src/HOL/Library/Convex.thy
author nipkow
Sat Apr 12 17:26:27 2014 +0200 (2014-04-12)
changeset 56544 b60d5d119489
parent 56541 0e3abadbef39
child 56571 f4635657d66f
permissions -rw-r--r--
made mult_pos_pos a simp rule
     1 (*  Title:      HOL/Library/Convex.thy
     2     Author:     Armin Heller, TU Muenchen
     3     Author:     Johannes Hoelzl, TU Muenchen
     4 *)
     5 
     6 header {* Convexity in real vector spaces *}
     7 
     8 theory Convex
     9 imports Product_Vector
    10 begin
    11 
    12 subsection {* Convexity. *}
    13 
    14 definition convex :: "'a::real_vector set \<Rightarrow> bool"
    15   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)"
    16 
    17 lemma convexI:
    18   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"
    19   shows "convex s"
    20   using assms unfolding convex_def by fast
    21 
    22 lemma convexD:
    23   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
    24   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
    25   using assms unfolding convex_def by fast
    26 
    27 lemma convex_alt:
    28   "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)"
    29   (is "_ \<longleftrightarrow> ?alt")
    30 proof
    31   assume alt[rule_format]: ?alt
    32   { fix x y and u v :: real assume mem: "x \<in> s" "y \<in> s"
    33     assume "0 \<le> u" "0 \<le> v"
    34     moreover assume "u + v = 1" then have "u = 1 - v" by auto
    35     ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s" using alt[OF mem] by auto }
    36   then show "convex s" unfolding convex_def by auto
    37 qed (auto simp: convex_def)
    38 
    39 lemma mem_convex:
    40   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
    41   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
    42   using assms unfolding convex_alt by auto
    43 
    44 lemma convex_empty[intro]: "convex {}"
    45   unfolding convex_def by simp
    46 
    47 lemma convex_singleton[intro]: "convex {a}"
    48   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
    49 
    50 lemma convex_UNIV[intro]: "convex UNIV"
    51   unfolding convex_def by auto
    52 
    53 lemma convex_Inter: "(\<forall>s\<in>f. convex s) ==> convex(\<Inter> f)"
    54   unfolding convex_def by auto
    55 
    56 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
    57   unfolding convex_def by auto
    58 
    59 lemma convex_INT: "\<forall>i\<in>A. convex (B i) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
    60   unfolding convex_def by auto
    61 
    62 lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
    63   unfolding convex_def by auto
    64 
    65 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
    66   unfolding convex_def
    67   by (auto simp: inner_add intro!: convex_bound_le)
    68 
    69 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
    70 proof -
    71   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" by auto
    72   show ?thesis unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
    73 qed
    74 
    75 lemma convex_hyperplane: "convex {x. inner a x = b}"
    76 proof -
    77   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" by auto
    78   show ?thesis using convex_halfspace_le convex_halfspace_ge
    79     by (auto intro!: convex_Int simp: *)
    80 qed
    81 
    82 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
    83   unfolding convex_def
    84   by (auto simp: convex_bound_lt inner_add)
    85 
    86 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
    87    using convex_halfspace_lt[of "-a" "-b"] by auto
    88 
    89 lemma convex_real_interval:
    90   fixes a b :: "real"
    91   shows "convex {a..}" and "convex {..b}"
    92     and "convex {a<..}" and "convex {..<b}"
    93     and "convex {a..b}" and "convex {a<..b}"
    94     and "convex {a..<b}" and "convex {a<..<b}"
    95 proof -
    96   have "{a..} = {x. a \<le> inner 1 x}" by auto
    97   then show 1: "convex {a..}" by (simp only: convex_halfspace_ge)
    98   have "{..b} = {x. inner 1 x \<le> b}" by auto
    99   then show 2: "convex {..b}" by (simp only: convex_halfspace_le)
   100   have "{a<..} = {x. a < inner 1 x}" by auto
   101   then show 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
   102   have "{..<b} = {x. inner 1 x < b}" by auto
   103   then show 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
   104   have "{a..b} = {a..} \<inter> {..b}" by auto
   105   then show "convex {a..b}" by (simp only: convex_Int 1 2)
   106   have "{a<..b} = {a<..} \<inter> {..b}" by auto
   107   then show "convex {a<..b}" by (simp only: convex_Int 3 2)
   108   have "{a..<b} = {a..} \<inter> {..<b}" by auto
   109   then show "convex {a..<b}" by (simp only: convex_Int 1 4)
   110   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
   111   then show "convex {a<..<b}" by (simp only: convex_Int 3 4)
   112 qed
   113 
   114 subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
   115 
   116 lemma convex_setsum:
   117   fixes C :: "'a::real_vector set"
   118   assumes "finite s" and "convex C" and "(\<Sum> i \<in> s. a i) = 1"
   119   assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
   120   shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
   121   using assms(1,3,4,5)
   122 proof (induct arbitrary: a set: finite)
   123   case empty
   124   then show ?case by simp
   125 next
   126   case (insert i s) note IH = this(3)
   127   have "a i + setsum a s = 1" and "0 \<le> a i" and "\<forall>j\<in>s. 0 \<le> a j" and "y i \<in> C" and "\<forall>j\<in>s. y j \<in> C"
   128     using insert.hyps(1,2) insert.prems by simp_all
   129   then have "0 \<le> setsum a s" by (simp add: setsum_nonneg)
   130   have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
   131   proof (cases)
   132     assume z: "setsum a s = 0"
   133     with `a i + setsum a s = 1` have "a i = 1" by simp
   134     from setsum_nonneg_0 [OF `finite s` _ z] `\<forall>j\<in>s. 0 \<le> a j` have "\<forall>j\<in>s. a j = 0" by simp
   135     show ?thesis using `a i = 1` and `\<forall>j\<in>s. a j = 0` and `y i \<in> C` by simp
   136   next
   137     assume nz: "setsum a s \<noteq> 0"
   138     with `0 \<le> setsum a s` have "0 < setsum a s" by simp
   139     then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
   140       using `\<forall>j\<in>s. 0 \<le> a j` and `\<forall>j\<in>s. y j \<in> C`
   141       by (simp add: IH divide_nonneg_pos setsum_divide_distrib [symmetric])
   142     from `convex C` and `y i \<in> C` and this and `0 \<le> a i`
   143       and `0 \<le> setsum a s` and `a i + setsum a s = 1`
   144     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"
   145       by (rule convexD)
   146     then show ?thesis by (simp add: scaleR_setsum_right nz)
   147   qed
   148   then show ?case using `finite s` and `i \<notin> s` by simp
   149 qed
   150 
   151 lemma convex:
   152   "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)
   153       \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
   154 proof safe
   155   fix k :: nat
   156   fix u :: "nat \<Rightarrow> real"
   157   fix x
   158   assume "convex s"
   159     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
   160     "setsum u {1..k} = 1"
   161   from this convex_setsum[of "{1 .. k}" s]
   162   show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" by auto
   163 next
   164   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
   165     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
   166   { fix \<mu> :: real
   167     fix x y :: 'a
   168     assume xy: "x \<in> s" "y \<in> s"
   169     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
   170     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
   171     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
   172     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}" by auto
   173     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" by simp
   174     then have "setsum ?u {1 .. 2} = 1"
   175       using setsum_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
   176       by auto
   177     with asm[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
   178       using mu xy by auto
   179     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
   180       using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
   181     from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
   182     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" by auto
   183     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute)
   184   }
   185   then show "convex s" unfolding convex_alt by auto
   186 qed
   187 
   188 
   189 lemma convex_explicit:
   190   fixes s :: "'a::real_vector set"
   191   shows "convex s \<longleftrightarrow>
   192     (\<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)"
   193 proof safe
   194   fix t
   195   fix u :: "'a \<Rightarrow> real"
   196   assume "convex s" "finite t"
   197     "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
   198   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
   199     using convex_setsum[of t s u "\<lambda> x. x"] by auto
   200 next
   201   assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x)
   202     \<and> setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
   203   show "convex s"
   204     unfolding convex_alt
   205   proof safe
   206     fix x y
   207     fix \<mu> :: real
   208     assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
   209     { assume "x \<noteq> y"
   210       then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
   211         using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
   212           asm by auto }
   213     moreover
   214     { assume "x = y"
   215       then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
   216         using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
   217           asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) }
   218     ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast
   219   qed
   220 qed
   221 
   222 lemma convex_finite:
   223   assumes "finite s"
   224   shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
   225                       \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
   226   unfolding convex_explicit
   227 proof safe
   228   fix t u
   229   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"
   230     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
   231   have *: "s \<inter> t = t" using as(2) by auto
   232   have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
   233     by simp
   234   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
   235    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
   236    by (auto simp: assms setsum_cases if_distrib if_distrib_arg)
   237 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
   238 
   239 subsection {* Functions that are convex on a set *}
   240 
   241 definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
   242   where "convex_on s f \<longleftrightarrow>
   243     (\<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)"
   244 
   245 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
   246   unfolding convex_on_def by auto
   247 
   248 lemma convex_on_add [intro]:
   249   assumes "convex_on s f" "convex_on s g"
   250   shows "convex_on s (\<lambda>x. f x + g x)"
   251 proof -
   252   { fix x y
   253     assume "x\<in>s" "y\<in>s"
   254     moreover
   255     fix u v :: real
   256     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   257     ultimately
   258     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)"
   259       using assms unfolding convex_on_def by (auto simp add: add_mono)
   260     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)"
   261       by (simp add: field_simps)
   262   }
   263   then show ?thesis unfolding convex_on_def by auto
   264 qed
   265 
   266 lemma convex_on_cmul [intro]:
   267   assumes "0 \<le> (c::real)" "convex_on s f"
   268   shows "convex_on s (\<lambda>x. c * f x)"
   269 proof-
   270   have *: "\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
   271     by (simp add: field_simps)
   272   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
   273     unfolding convex_on_def and * by auto
   274 qed
   275 
   276 lemma convex_lower:
   277   assumes "convex_on s f"  "x\<in>s"  "y \<in> s"  "0 \<le> u"  "0 \<le> v"  "u + v = 1"
   278   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
   279 proof-
   280   let ?m = "max (f x) (f y)"
   281   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
   282     using assms(4,5) by (auto simp add: mult_left_mono add_mono)
   283   also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[symmetric] by auto
   284   finally show ?thesis
   285     using assms unfolding convex_on_def by fastforce
   286 qed
   287 
   288 lemma convex_on_dist [intro]:
   289   fixes s :: "'a::real_normed_vector set"
   290   shows "convex_on s (\<lambda>x. dist a x)"
   291 proof (auto simp add: convex_on_def dist_norm)
   292   fix x y
   293   assume "x\<in>s" "y\<in>s"
   294   fix u v :: real
   295   assume "0 \<le> u" "0 \<le> v" "u + v = 1"
   296   have "a = u *\<^sub>R a + v *\<^sub>R a"
   297     unfolding scaleR_left_distrib[symmetric] and `u+v=1` by simp
   298   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
   299     by (auto simp add: algebra_simps)
   300   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
   301     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
   302     using `0 \<le> u` `0 \<le> v` by auto
   303 qed
   304 
   305 
   306 subsection {* Arithmetic operations on sets preserve convexity. *}
   307 
   308 lemma convex_linear_image:
   309   assumes "linear f" and "convex s" shows "convex (f ` s)"
   310 proof -
   311   interpret f: linear f by fact
   312   from `convex s` show "convex (f ` s)"
   313     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
   314 qed
   315 
   316 lemma convex_linear_vimage:
   317   assumes "linear f" and "convex s" shows "convex (f -` s)"
   318 proof -
   319   interpret f: linear f by fact
   320   from `convex s` show "convex (f -` s)"
   321     by (simp add: convex_def f.add f.scaleR)
   322 qed
   323 
   324 lemma convex_scaling:
   325   assumes "convex s" shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
   326 proof -
   327   have "linear (\<lambda>x. c *\<^sub>R x)" by (simp add: linearI scaleR_add_right)
   328   then show ?thesis using `convex s` by (rule convex_linear_image)
   329 qed
   330 
   331 lemma convex_negations:
   332   assumes "convex s" shows "convex ((\<lambda>x. - x) ` s)"
   333 proof -
   334   have "linear (\<lambda>x. - x)" by (simp add: linearI)
   335   then show ?thesis using `convex s` by (rule convex_linear_image)
   336 qed
   337 
   338 lemma convex_sums:
   339   assumes "convex s" and "convex t"
   340   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
   341 proof -
   342   have "linear (\<lambda>(x, y). x + y)"
   343     by (auto intro: linearI simp add: scaleR_add_right)
   344   with assms have "convex ((\<lambda>(x, y). x + y) ` (s \<times> t))"
   345     by (intro convex_linear_image convex_Times)
   346   also have "((\<lambda>(x, y). x + y) ` (s \<times> t)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
   347     by auto
   348   finally show ?thesis .
   349 qed
   350 
   351 lemma convex_differences:
   352   assumes "convex s" "convex t"
   353   shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
   354 proof -
   355   have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}"
   356     by (auto simp add: diff_conv_add_uminus simp del: add_uminus_conv_diff)
   357   then show ?thesis
   358     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
   359 qed
   360 
   361 lemma convex_translation:
   362   assumes "convex s"
   363   shows "convex ((\<lambda>x. a + x) ` s)"
   364 proof -
   365   have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
   366   then show ?thesis
   367     using convex_sums[OF convex_singleton[of a] assms] by auto
   368 qed
   369 
   370 lemma convex_affinity:
   371   assumes "convex s"
   372   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
   373 proof -
   374   have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s" by auto
   375   then show ?thesis
   376     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
   377 qed
   378 
   379 lemma pos_is_convex: "convex {0 :: real <..}"
   380   unfolding convex_alt
   381 proof safe
   382   fix y x \<mu> :: real
   383   assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   384   { assume "\<mu> = 0"
   385     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
   386     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
   387   moreover
   388   { assume "\<mu> = 1"
   389     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
   390   moreover
   391   { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
   392     then have "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
   393     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
   394       by (auto simp add: add_pos_pos) }
   395   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" using assms by fastforce
   396 qed
   397 
   398 lemma convex_on_setsum:
   399   fixes a :: "'a \<Rightarrow> real"
   400     and y :: "'a \<Rightarrow> 'b::real_vector"
   401     and f :: "'b \<Rightarrow> real"
   402   assumes "finite s" "s \<noteq> {}"
   403     and "convex_on C f"
   404     and "convex C"
   405     and "(\<Sum> i \<in> s. a i) = 1"
   406     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
   407     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
   408   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
   409   using assms
   410 proof (induct s arbitrary: a rule: finite_ne_induct)
   411   case (singleton i)
   412   then have ai: "a i = 1" by auto
   413   then show ?case by auto
   414 next
   415   case (insert i s) note asms = this
   416   then have "convex_on C f" by simp
   417   from this[unfolded convex_on_def, rule_format]
   418   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1
   419       \<Longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
   420     by simp
   421   { assume "a i = 1"
   422     then have "(\<Sum> j \<in> s. a j) = 0"
   423       using asms by auto
   424     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
   425       using setsum_nonneg_0[where 'b=real] asms by fastforce
   426     then have ?case using asms by auto }
   427   moreover
   428   { assume asm: "a i \<noteq> 1"
   429     from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
   430     have fis: "finite (insert i s)" using asms by auto
   431     then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
   432     then have "a i < 1" using asm by auto
   433     then have i0: "1 - a i > 0" by auto
   434     let ?a = "\<lambda>j. a j / (1 - a i)"
   435     { fix j assume "j \<in> s"
   436       then have "?a j \<ge> 0"
   437         using i0 asms divide_nonneg_pos
   438         by fastforce }
   439     note a_nonneg = this
   440     have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
   441     then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce
   442     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
   443     then have a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp
   444     have "convex C" using asms by auto
   445     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
   446       using asms convex_setsum[OF `finite s`
   447         `convex C` a1 a_nonneg] by auto
   448     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
   449       using a_nonneg a1 asms by blast
   450     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)"
   451       using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF `finite s` `i \<notin> s`] asms
   452       by (auto simp only:add_commute)
   453     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)"
   454       using i0 by auto
   455     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)"
   456       using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
   457       by (auto simp:algebra_simps)
   458     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)"
   459       by (auto simp: divide_inverse)
   460     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)"
   461       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
   462       by (auto simp add:add_commute)
   463     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
   464       using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
   465         OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
   466     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
   467       unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
   468     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)" using i0 by auto
   469     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))" using asms by auto
   470     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))"
   471       by simp }
   472   ultimately show ?case by auto
   473 qed
   474 
   475 lemma convex_on_alt:
   476   fixes C :: "'a::real_vector set"
   477   assumes "convex C"
   478   shows "convex_on C f =
   479   (\<forall> x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1
   480       \<longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
   481 proof safe
   482   fix x y
   483   fix \<mu> :: real
   484   assume asms: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
   485   from this[unfolded convex_on_def, rule_format]
   486   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
   487   from this[of "\<mu>" "1 - \<mu>", simplified] asms
   488   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" by auto
   489 next
   490   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"
   491   { fix x y
   492     fix u v :: real
   493     assume lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
   494     then have[simp]: "1 - u = v" by auto
   495     from asm[rule_format, of x y u]
   496     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using lasm by auto
   497   }
   498   then show "convex_on C f" unfolding convex_on_def by auto
   499 qed
   500 
   501 lemma convex_on_diff:
   502   fixes f :: "real \<Rightarrow> real"
   503   assumes f: "convex_on I f" and I: "x\<in>I" "y\<in>I" and t: "x < t" "t < y"
   504   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
   505     "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
   506 proof -
   507   def a \<equiv> "(t - y) / (x - y)"
   508   with t have "0 \<le> a" "0 \<le> 1 - a" by (auto simp: field_simps)
   509   with f `x \<in> I` `y \<in> I` have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
   510     by (auto simp: convex_on_def)
   511   have "a * x + (1 - a) * y = a * (x - y) + y" by (simp add: field_simps)
   512   also have "\<dots> = t" unfolding a_def using `x < t` `t < y` by simp
   513   finally have "f t \<le> a * f x + (1 - a) * f y" using cvx by simp
   514   also have "\<dots> = a * (f x - f y) + f y" by (simp add: field_simps)
   515   finally have "f t - f y \<le> a * (f x - f y)" by simp
   516   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
   517     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
   518   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
   519     by (simp add: le_divide_eq divide_le_eq field_simps)
   520 qed
   521 
   522 lemma pos_convex_function:
   523   fixes f :: "real \<Rightarrow> real"
   524   assumes "convex C"
   525     and leq: "\<And>x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
   526   shows "convex_on C f"
   527   unfolding convex_on_alt[OF assms(1)]
   528   using assms
   529 proof safe
   530   fix x y \<mu> :: real
   531   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
   532   assume asm: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   533   then have "1 - \<mu> \<ge> 0" by auto
   534   then have xpos: "?x \<in> C" using asm unfolding convex_alt by fastforce
   535   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x)
   536             \<ge> \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
   537     using add_mono[OF mult_left_mono[OF leq[OF xpos asm(2)] `\<mu> \<ge> 0`]
   538       mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
   539   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
   540     by (auto simp add: field_simps)
   541   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
   542     using convex_on_alt by auto
   543 qed
   544 
   545 lemma atMostAtLeast_subset_convex:
   546   fixes C :: "real set"
   547   assumes "convex C"
   548     and "x \<in> C" "y \<in> C" "x < y"
   549   shows "{x .. y} \<subseteq> C"
   550 proof safe
   551   fix z assume zasm: "z \<in> {x .. y}"
   552   { assume asm: "x < z" "z < y"
   553     let ?\<mu> = "(y - z) / (y - x)"
   554     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add: field_simps)
   555     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
   556       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
   557       by (simp add: algebra_simps)
   558     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
   559       by (auto simp add: field_simps)
   560     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
   561       using assms unfolding add_divide_distrib by (auto simp: field_simps)
   562     also have "\<dots> = z"
   563       using assms by (auto simp: field_simps)
   564     finally have "z \<in> C"
   565       using comb by auto }
   566   note less = this
   567   show "z \<in> C" using zasm less assms
   568     unfolding atLeastAtMost_iff le_less by auto
   569 qed
   570 
   571 lemma f''_imp_f':
   572   fixes f :: "real \<Rightarrow> real"
   573   assumes "convex C"
   574     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
   575     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
   576     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
   577     and "x \<in> C" "y \<in> C"
   578   shows "f' x * (y - x) \<le> f y - f x"
   579   using assms
   580 proof -
   581   { fix x y :: real
   582     assume asm: "x \<in> C" "y \<in> C" "y > x"
   583     then have ge: "y - x > 0" "y - x \<ge> 0" by auto
   584     from asm have le: "x - y < 0" "x - y \<le> 0" by auto
   585     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
   586       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `y \<in> C` `x < y`],
   587         THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
   588       by auto
   589     then have "z1 \<in> C" using atMostAtLeast_subset_convex
   590       `convex C` `x \<in> C` `y \<in> C` `x < y` by fastforce
   591     from z1 have z1': "f x - f y = (x - y) * f' z1"
   592       by (simp add:field_simps)
   593     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
   594       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `x \<in> C` `z1 \<in> C` `x < z1`],
   595         THEN f'', THEN MVT2[OF `x < z1`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
   596       by auto
   597     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
   598       using subsetD[OF atMostAtLeast_subset_convex[OF `convex C` `z1 \<in> C` `y \<in> C` `z1 < y`],
   599         THEN f'', THEN MVT2[OF `z1 < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
   600       by auto
   601     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
   602       using asm z1' by auto
   603     also have "\<dots> = (y - z1) * f'' z3" using z3 by auto
   604     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" by simp
   605     have A': "y - z1 \<ge> 0" using z1 by auto
   606     have "z3 \<in> C" using z3 asm atMostAtLeast_subset_convex
   607       `convex C` `x \<in> C` `z1 \<in> C` `x < z1` by fastforce
   608     then have B': "f'' z3 \<ge> 0" using assms by auto
   609     from A' B' have "(y - z1) * f'' z3 \<ge> 0" by auto
   610     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0" by auto
   611     from mult_right_mono_neg[OF this le(2)]
   612     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
   613       by (simp add: algebra_simps)
   614     then have "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
   615     then have res: "f' y * (x - y) \<le> f x - f y" by auto
   616     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
   617       using asm z1 by auto
   618     also have "\<dots> = (z1 - x) * f'' z2" using z2 by auto
   619     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" by simp
   620     have A: "z1 - x \<ge> 0" using z1 by auto
   621     have "z2 \<in> C" using z2 z1 asm atMostAtLeast_subset_convex
   622       `convex C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastforce
   623     then have B: "f'' z2 \<ge> 0" using assms by auto
   624     from A B have "(z1 - x) * f'' z2 \<ge> 0" by auto
   625     from cool this have "(f y - f x) / (y - x) - f' x \<ge> 0" by auto
   626     from mult_right_mono[OF this ge(2)]
   627     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
   628       by (simp add: algebra_simps)
   629     then have "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
   630     then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
   631       using res by auto } note less_imp = this
   632   { fix x y :: real
   633     assume "x \<in> C" "y \<in> C" "x \<noteq> y"
   634     then have"f y - f x \<ge> f' x * (y - x)"
   635     unfolding neq_iff using less_imp by auto } note neq_imp = this
   636   moreover
   637   { fix x y :: real
   638     assume asm: "x \<in> C" "y \<in> C" "x = y"
   639     then have "f y - f x \<ge> f' x * (y - x)" by auto }
   640   ultimately show ?thesis using assms by blast
   641 qed
   642 
   643 lemma f''_ge0_imp_convex:
   644   fixes f :: "real \<Rightarrow> real"
   645   assumes conv: "convex C"
   646     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
   647     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
   648     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
   649   shows "convex_on C f"
   650 using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function by fastforce
   651 
   652 lemma minus_log_convex:
   653   fixes b :: real
   654   assumes "b > 1"
   655   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
   656 proof -
   657   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" using DERIV_log by auto
   658   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
   659     by (auto simp: DERIV_minus)
   660   have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
   661     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
   662   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
   663   have "\<And>z :: real. z > 0 \<Longrightarrow>
   664     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
   665     by auto
   666   then have f''0: "\<And>z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
   667     unfolding inverse_eq_divide by (auto simp add: mult_assoc)
   668   have f''_ge0: "\<And>z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
   669     using `b > 1` by (auto intro!:less_imp_le)
   670   from f''_ge0_imp_convex[OF pos_is_convex,
   671     unfolded greaterThan_iff, OF f' f''0 f''_ge0]
   672   show ?thesis by auto
   673 qed
   674 
   675 end