--- a/src/HOL/Analysis/Convex.thy Tue Oct 06 16:55:56 2020 +0200
+++ b/src/HOL/Analysis/Convex.thy Tue Oct 06 20:34:38 2020 +0100
@@ -59,10 +59,8 @@
lemma mem_convex_alt:
assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
- apply (rule convexD)
using assms
- apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
- done
+ by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric])
lemma convex_empty[intro,simp]: "convex {}"
unfolding convex_def by simp
@@ -193,72 +191,72 @@
lemma convex_sum:
fixes C :: "'a::real_vector set"
- assumes "finite s"
+ 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"
+ 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"
+ 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 "\<forall>j\<in>S. 0 \<le> a j"
and "y i \<in> C"
- and "\<forall>j\<in>s. y j \<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"
+ 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")
+ 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"
+ 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"
+ 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>
+ 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"
+ 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>
+ 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"
+ 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>
+ 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)"
+ "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"
+ 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"
+ 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"
+ 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 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"
@@ -269,43 +267,43 @@
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"
+ 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 have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> S"
+ using S by (auto simp: add.commute)
}
- then show "convex s"
+ 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)"
+ 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"
+ 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
+ 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"
+ 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"
+ 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
@@ -321,30 +319,30 @@
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)"
- unfolding convex_explicit
- apply safe
- subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
- subgoal for t u
- 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)"
+ 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
- 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"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
- qed
- done
+ 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)"
+ 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>
@@ -388,17 +386,17 @@
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"
+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)"
+ 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"
+ assume "x \<in> S" "y \<in> S"
moreover
fix u v :: real
assume "0 \<le> u" "0 \<le> v" "u + v = 1"
@@ -415,8 +413,8 @@
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)"
+ 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
@@ -426,9 +424,9 @@
qed
lemma convex_lower:
- assumes "convex_on s f"
- and "x \<in> s"
- and "y \<in> s"
+ 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"
@@ -444,11 +442,11 @@
qed
lemma convex_on_dist [intro]:
- fixes s :: "'a::real_normed_vector set"
- shows "convex_on s (\<lambda>x. dist a x)"
+ 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"
+ assume "x \<in> S" "y \<in> S"
fix u v :: real
assume "0 \<le> u"
assume "0 \<le> v"
@@ -467,32 +465,32 @@
lemma convex_linear_image:
assumes "linear f"
- and "convex s"
- shows "convex (f ` s)"
+ and "convex S"
+ shows "convex (f ` S)"
proof -
interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f ` s)"
+ 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)"
+ and "convex S"
+ shows "convex (f -` S)"
proof -
interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f -` s)"
+ 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)"
+ 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)
+ using \<open>convex S\<close> by (rule convex_linear_image)
qed
lemma convex_scaled:
@@ -656,7 +654,6 @@
lemma convex_on_alt:
fixes C :: "'a::real_vector set"
- assumes "convex C"
shows "convex_on C f \<longleftrightarrow>
(\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
@@ -720,7 +717,7 @@
assumes "convex C"
and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
shows "convex_on C f"
- unfolding convex_on_alt[OF assms(1)]
+ unfolding convex_on_alt
using assms
proof safe
fix x y \<mu> :: real
@@ -737,7 +734,7 @@
then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
by (auto simp: field_simps)
then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- using convex_on_alt by auto
+ by auto
qed
lemma atMostAtLeast_subset_convex:
@@ -1081,15 +1078,11 @@
subsubsection \<open>Conic hull\<close>
-lemma cone_cone_hull: "cone (cone hull s)"
+lemma cone_cone_hull: "cone (cone hull S)"
unfolding hull_def by auto
-lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
- apply (rule hull_eq)
- using cone_Inter
- unfolding subset_eq
- apply auto
- done
+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"
@@ -1099,15 +1092,7 @@
lemma cone_contains_0:
assumes "cone S"
shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
-proof -
- {
- assume "S \<noteq> {}"
- then obtain a where "a \<in> S" by auto
- then have "0 \<in> S"
- using assms mem_cone[of S a 0] by auto
- }
- then show ?thesis by auto
-qed
+ using assms mem_cone by fastforce
lemma cone_0: "cone {0}"
unfolding cone_def by auto
@@ -1138,8 +1123,7 @@
fix x
assume "x \<in> ((*\<^sub>R) c) ` S"
then have "x \<in> S"
- using \<open>cone S\<close> \<open>c > 0\<close>
- unfolding cone_def image_def \<open>c > 0\<close> by auto
+ using \<open>0 < c\<close> \<open>cone S\<close> mem_cone by fastforce
}
ultimately have "((*\<^sub>R) c) ` S = S" by blast
}
@@ -1202,9 +1186,7 @@
fix x
assume "x \<in> S"
then have "1 *\<^sub>R x \<in> ?rhs"
- apply auto
- apply (rule_tac x = 1 in exI, auto)
- done
+ using zero_le_one by blast
then have "x \<in> ?rhs" by auto
}
then have "S \<subseteq> ?rhs" by auto
@@ -1477,7 +1459,7 @@
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
- by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
+ using diff_add_cancel diff_ge_0_iff_ge by blast
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expression for convex hull\<close>
@@ -1580,13 +1562,9 @@
{
fix j
assume "j\<in>{1..k}"
- then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
+ 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)
- apply simp
- apply (rule sum_nonneg)
- using obt(1)
- apply auto
- done
+ 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"
@@ -1611,7 +1589,6 @@
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}"
@@ -1627,9 +1604,7 @@
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}"
- apply auto
- using f(1)[unfolded inj_on_def]
- by (metis One_nat_def atLeastAtMost_iff)
+ 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"
@@ -1667,71 +1642,65 @@
"(\<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 (rule, case_tac[!] "a\<in>S")
- assume "a \<in> S"
- then have *: "insert a S = S" by auto
- assume ?lhs
- then show ?rhs
- unfolding * by (rule_tac x=0 in exI, auto)
-next
- 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
- assume "a \<notin> S"
- then show ?rhs
- apply (rule_tac x="u a" in exI)
- using u(1)[THEN bspec[where x=a]]
- apply simp
- apply (rule_tac x=u in exI)
- using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
- apply auto
- done
-next
- assume "a \<in> S"
+proof (cases "a \<in> S")
+ case True
then have *: "insert a S = S" by auto
- 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
- show ?lhs
- apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
- unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
- unfolding sum_clauses(2)[OF assms]
- using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
- apply auto
- done
+ 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
- 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 assume "a \<notin> S"
- 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 \<open>a \<notin> S\<close>
- by (auto simp: intro!: sum.cong)
- ultimately show ?lhs
- by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
+ 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}"
+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 *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
- by auto
have **: "finite {b}" by auto
- show ?thesis
- apply (simp add: convex_hull_finite)
- unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
- apply auto
- apply (rule_tac x=v in exI)
- apply (rule_tac x="1 - v" in exI, simp)
+ 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}"
@@ -1742,11 +1711,8 @@
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)"
- unfolding *
- apply auto
- apply (rule_tac[!] x=u in exI)
- apply (auto simp: algebra_simps)
- done
+ apply (simp add: *)
+ by (rule ex_cong1) (auto simp: algebra_simps)
qed
lemma convex_hull_3:
@@ -1818,54 +1784,51 @@
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}"
+ {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"
+ 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"
- apply (rule_tac ex_least_nat_le, auto)
- done
+ 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
+ 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"
+ 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"
+ 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"
+ 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 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"
+ 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"
- apply (rule_tac sum_nonneg, auto)
- done
- then have "sum w s > 0"
- unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
- using as[THEN bspec[where x=v]] \<open>v\<in>s\<close> \<open>w v \<noteq> 0\<close> by auto
+ 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)
+ 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"
+ have t: "\<forall>v\<in>S. u v + t * w v \<ge> 0"
proof
fix v
- assume "v \<in> s"
+ 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"
@@ -1875,28 +1838,24 @@
next
case True
then have "t \<le> u v / (- w v)"
- using \<open>v\<in>s\<close> unfolding t_def i_def
- apply (rule_tac Min_le)
- using obt(1) apply auto
- done
+ 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 pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
- by auto
+ 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"
+ 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"
+ 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"
+ 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="(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)
@@ -1904,69 +1863,48 @@
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"
+ 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}"
+ 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
- show "?lhs \<subseteq> ?rhs"
- apply (subst convex_hull_caratheodory_aff_dim, clarify)
- apply (rule_tac x=s in exI)
- apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
- done
-next
- show "?rhs \<subseteq> ?lhs"
- using hull_mono by blast
-qed
+ 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}"
+ {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"
- apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
- apply (erule ex_forward)+
- using aff_dim_le_DIM [of p]
- apply simp
- done
-next
- fix x
- assume "x \<in> ?rhs" then show "x \<in> ?lhs"
- by (auto simp: convex_hull_explicit)
-qed
+ 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}"
+ {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"
+ 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"
- apply (rule_tac x=s in exI)
- using hull_subset[of s convex]
- using convex_convex_hull[simplified convex_explicit, of s,
- THEN spec[where x=s], THEN spec[where x=u]]
- apply auto
- done
-next
- fix x s
- assume "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
- then show "x \<in> convex hull p"
- using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
-qed
+ 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>
@@ -1989,7 +1927,7 @@
lemma convex_hull_set_plus:
"convex hull (S + T) = convex hull S + convex hull T"
- unfolding set_plus_image
+ 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)
@@ -2009,7 +1947,7 @@
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(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
+ by (metis convex_hull_scaling convex_hull_translation image_image)
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of cone hulls\<close>
@@ -2089,20 +2027,19 @@
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"
+ 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 and sum.inter_restrict[OF assms(1), symmetric]
- apply (auto simp: Int_absorb1)
- done
+ 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}"
+ 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
@@ -2114,10 +2051,10 @@
qed
lemma Radon_v_lemma:
- assumes "finite s"
- and "sum f s = 0"
+ 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}"
+ 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
@@ -2130,79 +2067,63 @@
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> {}"
+ 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"
+ 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}"
+ 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"
+ 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")
+ 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"
- apply (rule_tac sum_mono, auto)
- done
+ 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
- apply (rule_tac conjI)
- apply (rule_tac sum_nonneg, auto)
- done
- 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)"
+ 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)
- apply (rule_tac[!] sum.mono_neutral_left, auto)
- done
- 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)"
+ 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"
- apply rule
- apply (rule mult_nonneg_nonneg)
- using *
- apply auto
- done
- ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
+ 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] and z_def
- apply (auto simp: sum_negf sum_distrib_left[symmetric])
- done
- moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
- apply rule
- apply (rule mult_nonneg_nonneg)
- using *
- apply auto
- done
- then have "z \<in> convex hull {v \<in> c. u v > 0}"
+ 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)
+ 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] and z_def
- using *
- apply (auto simp: sum_negf sum_distrib_left[symmetric])
- done
+ 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)
+ 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
@@ -2211,19 +2132,16 @@
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"
+ 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"
+ 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> {}"
+ obtain m p where "m \<inter> p = {}" "m \<union> p = S" "convex hull m \<inter> convex hull p \<noteq> {}"
by blast
- then show ?thesis
- apply (rule_tac that[of p m])
- using s
- apply auto
- done
+ with S show ?thesis
+ by (force intro: that[of p m])
qed
@@ -2288,10 +2206,7 @@
unfolding mp(2)[unfolded image_Un[symmetric] gh]
by auto
have *: "g \<inter> h = {}"
- using mp(1)
- unfolding gh
- using inj_on_image_Int[OF True gh(3,4)]
- by auto
+ 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
@@ -2305,10 +2220,7 @@
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> {}"
- apply (rule Helly_induct)
- using assms
- apply auto
- done
+ using Helly_induct assms by blast
subsection \<open>Epigraphs of convex functions\<close>
@@ -2323,14 +2235,7 @@
then show "convex_on S f"
by (auto simp: convex_def convex_on_def epigraph_def)
show "convex S"
- using L
- apply (clarsimp simp: convex_def convex_on_def epigraph_def)
- apply (erule_tac x=x in allE)
- apply (erule_tac x="f x" in allE, safe)
- apply (erule_tac x=y in allE)
- apply (erule_tac x="f y" in allE)
- apply (auto simp: )
- done
+ 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)"
@@ -2350,53 +2255,59 @@
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
- unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
- unfolding fst_sum snd_sum fst_scaleR snd_scaleR
- 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
- using assms[unfolded convex] apply simp
- apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
- apply (rule sum_mono)
- apply (erule_tac x=i in allE)
- unfolding real_scaleR_def
- apply (rule mult_left_mono)
- using assms[unfolded convex] apply auto
- done
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"
+ 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"
+ assume "x \<in> convex hull S"
then obtain k u v where
- obt: "\<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"
+ 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] obt(2) mult_1
- apply (drule_tac meta_mp)
- apply (rule mult_left_mono)
- using assms(2) obt(1)
- apply auto
- done
+ 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]
- unfolding obt(2-3)
- using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
- by auto
+ using hull_inc u by fastforce
qed
lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
@@ -2425,22 +2336,29 @@
using assms by (induct set: finite, simp, simp add: finite_set_plus)
lemma box_eq_set_sum_Basis:
- shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
- apply (subst set_sum_alt [OF finite_Basis], safe)
- apply (fast intro: euclidean_representation [symmetric])
- apply (subst inner_sum_left)
-apply (rename_tac f)
- apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
- apply (drule (1) bspec)
- apply clarsimp
- apply (frule sum.remove [OF finite_Basis])
- apply (erule trans, simp)
- apply (rule sum.neutral, clarsimp)
- apply (frule_tac x=i in bspec, assumption)
- apply (drule_tac x=x in bspec, assumption, clarsimp)
- apply (cut_tac u=x and v=i in inner_Basis, assumption+)
- apply (rule ccontr, simp)
- done
+ "{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))"