. log b x)"
  using assms
  by (intro f''_le0_imp_concave derivative_eq_intros | simp)+

lemma ln_concave: "concave_on {0<..} ln"
  unfolding log_ln by (simp add: log_concave)

lemma minus_log_convex:
  fixes b :: real
  assumes "b > 1"
  shows "convex_on {0 <..} (λ x. - log b x)"
  using assms concave_on_def log_concave by blast

lemma powr_convex: 
  assumes "p  1" shows "convex_on {0<..} (λx. x powr p)"
  using assms
  by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+

lemma exp_convex: "convex_on UNIV exp"
  by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+

text ‹The AM-GM inequality: the arithmetic mean exceeds the geometric mean.›
lemma arith_geom_mean:
  fixes x :: "'a  real"
  assumes "finite S" "S  {}"
    and x: "i. i  S  x i  0"
  shows "(i  S. x i / card S)  (i  S. x i) powr (1 / card S)"
proof (cases "iS. x i = 0")
  case True
  then have "(i  S. x i) = 0"
    by (simp add: finite S)
  moreover have "(i  S. x i / card S)  0"
    by (simp add: sum_nonneg x)
  ultimately show ?thesis
    by simp
next
  case False
  have "ln (i  S. (1 / card S) *R x i)  (i  S. (1 / card S) * ln (x i))"
  proof (intro concave_on_sum)
    show "concave_on {0<..} ln"
      by (simp add: ln_concave)
    show "i. iS  x i  {0<..}"
      using False x by fastforce
  qed (use assms False in auto)
  moreover have "(i  S. (1 / card S) *R x i) > 0"
    using False assms by (simp add: card_gt_0_iff less_eq_real_def sum_pos)
  ultimately have "(i  S. (1 / card S) *R x i)  exp (i  S. (1 / card S) * ln (x i))"
    using ln_ge_iff by blast
  then have "(i  S. x i / card S)  exp (i  S. ln (x i) / card S)"
    by (simp add: divide_simps)
  then show ?thesis
    using assms False
    by (smt (verit, ccfv_SIG) divide_inverse exp_ln exp_powr_real exp_sum inverse_eq_divide prod.cong prod_powr_distrib) 
qed

subsectiontag unimportant› ‹Convexity of real functions›

lemma convex_on_realI:
  assumes "connected A"
    and "x. x  A  (f has_real_derivative f' x) (at x)"
    and "x y. x  A  y  A  x  y  f' x  f' y"
  shows "convex_on A f"
proof (rule convex_on_linorderI)
  show "convex A"
    using connected A convex_real_interval interval_cases
    by (smt (verit, ccfv_SIG) connectedD_interval convex_UNIV convex_empty)
      ― ‹the equivalence of "connected" and "convex" for real intervals is proved later›
next
  fix t x y :: real
  assume t: "t > 0" "t < 1"
  assume xy: "x  A" "y  A" "x < y"
  define z where "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
  then show "(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:
  fixes A :: "real set"
  assumes "A  {0<..}" "convex A"
  shows "convex_on A inverse"
proof -
  have "convex_on {0::real<..} inverse"
  proof (intro convex_on_realI)
    fix u v :: real
    assume "u  {0<..}" "v  {0<..}" "u  v"
    with assms show "-inverse (u^2)  -inverse (v^2)"
      by simp
  next
    show "x. x  {0<..}  (inverse has_real_derivative - inverse (x2)) (at x)"
      by (rule derivative_eq_intros | simp add: power2_eq_square)+
  qed auto
  then show ?thesis
    using assms convex_on_subset by blast
qed

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 x y rule: linorder_cases)
  case less
  then have 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 field_split_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 (use assms in auto)

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 x y rule: linorder_cases)
  case less
  then have 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 field_split_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 (use assms in auto)

lemma concave_onD_Icc:
  assumes "concave_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 concave_onD [OF assms(1)]) simp_all

lemma concave_onD_Icc':
  assumes "concave_on {x..y} f" "c  {x..y}"
  defines "d  y - x"
  shows "f c  (f y - f x) / d * (c - x) + f x"
proof -
  have "- f c  (f x - f y) / d * (c - x) - f x"
    using assms convex_onD_Icc' [of x y "λx. - f x" c]
    by (simp add: concave_on_def)
  then show ?thesis
    by (smt (verit, best) divide_minus_left mult_minus_left)
qed

lemma concave_onD_Icc'':
  assumes "concave_on {x..y} f" "c  {x..y}"
  defines "d  y - x"
  shows "f c  (f x - f y) / d * (y - c) + f y"
proof -
  have "- f c  (f y - f x) / d * (y - c) - f y"
    using assms convex_onD_Icc'' [of x y "λx. - f x" c]
    by (simp add: concave_on_def)
  then show ?thesis
    by (smt (verit, best) divide_minus_left mult_minus_left)
qed

lemma convex_on_le_max:
  fixes a::real
  assumes "convex_on {x..y} f" and a: "a  {x..y}"
  shows "f a  max (f x) (f y)"
proof -
  have *: "(f y - f x) * (a - x)  (f y - f x) * (y - x)" if "f x  f y"
    using a that by (intro mult_left_mono) auto
  have "f a  (f y - f x) / (y - x) * (a - x) + f x" 
    using assms convex_onD_Icc' by blast
  also have "  max (f x) (f y)"
    using a *
    by (simp add: divide_le_0_iff mult_le_0_iff zero_le_mult_iff max_def add.commute mult.commute scaling_mono)
  finally show ?thesis .
qed

lemma concave_on_ge_min:
  fixes a::real
  assumes "concave_on {x..y} f" and a: "a  {x..y}"
  shows "f a  min (f x) (f y)"
proof -
  have *: "(f y - f x) * (a - x)  (f y - f x) * (y - x)" if "f x  f y"
    using a that by (intro mult_left_mono_neg) auto
  have "min (f x) (f y)  (f y - f x) / (y - x) * (a - x) + f x"
    using a * apply (simp add: zero_le_divide_iff mult_le_0_iff zero_le_mult_iff min_def)
    by (smt (verit, best) nonzero_eq_divide_eq pos_divide_le_eq)
  also have "  f a"
    using assms concave_onD_Icc' by blast
  finally show ?thesis .
qed

subsection ‹Some inequalities: Applications of convexity›

lemma Youngs_inequality_0:
  fixes a::real
  assumes "0  α" "0  β" "α+β = 1" "a>0" "b>0"
  shows "a powr α * b powr β  α*a + β*b"
proof -
  have "α * ln a + β * ln b  ln (α * a + β * b)"
    using assms ln_concave by (simp add: concave_on_iff)
  moreover have "0 < α * a + β * b"
    using assms by (smt (verit) mult_pos_pos split_mult_pos_le)
  ultimately show ?thesis
    using assms by (simp add: powr_def mult_exp_exp flip: ln_ge_iff)
qed

lemma Youngs_inequality:
  fixes p::real
  assumes "p>1" "q>1" "1/p + 1/q = 1" "a0" "b0"
  shows "a * b  a powr p / p + b powr q / q"
proof (cases "a=0  b=0")
  case False
  then show ?thesis 
  using Youngs_inequality_0 [of "1/p" "1/q" "a powr p" "b powr q"] assms
  by (simp add: powr_powr)
qed (use assms in auto)

lemma Cauchy_Schwarz_ineq_sum:
  fixes a :: "'a  'b::linordered_field"
  shows "(iI. a i * b i)2  (iI. (a i)2) * (iI. (b i)2)"
proof (cases "(iI. (b i)2) > 0")
  case False
  then consider "i. iI  b i = 0" | "infinite I"
    by (metis (mono_tags, lifting) sum_pos2 zero_le_power2 zero_less_power2)
  thus ?thesis
    by fastforce
next
  case True
  define r where "r  (iI. a i * b i) / (iI. (b i)2)"
  have "0  (iI. (a i - r * b i)2)"
    by (simp add: sum_nonneg)
  also have "... = (iI. (a i)2) - 2 * r * (iI. a i * b i) + r2 * (iI. (b i)2)"
    by (simp add: algebra_simps power2_eq_square sum_distrib_left flip: sum.distrib)
  also have " = (iI. (a i)2) - ((iI. a i * b i))2 / (iI. (b i)2)"
    by (simp add: r_def power2_eq_square)
  finally have "0  (iI. (a i)2) - ((iI. a i * b i))2 / (iI. (b i)2)" .
  hence "((iI. a i * b i))2 / (iI. (b i)2)  (iI. (a i)2)"
    by (simp add: le_diff_eq)
  thus "((iI. a i * b i))2  (iI. (a i)2) * (iI. (b i)2)"
    by (simp add: pos_divide_le_eq True)
qed

lemma sum_squared_le_sum_of_squares:
  fixes f :: "'a  real"
  assumes "i. iI  f i  0" "finite I" "I  {}"
  shows "(iI. f i)2  (yI. (f y)2) * card I"
proof (cases "finite I  I  {}")
  case True
  have "(iI. f i / real (card I))2  (iI. (f i)2 / real (card I))"
    using assms convex_on_sum [OF _ _ convex_power2, where a = "λx. 1 / real(card I)" and S=I]
    by simp
  then show ?thesis
    using assms  
    by (simp add: divide_simps power2_eq_square split: if_split_asm flip: sum_divide_distrib)
qed auto

subsection ‹Misc related lemmas›

lemma convex_translation_eq [simp]:
  "convex ((+) a ` s)  convex s"
  by (metis convex_translation translation_galois)

lemma convex_translation_subtract_eq [simp]:
  "convex ((λb. b - a) ` s)  convex s"
  using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp)

lemma convex_linear_image_eq [simp]:
    fixes f :: "'a::real_vector  'b::real_vector"
    shows "linear f; inj f  convex (f ` s)  convex s"
    by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)

lemma vector_choose_size:
  assumes "0  c"
  obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
proof -
  obtain a::'a where "a  0"
    using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
  then show ?thesis
    by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
qed

lemma vector_choose_dist:
  assumes "0  c"
  obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)

lemma sum_delta'':
  fixes s::"'a::real_vector set"
  assumes "finite s"
  shows "(xs. (if y = x then f x else 0) *R x) = (if ys then (f y) *R y else 0)"
proof -
  have *: "x y. (if y = x then f x else (