author | hoelzl |
Thu, 17 Jan 2013 11:59:12 +0100 | |
changeset 50936 | b28f258ebc1a |
parent 49609 | 89e10ed7668b |
child 51642 | 400ec5ae7f8f |
permissions | -rw-r--r-- |
36648 | 1 |
(* Title: HOL/Library/Convex.thy |
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Author: Armin Heller, TU Muenchen |
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Author: Johannes Hoelzl, TU Muenchen |
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*) |
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header {* Convexity in real vector spaces *} |
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36623 | 8 |
theory Convex |
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imports Product_Vector |
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begin |
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subsection {* Convexity. *} |
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definition convex :: "'a::real_vector set \<Rightarrow> bool" |
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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)" |
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36623 | 16 |
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lemma convex_alt: |
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"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)" |
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(is "_ \<longleftrightarrow> ?alt") |
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proof |
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assume alt[rule_format]: ?alt |
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{ fix x y and u v :: real assume mem: "x \<in> s" "y \<in> s" |
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assume "0 \<le> u" "0 \<le> v" |
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moreover assume "u + v = 1" then have "u = 1 - v" by auto |
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ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s" using alt[OF mem] by auto } |
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then show "convex s" unfolding convex_def by auto |
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qed (auto simp: convex_def) |
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lemma mem_convex: |
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assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1" |
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shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s" |
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using assms unfolding convex_alt by auto |
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lemma convex_empty[intro]: "convex {}" |
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unfolding convex_def by simp |
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lemma convex_singleton[intro]: "convex {a}" |
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unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric]) |
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lemma convex_UNIV[intro]: "convex UNIV" |
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unfolding convex_def by auto |
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lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)" |
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unfolding convex_def by auto |
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lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)" |
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unfolding convex_def by auto |
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lemma convex_halfspace_le: "convex {x. inner a x \<le> b}" |
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unfolding convex_def |
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44142 | 51 |
by (auto simp: inner_add intro!: convex_bound_le) |
36623 | 52 |
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lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}" |
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proof - |
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49609 | 55 |
have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto |
36623 | 56 |
show ?thesis unfolding * using convex_halfspace_le[of "-a" "-b"] by auto |
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qed |
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lemma convex_hyperplane: "convex {x. inner a x = b}" |
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49609 | 60 |
proof - |
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have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto |
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36623 | 62 |
show ?thesis using convex_halfspace_le convex_halfspace_ge |
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by (auto intro!: convex_Int simp: *) |
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qed |
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lemma convex_halfspace_lt: "convex {x. inner a x < b}" |
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unfolding convex_def |
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by (auto simp: convex_bound_lt inner_add) |
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lemma convex_halfspace_gt: "convex {x. inner a x > b}" |
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using convex_halfspace_lt[of "-a" "-b"] by auto |
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lemma convex_real_interval: |
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fixes a b :: "real" |
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shows "convex {a..}" and "convex {..b}" |
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49609 | 76 |
and "convex {a<..}" and "convex {..<b}" |
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and "convex {a..b}" and "convex {a<..b}" |
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and "convex {a..<b}" and "convex {a<..<b}" |
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proof - |
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have "{a..} = {x. a \<le> inner 1 x}" by auto |
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then show 1: "convex {a..}" by (simp only: convex_halfspace_ge) |
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have "{..b} = {x. inner 1 x \<le> b}" by auto |
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then show 2: "convex {..b}" by (simp only: convex_halfspace_le) |
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have "{a<..} = {x. a < inner 1 x}" by auto |
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then show 3: "convex {a<..}" by (simp only: convex_halfspace_gt) |
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have "{..<b} = {x. inner 1 x < b}" by auto |
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then show 4: "convex {..<b}" by (simp only: convex_halfspace_lt) |
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have "{a..b} = {a..} \<inter> {..b}" by auto |
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then show "convex {a..b}" by (simp only: convex_Int 1 2) |
36623 | 90 |
have "{a<..b} = {a<..} \<inter> {..b}" by auto |
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then show "convex {a<..b}" by (simp only: convex_Int 3 2) |
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have "{a..<b} = {a..} \<inter> {..<b}" by auto |
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then show "convex {a..<b}" by (simp only: convex_Int 1 4) |
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have "{a<..<b} = {a<..} \<inter> {..<b}" by auto |
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then show "convex {a<..<b}" by (simp only: convex_Int 3 4) |
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qed |
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subsection {* Explicit expressions for convexity in terms of arbitrary sums. *} |
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lemma convex_setsum: |
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fixes C :: "'a::real_vector set" |
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assumes "finite s" and "convex C" and "(\<Sum> i \<in> s. a i) = 1" |
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assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C" |
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shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" |
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using assms |
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proof (induct s arbitrary:a rule: finite_induct) |
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case empty |
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then show ?case by auto |
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next |
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case (insert i s) note asms = this |
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{ assume "a i = 1" |
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then have "(\<Sum> j \<in> s. a j) = 0" |
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using asms by auto |
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then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0" |
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using setsum_nonneg_0[where 'b=real] asms by fastforce |
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then have ?case using asms by auto } |
36623 | 118 |
moreover |
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{ assume asm: "a i \<noteq> 1" |
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from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto |
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have fis: "finite (insert i s)" using asms by auto |
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49609 | 122 |
then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a 1] asms by simp |
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then have "a i < 1" using asm by auto |
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then have i0: "1 - a i > 0" by auto |
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let ?a = "\<lambda>j. a j / (1 - a i)" |
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36623 | 126 |
{ fix j assume "j \<in> s" |
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then have "?a j \<ge> 0" |
36623 | 128 |
using i0 asms divide_nonneg_pos |
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by fastforce |
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} note a_nonneg = this |
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36623 | 131 |
have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto |
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then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce |
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then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto |
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then have a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp |
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with asms have "(\<Sum>j\<in>s. ?a j *\<^sub>R y j) \<in> C" using a_nonneg by fastforce |
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then have "a i *\<^sub>R y i + (1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C" |
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36623 | 137 |
using asms[unfolded convex_def, rule_format] yai ai1 by auto |
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then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. (1 - a i) *\<^sub>R (?a j *\<^sub>R y j)) \<in> C" |
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using scaleR_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j *\<^sub>R y j" s] by auto |
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then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" using i0 by auto |
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then have ?case using setsum.insert asms by auto |
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} |
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36623 | 143 |
ultimately show ?case by auto |
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qed |
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lemma convex: |
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49609 | 147 |
"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) |
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\<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)" |
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36623 | 149 |
proof safe |
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fix k :: nat |
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fix u :: "nat \<Rightarrow> real" |
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fix x |
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36623 | 153 |
assume "convex s" |
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"\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s" |
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"setsum u {1..k} = 1" |
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from this convex_setsum[of "{1 .. k}" s] |
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show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" by auto |
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next |
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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 |
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\<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s" |
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49609 | 161 |
{ fix \<mu> :: real |
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fix x y :: 'a |
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assume xy: "x \<in> s" "y \<in> s" |
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assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1" |
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let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>" |
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let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y" |
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have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}" by auto |
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then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" by simp |
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then have "setsum ?u {1 .. 2} = 1" |
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36623 | 170 |
using setsum_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"] |
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by auto |
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49609 | 172 |
with asm[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s" |
36623 | 173 |
using mu xy by auto |
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have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y" |
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using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto |
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from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this] |
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have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" by auto |
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then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute) |
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} |
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then show "convex s" unfolding convex_alt by auto |
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qed |
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lemma convex_explicit: |
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fixes s :: "'a::real_vector set" |
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shows "convex s \<longleftrightarrow> |
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(\<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)" |
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proof safe |
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fix t |
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fix u :: "'a \<Rightarrow> real" |
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assume "convex s" "finite t" |
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"t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" |
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then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
36623 | 194 |
using convex_setsum[of t s u "\<lambda> x. x"] by auto |
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next |
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assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) |
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\<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
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show "convex s" |
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unfolding convex_alt |
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proof safe |
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fix x y |
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fix \<mu> :: real |
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assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1" |
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{ assume "x \<noteq> y" |
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then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" |
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using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] |
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asm by auto } |
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moreover |
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{ assume "x = y" |
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then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" |
36623 | 211 |
using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"] |
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asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) } |
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ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast |
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qed |
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qed |
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lemma convex_finite: |
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assumes "finite s" |
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36623 | 219 |
shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 |
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\<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)" |
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unfolding convex_explicit |
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proof safe |
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fix t u |
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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" |
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36623 | 225 |
and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)" |
49609 | 226 |
have *: "s \<inter> t = t" using as(2) by auto |
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have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)" |
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by simp |
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36623 | 229 |
show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
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using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as * |
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by (auto simp: assms setsum_cases if_distrib if_distrib_arg) |
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qed (erule_tac x=s in allE, erule_tac x=u in allE, auto) |
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definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" |
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where "convex_on s f \<longleftrightarrow> |
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(\<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)" |
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36623 | 237 |
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lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f" |
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unfolding convex_on_def by auto |
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lemma convex_add[intro]: |
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assumes "convex_on s f" "convex_on s g" |
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shows "convex_on s (\<lambda>x. f x + g x)" |
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49609 | 244 |
proof - |
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{ fix x y |
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assume "x\<in>s" "y\<in>s" |
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moreover |
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fix u v :: real |
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assume "0 \<le> u" "0 \<le> v" "u + v = 1" |
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ultimately |
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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)" |
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using assms unfolding convex_on_def by (auto simp add: add_mono) |
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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)" |
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by (simp add: field_simps) |
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} |
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then show ?thesis unfolding convex_on_def by auto |
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36623 | 257 |
qed |
258 |
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lemma convex_cmul[intro]: |
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260 |
assumes "0 \<le> (c::real)" "convex_on s f" |
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shows "convex_on s (\<lambda>x. c * f x)" |
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proof- |
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49609 | 263 |
have *: "\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" |
264 |
by (simp add: field_simps) |
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265 |
show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)] |
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unfolding convex_on_def and * by auto |
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36623 | 267 |
qed |
268 |
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269 |
lemma convex_lower: |
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270 |
assumes "convex_on s f" "x\<in>s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1" |
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271 |
shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)" |
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272 |
proof- |
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273 |
let ?m = "max (f x) (f y)" |
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274 |
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" |
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275 |
using assms(4,5) by (auto simp add: mult_left_mono add_mono) |
49609 | 276 |
also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[symmetric] by auto |
36623 | 277 |
finally show ?thesis |
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278 |
using assms unfolding convex_on_def by fastforce |
36623 | 279 |
qed |
280 |
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281 |
lemma convex_distance[intro]: |
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282 |
fixes s :: "'a::real_normed_vector set" |
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283 |
shows "convex_on s (\<lambda>x. dist a x)" |
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49609 | 284 |
proof (auto simp add: convex_on_def dist_norm) |
285 |
fix x y |
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286 |
assume "x\<in>s" "y\<in>s" |
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287 |
fix u v :: real |
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288 |
assume "0 \<le> u" "0 \<le> v" "u + v = 1" |
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289 |
have "a = u *\<^sub>R a + v *\<^sub>R a" |
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290 |
unfolding scaleR_left_distrib[symmetric] and `u+v=1` by simp |
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291 |
then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))" |
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36623 | 292 |
by (auto simp add: algebra_simps) |
293 |
show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)" |
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294 |
unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"] |
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295 |
using `0 \<le> u` `0 \<le> v` by auto |
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296 |
qed |
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49609 | 298 |
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36623 | 299 |
subsection {* Arithmetic operations on sets preserve convexity. *} |
49609 | 300 |
|
36623 | 301 |
lemma convex_scaling: |
302 |
assumes "convex s" |
|
303 |
shows"convex ((\<lambda>x. c *\<^sub>R x) ` s)" |
|
49609 | 304 |
using assms unfolding convex_def image_iff |
36623 | 305 |
proof safe |
49609 | 306 |
fix x xa y xb :: "'a::real_vector" |
307 |
fix u v :: real |
|
36623 | 308 |
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" |
309 |
"xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1" |
|
310 |
show "\<exists>x\<in>s. u *\<^sub>R c *\<^sub>R xa + v *\<^sub>R c *\<^sub>R xb = c *\<^sub>R x" |
|
311 |
using bexI[of _ "u *\<^sub>R xa +v *\<^sub>R xb"] asm by (auto simp add: algebra_simps) |
|
312 |
qed |
|
313 |
||
314 |
lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)" |
|
49609 | 315 |
using assms unfolding convex_def image_iff |
36623 | 316 |
proof safe |
49609 | 317 |
fix x xa y xb :: "'a::real_vector" |
318 |
fix u v :: real |
|
36623 | 319 |
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" |
320 |
"xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1" |
|
321 |
show "\<exists>x\<in>s. u *\<^sub>R - xa + v *\<^sub>R - xb = - x" |
|
322 |
using bexI[of _ "u *\<^sub>R xa +v *\<^sub>R xb"] asm by auto |
|
323 |
qed |
|
324 |
||
325 |
lemma convex_sums: |
|
326 |
assumes "convex s" "convex t" |
|
327 |
shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}" |
|
49609 | 328 |
using assms unfolding convex_def image_iff |
36623 | 329 |
proof safe |
49609 | 330 |
fix xa xb ya yb |
331 |
assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t" |
|
332 |
fix u v :: real |
|
333 |
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1" |
|
36623 | 334 |
show "\<exists>x y. u *\<^sub>R (xa + ya) + v *\<^sub>R (xb + yb) = x + y \<and> x \<in> s \<and> y \<in> t" |
335 |
using exI[of _ "u *\<^sub>R xa + v *\<^sub>R xb"] exI[of _ "u *\<^sub>R ya + v *\<^sub>R yb"] |
|
336 |
assms[unfolded convex_def] uv xy by (auto simp add:scaleR_right_distrib) |
|
337 |
qed |
|
338 |
||
339 |
lemma convex_differences: |
|
340 |
assumes "convex s" "convex t" |
|
341 |
shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}" |
|
342 |
proof - |
|
343 |
have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}" |
|
344 |
proof safe |
|
49609 | 345 |
fix x x' y |
346 |
assume "x' \<in> s" "y \<in> t" |
|
347 |
then show "\<exists>x y'. x' - y = x + y' \<and> x \<in> s \<and> y' \<in> uminus ` t" |
|
36623 | 348 |
using exI[of _ x'] exI[of _ "-y"] by auto |
349 |
next |
|
49609 | 350 |
fix x x' y y' |
351 |
assume "x' \<in> s" "y' \<in> t" |
|
352 |
then show "\<exists>x y. x' + - y' = x - y \<and> x \<in> s \<and> y \<in> t" |
|
36623 | 353 |
using exI[of _ x'] exI[of _ y'] by auto |
354 |
qed |
|
49609 | 355 |
then show ?thesis |
356 |
using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto |
|
36623 | 357 |
qed |
358 |
||
49609 | 359 |
lemma convex_translation: |
360 |
assumes "convex s" |
|
361 |
shows "convex ((\<lambda>x. a + x) ` s)" |
|
362 |
proof - |
|
363 |
have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto |
|
364 |
then show ?thesis |
|
365 |
using convex_sums[OF convex_singleton[of a] assms] by auto |
|
366 |
qed |
|
36623 | 367 |
|
49609 | 368 |
lemma convex_affinity: |
369 |
assumes "convex s" |
|
370 |
shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)" |
|
371 |
proof - |
|
372 |
have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto |
|
373 |
then show ?thesis |
|
374 |
using convex_translation[OF convex_scaling[OF assms], of a c] by auto |
|
375 |
qed |
|
36623 | 376 |
|
377 |
lemma convex_linear_image: |
|
378 |
assumes c:"convex s" and l:"bounded_linear f" |
|
379 |
shows "convex(f ` s)" |
|
49609 | 380 |
proof (auto simp add: convex_def) |
36623 | 381 |
interpret f: bounded_linear f by fact |
49609 | 382 |
fix x y |
383 |
assume xy: "x \<in> s" "y \<in> s" |
|
384 |
fix u v :: real |
|
385 |
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1" |
|
36623 | 386 |
show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff |
387 |
using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR |
|
388 |
c[unfolded convex_def] xy uv by auto |
|
389 |
qed |
|
390 |
||
391 |
||
49609 | 392 |
lemma pos_is_convex: "convex {0 :: real <..}" |
393 |
unfolding convex_alt |
|
36623 | 394 |
proof safe |
395 |
fix y x \<mu> :: real |
|
396 |
assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1" |
|
397 |
{ assume "\<mu> = 0" |
|
49609 | 398 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp |
399 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp } |
|
36623 | 400 |
moreover |
401 |
{ assume "\<mu> = 1" |
|
49609 | 402 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp } |
36623 | 403 |
moreover |
404 |
{ assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0" |
|
49609 | 405 |
then have "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto |
406 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36648
diff
changeset
|
407 |
by (auto simp add: add_pos_pos mult_pos_pos) } |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
408 |
ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" using assms by fastforce |
36623 | 409 |
qed |
410 |
||
411 |
lemma convex_on_setsum: |
|
412 |
fixes a :: "'a \<Rightarrow> real" |
|
49609 | 413 |
and y :: "'a \<Rightarrow> 'b::real_vector" |
414 |
and f :: "'b \<Rightarrow> real" |
|
36623 | 415 |
assumes "finite s" "s \<noteq> {}" |
49609 | 416 |
and "convex_on C f" |
417 |
and "convex C" |
|
418 |
and "(\<Sum> i \<in> s. a i) = 1" |
|
419 |
and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" |
|
420 |
and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C" |
|
36623 | 421 |
shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))" |
49609 | 422 |
using assms |
423 |
proof (induct s arbitrary: a rule: finite_ne_induct) |
|
36623 | 424 |
case (singleton i) |
49609 | 425 |
then have ai: "a i = 1" by auto |
426 |
then show ?case by auto |
|
36623 | 427 |
next |
428 |
case (insert i s) note asms = this |
|
49609 | 429 |
then have "convex_on C f" by simp |
36623 | 430 |
from this[unfolded convex_on_def, rule_format] |
49609 | 431 |
have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 |
432 |
\<Longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
|
36623 | 433 |
by simp |
434 |
{ assume "a i = 1" |
|
49609 | 435 |
then have "(\<Sum> j \<in> s. a j) = 0" |
36623 | 436 |
using asms by auto |
49609 | 437 |
then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
438 |
using setsum_nonneg_0[where 'b=real] asms by fastforce |
49609 | 439 |
then have ?case using asms by auto } |
36623 | 440 |
moreover |
441 |
{ assume asm: "a i \<noteq> 1" |
|
442 |
from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto |
|
443 |
have fis: "finite (insert i s)" using asms by auto |
|
49609 | 444 |
then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp |
445 |
then have "a i < 1" using asm by auto |
|
446 |
then have i0: "1 - a i > 0" by auto |
|
447 |
let ?a = "\<lambda>j. a j / (1 - a i)" |
|
36623 | 448 |
{ fix j assume "j \<in> s" |
49609 | 449 |
then have "?a j \<ge> 0" |
36623 | 450 |
using i0 asms divide_nonneg_pos |
49609 | 451 |
by fastforce } |
452 |
note a_nonneg = this |
|
36623 | 453 |
have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto |
49609 | 454 |
then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce |
455 |
then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto |
|
456 |
then have a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp |
|
36623 | 457 |
have "convex C" using asms by auto |
49609 | 458 |
then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C" |
36623 | 459 |
using asms convex_setsum[OF `finite s` |
460 |
`convex C` a1 a_nonneg] by auto |
|
461 |
have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))" |
|
462 |
using a_nonneg a1 asms by blast |
|
463 |
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)" |
|
464 |
using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF `finite s` `i \<notin> s`] asms |
|
465 |
by (auto simp only:add_commute) |
|
466 |
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)" |
|
467 |
using i0 by auto |
|
468 |
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)" |
|
49609 | 469 |
using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric] |
470 |
by (auto simp:algebra_simps) |
|
36623 | 471 |
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)" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36648
diff
changeset
|
472 |
by (auto simp: divide_inverse) |
36623 | 473 |
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)" |
474 |
using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1] |
|
475 |
by (auto simp add:add_commute) |
|
476 |
also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)" |
|
477 |
using add_right_mono[OF mult_left_mono[of _ _ "1 - a i", |
|
478 |
OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp |
|
479 |
also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44142
diff
changeset
|
480 |
unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto |
36623 | 481 |
also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)" using i0 by auto |
482 |
also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))" using asms by auto |
|
483 |
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))" |
|
484 |
by simp } |
|
485 |
ultimately show ?case by auto |
|
486 |
qed |
|
487 |
||
488 |
lemma convex_on_alt: |
|
489 |
fixes C :: "'a::real_vector set" |
|
490 |
assumes "convex C" |
|
491 |
shows "convex_on C f = |
|
492 |
(\<forall> x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 |
|
493 |
\<longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)" |
|
494 |
proof safe |
|
49609 | 495 |
fix x y |
496 |
fix \<mu> :: real |
|
36623 | 497 |
assume asms: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1" |
498 |
from this[unfolded convex_on_def, rule_format] |
|
49609 | 499 |
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 |
36623 | 500 |
from this[of "\<mu>" "1 - \<mu>", simplified] asms |
49609 | 501 |
show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" by auto |
36623 | 502 |
next |
503 |
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" |
|
49609 | 504 |
{ fix x y |
505 |
fix u v :: real |
|
36623 | 506 |
assume lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1" |
49609 | 507 |
then have[simp]: "1 - u = v" by auto |
36623 | 508 |
from asm[rule_format, of x y u] |
49609 | 509 |
have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using lasm by auto |
510 |
} |
|
511 |
then show "convex_on C f" unfolding convex_on_def by auto |
|
36623 | 512 |
qed |
513 |
||
43337 | 514 |
lemma convex_on_diff: |
515 |
fixes f :: "real \<Rightarrow> real" |
|
516 |
assumes f: "convex_on I f" and I: "x\<in>I" "y\<in>I" and t: "x < t" "t < y" |
|
49609 | 517 |
shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)" |
518 |
"(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)" |
|
43337 | 519 |
proof - |
520 |
def a \<equiv> "(t - y) / (x - y)" |
|
521 |
with t have "0 \<le> a" "0 \<le> 1 - a" by (auto simp: field_simps) |
|
522 |
with f `x \<in> I` `y \<in> I` have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y" |
|
523 |
by (auto simp: convex_on_def) |
|
524 |
have "a * x + (1 - a) * y = a * (x - y) + y" by (simp add: field_simps) |
|
525 |
also have "\<dots> = t" unfolding a_def using `x < t` `t < y` by simp |
|
526 |
finally have "f t \<le> a * f x + (1 - a) * f y" using cvx by simp |
|
527 |
also have "\<dots> = a * (f x - f y) + f y" by (simp add: field_simps) |
|
528 |
finally have "f t - f y \<le> a * (f x - f y)" by simp |
|
529 |
with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)" |
|
44142 | 530 |
by (simp add: le_divide_eq divide_le_eq field_simps a_def) |
43337 | 531 |
with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)" |
44142 | 532 |
by (simp add: le_divide_eq divide_le_eq field_simps) |
43337 | 533 |
qed |
36623 | 534 |
|
535 |
lemma pos_convex_function: |
|
536 |
fixes f :: "real \<Rightarrow> real" |
|
537 |
assumes "convex C" |
|
49609 | 538 |
and leq: "\<And>x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x" |
36623 | 539 |
shows "convex_on C f" |
49609 | 540 |
unfolding convex_on_alt[OF assms(1)] |
541 |
using assms |
|
36623 | 542 |
proof safe |
543 |
fix x y \<mu> :: real |
|
544 |
let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" |
|
545 |
assume asm: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1" |
|
49609 | 546 |
then have "1 - \<mu> \<ge> 0" by auto |
547 |
then have xpos: "?x \<in> C" using asm unfolding convex_alt by fastforce |
|
36623 | 548 |
have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) |
549 |
\<ge> \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)" |
|
38642
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
36778
diff
changeset
|
550 |
using add_mono[OF mult_left_mono[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`] |
8fa437809c67
dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents:
36778
diff
changeset
|
551 |
mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto |
49609 | 552 |
then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0" |
553 |
by (auto simp add: field_simps) |
|
554 |
then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
|
36623 | 555 |
using convex_on_alt by auto |
556 |
qed |
|
557 |
||
558 |
lemma atMostAtLeast_subset_convex: |
|
559 |
fixes C :: "real set" |
|
560 |
assumes "convex C" |
|
49609 | 561 |
and "x \<in> C" "y \<in> C" "x < y" |
36623 | 562 |
shows "{x .. y} \<subseteq> C" |
563 |
proof safe |
|
564 |
fix z assume zasm: "z \<in> {x .. y}" |
|
565 |
{ assume asm: "x < z" "z < y" |
|
49609 | 566 |
let ?\<mu> = "(y - z) / (y - x)" |
567 |
have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add: field_simps) |
|
568 |
then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C" |
|
569 |
using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>] |
|
570 |
by (simp add: algebra_simps) |
|
36623 | 571 |
have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y" |
49609 | 572 |
by (auto simp add: field_simps) |
36623 | 573 |
also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)" |
49609 | 574 |
using assms unfolding add_divide_distrib by (auto simp: field_simps) |
36623 | 575 |
also have "\<dots> = z" |
49609 | 576 |
using assms by (auto simp: field_simps) |
36623 | 577 |
finally have "z \<in> C" |
49609 | 578 |
using comb by auto } |
579 |
note less = this |
|
36623 | 580 |
show "z \<in> C" using zasm less assms |
581 |
unfolding atLeastAtMost_iff le_less by auto |
|
582 |
qed |
|
583 |
||
584 |
lemma f''_imp_f': |
|
585 |
fixes f :: "real \<Rightarrow> real" |
|
586 |
assumes "convex C" |
|
49609 | 587 |
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)" |
588 |
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)" |
|
589 |
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0" |
|
590 |
and "x \<in> C" "y \<in> C" |
|
36623 | 591 |
shows "f' x * (y - x) \<le> f y - f x" |
49609 | 592 |
using assms |
36623 | 593 |
proof - |
49609 | 594 |
{ fix x y :: real |
595 |
assume asm: "x \<in> C" "y \<in> C" "y > x" |
|
596 |
then have ge: "y - x > 0" "y - x \<ge> 0" by auto |
|
36623 | 597 |
from asm have le: "x - y < 0" "x - y \<le> 0" by auto |
598 |
then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1" |
|
599 |
using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `y \<in> C` `x < y`], |
|
600 |
THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] |
|
601 |
by auto |
|
49609 | 602 |
then have "z1 \<in> C" using atMostAtLeast_subset_convex |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
603 |
`convex C` `x \<in> C` `y \<in> C` `x < y` by fastforce |
36623 | 604 |
from z1 have z1': "f x - f y = (x - y) * f' z1" |
605 |
by (simp add:field_simps) |
|
606 |
obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2" |
|
607 |
using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `z1 \<in> C` `x < z1`], |
|
608 |
THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 |
|
609 |
by auto |
|
610 |
obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3" |
|
611 |
using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `z1 \<in> C` `y \<in> C` `z1 < y`], |
|
612 |
THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 |
|
613 |
by auto |
|
614 |
have "f' y - (f x - f y) / (x - y) = f' y - f' z1" |
|
615 |
using asm z1' by auto |
|
616 |
also have "\<dots> = (y - z1) * f'' z3" using z3 by auto |
|
617 |
finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" by simp |
|
618 |
have A': "y - z1 \<ge> 0" using z1 by auto |
|
619 |
have "z3 \<in> C" using z3 asm atMostAtLeast_subset_convex |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
620 |
`convex C` `x \<in> C` `z1 \<in> C` `x < z1` by fastforce |
49609 | 621 |
then have B': "f'' z3 \<ge> 0" using assms by auto |
36623 | 622 |
from A' B' have "(y - z1) * f'' z3 \<ge> 0" using mult_nonneg_nonneg by auto |
623 |
from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0" by auto |
|
624 |
from mult_right_mono_neg[OF this le(2)] |
|
625 |
have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)" |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36648
diff
changeset
|
626 |
by (simp add: algebra_simps) |
49609 | 627 |
then have "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto |
628 |
then have res: "f' y * (x - y) \<le> f x - f y" by auto |
|
36623 | 629 |
have "(f y - f x) / (y - x) - f' x = f' z1 - f' x" |
630 |
using asm z1 by auto |
|
631 |
also have "\<dots> = (z1 - x) * f'' z2" using z2 by auto |
|
632 |
finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" by simp |
|
633 |
have A: "z1 - x \<ge> 0" using z1 by auto |
|
634 |
have "z2 \<in> C" using z2 z1 asm atMostAtLeast_subset_convex |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
635 |
`convex C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastforce |
49609 | 636 |
then have B: "f'' z2 \<ge> 0" using assms by auto |
36623 | 637 |
from A B have "(z1 - x) * f'' z2 \<ge> 0" using mult_nonneg_nonneg by auto |
638 |
from cool this have "(f y - f x) / (y - x) - f' x \<ge> 0" by auto |
|
639 |
from mult_right_mono[OF this ge(2)] |
|
640 |
have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)" |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36648
diff
changeset
|
641 |
by (simp add: algebra_simps) |
49609 | 642 |
then have "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto |
643 |
then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y" |
|
36623 | 644 |
using res by auto } note less_imp = this |
49609 | 645 |
{ fix x y :: real |
646 |
assume "x \<in> C" "y \<in> C" "x \<noteq> y" |
|
647 |
then have"f y - f x \<ge> f' x * (y - x)" |
|
36623 | 648 |
unfolding neq_iff using less_imp by auto } note neq_imp = this |
649 |
moreover |
|
49609 | 650 |
{ fix x y :: real |
651 |
assume asm: "x \<in> C" "y \<in> C" "x = y" |
|
652 |
then have "f y - f x \<ge> f' x * (y - x)" by auto } |
|
36623 | 653 |
ultimately show ?thesis using assms by blast |
654 |
qed |
|
655 |
||
656 |
lemma f''_ge0_imp_convex: |
|
657 |
fixes f :: "real \<Rightarrow> real" |
|
658 |
assumes conv: "convex C" |
|
49609 | 659 |
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)" |
660 |
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)" |
|
661 |
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0" |
|
36623 | 662 |
shows "convex_on C f" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44282
diff
changeset
|
663 |
using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function by fastforce |
36623 | 664 |
|
665 |
lemma minus_log_convex: |
|
666 |
fixes b :: real |
|
667 |
assumes "b > 1" |
|
668 |
shows "convex_on {0 <..} (\<lambda> x. - log b x)" |
|
669 |
proof - |
|
49609 | 670 |
have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" using DERIV_log by auto |
671 |
then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)" |
|
36623 | 672 |
using DERIV_minus by auto |
49609 | 673 |
have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))" |
36623 | 674 |
using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto |
675 |
from this[THEN DERIV_cmult, of _ "- 1 / ln b"] |
|
49609 | 676 |
have "\<And>z :: real. z > 0 \<Longrightarrow> |
677 |
DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))" |
|
36623 | 678 |
by auto |
49609 | 679 |
then have f''0: "\<And>z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36648
diff
changeset
|
680 |
unfolding inverse_eq_divide by (auto simp add: mult_assoc) |
49609 | 681 |
have f''_ge0: "\<And>z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0" |
682 |
using `b > 1` by (auto intro!:less_imp_le simp add: divide_pos_pos[of 1] mult_pos_pos) |
|
36623 | 683 |
from f''_ge0_imp_convex[OF pos_is_convex, |
684 |
unfolded greaterThan_iff, OF f' f''0 f''_ge0] |
|
685 |
show ?thesis by auto |
|
686 |
qed |
|
687 |
||
688 |
end |