(* 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 \<open>Convex Sets and Functions\<close>
theory Convex
imports
Affine
"HOL-Library.Set_Algebras"
begin
subsection \<open>Convex Sets\<close>
definition\<^marker>\<open>tag important\<close> convex :: "'a::real_vector set \<Rightarrow> bool"
where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
lemma convexI:
assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
shows "convex s"
using assms unfolding convex_def by fast
lemma convexD:
assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
using assms unfolding convex_def by fast
lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
(is "_ \<longleftrightarrow> ?alt")
proof
show "convex s" if alt: ?alt
proof -
{
fix x y and u v :: real
assume mem: "x \<in> s" "y \<in> s"
assume "0 \<le> u" "0 \<le> v"
moreover
assume "u + v = 1"
then have "u = 1 - v" by auto
ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
using alt [rule_format, OF mem] by auto
}
then show ?thesis
unfolding convex_def by auto
qed
show ?alt if "convex s"
using that by (auto simp: convex_def)
qed
lemma convexD_alt:
assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
using assms unfolding convex_alt by auto
lemma mem_convex_alt:
assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
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: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
unfolding convex_def by auto
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
unfolding convex_def by auto
lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
unfolding convex_def by auto
lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
unfolding convex_def by auto
lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
unfolding convex_def
by (auto simp: inner_add intro!: convex_bound_le)
lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
proof -
have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
by auto
show ?thesis
unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
qed
lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
proof -
have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> 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 \<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_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
using convex_halfspace_ge[of b "1::complex"] by simp
lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
using convex_halfspace_le[of "1::complex" b] by simp
lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
using convex_halfspace_ge[of b \<i>] by simp
lemma convex_halfspace_Im_le: "convex {x. Im x \<le> 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 \<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\<^marker>\<open>tag unimportant\<close> \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
lemma convex_sum:
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 + sum 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> sum a S"
by (simp add: sum_nonneg)
have "a i *\<^sub>R y i + (\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
proof (cases "sum a S = 0")
case True
with \<open>a i + sum a S = 1\<close> have "a i = 1"
by simp
from sum_nonneg_0 [OF \<open>finite S\<close> _ True] \<open>\<forall>j\<in>S. 0 \<le> a j\<close> have "\<forall>j\<in>S. a j = 0"
by simp
show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>S. a j = 0\<close> and \<open>y i \<in> C\<close>
by simp
next
case False
with \<open>0 \<le> sum a S\<close> have "0 < sum a S"
by simp
then have "(\<Sum>j\<in>S. (a j / sum 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 sum_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> sum a S\<close> and \<open>a i + sum a S = 1\<close>
have "a i *\<^sub>R y i + sum a S *\<^sub>R (\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C"
by (rule convexD)
then show ?thesis
by (simp add: scaleR_sum_right False)
qed
then show ?case using \<open>finite S\<close> and \<open>i \<notin> S\<close>
by simp
qed
lemma convex:
"convex S \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>S) \<and> (sum u {1..k} = 1)
\<longrightarrow> sum (\<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"
"sum u {1..k} = 1"
with convex_sum[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> sum 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 "sum ?u {1 .. 2} = 1"
using sum.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 sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
from sum.atLeast_Suc_atMost[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> sum u t = 1 \<longrightarrow> sum (\<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" "sum u t = 1"
then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S"
using convex_sum[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>
sum 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> sum u S = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) S \<in> S)"
(is "?lhs = ?rhs")
proof
{ have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
by simp
fix T :: "'a set" and u :: "'a \<Rightarrow> real"
assume sum: "\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> S"
assume *: "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1"
assume "T \<subseteq> S"
then have "S \<inter> T = T" by auto
with sum[THEN spec[where x="\<lambda>x. if x\<in>T then u x else 0"]] * have "(\<Sum>x\<in>T. u x *\<^sub>R x) \<in> 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 \<open>Convex Functions on a Set\<close>
definition\<^marker>\<open>tag important\<close> convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
where "convex_on S f \<longleftrightarrow>
(\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
lemma convex_onI [intro?]:
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
shows "convex_on A f"
unfolding convex_on_def
proof clarify
fix x y
fix u v :: real
assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
from A(5) have [simp]: "v = 1 - u"
by (simp add: algebra_simps)
from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
using assms[of u y x]
by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
qed
lemma convex_on_linorderI [intro?]:
fixes A :: "('a::{linorder,real_vector}) set"
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
shows "convex_on A f"
proof
fix x y
fix t :: real
assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
with assms [of t x y] assms [of "1 - t" y x]
show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
qed
lemma convex_onD:
assumes "convex_on A f"
shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
using assms by (auto simp: convex_on_def)
lemma convex_onD_Icc:
assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
lemma convex_on_subset: "convex_on t f \<Longrightarrow> S \<subseteq> t \<Longrightarrow> convex_on S f"
unfolding convex_on_def by auto
lemma convex_on_add [intro]:
assumes "convex_on S f"
and "convex_on S g"
shows "convex_on S (\<lambda>x. f x + g x)"
proof -
{
fix x y
assume "x \<in> S" "y \<in> S"
moreover
fix u v :: real
assume "0 \<le> u" "0 \<le> v" "u + v = 1"
ultimately
have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
using assms unfolding convex_on_def by (auto simp: add_mono)
then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
by (simp add: field_simps)
}
then show ?thesis
unfolding convex_on_def by auto
qed
lemma convex_on_cmul [intro]:
fixes c :: real
assumes "0 \<le> c"
and "convex_on S f"
shows "convex_on S (\<lambda>x. c * f x)"
proof -
have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
for u c fx v fy :: real
by (simp add: field_simps)
show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
unfolding convex_on_def and * by auto
qed
lemma convex_lower:
assumes "convex_on S f"
and "x \<in> S"
and "y \<in> S"
and "0 \<le> u"
and "0 \<le> v"
and "u + v = 1"
shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
proof -
let ?m = "max (f x) (f y)"
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
using assms(4,5) by (auto simp: mult_left_mono add_mono)
also have "\<dots> = max (f x) (f y)"
using assms(6) by (simp add: distrib_right [symmetric])
finally show ?thesis
using assms unfolding convex_on_def by fastforce
qed
lemma convex_on_dist [intro]:
fixes S :: "'a::real_normed_vector set"
shows "convex_on S (\<lambda>x. dist a x)"
proof (auto simp: convex_on_def dist_norm)
fix x y
assume "x \<in> S" "y \<in> S"
fix u v :: real
assume "0 \<le> u"
assume "0 \<le> v"
assume "u + v = 1"
have "a = u *\<^sub>R a + v *\<^sub>R a"
unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
by (auto simp: algebra_simps)
show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
qed
subsection\<^marker>\<open>tag unimportant\<close> \<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 (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
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)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
by auto
finally show ?thesis .
qed
lemma convex_differences:
assumes "convex S" "convex T"
shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
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:
"convex ((+) a ` S)" if "convex S"
proof -
have "(\<Union> x\<in> {a}. \<Union>y \<in> 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 ((\<lambda>b. 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 ((\<lambda>x. a + c *\<^sub>R x) ` S)"
proof -
have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>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 \<Rightarrow> real"
and y :: "'a \<Rightarrow> 'b::real_vector"
and f :: "'b \<Rightarrow> real"
assumes "finite s" "s \<noteq> {}"
and "convex_on C f"
and "convex C"
and "(\<Sum> i \<in> s. a i) = 1"
and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
using assms
proof (induct s arbitrary: a rule: finite_ne_induct)
case (singleton i)
then have ai: "a i = 1"
by auto
then show ?case
by auto
next
case (insert i s)
then have "convex_on C f"
by simp
from this[unfolded convex_on_def, rule_format]
have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
by simp
show ?case
proof (cases "a i = 1")
case True
then have "(\<Sum> j \<in> s. a j) = 0"
using insert by auto
then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
using insert by (fastforce simp: sum_nonneg_eq_0_iff)
then show ?thesis
using insert by auto
next
case False
from insert have yai: "y i \<in> C" "a i \<ge> 0"
by auto
have fis: "finite (insert i s)"
using insert by auto
then have ai1: "a i \<le> 1"
using sum_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 sum.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 sum_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_sum [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 sum.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.sum[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 sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
using i0 by auto
also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
using i0 by auto
also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
using insert by auto
finally show ?thesis
by simp
qed
qed
lemma convex_on_alt:
fixes C :: "'a::real_vector set"
shows "convex_on C f \<longleftrightarrow>
(\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
proof safe
fix x y
fix \<mu> :: real
assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
from this[unfolded convex_on_def, rule_format]
have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
by auto
from this [of "\<mu>" "1 - \<mu>", simplified] *
show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
by auto
next
assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
{
fix x y
fix u v :: real
assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
then have[simp]: "1 - u = v" by auto
from *[rule_format, of x y u]
have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
using ** by auto
}
then show "convex_on C f"
unfolding convex_on_def by auto
qed
lemma convex_on_diff:
fixes f :: "real \<Rightarrow> real"
assumes f: "convex_on I f"
and I: "x \<in> I" "y \<in> I"
and t: "x < t" "t < y"
shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
proof -
define a where "a \<equiv> (t - y) / (x - y)"
with t have "0 \<le> a" "0 \<le> 1 - a"
by (auto simp: field_simps)
with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
by (auto simp: convex_on_def)
have "a * x + (1 - a) * y = a * (x - y) + y"
by (simp add: field_simps)
also have "\<dots> = t"
unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
finally have "f t \<le> a * f x + (1 - a) * f y"
using cvx by simp
also have "\<dots> = a * (f x - f y) + f y"
by (simp add: field_simps)
finally have "f t - f y \<le> a * (f x - f y)"
by simp
with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
by (simp add: le_divide_eq divide_le_eq field_simps a_def)
with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
by (simp add: le_divide_eq divide_le_eq field_simps)
qed
lemma pos_convex_function:
fixes f :: "real \<Rightarrow> real"
assumes "convex C"
and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
shows "convex_on C f"
unfolding convex_on_alt
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"
by auto
qed
lemma atMostAtLeast_subset_convex:
fixes C :: "real set"
assumes "convex C"
and "x \<in> C" "y \<in> C" "x < y"
shows "{x .. y} \<subseteq> C"
proof safe
fix z assume z: "z \<in> {x .. y}"
have less: "z \<in> C" if *: "x < z" "z < y"
proof -
let ?\<mu> = "(y - z) / (y - x)"
have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
using assms * by (auto simp: field_simps)
then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
by (simp add: algebra_simps)
have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
by (auto simp: field_simps)
also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
also have "\<dots> = z"
using assms by (auto simp: field_simps)
finally show ?thesis
using comb by auto
qed
show "z \<in> C"
using z less assms by (auto simp: le_less)
qed
lemma f''_imp_f':
fixes f :: "real \<Rightarrow> real"
assumes "convex C"
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
and x: "x \<in> C"
and y: "y \<in> C"
shows "f' x * (y - x) \<le> f y - f x"
using assms
proof -
have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
proof -
from * have ge: "y - x > 0" "y - x \<ge> 0"
by auto
from * have le: "x - y < 0" "x - y \<le> 0"
by auto
then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
by auto
then have "z1 \<in> C"
using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
by fastforce
from z1 have z1': "f x - f y = (x - y) * f' z1"
by (simp add: field_simps)
obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
by auto
obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
by auto
have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
using * z1' by auto
also have "\<dots> = (y - z1) * f'' z3"
using z3 by auto
finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
by simp
have A': "y - z1 \<ge> 0"
using z1 by auto
have "z3 \<in> C"
using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
by fastforce
then have B': "f'' z3 \<ge> 0"
using assms by auto
from A' B' have "(y - z1) * f'' z3 \<ge> 0"
by auto
from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
by auto
from mult_right_mono_neg[OF this le(2)]
have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
by (simp add: algebra_simps)
then have "f' y * (x - y) - (f x - f y) \<le> 0"
using le by auto
then have res: "f' y * (x - y) \<le> f x - f y"
by auto
have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
using * z1 by auto
also have "\<dots> = (z1 - x) * f'' z2"
using z2 by auto
finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
by simp
have A: "z1 - x \<ge> 0"
using z1 by auto
have "z2 \<in> C"
using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
by fastforce
then have B: "f'' z2 \<ge> 0"
using assms by auto
from A B have "(z1 - x) * f'' z2 \<ge> 0"
by auto
with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
by auto
from mult_right_mono[OF this ge(2)]
have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
by (simp add: algebra_simps)
then have "f y - f x - f' x * (y - x) \<ge> 0"
using ge by auto
then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
using res by auto
qed
show ?thesis
proof (cases "x = y")
case True
with x y show ?thesis by auto
next
case False
with less_imp x y show ?thesis
by (auto simp: neq_iff)
qed
qed
lemma f''_ge0_imp_convex:
fixes f :: "real \<Rightarrow> real"
assumes conv: "convex C"
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
shows "convex_on C f"
using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
by fastforce
lemma minus_log_convex:
fixes b :: real
assumes "b > 1"
shows "convex_on {0 <..} (\<lambda> x. - log b x)"
proof -
have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
using DERIV_log by auto
then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
by (auto simp: DERIV_minus)
have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
have "\<And>z::real. z > 0 \<Longrightarrow>
DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
by auto
then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
unfolding inverse_eq_divide by (auto simp: mult.assoc)
have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
using \<open>b > 1\<close> by (auto intro!: less_imp_le)
from f''_ge0_imp_convex[OF convex_real_interval(3), unfolded greaterThan_iff, OF f' f''0 f''_ge0]
show ?thesis
by auto
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of real functions\<close>
lemma convex_on_realI:
assumes "connected A"
and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
shows "convex_on A f"
proof (rule convex_on_linorderI)
fix t x y :: real
assume t: "t > 0" "t < 1"
assume xy: "x \<in> A" "y \<in> A" "x < y"
define z where "z = (1 - t) * x + t * y"
with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
using connected_contains_Icc by blast
from xy t have xz: "z > x"
by (simp add: z_def algebra_simps)
have "y - z = (1 - t) * (y - x)"
by (simp add: z_def algebra_simps)
also from xy t have "\<dots> > 0"
by (intro mult_pos_pos) simp_all
finally have yz: "z < y"
by simp
from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
by (intro MVT2) (auto intro!: assms(2))
then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
by auto
from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
by (intro MVT2) (auto intro!: assms(2))
then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
by auto
from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
by auto
with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
by (intro assms(3)) auto
also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
using xz yz by (simp add: field_simps)
also have "z - x = t * (y - x)"
by (simp add: z_def algebra_simps)
also have "y - z = (1 - t) * (y - x)"
by (simp add: z_def algebra_simps)
finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
using xy by simp
then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
by (simp add: z_def algebra_simps)
qed
lemma convex_on_inverse:
assumes "A \<subseteq> {0<..}"
shows "convex_on A (inverse :: real \<Rightarrow> real)"
proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
fix u v :: real
assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
with assms show "-inverse (u^2) \<le> -inverse (v^2)"
by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square)
lemma convex_onD_Icc':
assumes "convex_on {x..y} f" "c \<in> {x..y}"
defines "d \<equiv> y - x"
shows "f c \<le> (f y - f x) / d * (c - x) + f x"
proof (cases x y rule: linorder_cases)
case less
then have d: "d > 0"
by (simp add: d_def)
from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
by (simp_all add: d_def 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) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
by (simp add: field_simps)
also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
using assms less by (intro convex_onD_Icc) simp_all
also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
by (simp add: field_simps)
finally show ?thesis .
qed (insert assms(2), simp_all)
lemma convex_onD_Icc'':
assumes "convex_on {x..y} f" "c \<in> {x..y}"
defines "d \<equiv> y - x"
shows "f c \<le> (f x - f y) / d * (y - c) + f y"
proof (cases x y rule: linorder_cases)
case less
then have d: "d > 0"
by (simp add: d_def)
from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
by (simp_all add: d_def 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) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
by (simp add: field_simps)
also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
by (simp add: field_simps)
finally show ?thesis .
qed (insert assms(2), simp_all)
lemma convex_translation_eq [simp]:
"convex ((+) a ` s) \<longleftrightarrow> convex s"
by (metis convex_translation translation_galois)
lemma convex_translation_subtract_eq [simp]:
"convex ((\<lambda>b. b - a) ` s) \<longleftrightarrow> 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 \<Rightarrow> 'b::real_vector"
shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
unfolding linear_iff by (simp add: algebra_simps)
lemma vector_choose_size:
assumes "0 \<le> c"
obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
proof -
obtain a::'a where "a \<noteq> 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 \<le> 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 "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
proof -
have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
by auto
show ?thesis
unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
qed
lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z \<longleftrightarrow>
norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
proof -
have *: "x - y + (y - z) = x - z" by auto
show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
by (auto simp:norm_minus_commute)
qed
subsection \<open>Cones\<close>
definition\<^marker>\<open>tag important\<close> cone :: "'a::real_vector set \<Rightarrow> bool"
where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
lemma cone_empty[intro, simp]: "cone {}"
unfolding cone_def by auto
lemma cone_univ[intro, simp]: "cone UNIV"
unfolding cone_def by auto
lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
unfolding cone_def by auto
lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
by (simp add: cone_def subspace_scale)
subsubsection \<open>Conic hull\<close>
lemma cone_cone_hull: "cone (cone hull S)"
unfolding hull_def by auto
lemma cone_hull_eq: "cone hull S = S \<longleftrightarrow> cone S"
by (metis cone_cone_hull hull_same)
lemma mem_cone:
assumes "cone S" "x \<in> S" "c \<ge> 0"
shows "c *\<^sub>R x \<in> S"
using assms cone_def[of S] by auto
lemma cone_contains_0:
assumes "cone S"
shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
using assms mem_cone by fastforce
lemma cone_0: "cone {0}"
unfolding cone_def by auto
lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
unfolding cone_def by blast
lemma cone_iff:
assumes "S \<noteq> {}"
shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
proof -
{
assume "cone S"
{
fix c :: real
assume "c > 0"
{
fix x
assume "x \<in> S"
then have "x \<in> ((*\<^sub>R) c) ` S"
unfolding image_def
using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
by auto
}
moreover
{
fix x
assume "x \<in> ((*\<^sub>R) c) ` S"
then have "x \<in> S"
using \<open>0 < c\<close> \<open>cone S\<close> mem_cone by fastforce
}
ultimately have "((*\<^sub>R) c) ` S = S" by blast
}
then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
using \<open>cone S\<close> cone_contains_0[of S] assms by auto
}
moreover
{
assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
{
fix x
assume "x \<in> S"
fix c1 :: real
assume "c1 \<ge> 0"
then have "c1 = 0 \<or> c1 > 0" by auto
then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
}
then have "cone S" unfolding cone_def by auto
}
ultimately show ?thesis by blast
qed
lemma cone_hull_empty: "cone hull {} = {}"
by (metis cone_empty cone_hull_eq)
lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
by (metis bot_least cone_hull_empty hull_subset xtrans(5))
lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
by auto
lemma mem_cone_hull:
assumes "x \<in> S" "c \<ge> 0"
shows "c *\<^sub>R x \<in> cone hull S"
by (metis assms cone_cone_hull hull_inc mem_cone)
proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x \<in> ?rhs"
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
by auto
fix c :: real
assume c: "c \<ge> 0"
then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
using x by (simp add: algebra_simps)
moreover
have "c * cx \<ge> 0" using c x by auto
ultimately
have "c *\<^sub>R x \<in> ?rhs" using x by auto
}
then have "cone ?rhs"
unfolding cone_def by auto
then have "?rhs \<in> Collect cone"
unfolding mem_Collect_eq by auto
{
fix x
assume "x \<in> S"
then have "1 *\<^sub>R x \<in> ?rhs"
using zero_le_one by blast
then have "x \<in> ?rhs" by auto
}
then have "S \<subseteq> ?rhs" by auto
then have "?lhs \<subseteq> ?rhs"
using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
moreover
{
fix x
assume "x \<in> ?rhs"
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
by auto
then have "xx \<in> cone hull S"
using hull_subset[of S] by auto
then have "x \<in> ?lhs"
using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
}
ultimately show ?thesis by auto
qed
lemma convex_cone:
"convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
(is "?lhs = ?rhs")
proof -
{
fix x y
assume "x\<in>s" "y\<in>s" and ?lhs
then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
unfolding cone_def by auto
then have "x + y \<in> s"
using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
apply (erule_tac x="2*\<^sub>R x" in ballE)
apply (erule_tac x="2*\<^sub>R y" in ballE)
apply (erule_tac x="1/2" in allE, simp)
apply (erule_tac x="1/2" in allE, auto)
done
}
then show ?thesis
unfolding convex_def cone_def by blast
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close>
lemma convex_connected:
fixes S :: "'a::real_normed_vector set"
assumes "convex S"
shows "connected S"
proof (rule connectedI)
fix A B
assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
moreover
assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
then have "continuous_on {0 .. 1} f"
by (auto intro!: continuous_intros)
then have "connected (f ` {0 .. 1})"
by (auto intro!: connected_continuous_image)
note connectedD[OF this, of A B]
moreover have "a \<in> A \<inter> f ` {0 .. 1}"
using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
moreover have "b \<in> B \<inter> f ` {0 .. 1}"
using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
moreover have "f ` {0 .. 1} \<subseteq> S"
using \<open>convex S\<close> a b unfolding convex_def f_def by auto
ultimately show False by auto
qed
corollary%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by (simp add: convex_connected)
lemma convex_prod:
assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
using assms unfolding convex_def
by (auto simp: inner_add_left)
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
by (rule convex_prod) (simp flip: atLeast_def)
subsection \<open>Convex hull\<close>
lemma convex_convex_hull [iff]: "convex (convex hull s)"
unfolding hull_def
using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
by auto
lemma convex_hull_subset:
"s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
by (simp add: subset_hull)
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
by (metis convex_convex_hull hull_same)
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Convex hull is "preserved" by a linear function\<close>
lemma convex_hull_linear_image:
assumes f: "linear f"
shows "f ` (convex hull s) = convex hull (f ` s)"
proof
show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
show "s \<subseteq> f -` (convex hull (f ` s))"
by (fast intro: hull_inc)
show "convex (f -` (convex hull (f ` s)))"
by (intro convex_linear_vimage [OF f] convex_convex_hull)
qed
qed
lemma in_convex_hull_linear_image:
assumes "linear f"
and "x \<in> convex hull s"
shows "f x \<in> convex hull (f ` s)"
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
lemma convex_hull_Times:
"convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
proof
show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
proof (rule hull_induct [OF x], rule hull_induct [OF y])
fix x y assume "x \<in> s" and "y \<in> t"
then show "(x, y) \<in> convex hull (s \<times> t)"
by (simp add: hull_inc)
next
fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
by (auto simp: image_def Bex_def)
finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
next
show "convex {x. (x, y) \<in> convex hull s \<times> t}"
proof -
fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
have "convex ?S"
by (intro convex_linear_vimage convex_translation convex_convex_hull,
simp add: linear_iff)
also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
by (auto simp: image_def Bex_def)
finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
qed
qed
then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
unfolding subset_eq split_paired_Ball_Sigma by blast
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems for convex hulls of finite sets\<close>
lemma convex_hull_empty[simp]: "convex hull {} = {}"
by (rule hull_unique) auto
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
by (rule hull_unique) auto
lemma convex_hull_insert:
fixes S :: "'a::real_vector set"
assumes "S \<noteq> {}"
shows "convex hull (insert a S) =
{x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
(is "_ = ?hull")
proof (intro equalityI hull_minimal subsetI)
fix x
assume "x \<in> insert a S"
then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
unfolding insert_iff
proof
assume "x = a"
then show ?thesis
by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
next
assume "x \<in> S"
with hull_subset[of S convex] show ?thesis
by force
qed
then show "x \<in> ?hull"
by simp
next
fix x
assume "x \<in> ?hull"
then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
by auto
have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
by auto
then show "x \<in> convex hull insert a S"
unfolding obt(5) using obt(1-3)
by (rule convexD [OF convex_convex_hull])
next
show "convex ?hull"
proof (rule convexI)
fix x y u v
assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
from x obtain u1 v1 b1 where
obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
by auto
from y obtain u2 v2 b2 where
obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
by auto
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
by (auto simp: algebra_simps)
have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
(u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
proof (cases "u * v1 + v * v2 = 0")
case True
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
by (auto simp: algebra_simps)
have eq0: "u * v1 = 0" "v * v2 = 0"
using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
by arith+
then have "u * u1 + v * u2 = 1"
using as(3) obt1(3) obt2(3) by auto
then show ?thesis
using "*" eq0 as obt1(4) xeq yeq by auto
next
case False
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
also have "\<dots> = u * v1 + v * v2"
by simp
finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
using as(1,2) obt1(1,2) obt2(1,2) by auto
show ?thesis
proof
show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
unfolding xeq yeq * **
using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
show "?b \<in> convex hull S"
using False zeroes obt1(4) obt2(4)
by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib add_divide_distrib[symmetric] zero_le_divide_iff)
qed
qed
then obtain b where b: "b \<in> convex hull S"
"u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
have u1: "u1 \<le> 1"
unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
have u2: "u2 \<le> 1"
unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
proof (rule add_mono)
show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
by (simp_all add: as mult_right_mono)
qed
also have "\<dots> \<le> 1"
unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
finally have le1: "u1 * u + u2 * v \<le> 1" .
show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
proof (intro CollectI exI conjI)
show "0 \<le> u * u1 + v * u2"
by (simp add: as(1) as(2) obt1(1) obt2(1))
show "0 \<le> 1 - u * u1 - v * u2"
by (simp add: le1 diff_diff_add mult.commute)
qed (use b in \<open>auto simp: algebra_simps\<close>)
qed
qed
lemma convex_hull_insert_alt:
"convex hull (insert a S) =
(if S = {} then {a}
else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
apply (auto simp: convex_hull_insert)
using diff_eq_eq apply fastforce
using diff_add_cancel diff_ge_0_iff_ge by blast
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expression for convex hull\<close>
proposition convex_hull_indexed:
fixes S :: "'a::real_vector set"
shows "convex hull S =
{y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
(sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
(is "?xyz = ?hull")
proof (rule hull_unique [OF _ convexI])
show "S \<subseteq> ?hull"
by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
next
fix T
assume "S \<subseteq> T" "convex T"
then show "?hull \<subseteq> T"
by (blast intro: convex_sum)
next
fix x y u v
assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
assume xy: "x \<in> ?hull" "y \<in> ?hull"
from xy obtain k1 u1 x1 where
x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S"
"sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
by auto
from xy obtain k2 u2 x2 where
y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S"
"sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
by auto
have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
"{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
by auto
have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
unfolding inj_on_def by auto
let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
proof (intro CollectI exI conjI ballI)
show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1" "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
sum.reindex[OF inj] Collect_mem_eq o_def
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
by (simp_all add: sum_distrib_left[symmetric] x(2,3) y(2,3) uv(3))
qed
qed
lemma convex_hull_finite:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
(is "?HULL = _")
proof (rule hull_unique [OF _ convexI]; clarify)
fix x
assume "x \<in> S"
then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x"
by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
next
fix u v :: real
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
by (simp add: that uv ux(1) uy(1))
moreover
have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
using uv(3) by auto
moreover
have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
by auto
ultimately
show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
(\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
qed (use assms in \<open>auto simp: convex_explicit\<close>)
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Another formulation\<close>
text "Formalized by Lars Schewe."
lemma convex_hull_explicit:
fixes p :: "'a::real_vector set"
shows "convex hull p =
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "?lhs = ?rhs")
proof -
{
fix x
assume "x\<in>?lhs"
then obtain k u y where
obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
unfolding convex_hull_indexed by auto
have fin: "finite {1..k}" by auto
have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
{
fix j
assume "j\<in>{1..k}"
then have "y j \<in> p \<and> 0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
using obt(1)[THEN bspec[where x=j]] and obt(2)
by (metis (no_types, lifting) One_nat_def atLeastAtMost_iff mem_Collect_eq obt(1) sum_nonneg)
}
moreover
have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
unfolding scaleR_left.sum using obt(3) by auto
ultimately
have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
apply (rule_tac x="y ` {1..k}" in exI)
apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
done
then have "x\<in>?rhs" by auto
}
moreover
{
fix y
assume "y\<in>?rhs"
then obtain S u where
obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
by auto
obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
{
fix i :: nat
assume "i\<in>{1..card S}"
then have "f i \<in> S"
using f(2) by blast
then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
}
moreover have *: "finite {1..card S}" by auto
{
fix y
assume "y\<in>S"
then obtain i where "i\<in>{1..card S}" "f i = y"
using f using image_iff[of y f "{1..card S}"]
by auto
then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
using f(1) inj_onD by fastforce
then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
"(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
by (auto simp: sum_constant_scaleR)
}
then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
unfolding f
using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
unfolding obt(4,5)
by auto
ultimately
have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
(\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
apply (rule_tac x="card S" in exI)
apply (rule_tac x="u \<circ> f" in exI)
apply (rule_tac x=f in exI, fastforce)
done
then have "y \<in> ?lhs"
unfolding convex_hull_indexed by auto
}
ultimately show ?thesis
unfolding set_eq_iff by blast
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>A stepping theorem for that expansion\<close>
lemma convex_hull_finite_step:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows
"(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
\<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
(is "?lhs = ?rhs")
proof (cases "a \<in> S")
case True
then have *: "insert a S = S" by auto
show ?thesis
proof
assume ?lhs
then show ?rhs
unfolding * by force
next
have fin: "finite (insert a S)" using assms by auto
assume ?rhs
then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
by auto
then show ?lhs
using uv True assms
apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
apply (auto simp: sum_clauses scaleR_left_distrib sum.distrib sum_delta''[OF fin])
done
qed
next
case False
show ?thesis
proof
assume ?lhs
then obtain u where u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
by auto
then show ?rhs
using u \<open>a\<notin>S\<close> by (rule_tac x="u a" in exI) (auto simp: sum_clauses assms)
next
assume ?rhs
then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
by auto
moreover
have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S" "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
using False by (auto intro!: sum.cong)
ultimately show ?lhs
using False by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
qed
qed
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence some special cases\<close>
lemma convex_hull_2: "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
(is "?lhs = ?rhs")
proof -
have **: "finite {b}" by auto
have "\<And>x v u. \<lbrakk>0 \<le> v; v \<le> 1; (1 - v) *\<^sub>R b = x - v *\<^sub>R a\<rbrakk>
\<Longrightarrow> \<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1"
by (metis add.commute diff_add_cancel diff_ge_0_iff_ge)
moreover
have "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk>
\<Longrightarrow> \<exists>p\<ge>0. \<exists>q. 0 \<le> q b \<and> q b = 1 - p \<and> q b *\<^sub>R b = u *\<^sub>R a + v *\<^sub>R b - p *\<^sub>R a"
apply (rule_tac x=u in exI, simp)
apply (rule_tac x="\<lambda>x. v" in exI, simp)
done
ultimately show ?thesis
using convex_hull_finite_step[OF **, of a 1]
by (auto simp add: convex_hull_finite)
qed
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}"
unfolding convex_hull_2
proof (rule Collect_cong)
have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
by auto
fix x
show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
(\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
apply (simp add: *)
by (rule ex_cong1) (auto simp: algebra_simps)
qed
lemma convex_hull_3:
"convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
proof -
have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
by auto
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
by (auto simp: field_simps)
show ?thesis
unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
unfolding convex_hull_finite_step[OF fin(3)]
apply (rule Collect_cong, simp)
apply auto
apply (rule_tac x=va in exI)
apply (rule_tac x="u c" in exI, simp)
apply (rule_tac x="1 - v - w" in exI, simp)
apply (rule_tac x=v in exI, simp)
apply (rule_tac x="\<lambda>x. w" in exI, simp)
done
qed
lemma convex_hull_3_alt:
"convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
proof -
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
by auto
show ?thesis
unfolding convex_hull_3
apply (auto simp: *)
apply (rule_tac x=v in exI)
apply (rule_tac x=w in exI)
apply (simp add: algebra_simps)
apply (rule_tac x=u in exI)
apply (rule_tac x=v in exI)
apply (simp add: algebra_simps)
done
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>Relations among closure notions and corresponding hulls\<close>
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
unfolding affine_def convex_def by auto
lemma convex_affine_hull [simp]: "convex (affine hull S)"
by (simp add: affine_imp_convex)
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
using subspace_imp_affine affine_imp_convex by auto
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
lemma aff_dim_convex_hull:
fixes S :: "'n::euclidean_space set"
shows "aff_dim (convex hull S) = aff_dim S"
using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
aff_dim_subset[of "convex hull S" "affine hull S"]
by auto
subsection \<open>Caratheodory's theorem\<close>
lemma convex_hull_caratheodory_aff_dim:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p =
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
proof (intro allI iffI)
fix y
let ?P = "\<lambda>n. \<exists>S u. finite S \<and> card S = n \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and>
sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
assume "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
then obtain N where "?P N" by auto
then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
by (rule_tac ex_least_nat_le, auto)
then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
by blast
then obtain S u where obt: "finite S" "card S = n" "S\<subseteq>p" "\<forall>x\<in>S. 0 \<le> u x"
"sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y" by auto
have "card S \<le> aff_dim p + 1"
proof (rule ccontr, simp only: not_le)
assume "aff_dim p + 1 < card S"
then have "affine_dependent S"
using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
by blast
then obtain w v where wv: "sum w S = 0" "v\<in>S" "w v \<noteq> 0" "(\<Sum>v\<in>S. w v *\<^sub>R v) = 0"
using affine_dependent_explicit_finite[OF obt(1)] by auto
define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>S. w v < 0}"
define t where "t = Min i"
have "\<exists>x\<in>S. w x < 0"
proof (rule ccontr, simp add: not_less)
assume as:"\<forall>x\<in>S. 0 \<le> w x"
then have "sum w (S - {v}) \<ge> 0"
by (meson Diff_iff sum_nonneg)
then have "sum w S > 0"
using as obt(1) sum_nonneg_eq_0_iff wv by blast
then show False using wv(1) by auto
qed
then have "i \<noteq> {}" unfolding i_def by auto
then have "t \<ge> 0"
using Min_ge_iff[of i 0] and obt(1)
unfolding t_def i_def
using obt(4)[unfolded le_less]
by (auto simp: divide_le_0_iff)
have t: "\<forall>v\<in>S. u v + t * w v \<ge> 0"
proof
fix v
assume "v \<in> S"
then have v: "0 \<le> u v"
using obt(4)[THEN bspec[where x=v]] by auto
show "0 \<le> u v + t * w v"
proof (cases "w v < 0")
case False
thus ?thesis using v \<open>t\<ge>0\<close> by auto
next
case True
then have "t \<le> u v / (- w v)"
using \<open>v\<in>S\<close> obt unfolding t_def i_def by (auto intro: Min_le)
then show ?thesis
unfolding real_0_le_add_iff
using True neg_le_minus_divide_eq by auto
qed
qed
obtain a where "a \<in> S" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
then have a: "a \<in> S" "u a + t * w a = 0" by auto
have *: "\<And>f. sum f (S - {a}) = sum f S - ((f a)::'b::ab_group_add)"
unfolding sum.remove[OF obt(1) \<open>a\<in>S\<close>] by auto
have "(\<Sum>v\<in>S. u v + t * w v) = 1"
unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
moreover have "(\<Sum>v\<in>S. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
ultimately have "?P (n - 1)"
apply (rule_tac x="(S - {a})" in exI)
apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
using obt(1-3) and t and a
apply (auto simp: * scaleR_left_distrib)
done
then show False
using smallest[THEN spec[where x="n - 1"]] by auto
qed
then show "\<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
using obt by auto
qed auto
lemma caratheodory_aff_dim:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
(is "?lhs = ?rhs")
proof
have "\<And>x S u. \<lbrakk>finite S; S \<subseteq> p; int (card S) \<le> aff_dim p + 1; \<forall>x\<in>S. 0 \<le> u x; sum u S = 1\<rbrakk>
\<Longrightarrow> (\<Sum>v\<in>S. u v *\<^sub>R v) \<in> convex hull S"
by (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
then show "?lhs \<subseteq> ?rhs"
by (subst convex_hull_caratheodory_aff_dim, auto)
qed (use hull_mono in auto)
lemma convex_hull_caratheodory:
fixes p :: "('a::euclidean_space) set"
shows "convex hull p =
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and>
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "?lhs = ?rhs")
proof (intro set_eqI iffI)
fix x
assume "x \<in> ?lhs" then show "x \<in> ?rhs"
unfolding convex_hull_caratheodory_aff_dim
using aff_dim_le_DIM [of p] by fastforce
qed (auto simp: convex_hull_explicit)
theorem caratheodory:
"convex hull p =
{x::'a::euclidean_space. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}"
proof safe
fix x
assume "x \<in> convex hull p"
then obtain S u where "finite S" "S \<subseteq> p" "card S \<le> DIM('a) + 1"
"\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = x"
unfolding convex_hull_caratheodory by auto
then show "\<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S"
using convex_hull_finite by fastforce
qed (use hull_mono in force)
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some Properties of subset of standard basis\<close>
lemma affine_hull_substd_basis:
assumes "d \<subseteq> Basis"
shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
(is "affine hull (insert 0 ?A) = ?B")
proof -
have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
by auto
show ?thesis
unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
qed
lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Moving and scaling convex hulls\<close>
lemma convex_hull_set_plus:
"convex hull (S + T) = convex hull S + convex hull T"
unfolding set_plus_image
apply (subst convex_hull_linear_image [symmetric])
apply (simp add: linear_iff scaleR_right_distrib)
apply (simp add: convex_hull_Times)
done
lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
unfolding set_plus_def by auto
lemma convex_hull_translation:
"convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
unfolding translation_eq_singleton_plus
by (simp only: convex_hull_set_plus convex_hull_singleton)
lemma convex_hull_scaling:
"convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
using linear_scaleR by (rule convex_hull_linear_image [symmetric])
lemma convex_hull_affinity:
"convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
by (metis convex_hull_scaling convex_hull_translation image_image)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of cone hulls\<close>
lemma convex_cone_hull:
assumes "convex S"
shows "convex (cone hull S)"
proof (rule convexI)
fix x y
assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
then have "S \<noteq> {}"
using cone_hull_empty_iff[of S] by auto
fix u v :: real
assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
using cone_hull_expl[of S] by auto
from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
using cone_hull_expl[of S] by auto
{
assume "cx + cy \<le> 0"
then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
using x y by auto
then have "u *\<^sub>R x + v *\<^sub>R y = 0"
by auto
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
}
moreover
{
assume "cx + cy > 0"
then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
using assms mem_convex_alt[of S xx yy cx cy] x y by auto
then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
by (auto simp: scaleR_right_distrib)
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
using x y by auto
}
moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
qed
lemma cone_convex_hull:
assumes "cone S"
shows "cone (convex hull S)"
proof (cases "S = {}")
case True
then show ?thesis by auto
next
case False
then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
using cone_iff[of S] assms by auto
{
fix c :: real
assume "c > 0"
then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
using convex_hull_scaling[of _ S] by auto
also have "\<dots> = convex hull S"
using * \<open>c > 0\<close> by auto
finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
by auto
}
then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
using * hull_subset[of S convex] by auto
then show ?thesis
using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
qed
subsection \<open>Radon's theorem\<close>
text "Formalized by Lars Schewe."
lemma Radon_ex_lemma:
assumes "finite c" "affine_dependent c"
shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
proof -
from assms(2)[unfolded affine_dependent_explicit]
obtain S u where
"finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
by blast
then show ?thesis
apply (rule_tac x="\<lambda>v. if v\<in>S then u v else 0" in exI)
unfolding if_smult scaleR_zero_left
by (auto simp: Int_absorb1 sum.inter_restrict[OF \<open>finite c\<close>, symmetric])
qed
lemma Radon_s_lemma:
assumes "finite S"
and "sum f S = (0::real)"
shows "sum f {x\<in>S. 0 < f x} = - sum f {x\<in>S. f x < 0}"
proof -
have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
by auto
show ?thesis
unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
and sum.distrib[symmetric] and *
using assms(2)
by assumption
qed
lemma Radon_v_lemma:
assumes "finite S"
and "sum f S = 0"
and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
shows "(sum f {x\<in>S. 0 < g x}) = - sum f {x\<in>S. g x < 0}"
proof -
have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
using assms(3) by auto
show ?thesis
unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
and sum.distrib[symmetric] and *
using assms(2)
apply assumption
done
qed
lemma Radon_partition:
assumes "finite C" "affine_dependent C"
shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = C \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
proof -
obtain u v where uv: "sum u C = 0" "v\<in>C" "u v \<noteq> 0" "(\<Sum>v\<in>C. u v *\<^sub>R v) = 0"
using Radon_ex_lemma[OF assms] by auto
have fin: "finite {x \<in> C. 0 < u x}" "finite {x \<in> C. 0 > u x}"
using assms(1) by auto
define z where "z = inverse (sum u {x\<in>C. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>C. u x > 0}"
have "sum u {x \<in> C. 0 < u x} \<noteq> 0"
proof (cases "u v \<ge> 0")
case False
then have "u v < 0" by auto
then show ?thesis
proof (cases "\<exists>w\<in>{x \<in> C. 0 < u x}. u w > 0")
case True
then show ?thesis
using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
next
case False
then have "sum u C \<le> sum (\<lambda>x. if x=v then u v else 0) C"
by (rule_tac sum_mono, auto)
then show ?thesis
unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
qed
qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
then have *: "sum u {x\<in>C. u x > 0} > 0"
unfolding less_le by (metis (no_types, lifting) mem_Collect_eq sum_nonneg)
moreover have "sum u ({x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}) = sum u C"
"(\<Sum>x\<in>{x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>C. u x *\<^sub>R x)"
using assms(1)
by (rule_tac[!] sum.mono_neutral_left, auto)
then have "sum u {x \<in> C. 0 < u x} = - sum u {x \<in> C. 0 > u x}"
"(\<Sum>x\<in>{x \<in> C. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> C. 0 > u x}. u x *\<^sub>R x)"
unfolding eq_neg_iff_add_eq_0
using uv(1,4)
by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
moreover have "\<forall>x\<in>{v \<in> C. u v < 0}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * - u x"
using * by (fastforce intro: mult_nonneg_nonneg)
ultimately have "z \<in> convex hull {v \<in> C. u v \<le> 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v \<in> C. u v < 0}" in exI)
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
moreover have "\<forall>x\<in>{v \<in> C. 0 < u v}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * u x"
using * by (fastforce intro: mult_nonneg_nonneg)
then have "z \<in> convex hull {v \<in> C. u v > 0}"
unfolding convex_hull_explicit mem_Collect_eq
apply (rule_tac x="{v \<in> C. 0 < u v}" in exI)
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * u y" in exI)
using assms(1)
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
using * by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
ultimately show ?thesis
apply (rule_tac x="{v\<in>C. u v \<le> 0}" in exI)
apply (rule_tac x="{v\<in>C. u v > 0}" in exI, auto)
done
qed
theorem Radon:
assumes "affine_dependent c"
obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
proof -
from assms[unfolded affine_dependent_explicit]
obtain S u where
"finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
by blast
then have *: "finite S" "affine_dependent S" and S: "S \<subseteq> c"
unfolding affine_dependent_explicit by auto
from Radon_partition[OF *]
obtain m p where "m \<inter> p = {}" "m \<union> p = S" "convex hull m \<inter> convex hull p \<noteq> {}"
by blast
with S show ?thesis
by (force intro: that[of p m])
qed
subsection \<open>Helly's theorem\<close>
lemma Helly_induct:
fixes f :: "'a::euclidean_space set set"
assumes "card f = n"
and "n \<ge> DIM('a) + 1"
and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
shows "\<Inter>f \<noteq> {}"
using assms
proof (induction n arbitrary: f)
case 0
then show ?case by auto
next
case (Suc n)
have "finite f"
using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
show "\<Inter>f \<noteq> {}"
proof (cases "n = DIM('a)")
case True
then show ?thesis
by (simp add: Suc.prems(1) Suc.prems(4))
next
case False
have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
proof (rule Suc.IH[rule_format])
show "card (f - {s}) = n"
by (simp add: Suc.prems(1) \<open>finite f\<close> that)
show "DIM('a) + 1 \<le> n"
using False Suc.prems(2) by linarith
show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
by (simp add: Suc.prems(4) subset_Diff_insert)
qed (use Suc in auto)
then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
by blast
then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
by metis
show ?thesis
proof (cases "inj_on X f")
case False
then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
unfolding inj_on_def by auto
then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
show ?thesis
by (metis "*" X disjoint_iff_not_equal st)
next
case True
then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
unfolding card_image[OF True] and \<open>card f = Suc n\<close>
using Suc(3) \<open>finite f\<close> and False
by auto
have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
using mp(2) by auto
then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
unfolding subset_image_iff by auto
then have "f \<union> (g \<union> h) = f" by auto
then have f: "f = g \<union> h"
using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
unfolding mp(2)[unfolded image_Un[symmetric] gh]
by auto
have *: "g \<inter> h = {}"
using gh(1) gh(2) local.mp(1) by blast
have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
then show ?thesis
unfolding f using mp(3)[unfolded gh] by blast
qed
qed
qed
theorem Helly:
fixes f :: "'a::euclidean_space set set"
assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
shows "\<Inter>f \<noteq> {}"
using Helly_induct assms by blast
subsection \<open>Epigraphs of convex functions\<close>
definition\<^marker>\<open>tag important\<close> "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
unfolding epigraph_def by auto
lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
proof safe
assume L: "convex (epigraph S f)"
then show "convex_on S f"
by (auto simp: convex_def convex_on_def epigraph_def)
show "convex S"
using L by (fastforce simp: convex_def convex_on_def epigraph_def)
next
assume "convex_on S f" "convex S"
then show "convex (epigraph S f)"
unfolding convex_def convex_on_def epigraph_def
apply safe
apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
apply (auto intro!:mult_left_mono add_mono)
done
qed
lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
unfolding convex_epigraph by auto
lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
by (simp add: convex_epigraph)
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Use this to derive general bound property of convex function\<close>
lemma convex_on:
assumes "convex S"
shows "convex_on S f \<longleftrightarrow>
(\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
(is "?lhs = (\<forall>k u x. ?rhs k u x)")
proof
assume ?lhs
then have \<section>: "convex {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
by (metis assms convex_epigraph epigraph_def)
show "\<forall>k u x. ?rhs k u x"
proof (intro allI)
fix k u x
show "?rhs k u x"
using \<section>
unfolding convex mem_Collect_eq fst_sum snd_sum
apply safe
apply (drule_tac x=k in spec)
apply (drule_tac x=u in spec)
apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
apply simp
done
qed
next
assume "\<forall>k u x. ?rhs k u x"
then show ?lhs
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq fst_sum snd_sum
using assms[unfolded convex] apply clarsimp
apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
by (auto simp add: mult_left_mono intro: sum_mono)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within a convex hull\<close>
lemma convex_on_convex_hull_bound:
assumes "convex_on (convex hull S) f"
and "\<forall>x\<in>S. f x \<le> b"
shows "\<forall>x\<in> convex hull S. f x \<le> b"
proof
fix x
assume "x \<in> convex hull S"
then obtain k u v where
u: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> S" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
unfolding sum_distrib_right[symmetric] u(2) mult_1
using assms(2) mult_left_mono u(1) by blast
then show "f x \<le> b"
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
using hull_inc u by fastforce
qed
lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
by (simp add: inner_sum_left sum.If_cases inner_Basis)
lemma convex_set_plus:
assumes "convex S" and "convex T" shows "convex (S + T)"
proof -
have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
using assms by (rule convex_sums)
moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
unfolding set_plus_def by auto
finally show "convex (S + T)" .
qed
lemma convex_set_sum:
assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
shows "convex (\<Sum>i\<in>A. B i)"
proof (cases "finite A")
case True then show ?thesis using assms
by induct (auto simp: convex_set_plus)
qed auto
lemma finite_set_sum:
assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
using assms by (induct set: finite, simp, simp add: finite_set_plus)
lemma box_eq_set_sum_Basis:
"{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (B i))" (is "?lhs = ?rhs")
proof -
have "\<And>x. \<forall>i\<in>Basis. x \<bullet> i \<in> B i \<Longrightarrow>
\<exists>s. x = sum s Basis \<and> (\<forall>i\<in>Basis. s i \<in> (\<lambda>x. x *\<^sub>R i) ` B i)"
by (metis (mono_tags, lifting) euclidean_representation image_iff)
moreover
have "sum f Basis \<bullet> i \<in> B i" if "i \<in> Basis" and f: "\<forall>i\<in>Basis. f i \<in> (\<lambda>x. x *\<^sub>R i) ` B i" for i f
proof -
have "(\<Sum>x\<in>Basis - {i}. f x \<bullet> i) = 0"
proof (rule sum.neutral, intro strip)
show "f x \<bullet> i = 0" if "x \<in> Basis - {i}" for x
using that f \<open>i \<in> Basis\<close> inner_Basis that by fastforce
qed
then have "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i"
by (metis (no_types) \<open>i \<in> Basis\<close> add.right_neutral sum.remove [OF finite_Basis])
then have "(\<Sum>x\<in>Basis. f x \<bullet> i) \<in> B i"
using f that(1) by auto
then show ?thesis
by (simp add: inner_sum_left)
qed
ultimately show ?thesis
by (subst set_sum_alt [OF finite_Basis]) auto
qed
lemma convex_hull_set_sum:
"convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
proof (cases "finite A")
assume "finite A" then show ?thesis
by (induct set: finite, simp, simp add: convex_hull_set_plus)
qed simp
end