
diff -r ce1c34c98eeb -r 89e10ed7668b src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Thu Sep 27 14:50:06 2012 +0200
+++ b/src/HOL/Library/Convex.thy	Thu Sep 27 14:52:50 2012 +0200
@@ -11,9 +11,8 @@
 
 subsection {* Convexity. *}
 
-definition
-  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)"
+definition 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 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)"
@@ -21,10 +20,10 @@
 proof
   assume alt[rule_format]: ?alt
   { fix x y and u v :: real assume mem: "x \<in> s" "y \<in> s"
-    assume "0 \<le> u" "0 \<le> v" "u + v = 1"
-    moreover hence "u = 1 - v" by auto
+    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[OF mem] by auto }
-  thus "convex s" unfolding convex_def by auto
+  then show "convex s" unfolding convex_def by auto
 qed (auto simp: convex_def)
 
 lemma mem_convex:
@@ -53,13 +52,13 @@
 
 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
+  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_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
+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
@@ -74,78 +73,83 @@
 lemma convex_real_interval:
   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}"
+    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
-  thus 1: "convex {a..}" by (simp only: convex_halfspace_ge)
+  then show 1: "convex {a..}" by (simp only: convex_halfspace_ge)
   have "{..b} = {x. inner 1 x \<le> b}" by auto
-  thus 2: "convex {..b}" by (simp only: convex_halfspace_le)
+  then show 2: "convex {..b}" by (simp only: convex_halfspace_le)
   have "{a<..} = {x. a < inner 1 x}" by auto
-  thus 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
+  then show 3: "convex {a<..}" by (simp only: convex_halfspace_gt)
   have "{..<b} = {x. inner 1 x < b}" by auto
-  thus 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
+  then show 4: "convex {..<b}" by (simp only: convex_halfspace_lt)
   have "{a..b} = {a..} \<inter> {..b}" by auto
-  thus "convex {a..b}" by (simp only: convex_Int 1 2)
+  then show "convex {a..b}" by (simp only: convex_Int 1 2)
   have "{a<..b} = {a<..} \<inter> {..b}" by auto
-  thus "convex {a<..b}" by (simp only: convex_Int 3 2)
+  then show "convex {a<..b}" by (simp only: convex_Int 3 2)
   have "{a..<b} = {a..} \<inter> {..<b}" by auto
-  thus "convex {a..<b}" by (simp only: convex_Int 1 4)
+  then show "convex {a..<b}" by (simp only: convex_Int 1 4)
   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
-  thus "convex {a<..<b}" by (simp only: convex_Int 3 4)
+  then show "convex {a<..<b}" by (simp only: convex_Int 3 4)
 qed
 
+
 subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}
 
 lemma convex_setsum:
   fixes C :: "'a::real_vector set"
   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"
+  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
-proof (induct s arbitrary:a rule:finite_induct)
-  case empty thus ?case by auto
+  using assms
+proof (induct s arbitrary:a rule: finite_induct)
+  case empty
+  then show ?case by auto
 next
   case (insert i s) note asms = this
   { assume "a i = 1"
-    hence "(\<Sum> j \<in> s. a j) = 0"
+    then have "(\<Sum> j \<in> s. a j) = 0"
       using asms by auto
-    hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
+    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
       using setsum_nonneg_0[where 'b=real] asms by fastforce
-    hence ?case using asms by auto }
+    then have ?case using asms by auto }
   moreover
   { assume asm: "a i \<noteq> 1"
     from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
     have fis: "finite (insert i s)" using asms by auto
-    hence ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a 1] asms by simp
-    hence "a i < 1" using asm by auto
-    hence i0: "1 - a i > 0" by auto
-    let "?a j" = "a j / (1 - a i)"
+    then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a 1] asms by simp
+    then have "a i < 1" using asm by auto
+    then have i0: "1 - a i > 0" by auto
+    let ?a = "\<lambda>j. a j / (1 - a i)"
     { fix j assume "j \<in> s"
-      hence "?a j \<ge> 0"
+      then have "?a j \<ge> 0"
         using i0 asms divide_nonneg_pos
-        by fastforce } note a_nonneg = this
+        by fastforce
+    } note a_nonneg = this
     have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
-    hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce
-    hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
-    hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp
-    from this asms
-    have "(\<Sum>j\<in>s. ?a j *\<^sub>R y j) \<in> C" using a_nonneg by fastforce
-    hence "a i *\<^sub>R y i + (1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
+    then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms 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 setsum_divide_distrib by simp
+    with asms have "(\<Sum>j\<in>s. ?a j *\<^sub>R y j) \<in> C" using a_nonneg by fastforce
+    then have "a i *\<^sub>R y i + (1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
       using asms[unfolded convex_def, rule_format] yai ai1 by auto
-    hence "a i *\<^sub>R y i + (\<Sum> j \<in> s. (1 - a i) *\<^sub>R (?a j *\<^sub>R y j)) \<in> C"
+    then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. (1 - a i) *\<^sub>R (?a j *\<^sub>R y j)) \<in> C"
       using scaleR_right.setsum[of "(1 - a i)" "\<lambda> j. ?a j *\<^sub>R y j" s] by auto
-    hence "a i *\<^sub>R y i + (\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" using i0 by auto
-    hence ?case using setsum.insert asms by auto }
+    then have "a i *\<^sub>R y i + (\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" using i0 by auto
+    then have ?case using setsum.insert asms by auto
+  }
   ultimately show ?case by auto
 qed
 
 lemma convex:
-  shows "convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (setsum u {1..k} = 1)
-           \<longrightarrow> setsum (\<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> (setsum u {1..k} = 1)
+      \<longrightarrow> setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)"
 proof safe
-  fix k :: nat fix u :: "nat \<Rightarrow> real" fix x
+  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"
     "setsum u {1..k} = 1"
@@ -154,35 +158,39 @@
 next
   assume asm: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1
     \<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 mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
-    let "?u i" = "if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
-    let "?x i" = "if (i :: nat) = 1 then x else y"
+  { fix \<mu> :: real
+    fix x y :: 'a
+    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
-    hence card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" by simp
-    hence "setsum ?u {1 .. 2} = 1"
+    then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" by simp
+    then have "setsum ?u {1 .. 2} = 1"
       using setsum_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
       by auto
-    from this asm[rule_format, of "2" ?u ?x]
-    have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s"
+    with asm[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 setsum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
     from setsum_head_Suc[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
-    hence "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute) }
-  thus "convex s" unfolding convex_alt by auto
+    then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" using s by (auto simp:add_commute)
+  }
+  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> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
+    (\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> setsum u t = 1 \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) t \<in> s)"
 proof safe
-  fix t fix u :: "'a \<Rightarrow> real"
+  fix t
+  fix u :: "'a \<Rightarrow> real"
   assume "convex s" "finite t"
     "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1"
-  thus "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
+  then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
     using convex_setsum[of t s u "\<lambda> x. x"] by auto
 next
   assume asm0: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x)
@@ -190,39 +198,42 @@
   show "convex s"
     unfolding convex_alt
   proof safe
-    fix x y fix \<mu> :: real
+    fix x y
+    fix \<mu> :: real
     assume asm: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1"
     { assume "x \<noteq> y"
-      hence "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+      then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
         using asm0[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"]
           asm by auto }
     moreover
     { assume "x = y"
-      hence "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
+      then have "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s"
         using asm0[rule_format, of "{x, y}" "\<lambda> z. 1"]
           asm by (auto simp:field_simps real_vector.scale_left_diff_distrib) }
     ultimately show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" by blast
   qed
 qed
 
-lemma convex_finite: assumes "finite s"
+lemma convex_finite:
+  assumes "finite s"
   shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1
                       \<longrightarrow> setsum (\<lambda>x. u x *\<^sub>R x) s \<in> s)"
   unfolding convex_explicit
-proof (safe)
-  fix t u assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
+proof safe
+  fix t u
+  assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s"
     and as: "finite t" "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = (1::real)"
-  have *:"s \<inter> t = t" using as(2) by auto
-  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
+  have *: "s \<inter> t = t" using as(2) by auto
+  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
   show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
    using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
    by (auto simp: assms setsum_cases if_distrib if_distrib_arg)
 qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
 
-definition
-  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)"
+definition 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)"
 
 lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
   unfolding convex_on_def by auto
@@ -230,21 +241,29 @@
 lemma convex_add[intro]:
   assumes "convex_on s f" "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" 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:add_mono)
-    hence "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)  }
-  thus ?thesis unfolding convex_on_def by auto
+proof -
+  { fix x y
+    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: 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_cmul[intro]:
   assumes "0 \<le> (c::real)" "convex_on s f"
   shows "convex_on s (\<lambda>x. c * f x)"
 proof-
-  have *:"\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" by (simp add: field_simps)
-  show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)] unfolding convex_on_def and * by auto
+  have *: "\<And>u c fx v fy ::real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+    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:
@@ -254,7 +273,7 @@
   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 add: mult_left_mono add_mono)
-  also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[THEN sym] by auto
+  also have "\<dots> = max (f x) (f y)" using assms(6) unfolding distrib[symmetric] by auto
   finally show ?thesis
     using assms unfolding convex_on_def by fastforce
 qed
@@ -262,24 +281,30 @@
 lemma convex_distance[intro]:
   fixes s :: "'a::real_normed_vector set"
   shows "convex_on s (\<lambda>x. dist a x)"
-proof(auto simp add: convex_on_def dist_norm)
-  fix x y assume "x\<in>s" "y\<in>s"
-  fix u v ::real assume "0 \<le> u" "0 \<le> v" "u + v = 1"
-  have "a = u *\<^sub>R a + v *\<^sub>R a" unfolding scaleR_left_distrib[THEN sym] and `u+v=1` by simp
-  hence *:"a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
+proof (auto simp add: convex_on_def dist_norm)
+  fix x y
+  assume "x\<in>s" "y\<in>s"
+  fix u v :: real
+  assume "0 \<le> u" "0 \<le> v" "u + v = 1"
+  have "a = u *\<^sub>R a + v *\<^sub>R a"
+    unfolding scaleR_left_distrib[symmetric] and `u+v=1` 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 add: 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 `0 \<le> u` `0 \<le> v` by auto
 qed
 
+
 subsection {* Arithmetic operations on sets preserve convexity. *}
+
 lemma convex_scaling:
   assumes "convex s"
   shows"convex ((\<lambda>x. c *\<^sub>R x) ` s)"
-using assms unfolding convex_def image_iff
+  using assms unfolding convex_def image_iff
 proof safe
-  fix x xa y xb :: "'a::real_vector" fix u v :: real
+  fix x xa y xb :: "'a::real_vector"
+  fix u v :: real
   assume asm: "\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
     "xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
   show "\<exists>x\<in>s. u *\<^sub>R c *\<^sub>R xa + v *\<^sub>R c *\<^sub>R xb = c *\<^sub>R x"
@@ -287,9 +312,10 @@
 qed
 
 lemma convex_negations: "convex s \<Longrightarrow> convex ((\<lambda>x. -x)` s)"
-using assms unfolding convex_def image_iff
+  using assms unfolding convex_def image_iff
 proof safe
-  fix x xa y xb :: "'a::real_vector" fix u v :: real
+  fix x xa y xb :: "'a::real_vector"
+  fix u v :: real
   assume asm: "\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
     "xa \<in> s" "xb \<in> s" "0 \<le> u" "0 \<le> v" "u + v = 1"
   show "\<exists>x\<in>s. u *\<^sub>R - xa + v *\<^sub>R - xb = - x"
@@ -299,10 +325,12 @@
 lemma convex_sums:
   assumes "convex s" "convex t"
   shows "convex {x + y| x y. x \<in> s \<and> y \<in> t}"
-using assms unfolding convex_def image_iff
+  using assms unfolding convex_def image_iff
 proof safe
-  fix xa xb ya yb assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
-  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
+  fix xa xb ya yb
+  assume xy:"xa\<in>s" "xb\<in>s" "ya\<in>t" "yb\<in>t"
+  fix u v :: real
+  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
   show "\<exists>x y. u *\<^sub>R (xa + ya) + v *\<^sub>R (xb + yb) = x + y \<and> x \<in> s \<and> y \<in> t"
     using exI[of _ "u *\<^sub>R xa + v *\<^sub>R xb"] exI[of _ "u *\<^sub>R ya + v *\<^sub>R yb"]
       assms[unfolded convex_def] uv xy by (auto simp add:scaleR_right_distrib)
@@ -314,105 +342,120 @@
 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}"
   proof safe
-    fix x x' y assume "x' \<in> s" "y \<in> t"
-    thus "\<exists>x y'. x' - y = x + y' \<and> x \<in> s \<and> y' \<in> uminus ` t"
+    fix x x' y
+    assume "x' \<in> s" "y \<in> t"
+    then show "\<exists>x y'. x' - y = x + y' \<and> x \<in> s \<and> y' \<in> uminus ` t"
       using exI[of _ x'] exI[of _ "-y"] by auto
   next
-    fix x x' y y' assume "x' \<in> s" "y' \<in> t"
-    thus "\<exists>x y. x' + - y' = x - y \<and> x \<in> s \<and> y \<in> t"
+    fix x x' y y'
+    assume "x' \<in> s" "y' \<in> t"
+    then show "\<exists>x y. x' + - y' = x - y \<and> x \<in> s \<and> y \<in> t"
       using exI[of _ x'] exI[of _ y'] by auto
   qed
-  thus ?thesis using convex_sums[OF assms(1)  convex_negations[OF assms(2)]] by auto
+  then show ?thesis
+    using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
 qed
 
-lemma convex_translation: assumes "convex s" shows "convex ((\<lambda>x. a + x) ` s)"
-proof- have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
-  thus ?thesis using convex_sums[OF convex_singleton[of a] assms] by auto qed
+lemma convex_translation:
+  assumes "convex s"
+  shows "convex ((\<lambda>x. a + x) ` s)"
+proof -
+  have "{a + y |y. y \<in> s} = (\<lambda>x. a + x) ` s" by auto
+  then show ?thesis
+    using convex_sums[OF convex_singleton[of a] assms] by auto
+qed
 
-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 = op + a ` op *\<^sub>R c ` s" by auto
-  thus ?thesis using convex_translation[OF convex_scaling[OF assms], of a c] by auto qed
+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 = op + a ` op *\<^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_linear_image:
   assumes c:"convex s" and l:"bounded_linear f"
   shows "convex(f ` s)"
-proof(auto simp add: convex_def)
+proof (auto simp add: convex_def)
   interpret f: bounded_linear f by fact
-  fix x y assume xy:"x \<in> s" "y \<in> s"
-  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
+  fix x y
+  assume xy: "x \<in> s" "y \<in> s"
+  fix u v :: real
+  assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
   show "u *\<^sub>R f x + v *\<^sub>R f y \<in> f ` s" unfolding image_iff
     using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
       c[unfolded convex_def] xy uv by auto
 qed
 
 
-lemma pos_is_convex:
-  shows "convex {0 :: real <..}"
-unfolding convex_alt
+lemma pos_is_convex: "convex {0 :: real <..}"
+  unfolding convex_alt
 proof safe
   fix y x \<mu> :: real
   assume asms: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   { assume "\<mu> = 0"
-    hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
-    hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
   moreover
   { assume "\<mu> = 1"
-    hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms by simp }
   moreover
   { assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
-    hence "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
-    hence "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
+    then have "\<mu> > 0" "(1 - \<mu>) > 0" using asms by auto
+    then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using asms
       by (auto simp add: add_pos_pos mult_pos_pos) }
   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" using assms by fastforce
 qed
 
 lemma convex_on_setsum:
   fixes a :: "'a \<Rightarrow> real"
-  fixes y :: "'a \<Rightarrow> 'b::real_vector"
-  fixes f :: "'b \<Rightarrow> real"
+    and y :: "'a \<Rightarrow> 'b::real_vector"
+    and f :: "'b \<Rightarrow> real"
   assumes "finite s" "s \<noteq> {}"
-  assumes "convex_on C f"
-  assumes "convex C"
-  assumes "(\<Sum> i \<in> s. a i) = 1"
-  assumes "\<And> i. i \<in> s \<Longrightarrow> a i \<ge> 0"
-  assumes "\<And> i. i \<in> s \<Longrightarrow> y i \<in> C"
+    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)
+  using assms
+proof (induct s arbitrary: a rule: finite_ne_induct)
   case (singleton i)
-  hence ai: "a i = 1" by auto
-  thus ?case by auto
+  then have ai: "a i = 1" by auto
+  then show ?case by auto
 next
   case (insert i s) note asms = this
-  hence "convex_on C f" by simp
+  then have "convex_on C f" by simp
   from this[unfolded convex_on_def, rule_format]
-  have conv: "\<And> x y \<mu>. \<lbrakk>x \<in> C; y \<in> C; 0 \<le> \<mu>; \<mu> \<le> 1\<rbrakk>
-  \<Longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+  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
   { assume "a i = 1"
-    hence "(\<Sum> j \<in> s. a j) = 0"
+    then have "(\<Sum> j \<in> s. a j) = 0"
       using asms by auto
-    hence "\<And> j. j \<in> s \<Longrightarrow> a j = 0"
+    then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
       using setsum_nonneg_0[where 'b=real] asms by fastforce
-    hence ?case using asms by auto }
+    then have ?case using asms by auto }
   moreover
   { assume asm: "a i \<noteq> 1"
     from asms have yai: "y i \<in> C" "a i \<ge> 0" by auto
     have fis: "finite (insert i s)" using asms by auto
-    hence ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
-    hence "a i < 1" using asm by auto
-    hence i0: "1 - a i > 0" by auto
-    let "?a j" = "a j / (1 - a i)"
+    then have ai1: "a i \<le> 1" using setsum_nonneg_leq_bound[of "insert i s" a] asms by simp
+    then have "a i < 1" using asm by auto
+    then have i0: "1 - a i > 0" by auto
+    let ?a = "\<lambda>j. a j / (1 - a i)"
     { fix j assume "j \<in> s"
-      hence "?a j \<ge> 0"
+      then have "?a j \<ge> 0"
         using i0 asms divide_nonneg_pos
-        by fastforce } note a_nonneg = this
+        by fastforce }
+    note a_nonneg = this
     have "(\<Sum> j \<in> insert i s. a j) = 1" using asms by auto
-    hence "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms by fastforce
-    hence "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" using i0 by auto
-    hence a1: "(\<Sum> j \<in> s. ?a j) = 1" unfolding setsum_divide_distrib by simp
+    then have "(\<Sum> j \<in> s. a j) = 1 - a i" using setsum.insert asms 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 setsum_divide_distrib by simp
     have "convex C" using asms by auto
-    hence asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
+    then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
       using asms convex_setsum[OF `finite s`
         `convex C` 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))"
@@ -423,7 +466,8 @@
     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.setsum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric] by (auto simp:algebra_simps)
+      using scaleR_right.setsum[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)"
@@ -448,27 +492,30 @@
   (\<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
+  fix x y
+  fix \<mu> :: real
   assume asms: "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 "\<And> u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" by auto
+  have "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" by auto
   from this[of "\<mu>" "1 - \<mu>", simplified] asms
-  show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y)
-          \<le> \<mu> * f x + (1 - \<mu>) * f y" by auto
+  show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" by auto
 next
   assume asm: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow> f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
-  {fix x y fix u v :: real
+  { fix x y
+    fix u v :: real
     assume lasm: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
-    hence[simp]: "1 - u = v" by auto
+    then have[simp]: "1 - u = v" by auto
     from asm[rule_format, of x y u]
-    have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using lasm by auto }
-  thus "convex_on C f" unfolding convex_on_def by auto
+    have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using lasm 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)" "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
+  shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
+    "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
 proof -
   def a \<equiv> "(t - y) / (x - y)"
   with t have "0 \<le> a" "0 \<le> 1 - a" by (auto simp: field_simps)
@@ -488,46 +535,48 @@
 lemma pos_convex_function:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
-  assumes leq: "\<And> x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
+    and leq: "\<And>x y. \<lbrakk>x \<in> C ; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
   shows "convex_on C f"
-unfolding convex_on_alt[OF assms(1)]
-using assms
+  unfolding convex_on_alt[OF assms(1)]
+  using assms
 proof safe
   fix x y \<mu> :: real
   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
   assume asm: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
-  hence "1 - \<mu> \<ge> 0" by auto
-  hence xpos: "?x \<in> C" using asm unfolding convex_alt by fastforce
+  then have "1 - \<mu> \<ge> 0" by auto
+  then have xpos: "?x \<in> C" using asm 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 asm(2)] `\<mu> \<ge> 0`]
       mult_left_mono[OF leq[OF xpos asm(3)] `1 - \<mu> \<ge> 0`]] by auto
-  hence "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
-    by (auto simp add:field_simps)
-  thus "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
+  then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
+    by (auto simp add: 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
 qed
 
 lemma atMostAtLeast_subset_convex:
   fixes C :: "real set"
   assumes "convex C"
-  assumes "x \<in> C" "y \<in> C" "x < y"
+    and "x \<in> C" "y \<in> C" "x < y"
   shows "{x .. y} \<subseteq> C"
 proof safe
   fix z assume zasm: "z \<in> {x .. y}"
   { assume asm: "x < z" "z < y"
-    let "?\<mu>" = "(y - z) / (y - x)"
-    have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add:field_simps)
-    hence 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)
+    let ?\<mu> = "(y - z) / (y - x)"
+    have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" using assms asm by (auto simp add: 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 add:field_simps)
+      by (auto simp add: field_simps)
     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
-      using assms unfolding add_divide_distrib by (auto simp:field_simps)
+      using assms unfolding add_divide_distrib by (auto simp: field_simps)
     also have "\<dots> = z"
-      using assms by (auto simp:field_simps)
+      using assms by (auto simp: field_simps)
     finally have "z \<in> C"
-      using comb by auto } note less = this
+      using comb by auto }
+  note less = this
   show "z \<in> C" using zasm less assms
     unfolding atLeastAtMost_iff le_less by auto
 qed
@@ -535,21 +584,22 @@
 lemma f''_imp_f':
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
-  assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
-  assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
-  assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
-  assumes "x \<in> C" "y \<in> 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 \<in> C" "y \<in> C"
   shows "f' x * (y - x) \<le> f y - f x"
-using assms
+  using assms
 proof -
-  { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "y > x"
-    hence ge: "y - x > 0" "y - x \<ge> 0" by auto
+  { fix x y :: real
+    assume asm: "x \<in> C" "y \<in> C" "y > x"
+    then have ge: "y - x > 0" "y - x \<ge> 0" by auto
     from asm 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 `convex C` `x \<in> C` `y \<in> C` `x < y`],
         THEN f', THEN MVT2[OF `x < y`, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
       by auto
-    hence "z1 \<in> C" using atMostAtLeast_subset_convex
+    then have "z1 \<in> C" using atMostAtLeast_subset_convex
       `convex C` `x \<in> C` `y \<in> C` `x < y` by fastforce
     from z1 have z1': "f x - f y = (x - y) * f' z1"
       by (simp add:field_simps)
@@ -568,14 +618,14 @@
     have A': "y - z1 \<ge> 0" using z1 by auto
     have "z3 \<in> C" using z3 asm atMostAtLeast_subset_convex
       `convex C` `x \<in> C` `z1 \<in> C` `x < z1` by fastforce
-    hence B': "f'' z3 \<ge> 0" using assms by auto
+    then have B': "f'' z3 \<ge> 0" using assms by auto
     from A' B' have "(y - z1) * f'' z3 \<ge> 0" using mult_nonneg_nonneg 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)
-    hence "f' y * (x - y) - (f x - f y) \<le> 0" using le by auto
-    hence res: "f' y * (x - y) \<le> f x - f y" by auto
+    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 asm z1 by auto
     also have "\<dots> = (z1 - x) * f'' z2" using z2 by auto
@@ -583,30 +633,32 @@
     have A: "z1 - x \<ge> 0" using z1 by auto
     have "z2 \<in> C" using z2 z1 asm atMostAtLeast_subset_convex
       `convex C` `z1 \<in> C` `y \<in> C` `z1 < y` by fastforce
-    hence B: "f'' z2 \<ge> 0" using assms by auto
+    then have B: "f'' z2 \<ge> 0" using assms by auto
     from A B have "(z1 - x) * f'' z2 \<ge> 0" using mult_nonneg_nonneg by auto
     from cool this 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)
-    hence "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
-    hence "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+    then have "f y - f x - f' x * (y - x) \<ge> 0" using ge by auto
+    then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
       using res by auto } note less_imp = this
-  { fix x y :: real assume "x \<in> C" "y \<in> C" "x \<noteq> y"
-    hence"f y - f x \<ge> f' x * (y - x)"
+  { fix x y :: real
+    assume "x \<in> C" "y \<in> C" "x \<noteq> y"
+    then have"f y - f x \<ge> f' x * (y - x)"
     unfolding neq_iff using less_imp by auto } note neq_imp = this
   moreover
-  { fix x y :: real assume asm: "x \<in> C" "y \<in> C" "x = y"
-    hence "f y - f x \<ge> f' x * (y - x)" by auto }
+  { fix x y :: real
+    assume asm: "x \<in> C" "y \<in> C" "x = y"
+    then have "f y - f x \<ge> f' x * (y - x)" by auto }
   ultimately show ?thesis using assms by blast
 qed
 
 lemma f''_ge0_imp_convex:
   fixes f :: "real \<Rightarrow> real"
   assumes conv: "convex C"
-  assumes f': "\<And> x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
-  assumes f'': "\<And> x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
-  assumes pos: "\<And> x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
+    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
 
@@ -615,18 +667,19 @@
   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
-  hence f': "\<And> z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
+  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)"
     using DERIV_minus by auto
-  have "\<And> z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
+  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)))"
+  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
-  hence f''0: "\<And> z :: real. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
+  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 add: mult_assoc)
-  have f''_ge0: "\<And> z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
-    using `b > 1` by (auto intro!:less_imp_le simp add:divide_pos_pos[of 1] mult_pos_pos)
+  have f''_ge0: "\<And>z :: real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
+    using `b > 1` by (auto intro!:less_imp_le simp add: divide_pos_pos[of 1] mult_pos_pos)
   from f''_ge0_imp_convex[OF pos_is_convex,
     unfolded greaterThan_iff, OF f' f''0 f''_ge0]
   show ?thesis by auto




