--- a/src/HOL/Library/Convex.thy Tue Apr 29 21:54:26 2014 +0200
+++ b/src/HOL/Library/Convex.thy Tue Apr 29 22:50:55 2014 +0200
@@ -29,11 +29,18 @@
(is "_ \<longleftrightarrow> ?alt")
proof
assume alt[rule_format]: ?alt
- { fix x y and u v :: real assume mem: "x \<in> s" "y \<in> s"
+ {
+ fix x y and u v :: real
+ assume mem: "x \<in> s" "y \<in> s"
assume "0 \<le> u" "0 \<le> v"
- moreover assume "u + v = 1" then have "u = 1 - v" by auto
- ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s" using alt[OF mem] by auto }
- then show "convex s" unfolding convex_def by auto
+ moreover
+ assume "u + v = 1"
+ then have "u = 1 - v" by auto
+ ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ using alt[OF mem] by auto
+ }
+ then show "convex s"
+ unfolding convex_def by auto
qed (auto simp: convex_def)
lemma mem_convex:
@@ -50,7 +57,7 @@
lemma convex_UNIV[intro]: "convex UNIV"
unfolding convex_def by auto
-lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
+lemma convex_Inter: "(\<forall>s\<in>f. convex s) \<Longrightarrow> convex(\<Inter> f)"
unfolding convex_def by auto
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
@@ -68,13 +75,16 @@
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
+ have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
+ by auto
+ show ?thesis
+ unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
qed
lemma convex_hyperplane: "convex {x. inner a x = b}"
proof -
- have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
+ 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
@@ -115,8 +125,11 @@
lemma convex_setsum:
fixes C :: "'a::real_vector set"
- assumes "finite s" and "convex C" and "(\<Sum> i \<in> s. a i) = 1"
- assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
+ 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)
@@ -124,18 +137,27 @@
then show ?case by simp
next
case (insert i s) note IH = this(3)
- have "a i + setsum 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"
+ have "a i + setsum 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> setsum a s" by (simp add: setsum_nonneg)
+ then have "0 \<le> setsum a s"
+ by (simp add: setsum_nonneg)
have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
proof (cases)
assume z: "setsum a s = 0"
- with `a i + setsum a s = 1` have "a i = 1" by simp
- from setsum_nonneg_0 [OF `finite s` _ z] `\<forall>j\<in>s. 0 \<le> a j` have "\<forall>j\<in>s. a j = 0" by simp
- show ?thesis using `a i = 1` and `\<forall>j\<in>s. a j = 0` and `y i \<in> C` by simp
+ with `a i + setsum a s = 1` have "a i = 1"
+ by simp
+ from setsum_nonneg_0 [OF `finite s` _ z] `\<forall>j\<in>s. 0 \<le> a j` have "\<forall>j\<in>s. a j = 0"
+ by simp
+ show ?thesis using `a i = 1` and `\<forall>j\<in>s. a j = 0` and `y i \<in> C`
+ by simp
next
assume nz: "setsum a s \<noteq> 0"
- with `0 \<le> setsum a s` have "0 < setsum a s" by simp
+ with `0 \<le> setsum a s` have "0 < setsum a s"
+ by simp
then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
using `\<forall>j\<in>s. 0 \<le> a j` and `\<forall>j\<in>s. y j \<in> C`
by (simp add: IH setsum_divide_distrib [symmetric])
@@ -143,9 +165,11 @@
and `0 \<le> setsum a s` and `a i + setsum a s = 1`
have "a i *\<^sub>R y i + setsum a s *\<^sub>R (\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
by (rule convexD)
- then show ?thesis by (simp add: scaleR_setsum_right nz)
+ then show ?thesis
+ by (simp add: scaleR_setsum_right nz)
qed
- then show ?case using `finite s` and `i \<notin> s` by simp
+ then show ?case using `finite s` and `i \<notin> s`
+ by simp
qed
lemma convex:
@@ -159,18 +183,22 @@
"\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
"setsum u {1..k} = 1"
from this convex_setsum[of "{1 .. k}" s]
- show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" by auto
+ show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
+ by auto
next
assume asm: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
\<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
- { fix \<mu> :: real
+ {
+ fix \<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
+ have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
+ by auto
+ then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
+ by simp
then have "setsum ?u {1 .. 2} = 1"
using setsum_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
by auto
@@ -179,10 +207,13 @@
have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
- have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" by auto
- then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute)
+ 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
+ then show "convex s"
+ unfolding convex_alt by auto
qed
@@ -193,42 +224,48 @@
proof safe
fix t
fix u :: "'a \<Rightarrow> real"
- assume "convex s" "finite t"
- "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
+ assume "convex s"
+ and "finite t"
+ and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
using convex_setsum[of t s u "\<lambda> x. x"] by auto
next
- assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x)
- \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
+ setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
show "convex s"
unfolding convex_alt
proof safe
fix x y
fix \<mu> :: real
assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
- { assume "x \<noteq> y"
+ {
+ assume "x \<noteq> y"
then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
- asm by auto }
+ asm by auto
+ }
moreover
- { assume "x = y"
+ {
+ assume "x = y"
then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
- asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) }
- ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast
+ asm by (auto simp: field_simps real_vector.scale_left_diff_distrib)
+ }
+ ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+ by blast
qed
qed
lemma convex_finite:
assumes "finite s"
- shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
- \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
+ shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
unfolding convex_explicit
proof safe
fix t u
assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
- have *: "s \<inter> t = t" using as(2) by auto
+ have *: "s \<inter> t = t"
+ using as(2) by auto
have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
by simp
show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
@@ -236,6 +273,7 @@
by (auto simp: assms setsum_cases if_distrib if_distrib_arg)
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
+
subsection {* Functions that are convex on a set *}
definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
@@ -246,11 +284,13 @@
unfolding convex_on_def by auto
lemma convex_on_add [intro]:
- assumes "convex_on s f" "convex_on s g"
+ 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"
+ {
+ 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"
@@ -260,13 +300,16 @@
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
+ then show ?thesis
+ unfolding convex_on_def by auto
qed
lemma convex_on_cmul [intro]:
- assumes "0 \<le> (c::real)" "convex_on s f"
+ fixes c :: real
+ assumes "0 \<le> c"
+ and "convex_on s f"
shows "convex_on s (\<lambda>x. c * f x)"
-proof-
+proof -
have *: "\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
by (simp add: field_simps)
show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
@@ -274,13 +317,19 @@
qed
lemma convex_lower:
- assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
+ 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-
+proof -
let ?m = "max (f x) (f y)"
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
using assms(4,5) by (auto simp add: mult_left_mono add_mono)
- also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[symmetric] by auto
+ also have "\<dots> = max (f x) (f y)"
+ using assms(6) unfolding distrib[symmetric] by auto
finally show ?thesis
using assms unfolding convex_on_def by fastforce
qed
@@ -290,11 +339,13 @@
shows "convex_on s (\<lambda>x. dist a x)"
proof (auto simp add: convex_on_def dist_norm)
fix x y
- assume "x\<in>s" "y\<in>s"
+ assume "x \<in> s" "y \<in> s"
fix u v :: real
- assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+ 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 `u+v=1` by simp
+ unfolding scaleR_left_distrib[symmetric] and `u + v = 1` by simp
then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
by (auto simp add: algebra_simps)
show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
@@ -306,7 +357,9 @@
subsection {* Arithmetic operations on sets preserve convexity. *}
lemma convex_linear_image:
- assumes "linear f" and "convex s" shows "convex (f ` s)"
+ assumes "linear f"
+ and "convex s"
+ shows "convex (f ` s)"
proof -
interpret f: linear f by fact
from `convex s` show "convex (f ` s)"
@@ -314,7 +367,9 @@
qed
lemma convex_linear_vimage:
- assumes "linear f" and "convex s" shows "convex (f -` s)"
+ assumes "linear f"
+ and "convex s"
+ shows "convex (f -` s)"
proof -
interpret f: linear f by fact
from `convex s` show "convex (f -` s)"
@@ -322,21 +377,28 @@
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 `convex s` by (rule convex_linear_image)
+ have "linear (\<lambda>x. c *\<^sub>R x)"
+ by (simp add: linearI scaleR_add_right)
+ then show ?thesis
+ using `convex s` by (rule convex_linear_image)
qed
lemma convex_negations:
- assumes "convex s" shows "convex ((\<lambda>x. - x) ` s)"
+ assumes "convex s"
+ shows "convex ((\<lambda>x. - x) ` s)"
proof -
- have "linear (\<lambda>x. - x)" by (simp add: linearI)
- then show ?thesis using `convex s` by (rule convex_linear_image)
+ have "linear (\<lambda>x. - x)"
+ by (simp add: linearI)
+ then show ?thesis
+ using `convex s` by (rule convex_linear_image)
qed
lemma convex_sums:
- assumes "convex s" and "convex t"
+ assumes "convex s"
+ and "convex t"
shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
proof -
have "linear (\<lambda>(x, y). x + y)"
@@ -362,7 +424,8 @@
assumes "convex s"
shows "convex ((\<lambda>x. a + x) ` s)"
proof -
- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
+ have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s"
+ by auto
then show ?thesis
using convex_sums[OF convex_singleton[of a] assms] by auto
qed
@@ -371,7 +434,8 @@
assumes "convex s"
shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
proof -
- have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto
+ have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s"
+ by auto
then show ?thesis
using convex_translation[OF convex_scaling[OF assms], of a c] by auto
qed
@@ -381,18 +445,25 @@
proof safe
fix y x \<mu> :: real
assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- { assume "\<mu> = 0"
+ {
+ assume "\<mu> = 0"
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp
+ }
moreover
- { assume "\<mu> = 1"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+ {
+ assume "\<mu> = 1"
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp
+ }
moreover
- { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
+ {
+ assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
then have "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
- by (auto simp add: add_pos_pos) }
- ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" using assms by fastforce
+ by (auto simp add: add_pos_pos)
+ }
+ ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
+ using assms by fastforce
qed
lemma convex_on_setsum:
@@ -415,25 +486,32 @@
case (insert i s) note asms = this
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"
+ 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
- { assume "a i = 1"
+ {
+ assume "a i = 1"
then have "(\<Sum> j \<in> s. a j) = 0"
using asms by auto
then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
using setsum_nonneg_0[where 'b=real] asms by fastforce
- then have ?case using asms by auto }
+ then have ?case using asms by auto
+ }
moreover
- { assume asm: "a i \<noteq> 1"
+ {
+ assume asm: "a i \<noteq> 1"
from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
have fis: "finite (insert i s)" using asms by auto
then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
then have "a i < 1" using asm by auto
then have i0: "1 - a i > 0" by auto
let ?a = "\<lambda>j. a j / (1 - a i)"
- { fix j assume "j \<in> s" with i0 asms have "?a j \<ge> 0"
- by fastforce }
+ {
+ fix j
+ assume "j \<in> s"
+ with i0 asms have "?a j \<ge> 0"
+ by fastforce
+ }
note a_nonneg = this
have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce
@@ -466,51 +544,66 @@
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 asms by auto
finally have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) \<le> (\<Sum> j \<in> insert i s. a j * f (y j))"
- by simp }
+ by simp
+ }
ultimately show ?case by auto
qed
lemma convex_on_alt:
fixes C :: "'a::real_vector set"
assumes "convex C"
- shows "convex_on C f =
- (\<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)"
+ 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 asms: "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 "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" by auto
+ have "\<And>u v. 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"
+ by auto
from this[of "\<mu>" "1 - \<mu>", simplified] asms
- show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" by auto
+ show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ by auto
next
- assume asm: "\<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
+ assume asm: "\<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 lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
then have[simp]: "1 - u = v" by auto
from asm[rule_format, of x y u]
- have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using lasm by auto
+ have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+ using lasm by auto
}
- then show "convex_on C f" unfolding convex_on_def 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"
+ 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)"
- "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
+ and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
proof -
def a \<equiv> "(t - y) / (x - y)"
- with t have "0 \<le> a" "0 \<le> 1 - a" by (auto simp: field_simps)
+ with t have "0 \<le> a" "0 \<le> 1 - a"
+ by (auto simp: field_simps)
with f `x \<in> I` `y \<in> I` 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 `x < t` `t < y` 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
+ have "a * x + (1 - a) * y = a * (x - y) + y"
+ by (simp add: field_simps)
+ also have "\<dots> = t"
+ unfolding a_def using `x < t` `t < y` 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)"
@@ -520,7 +613,7 @@
lemma pos_convex_function:
fixes f :: "real \<Rightarrow> real"
assumes "convex C"
- and leq: "\<And>x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
+ 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)]
using assms
@@ -529,11 +622,13 @@
let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
assume asm: "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 asm 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)"
+ then have xpos: "?x \<in> C"
+ using asm 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 asm(2)] `\<mu> \<ge> 0`]
- mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
+ mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]]
+ by auto
then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
by (auto simp add: field_simps)
then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
@@ -547,9 +642,11 @@
shows "{x .. y} \<subseteq> C"
proof safe
fix z assume zasm: "z \<in> {x .. y}"
- { assume asm: "x < z" "z < y"
+ {
+ assume asm: "x < z" "z < y"
let ?\<mu> = "(y - z) / (y - x)"
- have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add: field_simps)
+ have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
+ using assms asm by (auto simp add: 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)
@@ -560,7 +657,8 @@
also have "\<dots> = z"
using assms by (auto simp: field_simps)
finally have "z \<in> C"
- using comb by auto }
+ using comb by auto
+ }
note less = this
show "z \<in> C" using zasm less assms
unfolding atLeastAtMost_iff le_less by auto
@@ -576,7 +674,8 @@
shows "f' x * (y - x) \<le> f y - f x"
using assms
proof -
- { fix x y :: real
+ {
+ fix x y :: real
assume asm: "x \<in> C" "y \<in> C" "y > x"
then have ge: "y - x > 0" "y - x \<ge> 0" by auto
from asm have le: "x - y < 0" "x - y \<le> 0" by auto
@@ -627,14 +726,18 @@
then have "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
using res by auto } note less_imp = this
- { fix x y :: real
+ {
+ fix x y :: real
assume "x \<in> C" "y \<in> C" "x \<noteq> y"
then have"f y - f x \<ge> f' x * (y - x)"
- unfolding neq_iff using less_imp by auto } note neq_imp = this
+ unfolding neq_iff using less_imp by auto
+ }
moreover
- { fix x y :: real
+ {
+ fix x y :: real
assume asm: "x \<in> C" "y \<in> C" "x = y"
- then have "f y - f x \<ge> f' x * (y - x)" by auto }
+ then have "f y - f x \<ge> f' x * (y - x)" by auto
+ }
ultimately show ?thesis using assms by blast
qed
@@ -645,14 +748,16 @@
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
+ 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
+ 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))"
@@ -661,9 +766,10 @@
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)"
+ 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 add: mult_assoc)
- have f''_ge0: "\<And>z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
+ have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
using `b > 1` by (auto intro!:less_imp_le)
from f''_ge0_imp_convex[OF pos_is_convex,
unfolded greaterThan_iff, OF f' f''0 f''_ge0]