
hoelzl@36648: (*  Title:      HOL/Library/Convex.thy
hoelzl@36648:     Author:     Armin Heller, TU Muenchen
hoelzl@36648:     Author:     Johannes Hoelzl, TU Muenchen
hoelzl@36648: *)
hoelzl@36648: 
hoelzl@36648: header {* Convexity in real vector spaces *}
hoelzl@36648: 
hoelzl@36623: theory Convex
hoelzl@36623: imports Product_Vector
hoelzl@36623: begin
hoelzl@36623: 
hoelzl@36623: subsection {* Convexity. *}
hoelzl@36623: 
wenzelm@49609: definition convex :: "'a::real_vector set \<Rightarrow> bool"
wenzelm@49609:   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)"
hoelzl@36623: 
huffman@53676: lemma convexI:
huffman@53676:   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"
huffman@53676:   shows "convex s"
huffman@53676:   using assms unfolding convex_def by fast
huffman@53676: 
huffman@53676: lemma convexD:
huffman@53676:   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
huffman@53676:   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
huffman@53676:   using assms unfolding convex_def by fast
huffman@53676: 
hoelzl@36623: lemma convex_alt:
hoelzl@36623:   "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)"
hoelzl@36623:   (is "_ \<longleftrightarrow> ?alt")
hoelzl@36623: proof
hoelzl@36623:   assume alt[rule_format]: ?alt
wenzelm@56796:   {
wenzelm@56796:     fix x y and u v :: real
wenzelm@56796:     assume mem: "x \<in> s" "y \<in> s"
wenzelm@49609:     assume "0 \<le> u" "0 \<le> v"
wenzelm@56796:     moreover
wenzelm@56796:     assume "u + v = 1"
wenzelm@56796:     then have "u = 1 - v" by auto
wenzelm@56796:     ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
wenzelm@56796:       using alt[OF mem] by auto
wenzelm@56796:   }
wenzelm@56796:   then show "convex s"
wenzelm@56796:     unfolding convex_def by auto
hoelzl@36623: qed (auto simp: convex_def)
hoelzl@36623: 
hoelzl@36623: lemma mem_convex:
hoelzl@36623:   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
hoelzl@36623:   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
hoelzl@36623:   using assms unfolding convex_alt by auto
hoelzl@36623: 
hoelzl@36623: lemma convex_empty[intro]: "convex {}"
hoelzl@36623:   unfolding convex_def by simp
hoelzl@36623: 
hoelzl@36623: lemma convex_singleton[intro]: "convex {a}"
hoelzl@36623:   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
hoelzl@36623: 
hoelzl@36623: lemma convex_UNIV[intro]: "convex UNIV"
hoelzl@36623:   unfolding convex_def by auto
hoelzl@36623: 
wenzelm@56796: lemma convex_Inter: "(\<forall>s\<in>f. convex s) \<Longrightarrow> convex(\<Inter> f)"
hoelzl@36623:   unfolding convex_def by auto
hoelzl@36623: 
hoelzl@36623: lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
hoelzl@36623:   unfolding convex_def by auto
hoelzl@36623: 
huffman@53596: lemma convex_INT: "\<forall>i\<in>A. convex (B i) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
huffman@53596:   unfolding convex_def by auto
huffman@53596: 
huffman@53596: lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
huffman@53596:   unfolding convex_def by auto
huffman@53596: 
hoelzl@36623: lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
hoelzl@36623:   unfolding convex_def
huffman@44142:   by (auto simp: inner_add intro!: convex_bound_le)
hoelzl@36623: 
hoelzl@36623: lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
hoelzl@36623: proof -
wenzelm@56796:   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
wenzelm@56796:     by auto
wenzelm@56796:   show ?thesis
wenzelm@56796:     unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_hyperplane: "convex {x. inner a x = b}"
wenzelm@49609: proof -
wenzelm@56796:   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
wenzelm@56796:     by auto
hoelzl@36623:   show ?thesis using convex_halfspace_le convex_halfspace_ge
hoelzl@36623:     by (auto intro!: convex_Int simp: *)
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_halfspace_lt: "convex {x. inner a x < b}"
hoelzl@36623:   unfolding convex_def
hoelzl@36623:   by (auto simp: convex_bound_lt inner_add)
hoelzl@36623: 
hoelzl@36623: lemma convex_halfspace_gt: "convex {x. inner a x > b}"
hoelzl@36623:    using convex_halfspace_lt[of "-a" "-b"] by auto
hoelzl@36623: 
hoelzl@36623: lemma convex_real_interval:
hoelzl@36623:   fixes a b :: "real"
hoelzl@36623:   shows "convex {a..}" and "convex {..b}"
wenzelm@49609:     and "convex {a<..}" and "convex {..<b}"
wenzelm@49609:     and "convex {a..b}" and "convex {a<..b}"
wenzelm@49609:     and "convex {a..<b}" and "convex {a<..<b}"
hoelzl@36623: proof -
hoelzl@36623:   have "{a..} = {x. a \<le> inner 1 x}" by auto
wenzelm@49609:   then show 1: "convex {a..}" by (simp only: convex_halfspace_ge)
hoelzl@36623:   have "{..b} = {x. inner 1 x \<le> b}" by auto
wenzelm@49609:   then show 2: "convex {..b}" by (simp only: convex_halfspace_le)
hoelzl@36623:   have "{a<..} = {x. a < inner 1 x}" by auto
wenzelm@49609:   then show 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
hoelzl@36623:   have "{..<b} = {x. inner 1 x < b}" by auto
wenzelm@49609:   then show 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
hoelzl@36623:   have "{a..b} = {a..} \<inter> {..b}" by auto
wenzelm@49609:   then show "convex {a..b}" by (simp only: convex_Int 1 2)
hoelzl@36623:   have "{a<..b} = {a<..} \<inter> {..b}" by auto
wenzelm@49609:   then show "convex {a<..b}" by (simp only: convex_Int 3 2)
hoelzl@36623:   have "{a..<b} = {a..} \<inter> {..<b}" by auto
wenzelm@49609:   then show "convex {a..<b}" by (simp only: convex_Int 1 4)
hoelzl@36623:   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
wenzelm@49609:   then show "convex {a<..<b}" by (simp only: convex_Int 3 4)
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
hoelzl@36623: 
hoelzl@36623: lemma convex_setsum:
hoelzl@36623:   fixes C :: "'a::real_vector set"
wenzelm@56796:   assumes "finite s"
wenzelm@56796:     and "convex C"
wenzelm@56796:     and "(\<Sum> i \<in> s. a i) = 1"
wenzelm@56796:   assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
wenzelm@56796:     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
hoelzl@36623:   shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
huffman@55909:   using assms(1,3,4,5)
huffman@55909: proof (induct arbitrary: a set: finite)
wenzelm@49609:   case empty
huffman@55909:   then show ?case by simp
hoelzl@36623: next
huffman@55909:   case (insert i s) note IH = this(3)
wenzelm@56796:   have "a i + setsum a s = 1"
wenzelm@56796:     and "0 \<le> a i"
wenzelm@56796:     and "\<forall>j\<in>s. 0 \<le> a j"
wenzelm@56796:     and "y i \<in> C"
wenzelm@56796:     and "\<forall>j\<in>s. y j \<in> C"
huffman@55909:     using insert.hyps(1,2) insert.prems by simp_all
wenzelm@56796:   then have "0 \<le> setsum a s"
wenzelm@56796:     by (simp add: setsum_nonneg)
huffman@55909:   have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
huffman@55909:   proof (cases)
huffman@55909:     assume z: "setsum a s = 0"
wenzelm@56796:     with `a i + setsum a s = 1` have "a i = 1"
wenzelm@56796:       by simp
wenzelm@56796:     from setsum_nonneg_0 [OF `finite s` _ z] `\<forall>j\<in>s. 0 \<le> a j` have "\<forall>j\<in>s. a j = 0"
wenzelm@56796:       by simp
wenzelm@56796:     show ?thesis using `a i = 1` and `\<forall>j\<in>s. a j = 0` and `y i \<in> C`
wenzelm@56796:       by simp
huffman@55909:   next
huffman@55909:     assume nz: "setsum a s \<noteq> 0"
wenzelm@56796:     with `0 \<le> setsum a s` have "0 < setsum a s"
wenzelm@56796:       by simp
huffman@55909:     then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
huffman@55909:       using `\<forall>j\<in>s. 0 \<le> a j` and `\<forall>j\<in>s. y j \<in> C`
hoelzl@56571:       by (simp add: IH setsum_divide_distrib [symmetric])
huffman@55909:     from `convex C` and `y i \<in> C` and this and `0 \<le> a i`
huffman@55909:       and `0 \<le> setsum a s` and `a i + setsum a s = 1`
huffman@55909:     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"
huffman@55909:       by (rule convexD)
wenzelm@56796:     then show ?thesis
wenzelm@56796:       by (simp add: scaleR_setsum_right nz)
huffman@55909:   qed
wenzelm@56796:   then show ?case using `finite s` and `i \<notin> s`
wenzelm@56796:     by simp
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex:
wenzelm@49609:   "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)
wenzelm@49609:       \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
hoelzl@36623: proof safe
wenzelm@49609:   fix k :: nat
wenzelm@49609:   fix u :: "nat \<Rightarrow> real"
wenzelm@49609:   fix x
hoelzl@36623:   assume "convex s"
hoelzl@36623:     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
hoelzl@36623:     "setsum u {1..k} = 1"
hoelzl@36623:   from this convex_setsum[of "{1 .. k}" s]
wenzelm@56796:   show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
wenzelm@56796:     by auto
hoelzl@36623: next
hoelzl@36623:   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
hoelzl@36623:     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
wenzelm@56796:   {
wenzelm@56796:     fix \<mu> :: real
wenzelm@49609:     fix x y :: 'a
wenzelm@49609:     assume xy: "x \<in> s" "y \<in> s"
wenzelm@49609:     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
wenzelm@49609:     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
wenzelm@49609:     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
wenzelm@56796:     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
wenzelm@56796:       by auto
wenzelm@56796:     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
wenzelm@56796:       by simp
wenzelm@49609:     then have "setsum ?u {1 .. 2} = 1"
haftmann@57418:       using setsum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
hoelzl@36623:       by auto
wenzelm@49609:     with asm[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
hoelzl@36623:       using mu xy by auto
hoelzl@36623:     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
hoelzl@36623:       using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
hoelzl@36623:     from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
wenzelm@56796:     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
wenzelm@56796:       by auto
wenzelm@56796:     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
haftmann@57512:       using s by (auto simp:add.commute)
wenzelm@49609:   }
wenzelm@56796:   then show "convex s"
wenzelm@56796:     unfolding convex_alt by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: 
hoelzl@36623: lemma convex_explicit:
hoelzl@36623:   fixes s :: "'a::real_vector set"
hoelzl@36623:   shows "convex s \<longleftrightarrow>
wenzelm@49609:     (\<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)"
hoelzl@36623: proof safe
wenzelm@49609:   fix t
wenzelm@49609:   fix u :: "'a \<Rightarrow> real"
wenzelm@56796:   assume "convex s"
wenzelm@56796:     and "finite t"
wenzelm@56796:     and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
wenzelm@49609:   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
hoelzl@36623:     using convex_setsum[of t s u "\<lambda> x. x"] by auto
hoelzl@36623: next
wenzelm@56796:   assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
wenzelm@56796:     setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
hoelzl@36623:   show "convex s"
hoelzl@36623:     unfolding convex_alt
hoelzl@36623:   proof safe
wenzelm@49609:     fix x y
wenzelm@49609:     fix \<mu> :: real
hoelzl@36623:     assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
wenzelm@56796:     {
wenzelm@56796:       assume "x \<noteq> y"
wenzelm@49609:       then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
hoelzl@36623:         using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
wenzelm@56796:           asm by auto
wenzelm@56796:     }
hoelzl@36623:     moreover
wenzelm@56796:     {
wenzelm@56796:       assume "x = y"
wenzelm@49609:       then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
hoelzl@36623:         using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
wenzelm@56796:           asm by (auto simp: field_simps real_vector.scale_left_diff_distrib)
wenzelm@56796:     }
wenzelm@56796:     ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
wenzelm@56796:       by blast
hoelzl@36623:   qed
hoelzl@36623: qed
hoelzl@36623: 
wenzelm@49609: lemma convex_finite:
wenzelm@49609:   assumes "finite s"
wenzelm@56796:   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)"
hoelzl@36623:   unfolding convex_explicit
wenzelm@49609: proof safe
wenzelm@49609:   fix t u
wenzelm@49609:   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"
hoelzl@36623:     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
wenzelm@56796:   have *: "s \<inter> t = t"
wenzelm@56796:     using as(2) by auto
wenzelm@49609:   have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
wenzelm@49609:     by simp
hoelzl@36623:   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
hoelzl@36623:    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
haftmann@57418:    by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
hoelzl@36623: qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
hoelzl@36623: 
wenzelm@56796: 
huffman@55909: subsection {* Functions that are convex on a set *}
huffman@55909: 
wenzelm@49609: definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
wenzelm@49609:   where "convex_on s f \<longleftrightarrow>
wenzelm@49609:     (\<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)"
hoelzl@36623: 
hoelzl@36623: lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
hoelzl@36623:   unfolding convex_on_def by auto
hoelzl@36623: 
huffman@53620: lemma convex_on_add [intro]:
wenzelm@56796:   assumes "convex_on s f"
wenzelm@56796:     and "convex_on s g"
hoelzl@36623:   shows "convex_on s (\<lambda>x. f x + g x)"
wenzelm@49609: proof -
wenzelm@56796:   {
wenzelm@56796:     fix x y
wenzelm@56796:     assume "x \<in> s" "y \<in> s"
wenzelm@49609:     moreover
wenzelm@49609:     fix u v :: real
wenzelm@49609:     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
wenzelm@49609:     ultimately
wenzelm@49609:     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)"
wenzelm@49609:       using assms unfolding convex_on_def by (auto simp add: add_mono)
wenzelm@49609:     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)"
wenzelm@49609:       by (simp add: field_simps)
wenzelm@49609:   }
wenzelm@56796:   then show ?thesis
wenzelm@56796:     unfolding convex_on_def by auto
hoelzl@36623: qed
hoelzl@36623: 
huffman@53620: lemma convex_on_cmul [intro]:
wenzelm@56796:   fixes c :: real
wenzelm@56796:   assumes "0 \<le> c"
wenzelm@56796:     and "convex_on s f"
hoelzl@36623:   shows "convex_on s (\<lambda>x. c * f x)"
wenzelm@56796: proof -
wenzelm@49609:   have *: "\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
wenzelm@49609:     by (simp add: field_simps)
wenzelm@49609:   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
wenzelm@49609:     unfolding convex_on_def and * by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_lower:
wenzelm@56796:   assumes "convex_on s f"
wenzelm@56796:     and "x \<in> s"
wenzelm@56796:     and "y \<in> s"
wenzelm@56796:     and "0 \<le> u"
wenzelm@56796:     and "0 \<le> v"
wenzelm@56796:     and "u + v = 1"
hoelzl@36623:   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
wenzelm@56796: proof -
hoelzl@36623:   let ?m = "max (f x) (f y)"
hoelzl@36623:   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
haftmann@38642:     using assms(4,5) by (auto simp add: mult_left_mono add_mono)
wenzelm@56796:   also have "\<dots> = max (f x) (f y)"
wenzelm@56796:     using assms(6) unfolding distrib[symmetric] by auto
hoelzl@36623:   finally show ?thesis
nipkow@44890:     using assms unfolding convex_on_def by fastforce
hoelzl@36623: qed
hoelzl@36623: 
huffman@53620: lemma convex_on_dist [intro]:
hoelzl@36623:   fixes s :: "'a::real_normed_vector set"
hoelzl@36623:   shows "convex_on s (\<lambda>x. dist a x)"
wenzelm@49609: proof (auto simp add: convex_on_def dist_norm)
wenzelm@49609:   fix x y
wenzelm@56796:   assume "x \<in> s" "y \<in> s"
wenzelm@49609:   fix u v :: real
wenzelm@56796:   assume "0 \<le> u"
wenzelm@56796:   assume "0 \<le> v"
wenzelm@56796:   assume "u + v = 1"
wenzelm@49609:   have "a = u *\<^sub>R a + v *\<^sub>R a"
wenzelm@56796:     unfolding scaleR_left_distrib[symmetric] and `u + v = 1` by simp
wenzelm@49609:   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
hoelzl@36623:     by (auto simp add: algebra_simps)
hoelzl@36623:   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
hoelzl@36623:     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
hoelzl@36623:     using `0 \<le> u` `0 \<le> v` by auto
hoelzl@36623: qed
hoelzl@36623: 
wenzelm@49609: 
hoelzl@36623: subsection {* Arithmetic operations on sets preserve convexity. *}
wenzelm@49609: 
huffman@53620: lemma convex_linear_image:
wenzelm@56796:   assumes "linear f"
wenzelm@56796:     and "convex s"
wenzelm@56796:   shows "convex (f ` s)"
huffman@53620: proof -
huffman@53620:   interpret f: linear f by fact
huffman@53620:   from `convex s` show "convex (f ` s)"
huffman@53620:     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
hoelzl@36623: qed
hoelzl@36623: 
huffman@53620: lemma convex_linear_vimage:
wenzelm@56796:   assumes "linear f"
wenzelm@56796:     and "convex s"
wenzelm@56796:   shows "convex (f -` s)"
huffman@53620: proof -
huffman@53620:   interpret f: linear f by fact
huffman@53620:   from `convex s` show "convex (f -` s)"
huffman@53620:     by (simp add: convex_def f.add f.scaleR)
huffman@53620: qed
huffman@53620: 
huffman@53620: lemma convex_scaling:
wenzelm@56796:   assumes "convex s"
wenzelm@56796:   shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
huffman@53620: proof -
wenzelm@56796:   have "linear (\<lambda>x. c *\<^sub>R x)"
wenzelm@56796:     by (simp add: linearI scaleR_add_right)
wenzelm@56796:   then show ?thesis
wenzelm@56796:     using `convex s` by (rule convex_linear_image)
huffman@53620: qed
huffman@53620: 
huffman@53620: lemma convex_negations:
wenzelm@56796:   assumes "convex s"
wenzelm@56796:   shows "convex ((\<lambda>x. - x) ` s)"
huffman@53620: proof -
wenzelm@56796:   have "linear (\<lambda>x. - x)"
wenzelm@56796:     by (simp add: linearI)
wenzelm@56796:   then show ?thesis
wenzelm@56796:     using `convex s` by (rule convex_linear_image)
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_sums:
wenzelm@56796:   assumes "convex s"
wenzelm@56796:     and "convex t"
hoelzl@36623:   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
huffman@53620: proof -
huffman@53620:   have "linear (\<lambda>(x, y). x + y)"
huffman@53620:     by (auto intro: linearI simp add: scaleR_add_right)
huffman@53620:   with assms have "convex ((\<lambda>(x, y). x + y) ` (s \<times> t))"
huffman@53620:     by (intro convex_linear_image convex_Times)
huffman@53620:   also have "((\<lambda>(x, y). x + y) ` (s \<times> t)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
huffman@53620:     by auto
huffman@53620:   finally show ?thesis .
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_differences:
hoelzl@36623:   assumes "convex s" "convex t"
hoelzl@36623:   shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
hoelzl@36623: proof -
hoelzl@36623:   have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}"
haftmann@54230:     by (auto simp add: diff_conv_add_uminus simp del: add_uminus_conv_diff)
wenzelm@49609:   then show ?thesis
wenzelm@49609:     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
hoelzl@36623: qed
hoelzl@36623: 
wenzelm@49609: lemma convex_translation:
wenzelm@49609:   assumes "convex s"
wenzelm@49609:   shows "convex ((\<lambda>x. a + x) ` s)"
wenzelm@49609: proof -
wenzelm@56796:   have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s"
wenzelm@56796:     by auto
wenzelm@49609:   then show ?thesis
wenzelm@49609:     using convex_sums[OF convex_singleton[of a] assms] by auto
wenzelm@49609: qed
hoelzl@36623: 
wenzelm@49609: lemma convex_affinity:
wenzelm@49609:   assumes "convex s"
wenzelm@49609:   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
wenzelm@49609: proof -
wenzelm@56796:   have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s"
wenzelm@56796:     by auto
wenzelm@49609:   then show ?thesis
wenzelm@49609:     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
wenzelm@49609: qed
hoelzl@36623: 
wenzelm@49609: lemma pos_is_convex: "convex {0 :: real <..}"
wenzelm@49609:   unfolding convex_alt
hoelzl@36623: proof safe
hoelzl@36623:   fix y x \<mu> :: real
hoelzl@36623:   assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
wenzelm@56796:   {
wenzelm@56796:     assume "\<mu> = 0"
wenzelm@49609:     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
wenzelm@56796:     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp
wenzelm@56796:   }
hoelzl@36623:   moreover
wenzelm@56796:   {
wenzelm@56796:     assume "\<mu> = 1"
wenzelm@56796:     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp
wenzelm@56796:   }
hoelzl@36623:   moreover
wenzelm@56796:   {
wenzelm@56796:     assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
wenzelm@49609:     then have "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
wenzelm@49609:     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
wenzelm@56796:       by (auto simp add: add_pos_pos)
wenzelm@56796:   }
wenzelm@56796:   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
wenzelm@56796:     using assms by fastforce
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_on_setsum:
hoelzl@36623:   fixes a :: "'a \<Rightarrow> real"
wenzelm@49609:     and y :: "'a \<Rightarrow> 'b::real_vector"
wenzelm@49609:     and f :: "'b \<Rightarrow> real"
hoelzl@36623:   assumes "finite s" "s \<noteq> {}"
wenzelm@49609:     and "convex_on C f"
wenzelm@49609:     and "convex C"
wenzelm@49609:     and "(\<Sum> i \<in> s. a i) = 1"
wenzelm@49609:     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
wenzelm@49609:     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
hoelzl@36623:   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
wenzelm@49609:   using assms
wenzelm@49609: proof (induct s arbitrary: a rule: finite_ne_induct)
hoelzl@36623:   case (singleton i)
wenzelm@49609:   then have ai: "a i = 1" by auto
wenzelm@49609:   then show ?case by auto
hoelzl@36623: next
hoelzl@36623:   case (insert i s) note asms = this
wenzelm@49609:   then have "convex_on C f" by simp
hoelzl@36623:   from this[unfolded convex_on_def, rule_format]
wenzelm@56796:   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
wenzelm@56796:       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
hoelzl@36623:     by simp
wenzelm@56796:   {
wenzelm@56796:     assume "a i = 1"
wenzelm@49609:     then have "(\<Sum> j \<in> s. a j) = 0"
hoelzl@36623:       using asms by auto
wenzelm@49609:     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
nipkow@44890:       using setsum_nonneg_0[where 'b=real] asms by fastforce
wenzelm@56796:     then have ?case using asms by auto
wenzelm@56796:   }
hoelzl@36623:   moreover
wenzelm@56796:   {
wenzelm@56796:     assume asm: "a i \<noteq> 1"
hoelzl@36623:     from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
hoelzl@36623:     have fis: "finite (insert i s)" using asms by auto
wenzelm@49609:     then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
wenzelm@49609:     then have "a i < 1" using asm by auto
wenzelm@49609:     then have i0: "1 - a i > 0" by auto
wenzelm@49609:     let ?a = "\<lambda>j. a j / (1 - a i)"
wenzelm@56796:     {
wenzelm@56796:       fix j
wenzelm@56796:       assume "j \<in> s"
wenzelm@56796:       with i0 asms have "?a j \<ge> 0"
wenzelm@56796:         by fastforce
wenzelm@56796:     }
wenzelm@49609:     note a_nonneg = this
hoelzl@36623:     have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
wenzelm@49609:     then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce
wenzelm@49609:     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
wenzelm@49609:     then have a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp
hoelzl@36623:     have "convex C" using asms by auto
wenzelm@49609:     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
hoelzl@36623:       using asms convex_setsum[OF `finite s`
hoelzl@36623:         `convex C` a1 a_nonneg] by auto
hoelzl@36623:     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
hoelzl@36623:       using a_nonneg a1 asms by blast
hoelzl@36623:     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)"
hoelzl@36623:       using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF `finite s` `i \<notin> s`] asms
haftmann@57512:       by (auto simp only:add.commute)
hoelzl@36623:     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)"
hoelzl@36623:       using i0 by auto
hoelzl@36623:     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)"
wenzelm@49609:       using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
wenzelm@49609:       by (auto simp:algebra_simps)
hoelzl@36623:     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)"
huffman@36778:       by (auto simp: divide_inverse)
hoelzl@36623:     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)"
hoelzl@36623:       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
haftmann@57512:       by (auto simp add:add.commute)
hoelzl@36623:     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
hoelzl@36623:       using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
hoelzl@36623:         OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
hoelzl@36623:     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
huffman@44282:       unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
hoelzl@36623:     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)" using i0 by auto
hoelzl@36623:     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))" using asms by auto
hoelzl@36623:     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))"
wenzelm@56796:       by simp
wenzelm@56796:   }
hoelzl@36623:   ultimately show ?case by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma convex_on_alt:
hoelzl@36623:   fixes C :: "'a::real_vector set"
hoelzl@36623:   assumes "convex C"
wenzelm@56796:   shows "convex_on C f \<longleftrightarrow>
wenzelm@56796:     (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
wenzelm@56796:       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
hoelzl@36623: proof safe
wenzelm@49609:   fix x y
wenzelm@49609:   fix \<mu> :: real
hoelzl@36623:   assume asms: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
hoelzl@36623:   from this[unfolded convex_on_def, rule_format]
wenzelm@56796:   have "\<And>u v. 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
wenzelm@56796:     by auto
hoelzl@36623:   from this[of "\<mu>" "1 - \<mu>", simplified] asms
wenzelm@56796:   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
wenzelm@56796:     by auto
hoelzl@36623: next
wenzelm@56796:   assume asm: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
wenzelm@56796:     f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
wenzelm@56796:   {
wenzelm@56796:     fix x y
wenzelm@49609:     fix u v :: real
hoelzl@36623:     assume lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
wenzelm@49609:     then have[simp]: "1 - u = v" by auto
hoelzl@36623:     from asm[rule_format, of x y u]
wenzelm@56796:     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
wenzelm@56796:       using lasm by auto
wenzelm@49609:   }
wenzelm@56796:   then show "convex_on C f"
wenzelm@56796:     unfolding convex_on_def by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@43337: lemma convex_on_diff:
hoelzl@43337:   fixes f :: "real \<Rightarrow> real"
wenzelm@56796:   assumes f: "convex_on I f"
wenzelm@56796:     and I: "x \<in> I" "y \<in> I"
wenzelm@56796:     and t: "x < t" "t < y"
wenzelm@49609:   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
wenzelm@56796:     and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
hoelzl@43337: proof -
hoelzl@43337:   def a \<equiv> "(t - y) / (x - y)"
wenzelm@56796:   with t have "0 \<le> a" "0 \<le> 1 - a"
wenzelm@56796:     by (auto simp: field_simps)
hoelzl@43337:   with f `x \<in> I` `y \<in> I` have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
hoelzl@43337:     by (auto simp: convex_on_def)
wenzelm@56796:   have "a * x + (1 - a) * y = a * (x - y) + y"
wenzelm@56796:     by (simp add: field_simps)
wenzelm@56796:   also have "\<dots> = t"
wenzelm@56796:     unfolding a_def using `x < t` `t < y` by simp
wenzelm@56796:   finally have "f t \<le> a * f x + (1 - a) * f y"
wenzelm@56796:     using cvx by simp
wenzelm@56796:   also have "\<dots> = a * (f x - f y) + f y"
wenzelm@56796:     by (simp add: field_simps)
wenzelm@56796:   finally have "f t - f y \<le> a * (f x - f y)"
wenzelm@56796:     by simp
hoelzl@43337:   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
huffman@44142:     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
hoelzl@43337:   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
huffman@44142:     by (simp add: le_divide_eq divide_le_eq field_simps)
hoelzl@43337: qed
hoelzl@36623: 
hoelzl@36623: lemma pos_convex_function:
hoelzl@36623:   fixes f :: "real \<Rightarrow> real"
hoelzl@36623:   assumes "convex C"
wenzelm@56796:     and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
hoelzl@36623:   shows "convex_on C f"
wenzelm@49609:   unfolding convex_on_alt[OF assms(1)]
wenzelm@49609:   using assms
hoelzl@36623: proof safe
hoelzl@36623:   fix x y \<mu> :: real
hoelzl@36623:   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
hoelzl@36623:   assume asm: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
wenzelm@49609:   then have "1 - \<mu> \<ge> 0" by auto
wenzelm@56796:   then have xpos: "?x \<in> C"
wenzelm@56796:     using asm unfolding convex_alt by fastforce
wenzelm@56796:   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
wenzelm@56796:       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
haftmann@38642:     using add_mono[OF mult_left_mono[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`]
wenzelm@56796:       mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]]
wenzelm@56796:     by auto
wenzelm@49609:   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
wenzelm@49609:     by (auto simp add: field_simps)
wenzelm@49609:   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
hoelzl@36623:     using convex_on_alt by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma atMostAtLeast_subset_convex:
hoelzl@36623:   fixes C :: "real set"
hoelzl@36623:   assumes "convex C"
wenzelm@49609:     and "x \<in> C" "y \<in> C" "x < y"
hoelzl@36623:   shows "{x .. y} \<subseteq> C"
hoelzl@36623: proof safe
hoelzl@36623:   fix z assume zasm: "z \<in> {x .. y}"
wenzelm@56796:   {
wenzelm@56796:     assume asm: "x < z" "z < y"
wenzelm@49609:     let ?\<mu> = "(y - z) / (y - x)"
wenzelm@56796:     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
wenzelm@56796:       using assms asm by (auto simp add: field_simps)
wenzelm@49609:     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
wenzelm@49609:       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
wenzelm@49609:       by (simp add: algebra_simps)
hoelzl@36623:     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
wenzelm@49609:       by (auto simp add: field_simps)
hoelzl@36623:     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
wenzelm@49609:       using assms unfolding add_divide_distrib by (auto simp: field_simps)
hoelzl@36623:     also have "\<dots> = z"
wenzelm@49609:       using assms by (auto simp: field_simps)
hoelzl@36623:     finally have "z \<in> C"
wenzelm@56796:       using comb by auto
wenzelm@56796:   }
wenzelm@49609:   note less = this
hoelzl@36623:   show "z \<in> C" using zasm less assms
hoelzl@36623:     unfolding atLeastAtMost_iff le_less by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma f''_imp_f':
hoelzl@36623:   fixes f :: "real \<Rightarrow> real"
hoelzl@36623:   assumes "convex C"
wenzelm@49609:     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
wenzelm@49609:     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
wenzelm@49609:     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
wenzelm@49609:     and "x \<in> C" "y \<in> C"
hoelzl@36623:   shows "f' x * (y - x) \<le> f y - f x"
wenzelm@49609:   using assms
hoelzl@36623: proof -
wenzelm@56796:   {
wenzelm@56796:     fix x y :: real
wenzelm@49609:     assume asm: "x \<in> C" "y \<in> C" "y > x"
wenzelm@49609:     then have ge: "y - x > 0" "y - x \<ge> 0" by auto
hoelzl@36623:     from asm have le: "x - y < 0" "x - y \<le> 0" by auto
hoelzl@36623:     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
hoelzl@36623:       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `y \<in> C` `x < y`],
hoelzl@36623:         THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
hoelzl@36623:       by auto
wenzelm@49609:     then have "z1 \<in> C" using atMostAtLeast_subset_convex
nipkow@44890:       `convex C` `x \<in> C` `y \<in> C` `x < y` by fastforce
hoelzl@36623:     from z1 have z1': "f x - f y = (x - y) * f' z1"
hoelzl@36623:       by (simp add:field_simps)
hoelzl@36623:     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
hoelzl@36623:       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `z1 \<in> C` `x < z1`],
hoelzl@36623:         THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
hoelzl@36623:       by auto
hoelzl@36623:     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
hoelzl@36623:       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `z1 \<in> C` `y \<in> C` `z1 < y`],
hoelzl@36623:         THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
hoelzl@36623:       by auto
hoelzl@36623:     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
hoelzl@36623:       using asm z1' by auto
hoelzl@36623:     also have "\<dots> = (y - z1) * f'' z3" using z3 by auto
hoelzl@36623:     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" by simp
hoelzl@36623:     have A': "y - z1 \<ge> 0" using z1 by auto
hoelzl@36623:     have "z3 \<in> C" using z3 asm atMostAtLeast_subset_convex
nipkow@44890:       `convex C` `x \<in> C` `z1 \<in> C` `x < z1` by fastforce
wenzelm@49609:     then have B': "f'' z3 \<ge> 0" using assms by auto
nipkow@56536:     from A' B' have "(y - z1) * f'' z3 \<ge> 0" by auto
hoelzl@36623:     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0" by auto
hoelzl@36623:     from mult_right_mono_neg[OF this le(2)]
hoelzl@36623:     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
huffman@36778:       by (simp add: algebra_simps)
wenzelm@49609:     then have "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
wenzelm@49609:     then have res: "f' y * (x - y) \<le> f x - f y" by auto
hoelzl@36623:     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
hoelzl@36623:       using asm z1 by auto
hoelzl@36623:     also have "\<dots> = (z1 - x) * f'' z2" using z2 by auto
hoelzl@36623:     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" by simp
hoelzl@36623:     have A: "z1 - x \<ge> 0" using z1 by auto
hoelzl@36623:     have "z2 \<in> C" using z2 z1 asm atMostAtLeast_subset_convex
nipkow@44890:       `convex C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastforce
wenzelm@49609:     then have B: "f'' z2 \<ge> 0" using assms by auto
nipkow@56536:     from A B have "(z1 - x) * f'' z2 \<ge> 0" by auto
hoelzl@36623:     from cool this have "(f y - f x) / (y - x) - f' x \<ge> 0" by auto
hoelzl@36623:     from mult_right_mono[OF this ge(2)]
hoelzl@36623:     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
huffman@36778:       by (simp add: algebra_simps)
wenzelm@49609:     then have "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
wenzelm@49609:     then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
hoelzl@36623:       using res by auto } note less_imp = this
wenzelm@56796:   {
wenzelm@56796:     fix x y :: real
wenzelm@49609:     assume "x \<in> C" "y \<in> C" "x \<noteq> y"
wenzelm@49609:     then have"f y - f x \<ge> f' x * (y - x)"
wenzelm@56796:     unfolding neq_iff using less_imp by auto
wenzelm@56796:   }
hoelzl@36623:   moreover
wenzelm@56796:   {
wenzelm@56796:     fix x y :: real
wenzelm@49609:     assume asm: "x \<in> C" "y \<in> C" "x = y"
wenzelm@56796:     then have "f y - f x \<ge> f' x * (y - x)" by auto
wenzelm@56796:   }
hoelzl@36623:   ultimately show ?thesis using assms by blast
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: lemma f''_ge0_imp_convex:
hoelzl@36623:   fixes f :: "real \<Rightarrow> real"
hoelzl@36623:   assumes conv: "convex C"
wenzelm@49609:     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
wenzelm@49609:     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
wenzelm@49609:     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
hoelzl@36623:   shows "convex_on C f"
wenzelm@56796:   using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
wenzelm@56796:   by fastforce
hoelzl@36623: 
hoelzl@36623: lemma minus_log_convex:
hoelzl@36623:   fixes b :: real
hoelzl@36623:   assumes "b > 1"
hoelzl@36623:   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
hoelzl@36623: proof -
wenzelm@56796:   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
wenzelm@56796:     using DERIV_log by auto
wenzelm@49609:   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
hoelzl@56479:     by (auto simp: DERIV_minus)
wenzelm@49609:   have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
hoelzl@36623:     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
hoelzl@36623:   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
wenzelm@49609:   have "\<And>z :: real. z > 0 \<Longrightarrow>
wenzelm@49609:     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
hoelzl@36623:     by auto
wenzelm@56796:   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
wenzelm@56796:     DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
haftmann@57512:     unfolding inverse_eq_divide by (auto simp add: mult.assoc)
wenzelm@56796:   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
nipkow@56544:     using `b > 1` by (auto intro!:less_imp_le)
hoelzl@36623:   from f''_ge0_imp_convex[OF pos_is_convex,
hoelzl@36623:     unfolded greaterThan_iff, OF f' f''0 f''_ge0]
hoelzl@36623:   show ?thesis by auto
hoelzl@36623: qed
hoelzl@36623: 
hoelzl@36623: end




