diff --git a/src/HOL/Analysis/Convex.thy b/src/HOL/Analysis/Convex.thy
--- a/src/HOL/Analysis/Convex.thy
+++ b/src/HOL/Analysis/Convex.thy
@@ -1,2453 +1,2371 @@
 (* Title:      HOL/Analysis/Convex.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 
 section \<open>Convex Sets and Functions\<close>
 
 theory Convex
 imports
   Affine
   "HOL-Library.Set_Algebras"
 begin
 
 subsection \<open>Convex Sets\<close>
 
 definition\<^marker>\<open>tag important\<close> convex :: "'a::real_vector set \<Rightarrow> bool"
   where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
 
 lemma convexI:
   assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
   shows "convex s"
   using assms unfolding convex_def by fast
 
 lemma convexD:
   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
   using assms unfolding convex_def by fast
 
 lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
   (is "_ \<longleftrightarrow> ?alt")
 proof
   show "convex s" if alt: ?alt
   proof -
     {
       fix x y and u v :: real
       assume mem: "x \<in> s" "y \<in> s"
       assume "0 \<le> u" "0 \<le> v"
       moreover
       assume "u + v = 1"
       then have "u = 1 - v" by auto
       ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
         using alt [rule_format, OF mem] by auto
     }
     then show ?thesis
       unfolding convex_def by auto
   qed
   show ?alt if "convex s"
     using that by (auto simp: convex_def)
 qed
 
 lemma convexD_alt:
   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
   using assms unfolding convex_alt by auto
 
 lemma mem_convex_alt:
   assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
-  apply (rule convexD)
   using assms
-       apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
-  done
+  by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric])
 
 lemma convex_empty[intro,simp]: "convex {}"
   unfolding convex_def by simp
 
 lemma convex_singleton[intro,simp]: "convex {a}"
   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
 
 lemma convex_UNIV[intro,simp]: "convex UNIV"
   unfolding convex_def by auto
 
 lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
   unfolding convex_def by auto
 
 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
   unfolding convex_def by auto
 
 lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
   unfolding convex_def by auto
 
 lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
   unfolding convex_def by auto
 
 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
   unfolding convex_def
   by (auto simp: inner_add intro!: convex_bound_le)
 
 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
 proof -
   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
     by auto
   show ?thesis
     unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
 qed
 
 lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
 proof -
   have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
     by auto
   show ?thesis
     unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
 qed
 
 lemma convex_hyperplane: "convex {x. inner a x = b}"
 proof -
   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
     by auto
   show ?thesis using convex_halfspace_le convex_halfspace_ge
     by (auto intro!: convex_Int simp: *)
 qed
 
 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
   unfolding convex_def
   by (auto simp: convex_bound_lt inner_add)
 
 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
   using convex_halfspace_lt[of "-a" "-b"] by auto
 
 lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
   using convex_halfspace_ge[of b "1::complex"] by simp
 
 lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
   using convex_halfspace_le[of "1::complex" b] by simp
 
 lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
   using convex_halfspace_ge[of b \<i>] by simp
 
 lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
   using convex_halfspace_le[of \<i> b] by simp
 
 lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
   using convex_halfspace_gt[of b "1::complex"] by simp
 
 lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
   using convex_halfspace_lt[of "1::complex" b] by simp
 
 lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
   using convex_halfspace_gt[of b \<i>] by simp
 
 lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
   using convex_halfspace_lt[of \<i> b] by simp
 
 lemma convex_real_interval [iff]:
   fixes a b :: "real"
   shows "convex {a..}" and "convex {..b}"
     and "convex {a<..}" and "convex {..<b}"
     and "convex {a..b}" and "convex {a<..b}"
     and "convex {a..<b}" and "convex {a<..<b}"
 proof -
   have "{a..} = {x. a \<le> inner 1 x}"
     by auto
   then show 1: "convex {a..}"
     by (simp only: convex_halfspace_ge)
   have "{..b} = {x. inner 1 x \<le> b}"
     by auto
   then show 2: "convex {..b}"
     by (simp only: convex_halfspace_le)
   have "{a<..} = {x. a < inner 1 x}"
     by auto
   then show 3: "convex {a<..}"
     by (simp only: convex_halfspace_gt)
   have "{..<b} = {x. inner 1 x < b}"
     by auto
   then show 4: "convex {..<b}"
     by (simp only: convex_halfspace_lt)
   have "{a..b} = {a..} \<inter> {..b}"
     by auto
   then show "convex {a..b}"
     by (simp only: convex_Int 1 2)
   have "{a<..b} = {a<..} \<inter> {..b}"
     by auto
   then show "convex {a<..b}"
     by (simp only: convex_Int 3 2)
   have "{a..<b} = {a..} \<inter> {..<b}"
     by auto
   then show "convex {a..<b}"
     by (simp only: convex_Int 1 4)
   have "{a<..<b} = {a<..} \<inter> {..<b}"
     by auto
   then show "convex {a<..<b}"
     by (simp only: convex_Int 3 4)
 qed
 
 lemma convex_Reals: "convex \<real>"
   by (simp add: convex_def scaleR_conv_of_real)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
 
 lemma convex_sum:
   fixes C :: "'a::real_vector set"
-  assumes "finite s"
+  assumes "finite S"
     and "convex C"
-    and "(\<Sum> i \<in> s. a i) = 1"
-  assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
-    and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
-  shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C"
+    and "(\<Sum> i \<in> S. a i) = 1"
+  assumes "\<And>i. i \<in> S \<Longrightarrow> a i \<ge> 0"
+    and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C"
+  shows "(\<Sum> j \<in> S. a j *\<^sub>R y j) \<in> C"
   using assms(1,3,4,5)
 proof (induct arbitrary: a set: finite)
   case empty
   then show ?case by simp
 next
-  case (insert i s) note IH = this(3)
-  have "a i + sum a s = 1"
+  case (insert i S) note IH = this(3)
+  have "a i + sum a S = 1"
     and "0 \<le> a i"
-    and "\<forall>j\<in>s. 0 \<le> a j"
+    and "\<forall>j\<in>S. 0 \<le> a j"
     and "y i \<in> C"
-    and "\<forall>j\<in>s. y j \<in> C"
+    and "\<forall>j\<in>S. y j \<in> C"
     using insert.hyps(1,2) insert.prems by simp_all
-  then have "0 \<le> sum a s"
+  then have "0 \<le> sum a S"
     by (simp add: sum_nonneg)
-  have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C"
-  proof (cases "sum a s = 0")
+  have "a i *\<^sub>R y i + (\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
+  proof (cases "sum a S = 0")
     case True
-    with \<open>a i + sum a s = 1\<close> have "a i = 1"
+    with \<open>a i + sum a S = 1\<close> have "a i = 1"
       by simp
-    from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0"
+    from sum_nonneg_0 [OF \<open>finite S\<close> _ True] \<open>\<forall>j\<in>S. 0 \<le> a j\<close> have "\<forall>j\<in>S. a j = 0"
       by simp
-    show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close>
+    show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>S. a j = 0\<close> and \<open>y i \<in> C\<close>
       by simp
   next
     case False
-    with \<open>0 \<le> sum a s\<close> have "0 < sum a s"
+    with \<open>0 \<le> sum a S\<close> have "0 < sum a S"
       by simp
-    then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
-      using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close>
+    then have "(\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C"
+      using \<open>\<forall>j\<in>S. 0 \<le> a j\<close> and \<open>\<forall>j\<in>S. y j \<in> C\<close>
       by (simp add: IH sum_divide_distrib [symmetric])
     from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
-      and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close>
-    have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C"
+      and \<open>0 \<le> sum a S\<close> and \<open>a i + sum a S = 1\<close>
+    have "a i *\<^sub>R y i + sum a S *\<^sub>R (\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C"
       by (rule convexD)
     then show ?thesis
       by (simp add: scaleR_sum_right False)
   qed
-  then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close>
+  then show ?case using \<open>finite S\<close> and \<open>i \<notin> S\<close>
     by simp
 qed
 
 lemma convex:
-  "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (sum u {1..k} = 1)
-      \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
+  "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> (sum u {1..k} = 1)
+      \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> S)"
 proof safe
   fix k :: nat
   fix u :: "nat \<Rightarrow> real"
   fix x
-  assume "convex s"
-    "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s"
+  assume "convex S"
+    "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S"
     "sum u {1..k} = 1"
-  with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s"
+  with convex_sum[of "{1 .. k}" S] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> S"
     by auto
 next
-  assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1
-    \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s"
+  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> sum u {1..k} = 1
+    \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> S"
   {
     fix \<mu> :: real
     fix x y :: 'a
-    assume xy: "x \<in> s" "y \<in> s"
+    assume xy: "x \<in> S" "y \<in> S"
     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
       by auto
     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
       by simp
     then have "sum ?u {1 .. 2} = 1"
       using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
       by auto
-    with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
+    with *[rule_format, of "2" ?u ?x] have S: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> S"
       using mu xy by auto
     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
       using sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
     from sum.atLeast_Suc_atMost[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
       by auto
-    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s"
-      using s by (auto simp: add.commute)
+    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> S"
+      using S by (auto simp: add.commute)
   }
-  then show "convex s"
+  then show "convex S"
     unfolding convex_alt by auto
 qed
 
 
 lemma convex_explicit:
-  fixes s :: "'a::real_vector set"
-  shows "convex s \<longleftrightarrow>
-    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
+  fixes S :: "'a::real_vector set"
+  shows "convex S \<longleftrightarrow>
+    (\<forall>t u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> S)"
 proof safe
   fix t
   fix u :: "'a \<Rightarrow> real"
-  assume "convex s"
+  assume "convex S"
     and "finite t"
-    and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
-  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-    using convex_sum[of t s u "\<lambda> x. x"] by auto
+    and "t \<subseteq> S" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
+  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S"
+    using convex_sum[of t S u "\<lambda> x. x"] by auto
 next
-  assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
-    sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-  show "convex s"
+  assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
+    sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S"
+  show "convex S"
     unfolding convex_alt
   proof safe
     fix x y
     fix \<mu> :: real
-    assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
-    show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+    assume **: "x \<in> S" "y \<in> S" "0 \<le> \<mu>" "\<mu> \<le> 1"
+    show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> S"
     proof (cases "x = y")
       case False
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
         by auto
     next
       case True
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
         by (auto simp: field_simps real_vector.scale_left_diff_distrib)
     qed
   qed
 qed
 
 lemma convex_finite:
-  assumes "finite s"
-  shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
-  unfolding convex_explicit
-  apply safe
-  subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
-  subgoal for t u
-  proof -
-    have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
+  assumes "finite S"
+  shows "convex S \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) S \<in> S)"
+       (is "?lhs = ?rhs")
+proof 
+  { have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
       by simp
-    assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
-    assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
-    assume "t \<subseteq> s"
-    then have "s \<inter> t = t" by auto
-    with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
-      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
-  qed
-  done
+    fix T :: "'a set" and u :: "'a \<Rightarrow> real"
+    assume sum: "\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> S"
+    assume *: "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1"
+    assume "T \<subseteq> S"
+    then have "S \<inter> T = T" by auto
+    with sum[THEN spec[where x="\<lambda>x. if x\<in>T then u x else 0"]] * have "(\<Sum>x\<in>T. u x *\<^sub>R x) \<in> S"
+      by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) }
+  moreover assume ?rhs
+  ultimately show ?lhs
+    unfolding convex_explicit by auto
+qed (auto simp: convex_explicit assms)
 
 
 subsection \<open>Convex Functions on a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
-  where "convex_on s f \<longleftrightarrow>
-    (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
+  where "convex_on S f \<longleftrightarrow>
+    (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
 
 lemma convex_onI [intro?]:
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   shows "convex_on A f"
   unfolding convex_on_def
 proof clarify
   fix x y
   fix u v :: real
   assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
   from A(5) have [simp]: "v = 1 - u"
     by (simp add: algebra_simps)
   from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
     using assms[of u y x]
     by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
 qed
 
 lemma convex_on_linorderI [intro?]:
   fixes A :: "('a::{linorder,real_vector}) set"
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   shows "convex_on A f"
 proof
   fix x y
   fix t :: real
   assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
   with assms [of t x y] assms [of "1 - t" y x]
   show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
     by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
 qed
 
 lemma convex_onD:
   assumes "convex_on A f"
   shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   using assms by (auto simp: convex_on_def)
 
 lemma convex_onD_Icc:
   assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
   shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
 
-lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
+lemma convex_on_subset: "convex_on t f \<Longrightarrow> S \<subseteq> t \<Longrightarrow> convex_on S f"
   unfolding convex_on_def by auto
 
 lemma convex_on_add [intro]:
-  assumes "convex_on s f"
-    and "convex_on s g"
-  shows "convex_on s (\<lambda>x. f x + g x)"
+  assumes "convex_on S f"
+    and "convex_on S g"
+  shows "convex_on S (\<lambda>x. f x + g x)"
 proof -
   {
     fix x y
-    assume "x \<in> s" "y \<in> s"
+    assume "x \<in> S" "y \<in> S"
     moreover
     fix u v :: real
     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
     ultimately
     have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
       using assms unfolding convex_on_def by (auto simp: add_mono)
     then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
       by (simp add: field_simps)
   }
   then show ?thesis
     unfolding convex_on_def by auto
 qed
 
 lemma convex_on_cmul [intro]:
   fixes c :: real
   assumes "0 \<le> c"
-    and "convex_on s f"
-  shows "convex_on s (\<lambda>x. c * f x)"
+    and "convex_on S f"
+  shows "convex_on S (\<lambda>x. c * f x)"
 proof -
   have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
     for u c fx v fy :: real
     by (simp add: field_simps)
   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
     unfolding convex_on_def and * by auto
 qed
 
 lemma convex_lower:
-  assumes "convex_on s f"
-    and "x \<in> s"
-    and "y \<in> s"
+  assumes "convex_on S f"
+    and "x \<in> S"
+    and "y \<in> S"
     and "0 \<le> u"
     and "0 \<le> v"
     and "u + v = 1"
   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
 proof -
   let ?m = "max (f x) (f y)"
   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
     using assms(4,5) by (auto simp: mult_left_mono add_mono)
   also have "\<dots> = max (f x) (f y)"
     using assms(6) by (simp add: distrib_right [symmetric])
   finally show ?thesis
     using assms unfolding convex_on_def by fastforce
 qed
 
 lemma convex_on_dist [intro]:
-  fixes s :: "'a::real_normed_vector set"
-  shows "convex_on s (\<lambda>x. dist a x)"
+  fixes S :: "'a::real_normed_vector set"
+  shows "convex_on S (\<lambda>x. dist a x)"
 proof (auto simp: convex_on_def dist_norm)
   fix x y
-  assume "x \<in> s" "y \<in> s"
+  assume "x \<in> S" "y \<in> S"
   fix u v :: real
   assume "0 \<le> u"
   assume "0 \<le> v"
   assume "u + v = 1"
   have "a = u *\<^sub>R a + v *\<^sub>R a"
     unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
     by (auto simp: algebra_simps)
   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
     using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic operations on sets preserve convexity\<close>
 
 lemma convex_linear_image:
   assumes "linear f"
-    and "convex s"
-  shows "convex (f ` s)"
+    and "convex S"
+  shows "convex (f ` S)"
 proof -
   interpret f: linear f by fact
-  from \<open>convex s\<close> show "convex (f ` s)"
+  from \<open>convex S\<close> show "convex (f ` S)"
     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
 qed
 
 lemma convex_linear_vimage:
   assumes "linear f"
-    and "convex s"
-  shows "convex (f -` s)"
+    and "convex S"
+  shows "convex (f -` S)"
 proof -
   interpret f: linear f by fact
-  from \<open>convex s\<close> show "convex (f -` s)"
+  from \<open>convex S\<close> show "convex (f -` S)"
     by (simp add: convex_def f.add f.scaleR)
 qed
 
 lemma convex_scaling:
-  assumes "convex s"
-  shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)"
+  assumes "convex S"
+  shows "convex ((\<lambda>x. c *\<^sub>R x) ` S)"
 proof -
   have "linear (\<lambda>x. c *\<^sub>R x)"
     by (simp add: linearI scaleR_add_right)
   then show ?thesis
-    using \<open>convex s\<close> by (rule convex_linear_image)
+    using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_scaled:
   assumes "convex S"
   shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
 proof -
   have "linear (\<lambda>x. x *\<^sub>R c)"
     by (simp add: linearI scaleR_add_left)
   then show ?thesis
     using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_negations:
   assumes "convex S"
   shows "convex ((\<lambda>x. - x) ` S)"
 proof -
   have "linear (\<lambda>x. - x)"
     by (simp add: linearI)
   then show ?thesis
     using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_sums:
   assumes "convex S"
     and "convex T"
   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   have "linear (\<lambda>(x, y). x + y)"
     by (auto intro: linearI simp: scaleR_add_right)
   with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
     by (intro convex_linear_image convex_Times)
   also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by auto
   finally show ?thesis .
 qed
 
 lemma convex_differences:
   assumes "convex S" "convex T"
   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
     by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
   then show ?thesis
     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
 qed
 
 lemma convex_translation:
   "convex ((+) a ` S)" if "convex S"
 proof -
   have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (+) a ` S"
     by auto
   then show ?thesis
     using convex_sums [OF convex_singleton [of a] that] by auto
 qed
 
 lemma convex_translation_subtract:
   "convex ((\<lambda>b. b - a) ` S)" if "convex S"
   using convex_translation [of S "- a"] that by (simp cong: image_cong_simp)
 
 lemma convex_affinity:
   assumes "convex S"
   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
   have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
     by auto
   then show ?thesis
     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
 qed
 
 lemma convex_on_sum:
   fixes a :: "'a \<Rightarrow> real"
     and y :: "'a \<Rightarrow> 'b::real_vector"
     and f :: "'b \<Rightarrow> real"
   assumes "finite s" "s \<noteq> {}"
     and "convex_on C f"
     and "convex C"
     and "(\<Sum> i \<in> s. a i) = 1"
     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
   using assms
 proof (induct s arbitrary: a rule: finite_ne_induct)
   case (singleton i)
   then have ai: "a i = 1"
     by auto
   then show ?case
     by auto
 next
   case (insert i s)
   then have "convex_on C f"
     by simp
   from this[unfolded convex_on_def, rule_format]
   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by simp
   show ?case
   proof (cases "a i = 1")
     case True
     then have "(\<Sum> j \<in> s. a j) = 0"
       using insert by auto
     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
       using insert by (fastforce simp: sum_nonneg_eq_0_iff)
     then show ?thesis
       using insert by auto
   next
     case False
     from insert have yai: "y i \<in> C" "a i \<ge> 0"
       by auto
     have fis: "finite (insert i s)"
       using insert by auto
     then have ai1: "a i \<le> 1"
       using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
     then have "a i < 1"
       using False by auto
     then have i0: "1 - a i > 0"
       by auto
     let ?a = "\<lambda>j. a j / (1 - a i)"
     have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
       using i0 insert that by fastforce
     have "(\<Sum> j \<in> insert i s. a j) = 1"
       using insert by auto
     then have "(\<Sum> j \<in> s. a j) = 1 - a i"
       using sum.insert insert by fastforce
     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
       using i0 by auto
     then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
       unfolding sum_divide_distrib by simp
     have "convex C" using insert by auto
     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
       using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
       using a_nonneg a1 insert by blast
     have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
       by (auto simp only: add.commute)
     also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using i0 by auto
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
       using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
       by (auto simp: algebra_simps)
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       by (auto simp: divide_inverse)
     also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
       by (auto simp: add.commute)
     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
       using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
             OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
       by simp
     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
       unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
       using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
       using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
       using insert by auto
     finally show ?thesis
       by simp
   qed
 qed
 
 lemma convex_on_alt:
   fixes C :: "'a::real_vector set"
-  assumes "convex C"
   shows "convex_on C f \<longleftrightarrow>
     (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
 proof safe
   fix x y
   fix \<mu> :: real
   assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
   from this[unfolded convex_on_def, rule_format]
   have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
     by auto
   from this [of "\<mu>" "1 - \<mu>", simplified] *
   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by auto
 next
   assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
     f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
   {
     fix x y
     fix u v :: real
     assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
     then have[simp]: "1 - u = v" by auto
     from *[rule_format, of x y u]
     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
       using ** by auto
   }
   then show "convex_on C f"
     unfolding convex_on_def by auto
 qed
 
 lemma convex_on_diff:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "convex_on I f"
     and I: "x \<in> I" "y \<in> I"
     and t: "x < t" "t < y"
   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
     and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
 proof -
   define a where "a \<equiv> (t - y) / (x - y)"
   with t have "0 \<le> a" "0 \<le> 1 - a"
     by (auto simp: field_simps)
   with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
     by (auto simp: convex_on_def)
   have "a * x + (1 - a) * y = a * (x - y) + y"
     by (simp add: field_simps)
   also have "\<dots> = t"
     unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
   finally have "f t \<le> a * f x + (1 - a) * f y"
     using cvx by simp
   also have "\<dots> = a * (f x - f y) + f y"
     by (simp add: field_simps)
   finally have "f t - f y \<le> a * (f x - f y)"
     by simp
   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
     by (simp add: le_divide_eq divide_le_eq field_simps)
 qed
 
 lemma pos_convex_function:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
     and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
   shows "convex_on C f"
-  unfolding convex_on_alt[OF assms(1)]
+  unfolding convex_on_alt
   using assms
 proof safe
   fix x y \<mu> :: real
   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
   assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   then have "1 - \<mu> \<ge> 0" by auto
   then have xpos: "?x \<in> C"
     using * unfolding convex_alt by fastforce
   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
     using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
         mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
     by auto
   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
     by (auto simp: field_simps)
   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
-    using convex_on_alt by auto
+    by auto
 qed
 
 lemma atMostAtLeast_subset_convex:
   fixes C :: "real set"
   assumes "convex C"
     and "x \<in> C" "y \<in> C" "x < y"
   shows "{x .. y} \<subseteq> C"
 proof safe
   fix z assume z: "z \<in> {x .. y}"
   have less: "z \<in> C" if *: "x < z" "z < y"
   proof -
     let ?\<mu> = "(y - z) / (y - x)"
     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
       using assms * by (auto simp: field_simps)
     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
       by (simp add: algebra_simps)
     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
       by (auto simp: field_simps)
     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
       using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
     also have "\<dots> = z"
       using assms by (auto simp: field_simps)
     finally show ?thesis
       using comb by auto
   qed
   show "z \<in> C"
     using z less assms by (auto simp: le_less)
 qed
 
 lemma f''_imp_f':
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
     and x: "x \<in> C"
     and y: "y \<in> C"
   shows "f' x * (y - x) \<le> f y - f x"
   using assms
 proof -
   have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
     if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
   proof -
     from * have ge: "y - x > 0" "y - x \<ge> 0"
       by auto
     from * have le: "x - y < 0" "x - y \<le> 0"
       by auto
     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
           THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
       by auto
     then have "z1 \<in> C"
       using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
       by fastforce
     from z1 have z1': "f x - f y = (x - y) * f' z1"
       by (simp add: field_simps)
     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
           THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
           THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
       using * z1' by auto
     also have "\<dots> = (y - z1) * f'' z3"
       using z3 by auto
     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
       by simp
     have A': "y - z1 \<ge> 0"
       using z1 by auto
     have "z3 \<in> C"
       using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
       by fastforce
     then have B': "f'' z3 \<ge> 0"
       using assms by auto
     from A' B' have "(y - z1) * f'' z3 \<ge> 0"
       by auto
     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
       by auto
     from mult_right_mono_neg[OF this le(2)]
     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
       by (simp add: algebra_simps)
     then have "f' y * (x - y) - (f x - f y) \<le> 0"
       using le by auto
     then have res: "f' y * (x - y) \<le> f x - f y"
       by auto
     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
       using * z1 by auto
     also have "\<dots> = (z1 - x) * f'' z2"
       using z2 by auto
     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
       by simp
     have A: "z1 - x \<ge> 0"
       using z1 by auto
     have "z2 \<in> C"
       using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
       by fastforce
     then have B: "f'' z2 \<ge> 0"
       using assms by auto
     from A B have "(z1 - x) * f'' z2 \<ge> 0"
       by auto
     with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
       by auto
     from mult_right_mono[OF this ge(2)]
     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
       by (simp add: algebra_simps)
     then have "f y - f x - f' x * (y - x) \<ge> 0"
       using ge by auto
     then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
       using res by auto
   qed
   show ?thesis
   proof (cases "x = y")
     case True
     with x y show ?thesis by auto
   next
     case False
     with less_imp x y show ?thesis
       by (auto simp: neq_iff)
   qed
 qed
 
 lemma f''_ge0_imp_convex:
   fixes f :: "real \<Rightarrow> real"
   assumes conv: "convex C"
     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
   shows "convex_on C f"
   using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
   by fastforce
 
 lemma minus_log_convex:
   fixes b :: real
   assumes "b > 1"
   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
 proof -
   have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
     using DERIV_log by auto
   then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
     by (auto simp: DERIV_minus)
   have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
     using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
   from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
   have "\<And>z::real. z > 0 \<Longrightarrow>
     DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
     by auto
   then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
     DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
     unfolding inverse_eq_divide by (auto simp: mult.assoc)
   have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
     using \<open>b > 1\<close> by (auto intro!: less_imp_le)
   from f''_ge0_imp_convex[OF convex_real_interval(3), unfolded greaterThan_iff, OF f' f''0 f''_ge0]
   show ?thesis
     by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of real functions\<close>
 
 lemma convex_on_realI:
   assumes "connected A"
     and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
     and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
   shows "convex_on A f"
 proof (rule convex_on_linorderI)
   fix t x y :: real
   assume t: "t > 0" "t < 1"
   assume xy: "x \<in> A" "y \<in> A" "x < y"
   define z where "z = (1 - t) * x + t * y"
   with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
     using connected_contains_Icc by blast
 
   from xy t have xz: "z > x"
     by (simp add: z_def algebra_simps)
   have "y - z = (1 - t) * (y - x)"
     by (simp add: z_def algebra_simps)
   also from xy t have "\<dots> > 0"
     by (intro mult_pos_pos) simp_all
   finally have yz: "z < y"
     by simp
 
   from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
     by auto
   from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
     by auto
 
   from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
   also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
     by auto
   with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
     by (intro assms(3)) auto
   also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
   finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
     using xz yz by (simp add: field_simps)
   also have "z - x = t * (y - x)"
     by (simp add: z_def algebra_simps)
   also have "y - z = (1 - t) * (y - x)"
     by (simp add: z_def algebra_simps)
   finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
     using xy by simp
   then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
     by (simp add: z_def algebra_simps)
 qed
 
 lemma convex_on_inverse:
   assumes "A \<subseteq> {0<..}"
   shows "convex_on A (inverse :: real \<Rightarrow> real)"
 proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
   fix u v :: real
   assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
   with assms show "-inverse (u^2) \<le> -inverse (v^2)"
     by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
 qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square)
 
 lemma convex_onD_Icc':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
   shows "f c \<le> (f y - f x) / d * (c - x) + f x"
 proof (cases x y rule: linorder_cases)
   case less
   then have d: "d > 0"
     by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
     by (simp_all add: d_def field_split_simps)
   have "f c = f (x + (c - x) * 1)"
     by simp
   also from less have "1 = ((y - x) / d)"
     by (simp add: d_def)
   also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
     using assms less by (intro convex_onD_Icc) simp_all
   also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
     by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
 lemma convex_onD_Icc'':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
   shows "f c \<le> (f x - f y) / d * (y - c) + f y"
 proof (cases x y rule: linorder_cases)
   case less
   then have d: "d > 0"
     by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
     by (simp_all add: d_def field_split_simps)
   have "f c = f (y - (y - c) * 1)"
     by simp
   also from less have "1 = ((y - x) / d)"
     by (simp add: d_def)
   also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
     using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
   also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
     by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
 lemma convex_translation_eq [simp]:
   "convex ((+) a ` s) \<longleftrightarrow> convex s"
   by (metis convex_translation translation_galois)
 
 lemma convex_translation_subtract_eq [simp]:
   "convex ((\<lambda>b. b - a) ` s) \<longleftrightarrow> convex s"
   using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp)
 
 lemma convex_linear_image_eq [simp]:
     fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
     shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
     by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
 
 lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
   unfolding linear_iff by (simp add: algebra_simps)
 
 lemma vector_choose_size:
   assumes "0 \<le> c"
   obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
 proof -
   obtain a::'a where "a \<noteq> 0"
     using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
   then show ?thesis
     by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
 qed
 
 lemma vector_choose_dist:
   assumes "0 \<le> c"
   obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
 by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
 
 lemma sum_delta'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
 proof -
   have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
     by auto
   show ?thesis
     unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
 qed
 
 lemma dist_triangle_eq:
   fixes x y z :: "'a::real_inner"
   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
     norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
 proof -
   have *: "x - y + (y - z) = x - z" by auto
   show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
     by (auto simp:norm_minus_commute)
 qed
 
 
 
 
 subsection \<open>Cones\<close>
 
 definition\<^marker>\<open>tag important\<close> cone :: "'a::real_vector set \<Rightarrow> bool"
   where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
 
 lemma cone_empty[intro, simp]: "cone {}"
   unfolding cone_def by auto
 
 lemma cone_univ[intro, simp]: "cone UNIV"
   unfolding cone_def by auto
 
 lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
   unfolding cone_def by auto
 
 lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
   by (simp add: cone_def subspace_scale)
 
 
 subsubsection \<open>Conic hull\<close>
 
-lemma cone_cone_hull: "cone (cone hull s)"
+lemma cone_cone_hull: "cone (cone hull S)"
   unfolding hull_def by auto
 
-lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s"
-  apply (rule hull_eq)
-  using cone_Inter
-  unfolding subset_eq
-  apply auto
-  done
+lemma cone_hull_eq: "cone hull S = S \<longleftrightarrow> cone S"
+  by (metis cone_cone_hull hull_same)
 
 lemma mem_cone:
   assumes "cone S" "x \<in> S" "c \<ge> 0"
   shows "c *\<^sub>R x \<in> S"
   using assms cone_def[of S] by auto
 
 lemma cone_contains_0:
   assumes "cone S"
   shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
-proof -
-  {
-    assume "S \<noteq> {}"
-    then obtain a where "a \<in> S" by auto
-    then have "0 \<in> S"
-      using assms mem_cone[of S a 0] by auto
-  }
-  then show ?thesis by auto
-qed
+  using assms mem_cone by fastforce
 
 lemma cone_0: "cone {0}"
   unfolding cone_def by auto
 
 lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
   unfolding cone_def by blast
 
 lemma cone_iff:
   assumes "S \<noteq> {}"
   shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
 proof -
   {
     assume "cone S"
     {
       fix c :: real
       assume "c > 0"
       {
         fix x
         assume "x \<in> S"
         then have "x \<in> ((*\<^sub>R) c) ` S"
           unfolding image_def
           using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
             exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
           by auto
       }
       moreover
       {
         fix x
         assume "x \<in> ((*\<^sub>R) c) ` S"
         then have "x \<in> S"
-          using \<open>cone S\<close> \<open>c > 0\<close>
-          unfolding cone_def image_def \<open>c > 0\<close> by auto
+          using \<open>0 < c\<close> \<open>cone S\<close> mem_cone by fastforce
       }
       ultimately have "((*\<^sub>R) c) ` S = S" by blast
     }
     then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
       using \<open>cone S\<close> cone_contains_0[of S] assms by auto
   }
   moreover
   {
     assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
     {
       fix x
       assume "x \<in> S"
       fix c1 :: real
       assume "c1 \<ge> 0"
       then have "c1 = 0 \<or> c1 > 0" by auto
       then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
     }
     then have "cone S" unfolding cone_def by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma cone_hull_empty: "cone hull {} = {}"
   by (metis cone_empty cone_hull_eq)
 
 lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
   by (metis bot_least cone_hull_empty hull_subset xtrans(5))
 
 lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
   using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
   by auto
 
 lemma mem_cone_hull:
   assumes "x \<in> S" "c \<ge> 0"
   shows "c *\<^sub>R x \<in> cone hull S"
   by (metis assms cone_cone_hull hull_inc mem_cone)
 
 proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
       by auto
     fix c :: real
     assume c: "c \<ge> 0"
     then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
       using x by (simp add: algebra_simps)
     moreover
     have "c * cx \<ge> 0" using c x by auto
     ultimately
     have "c *\<^sub>R x \<in> ?rhs" using x by auto
   }
   then have "cone ?rhs"
     unfolding cone_def by auto
   then have "?rhs \<in> Collect cone"
     unfolding mem_Collect_eq by auto
   {
     fix x
     assume "x \<in> S"
     then have "1 *\<^sub>R x \<in> ?rhs"
-      apply auto
-      apply (rule_tac x = 1 in exI, auto)
-      done
+      using zero_le_one by blast
     then have "x \<in> ?rhs" by auto
   }
   then have "S \<subseteq> ?rhs" by auto
   then have "?lhs \<subseteq> ?rhs"
     using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
       by auto
     then have "xx \<in> cone hull S"
       using hull_subset[of S] by auto
     then have "x \<in> ?lhs"
       using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_cone:
   "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x y
     assume "x\<in>s" "y\<in>s" and ?lhs
     then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
       unfolding cone_def by auto
     then have "x + y \<in> s"
       using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
       apply (erule_tac x="2*\<^sub>R x" in ballE)
       apply (erule_tac x="2*\<^sub>R y" in ballE)
       apply (erule_tac x="1/2" in allE, simp)
       apply (erule_tac x="1/2" in allE, auto)
       done
   }
   then show ?thesis
     unfolding convex_def cone_def by blast
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close>
 
 lemma convex_connected:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "connected S"
 proof (rule connectedI)
   fix A B
   assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
   moreover
   assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
   then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
   define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
   then have "continuous_on {0 .. 1} f"
     by (auto intro!: continuous_intros)
   then have "connected (f ` {0 .. 1})"
     by (auto intro!: connected_continuous_image)
   note connectedD[OF this, of A B]
   moreover have "a \<in> A \<inter> f ` {0 .. 1}"
     using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
   moreover have "b \<in> B \<inter> f ` {0 .. 1}"
     using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
   moreover have "f ` {0 .. 1} \<subseteq> S"
     using \<open>convex S\<close> a b unfolding convex_def f_def by auto
   ultimately show False by auto
 qed
 
 corollary%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
 by (simp add: convex_connected)
 
 lemma convex_prod:
   assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
   shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
   using assms unfolding convex_def
   by (auto simp: inner_add_left)
 
 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
 by (rule convex_prod) (simp flip: atLeast_def)
 
 subsection \<open>Convex hull\<close>
 
 lemma convex_convex_hull [iff]: "convex (convex hull s)"
   unfolding hull_def
   using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
   by auto
 
 lemma convex_hull_subset:
     "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
   by (simp add: subset_hull)
 
 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
   by (metis convex_convex_hull hull_same)
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Convex hull is "preserved" by a linear function\<close>
 
 lemma convex_hull_linear_image:
   assumes f: "linear f"
   shows "f ` (convex hull s) = convex hull (f ` s)"
 proof
   show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
     by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
   show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
   proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
     show "s \<subseteq> f -` (convex hull (f ` s))"
       by (fast intro: hull_inc)
     show "convex (f -` (convex hull (f ` s)))"
       by (intro convex_linear_vimage [OF f] convex_convex_hull)
   qed
 qed
 
 lemma in_convex_hull_linear_image:
   assumes "linear f"
     and "x \<in> convex hull s"
   shows "f x \<in> convex hull (f ` s)"
   using convex_hull_linear_image[OF assms(1)] assms(2) by auto
 
 lemma convex_hull_Times:
   "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
 proof
   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
   have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
   proof (rule hull_induct [OF x], rule hull_induct [OF y])
     fix x y assume "x \<in> s" and "y \<in> t"
     then show "(x, y) \<in> convex hull (s \<times> t)"
       by (simp add: hull_inc)
   next
     fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
     have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
     also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
       by (auto simp: image_def Bex_def)
     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
   next
     show "convex {x. (x, y) \<in> convex hull s \<times> t}"
     proof -
       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
       have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
       also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
         by (auto simp: image_def Bex_def)
       finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
     qed
   qed
   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
     unfolding subset_eq split_paired_Ball_Sigma by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems for convex hulls of finite sets\<close>
 
 lemma convex_hull_empty[simp]: "convex hull {} = {}"
   by (rule hull_unique) auto
 
 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
   by (rule hull_unique) auto
 
 lemma convex_hull_insert:
   fixes S :: "'a::real_vector set"
   assumes "S \<noteq> {}"
   shows "convex hull (insert a S) =
          {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
   (is "_ = ?hull")
 proof (intro equalityI hull_minimal subsetI)
   fix x
   assume "x \<in> insert a S"
   then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
   unfolding insert_iff
   proof
     assume "x = a"
     then show ?thesis
       by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
   next
     assume "x \<in> S"
     with hull_subset[of S convex] show ?thesis
       by force
   qed
   then show "x \<in> ?hull"
     by simp
 next
   fix x
   assume "x \<in> ?hull"
   then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
     by auto
   have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
     using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
     by auto
   then show "x \<in> convex hull insert a S"
     unfolding obt(5) using obt(1-3)
     by (rule convexD [OF convex_convex_hull])
 next
   show "convex ?hull"
   proof (rule convexI)
     fix x y u v
     assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
     from x obtain u1 v1 b1 where
       obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
       by auto
     from y obtain u2 v2 b2 where
       obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
       by auto
     have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
       by (auto simp: algebra_simps)
     have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
       (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
     proof (cases "u * v1 + v * v2 = 0")
       case True
       have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
         by (auto simp: algebra_simps)
       have eq0: "u * v1 = 0" "v * v2 = 0"
         using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
         by arith+
       then have "u * u1 + v * u2 = 1"
         using as(3) obt1(3) obt2(3) by auto
       then show ?thesis
         using "*" eq0 as obt1(4) xeq yeq by auto
     next
       case False
       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * v1 + v * v2"
         by simp
       finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
       let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
       have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
         using as(1,2) obt1(1,2) obt2(1,2) by auto
       show ?thesis
       proof
         show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
           unfolding xeq yeq * **
           using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
         show "?b \<in> convex hull S"
           using False zeroes obt1(4) obt2(4)
           by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
       qed
     qed
     then obtain b where b: "b \<in> convex hull S" 
        "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
 
     have u1: "u1 \<le> 1"
       unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
     have u2: "u2 \<le> 1"
       unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
     have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
     proof (rule add_mono)
       show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
         by (simp_all add: as mult_right_mono)
     qed
     also have "\<dots> \<le> 1"
       unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
     finally have le1: "u1 * u + u2 * v \<le> 1" .    
     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
     proof (intro CollectI exI conjI)
       show "0 \<le> u * u1 + v * u2"
         by (simp add: as(1) as(2) obt1(1) obt2(1))
       show "0 \<le> 1 - u * u1 - v * u2"
         by (simp add: le1 diff_diff_add mult.commute)
     qed (use b in \<open>auto simp: algebra_simps\<close>)
   qed
 qed
 
 lemma convex_hull_insert_alt:
    "convex hull (insert a S) =
      (if S = {} then {a}
       else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
   apply (auto simp: convex_hull_insert)
   using diff_eq_eq apply fastforce
-  by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
+  using diff_add_cancel diff_ge_0_iff_ge by blast
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expression for convex hull\<close>
 
 proposition convex_hull_indexed:
   fixes S :: "'a::real_vector set"
   shows "convex hull S =
     {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
                 (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
     (is "?xyz = ?hull")
 proof (rule hull_unique [OF _ convexI])
   show "S \<subseteq> ?hull" 
     by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
 next
   fix T
   assume "S \<subseteq> T" "convex T"
   then show "?hull \<subseteq> T"
     by (blast intro: convex_sum)
 next
   fix x y u v
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   assume xy: "x \<in> ?hull" "y \<in> ?hull"
   from xy obtain k1 u1 x1 where
     x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S" 
                       "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
     by auto
   from xy obtain k2 u2 x2 where
     y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S" 
                      "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
     by auto
   have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
           "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
     by auto
   have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
     unfolding inj_on_def by auto
   let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
   let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
   proof (intro CollectI exI conjI ballI)
     show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
       using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
     show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1"  "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
       unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
         sum.reindex[OF inj] Collect_mem_eq o_def
       unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
       by (simp_all add: sum_distrib_left[symmetric]  x(2,3) y(2,3) uv(3))
   qed 
 qed
 
 lemma convex_hull_finite:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
   (is "?HULL = _")
 proof (rule hull_unique [OF _ convexI]; clarify)
   fix x
   assume "x \<in> S"
   then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x"
     by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
 next
   fix u v :: real
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
   fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
   fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
   have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
     by (simp add: that uv ux(1) uy(1))
   moreover
   have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
     unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
     using uv(3) by auto
   moreover
   have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
     unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
     by auto
   ultimately
   show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
              (\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
     by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
 qed (use assms in \<open>auto simp: convex_explicit\<close>)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Another formulation\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma convex_hull_explicit:
   fixes p :: "'a::real_vector set"
   shows "convex hull p =
     {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x\<in>?lhs"
     then obtain k u y where
         obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
       unfolding convex_hull_indexed by auto
 
     have fin: "finite {1..k}" by auto
     have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
     {
       fix j
       assume "j\<in>{1..k}"
-      then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
+      then have "y j \<in> p \<and> 0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
         using obt(1)[THEN bspec[where x=j]] and obt(2)
-        apply simp
-        apply (rule sum_nonneg)
-        using obt(1)
-        apply auto
-        done
+        by (metis (no_types, lifting) One_nat_def atLeastAtMost_iff mem_Collect_eq obt(1) sum_nonneg)
     }
     moreover
     have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
       unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
     moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
       using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
       unfolding scaleR_left.sum using obt(3) by auto
     ultimately
     have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
       apply (rule_tac x="y ` {1..k}" in exI)
       apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
       done
     then have "x\<in>?rhs" by auto
   }
   moreover
   {
     fix y
     assume "y\<in>?rhs"
     then obtain S u where
       obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
       by auto
 
     obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
       using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
-
     {
       fix i :: nat
       assume "i\<in>{1..card S}"
       then have "f i \<in> S"
         using f(2) by blast
       then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
     }
     moreover have *: "finite {1..card S}" by auto
     {
       fix y
       assume "y\<in>S"
       then obtain i where "i\<in>{1..card S}" "f i = y"
         using f using image_iff[of y f "{1..card S}"]
         by auto
       then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
-        apply auto
-        using f(1)[unfolded inj_on_def]
-        by (metis One_nat_def atLeastAtMost_iff)
+        using f(1) inj_onD by fastforce
       then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
       then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
           "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
         by (auto simp: sum_constant_scaleR)
     }
     then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
       unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
         and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
       unfolding f
       using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
       using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
       unfolding obt(4,5)
       by auto
     ultimately
     have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
         (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
       apply (rule_tac x="card S" in exI)
       apply (rule_tac x="u \<circ> f" in exI)
       apply (rule_tac x=f in exI, fastforce)
       done
     then have "y \<in> ?lhs"
       unfolding convex_hull_indexed by auto
   }
   ultimately show ?thesis
     unfolding set_eq_iff by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>A stepping theorem for that expansion\<close>
 
 lemma convex_hull_finite_step:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows
     "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
       \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
   (is "?lhs = ?rhs")
-proof (rule, case_tac[!] "a\<in>S")
-  assume "a \<in> S"
+proof (cases "a \<in> S")
+  case True
   then have *: "insert a S = S" by auto
-  assume ?lhs
-  then show ?rhs
-    unfolding *  by (rule_tac x=0 in exI, auto)
-next
-  assume ?lhs
-  then obtain u where
-      u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
-    by auto
-  assume "a \<notin> S"
-  then show ?rhs
-    apply (rule_tac x="u a" in exI)
-    using u(1)[THEN bspec[where x=a]]
-    apply simp
-    apply (rule_tac x=u in exI)
-    using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
-    apply auto
-    done
+  show ?thesis
+  proof
+    assume ?lhs
+    then show ?rhs
+      unfolding * by force
+  next
+    have fin: "finite (insert a S)" using assms by auto
+    assume ?rhs
+    then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+      by auto
+    then show ?lhs
+      using uv True assms
+      apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
+      apply (auto simp: sum_clauses scaleR_left_distrib sum.distrib sum_delta''[OF fin])
+      done
+  qed
 next
-  assume "a \<in> S"
-  then have *: "insert a S = S" by auto
-  have fin: "finite (insert a S)" using assms by auto
-  assume ?rhs
-  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
-    by auto
-  show ?lhs
-    apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
-    unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
-    unfolding sum_clauses(2)[OF assms]
-    using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
-    apply auto
-    done
-next
-  assume ?rhs
-  then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
-    by auto
-  moreover assume "a \<notin> S"
-  moreover
-  have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S"  "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
-    using \<open>a \<notin> S\<close>
-    by (auto simp: intro!: sum.cong)
-  ultimately show ?lhs
-    by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
+  case False
+  show ?thesis
+  proof
+    assume ?lhs
+    then obtain u where u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
+      by auto
+    then show ?rhs
+      using u \<open>a\<notin>S\<close> by (rule_tac x="u a" in exI) (auto simp: sum_clauses assms)
+  next
+    assume ?rhs
+    then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
+      by auto
+    moreover
+    have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S"  "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
+      using False by (auto intro!: sum.cong)
+    ultimately show ?lhs
+      using False by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
+  qed
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence some special cases\<close>
 
-lemma convex_hull_2:
-  "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
+lemma convex_hull_2: "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
+       (is "?lhs = ?rhs")
 proof -
-  have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
-    by auto
   have **: "finite {b}" by auto
-  show ?thesis
-    apply (simp add: convex_hull_finite)
-    unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
-    apply auto
-    apply (rule_tac x=v in exI)
-    apply (rule_tac x="1 - v" in exI, simp)
+  have "\<And>x v u. \<lbrakk>0 \<le> v; v \<le> 1; (1 - v) *\<^sub>R b = x - v *\<^sub>R a\<rbrakk>
+                \<Longrightarrow> \<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1"
+    by (metis add.commute diff_add_cancel diff_ge_0_iff_ge)
+  moreover
+  have "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk>
+               \<Longrightarrow> \<exists>p\<ge>0. \<exists>q. 0 \<le> q b \<and> q b = 1 - p \<and> q b *\<^sub>R b = u *\<^sub>R a + v *\<^sub>R b - p *\<^sub>R a"
     apply (rule_tac x=u in exI, simp)
     apply (rule_tac x="\<lambda>x. v" in exI, simp)
     done
+  ultimately show ?thesis
+    using convex_hull_finite_step[OF **, of a 1]
+    by (auto simp add: convex_hull_finite)
 qed
 
 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
   unfolding convex_hull_2
 proof (rule Collect_cong)
   have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
     by auto
   fix x
   show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
     (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
-    unfolding *
-    apply auto
-    apply (rule_tac[!] x=u in exI)
-    apply (auto simp: algebra_simps)
-    done
+    apply (simp add: *)
+    by (rule ex_cong1) (auto simp: algebra_simps)
 qed
 
 lemma convex_hull_3:
   "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
 proof -
   have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
     by auto
   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
     by (auto simp: field_simps)
   show ?thesis
     unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
     unfolding convex_hull_finite_step[OF fin(3)]
     apply (rule Collect_cong, simp)
     apply auto
     apply (rule_tac x=va in exI)
     apply (rule_tac x="u c" in exI, simp)
     apply (rule_tac x="1 - v - w" in exI, simp)
     apply (rule_tac x=v in exI, simp)
     apply (rule_tac x="\<lambda>x. w" in exI, simp)
     done
 qed
 
 lemma convex_hull_3_alt:
   "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
 proof -
   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
     by auto
   show ?thesis
     unfolding convex_hull_3
     apply (auto simp: *)
     apply (rule_tac x=v in exI)
     apply (rule_tac x=w in exI)
     apply (simp add: algebra_simps)
     apply (rule_tac x=u in exI)
     apply (rule_tac x=v in exI)
     apply (simp add: algebra_simps)
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relations among closure notions and corresponding hulls\<close>
 
 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
   unfolding affine_def convex_def by auto
 
 lemma convex_affine_hull [simp]: "convex (affine hull S)"
   by (simp add: affine_imp_convex)
 
 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
   using subspace_imp_affine affine_imp_convex by auto
 
 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
 
 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
 
 lemma aff_dim_convex_hull:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (convex hull S) = aff_dim S"
   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
     hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
     aff_dim_subset[of "convex hull S" "affine hull S"]
   by auto
 
 
 subsection \<open>Caratheodory's theorem\<close>
 
 lemma convex_hull_caratheodory_aff_dim:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
-    {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
-      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
+    {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
+        (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
 proof (intro allI iffI)
   fix y
-  let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and>
-    sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
-  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+  let ?P = "\<lambda>n. \<exists>S u. finite S \<and> card S = n \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and>
+    sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
+  assume "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
   then obtain N where "?P N" by auto
   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
-    apply (rule_tac ex_least_nat_le, auto)
-    done
+    by (rule_tac ex_least_nat_le, auto)
   then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
     by blast
-  then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x"
-    "sum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
+  then obtain S u where obt: "finite S" "card S = n" "S\<subseteq>p" "\<forall>x\<in>S. 0 \<le> u x"
+    "sum u S = 1"  "(\<Sum>v\<in>S. u v *\<^sub>R v) = y" by auto
 
-  have "card s \<le> aff_dim p + 1"
+  have "card S \<le> aff_dim p + 1"
   proof (rule ccontr, simp only: not_le)
-    assume "aff_dim p + 1 < card s"
-    then have "affine_dependent s"
+    assume "aff_dim p + 1 < card S"
+    then have "affine_dependent S"
       using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
       by blast
-    then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
+    then obtain w v where wv: "sum w S = 0" "v\<in>S" "w v \<noteq> 0" "(\<Sum>v\<in>S. w v *\<^sub>R v) = 0"
       using affine_dependent_explicit_finite[OF obt(1)] by auto
-    define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"
+    define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>S. w v < 0}"
     define t where "t = Min i"
-    have "\<exists>x\<in>s. w x < 0"
+    have "\<exists>x\<in>S. w x < 0"
     proof (rule ccontr, simp add: not_less)
-      assume as:"\<forall>x\<in>s. 0 \<le> w x"
-      then have "sum w (s - {v}) \<ge> 0"
-        apply (rule_tac sum_nonneg, auto)
-        done
-      then have "sum w s > 0"
-        unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
-        using as[THEN bspec[where x=v]]  \<open>v\<in>s\<close>  \<open>w v \<noteq> 0\<close> by auto
+      assume as:"\<forall>x\<in>S. 0 \<le> w x"
+      then have "sum w (S - {v}) \<ge> 0"
+        by (meson Diff_iff sum_nonneg)
+      then have "sum w S > 0"
+        using as obt(1) sum_nonneg_eq_0_iff wv by blast
       then show False using wv(1) by auto
     qed
     then have "i \<noteq> {}" unfolding i_def by auto
     then have "t \<ge> 0"
-      using Min_ge_iff[of i 0 ] and obt(1)
+      using Min_ge_iff[of i 0] and obt(1)
       unfolding t_def i_def
       using obt(4)[unfolded le_less]
       by (auto simp: divide_le_0_iff)
-    have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0"
+    have t: "\<forall>v\<in>S. u v + t * w v \<ge> 0"
     proof
       fix v
-      assume "v \<in> s"
+      assume "v \<in> S"
       then have v: "0 \<le> u v"
         using obt(4)[THEN bspec[where x=v]] by auto
       show "0 \<le> u v + t * w v"
       proof (cases "w v < 0")
         case False
         thus ?thesis using v \<open>t\<ge>0\<close> by auto
       next
         case True
         then have "t \<le> u v / (- w v)"
-          using \<open>v\<in>s\<close> unfolding t_def i_def
-          apply (rule_tac Min_le)
-          using obt(1) apply auto
-          done
+          using \<open>v\<in>S\<close> obt unfolding t_def i_def by (auto intro: Min_le)
         then show ?thesis
           unfolding real_0_le_add_iff
-          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
-          by auto
+          using True neg_le_minus_divide_eq by auto
       qed
     qed
-    obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
+    obtain a where "a \<in> S" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
       using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
-    then have a: "a \<in> s" "u a + t * w a = 0" by auto
-    have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)"
-      unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
-    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
+    then have a: "a \<in> S" "u a + t * w a = 0" by auto
+    have *: "\<And>f. sum f (S - {a}) = sum f S - ((f a)::'b::ab_group_add)"
+      unfolding sum.remove[OF obt(1) \<open>a\<in>S\<close>] by auto
+    have "(\<Sum>v\<in>S. u v + t * w v) = 1"
       unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
-    moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
+    moreover have "(\<Sum>v\<in>S. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
       unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
     ultimately have "?P (n - 1)"
-      apply (rule_tac x="(s - {a})" in exI)
+      apply (rule_tac x="(S - {a})" in exI)
       apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
       using obt(1-3) and t and a
       apply (auto simp: * scaleR_left_distrib)
       done
     then show False
       using smallest[THEN spec[where x="n - 1"]] by auto
   qed
-  then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and>
-      (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
+  then show "\<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
+      (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
     using obt by auto
 qed auto
 
 lemma caratheodory_aff_dim:
   fixes p :: "('a::euclidean_space) set"
-  shows "convex hull p = {x. \<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> x \<in> convex hull s}"
+  shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
         (is "?lhs = ?rhs")
 proof
-  show "?lhs \<subseteq> ?rhs"
-    apply (subst convex_hull_caratheodory_aff_dim, clarify)
-    apply (rule_tac x=s in exI)
-    apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
-    done
-next
-  show "?rhs \<subseteq> ?lhs"
-    using hull_mono by blast
-qed
+  have "\<And>x S u. \<lbrakk>finite S; S \<subseteq> p; int (card S) \<le> aff_dim p + 1; \<forall>x\<in>S. 0 \<le> u x; sum u S = 1\<rbrakk>
+                \<Longrightarrow> (\<Sum>v\<in>S. u v *\<^sub>R v) \<in> convex hull S"
+    by (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
+  then show "?lhs \<subseteq> ?rhs"
+    by (subst convex_hull_caratheodory_aff_dim, auto)
+qed (use hull_mono in auto)
 
 lemma convex_hull_caratheodory:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
-            {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
-              (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}"
+            {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and>
+              (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
         (is "?lhs = ?rhs")
 proof (intro set_eqI iffI)
   fix x
   assume "x \<in> ?lhs" then show "x \<in> ?rhs"
-    apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
-    apply (erule ex_forward)+
-    using aff_dim_le_DIM [of p]
-    apply simp
-    done
-next
-  fix x
-  assume "x \<in> ?rhs" then show "x \<in> ?lhs"
-    by (auto simp: convex_hull_explicit)
-qed
+    unfolding convex_hull_caratheodory_aff_dim 
+    using aff_dim_le_DIM [of p] by fastforce
+qed (auto simp: convex_hull_explicit)
 
 theorem caratheodory:
   "convex hull p =
-    {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
-      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
+    {x::'a::euclidean_space. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}"
 proof safe
   fix x
   assume "x \<in> convex hull p"
-  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
-    "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
+  then obtain S u where "finite S" "S \<subseteq> p" "card S \<le> DIM('a) + 1"
+    "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = x"
     unfolding convex_hull_caratheodory by auto
-  then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
-    apply (rule_tac x=s in exI)
-    using hull_subset[of s convex]
-    using convex_convex_hull[simplified convex_explicit, of s,
-      THEN spec[where x=s], THEN spec[where x=u]]
-    apply auto
-    done
-next
-  fix x s
-  assume  "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
-  then show "x \<in> convex hull p"
-    using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
-qed
+  then show "\<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S"
+    using convex_hull_finite by fastforce
+qed (use hull_mono in force)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some Properties of subset of standard basis\<close>
 
 lemma affine_hull_substd_basis:
   assumes "d \<subseteq> Basis"
   shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   (is "affine hull (insert 0 ?A) = ?B")
 proof -
   have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
     by auto
   show ?thesis
     unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
 qed
 
 lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
   by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Moving and scaling convex hulls\<close>
 
 lemma convex_hull_set_plus:
   "convex hull (S + T) = convex hull S + convex hull T"
-  unfolding set_plus_image
+  unfolding set_plus_image 
   apply (subst convex_hull_linear_image [symmetric])
   apply (simp add: linear_iff scaleR_right_distrib)
   apply (simp add: convex_hull_Times)
   done
 
 lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
   unfolding set_plus_def by auto
 
 lemma convex_hull_translation:
   "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
   unfolding translation_eq_singleton_plus
   by (simp only: convex_hull_set_plus convex_hull_singleton)
 
 lemma convex_hull_scaling:
   "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
   using linear_scaleR by (rule convex_hull_linear_image [symmetric])
 
 lemma convex_hull_affinity:
   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
-  by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
+  by (metis convex_hull_scaling convex_hull_translation image_image)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of cone hulls\<close>
 
 lemma convex_cone_hull:
   assumes "convex S"
   shows "convex (cone hull S)"
 proof (rule convexI)
   fix x y
   assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
   then have "S \<noteq> {}"
     using cone_hull_empty_iff[of S] by auto
   fix u v :: real
   assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
   then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
     using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
   from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
     using cone_hull_expl[of S] by auto
   from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
     using cone_hull_expl[of S] by auto
   {
     assume "cx + cy \<le> 0"
     then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
       using x y by auto
     then have "u *\<^sub>R x + v *\<^sub>R y = 0"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
       using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
   }
   moreover
   {
     assume "cx + cy > 0"
     then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
       using assms mem_convex_alt[of S xx yy cx cy] x y by auto
     then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
       by (auto simp: scaleR_right_distrib)
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
       using x y by auto
   }
   moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
 qed
 
 lemma cone_convex_hull:
   assumes "cone S"
   shows "cone (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis by auto
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   {
     fix c :: real
     assume "c > 0"
     then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
       using convex_hull_scaling[of _ S] by auto
     also have "\<dots> = convex hull S"
       using * \<open>c > 0\<close> by auto
     finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
       by auto
   }
   then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
     using * hull_subset[of S convex] by auto
   then show ?thesis
     using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
 qed
 
 subsection \<open>Radon's theorem\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma Radon_ex_lemma:
   assumes "finite c" "affine_dependent c"
   shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
 proof -
   from assms(2)[unfolded affine_dependent_explicit]
-  obtain s u where
-      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+  obtain S u where
+      "finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by blast
   then show ?thesis
-    apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI)
-    unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric]
-    apply (auto simp: Int_absorb1)
-    done
+    apply (rule_tac x="\<lambda>v. if v\<in>S then u v else 0" in exI)
+    unfolding if_smult scaleR_zero_left 
+    by (auto simp: Int_absorb1 sum.inter_restrict[OF \<open>finite c\<close>, symmetric])
 qed
 
 lemma Radon_s_lemma:
-  assumes "finite s"
-    and "sum f s = (0::real)"
-  shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}"
+  assumes "finite S"
+    and "sum f S = (0::real)"
+  shows "sum f {x\<in>S. 0 < f x} = - sum f {x\<in>S. f x < 0}"
 proof -
   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
     by auto
   show ?thesis
     unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
       and sum.distrib[symmetric] and *
     using assms(2)
     by assumption
 qed
 
 lemma Radon_v_lemma:
-  assumes "finite s"
-    and "sum f s = 0"
+  assumes "finite S"
+    and "sum f S = 0"
     and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
-  shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}"
+  shows "(sum f {x\<in>S. 0 < g x}) = - sum f {x\<in>S. g x < 0}"
 proof -
   have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
     using assms(3) by auto
   show ?thesis
     unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
       and sum.distrib[symmetric] and *
     using assms(2)
     apply assumption
     done
 qed
 
 lemma Radon_partition:
-  assumes "finite c" "affine_dependent c"
-  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
+  assumes "finite C" "affine_dependent C"
+  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = C \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
 proof -
-  obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0"
+  obtain u v where uv: "sum u C = 0" "v\<in>C" "u v \<noteq> 0"  "(\<Sum>v\<in>C. u v *\<^sub>R v) = 0"
     using Radon_ex_lemma[OF assms] by auto
-  have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}"
+  have fin: "finite {x \<in> C. 0 < u x}" "finite {x \<in> C. 0 > u x}"
     using assms(1) by auto
-  define z  where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
-  have "sum u {x \<in> c. 0 < u x} \<noteq> 0"
+  define z  where "z = inverse (sum u {x\<in>C. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>C. u x > 0}"
+  have "sum u {x \<in> C. 0 < u x} \<noteq> 0"
   proof (cases "u v \<ge> 0")
     case False
     then have "u v < 0" by auto
     then show ?thesis
-    proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0")
+    proof (cases "\<exists>w\<in>{x \<in> C. 0 < u x}. u w > 0")
       case True
       then show ?thesis
         using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
     next
       case False
-      then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c"
-        apply (rule_tac sum_mono, auto)
-        done
+      then have "sum u C \<le> sum (\<lambda>x. if x=v then u v else 0) C"
+        by (rule_tac sum_mono, auto)
       then show ?thesis
         unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
     qed
   qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
 
-  then have *: "sum u {x\<in>c. u x > 0} > 0"
-    unfolding less_le
-    apply (rule_tac conjI)
-    apply (rule_tac sum_nonneg, auto)
-    done
-  moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c"
-    "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
+  then have *: "sum u {x\<in>C. u x > 0} > 0"
+    unfolding less_le by (metis (no_types, lifting) mem_Collect_eq sum_nonneg)
+  moreover have "sum u ({x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}) = sum u C"
+    "(\<Sum>x\<in>{x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>C. u x *\<^sub>R x)"
     using assms(1)
-    apply (rule_tac[!] sum.mono_neutral_left, auto)
-    done
-  then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}"
-    "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)"
+    by (rule_tac[!] sum.mono_neutral_left, auto)
+  then have "sum u {x \<in> C. 0 < u x} = - sum u {x \<in> C. 0 > u x}"
+    "(\<Sum>x\<in>{x \<in> C. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> C. 0 > u x}. u x *\<^sub>R x)"
     unfolding eq_neg_iff_add_eq_0
     using uv(1,4)
     by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
-  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x"
-    apply rule
-    apply (rule mult_nonneg_nonneg)
-    using *
-    apply auto
-    done
-  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}"
+  moreover have "\<forall>x\<in>{v \<in> C. u v < 0}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * - u x"
+    using * by (fastforce intro: mult_nonneg_nonneg)
+  ultimately have "z \<in> convex hull {v \<in> C. u v \<le> 0}"
     unfolding convex_hull_explicit mem_Collect_eq
-    apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
-    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI)
-    using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
-    apply (auto simp: sum_negf sum_distrib_left[symmetric])
-    done
-  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
-    apply rule
-    apply (rule mult_nonneg_nonneg)
-    using *
-    apply auto
-    done
-  then have "z \<in> convex hull {v \<in> c. u v > 0}"
+    apply (rule_tac x="{v \<in> C. u v < 0}" in exI)
+    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * - u y" in exI)
+    using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] 
+    by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
+  moreover have "\<forall>x\<in>{v \<in> C. 0 < u v}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * u x"
+    using * by (fastforce intro: mult_nonneg_nonneg)
+  then have "z \<in> convex hull {v \<in> C. u v > 0}"
     unfolding convex_hull_explicit mem_Collect_eq
-    apply (rule_tac x="{v \<in> c. 0 < u v}" in exI)
-    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI)
+    apply (rule_tac x="{v \<in> C. 0 < u v}" in exI)
+    apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * u y" in exI)
     using assms(1)
-    unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def
-    using *
-    apply (auto simp: sum_negf sum_distrib_left[symmetric])
-    done
+    unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
+    using * by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
   ultimately show ?thesis
-    apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
-    apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto)
+    apply (rule_tac x="{v\<in>C. u v \<le> 0}" in exI)
+    apply (rule_tac x="{v\<in>C. u v > 0}" in exI, auto)
     done
 qed
 
 theorem Radon:
   assumes "affine_dependent c"
   obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
 proof -
   from assms[unfolded affine_dependent_explicit]
-  obtain s u where
-      "finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
+  obtain S u where
+      "finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by blast
-  then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c"
+  then have *: "finite S" "affine_dependent S" and S: "S \<subseteq> c"
     unfolding affine_dependent_explicit by auto
   from Radon_partition[OF *]
-  obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}"
+  obtain m p where "m \<inter> p = {}" "m \<union> p = S" "convex hull m \<inter> convex hull p \<noteq> {}"
     by blast
-  then show ?thesis
-    apply (rule_tac that[of p m])
-    using s
-    apply auto
-    done
+  with S show ?thesis
+    by (force intro: that[of p m])
 qed
 
 
 subsection \<open>Helly's theorem\<close>
 
 lemma Helly_induct:
   fixes f :: "'a::euclidean_space set set"
   assumes "card f = n"
     and "n \<ge> DIM('a) + 1"
     and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
   using assms
 proof (induction n arbitrary: f)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   have "finite f"
     using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
   show "\<Inter>f \<noteq> {}"
   proof (cases "n = DIM('a)")
     case True
     then show ?thesis
       by (simp add: Suc.prems(1) Suc.prems(4))
   next
     case False
     have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
     proof (rule Suc.IH[rule_format])
       show "card (f - {s}) = n"
         by (simp add: Suc.prems(1) \<open>finite f\<close> that)
       show "DIM('a) + 1 \<le> n"
         using False Suc.prems(2) by linarith
       show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
         by (simp add: Suc.prems(4) subset_Diff_insert)
     qed (use Suc in auto)
     then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
       by blast
     then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
       by metis
     show ?thesis
     proof (cases "inj_on X f")
       case False
       then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
         unfolding inj_on_def by auto
       then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
       show ?thesis
         by (metis "*" X disjoint_iff_not_equal st)
     next
       case True
       then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
         using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
         unfolding card_image[OF True] and \<open>card f = Suc n\<close>
         using Suc(3) \<open>finite f\<close> and False
         by auto
       have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
         using mp(2) by auto
       then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
         unfolding subset_image_iff by auto
       then have "f \<union> (g \<union> h) = f" by auto
       then have f: "f = g \<union> h"
         using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
         unfolding mp(2)[unfolded image_Un[symmetric] gh]
         by auto
       have *: "g \<inter> h = {}"
-        using mp(1)
-        unfolding gh
-        using inj_on_image_Int[OF True gh(3,4)]
-        by auto
+        using gh(1) gh(2) local.mp(1) by blast
       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
         by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
       then show ?thesis
         unfolding f using mp(3)[unfolded gh] by blast
     qed
   qed 
 qed
 
 theorem Helly:
   fixes f :: "'a::euclidean_space set set"
   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
     and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
-  apply (rule Helly_induct)
-  using assms
-  apply auto
-  done
+  using Helly_induct assms by blast
 
 subsection \<open>Epigraphs of convex functions\<close>
 
 definition\<^marker>\<open>tag important\<close> "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
 
 lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
   unfolding epigraph_def by auto
 
 lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
 proof safe
   assume L: "convex (epigraph S f)"
   then show "convex_on S f"
     by (auto simp: convex_def convex_on_def epigraph_def)
   show "convex S"
-    using L
-    apply (clarsimp simp: convex_def convex_on_def epigraph_def)
-    apply (erule_tac x=x in allE)
-    apply (erule_tac x="f x" in allE, safe)
-    apply (erule_tac x=y in allE)
-    apply (erule_tac x="f y" in allE)
-    apply (auto simp: )
-    done
+    using L by (fastforce simp: convex_def convex_on_def epigraph_def)
 next
   assume "convex_on S f" "convex S"
   then show "convex (epigraph S f)"
     unfolding convex_def convex_on_def epigraph_def
     apply safe
      apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
       apply (auto intro!:mult_left_mono add_mono)
     done
 qed
 
 lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
   unfolding convex_epigraph by auto
 
 lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
   by (simp add: convex_epigraph)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Use this to derive general bound property of convex function\<close>
 
+
 lemma convex_on:
   assumes "convex S"
   shows "convex_on S f \<longleftrightarrow>
     (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
       f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
+  (is "?lhs = (\<forall>k u x. ?rhs k u x)")
+proof
+  assume ?lhs 
+  then have \<section>: "convex {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
+    by (metis assms convex_epigraph epigraph_def)
+  show "\<forall>k u x. ?rhs k u x"
+  proof (intro allI)
+    fix k u x
+    show "?rhs k u x"
+      using \<section>
+      unfolding  convex mem_Collect_eq fst_sum snd_sum 
+      apply safe
+      apply (drule_tac x=k in spec)
+      apply (drule_tac x=u in spec)
+      apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
+      apply simp
+      done
+  qed
+next
+  assume "\<forall>k u x. ?rhs k u x"
+  then show ?lhs
+  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq fst_sum snd_sum
+  using assms[unfolded convex] apply clarsimp
+  apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
+  by (auto simp add: mult_left_mono intro: sum_mono)
+qed
 
-  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
-  unfolding fst_sum snd_sum fst_scaleR snd_scaleR
-  apply safe
-    apply (drule_tac x=k in spec)
-    apply (drule_tac x=u in spec)
-    apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
-    apply simp
-  using assms[unfolded convex] apply simp
-  apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
-   apply (rule sum_mono)
-   apply (erule_tac x=i in allE)
-  unfolding real_scaleR_def
-   apply (rule mult_left_mono)
-  using assms[unfolded convex] apply auto
-  done
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within a convex hull\<close>
 
 lemma convex_on_convex_hull_bound:
-  assumes "convex_on (convex hull s) f"
-    and "\<forall>x\<in>s. f x \<le> b"
-  shows "\<forall>x\<in> convex hull s. f x \<le> b"
+  assumes "convex_on (convex hull S) f"
+    and "\<forall>x\<in>S. f x \<le> b"
+  shows "\<forall>x\<in> convex hull S. f x \<le> b"
 proof
   fix x
-  assume "x \<in> convex hull s"
+  assume "x \<in> convex hull S"
   then obtain k u v where
-    obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
+    u: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> S" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
     unfolding convex_hull_indexed mem_Collect_eq by auto
   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
     using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
-    unfolding sum_distrib_right[symmetric] obt(2) mult_1
-    apply (drule_tac meta_mp)
-    apply (rule mult_left_mono)
-    using assms(2) obt(1)
-    apply auto
-    done
+    unfolding sum_distrib_right[symmetric] u(2) mult_1
+    using assms(2) mult_left_mono u(1) by blast
   then show "f x \<le> b"
     using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
-    unfolding obt(2-3)
-    using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
-    by auto
+    using hull_inc u by fastforce
 qed
 
 lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
   by (simp add: inner_sum_left sum.If_cases inner_Basis)
 
 lemma convex_set_plus:
   assumes "convex S" and "convex T" shows "convex (S + T)"
 proof -
   have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     using assms by (rule convex_sums)
   moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
     unfolding set_plus_def by auto
   finally show "convex (S + T)" .
 qed
 
 lemma convex_set_sum:
   assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
   shows "convex (\<Sum>i\<in>A. B i)"
 proof (cases "finite A")
   case True then show ?thesis using assms
     by induct (auto simp: convex_set_plus)
 qed auto
 
 lemma finite_set_sum:
   assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
   using assms by (induct set: finite, simp, simp add: finite_set_plus)
 
 lemma box_eq_set_sum_Basis:
-  shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
-  apply (subst set_sum_alt [OF finite_Basis], safe)
-  apply (fast intro: euclidean_representation [symmetric])
-  apply (subst inner_sum_left)
-apply (rename_tac f)
-  apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
-  apply (drule (1) bspec)
-  apply clarsimp
-  apply (frule sum.remove [OF finite_Basis])
-  apply (erule trans, simp)
-  apply (rule sum.neutral, clarsimp)
-  apply (frule_tac x=i in bspec, assumption)
-  apply (drule_tac x=x in bspec, assumption, clarsimp)
-  apply (cut_tac u=x and v=i in inner_Basis, assumption+)
-  apply (rule ccontr, simp)
-  done
+  "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (B i))" (is "?lhs = ?rhs")
+proof -
+  have "\<And>x. \<forall>i\<in>Basis. x \<bullet> i \<in> B i \<Longrightarrow>
+         \<exists>s. x = sum s Basis \<and> (\<forall>i\<in>Basis. s i \<in> (\<lambda>x. x *\<^sub>R i) ` B i)"
+    by (metis (mono_tags, lifting) euclidean_representation image_iff)
+  moreover
+  have "sum f Basis \<bullet> i \<in> B i" if "i \<in> Basis" and f: "\<forall>i\<in>Basis. f i \<in> (\<lambda>x. x *\<^sub>R i) ` B i" for i f
+  proof -
+    have "(\<Sum>x\<in>Basis - {i}. f x \<bullet> i) = 0"
+    proof (rule sum.neutral, intro strip)
+      show "f x \<bullet> i = 0" if "x \<in> Basis - {i}" for x
+        using that f \<open>i \<in> Basis\<close> inner_Basis that by fastforce
+    qed
+    then have "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i"
+      by (metis (no_types) \<open>i \<in> Basis\<close> add.right_neutral sum.remove [OF finite_Basis])
+    then have "(\<Sum>x\<in>Basis. f x \<bullet> i) \<in> B i"
+      using f that(1) by auto
+    then show ?thesis
+      by (simp add: inner_sum_left)
+  qed
+  ultimately show ?thesis
+    by (subst set_sum_alt [OF finite_Basis]) auto
+qed
 
 lemma convex_hull_set_sum:
   "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
 proof (cases "finite A")
   assume "finite A" then show ?thesis
     by (induct set: finite, simp, simp add: convex_hull_set_plus)
 qed simp
 
 
 end
\ No newline at end of file