Moved Convex theory to library.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Convex.thy Mon May 03 14:35:10 2010 +0200
@@ -0,0 +1,610 @@
+theory Convex
+imports Product_Vector
+begin
+
+subsection {* Convexity. *}
+
+definition
+ convex :: "'a::real_vector set \<Rightarrow> bool" where
+ "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
+
+lemma convex_alt:
+ "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
+ (is "_ \<longleftrightarrow> ?alt")
+proof
+ assume alt[rule_format]: ?alt
+ { fix x y and u v :: real assume mem: "x \<in> s" "y \<in> s"
+ assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+ moreover hence "u = 1 - v" by auto
+ ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s" using alt[OF mem] by auto }
+ thus "convex s" unfolding convex_def by auto
+qed (auto simp: convex_def)
+
+lemma mem_convex:
+ assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
+ shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
+ using assms unfolding convex_alt by auto
+
+lemma convex_empty[intro]: "convex {}"
+ unfolding convex_def by simp
+
+lemma convex_singleton[intro]: "convex {a}"
+ unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
+
+lemma convex_UNIV[intro]: "convex UNIV"
+ unfolding convex_def by auto
+
+lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
+ unfolding convex_def by auto
+
+lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
+ unfolding convex_def by auto
+
+lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
+ unfolding convex_def
+ by (auto simp: inner_add inner_scaleR intro!: convex_bound_le)
+
+lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
+proof -
+ have *:"{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto
+ show ?thesis unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
+qed
+
+lemma convex_hyperplane: "convex {x. inner a x = b}"
+proof-
+ have *:"{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
+ show ?thesis using convex_halfspace_le convex_halfspace_ge
+ by (auto intro!: convex_Int simp: *)
+qed
+
+lemma convex_halfspace_lt: "convex {x. inner a x < b}"
+ unfolding convex_def
+ by (auto simp: convex_bound_lt inner_add)
+
+lemma convex_halfspace_gt: "convex {x. inner a x > b}"
+ using convex_halfspace_lt[of "-a" "-b"] by auto
+
+lemma convex_real_interval:
+ fixes a b :: "real"
+ shows "convex {a..}" and "convex {..b}"
+ and "convex {a<..}" and "convex {..<b}"
+ and "convex {a..b}" and "convex {a<..b}"
+ and "convex {a..<b}" and "convex {a<..<b}"
+proof -
+ have "{a..} = {x. a \<le> inner 1 x}" by auto
+ thus 1: "convex {a..}" by (simp only: convex_halfspace_ge)
+ have "{..b} = {x. inner 1 x \<le> b}" by auto
+ thus 2: "convex {..b}" by (simp only: convex_halfspace_le)
+ have "{a<..} = {x. a < inner 1 x}" by auto
+ thus 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
+ have "{..<b} = {x. inner 1 x < b}" by auto
+ thus 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
+ have "{a..b} = {a..} \<inter> {..b}" by auto
+ thus "convex {a..b}" by (simp only: convex_Int 1 2)
+ have "{a<..b} = {a<..} \<inter> {..b}" by auto
+ thus "convex {a<..b}" by (simp only: convex_Int 3 2)
+ have "{a..<b} = {a..} \<inter> {..<b}" by auto
+ thus "convex {a..<b}" by (simp only: convex_Int 1 4)
+ have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
+ thus "convex {a<..<b}" by (simp only: convex_Int 3 4)
+qed
+
+subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
+
+lemma convex_setsum:
+ fixes C :: "'a::real_vector set"
+ assumes "finite s" and "convex C" and "(\<Sum> i \<in> s. a i) = 1"
+ assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0" and "\<And> i. i \<in> s \<Longrightarrow> y i \<in> C"
+ shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
+using assms
+proof (induct s arbitrary:a rule:finite_induct)
+ case empty thus ?case by auto
+next
+ case (insert i s) note asms = this
+ { assume "a i = 1"
+ hence "(\<Sum> j \<in> s. a j) = 0"
+ using asms by auto
+ hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
+ using setsum_nonneg_0[where 'b=real] asms by fastsimp
+ hence ?case using asms by auto }
+ moreover
+ { assume asm: "a i \<noteq> 1"
+ from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
+ have fis: "finite (insert i s)" using asms by auto
+ hence ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a 1] asms by simp
+ hence "a i < 1" using asm by auto
+ hence i0: "1 - a i > 0" by auto
+ let "?a j" = "a j / (1 - a i)"
+ { fix j assume "j \<in> s"
+ hence "?a j \<ge> 0"
+ using i0 asms divide_nonneg_pos
+ by fastsimp } note a_nonneg = this
+ have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
+ hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
+ hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
+ hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
+ from this asms
+ have "(\<Sum>j\<in>s. ?a j *\<^sub>R y j) \<in> C" using a_nonneg by fastsimp
+ hence "a i *\<^sub>R y i + (1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
+ using asms[unfolded convex_def, rule_format] yai ai1 by auto
+ hence "a i *\<^sub>R y i + (\<Sum> j \<in> s. (1 - a i) *\<^sub>R (?a j *\<^sub>R y j)) \<in> C"
+ using scaleR_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j *\<^sub>R y j" s] by auto
+ hence "a i *\<^sub>R y i + (\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" using i0 by auto
+ hence ?case using setsum.insert asms by auto }
+ ultimately show ?case by auto
+qed
+
+lemma convex:
+ shows "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
+ \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
+proof safe
+ fix k :: nat fix u :: "nat \<Rightarrow> real" fix x
+ assume "convex s"
+ "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
+ "setsum u {1..k} = 1"
+ from this convex_setsum[of "{1 .. k}" s]
+ show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" by auto
+next
+ assume asm: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
+ \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
+ { fix \<mu> :: real fix x y :: 'a assume xy: "x \<in> s" "y \<in> s" assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
+ let "?u i" = "if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
+ let "?x i" = "if (i :: nat) = 1 then x else y"
+ have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}" by auto
+ hence card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" by simp
+ hence "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
+ from this asm[rule_format, of "2" ?u ?x]
+ have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
+ using mu xy by auto
+ have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
+ using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
+ from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
+ have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" by auto
+ hence "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute) }
+ thus "convex s" unfolding convex_alt by auto
+qed
+
+
+lemma convex_explicit:
+ fixes s :: "'a::real_vector set"
+ shows "convex s \<longleftrightarrow>
+ (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
+proof safe
+ fix t fix u :: "'a \<Rightarrow> real"
+ assume "convex s" "finite t"
+ "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
+ thus "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ using convex_setsum[of t s u "\<lambda> x. x"] by auto
+next
+ assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x)
+ \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ show "convex s"
+ unfolding convex_alt
+ proof safe
+ fix x y fix \<mu> :: real
+ assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
+ { assume "x \<noteq> y"
+ hence "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+ using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
+ asm by auto }
+ moreover
+ { assume "x = y"
+ hence "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+ using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
+ asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) }
+ ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast
+ qed
+qed
+
+lemma convex_finite: assumes "finite s"
+ shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
+ \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
+ unfolding convex_explicit
+proof (safe elim!: conjE)
+ fix t u assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
+ and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
+ have *:"s \<inter> t = t" using as(2) by auto
+ have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)" by simp
+ show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+ using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
+ by (auto simp: assms setsum_cases if_distrib if_distrib_arg)
+qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
+
+definition
+ convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
+ "convex_on s f \<longleftrightarrow>
+ (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
+
+lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
+ unfolding convex_on_def by auto
+
+lemma convex_add[intro]:
+ assumes "convex_on s f" "convex_on s g"
+ shows "convex_on s (\<lambda>x. f x + g x)"
+proof-
+ { fix x y assume "x\<in>s" "y\<in>s" moreover
+ fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+ ultimately have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
+ using assms unfolding convex_on_def by (auto simp add:add_mono)
+ hence "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) }
+ thus ?thesis unfolding convex_on_def by auto
+qed
+
+lemma convex_cmul[intro]:
+ assumes "0 \<le> (c::real)" "convex_on s f"
+ shows "convex_on s (\<lambda>x. c * f x)"
+proof-
+ have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: field_simps)
+ show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
+qed
+
+lemma convex_lower:
+ assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
+ shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
+proof-
+ let ?m = "max (f x) (f y)"
+ have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
+ using assms(4,5) by(auto simp add: mult_mono1 add_mono)
+ also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
+ finally show ?thesis
+ using assms unfolding convex_on_def by fastsimp
+qed
+
+lemma convex_distance[intro]:
+ fixes s :: "'a::real_normed_vector set"
+ shows "convex_on s (\<lambda>x. dist a x)"
+proof(auto simp add: convex_on_def dist_norm)
+ fix x y assume "x\<in>s" "y\<in>s"
+ fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+ have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
+ hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
+ by (auto simp add: algebra_simps)
+ show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
+ unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
+ using `0 \<le> u` `0 \<le> v` by auto
+qed
+
+subsection {* Arithmetic operations on sets preserve convexity. *}
+lemma convex_scaling:
+ assumes "convex s"
+ shows"convex ((\<lambda>x. c *\<^sub>R x) ` s)"
+using assms unfolding convex_def image_iff
+proof safe
+ fix x xa y xb :: "'a::real_vector" fix u v :: real
+ assume asm: "\<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"
+ "xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
+ show "\<exists>x\<in>s. u *\<^sub>R c *\<^sub>R xa + v *\<^sub>R c *\<^sub>R xb = c *\<^sub>R x"
+ using bexI[of _ "u *\<^sub>R xa +v *\<^sub>R xb"] asm by (auto simp add: algebra_simps)
+qed
+
+lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
+using assms unfolding convex_def image_iff
+proof safe
+ fix x xa y xb :: "'a::real_vector" fix u v :: real
+ assume asm: "\<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"
+ "xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
+ show "\<exists>x\<in>s. u *\<^sub>R - xa + v *\<^sub>R - xb = - x"
+ using bexI[of _ "u *\<^sub>R xa +v *\<^sub>R xb"] asm by auto
+qed
+
+lemma convex_sums:
+ assumes "convex s" "convex t"
+ shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
+using assms unfolding convex_def image_iff
+proof safe
+ fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
+ fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
+ show "\<exists>x y. u *\<^sub>R (xa + ya) + v *\<^sub>R (xb + yb) = x + y \<and> x \<in> s \<and> y \<in> t"
+ using exI[of _ "u *\<^sub>R xa + v *\<^sub>R xb"] exI[of _ "u *\<^sub>R ya + v *\<^sub>R yb"]
+ assms[unfolded convex_def] uv xy by (auto simp add:scaleR_right_distrib)
+qed
+
+lemma convex_differences:
+ assumes "convex s" "convex t"
+ shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
+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}"
+ proof safe
+ fix x x' y assume "x' \<in> s" "y \<in> t"
+ thus "\<exists>x y'. x' - y = x + y' \<and> x \<in> s \<and> y' \<in> uminus ` t"
+ using exI[of _ x'] exI[of _ "-y"] by auto
+ next
+ fix x x' y y' assume "x' \<in> s" "y' \<in> t"
+ thus "\<exists>x y. x' + - y' = x - y \<and> x \<in> s \<and> y \<in> t"
+ using exI[of _ x'] exI[of _ y'] by auto
+ qed
+ thus ?thesis using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
+qed
+
+lemma convex_translation: 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
+ thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
+
+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 = op + a ` op *\<^sub>R c ` s" by auto
+ thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
+
+lemma convex_linear_image:
+ assumes c:"convex s" and l:"bounded_linear f"
+ shows "convex(f ` s)"
+proof(auto simp add: convex_def)
+ interpret f: bounded_linear f by fact
+ fix x y assume xy:"x \<in> s" "y \<in> s"
+ fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
+ show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
+ using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
+ c[unfolded convex_def] xy uv by auto
+qed
+
+
+lemma pos_is_convex:
+ shows "convex {0 :: real <..}"
+unfolding convex_alt
+proof safe
+ fix y x \<mu> :: real
+ assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
+ { assume "\<mu> = 0"
+ hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
+ hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+ moreover
+ { assume "\<mu> = 1"
+ hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+ moreover
+ { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
+ hence "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
+ hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
+ using add_nonneg_pos[of "\<mu> *\<^sub>R x" "(1 - \<mu>) *\<^sub>R y"]
+ real_mult_order by auto fastsimp }
+ ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" using assms by fastsimp
+qed
+
+lemma convex_on_setsum:
+ fixes a :: "'a \<Rightarrow> real"
+ fixes y :: "'a \<Rightarrow> 'b::real_vector"
+ fixes f :: "'b \<Rightarrow> real"
+ assumes "finite s" "s \<noteq> {}"
+ assumes "convex_on C f"
+ assumes "convex C"
+ assumes "(\<Sum> i \<in> s. a i) = 1"
+ assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
+ assumes "\<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)
+ hence ai: "a i = 1" by auto
+ thus ?case by auto
+next
+ case (insert i s) note asms = this
+ hence "convex_on C f" by simp
+ from this[unfolded convex_on_def, rule_format]
+ have conv: "\<And> x y \<mu>. \<lbrakk>x \<in> C; y \<in> C; 0 \<le> \<mu>; \<mu> \<le> 1\<rbrakk>
+ \<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"
+ hence "(\<Sum> j \<in> s. a j) = 0"
+ using asms by auto
+ hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
+ using setsum_nonneg_0[where 'b=real] asms by fastsimp
+ hence ?case using asms by auto }
+ moreover
+ { assume asm: "a i \<noteq> 1"
+ from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
+ have fis: "finite (insert i s)" using asms by auto
+ hence ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
+ hence "a i < 1" using asm by auto
+ hence i0: "1 - a i > 0" by auto
+ let "?a j" = "a j / (1 - a i)"
+ { fix j assume "j \<in> s"
+ hence "?a j \<ge> 0"
+ using i0 asms divide_nonneg_pos
+ by fastsimp } note a_nonneg = this
+ have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
+ hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
+ hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
+ hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
+ have "convex C" using asms by auto
+ hence asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
+ using asms convex_setsum[OF `finite s`
+ `convex C` 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 asms 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 setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF `finite s` `i \<notin> s`] asms
+ 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.setsum[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:real_divide_def)
+ 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: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 mult_right.setsum[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 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 }
+ 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)"
+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
+ 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
+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 fix u v :: real
+ assume lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+ hence[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 }
+ thus "convex_on C f" unfolding convex_on_def by auto
+qed
+
+
+lemma pos_convex_function:
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex C"
+ assumes leq: "\<And> x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
+ shows "convex_on C f"
+unfolding convex_on_alt[OF assms(1)]
+using assms
+proof safe
+ fix x y \<mu> :: real
+ 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"
+ hence "1 - \<mu> \<ge> 0" by auto
+ hence xpos: "?x \<in> C" using asm unfolding convex_alt by fastsimp
+ 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_mono1[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`]
+ mult_mono1[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
+ hence "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
+ by (auto simp add:field_simps)
+ thus "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+ using convex_on_alt by auto
+qed
+
+lemma atMostAtLeast_subset_convex:
+ fixes C :: "real set"
+ assumes "convex C"
+ assumes "x \<in> C" "y \<in> C" "x < y"
+ shows "{x .. y} \<subseteq> C"
+proof safe
+ fix z assume zasm: "z \<in> {x .. 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)
+ hence 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 add:field_simps)
+ also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
+ using assms unfolding add_divide_distrib by (auto simp:field_simps)
+ also have "\<dots> = z"
+ using assms by (auto simp:field_simps)
+ finally have "z \<in> C"
+ using comb by auto } note less = this
+ show "z \<in> C" using zasm less assms
+ unfolding atLeastAtMost_iff le_less by auto
+qed
+
+lemma f''_imp_f':
+ fixes f :: "real \<Rightarrow> real"
+ assumes "convex C"
+ assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
+ assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
+ assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
+ assumes "x \<in> C" "y \<in> C"
+ shows "f' x * (y - x) \<le> f y - f x"
+using assms
+proof -
+ { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "y > x"
+ hence ge: "y - x > 0" "y - x \<ge> 0" by auto
+ from asm 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 `convex C` `x \<in> C` `y \<in> C` `x < y`],
+ THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
+ by auto
+ hence "z1 \<in> C" using atMostAtLeast_subset_convex
+ `convex C` `x \<in> C` `y \<in> C` `x < y` by fastsimp
+ 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 `convex C` `x \<in> C` `z1 \<in> C` `x < z1`],
+ THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ by auto
+ obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
+ using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `z1 \<in> C` `y \<in> C` `z1 < y`],
+ THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ by auto
+ have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
+ using asm 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 asm atMostAtLeast_subset_convex
+ `convex C` `x \<in> C` `z1 \<in> C` `x < z1` by fastsimp
+ hence B': "f'' z3 \<ge> 0" using assms by auto
+ from A' B' have "(y - z1) * f'' z3 \<ge> 0" using mult_nonneg_nonneg 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)"
+ unfolding diff_def using real_add_mult_distrib by auto
+ hence "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
+ hence 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 asm 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 asm atMostAtLeast_subset_convex
+ `convex C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastsimp
+ hence B: "f'' z2 \<ge> 0" using assms by auto
+ from A B have "(z1 - x) * f'' z2 \<ge> 0" using mult_nonneg_nonneg by auto
+ from cool this 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)"
+ unfolding diff_def using real_add_mult_distrib by auto
+ hence "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
+ hence "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 assume "x \<in> C" "y \<in> C" "x \<noteq> y"
+ hence"f y - f x \<ge> f' x * (y - x)"
+ unfolding neq_iff using less_imp by auto } note neq_imp = this
+ moreover
+ { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "x = y"
+ hence "f y - f x \<ge> f' x * (y - x)" by auto }
+ ultimately show ?thesis using assms by blast
+qed
+
+lemma f''_ge0_imp_convex:
+ fixes f :: "real \<Rightarrow> real"
+ assumes conv: "convex C"
+ assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
+ assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
+ assumes 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 fastsimp
+
+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
+ hence f': "\<And> z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
+ using DERIV_minus by auto
+ 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
+ hence 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:real_mult_assoc)
+ 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 simp add:divide_pos_pos[of 1] real_mult_order)
+ from f''_ge0_imp_convex[OF pos_is_convex,
+ unfolded greaterThan_iff, OF f' f''0 f''_ge0]
+ show ?thesis by auto
+qed
+
+end
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Tue Apr 20 17:58:34 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Mon May 03 14:35:10 2010 +0200
@@ -5,7 +5,7 @@
header {* Convex sets, functions and related things. *}
theory Convex_Euclidean_Space
-imports Topology_Euclidean_Space
+imports Topology_Euclidean_Space Convex
begin
@@ -315,176 +315,6 @@
shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
-subsection {* Convexity. *}
-
-definition
- convex :: "'a::real_vector set \<Rightarrow> bool" where
- "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
-
-lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
-proof- have *:"\<And>u v::real. u + v = 1 \<longleftrightarrow> u = 1 - v" by auto
- show ?thesis unfolding convex_def apply auto
- apply(erule_tac x=x in ballE) apply(erule_tac x=y in ballE) apply(erule_tac x="1 - u" in allE)
- by (auto simp add: *) qed
-
-lemma mem_convex:
- assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
- shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
- using assms unfolding convex_alt by auto
-
-lemma convex_empty[intro]: "convex {}"
- unfolding convex_def by simp
-
-lemma convex_singleton[intro]: "convex {a}"
- unfolding convex_def by (auto simp add:scaleR_left_distrib[THEN sym])
-
-lemma convex_UNIV[intro]: "convex UNIV"
- unfolding convex_def by auto
-
-lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
- unfolding convex_def by auto
-
-lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
- unfolding convex_def by auto
-
-lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
- unfolding convex_def apply auto
- unfolding inner_add inner_scaleR
- by (metis real_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 apply(unfold *) using convex_halfspace_le[of "-a" "-b"] by auto qed
-
-lemma convex_hyperplane: "convex {x. inner a x = b}"
-proof-
- have *:"{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
- show ?thesis unfolding * apply(rule convex_Int)
- using convex_halfspace_le convex_halfspace_ge by auto
-qed
-
-lemma convex_halfspace_lt: "convex {x. inner a x < b}"
- unfolding convex_def
- by(auto simp add: real_convex_bound_lt inner_add)
-
-lemma convex_halfspace_gt: "convex {x. inner a x > b}"
- using convex_halfspace_lt[of "-a" "-b"] by auto
-
-lemma convex_real_interval:
- fixes a b :: "real"
- shows "convex {a..}" and "convex {..b}"
- and "convex {a<..}" and "convex {..<b}"
- and "convex {a..b}" and "convex {a<..b}"
- and "convex {a..<b}" and "convex {a<..<b}"
-proof -
- have "{a..} = {x. a \<le> inner 1 x}" by auto
- thus 1: "convex {a..}" by (simp only: convex_halfspace_ge)
- have "{..b} = {x. inner 1 x \<le> b}" by auto
- thus 2: "convex {..b}" by (simp only: convex_halfspace_le)
- have "{a<..} = {x. a < inner 1 x}" by auto
- thus 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
- have "{..<b} = {x. inner 1 x < b}" by auto
- thus 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
- have "{a..b} = {a..} \<inter> {..b}" by auto
- thus "convex {a..b}" by (simp only: convex_Int 1 2)
- have "{a<..b} = {a<..} \<inter> {..b}" by auto
- thus "convex {a<..b}" by (simp only: convex_Int 3 2)
- have "{a..<b} = {a..} \<inter> {..<b}" by auto
- thus "convex {a..<b}" by (simp only: convex_Int 1 4)
- have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
- thus "convex {a<..<b}" by (simp only: convex_Int 3 4)
-qed
-
-lemma convex_box:
- assumes "\<And>i. convex {x. P i x}"
- shows "convex {x. \<forall>i. P i (x$i)}"
-using assms unfolding convex_def by auto
-
-lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
-by (rule convex_box, simp add: atLeast_def [symmetric] convex_real_interval)
-
-subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
-
-lemma convex: "convex s \<longleftrightarrow>
- (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
- \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
- unfolding convex_def apply rule apply(rule allI)+ defer apply(rule ballI)+ apply(rule allI)+ proof(rule,rule,rule,rule)
- fix x y u v assume as:"\<forall>(k::nat) u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> s"
- "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
- show "u *\<^sub>R x + v *\<^sub>R y \<in> s" using as(1)[THEN spec[where x=2], THEN spec[where x="\<lambda>n. if n=1 then u else v"], THEN spec[where x="\<lambda>n. if n=1 then x else y"]] and as(2-)
- by (auto simp add: setsum_head_Suc)
-next
- fix k u x assume as:"\<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"
- show "(\<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) \<in> s" apply(rule,erule conjE) proof(induct k arbitrary: u)
- case (Suc k) show ?case proof(cases "u (Suc k) = 1")
- case True hence "(\<Sum>i = Suc 0..k. u i *\<^sub>R x i) = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
- fix i assume i:"i \<in> {Suc 0..k}" "u i *\<^sub>R x i \<noteq> 0"
- hence ui:"u i \<noteq> 0" by auto
- hence "setsum (\<lambda>k. if k=i then u i else 0) {1 .. k} \<le> setsum u {1 .. k}" apply(rule_tac setsum_mono) using Suc(2) by auto
- hence "setsum u {1 .. k} \<ge> u i" using i(1) by(auto simp add: setsum_delta)
- hence "setsum u {1 .. k} > 0" using ui apply(rule_tac less_le_trans[of _ "u i"]) using Suc(2)[THEN spec[where x=i]] and i(1) by auto
- thus False using Suc(3) unfolding setsum_cl_ivl_Suc and True by simp qed
- thus ?thesis unfolding setsum_cl_ivl_Suc using True and Suc(2) by auto
- next
- have *:"setsum u {1..k} = 1 - u (Suc k)" using Suc(3)[unfolded setsum_cl_ivl_Suc] by auto
- have **:"u (Suc k) \<le> 1" unfolding not_le using Suc(3) using setsum_nonneg[of "{1..k}" u] using Suc(2) by auto
- have ***:"\<And>i k. (u i / (1 - u (Suc k))) *\<^sub>R x i = (inverse (1 - u (Suc k))) *\<^sub>R (u i *\<^sub>R x i)" unfolding real_divide_def by (auto simp add: algebra_simps)
- case False hence nn:"1 - u (Suc k) \<noteq> 0" by auto
- have "(\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) \<in> s" apply(rule Suc(1)) unfolding setsum_divide_distrib[THEN sym] and *
- apply(rule_tac allI) apply(rule,rule) apply(rule divide_nonneg_pos) using nn Suc(2) ** by auto
- hence "(1 - u (Suc k)) *\<^sub>R (\<Sum>i = 1..k. (u i / (1 - u (Suc k))) *\<^sub>R x i) + u (Suc k) *\<^sub>R x (Suc k) \<in> s"
- apply(rule as[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using Suc(2)[THEN spec[where x="Suc k"]] and ** by auto
- thus ?thesis unfolding setsum_cl_ivl_Suc and *** and scaleR_right.setsum [symmetric] using nn by auto qed qed auto qed
-
-
-lemma convex_explicit:
- fixes s :: "'a::real_vector set"
- shows "convex s \<longleftrightarrow>
- (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
- unfolding convex_def apply(rule,rule,rule) apply(subst imp_conjL,rule) defer apply(rule,rule,rule,rule,rule,rule,rule) proof-
- fix x y u v assume as:"\<forall>t 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" "x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
- show "u *\<^sub>R x + v *\<^sub>R y \<in> s" proof(cases "x=y")
- case True show ?thesis unfolding True and scaleR_left_distrib[THEN sym] using as(3,6) by auto next
- case False thus ?thesis using as(1)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>z. if z=x then u else v"]] and as(2-) by auto qed
-next
- fix t u assume asm:"\<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" "finite (t::'a set)"
- (*"finite t" "t \<subseteq> s" "\<forall>x\<in>t. (0::real) \<le> u x" "setsum u t = 1"*)
- from this(2) have "\<forall>u. 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" apply(induct t rule:finite_induct)
- prefer 2 apply (rule,rule) apply(erule conjE)+ proof-
- fix x f u assume ind:"\<forall>u. f \<subseteq> s \<and> (\<forall>x\<in>f. 0 \<le> u x) \<and> setsum u f = 1 \<longrightarrow> (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s"
- assume as:"finite f" "x \<notin> f" "insert x f \<subseteq> s" "\<forall>x\<in>insert x f. 0 \<le> u x" "setsum u (insert x f) = (1::real)"
- show "(\<Sum>x\<in>insert x f. u x *\<^sub>R x) \<in> s" proof(cases "u x = 1")
- case True hence "setsum (\<lambda>x. u x *\<^sub>R x) f = 0" apply(rule_tac setsum_0') apply(rule ccontr) unfolding ball_simps apply(erule bexE) proof-
- fix y assume y:"y \<in> f" "u y *\<^sub>R y \<noteq> 0"
- hence uy:"u y \<noteq> 0" by auto
- hence "setsum (\<lambda>k. if k=y then u y else 0) f \<le> setsum u f" apply(rule_tac setsum_mono) using as(4) by auto
- hence "setsum u f \<ge> u y" using y(1) and as(1) by(auto simp add: setsum_delta)
- hence "setsum u f > 0" using uy apply(rule_tac less_le_trans[of _ "u y"]) using as(4) and y(1) by auto
- thus False using as(2,5) unfolding setsum_clauses(2)[OF as(1)] and True by auto qed
- thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2,3) unfolding True by auto
- next
- have *:"setsum u f = setsum u (insert x f) - u x" using as(2) unfolding setsum_clauses(2)[OF as(1)] by auto
- have **:"u x \<le> 1" unfolding not_le using as(5)[unfolded setsum_clauses(2)[OF as(1)]] and as(2)
- using setsum_nonneg[of f u] and as(4) by auto
- case False hence "inverse (1 - u x) *\<^sub>R (\<Sum>x\<in>f. u x *\<^sub>R x) \<in> s" unfolding scaleR_right.setsum and scaleR_scaleR
- apply(rule_tac ind[THEN spec, THEN mp]) apply rule defer apply rule apply rule apply(rule mult_nonneg_nonneg)
- unfolding setsum_right_distrib[THEN sym] and * using as and ** by auto
- hence "u x *\<^sub>R x + (1 - u x) *\<^sub>R ((inverse (1 - u x)) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) f) \<in>s"
- apply(rule_tac asm(1)[THEN bspec, THEN bspec, THEN spec, THEN mp, THEN spec, THEN mp, THEN mp]) using as and ** False by auto
- thus ?thesis unfolding setsum_clauses(2)[OF as(1)] using as(2) and False by auto qed
- qed auto thus "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" by auto
-qed
-
-lemma convex_finite: assumes "finite s"
- shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
- \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
- unfolding convex_explicit apply(rule, rule, rule) defer apply(rule,rule,rule)apply(erule conjE)+ proof-
- fix t u assume as:"\<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" " 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(3) by auto
- show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" using as(1)[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]]
- unfolding if_smult and setsum_cases[OF assms] using as(2-) * by auto
-qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
-
subsection {* Cones. *}
definition
@@ -595,49 +425,15 @@
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
by(simp add: convex_connected convex_UNIV)
-subsection {* Convex functions into the reals. *}
-
-definition
- convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
- "convex_on s f \<longleftrightarrow>
- (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
-
-lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
- unfolding convex_on_def by auto
+subsection {* Balls, being convex, are connected. *}
-lemma convex_add[intro]:
- assumes "convex_on s f" "convex_on s g"
- shows "convex_on s (\<lambda>x. f x + g x)"
-proof-
- { fix x y assume "x\<in>s" "y\<in>s" moreover
- fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- ultimately have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
- using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
- using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
- apply - apply(rule add_mono) by auto
- hence "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) }
- thus ?thesis unfolding convex_on_def by auto
-qed
+lemma convex_box:
+ assumes "\<And>i. convex {x. P i x}"
+ shows "convex {x. \<forall>i. P i (x$i)}"
+using assms unfolding convex_def by auto
-lemma convex_cmul[intro]:
- assumes "0 \<le> (c::real)" "convex_on s f"
- shows "convex_on s (\<lambda>x. c * f x)"
-proof-
- have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: field_simps)
- show ?thesis using assms(2) and mult_mono1[OF _ assms(1)] unfolding convex_on_def and * by auto
-qed
-
-lemma convex_lower:
- assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
- shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
-proof-
- let ?m = "max (f x) (f y)"
- have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" apply(rule add_mono)
- using assms(4,5) by(auto simp add: mult_mono1)
- also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
- finally show ?thesis using assms(1)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x=u]]
- using assms(2-6) by auto
-qed
+lemma convex_positive_orthant: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
+ by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
lemma convex_local_global_minimum:
fixes s :: "'a::real_normed_vector set"
@@ -661,76 +457,6 @@
ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
qed
-lemma convex_distance[intro]:
- fixes s :: "'a::real_normed_vector set"
- shows "convex_on s (\<lambda>x. dist a x)"
-proof(auto simp add: convex_on_def dist_norm)
- fix x y assume "x\<in>s" "y\<in>s"
- fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
- hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
- by (auto simp add: algebra_simps)
- show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
- unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
- using `0 \<le> u` `0 \<le> v` by auto
-qed
-
-subsection {* Arithmetic operations on sets preserve convexity. *}
-
-lemma convex_scaling: "convex s \<Longrightarrow> convex ((\<lambda>x. c *\<^sub>R x) ` s)"
- unfolding convex_def and image_iff apply auto
- apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by (auto simp add: algebra_simps)
-
-lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
- unfolding convex_def and image_iff apply auto
- apply (rule_tac x="u *\<^sub>R x+v *\<^sub>R y" in bexI) by auto
-
-lemma convex_sums:
- assumes "convex s" "convex t"
- shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
-proof(auto simp add: convex_def image_iff scaleR_right_distrib)
- fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
- fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
- show "\<exists>x y. u *\<^sub>R xa + u *\<^sub>R ya + (v *\<^sub>R xb + v *\<^sub>R yb) = x + y \<and> x \<in> s \<and> y \<in> t"
- apply(rule_tac x="u *\<^sub>R xa + v *\<^sub>R xb" in exI) apply(rule_tac x="u *\<^sub>R ya + v *\<^sub>R yb" in exI)
- using assms(1)[unfolded convex_def, THEN bspec[where x=xa], THEN bspec[where x=xb]]
- using assms(2)[unfolded convex_def, THEN bspec[where x=ya], THEN bspec[where x=yb]]
- using uv xy by auto
-qed
-
-lemma convex_differences:
- assumes "convex s" "convex t"
- shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
-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}" unfolding image_iff apply auto
- apply(rule_tac x=xa in exI) apply(rule_tac x="-y" in exI) apply simp
- apply(rule_tac x=xa in exI) apply(rule_tac x=xb in exI) by simp
- thus ?thesis using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
-qed
-
-lemma convex_translation: 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
- thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
-
-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 = op + a ` op *\<^sub>R c ` s" by auto
- thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
-
-lemma convex_linear_image:
- assumes c:"convex s" and l:"bounded_linear f"
- shows "convex(f ` s)"
-proof(auto simp add: convex_def)
- interpret f: bounded_linear f by fact
- fix x y assume xy:"x \<in> s" "y \<in> s"
- fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
- show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
- apply(rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in bexI)
- unfolding f.add f.scaleR
- using c[unfolded convex_def] xy uv by auto
-qed
-
-subsection {* Balls, being convex, are connected. *}
-
lemma convex_ball:
fixes x :: "'a::real_normed_vector"
shows "convex (ball x e)"
@@ -739,7 +465,7 @@
fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
- thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using real_convex_bound_lt[OF yz uv] by auto
+ thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
qed
lemma convex_cball:
@@ -750,7 +476,7 @@
fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
- thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using real_convex_bound_le[OF yz uv] by auto
+ thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto
qed
lemma connected_ball:
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Tue Apr 20 17:58:34 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Mon May 03 14:35:10 2010 +0200
@@ -8,7 +8,7 @@
imports
Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
Finite_Cartesian_Product Infinite_Set Numeral_Type
- Inner_Product L2_Norm
+ Inner_Product L2_Norm Convex
uses "positivstellensatz.ML" ("normarith.ML")
begin
@@ -1411,40 +1411,6 @@
done
-lemma real_convex_bound_lt:
- assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
- and uv: "u + v = 1"
- shows "u * x + v * y < a"
-proof-
- have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
- have "a = a * (u + v)" unfolding uv by simp
- hence th: "u * a + v * a = a" by (simp add: field_simps)
- from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_strict_left_mono)
- from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_strict_left_mono)
- from xa ya u v have "u * x + v * y < u * a + v * a"
- apply (cases "u = 0", simp_all add: uv')
- apply(rule mult_strict_left_mono)
- using uv' apply simp_all
-
- apply (rule add_less_le_mono)
- apply(rule mult_strict_left_mono)
- apply simp_all
- apply (rule mult_left_mono)
- apply simp_all
- done
- thus ?thesis unfolding th .
-qed
-
-lemma real_convex_bound_le:
- assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
- and uv: "u + v = 1"
- shows "u * x + v * y \<le> a"
-proof-
- from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
- also have "\<dots> \<le> (u + v) * a" by (simp add: field_simps)
- finally show ?thesis unfolding uv by simp
-qed
-
lemma infinite_enumerate: assumes fS: "infinite S"
shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
unfolding subseq_def
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Tue Apr 20 17:58:34 2010 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon May 03 14:35:10 2010 +0200
@@ -6,7 +6,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space Product_Vector Glbs
+imports SEQ Euclidean_Space Glbs
begin
subsection{* General notion of a topology *}
--- a/src/HOL/Probability/Information.thy Tue Apr 20 17:58:34 2010 +0200
+++ b/src/HOL/Probability/Information.thy Mon May 03 14:35:10 2010 +0200
@@ -1,5 +1,5 @@
theory Information
-imports Probability_Space Product_Measure
+imports Probability_Space Product_Measure "../Multivariate_Analysis/Convex"
begin
lemma pos_neg_part_abs:
@@ -699,338 +699,6 @@
qed
(* --------------- upper bound on entropy for a rv ------------------------- *)
-definition convex_set :: "real set \<Rightarrow> bool"
-where
- "convex_set C \<equiv> (\<forall> x y \<mu>. x \<in> C \<and> y \<in> C \<and> 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow> \<mu> * x + (1 - \<mu>) * y \<in> C)"
-
-lemma pos_is_convex:
- shows "convex_set {0 <..}"
-unfolding convex_set_def
-proof safe
- fix x y \<mu> :: real
- assume asms: "\<mu> \<ge> 0" "\<mu> \<le> 1" "x > 0" "y > 0"
- { assume "\<mu> = 0"
- hence "\<mu> * x + (1 - \<mu>) * y = y" by simp
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms by simp }
- moreover
- { assume "\<mu> = 1"
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms by simp }
- moreover
- { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
- hence "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
- hence "\<mu> * x + (1 - \<mu>) * y > 0" using asms
- apply (subst add_nonneg_pos[of "\<mu> * x" "(1 - \<mu>) * y"])
- using real_mult_order by auto fastsimp }
- ultimately show "\<mu> * x + (1 - \<mu>) * y > 0" using assms by blast
-qed
-
-definition convex_fun :: "(real \<Rightarrow> real) \<Rightarrow> real set \<Rightarrow> bool"
-where
- "convex_fun f C \<equiv> (\<forall> x y \<mu>. convex_set C \<and> (x \<in> C \<and> y \<in> C \<and> 0 \<le> \<mu> \<and> \<mu> \<le> 1
- \<longrightarrow> f (\<mu> * x + (1 - \<mu>) * y) \<le> \<mu> * f x + (1 - \<mu>) * f y))"
-
-lemma pos_convex_function:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_set C"
- assumes leq: "\<And> x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
- shows "convex_fun f C"
-unfolding convex_fun_def
-using assms
-proof safe
- fix x y \<mu> :: real
- let ?x = "\<mu> * x + (1 - \<mu>) * y"
- assume asm: "convex_set C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- hence "1 - \<mu> \<ge> 0" by auto
- hence xpos: "?x \<in> C" using asm unfolding convex_set_def by auto
- 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_mono1[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`]
- mult_mono1[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
- hence "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
- by (auto simp add:field_simps)
- thus "\<mu> * f x + (1 - \<mu>) * f y \<ge> f ?x" by simp
-qed
-
-lemma atMostAtLeast_subset_convex:
- assumes "convex_set C"
- assumes "x \<in> C" "y \<in> C" "x < y"
- shows "{x .. y} \<subseteq> C"
-proof safe
- fix z assume zasm: "z \<in> {x .. 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)
- hence comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
- using assms[unfolded convex_set_def] by blast
- have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
- by (auto simp add:field_simps)
- also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
- using assms unfolding add_divide_distrib by (auto simp:field_simps)
- also have "\<dots> = z"
- using assms by (auto simp:field_simps)
- finally have "z \<in> C"
- using comb by auto } note less = this
- show "z \<in> C" using zasm less assms
- unfolding atLeastAtMost_iff le_less by auto
-qed
-
-lemma f''_imp_f':
- fixes f :: "real \<Rightarrow> real"
- assumes "convex_set C"
- assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- assumes "x \<in> C" "y \<in> C"
- shows "f' x * (y - x) \<le> f y - f x"
-using assms
-proof -
- { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "y > x"
- hence ge: "y - x > 0" "y - x \<ge> 0" by auto
- from asm 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 `convex_set C` `x \<in> C` `y \<in> C` `x < y`],
- THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
- by auto
- hence "z1 \<in> C" using atMostAtLeast_subset_convex
- `convex_set C` `x \<in> C` `y \<in> C` `x < y` by fastsimp
- 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 `convex_set C` `x \<in> C` `z1 \<in> C` `x < z1`],
- THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
- using subsetD[OF atMostAtLeast_subset_convex[OF `convex_set C` `z1 \<in> C` `y \<in> C` `z1 < y`],
- THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
- using asm 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 asm atMostAtLeast_subset_convex
- `convex_set C` `x \<in> C` `z1 \<in> C` `x < z1` by fastsimp
- hence B': "f'' z3 \<ge> 0" using assms by auto
- from A' B' have "(y - z1) * f'' z3 \<ge> 0" using mult_nonneg_nonneg 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)"
- unfolding diff_def using real_add_mult_distrib by auto
- hence "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
- hence 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 asm 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 asm atMostAtLeast_subset_convex
- `convex_set C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastsimp
- hence B: "f'' z2 \<ge> 0" using assms by auto
- from A B have "(z1 - x) * f'' z2 \<ge> 0" using mult_nonneg_nonneg by auto
- from cool this 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)"
- unfolding diff_def using real_add_mult_distrib by auto
- hence "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
- hence "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 assume "x \<in> C" "y \<in> C" "x \<noteq> y"
- hence"f y - f x \<ge> f' x * (y - x)"
- unfolding neq_iff apply safe
- using less_imp by auto } note neq_imp = this
- moreover
- { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "x = y"
- hence "f y - f x \<ge> f' x * (y - x)" by auto }
- ultimately show ?thesis using assms by blast
-qed
-
-lemma f''_ge0_imp_convex:
- fixes f :: "real \<Rightarrow> real"
- assumes conv: "convex_set C"
- assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- shows "convex_fun f C"
-using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function by fastsimp
-
-lemma minus_log_convex:
- fixes b :: real
- assumes "b > 1"
- shows "convex_fun (\<lambda> x. - log b x) {0 <..}"
-proof -
- have "\<And> z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" using DERIV_log by auto
- hence f': "\<And> z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
- using DERIV_minus by auto
- 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
- hence 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:real_mult_assoc)
- 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 simp add:divide_pos_pos[of 1] real_mult_order)
- from f''_ge0_imp_convex[OF pos_is_convex,
- unfolded greaterThan_iff, OF f' f''0 f''_ge0]
- show ?thesis by auto
-qed
-
-lemma setsum_nonneg_0:
- fixes f :: "'a \<Rightarrow> real"
- assumes "finite s"
- assumes "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
- assumes "(\<Sum> i \<in> s. f i) = 0"
- assumes "i \<in> s"
- shows "f i = 0"
-proof -
- { assume asm: "f i > 0"
- from assms have "\<forall> j \<in> s - {i}. f j \<ge> 0" by auto
- from setsum_nonneg[of "s - {i}" f, OF this]
- have "(\<Sum> j \<in> s - {i}. f j) \<ge> 0" by simp
- hence "(\<Sum> j \<in> s - {i}. f j) + f i > 0" using asm by auto
- from this setsum.remove[of s i f, OF `finite s` `i \<in> s`]
- have "(\<Sum> j \<in> s. f j) > 0" by auto
- hence "False" using assms by auto }
- thus ?thesis using assms by fastsimp
-qed
-
-lemma setsum_nonneg_leq_1:
- fixes f :: "'a \<Rightarrow> real"
- assumes "finite s"
- assumes "\<And> i. i \<in> s \<Longrightarrow> f i \<ge> 0"
- assumes "(\<Sum> i \<in> s. f i) = 1"
- assumes "i \<in> s"
- shows "f i \<le> 1"
-proof -
- { assume asm: "f i > 1"
- from assms have "\<forall> j \<in> s - {i}. f j \<ge> 0" by auto
- from setsum_nonneg[of "s - {i}" f, OF this]
- have "(\<Sum> j \<in> s - {i}. f j) \<ge> 0" by simp
- hence "(\<Sum> j \<in> s - {i}. f j) + f i > 1" using asm by auto
- from this setsum.remove[of s i f, OF `finite s` `i \<in> s`]
- have "(\<Sum> j \<in> s. f j) > 1" by auto
- hence "False" using assms by auto }
- thus ?thesis using assms by fastsimp
-qed
-
-lemma convex_set_setsum:
- assumes "finite s" "s \<noteq> {}"
- assumes "convex_set C"
- assumes "(\<Sum> i \<in> s. a i) = 1"
- assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
- assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> C"
- shows "(\<Sum> j \<in> s. a j * y j) \<in> C"
-using assms
-proof (induct s arbitrary:a rule:finite_ne_induct)
- case (singleton i) note asms = this
- hence "a i = 1" by auto
- thus ?case using asms by auto
-next
- case (insert i s) note asms = this
- { assume "a i = 1"
- hence "(\<Sum> j \<in> s. a j) = 0"
- using asms by auto
- hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
- using setsum_nonneg_0 asms by fastsimp
- hence ?case using asms by auto }
- moreover
- { assume asm: "a i \<noteq> 1"
- from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
- have fis: "finite (insert i s)" using asms by auto
- hence ai1: "a i \<le> 1" using setsum_nonneg_leq_1[of "insert i s" a] asms by simp
- hence "a i < 1" using asm by auto
- hence i0: "1 - a i > 0" by auto
- let "?a j" = "a j / (1 - a i)"
- { fix j assume "j \<in> s"
- hence "?a j \<ge> 0"
- using i0 asms divide_nonneg_pos
- by fastsimp } note a_nonneg = this
- have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
- hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
- hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
- hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
- from this asms
- have "(\<Sum>j\<in>s. ?a j * y j) \<in> C" using a_nonneg by fastsimp
- hence "a i * y i + (1 - a i) * (\<Sum> j \<in> s. ?a j * y j) \<in> C"
- using asms[unfolded convex_set_def, rule_format] yai ai1 by auto
- hence "a i * y i + (\<Sum> j \<in> s. (1 - a i) * (?a j * y j)) \<in> C"
- using mult_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j * y j" s] by auto
- hence "a i * y i + (\<Sum> j \<in> s. a j * y j) \<in> C" using i0 by auto
- hence ?case using setsum.insert asms by auto }
- ultimately show ?case by auto
-qed
-
-lemma convex_fun_setsum:
- fixes a :: "'a \<Rightarrow> real"
- assumes "finite s" "s \<noteq> {}"
- assumes "convex_fun f C"
- assumes "(\<Sum> i \<in> s. a i) = 1"
- assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
- assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> C"
- shows "f (\<Sum> i \<in> s. a i * 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)
- hence ai: "a i = 1" by auto
- thus ?case by auto
-next
- case (insert i s) note asms = this
- hence "convex_fun f C" by simp
- from this[unfolded convex_fun_def, rule_format]
- have conv: "\<And> x y \<mu>. \<lbrakk>x \<in> C; y \<in> C; 0 \<le> \<mu>; \<mu> \<le> 1\<rbrakk>
- \<Longrightarrow> f (\<mu> * x + (1 - \<mu>) * y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by simp
- { assume "a i = 1"
- hence "(\<Sum> j \<in> s. a j) = 0"
- using asms by auto
- hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
- using setsum_nonneg_0 asms by fastsimp
- hence ?case using asms by auto }
- moreover
- { assume asm: "a i \<noteq> 1"
- from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
- have fis: "finite (insert i s)" using asms by auto
- hence ai1: "a i \<le> 1" using setsum_nonneg_leq_1[of "insert i s" a] asms by simp
- hence "a i < 1" using asm by auto
- hence i0: "1 - a i > 0" by auto
- let "?a j" = "a j / (1 - a i)"
- { fix j assume "j \<in> s"
- hence "?a j \<ge> 0"
- using i0 asms divide_nonneg_pos
- by fastsimp } note a_nonneg = this
- have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
- hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastsimp
- hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
- hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding divide.setsum by simp
- have "convex_set C" using asms unfolding convex_fun_def by auto
- hence asum: "(\<Sum> j \<in> s. ?a j * y j) \<in> C"
- using asms convex_set_setsum[OF `finite s` `s \<noteq> {}`
- `convex_set C` a1 a_nonneg] by auto
- have asum_le: "f (\<Sum> j \<in> s. ?a j * y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
- using a_nonneg a1 asms by blast
- have "f (\<Sum> j \<in> insert i s. a j * y j) = f ((\<Sum> j \<in> s. a j * y j) + a i * y i)"
- using setsum.insert[of s i "\<lambda> j. a j * y j", OF `finite s` `i \<notin> s`] asms
- by (auto simp only:add_commute)
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. a j * y j) / (1 - a i) + a i * y i)"
- using i0 by auto
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. a j * y j / (1 - a i)) + a i * y i)"
- unfolding divide.setsum[of "\<lambda> j. a j * y j" s "1 - a i", symmetric] by auto
- also have "\<dots> = f ((1 - a i) * (\<Sum> j \<in> s. ?a j * y j) + a i * y i)" by auto
- also have "\<dots> \<le> (1 - a i) * f ((\<Sum> j \<in> s. ?a j * y j)) + a i * f (y i)"
- using conv[of "y i" "(\<Sum> j \<in> s. ?a j * y j)" "a i", OF yai(1) asum yai(2) ai1]
- by (auto simp only: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 mult_right.setsum[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 asms by auto
- finally have "f (\<Sum> j \<in> insert i s. a j * y j) \<le> (\<Sum> j \<in> insert i s. a j * f (y j))"
- by simp }
- ultimately show ?case by auto
-qed
-
lemma log_setsum:
assumes "finite s" "s \<noteq> {}"
assumes "b > 1"
@@ -1039,10 +707,10 @@
assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> {0 <..}"
shows "log b (\<Sum> i \<in> s. a i * y i) \<ge> (\<Sum> i \<in> s. a i * log b (y i))"
proof -
- have "convex_fun (\<lambda> x. - log b x) {0 <..}"
+ have "convex_on {0 <..} (\<lambda> x. - log b x)"
by (rule minus_log_convex[OF `b > 1`])
hence "- log b (\<Sum> i \<in> s. a i * y i) \<le> (\<Sum> i \<in> s. a i * - log b (y i))"
- using convex_fun_setsum assms by blast
+ using convex_on_setsum[of _ _ "\<lambda> x. - log b x"] assms pos_is_convex by fastsimp
thus ?thesis by (auto simp add:setsum_negf le_imp_neg_le)
qed