src/HOL/Library/Convex.thy
 author wenzelm Wed Jun 10 22:28:56 2015 +0200 (2015-06-10) changeset 60423 5035a2af185b parent 60303 00c06f1315d0 child 60449 229bad93377e permissions -rw-r--r--
misc tuning;
```     1 (*  Title:      HOL/Library/Convex.thy
```
```     2     Author:     Armin Heller, TU Muenchen
```
```     3     Author:     Johannes Hoelzl, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 section \<open>Convexity in real vector spaces\<close>
```
```     7
```
```     8 theory Convex
```
```     9 imports Product_Vector
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Convexity\<close>
```
```    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   {
```
```    33     fix x y and u v :: real
```
```    34     assume mem: "x \<in> s" "y \<in> s"
```
```    35     assume "0 \<le> u" "0 \<le> v"
```
```    36     moreover
```
```    37     assume "u + v = 1"
```
```    38     then have "u = 1 - v" by auto
```
```    39     ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
```
```    40       using alt[OF mem] by auto
```
```    41   }
```
```    42   then show "convex s"
```
```    43     unfolding convex_def by auto
```
```    44 qed (auto simp: convex_def)
```
```    45
```
```    46 lemma mem_convex:
```
```    47   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
```
```    48   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
```
```    49   using assms unfolding convex_alt by auto
```
```    50
```
```    51 lemma convex_empty[intro,simp]: "convex {}"
```
```    52   unfolding convex_def by simp
```
```    53
```
```    54 lemma convex_singleton[intro,simp]: "convex {a}"
```
```    55   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
```
```    56
```
```    57 lemma convex_UNIV[intro,simp]: "convex UNIV"
```
```    58   unfolding convex_def by auto
```
```    59
```
```    60 lemma convex_Inter: "(\<forall>s\<in>f. convex s) \<Longrightarrow> convex(\<Inter>f)"
```
```    61   unfolding convex_def by auto
```
```    62
```
```    63 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
```
```    64   unfolding convex_def by auto
```
```    65
```
```    66 lemma convex_INT: "\<forall>i\<in>A. convex (B i) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
```
```    67   unfolding convex_def by auto
```
```    68
```
```    69 lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
```
```    70   unfolding convex_def by auto
```
```    71
```
```    72 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
```
```    73   unfolding convex_def
```
```    74   by (auto simp: inner_add intro!: convex_bound_le)
```
```    75
```
```    76 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
```
```    77 proof -
```
```    78   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
```
```    79     by auto
```
```    80   show ?thesis
```
```    81     unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
```
```    82 qed
```
```    83
```
```    84 lemma convex_hyperplane: "convex {x. inner a x = b}"
```
```    85 proof -
```
```    86   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
```
```    87     by auto
```
```    88   show ?thesis using convex_halfspace_le convex_halfspace_ge
```
```    89     by (auto intro!: convex_Int simp: *)
```
```    90 qed
```
```    91
```
```    92 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
```
```    93   unfolding convex_def
```
```    94   by (auto simp: convex_bound_lt inner_add)
```
```    95
```
```    96 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
```
```    97    using convex_halfspace_lt[of "-a" "-b"] by auto
```
```    98
```
```    99 lemma convex_real_interval:
```
```   100   fixes a b :: "real"
```
```   101   shows "convex {a..}" and "convex {..b}"
```
```   102     and "convex {a<..}" and "convex {..<b}"
```
```   103     and "convex {a..b}" and "convex {a<..b}"
```
```   104     and "convex {a..<b}" and "convex {a<..<b}"
```
```   105 proof -
```
```   106   have "{a..} = {x. a \<le> inner 1 x}"
```
```   107     by auto
```
```   108   then show 1: "convex {a..}"
```
```   109     by (simp only: convex_halfspace_ge)
```
```   110   have "{..b} = {x. inner 1 x \<le> b}"
```
```   111     by auto
```
```   112   then show 2: "convex {..b}"
```
```   113     by (simp only: convex_halfspace_le)
```
```   114   have "{a<..} = {x. a < inner 1 x}"
```
```   115     by auto
```
```   116   then show 3: "convex {a<..}"
```
```   117     by (simp only: convex_halfspace_gt)
```
```   118   have "{..<b} = {x. inner 1 x < b}"
```
```   119     by auto
```
```   120   then show 4: "convex {..<b}"
```
```   121     by (simp only: convex_halfspace_lt)
```
```   122   have "{a..b} = {a..} \<inter> {..b}"
```
```   123     by auto
```
```   124   then show "convex {a..b}"
```
```   125     by (simp only: convex_Int 1 2)
```
```   126   have "{a<..b} = {a<..} \<inter> {..b}"
```
```   127     by auto
```
```   128   then show "convex {a<..b}"
```
```   129     by (simp only: convex_Int 3 2)
```
```   130   have "{a..<b} = {a..} \<inter> {..<b}"
```
```   131     by auto
```
```   132   then show "convex {a..<b}"
```
```   133     by (simp only: convex_Int 1 4)
```
```   134   have "{a<..<b} = {a<..} \<inter> {..<b}"
```
```   135     by auto
```
```   136   then show "convex {a<..<b}"
```
```   137     by (simp only: convex_Int 3 4)
```
```   138 qed
```
```   139
```
```   140 lemma convex_Reals: "convex Reals"
```
```   141   by (simp add: convex_def scaleR_conv_of_real)
```
```   142
```
```   143
```
```   144 subsection \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
```
```   145
```
```   146 lemma convex_setsum:
```
```   147   fixes C :: "'a::real_vector set"
```
```   148   assumes "finite s"
```
```   149     and "convex C"
```
```   150     and "(\<Sum> i \<in> s. a i) = 1"
```
```   151   assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```   152     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
```
```   153   shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
```
```   154   using assms(1,3,4,5)
```
```   155 proof (induct arbitrary: a set: finite)
```
```   156   case empty
```
```   157   then show ?case by simp
```
```   158 next
```
```   159   case (insert i s) note IH = this(3)
```
```   160   have "a i + setsum a s = 1"
```
```   161     and "0 \<le> a i"
```
```   162     and "\<forall>j\<in>s. 0 \<le> a j"
```
```   163     and "y i \<in> C"
```
```   164     and "\<forall>j\<in>s. y j \<in> C"
```
```   165     using insert.hyps(1,2) insert.prems by simp_all
```
```   166   then have "0 \<le> setsum a s"
```
```   167     by (simp add: setsum_nonneg)
```
```   168   have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
```
```   169   proof (cases)
```
```   170     assume z: "setsum a s = 0"
```
```   171     with \<open>a i + setsum a s = 1\<close> have "a i = 1"
```
```   172       by simp
```
```   173     from setsum_nonneg_0 [OF \<open>finite s\<close> _ z] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
```
```   174       by simp
```
```   175     show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
```
```   176       by simp
```
```   177   next
```
```   178     assume nz: "setsum a s \<noteq> 0"
```
```   179     with \<open>0 \<le> setsum a s\<close> have "0 < setsum a s"
```
```   180       by simp
```
```   181     then have "(\<Sum>j\<in>s. (a j / setsum a s) *\<^sub>R y j) \<in> C"
```
```   182       using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
```
```   183       by (simp add: IH setsum_divide_distrib [symmetric])
```
```   184     from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
```
```   185       and \<open>0 \<le> setsum a s\<close> and \<open>a i + setsum a s = 1\<close>
```
```   186     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"
```
```   187       by (rule convexD)
```
```   188     then show ?thesis
```
```   189       by (simp add: scaleR_setsum_right nz)
```
```   190   qed
```
```   191   then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
```
```   192     by simp
```
```   193 qed
```
```   194
```
```   195 lemma convex:
```
```   196   "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)
```
```   197       \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
```
```   198 proof safe
```
```   199   fix k :: nat
```
```   200   fix u :: "nat \<Rightarrow> real"
```
```   201   fix x
```
```   202   assume "convex s"
```
```   203     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
```
```   204     "setsum u {1..k} = 1"
```
```   205   with convex_setsum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
```
```   206     by auto
```
```   207 next
```
```   208   assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
```
```   209     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
```
```   210   {
```
```   211     fix \<mu> :: real
```
```   212     fix x y :: 'a
```
```   213     assume xy: "x \<in> s" "y \<in> s"
```
```   214     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   215     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
```
```   216     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
```
```   217     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
```
```   218       by auto
```
```   219     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
```
```   220       by simp
```
```   221     then have "setsum ?u {1 .. 2} = 1"
```
```   222       using setsum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
```
```   223       by auto
```
```   224     with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
```
```   225       using mu xy by auto
```
```   226     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
```
```   227       using setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
```
```   228     from setsum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
```
```   229     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
```
```   230       by auto
```
```   231     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
```
```   232       using s by (auto simp: add.commute)
```
```   233   }
```
```   234   then show "convex s"
```
```   235     unfolding convex_alt by auto
```
```   236 qed
```
```   237
```
```   238
```
```   239 lemma convex_explicit:
```
```   240   fixes s :: "'a::real_vector set"
```
```   241   shows "convex s \<longleftrightarrow>
```
```   242     (\<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)"
```
```   243 proof safe
```
```   244   fix t
```
```   245   fix u :: "'a \<Rightarrow> real"
```
```   246   assume "convex s"
```
```   247     and "finite t"
```
```   248     and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
```
```   249   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
```
```   250     using convex_setsum[of t s u "\<lambda> x. x"] by auto
```
```   251 next
```
```   252   assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
```
```   253     setsum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
```
```   254   show "convex s"
```
```   255     unfolding convex_alt
```
```   256   proof safe
```
```   257     fix x y
```
```   258     fix \<mu> :: real
```
```   259     assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
```
```   260     show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
```
```   261     proof (cases "x = y")
```
```   262       case False
```
```   263       then show ?thesis
```
```   264         using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
```
```   265           by auto
```
```   266     next
```
```   267       case True
```
```   268       then show ?thesis
```
```   269         using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
```
```   270           by (auto simp: field_simps real_vector.scale_left_diff_distrib)
```
```   271     qed
```
```   272   qed
```
```   273 qed
```
```   274
```
```   275 lemma convex_finite:
```
```   276   assumes "finite s"
```
```   277   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)"
```
```   278   unfolding convex_explicit
```
```   279 proof safe
```
```   280   fix t u
```
```   281   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"
```
```   282     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
```
```   283   have *: "s \<inter> t = t"
```
```   284     using as(2) by auto
```
```   285   have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
```
```   286     by simp
```
```   287   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
```
```   288    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
```
```   289    by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
```
```   290 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
```
```   291
```
```   292
```
```   293 subsection \<open>Functions that are convex on a set\<close>
```
```   294
```
```   295 definition convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
```
```   296   where "convex_on s f \<longleftrightarrow>
```
```   297     (\<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)"
```
```   298
```
```   299 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
```
```   300   unfolding convex_on_def by auto
```
```   301
```
```   302 lemma convex_on_add [intro]:
```
```   303   assumes "convex_on s f"
```
```   304     and "convex_on s g"
```
```   305   shows "convex_on s (\<lambda>x. f x + g x)"
```
```   306 proof -
```
```   307   {
```
```   308     fix x y
```
```   309     assume "x \<in> s" "y \<in> s"
```
```   310     moreover
```
```   311     fix u v :: real
```
```   312     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
```
```   313     ultimately
```
```   314     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)"
```
```   315       using assms unfolding convex_on_def by (auto simp: add_mono)
```
```   316     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)"
```
```   317       by (simp add: field_simps)
```
```   318   }
```
```   319   then show ?thesis
```
```   320     unfolding convex_on_def by auto
```
```   321 qed
```
```   322
```
```   323 lemma convex_on_cmul [intro]:
```
```   324   fixes c :: real
```
```   325   assumes "0 \<le> c"
```
```   326     and "convex_on s f"
```
```   327   shows "convex_on s (\<lambda>x. c * f x)"
```
```   328 proof -
```
```   329   have *: "\<And>u c fx v fy :: real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
```
```   330     by (simp add: field_simps)
```
```   331   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
```
```   332     unfolding convex_on_def and * by auto
```
```   333 qed
```
```   334
```
```   335 lemma convex_lower:
```
```   336   assumes "convex_on s f"
```
```   337     and "x \<in> s"
```
```   338     and "y \<in> s"
```
```   339     and "0 \<le> u"
```
```   340     and "0 \<le> v"
```
```   341     and "u + v = 1"
```
```   342   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
```
```   343 proof -
```
```   344   let ?m = "max (f x) (f y)"
```
```   345   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
```
```   346     using assms(4,5) by (auto simp: mult_left_mono add_mono)
```
```   347   also have "\<dots> = max (f x) (f y)"
```
```   348     using assms(6) by (simp add: distrib_right [symmetric])
```
```   349   finally show ?thesis
```
```   350     using assms unfolding convex_on_def by fastforce
```
```   351 qed
```
```   352
```
```   353 lemma convex_on_dist [intro]:
```
```   354   fixes s :: "'a::real_normed_vector set"
```
```   355   shows "convex_on s (\<lambda>x. dist a x)"
```
```   356 proof (auto simp: convex_on_def dist_norm)
```
```   357   fix x y
```
```   358   assume "x \<in> s" "y \<in> s"
```
```   359   fix u v :: real
```
```   360   assume "0 \<le> u"
```
```   361   assume "0 \<le> v"
```
```   362   assume "u + v = 1"
```
```   363   have "a = u *\<^sub>R a + v *\<^sub>R a"
```
```   364     unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
```
```   365   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
```
```   366     by (auto simp: algebra_simps)
```
```   367   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
```
```   368     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
```
```   369     using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
```
```   370 qed
```
```   371
```
```   372
```
```   373 subsection \<open>Arithmetic operations on sets preserve convexity\<close>
```
```   374
```
```   375 lemma convex_linear_image:
```
```   376   assumes "linear f"
```
```   377     and "convex s"
```
```   378   shows "convex (f ` s)"
```
```   379 proof -
```
```   380   interpret f: linear f by fact
```
```   381   from \<open>convex s\<close> show "convex (f ` s)"
```
```   382     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
```
```   383 qed
```
```   384
```
```   385 lemma convex_linear_vimage:
```
```   386   assumes "linear f"
```
```   387     and "convex s"
```
```   388   shows "convex (f -` s)"
```
```   389 proof -
```
```   390   interpret f: linear f by fact
```
```   391   from \<open>convex s\<close> show "convex (f -` s)"
```
```   392     by (simp add: convex_def f.add f.scaleR)
```
```   393 qed
```
```   394
```
```   395 lemma convex_scaling:
```
```   396   assumes "convex s"
```
```   397   shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
```
```   398 proof -
```
```   399   have "linear (\<lambda>x. c *\<^sub>R x)"
```
```   400     by (simp add: linearI scaleR_add_right)
```
```   401   then show ?thesis
```
```   402     using \<open>convex s\<close> by (rule convex_linear_image)
```
```   403 qed
```
```   404
```
```   405 lemma convex_scaled:
```
```   406   assumes "convex s"
```
```   407   shows "convex ((\<lambda>x. x *\<^sub>R c) ` s)"
```
```   408 proof -
```
```   409   have "linear (\<lambda>x. x *\<^sub>R c)"
```
```   410     by (simp add: linearI scaleR_add_left)
```
```   411   then show ?thesis
```
```   412     using \<open>convex s\<close> by (rule convex_linear_image)
```
```   413 qed
```
```   414
```
```   415 lemma convex_negations:
```
```   416   assumes "convex s"
```
```   417   shows "convex ((\<lambda>x. - x) ` s)"
```
```   418 proof -
```
```   419   have "linear (\<lambda>x. - x)"
```
```   420     by (simp add: linearI)
```
```   421   then show ?thesis
```
```   422     using \<open>convex s\<close> by (rule convex_linear_image)
```
```   423 qed
```
```   424
```
```   425 lemma convex_sums:
```
```   426   assumes "convex s"
```
```   427     and "convex t"
```
```   428   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
```
```   429 proof -
```
```   430   have "linear (\<lambda>(x, y). x + y)"
```
```   431     by (auto intro: linearI simp: scaleR_add_right)
```
```   432   with assms have "convex ((\<lambda>(x, y). x + y) ` (s \<times> t))"
```
```   433     by (intro convex_linear_image convex_Times)
```
```   434   also have "((\<lambda>(x, y). x + y) ` (s \<times> t)) = {x + y| x y. x \<in> s \<and> y \<in> t}"
```
```   435     by auto
```
```   436   finally show ?thesis .
```
```   437 qed
```
```   438
```
```   439 lemma convex_differences:
```
```   440   assumes "convex s" "convex t"
```
```   441   shows "convex {x - y| x y. x \<in> s \<and> y \<in> t}"
```
```   442 proof -
```
```   443   have "{x - y| x y. x \<in> s \<and> y \<in> t} = {x + y |x y. x \<in> s \<and> y \<in> uminus ` t}"
```
```   444     by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
```
```   445   then show ?thesis
```
```   446     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
```
```   447 qed
```
```   448
```
```   449 lemma convex_translation:
```
```   450   assumes "convex s"
```
```   451   shows "convex ((\<lambda>x. a + x) ` s)"
```
```   452 proof -
```
```   453   have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s"
```
```   454     by auto
```
```   455   then show ?thesis
```
```   456     using convex_sums[OF convex_singleton[of a] assms] by auto
```
```   457 qed
```
```   458
```
```   459 lemma convex_affinity:
```
```   460   assumes "convex s"
```
```   461   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` s)"
```
```   462 proof -
```
```   463   have "(\<lambda>x. a + c *\<^sub>R x) ` s = op + a ` op *\<^sub>R c ` s"
```
```   464     by auto
```
```   465   then show ?thesis
```
```   466     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
```
```   467 qed
```
```   468
```
```   469 lemma pos_is_convex: "convex {0 :: real <..}"
```
```   470   unfolding convex_alt
```
```   471 proof safe
```
```   472   fix y x \<mu> :: real
```
```   473   assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   474   {
```
```   475     assume "\<mu> = 0"
```
```   476     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
```
```   477     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
```
```   478   }
```
```   479   moreover
```
```   480   {
```
```   481     assume "\<mu> = 1"
```
```   482     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
```
```   483   }
```
```   484   moreover
```
```   485   {
```
```   486     assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
```
```   487     then have "\<mu> > 0" "(1 - \<mu>) > 0" using * by auto
```
```   488     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using *
```
```   489       by (auto simp: add_pos_pos)
```
```   490   }
```
```   491   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
```
```   492     using assms by fastforce
```
```   493 qed
```
```   494
```
```   495 lemma convex_on_setsum:
```
```   496   fixes a :: "'a \<Rightarrow> real"
```
```   497     and y :: "'a \<Rightarrow> 'b::real_vector"
```
```   498     and f :: "'b \<Rightarrow> real"
```
```   499   assumes "finite s" "s \<noteq> {}"
```
```   500     and "convex_on C f"
```
```   501     and "convex C"
```
```   502     and "(\<Sum> i \<in> s. a i) = 1"
```
```   503     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
```
```   504     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
```
```   505   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
```
```   506   using assms
```
```   507 proof (induct s arbitrary: a rule: finite_ne_induct)
```
```   508   case (singleton i)
```
```   509   then have ai: "a i = 1" by auto
```
```   510   then show ?case by auto
```
```   511 next
```
```   512   case (insert i s)
```
```   513   then have "convex_on C f" by simp
```
```   514   from this[unfolded convex_on_def, rule_format]
```
```   515   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
```
```   516       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   517     by simp
```
```   518   show ?case
```
```   519   proof (cases "a i = 1")
```
```   520     case True
```
```   521     then have "(\<Sum> j \<in> s. a j) = 0"
```
```   522       using insert by auto
```
```   523     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
```
```   524       using setsum_nonneg_0[where 'b=real] insert by fastforce
```
```   525     then show ?thesis
```
```   526       using insert by auto
```
```   527   next
```
```   528     case False
```
```   529     from insert have yai: "y i \<in> C" "a i \<ge> 0"
```
```   530       by auto
```
```   531     have fis: "finite (insert i s)"
```
```   532       using insert by auto
```
```   533     then have ai1: "a i \<le> 1"
```
```   534       using setsum_nonneg_leq_bound[of "insert i s" a] insert by simp
```
```   535     then have "a i < 1"
```
```   536       using False by auto
```
```   537     then have i0: "1 - a i > 0"
```
```   538       by auto
```
```   539     let ?a = "\<lambda>j. a j / (1 - a i)"
```
```   540     have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
```
```   541       using i0 insert prems by fastforce
```
```   542     have "(\<Sum> j \<in> insert i s. a j) = 1"
```
```   543       using insert by auto
```
```   544     then have "(\<Sum> j \<in> s. a j) = 1 - a i"
```
```   545       using setsum.insert insert by fastforce
```
```   546     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
```
```   547       using i0 by auto
```
```   548     then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
```
```   549       unfolding setsum_divide_distrib by simp
```
```   550     have "convex C" using insert by auto
```
```   551     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
```
```   552       using insert convex_setsum[OF \<open>finite s\<close>
```
```   553         \<open>convex C\<close> a1 a_nonneg] by auto
```
```   554     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
```
```   555       using a_nonneg a1 insert by blast
```
```   556     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)"
```
```   557       using setsum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
```
```   558       by (auto simp only: add.commute)
```
```   559     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)"
```
```   560       using i0 by auto
```
```   561     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)"
```
```   562       using scaleR_right.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
```
```   563       by (auto simp: algebra_simps)
```
```   564     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)"
```
```   565       by (auto simp: divide_inverse)
```
```   566     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)"
```
```   567       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
```
```   568       by (auto simp: add.commute)
```
```   569     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
```
```   570       using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
```
```   571         OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
```
```   572     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
```
```   573       unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
```
```   574     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
```
```   575       using i0 by auto
```
```   576     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
```
```   577       using insert by auto
```
```   578     finally show ?thesis
```
```   579       by simp
```
```   580   qed
```
```   581 qed
```
```   582
```
```   583 lemma convex_on_alt:
```
```   584   fixes C :: "'a::real_vector set"
```
```   585   assumes "convex C"
```
```   586   shows "convex_on C f \<longleftrightarrow>
```
```   587     (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
```
```   588       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
```
```   589 proof safe
```
```   590   fix x y
```
```   591   fix \<mu> :: real
```
```   592   assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
```
```   593   from this[unfolded convex_on_def, rule_format]
```
```   594   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"
```
```   595     by auto
```
```   596   from this[of "\<mu>" "1 - \<mu>", simplified] *
```
```   597   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   598     by auto
```
```   599 next
```
```   600   assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
```
```   601     f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   602   {
```
```   603     fix x y
```
```   604     fix u v :: real
```
```   605     assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
```
```   606     then have[simp]: "1 - u = v" by auto
```
```   607     from *[rule_format, of x y u]
```
```   608     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
```
```   609       using ** by auto
```
```   610   }
```
```   611   then show "convex_on C f"
```
```   612     unfolding convex_on_def by auto
```
```   613 qed
```
```   614
```
```   615 lemma convex_on_diff:
```
```   616   fixes f :: "real \<Rightarrow> real"
```
```   617   assumes f: "convex_on I f"
```
```   618     and I: "x \<in> I" "y \<in> I"
```
```   619     and t: "x < t" "t < y"
```
```   620   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```   621     and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
```
```   622 proof -
```
```   623   def a \<equiv> "(t - y) / (x - y)"
```
```   624   with t have "0 \<le> a" "0 \<le> 1 - a"
```
```   625     by (auto simp: field_simps)
```
```   626   with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
```
```   627     by (auto simp: convex_on_def)
```
```   628   have "a * x + (1 - a) * y = a * (x - y) + y"
```
```   629     by (simp add: field_simps)
```
```   630   also have "\<dots> = t"
```
```   631     unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
```
```   632   finally have "f t \<le> a * f x + (1 - a) * f y"
```
```   633     using cvx by simp
```
```   634   also have "\<dots> = a * (f x - f y) + f y"
```
```   635     by (simp add: field_simps)
```
```   636   finally have "f t - f y \<le> a * (f x - f y)"
```
```   637     by simp
```
```   638   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
```
```   639     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
```
```   640   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
```
```   641     by (simp add: le_divide_eq divide_le_eq field_simps)
```
```   642 qed
```
```   643
```
```   644 lemma pos_convex_function:
```
```   645   fixes f :: "real \<Rightarrow> real"
```
```   646   assumes "convex C"
```
```   647     and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
```
```   648   shows "convex_on C f"
```
```   649   unfolding convex_on_alt[OF assms(1)]
```
```   650   using assms
```
```   651 proof safe
```
```   652   fix x y \<mu> :: real
```
```   653   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
```
```   654   assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
```
```   655   then have "1 - \<mu> \<ge> 0" by auto
```
```   656   then have xpos: "?x \<in> C"
```
```   657     using * unfolding convex_alt by fastforce
```
```   658   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
```
```   659       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
```
```   660     using add_mono[OF mult_left_mono[OF leq[OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
```
```   661       mult_left_mono[OF leq[OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
```
```   662     by auto
```
```   663   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
```
```   664     by (auto simp: field_simps)
```
```   665   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
```
```   666     using convex_on_alt by auto
```
```   667 qed
```
```   668
```
```   669 lemma atMostAtLeast_subset_convex:
```
```   670   fixes C :: "real set"
```
```   671   assumes "convex C"
```
```   672     and "x \<in> C" "y \<in> C" "x < y"
```
```   673   shows "{x .. y} \<subseteq> C"
```
```   674 proof safe
```
```   675   fix z assume z: "z \<in> {x .. y}"
```
```   676   have less: "z \<in> C" if *: "x < z" "z < y"
```
```   677   proof -
```
```   678     let ?\<mu> = "(y - z) / (y - x)"
```
```   679     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
```
```   680       using assms * by (auto simp: field_simps)
```
```   681     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
```
```   682       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
```
```   683       by (simp add: algebra_simps)
```
```   684     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
```
```   685       by (auto simp: field_simps)
```
```   686     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
```
```   687       using assms unfolding add_divide_distrib by (auto simp: field_simps)
```
```   688     also have "\<dots> = z"
```
```   689       using assms by (auto simp: field_simps)
```
```   690     finally show ?thesis
```
```   691       using comb by auto
```
```   692   qed
```
```   693   show "z \<in> C" using z less assms
```
```   694     unfolding atLeastAtMost_iff le_less by auto
```
```   695 qed
```
```   696
```
```   697 lemma f''_imp_f':
```
```   698   fixes f :: "real \<Rightarrow> real"
```
```   699   assumes "convex C"
```
```   700     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
```
```   701     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
```
```   702     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
```
```   703     and "x \<in> C" "y \<in> C"
```
```   704   shows "f' x * (y - x) \<le> f y - f x"
```
```   705   using assms
```
```   706 proof -
```
```   707   {
```
```   708     fix x y :: real
```
```   709     assume *: "x \<in> C" "y \<in> C" "y > x"
```
```   710     then have ge: "y - x > 0" "y - x \<ge> 0"
```
```   711       by auto
```
```   712     from * have le: "x - y < 0" "x - y \<le> 0"
```
```   713       by auto
```
```   714     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
```
```   715       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
```
```   716         THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
```
```   717       by auto
```
```   718     then have "z1 \<in> C"
```
```   719       using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
```
```   720       by fastforce
```
```   721     from z1 have z1': "f x - f y = (x - y) * f' z1"
```
```   722       by (simp add: field_simps)
```
```   723     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
```
```   724       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
```
```   725         THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
```
```   726       by auto
```
```   727     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
```
```   728       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
```
```   729         THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
```
```   730       by auto
```
```   731     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
```
```   732       using * z1' by auto
```
```   733     also have "\<dots> = (y - z1) * f'' z3"
```
```   734       using z3 by auto
```
```   735     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
```
```   736       by simp
```
```   737     have A': "y - z1 \<ge> 0"
```
```   738       using z1 by auto
```
```   739     have "z3 \<in> C"
```
```   740       using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
```
```   741       by fastforce
```
```   742     then have B': "f'' z3 \<ge> 0"
```
```   743       using assms by auto
```
```   744     from A' B' have "(y - z1) * f'' z3 \<ge> 0"
```
```   745       by auto
```
```   746     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
```
```   747       by auto
```
```   748     from mult_right_mono_neg[OF this le(2)]
```
```   749     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
```
```   750       by (simp add: algebra_simps)
```
```   751     then have "f' y * (x - y) - (f x - f y) \<le> 0"
```
```   752       using le by auto
```
```   753     then have res: "f' y * (x - y) \<le> f x - f y"
```
```   754       by auto
```
```   755     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
```
```   756       using * z1 by auto
```
```   757     also have "\<dots> = (z1 - x) * f'' z2"
```
```   758       using z2 by auto
```
```   759     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
```
```   760       by simp
```
```   761     have A: "z1 - x \<ge> 0"
```
```   762       using z1 by auto
```
```   763     have "z2 \<in> C"
```
```   764       using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
```
```   765       by fastforce
```
```   766     then have B: "f'' z2 \<ge> 0"
```
```   767       using assms by auto
```
```   768     from A B have "(z1 - x) * f'' z2 \<ge> 0"
```
```   769       by auto
```
```   770     with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
```
```   771       by auto
```
```   772     from mult_right_mono[OF this ge(2)]
```
```   773     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
```
```   774       by (simp add: algebra_simps)
```
```   775     then have "f y - f x - f' x * (y - x) \<ge> 0"
```
```   776       using ge by auto
```
```   777     then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
```
```   778       using res by auto
```
```   779   } note less_imp = this
```
```   780   {
```
```   781     fix x y :: real
```
```   782     assume "x \<in> C" "y \<in> C" "x \<noteq> y"
```
```   783     then have"f y - f x \<ge> f' x * (y - x)"
```
```   784     unfolding neq_iff using less_imp by auto
```
```   785   }
```
```   786   moreover
```
```   787   {
```
```   788     fix x y :: real
```
```   789     assume "x \<in> C" "y \<in> C" "x = y"
```
```   790     then have "f y - f x \<ge> f' x * (y - x)" by auto
```
```   791   }
```
```   792   ultimately show ?thesis using assms by blast
```
```   793 qed
```
```   794
```
```   795 lemma f''_ge0_imp_convex:
```
```   796   fixes f :: "real \<Rightarrow> real"
```
```   797   assumes conv: "convex C"
```
```   798     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
```
```   799     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
```
```   800     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
```
```   801   shows "convex_on C f"
```
```   802   using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
```
```   803   by fastforce
```
```   804
```
```   805 lemma minus_log_convex:
```
```   806   fixes b :: real
```
```   807   assumes "b > 1"
```
```   808   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
```
```   809 proof -
```
```   810   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
```
```   811     using DERIV_log by auto
```
```   812   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
```
```   813     by (auto simp: DERIV_minus)
```
```   814   have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
```
```   815     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
```
```   816   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
```
```   817   have "\<And>z :: real. z > 0 \<Longrightarrow>
```
```   818     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
```
```   819     by auto
```
```   820   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
```
```   821     DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
```
```   822     unfolding inverse_eq_divide by (auto simp: mult.assoc)
```
```   823   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
```
```   824     using \<open>b > 1\<close> by (auto intro!: less_imp_le)
```
```   825   from f''_ge0_imp_convex[OF pos_is_convex,
```
```   826     unfolded greaterThan_iff, OF f' f''0 f''_ge0]
```
```   827   show ?thesis by auto
```
```   828 qed
```
```   829
```
```   830 end
```