diff --git a/src/HOL/Analysis/Convex_Euclidean_Space.thy b/src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy
@@ -1,2274 +1,2126 @@
 (* Title:      HOL/Analysis/Convex_Euclidean_Space.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 on (Normed) Euclidean Spaces\<close>
 
 theory Convex_Euclidean_Space
 imports
   Convex
   Topology_Euclidean_Space
 begin
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Topological Properties of Convex Sets and Functions\<close>
 
 lemma aff_dim_cball:
   fixes a :: "'n::euclidean_space"
   assumes "e > 0"
   shows "aff_dim (cball a e) = int (DIM('n))"
 proof -
   have "(\<lambda>x. a + x) ` (cball 0 e) \<subseteq> cball a e"
     unfolding cball_def dist_norm by auto
   then have "aff_dim (cball (0 :: 'n::euclidean_space) e) \<le> aff_dim (cball a e)"
     using aff_dim_translation_eq[of a "cball 0 e"]
           aff_dim_subset[of "(+) a ` cball 0 e" "cball a e"]
     by auto
   moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))"
     using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"]
       centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms
     by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
   ultimately show ?thesis
     using aff_dim_le_DIM[of "cball a e"] by auto
 qed
 
 lemma aff_dim_open:
   fixes S :: "'n::euclidean_space set"
   assumes "open S"
     and "S \<noteq> {}"
   shows "aff_dim S = int (DIM('n))"
 proof -
   obtain x where "x \<in> S"
     using assms by auto
   then obtain e where e: "e > 0" "cball x e \<subseteq> S"
     using open_contains_cball[of S] assms by auto
   then have "aff_dim (cball x e) \<le> aff_dim S"
     using aff_dim_subset by auto
   with e show ?thesis
     using aff_dim_cball[of e x] aff_dim_le_DIM[of S] by auto
 qed
 
 lemma low_dim_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "\<not> aff_dim S = int (DIM('n))"
   shows "interior S = {}"
 proof -
   have "aff_dim(interior S) \<le> aff_dim S"
     using interior_subset aff_dim_subset[of "interior S" S] by auto
   then show ?thesis
     using aff_dim_open[of "interior S"] aff_dim_le_DIM[of S] assms by auto
 qed
 
 corollary empty_interior_lowdim:
   fixes S :: "'n::euclidean_space set"
   shows "dim S < DIM ('n) \<Longrightarrow> interior S = {}"
 by (metis low_dim_interior affine_hull_UNIV dim_affine_hull less_not_refl dim_UNIV)
 
 corollary aff_dim_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> aff_dim S = DIM('a)"
 by (metis low_dim_interior)
 
 
 subsection \<open>Relative interior of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_interior S =
   {x. \<exists>T. openin (top_of_set (affine hull S)) T \<and> x \<in> T \<and> T \<subseteq> S}"
 
 lemma rel_interior_mono:
    "\<lbrakk>S \<subseteq> T; affine hull S = affine hull T\<rbrakk>
    \<Longrightarrow> (rel_interior S) \<subseteq> (rel_interior T)"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior_maximal:
    "\<lbrakk>T \<subseteq> S; openin(top_of_set (affine hull S)) T\<rbrakk> \<Longrightarrow> T \<subseteq> (rel_interior S)"
   by (auto simp: rel_interior_def)
 
-lemma rel_interior:
-  "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
-  unfolding rel_interior_def[of S] openin_open[of "affine hull S"]
-  apply auto
-proof -
-  fix x T
-  assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
-  then have **: "x \<in> T \<inter> affine hull S"
-    using hull_inc by auto
-  show "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
-    apply (rule_tac x = "T \<inter> (affine hull S)" in exI)
-    using * **
-    apply auto
-    done
+lemma rel_interior: "rel_interior S = {x \<in> S. \<exists>T. open T \<and> x \<in> T \<and> T \<inter> affine hull S \<subseteq> S}"
+       (is "?lhs = ?rhs")
+proof
+  show "?lhs \<subseteq> ?rhs"
+    by (force simp add: rel_interior_def openin_open)
+  { fix x T
+    assume *: "x \<in> S" "open T" "x \<in> T" "T \<inter> affine hull S \<subseteq> S"
+    then have **: "x \<in> T \<inter> affine hull S"
+      using hull_inc by auto
+    with * have "\<exists>Tb. (\<exists>Ta. open Ta \<and> Tb = affine hull S \<inter> Ta) \<and> x \<in> Tb \<and> Tb \<subseteq> S"
+      by (rule_tac x = "T \<inter> (affine hull S)" in exI) auto
+  }
+  then show "?rhs \<subseteq> ?lhs"
+    by (force simp add: rel_interior_def openin_open)
 qed
 
 lemma mem_rel_interior: "x \<in> rel_interior S \<longleftrightarrow> (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
   by (auto simp: rel_interior)
 
 lemma mem_rel_interior_ball:
   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S)"
-  apply (simp add: rel_interior, safe)
-  apply (force simp: open_contains_ball)
-  apply (rule_tac x = "ball x e" in exI, simp)
-  done
+  (is "?lhs = ?rhs")
+proof
+  assume ?rhs then show ?lhs
+  by (simp add: rel_interior) (meson Elementary_Metric_Spaces.open_ball centre_in_ball)
+qed (force simp: rel_interior open_contains_ball)
 
 lemma rel_interior_ball:
   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> ball x e \<inter> affine hull S \<subseteq> S}"
   using mem_rel_interior_ball [of _ S] by auto
 
 lemma mem_rel_interior_cball:
   "x \<in> rel_interior S \<longleftrightarrow> x \<in> S \<and> (\<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S)"
-  apply (simp add: rel_interior, safe)
-  apply (force simp: open_contains_cball)
-  apply (rule_tac x = "ball x e" in exI)
-  apply (simp add: subset_trans [OF ball_subset_cball], auto)
-  done
+  (is "?lhs = ?rhs")
+proof
+  assume ?rhs then obtain e where "x \<in> S" "e > 0" "cball x e \<inter> affine hull S \<subseteq> S"
+    by (auto simp: rel_interior)
+  then have "ball x e \<inter> affine hull S \<subseteq> S"
+    by auto
+  then show ?lhs
+    using \<open>0 < e\<close> \<open>x \<in> S\<close> rel_interior_ball by auto
+qed (force simp: rel_interior open_contains_cball)
 
 lemma rel_interior_cball:
   "rel_interior S = {x \<in> S. \<exists>e. e > 0 \<and> cball x e \<inter> affine hull S \<subseteq> S}"
   using mem_rel_interior_cball [of _ S] by auto
 
 lemma rel_interior_empty [simp]: "rel_interior {} = {}"
    by (auto simp: rel_interior_def)
 
 lemma affine_hull_sing [simp]: "affine hull {a :: 'n::euclidean_space} = {a}"
   by (metis affine_hull_eq affine_sing)
 
 lemma rel_interior_sing [simp]:
-    fixes a :: "'n::euclidean_space"  shows "rel_interior {a} = {a}"
-  apply (auto simp: rel_interior_ball)
-  apply (rule_tac x=1 in exI, force)
-  done
+  fixes a :: "'n::euclidean_space"  shows "rel_interior {a} = {a}"
+proof -
+  have "\<exists>x::real. 0 < x"
+    using zero_less_one by blast
+  then show ?thesis
+    by (auto simp: rel_interior_ball)
+qed
 
 lemma subset_rel_interior:
   fixes S T :: "'n::euclidean_space set"
   assumes "S \<subseteq> T"
     and "affine hull S = affine hull T"
   shows "rel_interior S \<subseteq> rel_interior T"
   using assms by (auto simp: rel_interior_def)
 
 lemma rel_interior_subset: "rel_interior S \<subseteq> S"
   by (auto simp: rel_interior_def)
 
 lemma rel_interior_subset_closure: "rel_interior S \<subseteq> closure S"
   using rel_interior_subset by (auto simp: closure_def)
 
 lemma interior_subset_rel_interior: "interior S \<subseteq> rel_interior S"
   by (auto simp: rel_interior interior_def)
 
 lemma interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "aff_dim S = int(DIM('n))"
   shows "rel_interior S = interior S"
 proof -
   have "affine hull S = UNIV"
     using assms affine_hull_UNIV[of S] by auto
   then show ?thesis
     unfolding rel_interior interior_def by auto
 qed
 
 lemma rel_interior_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "affine hull S = UNIV"
   shows "rel_interior S = interior S"
   using assms unfolding rel_interior interior_def by auto
 
 lemma rel_interior_open:
   fixes S :: "'n::euclidean_space set"
   assumes "open S"
   shows "rel_interior S = S"
   by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
 
 lemma interior_rel_interior_gen:
   fixes S :: "'n::euclidean_space set"
   shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
   by (metis interior_rel_interior low_dim_interior)
 
 lemma rel_interior_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_interior S = interior S"
 by (metis interior_rel_interior_gen)
 
 lemma affine_hull_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> affine hull S = UNIV"
 by (metis affine_hull_UNIV interior_rel_interior_gen)
 
 lemma rel_interior_affine_hull [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "rel_interior (affine hull S) = affine hull S"
 proof -
   have *: "rel_interior (affine hull S) \<subseteq> affine hull S"
     using rel_interior_subset by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     define e :: real where "e = 1"
     then have "e > 0" "ball x e \<inter> affine hull (affine hull S) \<subseteq> affine hull S"
       using hull_hull[of _ S] by auto
     then have "x \<in> rel_interior (affine hull S)"
       using x rel_interior_ball[of "affine hull S"] by auto
   }
   then show ?thesis using * by auto
 qed
 
 lemma rel_interior_UNIV [simp]: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
   by (metis open_UNIV rel_interior_open)
 
 lemma rel_interior_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> rel_interior S"
     and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
     using assms(2) unfolding  mem_rel_interior_ball by auto
   {
     fix y
     assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d" "y \<in> affine hull S"
     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
       using \<open>e > 0\<close> by (auto simp: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "x \<in> affine hull S"
       using assms hull_subset[of S] by auto
     moreover have "1 / e + - ((1 - e) / e) = 1"
       using \<open>e > 0\<close> left_diff_distrib[of "1" "(1-e)" "1/e"] by auto
     ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> affine hull S"
       using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"]
       by (simp add: algebra_simps)
-    have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
-      unfolding dist_norm norm_scaleR[symmetric]
-      apply (rule arg_cong[where f=norm])
+    have "c - ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = (1 / e) *\<^sub>R (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       using \<open>e > 0\<close>
-      apply (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
-      done
+      by (auto simp: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
+    then have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
+      unfolding dist_norm norm_scaleR[symmetric] by auto
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
-    finally have "y \<in> S"
-      apply (subst *)
-      apply (rule assms(1)[unfolded convex_alt,rule_format])
-      apply (rule d[THEN subsetD])
-      unfolding mem_ball
-      using assms(3-5) **
-      apply auto
-      done
+    finally have "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x \<in> S"
+      using "**"  d by auto
+    then have "y \<in> S"
+      using * convexD [OF \<open>convex S\<close>] assms(3-5)
+      by (metis diff_add_cancel diff_ge_0_iff_ge le_add_same_cancel1 less_eq_real_def)
   }
   then have "ball (x - e *\<^sub>R (x - c)) (e*d) \<inter> affine hull S \<subseteq> S"
     by auto
   moreover have "e * d > 0"
     using \<open>e > 0\<close> \<open>d > 0\<close> by simp
   moreover have c: "c \<in> S"
     using assms rel_interior_subset by auto
   moreover from c have "x - e *\<^sub>R (x - c) \<in> S"
-    using convexD_alt[of S x c e]
-    apply (simp add: algebra_simps)
-    using assms
-    apply auto
-    done
+    using convexD_alt[of S x c e] assms
+    by (metis  diff_add_eq diff_diff_eq2 less_eq_real_def scaleR_diff_left scaleR_one scale_right_diff_distrib)
   ultimately show ?thesis
     using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] \<open>e > 0\<close> by auto
 qed
 
 lemma interior_real_atLeast [simp]:
   fixes a :: real
   shows "interior {a..} = {a<..}"
 proof -
   {
     fix y
-    assume "a < y"
-    then have "y \<in> interior {a..}"
-      apply (simp add: mem_interior)
-      apply (rule_tac x="(y-a)" in exI)
-      apply (auto simp: dist_norm)
-      done
+    have "ball y (y - a) \<subseteq> {a..}"
+      by (auto simp: dist_norm)
+    moreover assume "a < y"
+    ultimately have "y \<in> interior {a..}"
+      by (force simp add: mem_interior)
   }
   moreover
   {
     fix y
     assume "y \<in> interior {a..}"
     then obtain e where e: "e > 0" "cball y e \<subseteq> {a..}"
       using mem_interior_cball[of y "{a..}"] by auto
     moreover from e have "y - e \<in> cball y e"
       by (auto simp: cball_def dist_norm)
     ultimately have "a \<le> y - e" by blast
     then have "a < y" using e by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma continuous_ge_on_Ioo:
   assumes "continuous_on {c..d} g" "\<And>x. x \<in> {c<..<d} \<Longrightarrow> g x \<ge> a" "c < d" "x \<in> {c..d}"
   shows "g (x::real) \<ge> (a::real)"
 proof-
   from assms(3) have "{c..d} = closure {c<..<d}" by (rule closure_greaterThanLessThan[symmetric])
   also from assms(2) have "{c<..<d} \<subseteq> (g -` {a..} \<inter> {c..d})" by auto
   hence "closure {c<..<d} \<subseteq> closure (g -` {a..} \<inter> {c..d})" by (rule closure_mono)
   also from assms(1) have "closed (g -` {a..} \<inter> {c..d})"
     by (auto simp: continuous_on_closed_vimage)
   hence "closure (g -` {a..} \<inter> {c..d}) = g -` {a..} \<inter> {c..d}" by simp
   finally show ?thesis using \<open>x \<in> {c..d}\<close> by auto
 qed
 
 lemma interior_real_atMost [simp]:
   fixes a :: real
   shows "interior {..a} = {..<a}"
 proof -
   {
     fix y
-    assume "a > y"
-    then have "y \<in> interior {..a}"
-      apply (simp add: mem_interior)
-      apply (rule_tac x="(a-y)" in exI)
-      apply (auto simp: dist_norm)
-      done
+    have "ball y (a - y) \<subseteq> {..a}"
+      by (auto simp: dist_norm)
+    moreover assume "a > y"
+    ultimately have "y \<in> interior {..a}"
+      by (force simp add: mem_interior)
   }
   moreover
   {
     fix y
     assume "y \<in> interior {..a}"
     then obtain e where e: "e > 0" "cball y e \<subseteq> {..a}"
       using mem_interior_cball[of y "{..a}"] by auto
     moreover from e have "y + e \<in> cball y e"
       by (auto simp: cball_def dist_norm)
     ultimately have "a \<ge> y + e" by auto
     then have "a > y" using e by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma interior_atLeastAtMost_real [simp]: "interior {a..b} = {a<..<b :: real}"
 proof-
   have "{a..b} = {a..} \<inter> {..b}" by auto
   also have "interior \<dots> = {a<..} \<inter> {..<b}"
     by (simp)
   also have "\<dots> = {a<..<b}" by auto
   finally show ?thesis .
 qed
 
 lemma interior_atLeastLessThan [simp]:
   fixes a::real shows "interior {a..<b} = {a<..<b}"
   by (metis atLeastLessThan_def greaterThanLessThan_def interior_atLeastAtMost_real interior_Int interior_interior interior_real_atLeast)
 
 lemma interior_lessThanAtMost [simp]:
   fixes a::real shows "interior {a<..b} = {a<..<b}"
   by (metis atLeastAtMost_def greaterThanAtMost_def interior_atLeastAtMost_real interior_Int
             interior_interior interior_real_atLeast)
 
 lemma interior_greaterThanLessThan_real [simp]: "interior {a<..<b} = {a<..<b :: real}"
   by (metis interior_atLeastAtMost_real interior_interior)
 
 lemma frontier_real_atMost [simp]:
   fixes a :: real
   shows "frontier {..a} = {a}"
-  unfolding frontier_def by (auto simp: interior_real_atMost)
+  unfolding frontier_def by auto
 
 lemma frontier_real_atLeast [simp]: "frontier {a..} = {a::real}"
   by (auto simp: frontier_def)
 
 lemma frontier_real_greaterThan [simp]: "frontier {a<..} = {a::real}"
   by (auto simp: interior_open frontier_def)
 
 lemma frontier_real_lessThan [simp]: "frontier {..<a} = {a::real}"
   by (auto simp: interior_open frontier_def)
 
 lemma rel_interior_real_box [simp]:
   fixes a b :: real
   assumes "a < b"
   shows "rel_interior {a .. b} = {a <..< b}"
 proof -
   have "box a b \<noteq> {}"
     using assms
     unfolding set_eq_iff
     by (auto intro!: exI[of _ "(a + b) / 2"] simp: box_def)
   then show ?thesis
     using interior_rel_interior_gen[of "cbox a b", symmetric]
-    by (simp split: if_split_asm del: box_real add: box_real[symmetric] interior_cbox)
+    by (simp split: if_split_asm del: box_real add: box_real[symmetric])
 qed
 
 lemma rel_interior_real_semiline [simp]:
   fixes a :: real
   shows "rel_interior {a..} = {a<..}"
 proof -
   have *: "{a<..} \<noteq> {}"
     unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
   then show ?thesis using interior_real_atLeast interior_rel_interior_gen[of "{a..}"]
     by (auto split: if_split_asm)
 qed
 
 subsubsection \<open>Relative open sets\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_open S \<longleftrightarrow> rel_interior S = S"
 
-lemma rel_open: "rel_open S \<longleftrightarrow> openin (top_of_set (affine hull S)) S"
-  unfolding rel_open_def rel_interior_def
-  apply auto
-  using openin_subopen[of "top_of_set (affine hull S)" S]
-  apply auto
-  done
+lemma rel_open: "rel_open S \<longleftrightarrow> openin (top_of_set (affine hull S)) S" (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    unfolding rel_open_def rel_interior_def
+    using openin_subopen[of "top_of_set (affine hull S)" S] by auto
+qed (auto simp:  rel_open_def rel_interior_def)
 
 lemma openin_rel_interior: "openin (top_of_set (affine hull S)) (rel_interior S)"
-  apply (simp add: rel_interior_def)
-  apply (subst openin_subopen, blast)
-  done
+  using openin_subopen by (fastforce simp add: rel_interior_def)
 
 lemma openin_set_rel_interior:
    "openin (top_of_set S) (rel_interior S)"
 by (rule openin_subset_trans [OF openin_rel_interior rel_interior_subset hull_subset])
 
 lemma affine_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S"
   shows "rel_open S"
   unfolding rel_open_def
   using assms rel_interior_affine_hull[of S] affine_hull_eq[of S]
   by metis
 
 lemma affine_closed:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S"
   shows "closed S"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain L where L: "subspace L" "affine_parallel S L"
       using assms affine_parallel_subspace[of S] by auto
     then obtain a where a: "S = ((+) a ` L)"
       using affine_parallel_def[of L S] affine_parallel_commut by auto
     from L have "closed L" using closed_subspace by auto
     then have "closed S"
       using closed_translation a by auto
   }
   then show ?thesis by auto
 qed
 
 lemma closure_affine_hull:
   fixes S :: "'n::euclidean_space set"
   shows "closure S \<subseteq> affine hull S"
   by (intro closure_minimal hull_subset affine_closed affine_affine_hull)
 
 lemma closed_affine_hull [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (affine hull S)"
   by (metis affine_affine_hull affine_closed)
 
 lemma closure_same_affine_hull [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull (closure S) = affine hull S"
 proof -
   have "affine hull (closure S) \<subseteq> affine hull S"
     using hull_mono[of "closure S" "affine hull S" "affine"]
       closure_affine_hull[of S] hull_hull[of "affine" S]
     by auto
   moreover have "affine hull (closure S) \<supseteq> affine hull S"
     using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
   ultimately show ?thesis by auto
 qed
 
 lemma closure_aff_dim [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (closure S) = aff_dim S"
 proof -
   have "aff_dim S \<le> aff_dim (closure S)"
     using aff_dim_subset closure_subset by auto
   moreover have "aff_dim (closure S) \<le> aff_dim (affine hull S)"
     using aff_dim_subset closure_affine_hull by blast
   moreover have "aff_dim (affine hull S) = aff_dim S"
     using aff_dim_affine_hull by auto
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_closure_convex_shrink:
   fixes S :: "_::euclidean_space set"
   assumes "convex S"
     and "c \<in> rel_interior S"
     and "x \<in> closure S"
     and "e > 0"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<inter> affine hull S \<subseteq> S"
     using assms(2) unfolding mem_rel_interior_ball by auto
   have "\<exists>y \<in> S. norm (y - x) * (1 - e) < e * d"
   proof (cases "x \<in> S")
     case True
-    then show ?thesis using \<open>e > 0\<close> \<open>d > 0\<close>
-      apply (rule_tac bexI[where x=x], auto)
-      done
+    then show ?thesis using \<open>e > 0\<close> \<open>d > 0\<close> by force
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
-        apply (rule_tac x=y in bexI)
-        unfolding True
-        using \<open>d > 0\<close>
-        apply auto
-        done
+        unfolding True using \<open>d > 0\<close> by (force simp add: )
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
       then show ?thesis
-        apply (rule_tac x=y in bexI)
-        unfolding dist_norm
-        using pos_less_divide_eq[OF *]
-        apply auto
-        done
+        unfolding dist_norm using pos_less_divide_eq[OF *] by force
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
   have zball: "z \<in> ball c d"
     using mem_ball z_def dist_norm[of c]
     using y and assms(4,5)
     by (simp add: norm_minus_commute) (simp add: field_simps)
   have "x \<in> affine hull S"
     using closure_affine_hull assms by auto
   moreover have "y \<in> affine hull S"
     using \<open>y \<in> S\<close> hull_subset[of S] by auto
   moreover have "c \<in> affine hull S"
     using assms rel_interior_subset hull_subset[of S] by auto
   ultimately have "z \<in> affine hull S"
     using z_def affine_affine_hull[of S]
       mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"]
       assms
     by simp
   then have "z \<in> S" using d zball by auto
   obtain d1 where "d1 > 0" and d1: "ball z d1 \<le> ball c d"
     using zball open_ball[of c d] openE[of "ball c d" z] by auto
   then have "ball z d1 \<inter> affine hull S \<subseteq> ball c d \<inter> affine hull S"
     by auto
   then have "ball z d1 \<inter> affine hull S \<subseteq> S"
     using d by auto
   then have "z \<in> rel_interior S"
     using mem_rel_interior_ball using \<open>d1 > 0\<close> \<open>z \<in> S\<close> by auto
   then have "y - e *\<^sub>R (y - z) \<in> rel_interior S"
     using rel_interior_convex_shrink[of S z y e] assms \<open>y \<in> S\<close> by auto
   then show ?thesis using * by auto
 qed
 
 lemma rel_interior_eq:
    "rel_interior s = s \<longleftrightarrow> openin(top_of_set (affine hull s)) s"
 using rel_open rel_open_def by blast
 
 lemma rel_interior_openin:
    "openin(top_of_set (affine hull s)) s \<Longrightarrow> rel_interior s = s"
 by (simp add: rel_interior_eq)
 
 lemma rel_interior_affine:
   fixes S :: "'n::euclidean_space set"
   shows  "affine S \<Longrightarrow> rel_interior S = S"
 using affine_rel_open rel_open_def by auto
 
 lemma rel_interior_eq_closure:
   fixes S :: "'n::euclidean_space set"
   shows "rel_interior S = closure S \<longleftrightarrow> affine S"
 proof (cases "S = {}")
   case True
  then show ?thesis
     by auto
 next
   case False show ?thesis
   proof
     assume eq: "rel_interior S = closure S"
-    have "S = {} \<or> S = affine hull S"
-      apply (rule connected_clopen [THEN iffD1, rule_format])
-       apply (simp add: affine_imp_convex convex_connected)
-      apply (rule conjI)
-       apply (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
-      apply (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
-      done
+    have "openin (top_of_set (affine hull S)) S"
+      by (metis eq closure_subset openin_rel_interior rel_interior_subset subset_antisym)
+    moreover have "closedin (top_of_set (affine hull S)) S"
+      by (metis closed_subset closure_subset_eq eq hull_subset rel_interior_subset)
+    ultimately have "S = {} \<or> S = affine hull S"
+      using convex_connected connected_clopen convex_affine_hull by metis
     with False have "affine hull S = S"
       by auto
     then show "affine S"
       by (metis affine_hull_eq)
   next
     assume "affine S"
     then show "rel_interior S = closure S"
       by (simp add: rel_interior_affine affine_closed)
   qed
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Relative interior preserves under linear transformations\<close>
 
 lemma rel_interior_translation_aux:
   fixes a :: "'n::euclidean_space"
   shows "((\<lambda>x. a + x) ` rel_interior S) \<subseteq> rel_interior ((\<lambda>x. a + x) ` S)"
 proof -
   {
     fix x
     assume x: "x \<in> rel_interior S"
     then obtain T where "open T" "x \<in> T \<inter> S" "T \<inter> affine hull S \<subseteq> S"
       using mem_rel_interior[of x S] by auto
     then have "open ((\<lambda>x. a + x) ` T)"
       and "a + x \<in> ((\<lambda>x. a + x) ` T) \<inter> ((\<lambda>x. a + x) ` S)"
       and "((\<lambda>x. a + x) ` T) \<inter> affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` S"
       using affine_hull_translation[of a S] open_translation[of T a] x by auto
     then have "a + x \<in> rel_interior ((\<lambda>x. a + x) ` S)"
       using mem_rel_interior[of "a+x" "((\<lambda>x. a + x) ` S)"] by auto
   }
   then show ?thesis by auto
 qed
 
 lemma rel_interior_translation:
   fixes a :: "'n::euclidean_space"
   shows "rel_interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` rel_interior S"
 proof -
   have "(\<lambda>x. (-a) + x) ` rel_interior ((\<lambda>x. a + x) ` S) \<subseteq> rel_interior S"
     using rel_interior_translation_aux[of "-a" "(\<lambda>x. a + x) ` S"]
       translation_assoc[of "-a" "a"]
     by auto
   then have "((\<lambda>x. a + x) ` rel_interior S) \<supseteq> rel_interior ((\<lambda>x. a + x) ` S)"
     using translation_inverse_subset[of a "rel_interior ((+) a ` S)" "rel_interior S"]
     by auto
   then show ?thesis
     using rel_interior_translation_aux[of a S] by auto
 qed
 
 
 lemma affine_hull_linear_image:
   assumes "bounded_linear f"
   shows "f ` (affine hull s) = affine hull f ` s"
 proof -
   interpret f: bounded_linear f by fact
   have "affine {x. f x \<in> affine hull f ` s}"
     unfolding affine_def
     by (auto simp: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format])
   moreover have "affine {x. x \<in> f ` (affine hull s)}"
     using affine_affine_hull[unfolded affine_def, of s]
     unfolding affine_def by (auto simp: f.scaleR [symmetric] f.add [symmetric])
   ultimately show ?thesis
     by (auto simp: hull_inc elim!: hull_induct)
 qed 
 
 
 lemma rel_interior_injective_on_span_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
     and S :: "'m::euclidean_space set"
   assumes "bounded_linear f"
     and "inj_on f (span S)"
   shows "rel_interior (f ` S) = f ` (rel_interior S)"
 proof -
   {
     fix z
     assume z: "z \<in> rel_interior (f ` S)"
     then have "z \<in> f ` S"
       using rel_interior_subset[of "f ` S"] by auto
     then obtain x where x: "x \<in> S" "f x = z" by auto
     obtain e2 where e2: "e2 > 0" "cball z e2 \<inter> affine hull (f ` S) \<subseteq> (f ` S)"
       using z rel_interior_cball[of "f ` S"] by auto
     obtain K where K: "K > 0" "\<And>x. norm (f x) \<le> norm x * K"
      using assms Real_Vector_Spaces.bounded_linear.pos_bounded[of f] by auto
     define e1 where "e1 = 1 / K"
     then have e1: "e1 > 0" "\<And>x. e1 * norm (f x) \<le> norm x"
       using K pos_le_divide_eq[of e1] by auto
     define e where "e = e1 * e2"
     then have "e > 0" using e1 e2 by auto
     {
       fix y
       assume y: "y \<in> cball x e \<inter> affine hull S"
       then have h1: "f y \<in> affine hull (f ` S)"
         using affine_hull_linear_image[of f S] assms by auto
       from y have "norm (x-y) \<le> e1 * e2"
         using cball_def[of x e] dist_norm[of x y] e_def by auto
       moreover have "f x - f y = f (x - y)"
         using assms linear_diff[of f x y] linear_conv_bounded_linear[of f] by auto
       moreover have "e1 * norm (f (x-y)) \<le> norm (x - y)"
         using e1 by auto
       ultimately have "e1 * norm ((f x)-(f y)) \<le> e1 * e2"
         by auto
       then have "f y \<in> cball z e2"
         using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1 x by auto
       then have "f y \<in> f ` S"
         using y e2 h1 by auto
       then have "y \<in> S"
         using assms y hull_subset[of S] affine_hull_subset_span
           inj_on_image_mem_iff [OF \<open>inj_on f (span S)\<close>]
         by (metis Int_iff span_superset subsetCE)
     }
     then have "z \<in> f ` (rel_interior S)"
       using mem_rel_interior_cball[of x S] \<open>e > 0\<close> x by auto
   }
   moreover
   {
     fix x
     assume x: "x \<in> rel_interior S"
     then obtain e2 where e2: "e2 > 0" "cball x e2 \<inter> affine hull S \<subseteq> S"
       using rel_interior_cball[of S] by auto
     have "x \<in> S" using x rel_interior_subset by auto
     then have *: "f x \<in> f ` S" by auto
     have "\<forall>x\<in>span S. f x = 0 \<longrightarrow> x = 0"
       using assms subspace_span linear_conv_bounded_linear[of f]
         linear_injective_on_subspace_0[of f "span S"]
       by auto
     then obtain e1 where e1: "e1 > 0" "\<forall>x \<in> span S. e1 * norm x \<le> norm (f x)"
       using assms injective_imp_isometric[of "span S" f]
         subspace_span[of S] closed_subspace[of "span S"]
       by auto
     define e where "e = e1 * e2"
     hence "e > 0" using e1 e2 by auto
     {
       fix y
       assume y: "y \<in> cball (f x) e \<inter> affine hull (f ` S)"
       then have "y \<in> f ` (affine hull S)"
         using affine_hull_linear_image[of f S] assms by auto
       then obtain xy where xy: "xy \<in> affine hull S" "f xy = y" by auto
       with y have "norm (f x - f xy) \<le> e1 * e2"
         using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
       moreover have "f x - f xy = f (x - xy)"
         using assms linear_diff[of f x xy] linear_conv_bounded_linear[of f] by auto
       moreover have *: "x - xy \<in> span S"
         using subspace_diff[of "span S" x xy] subspace_span \<open>x \<in> S\<close> xy
           affine_hull_subset_span[of S] span_superset
         by auto
       moreover from * have "e1 * norm (x - xy) \<le> norm (f (x - xy))"
         using e1 by auto
       ultimately have "e1 * norm (x - xy) \<le> e1 * e2"
         by auto
       then have "xy \<in> cball x e2"
         using cball_def[of x e2] dist_norm[of x xy] e1 by auto
       then have "y \<in> f ` S"
         using xy e2 by auto
     }
     then have "f x \<in> rel_interior (f ` S)"
       using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * \<open>e > 0\<close> by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_injective_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "bounded_linear f"
     and "inj f"
   shows "rel_interior (f ` S) = f ` (rel_interior S)"
   using assms rel_interior_injective_on_span_linear_image[of f S]
     subset_inj_on[of f "UNIV" "span S"]
   by auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Openness and compactness are preserved by convex hull operation\<close>
 
 lemma open_convex_hull[intro]:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open (convex hull S)"
 proof (clarsimp simp: open_contains_cball convex_hull_explicit)
   fix T and u :: "'a\<Rightarrow>real"
   assume obt: "finite T" "T\<subseteq>S" "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1" 
 
   from assms[unfolded open_contains_cball] obtain b
     where b: "\<And>x. x\<in>S \<Longrightarrow> 0 < b x \<and> cball x (b x) \<subseteq> S" by metis
   have "b ` T \<noteq> {}"
     using obt by auto
   define i where "i = b ` T"
   let ?\<Phi> = "\<lambda>y. \<exists>F. finite F \<and> F \<subseteq> S \<and> (\<exists>u. (\<forall>x\<in>F. 0 \<le> u x) \<and> sum u F = 1 \<and> (\<Sum>v\<in>F. u v *\<^sub>R v) = y)"
   let ?a = "\<Sum>v\<in>T. u v *\<^sub>R v"
   show "\<exists>e > 0. cball ?a e \<subseteq> {y. ?\<Phi> y}"
   proof (intro exI subsetI conjI)
     show "0 < Min i"
       unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] \<open>b ` T\<noteq>{}\<close>]
       using b \<open>T\<subseteq>S\<close> by auto
   next
     fix y
     assume "y \<in> cball ?a (Min i)"
     then have y: "norm (?a - y) \<le> Min i"
       unfolding dist_norm[symmetric] by auto
     { fix x
       assume "x \<in> T"
       then have "Min i \<le> b x"
         by (simp add: i_def obt(1))
       then have "x + (y - ?a) \<in> cball x (b x)"
         using y unfolding mem_cball dist_norm by auto
       moreover have "x \<in> S"
         using \<open>x\<in>T\<close> \<open>T\<subseteq>S\<close> by auto
       ultimately have "x + (y - ?a) \<in> S"
         using y b by blast
     }
     moreover
     have *: "inj_on (\<lambda>v. v + (y - ?a)) T"
       unfolding inj_on_def by auto
     have "(\<Sum>v\<in>(\<lambda>v. v + (y - ?a)) ` T. u (v - (y - ?a)) *\<^sub>R v) = y"
       unfolding sum.reindex[OF *] o_def using obt(4)
       by (simp add: sum.distrib sum_subtractf scaleR_left.sum[symmetric] scaleR_right_distrib)
     ultimately show "y \<in> {y. ?\<Phi> y}"
     proof (intro CollectI exI conjI)
       show "finite ((\<lambda>v. v + (y - ?a)) ` T)"
         by (simp add: obt(1))
       show "sum (\<lambda>v. u (v - (y - ?a))) ((\<lambda>v. v + (y - ?a)) ` T) = 1"
         unfolding sum.reindex[OF *] o_def using obt(4) by auto
     qed (use obt(1, 3) in auto)
   qed
 qed
 
 lemma compact_convex_combinations:
   fixes S T :: "'a::real_normed_vector set"
   assumes "compact S" "compact T"
   shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T}"
 proof -
   let ?X = "{0..1} \<times> S \<times> T"
   let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
   have *: "{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> T} = ?h ` ?X"
     by force
   have "continuous_on ?X (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
   with assms show ?thesis
     by (simp add: * compact_Times compact_continuous_image)
 qed
 
 lemma finite_imp_compact_convex_hull:
   fixes S :: "'a::real_normed_vector set"
   assumes "finite S"
   shows "compact (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis by simp
 next
   case False
   with assms show ?thesis
   proof (induct rule: finite_ne_induct)
     case (singleton x)
     show ?case by simp
   next
     case (insert x A)
     let ?f = "\<lambda>(u, y::'a). u *\<^sub>R x + (1 - u) *\<^sub>R y"
     let ?T = "{0..1::real} \<times> (convex hull A)"
     have "continuous_on ?T ?f"
       unfolding split_def continuous_on by (intro ballI tendsto_intros)
     moreover have "compact ?T"
       by (intro compact_Times compact_Icc insert)
     ultimately have "compact (?f ` ?T)"
       by (rule compact_continuous_image)
     also have "?f ` ?T = convex hull (insert x A)"
       unfolding convex_hull_insert [OF \<open>A \<noteq> {}\<close>]
       apply safe
       apply (rule_tac x=a in exI, simp)
       apply (rule_tac x="1 - a" in exI, simp, fast)
       apply (rule_tac x="(u, b)" in image_eqI, simp_all)
       done
     finally show "compact (convex hull (insert x A))" .
   qed
 qed
 
 lemma compact_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S"
   shows "compact (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis using compact_empty by simp
 next
   case False
   then obtain w where "w \<in> S" by auto
   show ?thesis
     unfolding caratheodory[of S]
   proof (induct ("DIM('a) + 1"))
     case 0
     have *: "{x.\<exists>sa. finite sa \<and> sa \<subseteq> S \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
       using compact_empty by auto
     from 0 show ?case unfolding * by simp
   next
     case (Suc n)
     show ?case
     proof (cases "n = 0")
       case True
       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} = S"
         unfolding set_eq_iff and mem_Collect_eq
       proof (rule, rule)
         fix x
         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
           by auto
         show "x \<in> S"
         proof (cases "card T = 0")
           case True
           then show ?thesis
             using T(4) unfolding card_0_eq[OF T(1)] by simp
         next
           case False
           then have "card T = Suc 0" using T(3) \<open>n=0\<close> by auto
           then obtain a where "T = {a}" unfolding card_Suc_eq by auto
           then show ?thesis using T(2,4) by simp
         qed
       next
         fix x assume "x\<in>S"
         then show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
-          apply (rule_tac x="{x}" in exI)
-          unfolding convex_hull_singleton
-          apply auto
-          done
+          by (rule_tac x="{x}" in exI) (use convex_hull_singleton in auto)
       qed
       then show ?thesis using assms by simp
     next
       case False
       have "{x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T} =
         {(1 - u) *\<^sub>R x + u *\<^sub>R y | x y u.
           0 \<le> u \<and> u \<le> 1 \<and> x \<in> S \<and> y \<in> {x. \<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> x \<in> convex hull T}}"
         unfolding set_eq_iff and mem_Collect_eq
       proof (rule, rule)
         fix x
         assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
         then obtain u v c T where obt: "x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
           "0 \<le> c \<and> c \<le> 1" "u \<in> S" "finite T" "T \<subseteq> S" "card T \<le> n"  "v \<in> convex hull T"
           by auto
         moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u T"
-          apply (rule convexD_alt)
-          using obt(2) and convex_convex_hull and hull_subset[of "insert u T" convex]
-          using obt(7) and hull_mono[of T "insert u T"]
-          apply auto
-          done
+          by (meson convexD_alt convex_convex_hull hull_inc hull_mono in_mono insertCI obt(2) obt(7) subset_insertI)
         ultimately show "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
-          apply (rule_tac x="insert u T" in exI)
-          apply (auto simp: card_insert_if)
-          done
+          by (rule_tac x="insert u T" in exI) (auto simp: card_insert_if)
       next
         fix x
         assume "\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> Suc n \<and> x \<in> convex hull T"
         then obtain T where T: "finite T" "T \<subseteq> S" "card T \<le> Suc n" "x \<in> convex hull T"
           by auto
         show "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
           0 \<le> c \<and> c \<le> 1 \<and> u \<in> S \<and> (\<exists>T. finite T \<and> T \<subseteq> S \<and> card T \<le> n \<and> v \<in> convex hull T)"
         proof (cases "card T = Suc n")
           case False
           then have "card T \<le> n" using T(3) by auto
           then show ?thesis
-            apply (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI)
             using \<open>w\<in>S\<close> and T
-            apply (auto intro!: exI[where x=T])
-            done
+            by (rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) auto
         next
           case True
           then obtain a u where au: "T = insert a u" "a\<notin>u"
-            apply (drule_tac card_eq_SucD, auto)
-            done
+            by (metis card_le_Suc_iff order_refl)
           show ?thesis
           proof (cases "u = {}")
             case True
             then have "x = a" using T(4)[unfolded au] by auto
             show ?thesis unfolding \<open>x = a\<close>
-              apply (rule_tac x=a in exI)
-              apply (rule_tac x=a in exI)
-              apply (rule_tac x=1 in exI)
-              using T and \<open>n \<noteq> 0\<close>
-              unfolding au
-              apply (auto intro!: exI[where x="{a}"])
-              done
+              using T \<open>n \<noteq> 0\<close> unfolding au              
+              by (rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI) force
           next
             case False
             obtain ux vx b where obt: "ux\<ge>0" "vx\<ge>0" "ux + vx = 1"
               "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
               using T(4)[unfolded au convex_hull_insert[OF False]]
               by auto
             have *: "1 - vx = ux" using obt(3) by auto
             show ?thesis
-              apply (rule_tac x=a in exI)
-              apply (rule_tac x=b in exI)
-              apply (rule_tac x=vx in exI)
-              using obt and T(1-3)
-              unfolding au and * using card_insert_disjoint[OF _ au(2)]
-              apply (auto intro!: exI[where x=u])
-              done
+              using obt T(1-3) card_insert_disjoint[OF _ au(2)] unfolding au *  
+              by (rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI) force
           qed
         qed
       qed
       then show ?thesis
         using compact_convex_combinations[OF assms Suc] by simp
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Extremal points of a simplex are some vertices\<close>
 
 lemma dist_increases_online:
   fixes a b d :: "'a::real_inner"
   assumes "d \<noteq> 0"
   shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
 proof (cases "inner a d - inner b d > 0")
   case True
   then have "0 < inner d d + (inner a d * 2 - inner b d * 2)"
-    apply (rule_tac add_pos_pos)
     using assms
-    apply auto
-    done
+    by (intro add_pos_pos) auto
   then show ?thesis
-    apply (rule_tac disjI2)
     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
-    apply  (simp add: algebra_simps inner_commute)
-    done
+    by (simp add: algebra_simps inner_commute)
 next
   case False
   then have "0 < inner d d + (inner b d * 2 - inner a d * 2)"
-    apply (rule_tac add_pos_nonneg)
     using assms
-    apply auto
-    done
+    by (intro add_pos_nonneg) auto
   then show ?thesis
-    apply (rule_tac disjI1)
     unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
-    apply (simp add: algebra_simps inner_commute)
-    done
+    by (simp add: algebra_simps inner_commute)
 qed
 
 lemma norm_increases_online:
   fixes d :: "'a::real_inner"
   shows "d \<noteq> 0 \<Longrightarrow> norm (a + d) > norm a \<or> norm(a - d) > norm a"
   using dist_increases_online[of d a 0] unfolding dist_norm by auto
 
 lemma simplex_furthest_lt:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
   shows "\<forall>x \<in> convex hull S.  x \<notin> S \<longrightarrow> (\<exists>y \<in> convex hull S. norm (x - a) < norm(y - a))"
   using assms
 proof induct
   fix x S
   assume as: "finite S" "x\<notin>S" "\<forall>x\<in>convex hull S. x \<notin> S \<longrightarrow> (\<exists>y\<in>convex hull S. norm (x - a) < norm (y - a))"
   show "\<forall>xa\<in>convex hull insert x S. xa \<notin> insert x S \<longrightarrow>
     (\<exists>y\<in>convex hull insert x S. norm (xa - a) < norm (y - a))"
   proof (intro impI ballI, cases "S = {}")
     case False
     fix y
     assume y: "y \<in> convex hull insert x S" "y \<notin> insert x S"
     obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "y = u *\<^sub>R x + v *\<^sub>R b"
       using y(1)[unfolded convex_hull_insert[OF False]] by auto
     show "\<exists>z\<in>convex hull insert x S. norm (y - a) < norm (z - a)"
     proof (cases "y \<in> convex hull S")
       case True
       then obtain z where "z \<in> convex hull S" "norm (y - a) < norm (z - a)"
         using as(3)[THEN bspec[where x=y]] and y(2) by auto
       then show ?thesis
-        apply (rule_tac x=z in bexI)
-        unfolding convex_hull_insert[OF False]
-        apply auto
-        done
+        by (meson hull_mono subsetD subset_insertI)
     next
       case False
       show ?thesis
-        using obt(3)
-      proof (cases "u = 0", case_tac[!] "v = 0")
-        assume "u = 0" "v \<noteq> 0"
-        then have "y = b" using obt by auto
-        then show ?thesis using False and obt(4) by auto
+      proof (cases "u = 0 \<or> v = 0")
+        case True
+        with False show ?thesis
+          using obt y by auto
       next
-        assume "u \<noteq> 0" "v = 0"
-        then have "y = x" using obt by auto
-        then show ?thesis using y(2) by auto
-      next
-        assume "u \<noteq> 0" "v \<noteq> 0"
+        case False
         then obtain w where w: "w>0" "w<u" "w<v"
           using field_lbound_gt_zero[of u v] and obt(1,2) by auto
         have "x \<noteq> b"
         proof
           assume "x = b"
           then have "y = b" unfolding obt(5)
             using obt(3) by (auto simp: scaleR_left_distrib[symmetric])
-          then show False using obt(4) and False by simp
+          then show False using obt(4) and False
+            using \<open>x = b\<close> y(2) by blast
         qed
         then have *: "w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
         show ?thesis
           using dist_increases_online[OF *, of a y]
         proof (elim disjE)
           assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
           then have "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
             unfolding dist_commute[of a]
             unfolding dist_norm obt(5)
             by (simp add: algebra_simps)
           moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x S"
             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
           proof (intro CollectI conjI exI)
             show "u + w \<ge> 0" "v - w \<ge> 0"
               using obt(1) w by auto
           qed (use obt in auto)
           ultimately show ?thesis by auto
         next
           assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
           then have "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
             unfolding dist_commute[of a]
             unfolding dist_norm obt(5)
             by (simp add: algebra_simps)
           moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x S"
             unfolding convex_hull_insert[OF \<open>S\<noteq>{}\<close>]
           proof (intro CollectI conjI exI)
             show "u - w \<ge> 0" "v + w \<ge> 0"
               using obt(1) w by auto
           qed (use obt in auto)
           ultimately show ?thesis by auto
         qed
-      qed auto
+      qed
     qed
   qed auto
 qed (auto simp: assms)
 
 lemma simplex_furthest_le:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
     and "S \<noteq> {}"
   shows "\<exists>y\<in>S. \<forall>x\<in> convex hull S. norm (x - a) \<le> norm (y - a)"
 proof -
   have "convex hull S \<noteq> {}"
     using hull_subset[of S convex] and assms(2) by auto
   then obtain x where x: "x \<in> convex hull S" "\<forall>y\<in>convex hull S. norm (y - a) \<le> norm (x - a)"
     using distance_attains_sup[OF finite_imp_compact_convex_hull[OF \<open>finite S\<close>], of a]
     unfolding dist_commute[of a]
     unfolding dist_norm
     by auto
   show ?thesis
   proof (cases "x \<in> S")
     case False
     then obtain y where "y \<in> convex hull S" "norm (x - a) < norm (y - a)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1)
       by auto
     then show ?thesis
       using x(2)[THEN bspec[where x=y]] by auto
   next
     case True
     with x show ?thesis by auto
   qed
 qed
 
 lemma simplex_furthest_le_exists:
   fixes S :: "('a::real_inner) set"
   shows "finite S \<Longrightarrow> \<forall>x\<in>(convex hull S). \<exists>y\<in>S. norm (x - a) \<le> norm (y - a)"
   using simplex_furthest_le[of S] by (cases "S = {}") auto
 
 lemma simplex_extremal_le:
   fixes S :: "'a::real_inner set"
   assumes "finite S"
     and "S \<noteq> {}"
   shows "\<exists>u\<in>S. \<exists>v\<in>S. \<forall>x\<in>convex hull S. \<forall>y \<in> convex hull S. norm (x - y) \<le> norm (u - v)"
 proof -
   have "convex hull S \<noteq> {}"
     using hull_subset[of S convex] and assms(2) by auto
   then obtain u v where obt: "u \<in> convex hull S" "v \<in> convex hull S"
     "\<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
     using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]]
     by (auto simp: dist_norm)
   then show ?thesis
   proof (cases "u\<notin>S \<or> v\<notin>S", elim disjE)
     assume "u \<notin> S"
     then obtain y where "y \<in> convex hull S" "norm (u - v) < norm (y - v)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1)
       by auto
     then show ?thesis
       using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2)
       by auto
   next
     assume "v \<notin> S"
     then obtain y where "y \<in> convex hull S" "norm (v - u) < norm (y - u)"
       using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2)
       by auto
     then show ?thesis
       using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
       by (auto simp: norm_minus_commute)
   qed auto
 qed
 
 lemma simplex_extremal_le_exists:
   fixes S :: "'a::real_inner set"
   shows "finite S \<Longrightarrow> x \<in> convex hull S \<Longrightarrow> y \<in> convex hull S \<Longrightarrow>
     \<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
   using convex_hull_empty simplex_extremal_le[of S]
   by(cases "S = {}") auto
 
 
 subsection \<open>Closest point of a convex set is unique, with a continuous projection\<close>
 
 definition\<^marker>\<open>tag important\<close> closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a"
   where "closest_point S a = (SOME x. x \<in> S \<and> (\<forall>y\<in>S. dist a x \<le> dist a y))"
 
 lemma closest_point_exists:
   assumes "closed S"
     and "S \<noteq> {}"
   shows closest_point_in_set: "closest_point S a \<in> S"
     and "\<forall>y\<in>S. dist a (closest_point S a) \<le> dist a y"
   unfolding closest_point_def
-  apply(rule_tac[!] someI2_ex)
-  apply (auto intro: distance_attains_inf[OF assms(1,2), of a])
-  done
+  by (rule_tac someI2_ex, auto intro: distance_attains_inf[OF assms(1,2), of a])+
 
 lemma closest_point_le: "closed S \<Longrightarrow> x \<in> S \<Longrightarrow> dist a (closest_point S a) \<le> dist a x"
   using closest_point_exists[of S] by auto
 
 lemma closest_point_self:
   assumes "x \<in> S"
   shows "closest_point S x = x"
   unfolding closest_point_def
-  apply (rule some1_equality, rule ex1I[of _ x])
-  using assms
-  apply auto
-  done
+  by (rule some1_equality, rule ex1I[of _ x]) (use assms in auto)
 
 lemma closest_point_refl: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S x = x \<longleftrightarrow> x \<in> S"
   using closest_point_in_set[of S x] closest_point_self[of x S]
   by auto
 
 lemma closer_points_lemma:
   assumes "inner y z > 0"
   shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
 proof -
   have z: "inner z z > 0"
     unfolding inner_gt_zero_iff using assms by auto
   have "norm (v *\<^sub>R z - y) < norm y"
     if "0 < v" and "v \<le> inner y z / inner z z" for v
     unfolding norm_lt using z assms that
     by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ \<open>0<v\<close>])
   then show ?thesis
     using assms z
     by (rule_tac x = "inner y z / inner z z" in exI) auto
 qed
 
 lemma closer_point_lemma:
   assumes "inner (y - x) (z - x) > 0"
   shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
 proof -
   obtain u where "u > 0"
-    and u: "\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
+    and u: "\<And>v. \<lbrakk>0<v; v \<le> u\<rbrakk> \<Longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
     using closer_points_lemma[OF assms] by auto
   show ?thesis
-    apply (rule_tac x="min u 1" in exI)
-    using u[THEN spec[where x="min u 1"]] and \<open>u > 0\<close>
-    unfolding dist_norm by (auto simp: norm_minus_commute field_simps)
+    using u[of "min u 1"] and \<open>u > 0\<close>
+    by (metis diff_diff_add dist_commute dist_norm less_eq_real_def not_less u zero_less_one)
 qed
 
 lemma any_closest_point_dot:
   assumes "convex S" "closed S" "x \<in> S" "y \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
   shows "inner (a - x) (y - x) \<le> 0"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   then obtain u where u: "u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a"
     using closer_point_lemma[of a x y] by auto
   let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y"
   have "?z \<in> S"
     using convexD_alt[OF assms(1,3,4), of u] using u by auto
   then show False
     using assms(5)[THEN bspec[where x="?z"]] and u(3)
     by (auto simp: dist_commute algebra_simps)
 qed
 
 lemma any_closest_point_unique:
   fixes x :: "'a::real_inner"
   assumes "convex S" "closed S" "x \<in> S" "y \<in> S"
     "\<forall>z\<in>S. dist a x \<le> dist a z" "\<forall>z\<in>S. dist a y \<le> dist a z"
   shows "x = y"
   using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
   unfolding norm_pths(1) and norm_le_square
   by (auto simp: algebra_simps)
 
 lemma closest_point_unique:
   assumes "convex S" "closed S" "x \<in> S" "\<forall>z\<in>S. dist a x \<le> dist a z"
   shows "x = closest_point S a"
   using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point S a"]
   using closest_point_exists[OF assms(2)] and assms(3) by auto
 
 lemma closest_point_dot:
   assumes "convex S" "closed S" "x \<in> S"
   shows "inner (a - closest_point S a) (x - closest_point S a) \<le> 0"
-  apply (rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
-  using closest_point_exists[OF assms(2)] and assms(3)
-  apply auto
-  done
+  using any_closest_point_dot[OF assms(1,2) _ assms(3)]
+  by (metis assms(2) assms(3) closest_point_in_set closest_point_le empty_iff)
 
 lemma closest_point_lt:
   assumes "convex S" "closed S" "x \<in> S" "x \<noteq> closest_point S a"
   shows "dist a (closest_point S a) < dist a x"
-  apply (rule ccontr)
-  apply (rule_tac notE[OF assms(4)])
-  apply (rule closest_point_unique[OF assms(1-3), of a])
-  using closest_point_le[OF assms(2), of _ a]
-  apply fastforce
-  done
+  using closest_point_unique[where a=a] closest_point_le[where a=a] assms by fastforce
 
 lemma setdist_closest_point:
     "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> setdist {a} S = dist a (closest_point S a)"
-  apply (rule setdist_unique)
-  using closest_point_le
-  apply (auto simp: closest_point_in_set)
-  done
+  by (metis closest_point_exists(2) closest_point_in_set emptyE insert_iff setdist_unique)
 
 lemma closest_point_lipschitz:
   assumes "convex S"
     and "closed S" "S \<noteq> {}"
   shows "dist (closest_point S x) (closest_point S y) \<le> dist x y"
 proof -
   have "inner (x - closest_point S x) (closest_point S y - closest_point S x) \<le> 0"
     and "inner (y - closest_point S y) (closest_point S x - closest_point S y) \<le> 0"
-    apply (rule_tac[!] any_closest_point_dot[OF assms(1-2)])
-    using closest_point_exists[OF assms(2-3)]
-    apply auto
-    done
+    by (simp_all add: assms closest_point_dot closest_point_in_set)
   then show ?thesis unfolding dist_norm and norm_le
     using inner_ge_zero[of "(x - closest_point S x) - (y - closest_point S y)"]
     by (simp add: inner_add inner_diff inner_commute)
 qed
 
 lemma continuous_at_closest_point:
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
   shows "continuous (at x) (closest_point S)"
   unfolding continuous_at_eps_delta
   using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
 
 lemma continuous_on_closest_point:
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
   shows "continuous_on t (closest_point S)"
   by (metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
 
 proposition closest_point_in_rel_interior:
   assumes "closed S" "S \<noteq> {}" and x: "x \<in> affine hull S"
     shows "closest_point S x \<in> rel_interior S \<longleftrightarrow> x \<in> rel_interior S"
 proof (cases "x \<in> S")
   case True
   then show ?thesis
     by (simp add: closest_point_self)
 next
   case False
   then have "False" if asm: "closest_point S x \<in> rel_interior S"
   proof -
     obtain e where "e > 0" and clox: "closest_point S x \<in> S"
                and e: "cball (closest_point S x) e \<inter> affine hull S \<subseteq> S"
       using asm mem_rel_interior_cball by blast
     then have clo_notx: "closest_point S x \<noteq> x"
       using \<open>x \<notin> S\<close> by auto
     define y where "y \<equiv> closest_point S x -
                         (min 1 (e / norm(closest_point S x - x))) *\<^sub>R (closest_point S x - x)"
     have "x - y = (1 - min 1 (e / norm (closest_point S x - x))) *\<^sub>R (x - closest_point S x)"
       by (simp add: y_def algebra_simps)
     then have "norm (x - y) = abs ((1 - min 1 (e / norm (closest_point S x - x)))) * norm(x - closest_point S x)"
       by simp
     also have "\<dots> < norm(x - closest_point S x)"
       using clo_notx \<open>e > 0\<close>
       by (auto simp: mult_less_cancel_right2 field_split_simps)
     finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
     have "y \<in> affine hull S"
       unfolding y_def
       by (meson affine_affine_hull clox hull_subset mem_affine_3_minus2 subsetD x)
     moreover have "dist (closest_point S x) y \<le> e"
       using \<open>e > 0\<close> by (auto simp: y_def min_mult_distrib_right)
     ultimately have "y \<in> S"
       using subsetD [OF e] by simp
     then have "dist x (closest_point S x) \<le> dist x y"
       by (simp add: closest_point_le \<open>closed S\<close>)
     with no_less show False
       by (simp add: dist_norm)
   qed
   moreover have "x \<notin> rel_interior S"
     using rel_interior_subset False by blast
   ultimately show ?thesis by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Various point-to-set separating/supporting hyperplane theorems\<close>
 
 lemma supporting_hyperplane_closed_point:
   fixes z :: "'a::{real_inner,heine_borel}"
   assumes "convex S"
     and "closed S"
     and "S \<noteq> {}"
     and "z \<notin> S"
   shows "\<exists>a b. \<exists>y\<in>S. inner a z < b \<and> inner a y = b \<and> (\<forall>x\<in>S. inner a x \<ge> b)"
 proof -
   obtain y where "y \<in> S" and y: "\<forall>x\<in>S. dist z y \<le> dist z x"
     by (metis distance_attains_inf[OF assms(2-3)])
   show ?thesis
   proof (intro exI bexI conjI ballI)
     show "(y - z) \<bullet> z < (y - z) \<bullet> y"
       by (metis \<open>y \<in> S\<close> assms(4) diff_gt_0_iff_gt inner_commute inner_diff_left inner_gt_zero_iff right_minus_eq)
     show "(y - z) \<bullet> y \<le> (y - z) \<bullet> x" if "x \<in> S" for x
     proof (rule ccontr)
       have *: "\<And>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
         using assms(1)[unfolded convex_alt] and y and \<open>x\<in>S\<close> and \<open>y\<in>S\<close> by auto
       assume "\<not> (y - z) \<bullet> y \<le> (y - z) \<bullet> x"
       then obtain v where "v > 0" "v \<le> 1" "dist (y + v *\<^sub>R (x - y)) z < dist y z"
         using closer_point_lemma[of z y x] by (auto simp: inner_diff)
       then show False
         using *[of v] by (auto simp: dist_commute algebra_simps)
     qed
   qed (use \<open>y \<in> S\<close> in auto)
 qed
 
 lemma separating_hyperplane_closed_point:
   fixes z :: "'a::{real_inner,heine_borel}"
   assumes "convex S"
     and "closed S"
     and "z \<notin> S"
   shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>S. inner a x > b)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: gt_ex)
 next
   case False
   obtain y where "y \<in> S" and y: "\<And>x. x \<in> S \<Longrightarrow> dist z y \<le> dist z x"
     by (metis distance_attains_inf[OF assms(2) False])
   show ?thesis
   proof (intro exI conjI ballI)
     show "(y - z) \<bullet> z < inner (y - z) z + (norm (y - z))\<^sup>2 / 2"
       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
   next
     fix x
     assume "x \<in> S"
     have "False" if *: "0 < inner (z - y) (x - y)"
     proof -
       obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
         using * closer_point_lemma by blast
       then show False using y[of "y + u *\<^sub>R (x - y)"] convexD_alt [OF \<open>convex S\<close>]
         using \<open>x\<in>S\<close> \<open>y\<in>S\<close> by (auto simp: dist_commute algebra_simps)
     qed
     moreover have "0 < (norm (y - z))\<^sup>2"
       using \<open>y\<in>S\<close> \<open>z\<notin>S\<close> by auto
     then have "0 < inner (y - z) (y - z)"
       unfolding power2_norm_eq_inner by simp
     ultimately show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (force simp: field_simps power2_norm_eq_inner inner_commute inner_diff)
   qed 
 qed
 
 lemma separating_hyperplane_closed_0:
   assumes "convex (S::('a::euclidean_space) set)"
     and "closed S"
     and "0 \<notin> S"
   shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>S. inner a x > b)"
 proof (cases "S = {}")
   case True
   have "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
     by (metis Basis_zero SOME_Basis)
   then show ?thesis
     using True zero_less_one by blast
 next
   case False
   then show ?thesis
     using False using separating_hyperplane_closed_point[OF assms]
     by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Now set-to-set for closed/compact sets\<close>
 
 lemma separating_hyperplane_closed_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "closed S"
     and "convex T"
     and "compact T"
     and "T \<noteq> {}"
     and "S \<inter> T = {}"
   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
 proof (cases "S = {}")
   case True
   obtain b where b: "b > 0" "\<forall>x\<in>T. norm x \<le> b"
     using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
   obtain z :: 'a where z: "norm z = b + 1"
     using vector_choose_size[of "b + 1"] and b(1) by auto
   then have "z \<notin> T" using b(2)[THEN bspec[where x=z]] by auto
   then obtain a b where ab: "inner a z < b" "\<forall>x\<in>T. b < inner a x"
     using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z]
     by auto
   then show ?thesis
     using True by auto
 next
   case False
   then obtain y where "y \<in> S" by auto
-  obtain a b where "0 < b" "\<forall>x \<in> (\<Union>x\<in> S. \<Union>y \<in> T. {x - y}). b < inner a x"
+  obtain a b where "0 < b" and \<section>: "\<And>x. x \<in> (\<Union>x\<in> S. \<Union>y \<in> T. {x - y}) \<Longrightarrow> b < inner a x"
     using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
-    using closed_compact_differences[OF assms(2,4)]
-    using assms(6) by auto 
-  then have ab: "\<forall>x\<in>S. \<forall>y\<in>T. b + inner a y < inner a x"
-    apply -
-    apply rule
-    apply rule
-    apply (erule_tac x="x - y" in ballE)
-    apply (auto simp: inner_diff)
-    done
+    using closed_compact_differences assms by fastforce 
+  have ab: "b + inner a y < inner a x" if "x\<in>S" "y\<in>T" for x y
+    using \<section> [of "x-y"] that by (auto simp add: inner_diff_right less_diff_eq)
   define k where "k = (SUP x\<in>T. a \<bullet> x)"
-  show ?thesis
-    apply (rule_tac x="-a" in exI)
-    apply (rule_tac x="-(k + b / 2)" in exI)
-    apply (intro conjI ballI)
-    unfolding inner_minus_left and neg_less_iff_less
+  have "k + b / 2 < a \<bullet> x" if "x \<in> S" for x
   proof -
-    fix x assume "x \<in> T"
-    then have "inner a x - b / 2 < k"
+    have "k \<le> inner a x - b"
+      unfolding k_def
+      using \<open>T \<noteq> {}\<close> ab that by (fastforce intro: cSUP_least)
+    then show ?thesis
+      using \<open>0 < b\<close> by auto
+  qed
+  moreover
+  have "- (k + b / 2) < - a \<bullet> x" if "x \<in> T" for x
+  proof -
+    have "inner a x - b / 2 < k"
       unfolding k_def
     proof (subst less_cSUP_iff)
       show "T \<noteq> {}" by fact
       show "bdd_above ((\<bullet>) a ` T)"
         using ab[rule_format, of y] \<open>y \<in> S\<close>
         by (intro bdd_aboveI2[where M="inner a y - b"]) (auto simp: field_simps intro: less_imp_le)
-    qed (auto intro!: bexI[of _ x] \<open>0<b\<close>)
-    then show "inner a x < k + b / 2"
+      show "\<exists>y\<in>T. a \<bullet> x - b / 2 < a \<bullet> y"
+        using \<open>0 < b\<close> that by force
+    qed 
+    then show ?thesis
       by auto
-  next
-    fix x
-    assume "x \<in> S"
-    then have "k \<le> inner a x - b"
-      unfolding k_def
-      apply (rule_tac cSUP_least)
-      using assms(5)
-      using ab[THEN bspec[where x=x]]
-      apply auto
-      done
-    then show "k + b / 2 < inner a x"
-      using \<open>0 < b\<close> by auto
   qed
+  ultimately show ?thesis
+    by (metis inner_minus_left neg_less_iff_less)
 qed
 
 lemma separating_hyperplane_compact_closed:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "compact S"
     and "S \<noteq> {}"
     and "convex T"
     and "closed T"
     and "S \<inter> T = {}"
   shows "\<exists>a b. (\<forall>x\<in>S. inner a x < b) \<and> (\<forall>x\<in>T. inner a x > b)"
 proof -
   obtain a b where "(\<forall>x\<in>T. inner a x < b) \<and> (\<forall>x\<in>S. b < inner a x)"
-    using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6)
-    by auto
+    by (metis disjoint_iff_not_equal separating_hyperplane_closed_compact assms)
   then show ?thesis
-    apply (rule_tac x="-a" in exI)
-    apply (rule_tac x="-b" in exI, auto)
-    done
+    by (metis inner_minus_left neg_less_iff_less)
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>General case without assuming closure and getting non-strict separation\<close>
 
 lemma separating_hyperplane_set_0:
   assumes "convex S" "(0::'a::euclidean_space) \<notin> S"
   shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>S. 0 \<le> inner a x)"
 proof -
   let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
   have *: "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" if as: "f \<subseteq> ?k ` S" "finite f" for f
   proof -
     obtain c where c: "f = ?k ` c" "c \<subseteq> S" "finite c"
       using finite_subset_image[OF as(2,1)] by auto
     then obtain a b where ab: "a \<noteq> 0" "0 < b" "\<forall>x\<in>convex hull c. b < inner a x"
       using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
       using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
       using subset_hull[of convex, OF assms(1), symmetric, of c]
       by force
-    then have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
-      apply (rule_tac x = "inverse(norm a) *\<^sub>R a" in exI)
-      using hull_subset[of c convex]
-      unfolding subset_eq and inner_scaleR
-      by (auto simp: inner_commute del: ballE elim!: ballE)
+    have "norm (a /\<^sub>R norm a) = 1"
+      by (simp add: ab(1))
+    moreover have "(\<forall>y\<in>c. 0 \<le> y \<bullet> (a /\<^sub>R norm a))"
+      using hull_subset[of c convex] ab by (force simp: inner_commute)
+    ultimately have "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)"
+      by blast
     then show "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}"
       unfolding c(1) frontier_cball sphere_def dist_norm by auto
   qed
   have "frontier (cball 0 1) \<inter> (\<Inter>(?k ` S)) \<noteq> {}"
-    apply (rule compact_imp_fip)
-    apply (rule compact_frontier[OF compact_cball])
-    using * closed_halfspace_ge
-    by auto
+    by (rule compact_imp_fip) (use * closed_halfspace_ge in auto)
   then obtain x where "norm x = 1" "\<forall>y\<in>S. x\<in>?k y"
     unfolding frontier_cball dist_norm sphere_def by auto
   then show ?thesis
     by (metis inner_commute mem_Collect_eq norm_eq_zero zero_neq_one)
 qed
 
 lemma separating_hyperplane_sets:
   fixes S T :: "'a::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<noteq> {}"
     and "T \<noteq> {}"
     and "S \<inter> T = {}"
   shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>S. inner a x \<le> b) \<and> (\<forall>x\<in>T. inner a x \<ge> b)"
 proof -
   from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
   obtain a where "a \<noteq> 0" "\<forall>x\<in>{x - y |x y. x \<in> T \<and> y \<in> S}. 0 \<le> inner a x"
     using assms(3-5) by force
   then have *: "\<And>x y. x \<in> T \<Longrightarrow> y \<in> S \<Longrightarrow> inner a y \<le> inner a x"
     by (force simp: inner_diff)
   then have bdd: "bdd_above (((\<bullet>) a)`S)"
     using \<open>T \<noteq> {}\<close> by (auto intro: bdd_aboveI2[OF *])
   show ?thesis
     using \<open>a\<noteq>0\<close>
     by (intro exI[of _ a] exI[of _ "SUP x\<in>S. a \<bullet> x"])
        (auto intro!: cSUP_upper bdd cSUP_least \<open>a \<noteq> 0\<close> \<open>S \<noteq> {}\<close> *)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>More convexity generalities\<close>
 
 lemma convex_closure [intro,simp]:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "convex (closure S)"
   apply (rule convexI)
-  apply (unfold closure_sequential, elim exE)
-  apply (rule_tac x="\<lambda>n. u *\<^sub>R xa n + v *\<^sub>R xb n" in exI)
-  apply (rule,rule)
-  apply (rule convexD [OF assms])
-  apply (auto del: tendsto_const intro!: tendsto_intros)
+  unfolding closure_sequential
+  apply (elim exE)
+  subgoal for x y u v f g
+    by (rule_tac x="\<lambda>n. u *\<^sub>R f n + v *\<^sub>R g n" in exI) (force intro: tendsto_intros dest: convexD [OF assms])
   done
 
 lemma convex_interior [intro,simp]:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "convex (interior S)"
   unfolding convex_alt Ball_def mem_interior
 proof clarify
   fix x y u
   assume u: "0 \<le> u" "u \<le> (1::real)"
   fix e d
   assume ed: "ball x e \<subseteq> S" "ball y d \<subseteq> S" "0<d" "0<e"
   show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> S"
   proof (intro exI conjI subsetI)
     fix z
-    assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
-    then have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> S"
-      apply (rule_tac assms[unfolded convex_alt, rule_format])
-      using ed(1,2) and u
-      unfolding subset_eq mem_ball Ball_def dist_norm
-      apply (auto simp: algebra_simps)
-      done
+    assume z: "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
+    have "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> S"
+    proof (rule_tac assms[unfolded convex_alt, rule_format])
+       show "z - u *\<^sub>R (y - x) \<in> S" "z + (1 - u) *\<^sub>R (y - x) \<in> S"
+         using ed z u by (auto simp add: algebra_simps dist_norm)
+    qed (use u in auto)
     then show "z \<in> S"
       using u by (auto simp: algebra_simps)
-  qed(insert u ed(3-4), auto)
+  qed(use u ed in auto)
 qed
 
 lemma convex_hull_eq_empty[simp]: "convex hull S = {} \<longleftrightarrow> S = {}"
   using hull_subset[of S convex] convex_hull_empty by auto
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convex set as intersection of halfspaces\<close>
 
 lemma convex_halfspace_intersection:
-  fixes s :: "('a::euclidean_space) set"
-  assumes "closed s" "convex s"
-  shows "s = \<Inter>{h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
-  apply (rule set_eqI, rule)
-  unfolding Inter_iff Ball_def mem_Collect_eq
-  apply (rule,rule,erule conjE)
+  fixes S :: "('a::euclidean_space) set"
+  assumes "closed S" "convex S"
+  shows "S = \<Inter>{h. S \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
 proof -
-  fix x
-  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
-  then have "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}"
-    by blast
-  then show "x \<in> s"
-    apply (rule_tac ccontr)
-    apply (drule separating_hyperplane_closed_point[OF assms(2,1)])
-    apply (erule exE)+
-    apply (erule_tac x="-a" in allE)
-    apply (erule_tac x="-b" in allE, auto)
-    done
-qed auto
+  { fix z
+    assume "\<forall>T. S \<subseteq> T \<and> (\<exists>a b. T = {x. inner a x \<le> b}) \<longrightarrow> z \<in> T"  "z \<notin> S"
+    then have \<section>: "\<And>a b. S \<subseteq> {x. inner a x \<le> b} \<Longrightarrow> z \<in> {x. inner a x \<le> b}"
+      by blast
+    obtain a b where "inner a z < b" "(\<forall>x\<in>S. inner a x > b)"
+      using \<open>z \<notin> S\<close> assms separating_hyperplane_closed_point by blast
+    then have False
+      using \<section> [of "-a" "-b"] by fastforce
+  }
+  then show ?thesis
+    by force
+qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of general and special intervals\<close>
 
 lemma is_interval_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "is_interval S"
   shows "convex S"
 proof (rule convexI)
   fix x y and u v :: real
-  assume as: "x \<in> S" "y \<in> S" "0 \<le> u" "0 \<le> v" "u + v = 1"
+  assume "x \<in> S" "y \<in> S" and uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
   then have *: "u = 1 - v" "1 - v \<ge> 0" and **: "v = 1 - u" "1 - u \<ge> 0"
     by auto
   {
     fix a b
     assume "\<not> b \<le> u * a + v * b"
     then have "u * a < (1 - v) * b"
-      unfolding not_le using as(4) by (auto simp: field_simps)
+      unfolding not_le using \<open>0 \<le> v\<close>by (auto simp: field_simps)
     then have "a < b"
-      unfolding * using as(4) *(2)
-      apply (rule_tac mult_left_less_imp_less[of "1 - v"])
-      apply (auto simp: field_simps)
-      done
+      using "*"(1) less_eq_real_def uv(1) by auto
     then have "a \<le> u * a + v * b"
-      unfolding * using as(4)
+      unfolding * using \<open>0 \<le> v\<close>
       by (auto simp: field_simps intro!:mult_right_mono)
   }
   moreover
   {
     fix a b
     assume "\<not> u * a + v * b \<le> a"
     then have "v * b > (1 - u) * a"
-      unfolding not_le using as(4) by (auto simp: field_simps)
+      unfolding not_le using \<open>0 \<le> v\<close> by (auto simp: field_simps)
     then have "a < b"
-      unfolding * using as(4)
-      apply (rule_tac mult_left_less_imp_less)
-      apply (auto simp: field_simps)
-      done
+      unfolding * using \<open>0 \<le> v\<close>
+      by (rule_tac mult_left_less_imp_less) (auto simp: field_simps)
     then have "u * a + v * b \<le> b"
       unfolding **
-      using **(2) as(3)
-      by (auto simp: field_simps intro!:mult_right_mono)
+      using **(2) \<open>0 \<le> u\<close> by (auto simp: algebra_simps mult_right_mono)
   }
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> S"
-    apply -
-    apply (rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
-    using as(3-) DIM_positive[where 'a='a]
-    apply (auto simp: inner_simps)
-    done
+    using DIM_positive[where 'a='a]
+    by (intro mem_is_intervalI [OF assms \<open>x \<in> S\<close> \<open>y \<in> S\<close>]) (auto simp: inner_simps)
 qed
 
 lemma is_interval_connected:
   fixes S :: "'a::euclidean_space set"
   shows "is_interval S \<Longrightarrow> connected S"
   using is_interval_convex convex_connected by auto
 
 lemma convex_box [simp]: "convex (cbox a b)" "convex (box a (b::'a::euclidean_space))"
-  apply (rule_tac[!] is_interval_convex)+
-  using is_interval_box is_interval_cbox
-  apply auto
-  done
+  by (auto simp add: is_interval_convex)
 
 text\<open>A non-singleton connected set is perfect (i.e. has no isolated points). \<close>
 lemma connected_imp_perfect:
   fixes a :: "'a::metric_space"
   assumes "connected S" "a \<in> S" and S: "\<And>x. S \<noteq> {x}"
   shows "a islimpt S"
 proof -
   have False if "a \<in> T" "open T" "\<And>y. \<lbrakk>y \<in> S; y \<in> T\<rbrakk> \<Longrightarrow> y = a" for T
   proof -
     obtain e where "e > 0" and e: "cball a e \<subseteq> T"
       using \<open>open T\<close> \<open>a \<in> T\<close> by (auto simp: open_contains_cball)
     have "openin (top_of_set S) {a}"
       unfolding openin_open using that \<open>a \<in> S\<close> by blast
     moreover have "closedin (top_of_set S) {a}"
       by (simp add: assms)
     ultimately show "False"
       using \<open>connected S\<close> connected_clopen S by blast
   qed
   then show ?thesis
     unfolding islimpt_def by blast
 qed
 
 lemma connected_imp_perfect_aff_dim:
      "\<lbrakk>connected S; aff_dim S \<noteq> 0; a \<in> S\<rbrakk> \<Longrightarrow> a islimpt S"
   using aff_dim_sing connected_imp_perfect by blast
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>On \<open>real\<close>, \<open>is_interval\<close>, \<open>convex\<close> and \<open>connected\<close> are all equivalent\<close>
 
 lemma mem_is_interval_1_I:
   fixes a b c::real
   assumes "is_interval S"
   assumes "a \<in> S" "c \<in> S"
   assumes "a \<le> b" "b \<le> c"
   shows "b \<in> S"
   using assms is_interval_1 by blast
 
 lemma is_interval_connected_1:
-  fixes s :: "real set"
-  shows "is_interval s \<longleftrightarrow> connected s"
-  apply rule
-  apply (rule is_interval_connected, assumption)
-  unfolding is_interval_1
-  apply rule
-  apply rule
-  apply rule
-  apply rule
-  apply (erule conjE)
-  apply (rule ccontr)       
-proof -
-  fix a b x
-  assume as: "connected s" "a \<in> s" "b \<in> s" "a \<le> x" "x \<le> b" "x \<notin> s"
-  then have *: "a < x" "x < b"
-    unfolding not_le [symmetric] by auto
-  let ?halfl = "{..<x} "
-  let ?halfr = "{x<..}"
-  {
-    fix y
-    assume "y \<in> s"
-    with \<open>x \<notin> s\<close> have "x \<noteq> y" by auto
-    then have "y \<in> ?halfr \<union> ?halfl" by auto
-  }
-  moreover have "a \<in> ?halfl" "b \<in> ?halfr" using * by auto
-  then have "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"
-    using as(2-3) by auto
-  ultimately show False
-    apply (rule_tac notE[OF as(1)[unfolded connected_def]])
-    apply (rule_tac x = ?halfl in exI)
-    apply (rule_tac x = ?halfr in exI, rule)
-    apply (rule open_lessThan, rule)
-    apply (rule open_greaterThan, auto)
-    done
-qed
+  fixes S :: "real set"
+  shows "is_interval S \<longleftrightarrow> connected S"
+  by (meson connected_iff_interval is_interval_1)
 
 lemma is_interval_convex_1:
-  fixes s :: "real set"
-  shows "is_interval s \<longleftrightarrow> convex s"
+  fixes S :: "real set"
+  shows "is_interval S \<longleftrightarrow> convex S"
   by (metis is_interval_convex convex_connected is_interval_connected_1)
 
 lemma connected_compact_interval_1:
      "connected S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = {a..b::real})"
   by (auto simp: is_interval_connected_1 [symmetric] is_interval_compact)
 
 lemma connected_convex_1:
-  fixes s :: "real set"
-  shows "connected s \<longleftrightarrow> convex s"
+  fixes S :: "real set"
+  shows "connected S \<longleftrightarrow> convex S"
   by (metis is_interval_convex convex_connected is_interval_connected_1)
 
 lemma connected_convex_1_gen:
-  fixes s :: "'a :: euclidean_space set"
+  fixes S :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
-  shows "connected s \<longleftrightarrow> convex s"
+  shows "connected S \<longleftrightarrow> convex S"
 proof -
   obtain f:: "'a \<Rightarrow> real" where linf: "linear f" and "inj f"
     using subspace_isomorphism[OF subspace_UNIV subspace_UNIV,
         where 'a='a and 'b=real]
     unfolding Euclidean_Space.dim_UNIV
     by (auto simp: assms)
-  then have "f -` (f ` s) = s"
+  then have "f -` (f ` S) = S"
     by (simp add: inj_vimage_image_eq)
   then show ?thesis
     by (metis connected_convex_1 convex_linear_vimage linf convex_connected connected_linear_image)
 qed
 
 lemma [simp]:
   fixes r s::real
   shows is_interval_io: "is_interval {..<r}"
     and is_interval_oi: "is_interval {r<..}"
     and is_interval_oo: "is_interval {r<..<s}"
     and is_interval_oc: "is_interval {r<..s}"
     and is_interval_co: "is_interval {r..<s}"
   by (simp_all add: is_interval_convex_1)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Another intermediate value theorem formulation\<close>
 
 lemma ivt_increasing_component_on_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "a \<le> b"
     and "continuous_on {a..b} f"
     and "(f a)\<bullet>k \<le> y" "y \<le> (f b)\<bullet>k"
   shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
 proof -
   have "f a \<in> f ` cbox a b" "f b \<in> f ` cbox a b"
-    apply (rule_tac[!] imageI)
-    using assms(1)
-    apply auto
-    done
+    using \<open>a \<le> b\<close> by auto
   then show ?thesis
     using connected_ivt_component[of "f ` cbox a b" "f a" "f b" k y]
     by (simp add: connected_continuous_image assms)
 qed
 
 lemma ivt_increasing_component_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a..b}. continuous (at x) f \<Longrightarrow>
     f a\<bullet>k \<le> y \<Longrightarrow> y \<le> f b\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   by (rule ivt_increasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
 
 lemma ivt_decreasing_component_on_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "a \<le> b"
     and "continuous_on {a..b} f"
     and "(f b)\<bullet>k \<le> y"
     and "y \<le> (f a)\<bullet>k"
   shows "\<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
-  apply (subst neg_equal_iff_equal[symmetric])
-  using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"]
-  using assms using continuous_on_minus
-  apply auto
-  done
+  using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] neg_equal_iff_equal
+  using assms continuous_on_minus by force
 
 lemma ivt_decreasing_component_1:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a..b}. continuous (at x) f \<Longrightarrow>
     f b\<bullet>k \<le> y \<Longrightarrow> y \<le> f a\<bullet>k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)\<bullet>k = y"
   by (rule ivt_decreasing_component_on_1) (auto simp: continuous_at_imp_continuous_on)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within an interval\<close>
 
 lemma convex_hull_eq_real_cbox:
   fixes x y :: real assumes "x \<le> y"
   shows "convex hull {x, y} = cbox x y"
 proof (rule hull_unique)
   show "{x, y} \<subseteq> cbox x y" using \<open>x \<le> y\<close> by auto
   show "convex (cbox x y)"
     by (rule convex_box)
 next
   fix S assume "{x, y} \<subseteq> S" and "convex S"
   then show "cbox x y \<subseteq> S"
     unfolding is_interval_convex_1 [symmetric] is_interval_def Basis_real_def
     by - (clarify, simp (no_asm_use), fast)
 qed
 
 lemma unit_interval_convex_hull:
   "cbox (0::'a::euclidean_space) One = convex hull {x. \<forall>i\<in>Basis. (x\<bullet>i = 0) \<or> (x\<bullet>i = 1)}"
   (is "?int = convex hull ?points")
 proof -
   have One[simp]: "\<And>i. i \<in> Basis \<Longrightarrow> One \<bullet> i = 1"
     by (simp add: inner_sum_left sum.If_cases inner_Basis)
   have "?int = {x. \<forall>i\<in>Basis. x \<bullet> i \<in> cbox 0 1}"
     by (auto simp: cbox_def)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` cbox 0 1)"
     by (simp only: box_eq_set_sum_Basis)
   also have "\<dots> = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (convex hull {0, 1}))"
     by (simp only: convex_hull_eq_real_cbox zero_le_one)
   also have "\<dots> = (\<Sum>i\<in>Basis. convex hull ((\<lambda>x. x *\<^sub>R i) ` {0, 1}))"
     by (simp add: convex_hull_linear_image)
   also have "\<dots> = convex hull (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` {0, 1})"
     by (simp only: convex_hull_set_sum)
   also have "\<dots> = convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}}"
     by (simp only: box_eq_set_sum_Basis)
   also have "convex hull {x. \<forall>i\<in>Basis. x\<bullet>i \<in> {0, 1}} = convex hull ?points"
     by simp
   finally show ?thesis .
 qed
 
 text \<open>And this is a finite set of vertices.\<close>
 
 lemma unit_cube_convex_hull:
   obtains S :: "'a::euclidean_space set"
   where "finite S" and "cbox 0 (\<Sum>Basis) = convex hull S"
 proof
   show "finite {x::'a. \<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1}"
   proof (rule finite_subset, clarify)
     show "finite ((\<lambda>S. \<Sum>i\<in>Basis. (if i \<in> S then 1 else 0) *\<^sub>R i) ` Pow Basis)"
       using finite_Basis by blast
     fix x :: 'a
-    assume as: "\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
+    assume x: "\<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1"
     show "x \<in> (\<lambda>S. \<Sum>i\<in>Basis. (if i\<in>S then 1 else 0) *\<^sub>R i) ` Pow Basis"
-      apply (rule image_eqI[where x="{i. i\<in>Basis \<and> x\<bullet>i = 1}"])
-      using as
-       apply (subst euclidean_eq_iff, auto)
-      done
+      apply (rule image_eqI[where x="{i. i \<in> Basis \<and> x\<bullet>i = 1}"])
+      using x
+      by (subst euclidean_eq_iff, auto)
   qed
   show "cbox 0 One = convex hull {x. \<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1}"
     using unit_interval_convex_hull by blast
 qed 
 
 text \<open>Hence any cube (could do any nonempty interval).\<close>
 
 lemma cube_convex_hull:
   assumes "d > 0"
   obtains S :: "'a::euclidean_space set" where
     "finite S" and "cbox (x - (\<Sum>i\<in>Basis. d*\<^sub>Ri)) (x + (\<Sum>i\<in>Basis. d*\<^sub>Ri)) = convex hull S"
 proof -
   let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
   have *: "cbox (x - ?d) (x + ?d) = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 (\<Sum>Basis)"
   proof (intro set_eqI iffI)
     fix y
     assume "y \<in> cbox (x - ?d) (x + ?d)"
     then have "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> cbox 0 (\<Sum>Basis)"
       using assms by (simp add: mem_box inner_simps) (simp add: field_simps)
     with \<open>0 < d\<close> show "y \<in> (\<lambda>y. x - sum ((*\<^sub>R) d) Basis + (2 * d) *\<^sub>R y) ` cbox 0 One"
       by (auto intro: image_eqI[where x= "inverse (2 * d) *\<^sub>R (y - (x - ?d))"])
   next
     fix y
     assume "y \<in> (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` cbox 0 One"
     then obtain z where z: "z \<in> cbox 0 One" "y = x - ?d + (2*d) *\<^sub>R z"
       by auto
     then show "y \<in> cbox (x - ?d) (x + ?d)"
       using z assms by (auto simp: mem_box inner_simps)
   qed
   obtain S where "finite S" "cbox 0 (\<Sum>Basis::'a) = convex hull S"
     using unit_cube_convex_hull by auto
   then show ?thesis
     by (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` S"]) (auto simp: convex_hull_affinity *)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Representation of any interval as a finite convex hull\<close>
 
 lemma image_stretch_interval:
   "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k)) *\<^sub>R k) ` cbox a (b::'a::euclidean_space) =
   (if (cbox a b) = {} then {} else
     cbox (\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k::'a)
      (\<Sum>k\<in>Basis. (max (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k))"
 proof cases
   assume *: "cbox a b \<noteq> {}"
   show ?thesis
     unfolding box_ne_empty if_not_P[OF *]
     apply (simp add: cbox_def image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
     apply (subst choice_Basis_iff[symmetric])
   proof (intro allI ball_cong refl)
     fix x i :: 'a assume "i \<in> Basis"
     with * have a_le_b: "a \<bullet> i \<le> b \<bullet> i"
       unfolding box_ne_empty by auto
     show "(\<exists>xa. x \<bullet> i = m i * xa \<and> a \<bullet> i \<le> xa \<and> xa \<le> b \<bullet> i) \<longleftrightarrow>
         min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) \<le> x \<bullet> i \<and> x \<bullet> i \<le> max (m i * (a \<bullet> i)) (m i * (b \<bullet> i))"
     proof (cases "m i = 0")
       case True
       with a_le_b show ?thesis by auto
     next
       case False
       then have *: "\<And>a b. a = m i * b \<longleftrightarrow> b = a / m i"
         by (auto simp: field_simps)
       from False have
           "min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (a \<bullet> i) else m i * (b \<bullet> i))"
           "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (b \<bullet> i) else m i * (a \<bullet> i))"
         using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
       with False show ?thesis using a_le_b *
         by (simp add: le_divide_eq divide_le_eq) (simp add: ac_simps)
     qed
   qed
 qed simp
 
 lemma interval_image_stretch_interval:
   "\<exists>u v. (\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k) ` cbox a (b::'a::euclidean_space) = cbox u (v::'a::euclidean_space)"
   unfolding image_stretch_interval by auto
 
 lemma cbox_translation: "cbox (c + a) (c + b) = image (\<lambda>x. c + x) (cbox a b)"
   using image_affinity_cbox [of 1 c a b]
   using box_ne_empty [of "a+c" "b+c"]  box_ne_empty [of a b]
   by (auto simp: inner_left_distrib add.commute)
 
 lemma cbox_image_unit_interval:
   fixes a :: "'a::euclidean_space"
   assumes "cbox a b \<noteq> {}"
     shows "cbox a b =
            (+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` cbox 0 One"
 using assms
 apply (simp add: box_ne_empty image_stretch_interval cbox_translation [symmetric])
 apply (simp add: min_def max_def algebra_simps sum_subtractf euclidean_representation)
 done
 
 lemma closed_interval_as_convex_hull:
   fixes a :: "'a::euclidean_space"
   obtains S where "finite S" "cbox a b = convex hull S"
 proof (cases "cbox a b = {}")
   case True with convex_hull_empty that show ?thesis
     by blast
 next
   case False
   obtain S::"'a set" where "finite S" and eq: "cbox 0 One = convex hull S"
     by (blast intro: unit_cube_convex_hull)
-  have lin: "linear (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k)"
-    by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
-  have "finite ((+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` S)"
-    by (rule finite_imageI \<open>finite S\<close>)+
-  then show ?thesis
-    apply (rule that)
-    apply (simp add: convex_hull_translation convex_hull_linear_image [OF lin, symmetric])
-    apply (simp add: eq [symmetric] cbox_image_unit_interval [OF False])
-    done
+  let ?S = "((+) a ` (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k) ` S)"
+  show thesis
+  proof
+    show "finite ?S"
+      by (simp add: \<open>finite S\<close>)
+    have lin: "linear (\<lambda>x. \<Sum>k\<in>Basis. ((b \<bullet> k - a \<bullet> k) * (x \<bullet> k)) *\<^sub>R k)"
+      by (rule linear_compose_sum) (auto simp: algebra_simps linearI)
+    show "cbox a b = convex hull ?S"
+      using convex_hull_linear_image [OF lin] 
+      by (simp add: convex_hull_translation eq cbox_image_unit_interval [OF False])
+  qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounded convex function on open set is continuous\<close>
 
 lemma convex_on_bounded_continuous:
   fixes S :: "('a::real_normed_vector) set"
   assumes "open S"
     and "convex_on S f"
     and "\<forall>x\<in>S. \<bar>f x\<bar> \<le> b"
   shows "continuous_on S f"
-  apply (rule continuous_at_imp_continuous_on)
-  unfolding continuous_at_real_range
-proof (rule,rule,rule)
-  fix x and e :: real
-  assume "x \<in> S" "e > 0"
-  define B where "B = \<bar>b\<bar> + 1"
-  then have B:  "0 < B""\<And>x. x\<in>S \<Longrightarrow> \<bar>f x\<bar> \<le> B"
-    using assms(3) by auto 
-  obtain k where "k > 0" and k: "cball x k \<subseteq> S"
-    using \<open>x \<in> S\<close> assms(1) open_contains_cball_eq by blast
-  show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
-  proof (intro exI conjI allI impI)
-    fix y
-    assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
-    show "\<bar>f y - f x\<bar> < e"
-    proof (cases "y = x")
-      case False
-      define t where "t = k / norm (y - x)"
-      have "2 < t" "0<t"
-        unfolding t_def using as False and \<open>k>0\<close>
-        by (auto simp:field_simps)
-      have "y \<in> S"
-        apply (rule k[THEN subsetD])
-        unfolding mem_cball dist_norm
-        apply (rule order_trans[of _ "2 * norm (x - y)"])
-        using as
-        by (auto simp: field_simps norm_minus_commute)
-      {
-        define w where "w = x + t *\<^sub>R (y - x)"
-        have "w \<in> S"
-          using \<open>k>0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
-        have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x"
-          by (auto simp: algebra_simps)
-        also have "\<dots> = 0"
-          using \<open>t > 0\<close> by (auto simp:field_simps)
-        finally have w: "(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y"
-          unfolding w_def using False and \<open>t > 0\<close>
-          by (auto simp: algebra_simps)
-        have 2: "2 * B < e * t"
-          unfolding t_def using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
+proof -
+  have "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e" if "x \<in> S" "e > 0" for x and e :: real
+  proof -
+    define B where "B = \<bar>b\<bar> + 1"
+    then have B:  "0 < B""\<And>x. x\<in>S \<Longrightarrow> \<bar>f x\<bar> \<le> B"
+      using assms(3) by auto 
+    obtain k where "k > 0" and k: "cball x k \<subseteq> S"
+      using \<open>x \<in> S\<close> assms(1) open_contains_cball_eq by blast
+    show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
+    proof (intro exI conjI allI impI)
+      fix y
+      assume as: "norm (y - x) < min (k / 2) (e / (2 * B) * k)"
+      show "\<bar>f y - f x\<bar> < e"
+      proof (cases "y = x")
+        case False
+        define t where "t = k / norm (y - x)"
+        have "2 < t" "0<t"
+          unfolding t_def using as False and \<open>k>0\<close>
           by (auto simp:field_simps)
-        have "f y - f x \<le> (f w - f x) / t"
-          using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
-          using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>x \<in> S\<close> \<open>w \<in> S\<close>
-          by (auto simp:field_simps)
-        also have "... < e"
-          using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>x\<in>S\<close>] 2 \<open>t > 0\<close> by (auto simp: field_simps)
-        finally have th1: "f y - f x < e" .
-      }
-      moreover
-      {
-        define w where "w = x - t *\<^sub>R (y - x)"
-        have "w \<in> S"
-          using \<open>k > 0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
-        have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x"
-          by (auto simp: algebra_simps)
-        also have "\<dots> = x"
-          using \<open>t > 0\<close> by (auto simp:field_simps)
-        finally have w: "(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x"
-          unfolding w_def using False and \<open>t > 0\<close>
-          by (auto simp: algebra_simps)
-        have "2 * B < e * t"
-          unfolding t_def
-          using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
-          by (auto simp:field_simps)
-        then have *: "(f w - f y) / t < e"
-          using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>y\<in>S\<close>]
-          using \<open>t > 0\<close>
-          by (auto simp:field_simps)
-        have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
-          using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
-          using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>y \<in> S\<close> \<open>w \<in> S\<close>
-          by (auto simp:field_simps)
-        also have "\<dots> = (f w + t * f y) / (1 + t)"
-          using \<open>t > 0\<close> by (simp add: add_divide_distrib) 
-        also have "\<dots> < e + f y"
-          using \<open>t > 0\<close> * \<open>e > 0\<close> by (auto simp: field_simps)
-        finally have "f x - f y < e" by auto
-      }
-      ultimately show ?thesis by auto
-    qed (insert \<open>0<e\<close>, auto)
-  qed (insert \<open>0<e\<close> \<open>0<k\<close> \<open>0<B\<close>, auto simp: field_simps)
+        have "y \<in> S"
+          apply (rule k[THEN subsetD])
+          unfolding mem_cball dist_norm
+          apply (rule order_trans[of _ "2 * norm (x - y)"])
+          using as
+          by (auto simp: field_simps norm_minus_commute)
+        {
+          define w where "w = x + t *\<^sub>R (y - x)"
+          have "w \<in> S"
+            using \<open>k>0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
+          have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x"
+            by (auto simp: algebra_simps)
+          also have "\<dots> = 0"
+            using \<open>t > 0\<close> by (auto simp:field_simps)
+          finally have w: "(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y"
+            unfolding w_def using False and \<open>t > 0\<close>
+            by (auto simp: algebra_simps)
+          have 2: "2 * B < e * t"
+            unfolding t_def using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
+            by (auto simp:field_simps)
+          have "f y - f x \<le> (f w - f x) / t"
+            using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
+            using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>x \<in> S\<close> \<open>w \<in> S\<close>
+            by (auto simp:field_simps)
+          also have "... < e"
+            using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>x\<in>S\<close>] 2 \<open>t > 0\<close> by (auto simp: field_simps)
+          finally have th1: "f y - f x < e" .
+        }
+        moreover
+        {
+          define w where "w = x - t *\<^sub>R (y - x)"
+          have "w \<in> S"
+            using \<open>k > 0\<close> by (auto simp: dist_norm t_def w_def k[THEN subsetD])
+          have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x"
+            by (auto simp: algebra_simps)
+          also have "\<dots> = x"
+            using \<open>t > 0\<close> by (auto simp:field_simps)
+          finally have w: "(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x"
+            unfolding w_def using False and \<open>t > 0\<close>
+            by (auto simp: algebra_simps)
+          have "2 * B < e * t"
+            unfolding t_def
+            using \<open>0 < e\<close> \<open>0 < k\<close> \<open>B > 0\<close> and as and False
+            by (auto simp:field_simps)
+          then have *: "(f w - f y) / t < e"
+            using B(2)[OF \<open>w\<in>S\<close>] and B(2)[OF \<open>y\<in>S\<close>]
+            using \<open>t > 0\<close>
+            by (auto simp:field_simps)
+          have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y"
+            using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
+            using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>y \<in> S\<close> \<open>w \<in> S\<close>
+            by (auto simp:field_simps)
+          also have "\<dots> = (f w + t * f y) / (1 + t)"
+            using \<open>t > 0\<close> by (simp add: add_divide_distrib) 
+          also have "\<dots> < e + f y"
+            using \<open>t > 0\<close> * \<open>e > 0\<close> by (auto simp: field_simps)
+          finally have "f x - f y < e" by auto
+        }
+        ultimately show ?thesis by auto
+      qed (use \<open>0<e\<close> in auto)
+    qed (use \<open>0<e\<close> \<open>0<k\<close> \<open>0<B\<close> in \<open>auto simp: field_simps\<close>)
+  qed
+  then show ?thesis
+    by (metis continuous_on_iff dist_norm real_norm_def)
 qed
 
-
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Upper bound on a ball implies upper and lower bounds\<close>
 
 lemma convex_bounds_lemma:
   fixes x :: "'a::real_normed_vector"
   assumes "convex_on (cball x e) f"
-    and "\<forall>y \<in> cball x e. f y \<le> b"
-  shows "\<forall>y \<in> cball x e. \<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
-  apply rule
+    and "\<forall>y \<in> cball x e. f y \<le> b" and y: "y \<in> cball x e"
+  shows "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
 proof (cases "0 \<le> e")
   case True
-  fix y
-  assume y: "y \<in> cball x e"
   define z where "z = 2 *\<^sub>R x - y"
   have *: "x - (2 *\<^sub>R x - y) = y - x"
     by (simp add: scaleR_2)
   have z: "z \<in> cball x e"
     using y unfolding z_def mem_cball dist_norm * by (auto simp: norm_minus_commute)
   have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x"
     unfolding z_def by (auto simp: algebra_simps)
   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
     using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
     using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z]
     by (auto simp:field_simps)
 next
   case False
-  fix y
-  assume "y \<in> cball x e"
-  then have "dist x y < 0"
-    using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
+  have "dist x y < 0"
+    using False y unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
   then show "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>"
     using zero_le_dist[of x y] by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence a convex function on an open set is continuous\<close>
 
 lemma real_of_nat_ge_one_iff: "1 \<le> real (n::nat) \<longleftrightarrow> 1 \<le> n"
   by auto
 
 lemma convex_on_continuous:
   assumes "open (s::('a::euclidean_space) set)" "convex_on s f"
   shows "continuous_on s f"
   unfolding continuous_on_eq_continuous_at[OF assms(1)]
 proof
   note dimge1 = DIM_positive[where 'a='a]
   fix x
   assume "x \<in> s"
   then obtain e where e: "cball x e \<subseteq> s" "e > 0"
     using assms(1) unfolding open_contains_cball by auto
   define d where "d = e / real DIM('a)"
   have "0 < d"
     unfolding d_def using \<open>e > 0\<close> dimge1 by auto
   let ?d = "(\<Sum>i\<in>Basis. d *\<^sub>R i)::'a"
   obtain c
     where c: "finite c"
     and c1: "convex hull c \<subseteq> cball x e"
     and c2: "cball x d \<subseteq> convex hull c"
   proof
     define c where "c = (\<Sum>i\<in>Basis. (\<lambda>a. a *\<^sub>R i) ` {x\<bullet>i - d, x\<bullet>i + d})"
     show "finite c"
       unfolding c_def by (simp add: finite_set_sum)
-    have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
-      unfolding box_eq_set_sum_Basis
-      unfolding c_def convex_hull_set_sum
+    have "\<And>i. i \<in> Basis \<Longrightarrow> convex hull {x \<bullet> i - d, x \<bullet> i + d} = cbox (x \<bullet> i - d) (x \<bullet> i + d)"
+      using \<open>0 < d\<close> convex_hull_eq_real_cbox by auto
+    then have 1: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cbox (x \<bullet> i - d) (x \<bullet> i + d)}"
+      unfolding box_eq_set_sum_Basis c_def convex_hull_set_sum 
       apply (subst convex_hull_linear_image [symmetric])
-      apply (simp add: linear_iff scaleR_add_left)
-      apply (rule sum.cong [OF refl])
-      apply (rule image_cong [OF _ refl])
-      apply (rule convex_hull_eq_real_cbox)
-      apply (cut_tac \<open>0 < d\<close>, simp)
-      done
+      by (force simp add: linear_iff scaleR_add_left)+
     then have 2: "convex hull c = {a. \<forall>i\<in>Basis. a \<bullet> i \<in> cball (x \<bullet> i) d}"
       by (simp add: dist_norm abs_le_iff algebra_simps)
     show "cball x d \<subseteq> convex hull c"
       unfolding 2
       by (clarsimp simp: dist_norm) (metis inner_commute inner_diff_right norm_bound_Basis_le)
     have e': "e = (\<Sum>(i::'a)\<in>Basis. d)"
       by (simp add: d_def)
     show "convex hull c \<subseteq> cball x e"
       unfolding 2
-      apply clarsimp
-      apply (subst euclidean_dist_l2)
-      apply (rule order_trans [OF L2_set_le_sum])
-      apply (rule zero_le_dist)
-      unfolding e'
-      apply (rule sum_mono, simp)
-      done
+    proof clarsimp
+      show "dist x y \<le> e" if "\<forall>i\<in>Basis. dist (x \<bullet> i) (y \<bullet> i) \<le> d" for y
+      proof -
+        have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> dist (x \<bullet> i) (y \<bullet> i)"
+          by simp
+        have "(\<Sum>i\<in>Basis. dist (x \<bullet> i) (y \<bullet> i)) \<le> e"
+          using e' sum_mono that by fastforce
+        then show ?thesis
+          by (metis (mono_tags) euclidean_dist_l2 order_trans [OF L2_set_le_sum] zero_le_dist)
+      qed
+    qed
   qed
   define k where "k = Max (f ` c)"
   have "convex_on (convex hull c) f"
-    apply(rule convex_on_subset[OF assms(2)])
-    apply(rule subset_trans[OF c1 e(1)])
-    done
+    using assms(2) c1 convex_on_subset e(1) by blast
   then have k: "\<forall>y\<in>convex hull c. f y \<le> k"
-    apply (rule_tac convex_on_convex_hull_bound, assumption)
-    by (simp add: k_def c)
+    using c convex_on_convex_hull_bound k_def by fastforce
   have "e \<le> e * real DIM('a)"
     using e(2) real_of_nat_ge_one_iff by auto
   then have "d \<le> e"
     by (simp add: d_def field_split_simps)
   then have dsube: "cball x d \<subseteq> cball x e"
     by (rule subset_cball)
   have conv: "convex_on (cball x d) f"
     using \<open>convex_on (convex hull c) f\<close> c2 convex_on_subset by blast
-  then have "\<forall>y\<in>cball x d. \<bar>f y\<bar> \<le> k + 2 * \<bar>f x\<bar>"
+  then have "\<And>y. y\<in>cball x d \<Longrightarrow> \<bar>f y\<bar> \<le> k + 2 * \<bar>f x\<bar>"
     by (rule convex_bounds_lemma) (use c2 k in blast)
   then have "continuous_on (ball x d) f"
-    apply (rule_tac convex_on_bounded_continuous)
-    apply (rule open_ball, rule convex_on_subset[OF conv])
-    apply (rule ball_subset_cball, force)
-    done
+    by (meson Elementary_Metric_Spaces.open_ball ball_subset_cball conv convex_on_bounded_continuous 
+              convex_on_subset mem_ball_imp_mem_cball)
   then show "continuous (at x) f"
     unfolding continuous_on_eq_continuous_at[OF open_ball]
     using \<open>d > 0\<close> by auto
 qed
 
 end