src/HOL/Library/Convex.thy

generalized and simplified proofs of several theorems about convex sets

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

header {* Convexity in real vector spaces *}

theory Convex
imports Product_Vector
begin

subsection {* Convexity. *}

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 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 mem_convex:
  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 convex_empty[intro]: "convex {}"
  unfolding convex_def by simp

lemma convex_singleton[intro]: "convex {a}"
  unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])

lemma convex_UNIV[intro]: "convex UNIV"
  unfolding convex_def by auto

lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> 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:
  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


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 "(\<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
proof (induct s arbitrary:a rule: finite_induct)
  case empty
  then show ?case by auto
next
  case (insert i s) note asms = this
  { assume "a i = 1"
    then have "(\<Sum> j \<in> s. a j) = 0"
      using asms by auto
    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
      using setsum_nonneg_0[where 'b=real] asms by fastforce
    then have ?case using asms by auto }
  moreover
  { assume asm: "a i \<noteq> 1"
    from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
    have fis: "finite (insert i s)" using asms by auto
    then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a 1] asms by simp
    then have "a i < 1" using asm by auto
    then have i0: "1 - a i > 0" by auto
    let ?a = "\<lambda>j. a j / (1 - a i)"
    { fix j assume "j \<in> s"
      then have "?a j \<ge> 0"
        using i0 asms divide_nonneg_pos
        by fastforce
    } note a_nonneg = this
    have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
    then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms 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
    with asms have "(\<Sum>j\<in>s. ?a j *\<^sub>R y j) \<in> C" using a_nonneg by fastforce
    then have "a i *\<^sub>R y i + (1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
      using asms[unfolded convex_def, rule_format] yai ai1 by auto
    then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. (1 - a i) *\<^sub>R (?a j *\<^sub>R y j)) \<in> C"
      using scaleR_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j *\<^sub>R y j" s] by auto
    then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" using i0 by auto
    then have ?case using setsum.insert asms by auto
  }
  ultimately show ?case by auto
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"
  from this convex_setsum[of "{1 .. k}" s]
  show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" by auto
next
  assume asm: "\<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_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
      by auto
    with asm[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" "finite t"
    "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 asm0: "\<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 asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
    { assume "x \<noteq> y"
      then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
        using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
          asm by auto }
    moreover
    { assume "x = y"
      then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
        using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
          asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) }
    ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast
  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_cases if_distrib if_distrib_arg)
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)

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" "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: 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]:
  assumes "0 \<le> (c::real)" "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"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "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 add: mult_left_mono add_mono)
  also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[symmetric] by auto
  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 add: convex_on_def dist_norm)
  fix x y
  assume "x\<in>s" "y\<in>s"
  fix u v :: real
  assume "0 \<le> u" "0 \<le> v" "u + v = 1"
  have "a = u *\<^sub>R a + v *\<^sub>R a"
    unfolding scaleR_left_distrib[symmetric] and `u+v=1` 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 add: 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 `0 \<le> u` `0 \<le> 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 ((\<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 `convex s` 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)