--- a/src/HOL/Library/Convex.thy Fri Sep 30 11:35:39 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,999 +0,0 @@
-(* Title: HOL/Library/Convex.thy
- Author: Armin Heller, TU Muenchen
- Author: Johannes Hoelzl, TU Muenchen
-*)
-
-section \<open>Convexity in real vector spaces\<close>
-
-theory Convex
- imports Product_Vector
-begin
-
-subsection \<open>Convexity\<close>
-
-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 convexI:
- assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
- shows "convex s"
- using assms unfolding convex_def by fast
-
-lemma convexD:
- assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
- shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
- using assms unfolding convex_def by fast
-
-lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
- (is "_ \<longleftrightarrow> ?alt")
-proof
- show "convex s" if alt: ?alt
- proof -
- {
- fix x y and u v :: real
- assume mem: "x \<in> s" "y \<in> s"
- assume "0 \<le> u" "0 \<le> v"
- moreover
- assume "u + v = 1"
- then have "u = 1 - v" by auto
- ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
- using alt [rule_format, OF mem] by auto
- }
- then show ?thesis
- unfolding convex_def by auto
- qed
- show ?alt if "convex s"
- using that by (auto simp: convex_def)
-qed
-
-lemma convexD_alt:
- assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
- shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
- using assms unfolding convex_alt by auto
-
-lemma mem_convex_alt:
- assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
- shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
- apply (rule convexD)
- using assms
- apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
- done
-
-lemma convex_empty[intro,simp]: "convex {}"
- unfolding convex_def by simp
-
-lemma convex_singleton[intro,simp]: "convex {a}"
- unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
-
-lemma convex_UNIV[intro,simp]: "convex UNIV"
- unfolding convex_def by auto
-
-lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
- unfolding convex_def by auto
-
-lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
- unfolding convex_def by auto
-
-lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
- unfolding convex_def by auto
-
-lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
- unfolding convex_def by auto
-
-lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
- unfolding convex_def
- by (auto simp: inner_add intro!: convex_bound_le)
-
-lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
-proof -
- have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
- by auto
- show ?thesis
- unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
-qed
-
-lemma convex_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 [iff]:
- fixes a b :: "real"
- shows "convex {a..}" and "convex {..b}"
- and "convex {a<..}" and "convex {..<b}"
- and "convex {a..b}" and "convex {a<..b}"
- and "convex {a..<b}" and "convex {a<..<b}"
-proof -
- have "{a..} = {x. a \<le> inner 1 x}"
- by auto
- then show 1: "convex {a..}"
- by (simp only: convex_halfspace_ge)
- have "{..b} = {x. inner 1 x \<le> b}"
- by auto
- then show 2: "convex {..b}"
- by (simp only: convex_halfspace_le)
- have "{a<..} = {x. a < inner 1 x}"
- by auto
- then show 3: "convex {a<..}"
- by (simp only: convex_halfspace_gt)
- have "{..<b} = {x. inner 1 x < b}"
- by auto
- then show 4: "convex {..<b}"
- by (simp only: convex_halfspace_lt)
- have "{a..b} = {a..} \<inter> {..b}"
- by auto
- then show "convex {a..b}"
- by (simp only: convex_Int 1 2)
- have "{a<..b} = {a<..} \<inter> {..b}"
- by auto
- then show "convex {a<..b}"
- by (simp only: convex_Int 3 2)
- have "{a..<b} = {a..} \<inter> {..<b}"
- by auto
- then show "convex {a..<b}"
- by (simp only: convex_Int 1 4)
- have "{a<..<b} = {a<..} \<inter> {..<b}"
- by auto
- then show "convex {a<..<b}"
- by (simp only: convex_Int 3 4)
-qed
-
-lemma convex_Reals: "convex \<real>"
- by (simp add: convex_def scaleR_conv_of_real)
-
-
-subsection \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
-
-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(1,3,4,5)
-proof (induct arbitrary: a set: finite)
- case empty
- then show ?case by simp
-next
- case (insert i s) note IH = this(3)
- have "a i + 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)
- have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
- proof (cases "setsum a s = 0")
- case True
- with \<open>a i + setsum a s = 1\<close> have "a i = 1"
- by simp
- from setsum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
- by simp
- show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
- by simp
- next
- case False
- with \<open>0 \<le> setsum a s\<close> 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 \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
- by (simp add: IH setsum_divide_distrib [symmetric])
- from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
- and \<open>0 \<le> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
- 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 False)
- qed
- then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
- by simp
-qed
-
-lemma convex:
- "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (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"
- with convex_setsum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
- by auto
-next
- assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
- \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
- {
- fix \<mu> :: real
- fix x y :: 'a
- assume xy: "x \<in> s" "y \<in> s"
- assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
- let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
- let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
- have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
- by auto
- then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
- by simp
- then have "setsum ?u {1 .. 2} = 1"
- using setsum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
- by auto
- with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
- using mu xy by auto
- have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
- using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
- from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
- have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
- by auto
- then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
- using s by (auto simp: add.commute)
- }
- then show "convex s"
- unfolding convex_alt by auto
-qed
-
-
-lemma convex_explicit:
- fixes s :: "'a::real_vector set"
- shows "convex s \<longleftrightarrow>
- (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
-proof safe
- fix t
- fix u :: "'a \<Rightarrow> real"
- assume "convex s"
- 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 *: "\<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 **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
- show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
- proof (cases "x = y")
- case False
- then show ?thesis
- using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
- by auto
- next
- case True
- then show ?thesis
- using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
- by (auto simp: field_simps real_vector.scale_left_diff_distrib)
- qed
- qed
-qed
-
-lemma convex_finite:
- assumes "finite s"
- shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
- unfolding convex_explicit
- apply safe
- subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
- subgoal for t u
- proof -
- have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
- by simp
- 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"
- assume *: "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
- assume "t \<subseteq> s"
- then have "s \<inter> t = t" by auto
- with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
- qed
- done
-
-
-subsection \<open>Functions that are convex on a set\<close>
-
-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_onI [intro?]:
- assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
- unfolding convex_on_def
-proof clarify
- fix x y
- fix u v :: real
- assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
- from A(5) have [simp]: "v = 1 - u"
- by (simp add: algebra_simps)
- from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
- using assms[of u y x]
- by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
-qed
-
-lemma convex_on_linorderI [intro?]:
- fixes A :: "('a::{linorder,real_vector}) set"
- assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
-proof
- fix x y
- fix t :: real
- assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
- with assms [of t x y] assms [of "1 - t" y x]
- show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
-qed
-
-lemma convex_onD:
- assumes "convex_on A f"
- shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms by (auto simp: convex_on_def)
-
-lemma convex_onD_Icc:
- assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
- shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
-
-lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
- unfolding convex_on_def by auto
-
-lemma convex_on_add [intro]:
- assumes "convex_on s f"
- and "convex_on s g"
- shows "convex_on s (\<lambda>x. f x + g x)"
-proof -
- {
- fix x y
- assume "x \<in> s" "y \<in> s"
- moreover
- fix u v :: real
- assume "0 \<le> u" "0 \<le> v" "u + v = 1"
- ultimately
- have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
- using assms unfolding convex_on_def by (auto simp: add_mono)
- then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
- by (simp add: field_simps)
- }
- then show ?thesis
- unfolding convex_on_def by auto
-qed
-
-lemma convex_on_cmul [intro]:
- fixes c :: real
- assumes "0 \<le> c"
- and "convex_on s f"
- shows "convex_on s (\<lambda>x. c * f x)"
-proof -
- have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
- for u c fx v fy :: real
- by (simp add: field_simps)
- show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
- unfolding convex_on_def and * by auto
-qed
-
-lemma convex_lower:
- assumes "convex_on s f"
- and "x \<in> s"
- and "y \<in> s"
- and "0 \<le> u"
- and "0 \<le> v"
- and "u + v = 1"
- shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
-proof -
- let ?m = "max (f x) (f y)"
- have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
- using assms(4,5) by (auto simp: mult_left_mono add_mono)
- also have "\<dots> = max (f x) (f y)"
- using assms(6) by (simp add: distrib_right [symmetric])
- finally show ?thesis
- using assms unfolding convex_on_def by fastforce
-qed
-
-lemma convex_on_dist [intro]:
- fixes s :: "'a::real_normed_vector set"
- shows "convex_on s (\<lambda>x. dist a x)"
-proof (auto simp: convex_on_def dist_norm)
- fix x y
- assume "x \<in> s" "y \<in> s"
- fix u v :: real
- assume "0 \<le> u"
- assume "0 \<le> v"
- assume "u + v = 1"
- have "a = u *\<^sub>R a + v *\<^sub>R a"
- unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
- then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
- by (auto simp: algebra_simps)
- show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
- unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
- using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
-qed
-
-
-subsection \<open>Arithmetic operations on sets preserve convexity\<close>
-
-lemma convex_linear_image:
- assumes "linear f"
- and "convex s"
- shows "convex (f ` s)"
-proof -
- interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f ` s)"
- by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
-qed
-
-lemma convex_linear_vimage:
- assumes "linear f"
- and "convex s"
- shows "convex (f -` s)"
-proof -
- interpret f: linear f by fact
- from \<open>convex s\<close> show "convex (f -` s)"
- by (simp add: convex_def f.add f.scaleR)
-qed
-
-lemma convex_scaling:
- assumes "convex s"
- shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
- have "linear (\<lambda>x. c *\<^sub>R x)"
- by (simp add: linearI scaleR_add_right)
- then show ?thesis
- using \<open>convex s\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_scaled:
- assumes "convex s"
- shows "convex ((\<lambda>x. x *\<^sub>R c) ` s)"
-proof -
- have "linear (\<lambda>x. x *\<^sub>R c)"
- by (simp add: linearI scaleR_add_left)
- then show ?thesis
- using \<open>convex s\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_negations:
- assumes "convex s"
- shows "convex ((\<lambda>x. - x) ` s)"
-proof -
- have "linear (\<lambda>x. - x)"
- by (simp add: linearI)
- then show ?thesis
- using \<open>convex s\<close> by (rule convex_linear_image)
-qed
-
-lemma convex_sums:
- assumes "convex s"
- and "convex t"
- shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
-proof -
- have "linear (\<lambda>(x, y). x + y)"
- by (auto intro: linearI simp: scaleR_add_right)
- with assms have "convex ((\<lambda>(x, y). x + y) ` (s \<times> t))"
- by (intro convex_linear_image convex_Times)
- also have "((\<lambda>(x, y). x + y) ` (s \<times> t)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
- by auto
- finally show ?thesis .
-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}"
- by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
- then show ?thesis
- using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
-qed
-
-lemma convex_translation:
- 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
- then show ?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
- then show ?thesis
- using convex_translation[OF convex_scaling[OF assms], of a c] by auto
-qed
-
-lemma pos_is_convex: "convex {0 :: real <..}"
- unfolding convex_alt
-proof safe
- fix y x \<mu> :: real
- assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- {
- 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 * by simp
- }
- moreover
- {
- assume "\<mu> = 1"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
- using * by simp
- }
- moreover
- {
- assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
- then have "\<mu> > 0" "(1 - \<mu>) > 0"
- using * by auto
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
- using * by (auto simp: add_pos_pos)
- }
- ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
- by fastforce
-qed
-
-lemma convex_on_setsum:
- fixes a :: "'a \<Rightarrow> real"
- and y :: "'a \<Rightarrow> 'b::real_vector"
- and f :: "'b \<Rightarrow> real"
- assumes "finite s" "s \<noteq> {}"
- and "convex_on C f"
- and "convex C"
- and "(\<Sum> i \<in> s. a i) = 1"
- and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
- and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
- shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
- using assms
-proof (induct s arbitrary: a rule: finite_ne_induct)
- case (singleton i)
- then have ai: "a i = 1"
- by auto
- then show ?case
- by auto
-next
- case (insert i s)
- then have "convex_on C f"
- by simp
- from this[unfolded convex_on_def, rule_format]
- have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by simp
- show ?case
- proof (cases "a i = 1")
- case True
- then have "(\<Sum> j \<in> s. a j) = 0"
- using insert by auto
- then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
- using insert by (fastforce simp: setsum_nonneg_eq_0_iff)
- then show ?thesis
- using insert by auto
- next
- case False
- from insert have yai: "y i \<in> C" "a i \<ge> 0"
- by auto
- have fis: "finite (insert i s)"
- using insert by auto
- then have ai1: "a i \<le> 1"
- using setsum_nonneg_leq_bound[of "insert i s" a] insert by simp
- then have "a i < 1"
- using False by auto
- then have i0: "1 - a i > 0"
- by auto
- let ?a = "\<lambda>j. a j / (1 - a i)"
- have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
- using i0 insert that by fastforce
- have "(\<Sum> j \<in> insert i s. a j) = 1"
- using insert by auto
- then have "(\<Sum> j \<in> s. a j) = 1 - a i"
- using setsum.insert insert by fastforce
- then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
- using i0 by auto
- then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
- unfolding setsum_divide_distrib by simp
- have "convex C" using insert by auto
- then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
- using insert convex_setsum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
- have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
- using a_nonneg a1 insert by blast
- have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
- using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
- by (auto simp only: add.commute)
- also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
- using i0 by auto
- also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
- using scaleR_right.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: divide_inverse)
- also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
- using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
- by (auto simp: add.commute)
- also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
- using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
- OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
- by simp
- also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
- unfolding setsum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
- using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
- using i0 by auto
- also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
- using insert by auto
- finally show ?thesis
- by simp
- qed
-qed
-
-lemma convex_on_alt:
- fixes C :: "'a::real_vector set"
- assumes "convex C"
- shows "convex_on C f \<longleftrightarrow>
- (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
-proof safe
- fix x y
- fix \<mu> :: real
- assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
- from this[unfolded convex_on_def, rule_format]
- have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
- by auto
- from this [of "\<mu>" "1 - \<mu>", simplified] *
- show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- by auto
-next
- assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
- f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- {
- fix x y
- fix u v :: real
- assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
- then have[simp]: "1 - u = v" by auto
- from *[rule_format, of x y u]
- have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
- using ** by auto
- }
- then show "convex_on C f"
- unfolding convex_on_def by auto
-qed
-
-lemma convex_on_diff:
- fixes f :: "real \<Rightarrow> real"
- assumes f: "convex_on I f"
- and I: "x \<in> I" "y \<in> I"
- and t: "x < t" "t < y"
- shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
-proof -
- define a where "a \<equiv> (t - y) / (x - y)"
- with t have "0 \<le> a" "0 \<le> 1 - a"
- by (auto simp: field_simps)
- with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
- by (auto simp: convex_on_def)
- have "a * x + (1 - a) * y = a * (x - y) + y"
- by (simp add: field_simps)
- also have "\<dots> = t"
- unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
- finally have "f t \<le> a * f x + (1 - a) * f y"
- using cvx by simp
- also have "\<dots> = a * (f x - f y) + f y"
- by (simp add: field_simps)
- finally have "f t - f y \<le> a * (f x - f y)"
- by simp
- with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
- by (simp add: le_divide_eq divide_le_eq field_simps a_def)
- with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
- by (simp add: le_divide_eq divide_le_eq field_simps)
-qed
-
-lemma pos_convex_function:
- fixes f :: "real \<Rightarrow> real"
- assumes "convex C"
- and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
- shows "convex_on C f"
- unfolding convex_on_alt[OF assms(1)]
- using assms
-proof safe
- fix x y \<mu> :: real
- let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
- assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
- then have "1 - \<mu> \<ge> 0" by auto
- then have xpos: "?x \<in> C"
- using * unfolding convex_alt by fastforce
- have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
- \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
- using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
- mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
- by auto
- then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
- by (auto simp: field_simps)
- then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
- using convex_on_alt by auto
-qed
-
-lemma atMostAtLeast_subset_convex:
- fixes C :: "real set"
- assumes "convex C"
- and "x \<in> C" "y \<in> C" "x < y"
- shows "{x .. y} \<subseteq> C"
-proof safe
- fix z assume z: "z \<in> {x .. y}"
- have less: "z \<in> C" if *: "x < z" "z < y"
- proof -
- let ?\<mu> = "(y - z) / (y - x)"
- have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
- using assms * by (auto simp: field_simps)
- then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
- using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
- by (simp add: algebra_simps)
- have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
- by (auto simp: field_simps)
- also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
- using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
- also have "\<dots> = z"
- using assms by (auto simp: field_simps)
- finally show ?thesis
- using comb by auto
- qed
- show "z \<in> C"
- using z less assms by (auto simp: le_less)
-qed
-
-lemma f''_imp_f':
- fixes f :: "real \<Rightarrow> real"
- assumes "convex C"
- and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- and x: "x \<in> C"
- and y: "y \<in> C"
- shows "f' x * (y - x) \<le> f y - f x"
- using assms
-proof -
- have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
- if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
- proof -
- from * have ge: "y - x > 0" "y - x \<ge> 0"
- by auto
- from * have le: "x - y < 0" "x - y \<le> 0"
- by auto
- then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
- THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
- by auto
- then have "z1 \<in> C"
- using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
- by fastforce
- from z1 have z1': "f x - f y = (x - y) * f' z1"
- by (simp add: field_simps)
- obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
- THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
- using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
- THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
- by auto
- have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
- using * z1' by auto
- also have "\<dots> = (y - z1) * f'' z3"
- using z3 by auto
- finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
- by simp
- have A': "y - z1 \<ge> 0"
- using z1 by auto
- have "z3 \<in> C"
- using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
- by fastforce
- then have B': "f'' z3 \<ge> 0"
- using assms by auto
- from A' B' have "(y - z1) * f'' z3 \<ge> 0"
- by auto
- from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
- by auto
- from mult_right_mono_neg[OF this le(2)]
- have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
- by (simp add: algebra_simps)
- then have "f' y * (x - y) - (f x - f y) \<le> 0"
- using le by auto
- then have res: "f' y * (x - y) \<le> f x - f y"
- by auto
- have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
- using * z1 by auto
- also have "\<dots> = (z1 - x) * f'' z2"
- using z2 by auto
- finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
- by simp
- have A: "z1 - x \<ge> 0"
- using z1 by auto
- have "z2 \<in> C"
- using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
- by fastforce
- then have B: "f'' z2 \<ge> 0"
- using assms by auto
- from A B have "(z1 - x) * f'' z2 \<ge> 0"
- by auto
- with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
- by auto
- from mult_right_mono[OF this ge(2)]
- have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
- by (simp add: algebra_simps)
- then have "f y - f x - f' x * (y - x) \<ge> 0"
- using ge by auto
- then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
- using res by auto
- qed
- show ?thesis
- proof (cases "x = y")
- case True
- with x y show ?thesis by auto
- next
- case False
- with less_imp x y show ?thesis
- by (auto simp: neq_iff)
- qed
-qed
-
-lemma f''_ge0_imp_convex:
- fixes f :: "real \<Rightarrow> real"
- assumes conv: "convex C"
- and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
- and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
- and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- shows "convex_on C f"
- using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
- by fastforce
-
-lemma minus_log_convex:
- fixes b :: real
- assumes "b > 1"
- shows "convex_on {0 <..} (\<lambda> x. - log b x)"
-proof -
- have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
- using DERIV_log by auto
- then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
- by (auto simp: DERIV_minus)
- have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
- using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
- from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
- have "\<And>z::real. z > 0 \<Longrightarrow>
- DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
- by auto
- then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
- DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
- unfolding inverse_eq_divide by (auto simp: mult.assoc)
- have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
- using \<open>b > 1\<close> by (auto intro!: less_imp_le)
- from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
- show ?thesis
- by auto
-qed
-
-
-subsection \<open>Convexity of real functions\<close>
-
-lemma convex_on_realI:
- assumes "connected A"
- and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
- and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
- shows "convex_on A f"
-proof (rule convex_on_linorderI)
- fix t x y :: real
- assume t: "t > 0" "t < 1"
- assume xy: "x \<in> A" "y \<in> A" "x < y"
- define z where "z = (1 - t) * x + t * y"
- with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
- using connected_contains_Icc by blast
-
- from xy t have xz: "z > x"
- by (simp add: z_def algebra_simps)
- have "y - z = (1 - t) * (y - x)"
- by (simp add: z_def algebra_simps)
- also from xy t have "\<dots> > 0"
- by (intro mult_pos_pos) simp_all
- finally have yz: "z < y"
- by simp
-
- from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
- by (intro MVT2) (auto intro!: assms(2))
- then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
- by auto
- from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
- by (intro MVT2) (auto intro!: assms(2))
- then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
- by auto
-
- from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
- also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
- by auto
- with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
- by (intro assms(3)) auto
- also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
- finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
- using xz yz by (simp add: field_simps)
- also have "z - x = t * (y - x)"
- by (simp add: z_def algebra_simps)
- also have "y - z = (1 - t) * (y - x)"
- by (simp add: z_def algebra_simps)
- finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
- using xy by simp
- then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
- by (simp add: z_def algebra_simps)
-qed
-
-lemma convex_on_inverse:
- assumes "A \<subseteq> {0<..}"
- shows "convex_on A (inverse :: real \<Rightarrow> real)"
-proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
- fix u v :: real
- assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
- with assms show "-inverse (u^2) \<le> -inverse (v^2)"
- by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
-qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
-
-lemma convex_onD_Icc':
- assumes "convex_on {x..y} f" "c \<in> {x..y}"
- defines "d \<equiv> y - x"
- shows "f c \<le> (f y - f x) / d * (c - x) + f x"
-proof (cases x y rule: linorder_cases)
- case less
- then have d: "d > 0"
- by (simp add: d_def)
- from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
- have "f c = f (x + (c - x) * 1)"
- by simp
- also from less have "1 = ((y - x) / d)"
- by (simp add: d_def)
- also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
- by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
- using assms less by (intro convex_onD_Icc) simp_all
- also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
- by (simp add: field_simps)
- finally show ?thesis .
-qed (insert assms(2), simp_all)
-
-lemma convex_onD_Icc'':
- assumes "convex_on {x..y} f" "c \<in> {x..y}"
- defines "d \<equiv> y - x"
- shows "f c \<le> (f x - f y) / d * (y - c) + f y"
-proof (cases x y rule: linorder_cases)
- case less
- then have d: "d > 0"
- by (simp add: d_def)
- from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
- have "f c = f (y - (y - c) * 1)"
- by simp
- also from less have "1 = ((y - x) / d)"
- by (simp add: d_def)
- also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
- by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
- using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
- also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
- by (simp add: field_simps)
- finally show ?thesis .
-qed (insert assms(2), simp_all)
-
-end