# HG changeset patch
# User paulson
# Date 1525261688 -3600
# Node ID d2daeef3ce4722d315a0574f7b0a5844d1e53d4c
# Parent  539048827fe8ab4752f97a25f57682ad5018e054# Parent  b249fab48c76ee1d3ed0d54ce3961b6bcbd4fae4
merged

diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Cartesian_Euclidean_Space.thy
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy	Wed May 02 12:48:08 2018 +0100
@@ -1919,15 +1919,13 @@
 
 definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
 
-lemma vector_1: "(vector[x]) $1 = x"
+lemma vector_1 [simp]: "(vector[x]) $1 = x"
   unfolding vector_def by simp
 
-lemma vector_2:
- "(vector[x,y]) $1 = x"
- "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
+lemma vector_2 [simp]: "(vector[x,y]) $1 = x" "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
   unfolding vector_def by simp_all
 
-lemma vector_3:
+lemma vector_3 [simp]:
  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
@@ -1958,7 +1956,7 @@
   done
 
 lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
-  apply (rule bounded_linearI[where K=1])
+  apply (rule bounded_linear_intro[where K=1])
   using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
 
 lemma interval_split_cart:
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Complex_Analysis_Basics.thy
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy	Wed May 02 12:48:08 2018 +0100
@@ -837,15 +837,13 @@
     qed
   } note ** = this
   show ?thesis
-  unfolding has_field_derivative_def
+    unfolding has_field_derivative_def
   proof (rule has_derivative_sequence [OF cvs _ _ x])
-    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (( * ) (f' n x))) (at x within s)"
-      by (metis has_field_derivative_def df)
-  next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
+  show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
     by (rule tf)
-  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
+  next show "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
     by (blast intro: **)
-  qed
+  qed (metis has_field_derivative_def df)
 qed
 
 lemma has_complex_derivative_series:
@@ -884,7 +882,7 @@
       by (metis df has_field_derivative_def mult_commute_abs)
   next show " ((\<lambda>n. f n x) sums l)"
     by (rule sf)
-  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
+  next show "\<And>e. e>0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
     by (blast intro: **)
   qed
 qed
@@ -896,7 +894,7 @@
   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
-  shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
+  shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
 proof -
   from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     unfolding uniformly_convergent_on_def by blast
@@ -905,7 +903,6 @@
     by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
   then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
     "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
-  from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
   from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
     by (simp add: has_field_derivative_def s)
   have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
@@ -915,15 +912,6 @@
     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
 qed
 
-lemma field_differentiable_series':
-  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
-  assumes "convex s" "open s"
-  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
-  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
-  shows   "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x0)"
-  using field_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
-
 subsection\<open>Bound theorem\<close>
 
 lemma field_differentiable_bound:
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Wed May 02 12:48:08 2018 +0100
@@ -2470,8 +2470,8 @@
 proof
   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
-  have "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)"
-  proof (intro hull_induct)
+  have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
+  proof (rule hull_induct [OF x], rule hull_induct [OF y])
     fix x y assume "x \<in> s" and "y \<in> t"
     then show "(x, y) \<in> convex hull (s \<times> t)"
       by (simp add: hull_inc)
@@ -2484,8 +2484,8 @@
       by (auto simp: image_def Bex_def)
     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
   next
-    show "convex {x. \<forall>y\<in>convex hull t. (x, y) \<in> convex hull (s \<times> t)}"
-    proof (unfold Collect_ball_eq, rule convex_INT [rule_format])
+    show "convex {x. (x, y) \<in> convex hull s \<times> t}"
+    proof -
       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
       have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
@@ -2496,7 +2496,7 @@
     qed
   qed
   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
-    unfolding subset_eq split_paired_Ball_Sigma .
+    unfolding subset_eq split_paired_Ball_Sigma by blast
 qed
 
 
@@ -2512,16 +2512,13 @@
   fixes S :: "'a::real_vector set"
   assumes "S \<noteq> {}"
   shows "convex hull (insert a S) =
-    {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
+         {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
   (is "_ = ?hull")
-  apply (rule, rule hull_minimal, rule)
+proof (intro equalityI hull_minimal subsetI)
+  fix x
+  assume "x \<in> insert a S"
+  then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
   unfolding insert_iff
-  prefer 3
-  apply rule
-proof -
-  fix x
-  assume x: "x = a \<or> x \<in> S"
-  then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
   proof
     assume "x = a"
     then show ?thesis
@@ -2619,7 +2616,7 @@
 
 lemma convex_hull_insert_alt:
    "convex hull (insert a S) =
-      (if S = {} then {a}
+     (if S = {} then {a}
       else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
   apply (auto simp: convex_hull_insert)
   using diff_eq_eq apply fastforce
@@ -4371,7 +4368,7 @@
     finally have "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
-      apply (rule d[unfolded subset_eq,rule_format])
+      apply (rule d[THEN subsetD])
       unfolding mem_ball
       using assms(3-5) **
       apply auto
@@ -4751,22 +4748,17 @@
 lemma affine_hull_linear_image:
   assumes "bounded_linear f"
   shows "f ` (affine hull s) = affine hull f ` s"
-  apply rule
-  unfolding subset_eq ball_simps
-  apply (rule_tac[!] hull_induct, rule hull_inc)
-  prefer 3
-  apply (erule imageE)
-  apply (rule_tac x=xa in image_eqI, assumption)
-  apply (rule hull_subset[unfolded subset_eq, rule_format], assumption)
 proof -
   interpret f: bounded_linear f by fact
-  show "affine {x. f x \<in> affine hull f ` s}"
+  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])
-  show "affine {x. x \<in> f ` (affine hull s)}"
+  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])
-qed auto
+  ultimately show ?thesis
+    by (auto simp: hull_inc elim!: hull_induct)
+qed 
 
 
 lemma rel_interior_injective_on_span_linear_image:
@@ -4893,112 +4885,77 @@
 subsection%unimportant \<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)"
-  unfolding open_contains_cball convex_hull_explicit
-  unfolding mem_Collect_eq ball_simps(8)
-proof (rule, rule)
-  fix a
-  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
-  then obtain t u where obt: "finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a"
-    by auto
+  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: "\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
-    using bchoice[of s "\<lambda>x e. e > 0 \<and> cball x e \<subseteq> s"] by auto
-  have "b ` t \<noteq> {}"
+    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"
-
-  show "\<exists>e > 0.
-    cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
-    apply (rule_tac x = "Min i" in exI)
-    unfolding subset_eq
-    apply rule
-    defer
-    apply rule
-    unfolding mem_Collect_eq
-  proof -
+  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
-      apply simp
-      apply rule
-      apply (erule_tac x=x in ballE)
-      using \<open>t\<subseteq>s\<close>
-      apply auto
-      done
+      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"
+    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"
+    { fix x
+      assume "x \<in> T"
       then have "Min i \<le> b x"
-        unfolding i_def
-        apply (rule_tac Min_le)
-        using obt(1)
-        apply auto
-        done
-      then have "x + (y - a) \<in> cball x (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 from \<open>x\<in>t\<close> have "x \<in> s"
-        using obt(2) by auto
-      ultimately have "x + (y - a) \<in> s"
-        using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast
+      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"
+    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))) = 1"
-      unfolding sum.reindex[OF *] o_def using obt(4) by auto
-    moreover 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(4,5)
+    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 "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> sum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
-      apply (rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI)
-      apply (rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
-      using obt(1, 3)
-      apply auto
-      done
+    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}"
+  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 ?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"
-    apply (rule set_eqI)
-    unfolding image_iff mem_Collect_eq
-    apply rule
-    apply auto
-    apply (rule_tac x=u in rev_bexI, simp)
-    apply (erule rev_bexI)
-    apply (erule rev_bexI, simp)
-    apply auto
-    done
+  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)
-  then show ?thesis
-    unfolding *
-    apply (rule compact_continuous_image)
-    apply (intro compact_Times compact_Icc assms)
-    done
+  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 = {}")
+  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
@@ -5029,20 +4986,20 @@
 qed
 
 lemma compact_convex_hull:
-  fixes s :: "'a::euclidean_space set"
-  assumes "compact s"
-  shows "compact (convex hull s)"
-proof (cases "s = {}")
+  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
+  then obtain w where "w \<in> S" by auto
   show ?thesis
-    unfolding caratheodory[of s]
+    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} = {}"
+    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
@@ -5050,27 +5007,27 @@
     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"
+      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"
+        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")
+        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
+            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
+          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"
+        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
@@ -5079,56 +5036,56 @@
       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} =
+      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}}"
+          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"
+          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"
+        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"]
+          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
-        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)
+        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
       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"
+        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")
+          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 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])
+            using \<open>w\<in>S\<close> and T
+            apply (auto intro!: exI[where x=T])
             done
         next
           case True
-          then obtain a u where au: "t = insert a u" "a\<notin>u"
+          then obtain a u where au: "T = insert a u" "a\<notin>u"
             apply (drule_tac card_eq_SucD, auto)
             done
           show ?thesis
           proof (cases "u = {}")
             case True
-            then have "x = a" using t(4)[unfolded au] by auto
+            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>
+              using T and \<open>n \<noteq> 0\<close>
               unfolding au
               apply (auto intro!: exI[where x="{a}"])
               done
@@ -5136,14 +5093,14 @@
             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]]
+              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)
+              using obt and T(1-3)
               unfolding au and * using card_insert_disjoint[OF _ au(2)]
               apply (auto intro!: exI[where x=u])
               done
@@ -5195,25 +5152,25 @@
   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))"
+  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 (rule, rule, cases "s = {}")
+  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"
+    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")
+    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)"
+      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)
@@ -5252,14 +5209,12 @@
             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>] and mem_Collect_eq
-            apply (rule_tac x="u + w" in exI, rule)
-            defer
-            apply (rule_tac x="v - w" in exI)
-            using \<open>u \<ge> 0\<close> and w and obt(3,4)
-            apply auto
-            done
+          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))"
@@ -5267,14 +5222,12 @@
             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>] and mem_Collect_eq
-            apply (rule_tac x="u - w" in exI, rule)
-            defer
-            apply (rule_tac x="v + w" in exI)
-            using \<open>u \<ge> 0\<close> and w and obt(3,4)
-            apply auto
-            done
+          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
@@ -5283,22 +5236,22 @@
 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)"
+  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 assms(1)], of a]
+  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")
+  proof (cases "x \<in> S")
     case False
-    then obtain y where "y \<in> convex hull s" "norm (x - a) < norm (y - a)"
+    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
@@ -5310,34 +5263,34 @@
 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
+  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)"
+  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)"
+  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)"
+  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)"
+    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
@@ -5347,45 +5300,45 @@
 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
+  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%important 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))"
+  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 s a \<in> s"
-    and "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
+  assumes "closed S"
+    and "S \<noteq> {}"
+  shows "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
 
-lemma closest_point_in_set: "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> closest_point s a \<in> s"
+lemma closest_point_in_set: "closed S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> closest_point S a \<in> S"
   by (meson closest_point_exists)
 
-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_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"
+  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
 
-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]
+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:
@@ -5417,14 +5370,14 @@
 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"
+  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"
+  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)
@@ -5433,30 +5386,30 @@
 
 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"
+  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"]
+  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"
+  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
 
 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"
+  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])
@@ -5465,34 +5418,34 @@
   done
 
 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"
+  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"
+  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
   then show ?thesis unfolding dist_norm and norm_le
-    using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
+    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)"
+  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)"
+  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:
@@ -5543,119 +5496,84 @@
 
 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)"
+  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"
+  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
-    apply (rule_tac x="y - z" in exI)
-    apply (rule_tac x="inner (y - z) y" in exI)
-    apply (rule_tac x=y in bexI, rule)
-    defer
-    apply rule
-    defer
-    apply rule
-    apply (rule ccontr)
-    using \<open>y \<in> s\<close>
-  proof -
-    show "inner (y - z) z < inner (y - z) y"
-      apply (subst diff_gt_0_iff_gt [symmetric])
-      unfolding inner_diff_right[symmetric] and inner_gt_zero_iff
-      using \<open>y\<in>s\<close> \<open>z\<notin>s\<close>
-      apply auto
-      done
-  next
-    fix x
-    assume "x \<in> s"
-    have *: "\<forall>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> inner (y - z) y \<le> inner (y - z) 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 *[THEN spec[where x=v]] by (auto simp: dist_commute algebra_simps)
-  qed auto
+  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 = {}")
+  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
-    apply (rule_tac x="-z" in exI)
-    apply (rule_tac x=1 in exI)
-    using less_le_trans[OF _ inner_ge_zero[of z]]
-    apply auto
-    done
+    by (simp add: gt_ex)
 next
   case False
-  obtain y where "y \<in> s" and y: "\<forall>x\<in>s. dist z y \<le> dist z x"
+  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
-    apply (rule_tac x="y - z" in exI)
-    apply (rule_tac x="inner (y - z) z + (norm (y - z))\<^sup>2 / 2" in exI)
-    apply rule
-    defer
-    apply rule
-  proof -
+  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 "\<not> 0 < inner (z - y) (x - y)"
-      apply (rule notI)
-      apply (drule closer_point_lemma)
+    assume "x \<in> S"
+    have "False" if *: "0 < inner (z - y) (x - y)"
     proof -
-      assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
-      then obtain u where "u > 0" "u \<le> 1" "dist (y + u *\<^sub>R (x - y)) z < dist y z"
-        by auto
-      then show False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
-        using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
-        using \<open>x\<in>s\<close> \<open>y\<in>s\<close> by (auto simp: dist_commute algebra_simps)
+      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
+      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 "inner (y - z) z + (norm (y - z))\<^sup>2 / 2 < inner (y - z) x"
-      unfolding power2_norm_eq_inner and not_less
-      by (auto simp: field_simps inner_commute inner_diff)
-  qed (insert \<open>y\<in>s\<close> \<open>z\<notin>s\<close>, auto)
+    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 = {}")
+  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 "norm ((SOME i. i\<in>Basis)::'a) = 1" "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
-    defer
-    apply (subst norm_le_zero_iff[symmetric])
-    apply (auto simp: SOME_Basis)
-    done
+  have "(SOME i. i\<in>Basis) \<noteq> (0::'a)"
+    by (metis Basis_zero SOME_Basis)
   then show ?thesis
-    apply (rule_tac x="SOME i. i\<in>Basis" in exI)
-    apply (rule_tac x=1 in exI)
-    using True using DIM_positive[where 'a='a]
-    apply auto
-    done
+    using True zero_less_one by blast
 next
   case False
   then show ?thesis
     using False using separating_hyperplane_closed_point[OF assms]
-    apply (elim exE)
-    unfolding inner_zero_right
-    apply (rule_tac x=a in exI)
-    apply (rule_tac x=b in exI, auto)
-    done
+    by (metis all_not_in_conv inner_zero_left inner_zero_right less_eq_real_def not_le)
 qed
 
 
@@ -5826,19 +5744,13 @@
   assumes "convex S"
   shows "convex (interior S)"
   unfolding convex_alt Ball_def mem_interior
-  apply (rule,rule,rule,rule,rule,rule)
-  apply (elim exE conjE)
-proof -
+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"
-    apply (rule_tac x="min d e" in exI, rule)
-    unfolding subset_eq
-    defer
-    apply rule
-  proof -
+  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"
@@ -6129,7 +6041,7 @@
     and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
   using assms
-proof (induct n arbitrary: f)
+proof (induction n arbitrary: f)
   case 0
   then show ?case by auto
 next
@@ -6137,45 +6049,39 @@
   have "finite f"
     using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
   show "\<Inter>f \<noteq> {}"
-    apply (cases "n = DIM('a)")
-    apply (rule Suc(5)[rule_format])
-    unfolding \<open>card f = Suc n\<close>
-  proof -
-    assume ng: "n \<noteq> DIM('a)"
-    then have "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})"
-      apply (rule_tac bchoice)
-      unfolding ex_in_conv
-      apply (rule, rule Suc(1)[rule_format])
-      unfolding card_Diff_singleton_if[OF \<open>finite f\<close>] \<open>card f = Suc n\<close>
-      defer
-      defer
-      apply (rule Suc(4)[rule_format])
-      defer
-      apply (rule Suc(5)[rule_format])
-      using Suc(3) \<open>finite f\<close>
-      apply auto
-      done
-    then obtain X where X: "\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
+  proof (cases "n = DIM('a)")
+    case True
+    then show ?thesis
+      by (simp add: Suc.prems(1) Suc.prems(4))
+  next
+    case False
+    have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
+    proof (rule Suc.IH[rule_format])
+      show "card (f - {s}) = n"
+        by (simp add: Suc.prems(1) \<open>finite f\<close> that)
+      show "DIM('a) + 1 \<le> n"
+        using False Suc.prems(2) by linarith
+      show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
+        by (simp add: Suc.prems(4) subset_Diff_insert)
+    qed (use Suc in auto)
+    then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
+      by blast
+    then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
+      by metis
     show ?thesis
     proof (cases "inj_on X f")
       case False
-      then obtain s t where st: "s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t"
+      then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
         unfolding inj_on_def by auto
       then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
       show ?thesis
-        unfolding *
-        unfolding ex_in_conv[symmetric]
-        apply (rule_tac x="X s" in exI, rule)
-        apply (rule X[rule_format])
-        using X st
-        apply auto
-        done
+        by (metis "*" X disjoint_iff_not_equal st)
     next
       case True
       then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
         using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
         unfolding card_image[OF True] and \<open>card f = Suc n\<close>
-        using Suc(3) \<open>finite f\<close> and ng
+        using Suc(3) \<open>finite f\<close> and False
         by auto
       have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
         using mp(2) by auto
@@ -6192,40 +6098,17 @@
         using inj_on_image_Int[OF True gh(3,4)]
         by auto
       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
-        apply (rule_tac [!] hull_minimal)
-        using Suc gh(3-4)
-        unfolding subset_eq
-        apply (rule_tac [2] convex_Inter, rule_tac [4] convex_Inter)
-        prefer 3
-        apply rule
-      proof -
-        fix x
-        assume "x \<in> X ` g"
-        then obtain y where "y \<in> g" "x = X y"
-          unfolding image_iff ..
-        then show "x \<in> \<Inter>h"
-          using X[THEN bspec[where x=y]] using * f by auto
-      next
-        show "\<forall>x\<in>X ` h. x \<in> \<Inter>g"
-        proof
-        fix x
-        assume "x \<in> X ` h"
-        then obtain y where "y \<in> h" "x = X y"
-          unfolding image_iff ..
-        then show "x \<in> \<Inter>g"
-          using X[THEN bspec[where x=y]] using * f by auto
-      qed
-      qed auto
+        by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
       then show ?thesis
         unfolding f using mp(3)[unfolded gh] by blast
     qed
-  qed auto
+  qed 
 qed
 
 theorem%important helly:
   fixes f :: "'a::euclidean_space set set"
   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
-    and "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
+    and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
   apply%unimportant (rule helly_induct)
   using assms
@@ -6235,70 +6118,76 @@
 
 subsection \<open>Epigraphs of convex functions\<close>
 
-definition%important "epigraph s (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
-
-lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y"
+definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
+
+lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
   unfolding epigraph_def by auto
 
-lemma convex_epigraph: "convex (epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
-  unfolding convex_def convex_on_def
-  unfolding Ball_def split_paired_All epigraph_def
-  unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
-  apply safe
-  defer
-  apply (erule_tac x=x in allE)
-  apply (erule_tac x="f x" in allE, safe)
-  apply (erule_tac x=xa in allE)
-  apply (erule_tac x="f xa" in allE)
-  prefer 3
-  apply (rule_tac y="u * f a + v * f aa" in order_trans)
-  defer
-  apply (auto intro!:mult_left_mono add_mono)
-  done
-
-lemma convex_epigraphI: "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex (epigraph s f)"
+lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
+proof safe
+  assume L: "convex (epigraph S f)"
+  then show "convex_on S f"
+    by (auto simp: convex_def convex_on_def epigraph_def)
+  show "convex S"
+    using L
+    apply (clarsimp simp: convex_def convex_on_def epigraph_def)
+    apply (erule_tac x=x in allE)
+    apply (erule_tac x="f x" in allE, safe)
+    apply (erule_tac x=y in allE)
+    apply (erule_tac x="f y" in allE)
+    apply (auto simp: )
+    done
+next
+  assume "convex_on S f" "convex S"
+  then show "convex (epigraph S f)"
+    unfolding convex_def convex_on_def epigraph_def
+    apply safe
+     apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
+      apply (auto intro!:mult_left_mono add_mono)
+    done
+qed
+
+lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
   unfolding convex_epigraph by auto
 
-lemma convex_epigraph_convex: "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
+lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
   by (simp add: convex_epigraph)
 
 
 subsubsection%unimportant \<open>Use this to derive general bound property of convex function\<close>
 
 lemma convex_on:
-  assumes "convex s"
-  shows "convex_on s f \<longleftrightarrow>
-    (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1 \<longrightarrow>
+  assumes "convex S"
+  shows "convex_on S f \<longleftrightarrow>
+    (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
       f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
+
   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
   unfolding fst_sum snd_sum fst_scaleR snd_scaleR
   apply safe
-  apply (drule_tac x=k in spec)
-  apply (drule_tac x=u in spec)
-  apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
-  apply simp
-  using assms[unfolded convex]
-  apply simp
-  apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
-  defer
-  apply (rule sum_mono)
-  apply (erule_tac x=i in allE)
+    apply (drule_tac x=k in spec)
+    apply (drule_tac x=u in spec)
+    apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
+    apply simp
+  using assms[unfolded convex] apply simp
+  apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
+   apply (rule sum_mono)
+   apply (erule_tac x=i in allE)
   unfolding real_scaleR_def
-  apply (rule mult_left_mono)
-  using assms[unfolded convex]
-  apply auto
+   apply (rule mult_left_mono)
+  using assms[unfolded convex] apply auto
   done
 
 
 subsection%unimportant \<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"
+  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 as: "x \<in> S" "y \<in> S" "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
   {
@@ -6331,7 +6220,7 @@
       using **(2) as(3)
       by (auto simp: field_simps intro!:mult_right_mono)
   }
-  ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+  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]
@@ -6340,8 +6229,8 @@
 qed
 
 lemma is_interval_connected:
-  fixes s :: "'a::euclidean_space set"
-  shows "is_interval s \<Longrightarrow> connected s"
+  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))"
@@ -6587,8 +6476,8 @@
   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"
+  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
@@ -6619,66 +6508,53 @@
 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"
-  apply (rule that[of "{x::'a. \<forall>i\<in>Basis. x\<bullet>i=0 \<or> x\<bullet>i=1}"])
-  apply (rule finite_subset[of _ "(\<lambda>s. (\<Sum>i\<in>Basis. (if i\<in>s then 1 else 0) *\<^sub>R i)::'a) ` Pow Basis"])
-  prefer 3
-  apply (rule unit_interval_convex_hull, rule)
-  unfolding mem_Collect_eq
-proof -
-  fix x :: 'a
-  assume as: "\<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
-qed auto
+  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"
+    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
+  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"
+  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>Ri)::'a"
+  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)"
-    apply (rule set_eqI, rule)
-    unfolding image_iff
-    defer
-    apply (erule bexE)
-  proof -
+  proof (intro set_eqI iffI)
     fix y
-    assume as: "y\<in>cbox (x - ?d) (x + ?d)"
+    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 field_simps inner_simps)
-    with \<open>0 < d\<close> show "\<exists>z\<in>cbox 0 (\<Sum>Basis). y = x - ?d + (2 * d) *\<^sub>R z"
-      by (intro bexI[of _ "inverse (2 * d) *\<^sub>R (y - (x - ?d))"]) auto
+    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 z
-    assume as: "z\<in>cbox 0 (\<Sum>Basis)" "y = x - ?d + (2*d) *\<^sub>R z"
-    have "\<And>i. i\<in>Basis \<Longrightarrow> 0 \<le> d * (z \<bullet> i) \<and> d * (z \<bullet> i) \<le> d"
-      using assms as(1)[unfolded mem_box]
+    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)"
-      unfolding as(2) mem_box
-      apply -
-      apply rule
-      using as(1)[unfolded mem_box]
-      apply (erule_tac x=i in ballE)
-      using assms
-      apply (auto simp: inner_simps)
-      done
+      using z assms by (auto simp: mem_box inner_simps)
   qed
-  obtain s where "finite s" "cbox 0 (\<Sum>Basis::'a) = convex hull s"
+  obtain S where "finite S" "cbox 0 (\<Sum>Basis::'a) = convex hull S"
     using unit_cube_convex_hull by auto
   then show ?thesis
-    apply (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"])
-    unfolding * and convex_hull_affinity
-    apply auto
-    done
+    by (rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` S"]) (auto simp: convex_hull_affinity *)
 qed
 
 subsection%unimportant\<open>Representation of any interval as a finite convex hull\<close>
@@ -6738,18 +6614,18 @@
 
 lemma closed_interval_as_convex_hull:
   fixes a :: "'a::euclidean_space"
-  obtains s where "finite s" "cbox a b = convex hull s"
+  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"
+  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>)+
+  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])
@@ -6761,31 +6637,23 @@
 subsection%unimportant \<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"
+  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"
+  assume "x \<in> S" "e > 0"
   define B where "B = \<bar>b\<bar> + 1"
-  have B: "0 < B" "\<And>x. x\<in>s \<Longrightarrow> \<bar>f x\<bar> \<le> B"
-    unfolding B_def
-    defer
-    apply (drule assms(3)[rule_format])
-    apply auto
-    done
-  obtain k where "k > 0" and k: "cball x k \<subseteq> s"
-    using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]]
-    using \<open>x\<in>s\<close> by auto
+  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"
-    apply (rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI)
-    apply rule
-    defer
-  proof (rule, rule)
+  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"
@@ -6795,22 +6663,16 @@
       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[unfolded subset_eq,rule_format])
+      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"
-          unfolding w_def
-          apply (rule k[unfolded subset_eq,rule_format])
-          unfolding mem_cball dist_norm
-          unfolding t_def
-          using \<open>k>0\<close>
-          apply auto
-          done
+        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"
@@ -6818,32 +6680,22 @@
         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 * B < e * t"
+        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)
-        then have "(f w - f x) / t < e"
-          using B(2)[OF \<open>w\<in>s\<close>] and B(2)[OF \<open>x\<in>s\<close>]
-          using \<open>t > 0\<close> by (auto simp:field_simps)
-        then have th1: "f y - f x < e"
-          apply -
-          apply (rule le_less_trans)
-          defer
-          apply assumption
+        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>
+          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"
-          unfolding w_def
-          apply (rule k[unfolded subset_eq,rule_format])
-          unfolding mem_cball dist_norm
-          unfolding t_def
-          using \<open>k > 0\<close>
-          apply auto
-          done
+        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"
@@ -6856,12 +6708,12 @@
           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 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>
+          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 (auto simp: divide_simps)
@@ -6950,11 +6802,7 @@
       by (simp add: dist_norm abs_le_iff algebra_simps)
     show "cball x d \<subseteq> convex hull c"
       unfolding 2
-      apply clarsimp
-      apply (simp only: dist_norm)
-      apply (subst inner_diff_left [symmetric], simp)
-      apply (erule (1) order_trans [OF Basis_le_norm])
-      done
+      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 DIM_positive)
     show "convex hull c \<subseteq> cball x e"
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Derivative.thy
--- a/src/HOL/Analysis/Derivative.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Derivative.thy	Wed May 02 12:48:08 2018 +0100
@@ -373,77 +373,76 @@
 
 lemma frechet_derivative_unique_within:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "(f has_derivative f') (at x within s)"
-    and "(f has_derivative f'') (at x within s)"
-    and "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> (x + d *\<^sub>R i) \<in> s"
+  assumes "(f has_derivative f') (at x within S)"
+    and "(f has_derivative f'') (at x within S)"
+    and "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> (x + d *\<^sub>R i) \<in> S"
   shows "f' = f''"
 proof -
   note as = assms(1,2)[unfolded has_derivative_def]
   then interpret f': bounded_linear f' by auto
   from as interpret f'': bounded_linear f'' by auto
-  have "x islimpt s" unfolding islimpt_approachable
+  have "x islimpt S" unfolding islimpt_approachable
   proof (rule, rule)
     fix e :: real
     assume "e > 0"
-    obtain d where "0 < \<bar>d\<bar>" and "\<bar>d\<bar> < e" and "x + d *\<^sub>R (SOME i. i \<in> Basis) \<in> s"
+    obtain d where "0 < \<bar>d\<bar>" and "\<bar>d\<bar> < e" and "x + d *\<^sub>R (SOME i. i \<in> Basis) \<in> S"
       using assms(3) SOME_Basis \<open>e>0\<close> by blast
-    then show "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
+    then show "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
       apply (rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI)
       unfolding dist_norm
       apply (auto simp: SOME_Basis nonzero_Basis)
       done
   qed
-  then have *: "netlimit (at x within s) = x"
+  then have *: "netlimit (at x within S) = x"
     apply (auto intro!: netlimit_within)
     by (metis trivial_limit_within)
   show ?thesis
-    apply (rule linear_eq_stdbasis)
-    unfolding linear_conv_bounded_linear
-    apply (rule as(1,2)[THEN conjunct1])+
-  proof (rule, rule ccontr)
+  proof (rule linear_eq_stdbasis)
+    show "linear f'" "linear f''"
+      unfolding linear_conv_bounded_linear using as by auto
+  next
     fix i :: 'a
     assume i: "i \<in> Basis"
     define e where "e = norm (f' i - f'' i)"
-    assume "f' i \<noteq> f'' i"
-    then have "e > 0"
-      unfolding e_def by auto
-    obtain d where d:
-      "0 < d"
-      "(\<And>xa. xa\<in>s \<longrightarrow> 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
-          dist ((f xa - f x - f' (xa - x)) /\<^sub>R norm (xa - x) -
-              (f xa - f x - f'' (xa - x)) /\<^sub>R norm (xa - x)) (0 - 0) < e)"
-      using tendsto_diff [OF as(1,2)[THEN conjunct2]]
-      unfolding * Lim_within
-      using \<open>e>0\<close> by blast
-    obtain c where c: "0 < \<bar>c\<bar>" "\<bar>c\<bar> < d \<and> x + c *\<^sub>R i \<in> s"
-      using assms(3) i d(1) by blast
-    have *: "norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) =
+    show "f' i = f'' i"
+    proof (rule ccontr)
+      assume "f' i \<noteq> f'' i"
+      then have "e > 0"
+        unfolding e_def by auto
+      obtain d where d:
+        "0 < d"
+        "(\<And>y. y\<in>S \<longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow>
+          dist ((f y - f x - f' (y - x)) /\<^sub>R norm (y - x) -
+              (f y - f x - f'' (y - x)) /\<^sub>R norm (y - x)) (0 - 0) < e)"
+        using tendsto_diff [OF as(1,2)[THEN conjunct2]]
+        unfolding * Lim_within
+        using \<open>e>0\<close> by blast
+      obtain c where c: "0 < \<bar>c\<bar>" "\<bar>c\<bar> < d \<and> x + c *\<^sub>R i \<in> S"
+        using assms(3) i d(1) by blast
+      have *: "norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) =
         norm ((1 / \<bar>c\<bar>) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
-      unfolding scaleR_right_distrib by auto
-    also have "\<dots> = norm ((1 / \<bar>c\<bar>) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
-      unfolding f'.scaleR f''.scaleR
-      unfolding scaleR_right_distrib scaleR_minus_right
-      by auto
-    also have "\<dots> = e"
-      unfolding e_def
-      using c(1)
-      using norm_minus_cancel[of "f' i - f'' i"]
-      by auto
-    finally show False
-      using c
-      using d(2)[of "x + c *\<^sub>R i"]
-      unfolding dist_norm
-      unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
-        scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
-      using i
-      by (auto simp: inverse_eq_divide)
+        unfolding scaleR_right_distrib by auto
+      also have "\<dots> = norm ((1 / \<bar>c\<bar>) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
+        unfolding f'.scaleR f''.scaleR
+        unfolding scaleR_right_distrib scaleR_minus_right
+        by auto
+      also have "\<dots> = e"
+        unfolding e_def
+        using c(1)
+        using norm_minus_cancel[of "f' i - f'' i"]
+        by auto
+      finally show False
+        using c
+        using d(2)[of "x + c *\<^sub>R i"]
+        unfolding dist_norm
+        unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
+          scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
+        using i
+        by (auto simp: inverse_eq_divide)
+    qed
   qed
 qed
 
-lemma frechet_derivative_unique_at:
-  "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
-  by (rule has_derivative_unique)
-
 lemma frechet_derivative_unique_within_closed_interval:
   fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
@@ -506,12 +505,12 @@
   from assms(1) have *: "at x within box a b = at x"
     by (metis at_within_interior interior_open open_box)
   from assms(2,3) [unfolded *] show "f' = f''"
-    by (rule frechet_derivative_unique_at)
+    by (rule has_derivative_unique)
 qed
 
 lemma frechet_derivative_at:
   "(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
-  apply (rule frechet_derivative_unique_at[of f])
+  apply (rule has_derivative_unique[of f])
   apply assumption
   unfolding frechet_derivative_works[symmetric]
   using differentiable_def
@@ -1197,13 +1196,13 @@
 
 lemma has_derivative_inverse_basic:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes "(f has_derivative f') (at (g y))"
-    and "bounded_linear g'"
+  assumes derf: "(f has_derivative f') (at (g y))"
+    and ling': "bounded_linear g'"
     and "g' \<circ> f' = id"
-    and "continuous (at y) g"
-    and "open t"
-    and "y \<in> t"
-    and "\<forall>z\<in>t. f (g z) = z"
+    and contg: "continuous (at y) g"
+    and "open T"
+    and "y \<in> T"
+    and fg: "\<And>z. z \<in> T \<Longrightarrow> f (g z) = z"
   shows "(g has_derivative g') (at y)"
 proof -
   interpret f': bounded_linear f'
@@ -1214,70 +1213,48 @@
     using bounded_linear.pos_bounded[OF assms(2)] by blast
   have lem1: "\<forall>e>0. \<exists>d>0. \<forall>z.
     norm (z - y) < d \<longrightarrow> norm (g z - g y - g'(z - y)) \<le> e * norm (g z - g y)"
-  proof (rule, rule)
+  proof (intro allI impI)
     fix e :: real
     assume "e > 0"
     with C(1) have *: "e / C > 0" by auto
-    obtain d0 where d0:
-        "0 < d0"
-        "\<forall>ya. norm (ya - g y) < d0 \<longrightarrow> norm (f ya - f (g y) - f' (ya - g y)) \<le> e / C * norm (ya - g y)"
-      using assms(1)
-      unfolding has_derivative_at_alt
-      using * by blast
-    obtain d1 where d1:
-        "0 < d1"
-        "\<forall>x. 0 < dist x y \<and> dist x y < d1 \<longrightarrow> dist (g x) (g y) < d0"
-      using assms(4)
-      unfolding continuous_at Lim_at
-      using d0(1) by blast
-    obtain d2 where d2:
-        "0 < d2"
-        "\<forall>ya. dist ya y < d2 \<longrightarrow> ya \<in> t"
-      using assms(5)
-      unfolding open_dist
-      using assms(6) by blast
+    obtain d0 where  "0 < d0" and d0:
+        "\<And>u. norm (u - g y) < d0 \<Longrightarrow> norm (f u - f (g y) - f' (u - g y)) \<le> e / C * norm (u - g y)"
+      using derf * unfolding has_derivative_at_alt by blast
+    obtain d1 where "0 < d1" and d1: "\<And>x. \<lbrakk>0 < dist x y; dist x y < d1\<rbrakk> \<Longrightarrow> dist (g x) (g y) < d0"
+      using contg \<open>0 < d0\<close> unfolding continuous_at Lim_at by blast
+    obtain d2 where "0 < d2" and d2: "\<And>u. dist u y < d2 \<Longrightarrow> u \<in> T"
+      using \<open>open T\<close> \<open>y \<in> T\<close> unfolding open_dist by blast
     obtain d where d: "0 < d" "d < d1" "d < d2"
-      using real_lbound_gt_zero[OF d1(1) d2(1)] by blast
-    then show "\<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
-      apply (rule_tac x=d in exI)
-      apply rule
-      defer
-      apply rule
-      apply rule
-    proof -
+      using real_lbound_gt_zero[OF \<open>0 < d1\<close> \<open>0 < d2\<close>] by blast
+    show "\<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
+    proof (intro exI allI impI conjI)
       fix z
       assume as: "norm (z - y) < d"
-      then have "z \<in> t"
+      then have "z \<in> T"
         using d2 d unfolding dist_norm by auto
       have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
         unfolding g'.diff f'.diff
-        unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
-        unfolding assms(7)[rule_format,OF \<open>z\<in>t\<close>]
-        apply (subst norm_minus_cancel[symmetric])
-        apply auto
-        done
+        unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] fg[OF \<open>z\<in>T\<close>]
+        by (simp add: norm_minus_commute)
       also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
         by (rule C(2))
       also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
-        apply (rule mult_right_mono)
-        apply (rule d0(2)[rule_format,unfolded assms(7)[rule_format,OF \<open>y\<in>t\<close>]])
-        apply (cases "z = y")
-        defer
-        apply (rule d1(2)[unfolded dist_norm,rule_format])
-        using as d C d0
-        apply auto
-        done
+      proof -
+        have "norm (g z - g y) < d0"
+          by (metis as cancel_comm_monoid_add_class.diff_cancel d(2) \<open>0 < d0\<close> d1 diff_gt_0_iff_gt diff_strict_mono dist_norm dist_self zero_less_dist_iff)
+        then show ?thesis
+          by (metis C(1) \<open>y \<in> T\<close> d0 fg real_mult_le_cancel_iff1)
+      qed
       also have "\<dots> \<le> e * norm (g z - g y)"
         using C by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
         by simp
-    qed auto
+    qed (use d in auto)
   qed
   have *: "(0::real) < 1 / 2"
     by auto
-  obtain d where d:
-      "0 < d"
-      "\<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> 1 / 2 * norm (g z - g y)"
+  obtain d where "0 < d" and d:
+      "\<And>z. norm (z - y) < d \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> 1/2 * norm (g z - g y)"
     using lem1 * by blast
   define B where "B = C * 2"
   have "B > 0"
@@ -1287,51 +1264,37 @@
     have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
       by (rule norm_triangle_sub)
     also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
-      apply (rule add_left_mono)
-      using d and z
-      apply auto
-      done
+      by (rule add_left_mono) (use d z in auto)
     also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
-      apply (rule add_right_mono)
-      using C
-      apply auto
-      done
+      by (rule add_right_mono) (use C in auto)
     finally show "norm (g z - g y) \<le> B * norm (z - y)"
       unfolding B_def
       by (auto simp add: field_simps)
   qed
   show ?thesis
     unfolding has_derivative_at_alt
-    apply rule
-    apply (rule assms)
-    apply rule
-    apply rule
-  proof -
+  proof (intro conjI assms allI impI)
     fix e :: real
     assume "e > 0"
     then have *: "e / B > 0" by (metis \<open>B > 0\<close> divide_pos_pos)
-    obtain d' where d':
-        "0 < d'"
-        "\<forall>z. norm (z - y) < d' \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e / B * norm (g z - g y)"
+    obtain d' where "0 < d'" and d':
+        "\<And>z. norm (z - y) < d' \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> e / B * norm (g z - g y)"
       using lem1 * by blast
     obtain k where k: "0 < k" "k < d" "k < d'"
-      using real_lbound_gt_zero[OF d(1) d'(1)] by blast
+      using real_lbound_gt_zero[OF \<open>0 < d\<close> \<open>0 < d'\<close>] by blast
     show "\<exists>d>0. \<forall>ya. norm (ya - y) < d \<longrightarrow> norm (g ya - g y - g' (ya - y)) \<le> e * norm (ya - y)"
-      apply (rule_tac x=k in exI)
-      apply auto
-    proof -
+    proof (intro exI allI impI conjI)
       fix z
       assume as: "norm (z - y) < k"
       then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
         using d' k by auto
       also have "\<dots> \<le> e * norm (z - y)"
         unfolding times_divide_eq_left pos_divide_le_eq[OF \<open>B>0\<close>]
-        using lem2[of z]
-        using k as using \<open>e > 0\<close>
+        using lem2[of z] k as \<open>e > 0\<close>
         by (auto simp add: field_simps)
       finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
         by simp
-    qed(insert k, auto)
+    qed (use k in auto)
   qed
 qed
 
@@ -1344,25 +1307,22 @@
     and "g' \<circ> f' = id"
     and "continuous (at (f x)) g"
     and "g (f x) = x"
-    and "open t"
-    and "f x \<in> t"
-    and "\<forall>y\<in>t. f (g y) = y"
+    and "open T"
+    and "f x \<in> T"
+    and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"
   shows "(g has_derivative g') (at (f x))"
-  apply (rule has_derivative_inverse_basic)
-  using assms
-  apply auto
-  done
+  by (rule has_derivative_inverse_basic) (use assms in auto)
 
 text \<open>This is the version in Dieudonne', assuming continuity of f and g.\<close>
 
 lemma has_derivative_inverse_dieudonne:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes "open s"
-    and "open (f ` s)"
-    and "continuous_on s f"
-    and "continuous_on (f ` s) g"
-    and "\<forall>x\<in>s. g (f x) = x"
-    and "x \<in> s"
+  assumes "open S"
+    and "open (f ` S)"
+    and "continuous_on S f"
+    and "continuous_on (f ` S) g"
+    and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
+    and "x \<in> S"
     and "(f has_derivative f') (at x)"
     and "bounded_linear g'"
     and "g' \<circ> f' = id"
@@ -1377,11 +1337,11 @@
 
 lemma has_derivative_inverse:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes "compact s"
-    and "x \<in> s"
-    and "f x \<in> interior (f ` s)"
-    and "continuous_on s f"
-    and "\<forall>y\<in>s. g (f y) = y"
+  assumes "compact S"
+    and "x \<in> S"
+    and fx: "f x \<in> interior (f ` S)"
+    and "continuous_on S f"
+    and "\<And>y. y \<in> S \<Longrightarrow> g (f y) = y"
     and "(f has_derivative f') (at x)"
     and "bounded_linear g'"
     and "g' \<circ> f' = id"
@@ -1389,20 +1349,17 @@
 proof -
   {
     fix y
-    assume "y \<in> interior (f ` s)"
-    then obtain x where "x \<in> s" and *: "y = f x"
-      unfolding image_iff
-      using interior_subset
-      by auto
+    assume "y \<in> interior (f ` S)"
+    then obtain x where "x \<in> S" and *: "y = f x"
+      by (meson imageE interior_subset subsetCE)
     have "f (g y) = y"
-      unfolding * and assms(5)[rule_format,OF \<open>x\<in>s\<close>] ..
+      unfolding * and assms(5)[rule_format,OF \<open>x\<in>S\<close>] ..
   } note * = this
   show ?thesis
-    apply (rule has_derivative_inverse_basic_x[OF assms(6-8)])
-    apply (rule continuous_on_interior[OF _ assms(3)])
-    apply (rule continuous_on_inv[OF assms(4,1)])
-    apply (rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
-    apply (metis *)
+    apply (rule has_derivative_inverse_basic_x[OF assms(6-8), where T = "interior (f ` S)"])
+    apply (rule continuous_on_interior[OF _ fx])
+    apply (rule continuous_on_inv)
+    apply (simp_all add: assms *)
     done
 qed
 
@@ -1411,13 +1368,13 @@
 
 lemma brouwer_surjective:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
-  assumes "compact t"
-    and "convex t"
-    and "t \<noteq> {}"
-    and "continuous_on t f"
-    and "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t"
-    and "x \<in> s"
-  shows "\<exists>y\<in>t. f y = x"
+  assumes "compact T"
+    and "convex T"
+    and "T \<noteq> {}"
+    and "continuous_on T f"
+    and "\<And>x y. \<lbrakk>x\<in>S; y\<in>T\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> T"
+    and "x \<in> S"
+  shows "\<exists>y\<in>T. f y = x"
 proof -
   have *: "\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
     by (auto simp add: algebra_simps)
@@ -1432,10 +1389,10 @@
 
 lemma brouwer_surjective_cball:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
-  assumes "e > 0"
-    and "continuous_on (cball a e) f"
-    and "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e"
-    and "x \<in> s"
+  assumes "continuous_on (cball a e) f"
+    and "e > 0"
+    and "x \<in> S"
+    and "\<And>x y. \<lbrakk>x\<in>S; y\<in>cball a e\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> cball a e"
   shows "\<exists>y\<in>cball a e. f y = x"
   apply (rule brouwer_surjective)
   apply (rule compact_cball convex_cball)+
@@ -1448,14 +1405,14 @@
 
 lemma sussmann_open_mapping:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
-  assumes "open s"
-    and "continuous_on s f"
-    and "x \<in> s"
+  assumes "open S"
+    and "continuous_on S f"
+    and "x \<in> S"
     and "(f has_derivative f') (at x)"
     and "bounded_linear g'" "f' \<circ> g' = id"
-    and "t \<subseteq> s"
-    and "x \<in> interior t"
-  shows "f x \<in> interior (f ` t)"
+    and "T \<subseteq> S"
+    and "x \<in> interior T"
+  shows "f x \<in> interior (f ` T)"
 proof -
   interpret f': bounded_linear f'
     using assms
@@ -1473,31 +1430,17 @@
     using assms(4)
     unfolding has_derivative_at_alt
     using * by blast
-  obtain e1 where e1: "0 < e1" "cball x e1 \<subseteq> t"
+  obtain e1 where e1: "0 < e1" "cball x e1 \<subseteq> T"
     using assms(8)
     unfolding mem_interior_cball
     by blast
   have *: "0 < e0 / B" "0 < e1 / B" using e0 e1 B by auto
   obtain e where e: "0 < e" "e < e0 / B" "e < e1 / B"
     using real_lbound_gt_zero[OF *] by blast
-  have "\<forall>z\<in>cball (f x) (e / 2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
-    apply rule
-    apply (rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
-    prefer 3
-    apply rule
-    apply rule
-  proof-
-    show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
-      unfolding g'.diff
-      apply (rule continuous_on_compose[of _ _ f, unfolded o_def])
-      apply (rule continuous_intros linear_continuous_on[OF assms(5)])+
-      apply (rule continuous_on_subset[OF assms(2)])
-      apply rule
-      apply (unfold image_iff)
-      apply (erule bexE)
+  have lem: "\<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z" if "z\<in>cball (f x) (e / 2)" for z
+  proof (rule brouwer_surjective_cball)
+    have z: "z \<in> S" if as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))" for y z
     proof-
-      fix y z
-      assume as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
       have "dist x z = norm (g' (f x) - g' y)"
         unfolding as(2) and dist_norm by auto
       also have "\<dots> \<le> norm (f x - y) * B"
@@ -1516,9 +1459,16 @@
       finally have "z \<in> cball x e1"
         unfolding mem_cball
         by force
-      then show "z \<in> s"
+      then show "z \<in> S"
         using e1 assms(7) by auto
     qed
+    show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
+      unfolding g'.diff
+      apply (rule continuous_on_compose[of _ _ f, unfolded o_def])
+       apply (rule continuous_intros linear_continuous_on[OF assms(5)])+
+      apply (rule continuous_on_subset[OF assms(2)])
+      using z
+      by blast
   next
     fix y z
     assume as: "y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
@@ -1528,7 +1478,7 @@
       apply (rule mult_right_mono)
       using as(2)[unfolded mem_cball dist_norm] and B
       unfolding norm_minus_commute
-      apply auto
+       apply auto
       done
     also have "\<dots> < e0"
       using e and B
@@ -1563,7 +1513,7 @@
     finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
       unfolding mem_cball dist_norm
       by auto
-  qed (insert e, auto) note lem = this
+  qed (use e that in auto) 
   show ?thesis
     unfolding mem_interior
     apply (rule_tac x="e/2" in exI)
@@ -1589,13 +1539,13 @@
       done
     also have "\<dots> \<le> e1"
       using e B unfolding less_divide_eq by auto
-    finally have "x + g'(z - f x) \<in> t"
+    finally have "x + g'(z - f x) \<in> T"
       apply -
       apply (rule e1(2)[unfolded subset_eq,rule_format])
       unfolding mem_cball dist_norm
       apply auto
       done
-    then show "y \<in> f ` t"
+    then show "y \<in> f ` T"
       using z by auto
   qed (insert e, auto)
 qed
@@ -1750,9 +1700,9 @@
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "a \<in> s"
       and "open s"
-      and "bounded_linear g'"
+      and bling: "bounded_linear g'"
       and "g' \<circ> f' a = id"
-      and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x)"
+      and derf: "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x)"
       and "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
   obtains r where "r > 0" "ball a r \<subseteq> s" "inj_on f (ball a r)"
 proof -
@@ -1760,11 +1710,7 @@
     using assms by auto
   note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
   have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)"
-    defer
-    apply (subst euclidean_eq_iff)
-    using f'g'
-    apply auto
-    done
+    using f'g' by auto
   then have *: "0 < onorm g'"
     unfolding onorm_pos_lt[OF assms(3)]
     by fastforce
@@ -1812,17 +1758,11 @@
         have *: "(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
           unfolding o_def and diff
           using f'g' by auto
+        have blin: "bounded_linear (f' a)"
+          using \<open>a \<in> s\<close> derf by blast
         show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
-          unfolding ph' *
-          apply (simp add: comp_def)
-          apply (rule bounded_linear.has_derivative[OF assms(3)])
-          apply (rule derivative_intros)
-          defer
-          apply (rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
-          apply (rule has_derivative_at_withinI)
-          using assms(5) and \<open>u \<in> s\<close> \<open>a \<in> s\<close>
-          apply (auto intro!: derivative_intros bounded_linear.has_derivative[of _ "\<lambda>x. x"] has_derivative_bounded_linear)
-          done
+          unfolding ph' * comp_def
+          by (rule \<open>u \<in> s\<close> derivative_eq_intros has_derivative_at_withinI [OF derf] bounded_linear.has_derivative [OF blin]  bounded_linear.has_derivative [OF bling] |simp)+
         have **: "bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)"
           apply (rule_tac[!] bounded_linear_sub)
           apply (rule_tac[!] has_derivative_bounded_linear)
@@ -1896,21 +1836,20 @@
   qed
 qed
 
-lemma has_derivative_sequence_lipschitz:
+lemma has_derivative_sequence_Lipschitz:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes "convex s"
-    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
-  shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s.
+  assumes "convex S"
+    and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
+    and "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+    and "e > 0"
+  shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
     norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
-proof (rule, rule)
-  fix e :: real
-  assume "e > 0"
-  then have *: "2 * (1/2* e) = e" "1/2 * e >0"
-    by auto
-  obtain N where "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> 1 / 2 * e * norm h"
+proof -
+  have *: "2 * (1/2* e) = e" "1/2 * e >0"
+    using \<open>e>0\<close> by auto
+  obtain N where "\<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> 1 / 2 * e * norm h"
     using assms(3) *(2) by blast
-  then show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
+  then show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
     apply (rule_tac x=N in exI)
     apply (rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
     using assms \<open>e > 0\<close>
@@ -1920,22 +1859,22 @@
 
 lemma has_derivative_sequence:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
-  assumes "convex s"
-    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
-    and "x0 \<in> s"
+  assumes "convex S"
+    and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
+    and "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+    and "x0 \<in> S"
     and "((\<lambda>n. f n x0) \<longlongrightarrow> l) sequentially"
-  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)"
+  shows "\<exists>g. \<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g'(x)) (at x within S)"
 proof -
-  have lem1: "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s.
+  have lem1: "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
       norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
-    using assms(1,2,3) by (rule has_derivative_sequence_lipschitz)
-  have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
+    using assms(1,2,3) by (rule has_derivative_sequence_Lipschitz)
+  have "\<exists>g. \<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
     apply (rule bchoice)
     unfolding convergent_eq_Cauchy
   proof
     fix x
-    assume "x \<in> s"
+    assume "x \<in> S"
     show "Cauchy (\<lambda>n. f n x)"
     proof (cases "x = x0")
       case True
@@ -1945,7 +1884,7 @@
       case False
       show ?thesis
         unfolding Cauchy_def
-      proof (rule, rule)
+      proof (intro allI impI)
         fix e :: real
         assume "e > 0"
         hence *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0" using False by auto
@@ -1955,12 +1894,11 @@
           using *(1) by blast
         obtain N where N:
           "\<forall>m\<ge>N. \<forall>n\<ge>N.
-            \<forall>xa\<in>s. \<forall>y\<in>s. norm (f m xa - f n xa - (f m y - f n y)) \<le>
+            \<forall>xa\<in>S. \<forall>y\<in>S. norm (f m xa - f n xa - (f m y - f n y)) \<le>
               e / 2 / norm (x - x0) * norm (xa - y)"
         using lem1 *(2) by blast
         show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
-          apply (rule_tac x="max M N" in exI)
-        proof rule+
+        proof (intro exI allI impI)
           fix m n
           assume as: "max M N \<le>m" "max M N\<le>n"
           have "dist (f m x) (f n x) \<le>
@@ -1968,7 +1906,7 @@
             unfolding dist_norm
             by (rule norm_triangle_sub)
           also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
-            using N[rule_format,OF _ _ \<open>x\<in>s\<close> \<open>x0\<in>s\<close>, of m n] and as and False
+            using N[rule_format,OF _ _ \<open>x\<in>S\<close> \<open>x0\<in>S\<close>, of m n] and as and False
             by auto
           also have "\<dots> < e / 2 + e / 2"
             apply (rule add_strict_right_mono)
@@ -1982,27 +1920,24 @@
       qed
     qed
   qed
-  then obtain g where g: "\<forall>x\<in>s. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" ..
-  have lem2: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)"
+  then obtain g where g: "\<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" ..
+  have lem2: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)"
   proof (rule, rule)
     fix e :: real
     assume *: "e > 0"
     obtain N where
-      N: "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
+      N: "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
       using lem1 * by blast
-    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
+    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
       apply (rule_tac x=N in exI)
     proof rule+
       fix n x y
-      assume as: "N \<le> n" "x \<in> s" "y \<in> s"
+      assume as: "N \<le> n" "x \<in> S" "y \<in> S"
       have "((\<lambda>m. norm (f n x - f n y - (f m x - f m y))) \<longlongrightarrow> norm (f n x - f n y - (g x - g y))) sequentially"
         by (intro tendsto_intros g[rule_format] as)
       moreover have "eventually (\<lambda>m. norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)) sequentially"
         unfolding eventually_sequentially
-        apply (rule_tac x=N in exI)
-        apply rule
-        apply rule
-      proof -
+      proof (intro exI allI impI)
         fix m
         assume "N \<le> m"
         then show "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
@@ -2013,11 +1948,11 @@
         by (simp add: tendsto_upperbound)
     qed
   qed
-  have "\<forall>x\<in>s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g' x) (at x within s)"
+  have "\<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g' x) (at x within S)"
     unfolding has_derivative_within_alt2
   proof (intro ballI conjI)
     fix x
-    assume "x \<in> s"
+    assume "x \<in> S"
     then show "((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
       by (simp add: g)
     have lem3: "\<forall>u. ((\<lambda>n. f' n x u) \<longlongrightarrow> g' x u) sequentially"
@@ -2030,7 +1965,7 @@
       proof (cases "u = 0")
         case True
         have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e * norm u) sequentially"
-          using assms(3)[folded eventually_sequentially] and \<open>0 < e\<close> and \<open>x \<in> s\<close>
+          using assms(3)[folded eventually_sequentially] and \<open>0 < e\<close> and \<open>x \<in> S\<close>
           by (fast elim: eventually_mono)
         then show ?thesis
           using \<open>u = 0\<close> and \<open>0 < e\<close> by (auto elim: eventually_mono)
@@ -2038,7 +1973,7 @@
         case False
         with \<open>0 < e\<close> have "0 < e / norm u" by simp
         then have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e / norm u * norm u) sequentially"
-          using assms(3)[folded eventually_sequentially] and \<open>x \<in> s\<close>
+          using assms(3)[folded eventually_sequentially] and \<open>x \<in> S\<close>
           by (fast elim: eventually_mono)
         then show ?thesis
           using \<open>u \<noteq> 0\<close> by simp
@@ -2048,7 +1983,7 @@
     proof
       fix x' y z :: 'a
       fix c :: real
-      note lin = assms(2)[rule_format,OF \<open>x\<in>s\<close>,THEN has_derivative_bounded_linear]
+      note lin = assms(2)[rule_format,OF \<open>x\<in>S\<close>,THEN has_derivative_bounded_linear]
       show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
         apply (rule tendsto_unique[OF trivial_limit_sequentially])
         apply (rule lem3[rule_format])
@@ -2064,9 +1999,9 @@
         apply (rule lem3[rule_format])+
         done
       obtain N where N: "\<forall>h. norm (f' N x h - g' x h) \<le> 1 * norm h"
-        using assms(3) \<open>x \<in> s\<close> by (fast intro: zero_less_one)
+        using assms(3) \<open>x \<in> S\<close> by (fast intro: zero_less_one)
       have "bounded_linear (f' N x)"
-        using assms(2) \<open>x \<in> s\<close> by fast
+        using assms(2) \<open>x \<in> S\<close> by fast
       from bounded_linear.bounded [OF this]
       obtain K where K: "\<forall>h. norm (f' N x h) \<le> norm h * K" ..
       {
@@ -2082,36 +2017,36 @@
       }
       then show "\<exists>K. \<forall>h. norm (g' x h) \<le> norm h * K" by fast
     qed
-    show "\<forall>e>0. eventually (\<lambda>y. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)) (at x within s)"
+    show "\<forall>e>0. eventually (\<lambda>y. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)) (at x within S)"
     proof (rule, rule)
       fix e :: real
       assume "e > 0"
       then have *: "e / 3 > 0"
         by auto
-      obtain N1 where N1: "\<forall>n\<ge>N1. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e / 3 * norm h"
+      obtain N1 where N1: "\<forall>n\<ge>N1. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e / 3 * norm h"
         using assms(3) * by blast
       obtain N2 where
-          N2: "\<forall>n\<ge>N2. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e / 3 * norm (x - y)"
+          N2: "\<forall>n\<ge>N2. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e / 3 * norm (x - y)"
         using lem2 * by blast
       let ?N = "max N1 N2"
-      have "eventually (\<lambda>y. norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)) (at x within s)"
-        using assms(2)[unfolded has_derivative_within_alt2] and \<open>x \<in> s\<close> and * by fast
-      moreover have "eventually (\<lambda>y. y \<in> s) (at x within s)"
+      have "eventually (\<lambda>y. norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)) (at x within S)"
+        using assms(2)[unfolded has_derivative_within_alt2] and \<open>x \<in> S\<close> and * by fast
+      moreover have "eventually (\<lambda>y. y \<in> S) (at x within S)"
         unfolding eventually_at by (fast intro: zero_less_one)
-      ultimately show "\<forall>\<^sub>F y in at x within s. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
+      ultimately show "\<forall>\<^sub>F y in at x within S. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
       proof (rule eventually_elim2)
         fix y
-        assume "y \<in> s"
+        assume "y \<in> S"
         assume "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
         moreover have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e / 3 * norm (y - x)"
-          using N2[rule_format, OF _ \<open>y \<in> s\<close> \<open>x \<in> s\<close>]
+          using N2[rule_format, OF _ \<open>y \<in> S\<close> \<open>x \<in> S\<close>]
           by (simp add: norm_minus_commute)
         ultimately have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
           using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
           by (auto simp add: algebra_simps)
         moreover
         have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
-          using N1 \<open>x \<in> s\<close> by auto
+          using N1 \<open>x \<in> S\<close> by auto
         ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
           using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
           by (auto simp add: algebra_simps)
@@ -2125,77 +2060,63 @@
 
 lemma has_antiderivative_sequence:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
-  assumes "convex s"
-    and "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
-    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
-  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
-proof (cases "s = {}")
+  assumes "convex S"
+    and der: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
+    and no: "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+  shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
+proof (cases "S = {}")
   case False
-  then obtain a where "a \<in> s"
+  then obtain a where "a \<in> S"
     by auto
-  have *: "\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x"
+  have *: "\<And>P Q. \<exists>g. \<forall>x\<in>S. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>S. Q g x"
     by auto
   show ?thesis
     apply (rule *)
-    apply (rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
-    apply (metis assms(2) has_derivative_add_const)
-    apply (rule \<open>a \<in> s\<close>)
+    apply (rule has_derivative_sequence [OF \<open>convex S\<close> _ no, of "\<lambda>n x. f n x + (f 0 a - f n a)"])
+       apply (metis assms(2) has_derivative_add_const)
+    using \<open>a \<in> S\<close> 
     apply auto
     done
 qed auto
 
 lemma has_antiderivative_limit:
   fixes g' :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'b::banach"
-  assumes "convex s"
-    and "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s.
-      (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm (f' x h - g' x h) \<le> e * norm h)"
-  shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g' x) (at x within s)"
+  assumes "convex S"
+    and "\<And>e. e>0 \<Longrightarrow> \<exists>f f'. \<forall>x\<in>S.
+           (f has_derivative (f' x)) (at x within S) \<and> (\<forall>h. norm (f' x h - g' x h) \<le> e * norm h)"
+  shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
 proof -
-  have *: "\<forall>n. \<exists>f f'. \<forall>x\<in>s.
-    (f has_derivative (f' x)) (at x within s) \<and>
+  have *: "\<forall>n. \<exists>f f'. \<forall>x\<in>S.
+    (f has_derivative (f' x)) (at x within S) \<and>
     (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm h)"
     by (simp add: assms(2))
   obtain f where
-    *: "\<forall>x. \<exists>f'. \<forall>xa\<in>s. (f x has_derivative f' xa) (at xa within s) \<and>
-      (\<forall>h. norm (f' xa h - g' xa h) \<le> inverse (real (Suc x)) * norm h)"
-    using *[THEN choice] ..
+    *: "\<And>x. \<exists>f'. \<forall>xa\<in>S. (f x has_derivative f' xa) (at xa within S) \<and>
+        (\<forall>h. norm (f' xa h - g' xa h) \<le> inverse (real (Suc x)) * norm h)"
+    using * by metis
   obtain f' where
-    f: "\<forall>x. \<forall>xa\<in>s. (f x has_derivative f' x xa) (at xa within s) \<and>
-      (\<forall>h. norm (f' x xa h - g' xa h) \<le> inverse (real (Suc x)) * norm h)"
-    using *[THEN choice] ..
+    f': "\<And>x. \<forall>z\<in>S. (f x has_derivative f' x z) (at z within S) \<and>
+            (\<forall>h. norm (f' x z h - g' z h) \<le> inverse (real (Suc x)) * norm h)"
+    using * by metis
   show ?thesis
-    apply (rule has_antiderivative_sequence[OF assms(1), of f f'])
-    defer
-    apply rule
-    apply rule
-  proof -
+  proof (rule has_antiderivative_sequence[OF \<open>convex S\<close>, of f f'])
     fix e :: real
     assume "e > 0"
     obtain N where N: "inverse (real (Suc N)) < e"
       using reals_Archimedean[OF \<open>e>0\<close>] ..
-    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
-      apply (rule_tac x=N in exI)
-      apply rule
-      apply rule
-      apply rule
-      apply rule
-    proof -
+    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+    proof (intro exI allI ballI impI)
       fix n x h
-      assume n: "N \<le> n" and x: "x \<in> s"
+      assume n: "N \<le> n" and x: "x \<in> S"
       have *: "inverse (real (Suc n)) \<le> e"
         apply (rule order_trans[OF _ N[THEN less_imp_le]])
         using n
         apply (auto simp add: field_simps)
         done
       show "norm (f' n x h - g' x h) \<le> e * norm h"
-        using f[rule_format,THEN conjunct2, OF x, of n, THEN spec[where x=h]]
-        apply (rule order_trans)
-        using N *
-        apply (cases "h = 0")
-        apply auto
-        done
+        by (meson "*" mult_right_mono norm_ge_zero order.trans x f')
     qed
-  qed (insert f, auto)
+  qed (use f' in auto)
 qed
 
 
@@ -2203,12 +2124,12 @@
 
 lemma has_derivative_series:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
-  assumes "convex s"
-    and "\<And>n x. x \<in> s \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within s)"
-    and "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (sum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
-    and "x \<in> s"
+  assumes "convex S"
+    and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
+    and "\<And>e. e>0 \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (sum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
+    and "x \<in> S"
     and "(\<lambda>n. f n x) sums l"
-  shows "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums (g x) \<and> (g has_derivative g' x) (at x within s)"
+  shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums (g x) \<and> (g has_derivative g' x) (at x within S)"
   unfolding sums_def
   apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
   apply (metis assms(2) has_derivative_sum)
@@ -2219,20 +2140,19 @@
 
 lemma has_field_derivative_series:
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
-  assumes "convex s"
-  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
-  assumes "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
-  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
-  shows   "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
+  assumes "convex S"
+  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
+  assumes "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
+  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
+  shows   "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
 unfolding has_field_derivative_def
 proof (rule has_derivative_series)
-  show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h"
-  proof (intro allI impI)
-    fix e :: real assume "e > 0"
-    with assms(3) obtain N where N: "\<And>n x. n \<ge> N \<Longrightarrow> x \<in> s \<Longrightarrow> norm ((\<Sum>i<n. f' i x) - g' x) < e"
+  show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" if "e > 0" for e
+  proof -
+    from that assms(3) obtain N where N: "\<And>n x. n \<ge> N \<Longrightarrow> x \<in> S \<Longrightarrow> norm ((\<Sum>i<n. f' i x) - g' x) < e"
       unfolding uniform_limit_iff eventually_at_top_linorder dist_norm by blast
     {
-      fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> s"
+      fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> S"
       have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) = norm ((\<Sum>i<n. f' i x) - g' x) * norm h"
         by (simp add: norm_mult [symmetric] ring_distribs sum_distrib_right)
       also from N[OF nx] have "norm ((\<Sum>i<n. f' i x) - g' x) \<le> e" by simp
@@ -2240,55 +2160,55 @@
         by (intro mult_right_mono) simp_all
       finally have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" .
     }
-    thus "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" by blast
+    thus "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" by blast
   qed
-qed (insert assms, auto simp: has_field_derivative_def)
+qed (use assms in \<open>auto simp: has_field_derivative_def\<close>)
 
 lemma has_field_derivative_series':
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
-  assumes "convex s"
-  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
-  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
-  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" "x \<in> interior s"
+  assumes "convex S"
+  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
+  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
+  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" "x \<in> interior S"
   shows   "summable (\<lambda>n. f n x)" "((\<lambda>x. \<Sum>n. f n x) has_field_derivative (\<Sum>n. f' n x)) (at x)"
 proof -
-  from \<open>x \<in> interior s\<close> have "x \<in> s" using interior_subset by blast
+  from \<open>x \<in> interior S\<close> have "x \<in> S" using interior_subset by blast
   define g' where [abs_def]: "g' x = (\<Sum>i. f' i x)" for x
-  from assms(3) have "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
+  from assms(3) have "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     by (simp add: uniformly_convergent_uniform_limit_iff suminf_eq_lim g'_def)
   from has_field_derivative_series[OF assms(1,2) this assms(4,5)] obtain g where g:
-    "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
-    "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
-  from g(1)[OF \<open>x \<in> s\<close>] show "summable (\<lambda>n. f n x)" by (simp add: sums_iff)
-  from g(2)[OF \<open>x \<in> s\<close>] \<open>x \<in> interior s\<close> have "(g has_field_derivative g' x) (at x)"
-    by (simp add: at_within_interior[of x s])
+    "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
+    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
+  from g(1)[OF \<open>x \<in> S\<close>] show "summable (\<lambda>n. f n x)" by (simp add: sums_iff)
+  from g(2)[OF \<open>x \<in> S\<close>] \<open>x \<in> interior S\<close> have "(g has_field_derivative g' x) (at x)"
+    by (simp add: at_within_interior[of x S])
   also have "(g has_field_derivative g' x) (at x) \<longleftrightarrow>
                 ((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)"
-    using eventually_nhds_in_nhd[OF \<open>x \<in> interior s\<close>] interior_subset[of s] g(1)
+    using eventually_nhds_in_nhd[OF \<open>x \<in> interior S\<close>] interior_subset[of S] g(1)
     by (intro DERIV_cong_ev) (auto elim!: eventually_mono simp: sums_iff)
   finally show "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)" .
 qed
 
 lemma differentiable_series:
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
-  assumes "convex s" "open s"
-  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
-  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
+  assumes "convex S" "open S"
+  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
+  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)"
 proof -
-  from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
+  from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
     unfolding uniformly_convergent_on_def by blast
-  from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
-  have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
-    by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
-  then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
-    "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
+  from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
+  have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
+    by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
+  then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
+    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
   from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
-    by (simp add: has_field_derivative_def s)
+    by (simp add: has_field_derivative_def S)
   have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
-    by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
+    by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
        (insert g, auto simp: sums_iff)
   thus "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)" unfolding differentiable_def
     by (auto simp: summable_def differentiable_def has_field_derivative_def)
@@ -2296,32 +2216,32 @@
 
 lemma differentiable_series':
   fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
-  assumes "convex s" "open s"
-  assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-  assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
-  assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
+  assumes "convex S" "open S"
+  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
+  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
   shows   "(\<lambda>x. \<Sum>n. f n x) differentiable (at x0)"
-  using differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
+  using differentiable_series[OF assms, of x0] \<open>x0 \<in> S\<close> by blast+
 
 text \<open>Considering derivative @{typ "real \<Rightarrow> 'b::real_normed_vector"} as a vector.\<close>
 
 definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
 
 lemma vector_derivative_unique_within:
-  assumes not_bot: "at x within s \<noteq> bot"
-    and f': "(f has_vector_derivative f') (at x within s)"
-    and f'': "(f has_vector_derivative f'') (at x within s)"
+  assumes not_bot: "at x within S \<noteq> bot"
+    and f': "(f has_vector_derivative f') (at x within S)"
+    and f'': "(f has_vector_derivative f'') (at x within S)"
   shows "f' = f''"
 proof -
   have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
   proof (rule frechet_derivative_unique_within)
-    show "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> s"
+    show "\<forall>i\<in>Basis. \<forall>e>0. \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> S"
     proof clarsimp
       fix e :: real assume "0 < e"
-      with islimpt_approachable_real[of x s] not_bot
-      obtain x' where "x' \<in> s" "x' \<noteq> x" "\<bar>x' - x\<bar> < e"
+      with islimpt_approachable_real[of x S] not_bot
+      obtain x' where "x' \<in> S" "x' \<noteq> x" "\<bar>x' - x\<bar> < e"
         by (auto simp add: trivial_limit_within)
-      then show "\<exists>d. d \<noteq> 0 \<and> \<bar>d\<bar> < e \<and> x + d \<in> s"
+      then show "\<exists>d. d \<noteq> 0 \<and> \<bar>d\<bar> < e \<and> x + d \<in> S"
         by (intro exI[of _ "x' - x"]) auto
     qed
   qed (insert f' f'', auto simp: has_vector_derivative_def)
@@ -2351,8 +2271,8 @@
 qed (auto simp: vector_derivative_def has_vector_derivative_def differentiable_def)
 
 lemma vector_derivative_within:
-  assumes not_bot: "at x within s \<noteq> bot" and y: "(f has_vector_derivative y) (at x within s)"
-  shows "vector_derivative f (at x within s) = y"
+  assumes not_bot: "at x within S \<noteq> bot" and y: "(f has_vector_derivative y) (at x within S)"
+  shows "vector_derivative f (at x within S) = y"
   using y
   by (intro vector_derivative_unique_within[OF not_bot vector_derivative_works[THEN iffD1] y])
      (auto simp: differentiable_def has_vector_derivative_def)
@@ -2381,11 +2301,11 @@
   by (metis has_field_derivative_def has_real_derivative)
 
 lemma has_vector_derivative_cong_ev:
-  assumes *: "eventually (\<lambda>x. x \<in> s \<longrightarrow> f x = g x) (nhds x)" "f x = g x"
-  shows "(f has_vector_derivative f') (at x within s) = (g has_vector_derivative f') (at x within s)"
+  assumes *: "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x = g x) (nhds x)" "f x = g x"
+  shows "(f has_vector_derivative f') (at x within S) = (g has_vector_derivative f') (at x within S)"
   unfolding has_vector_derivative_def has_derivative_def
   using *
-  apply (cases "at x within s \<noteq> bot")
+  apply (cases "at x within S \<noteq> bot")
   apply (intro refl conj_cong filterlim_cong)
   apply (auto simp: netlimit_within eventually_at_filter elim: eventually_mono)
   done
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Fashoda_Theorem.thy
--- a/src/HOL/Analysis/Fashoda_Theorem.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Fashoda_Theorem.thy	Wed May 02 12:48:08 2018 +0100
@@ -109,21 +109,16 @@
   have lem1: "\<forall>z::real^2. infnorm (negatex z) = infnorm z"
     unfolding negatex_def infnorm_2 vector_2 by auto
   have lem2: "\<forall>z. z \<noteq> 0 \<longrightarrow> infnorm (sqprojection z) = 1"
-    unfolding sqprojection_def
-    unfolding infnorm_mul[unfolded scalar_mult_eq_scaleR]
-    unfolding abs_inverse real_abs_infnorm
-    apply (subst infnorm_eq_0[symmetric])
-    apply auto
-    done
+    unfolding sqprojection_def infnorm_mul[unfolded scalar_mult_eq_scaleR]
+    by (simp add: real_abs_infnorm infnorm_eq_0)
   let ?F = "\<lambda>w::real^2. (f \<circ> (\<lambda>x. x$1)) w - (g \<circ> (\<lambda>x. x$2)) w"
-  have *: "\<And>i. (\<lambda>x::real^2. x $ i) ` cbox (- 1) 1 = {-1 .. 1}"
-    apply (rule set_eqI)
-    unfolding image_iff Bex_def mem_box_cart interval_cbox_cart
-    apply rule
-    defer
-    apply (rule_tac x="vec x" in exI)
-    apply auto
-    done
+  have *: "\<And>i. (\<lambda>x::real^2. x $ i) ` cbox (- 1) 1 = {-1..1}"
+  proof 
+    show "(\<lambda>x::real^2. x $ i) ` cbox (- 1) 1 \<subseteq> {-1..1}" for i
+      by (auto simp: mem_box_cart)
+    show "{-1..1} \<subseteq> (\<lambda>x::real^2. x $ i) ` cbox (- 1) 1" for i
+      by (clarsimp simp: image_iff mem_box_cart Bex_def) (metis (no_types, hide_lams) vec_component)
+  qed
   {
     fix x
     assume "x \<in> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w) ` (cbox (- 1) (1::real^2))"
@@ -136,37 +131,19 @@
       unfolding mem_box_cart atLeastAtMost_iff
       by auto
   } note x0 = this
-  have 21: "\<And>i::2. i \<noteq> 1 \<Longrightarrow> i = 2"
-    using UNIV_2 by auto
   have 1: "box (- 1) (1::real^2) \<noteq> {}"
     unfolding interval_eq_empty_cart by auto
-  have 2: "continuous_on (cbox (- 1) 1) (negatex \<circ> sqprojection \<circ> ?F)"
+  have "negatex (x + y) $ i = (negatex x + negatex y) $ i \<and> negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
+    for i x y c
+    using exhaust_2 [of i] by (auto simp: negatex_def)
+  then have "bounded_linear negatex"
+    by (simp add: bounded_linearI' vec_eq_iff)
+  then have 2: "continuous_on (cbox (- 1) 1) (negatex \<circ> sqprojection \<circ> ?F)"
     apply (intro continuous_intros continuous_on_component)
-    unfolding *
-    apply (rule assms)+
-    apply (subst sqprojection_def)
-    apply (intro continuous_intros)
-    apply (simp add: infnorm_eq_0 x0)
-    apply (rule linear_continuous_on)
-  proof -
-    show "bounded_linear negatex"
-      apply (rule bounded_linearI')
-      unfolding vec_eq_iff
-    proof (rule_tac[!] allI)
-      fix i :: 2
-      fix x y :: "real^2"
-      fix c :: real
-      show "negatex (x + y) $ i =
-        (negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
-        apply -
-        apply (case_tac[!] "i\<noteq>1")
-        prefer 3
-        apply (drule_tac[1-2] 21)
-        unfolding negatex_def
-        apply (auto simp add:vector_2)
-        done
-    qed
-  qed
+    unfolding * sqprojection_def
+    apply (intro assms continuous_intros)+
+     apply (simp_all add: infnorm_eq_0 x0 linear_continuous_on)
+    done
   have 3: "(negatex \<circ> sqprojection \<circ> ?F) ` cbox (-1) 1 \<subseteq> cbox (-1) 1"
     unfolding subset_eq
   proof (rule, goal_cases)
@@ -176,29 +153,19 @@
         "x = (negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) y"
       unfolding image_iff ..
     have "?F y \<noteq> 0"
-      apply (rule x0)
-      using y(1)
-      apply auto
-      done
+      by (rule x0) (use y in auto)
     then have *: "infnorm (sqprojection (?F y)) = 1"
       unfolding y o_def
       by - (rule lem2[rule_format])
-    have "infnorm x = 1"
+    have inf1: "infnorm x = 1"
       unfolding *[symmetric] y o_def
       by (rule lem1[rule_format])
-    then show "x \<in> cbox (-1) 1"
+    show "x \<in> cbox (-1) 1"
       unfolding mem_box_cart interval_cbox_cart infnorm_2
-      apply -
-      apply rule
-    proof -
+    proof 
       fix i
-      assume "max \<bar>x $ 1\<bar> \<bar>x $ 2\<bar> = 1"
-      then show "(- 1) $ i \<le> x $ i \<and> x $ i \<le> 1 $ i"
-        apply (cases "i = 1")
-        defer
-        apply (drule 21)
-        apply auto
-        done
+      show "(- 1) $ i \<le> x $ i \<and> x $ i \<le> 1 $ i"
+        using exhaust_2 [of i] inf1 by (auto simp: infnorm_2)
     qed
   qed
   obtain x :: "real^2" where x:
@@ -211,10 +178,7 @@
     apply blast
     done
   have "?F x \<noteq> 0"
-    apply (rule x0)
-    using x(1)
-    apply auto
-    done
+    by (rule x0) (use x in auto)
   then have *: "infnorm (sqprojection (?F x)) = 1"
     unfolding o_def
     by (rule lem2[rule_format])
@@ -223,109 +187,73 @@
     unfolding *[symmetric] o_def
     apply (rule lem1[rule_format])
     done
-  have "\<forall>x i. x \<noteq> 0 \<longrightarrow> (0 < (sqprojection x)$i \<longleftrightarrow> 0 < x$i)"
-    and "\<forall>x i. x \<noteq> 0 \<longrightarrow> ((sqprojection x)$i < 0 \<longleftrightarrow> x$i < 0)"
-    apply -
-    apply (rule_tac[!] allI impI)+
+  have iff: "0 < sqprojection x$i \<longleftrightarrow> 0 < x$i" "sqprojection x$i < 0 \<longleftrightarrow> x$i < 0" if "x \<noteq> 0" for x i
   proof -
-    fix x :: "real^2"
-    fix i :: 2
-    assume x: "x \<noteq> 0"
     have "inverse (infnorm x) > 0"
-      using x[unfolded infnorm_pos_lt[symmetric]] by auto
+      by (simp add: infnorm_pos_lt that)
     then show "(0 < sqprojection x $ i) = (0 < x $ i)"
       and "(sqprojection x $ i < 0) = (x $ i < 0)"
       unfolding sqprojection_def vector_component_simps vector_scaleR_component real_scaleR_def
       unfolding zero_less_mult_iff mult_less_0_iff
       by (auto simp add: field_simps)
   qed
-  note lem3 = this[rule_format]
   have x1: "x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}"
     using x(1) unfolding mem_box_cart by auto
   then have nz: "f (x $ 1) - g (x $ 2) \<noteq> 0"
-    unfolding right_minus_eq
-    apply -
-    apply (rule as)
-    apply auto
-    done
-  have "x $ 1 = -1 \<or> x $ 1 = 1 \<or> x $ 2 = -1 \<or> x $ 2 = 1"
+    using as by auto
+  consider "x $ 1 = -1" | "x $ 1 = 1" | "x $ 2 = -1" | "x $ 2 = 1"
     using nx unfolding infnorm_eq_1_2 by auto
   then show False
-  proof -
-    fix P Q R S
-    presume "P \<or> Q \<or> R \<or> S"
-      and "P \<Longrightarrow> False"
-      and "Q \<Longrightarrow> False"
-      and "R \<Longrightarrow> False"
-      and "S \<Longrightarrow> False"
-    then show False by auto
-  next
-    assume as: "x$1 = 1"
-    then have *: "f (x $ 1) $ 1 = 1"
-      using assms(6) by auto
-    have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
-      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
-      unfolding as negatex_def vector_2
-      by auto
-    moreover
-    from x1 have "g (x $ 2) \<in> cbox (-1) 1"
-      using assms(2) by blast
-    ultimately show False
-      unfolding lem3[OF nz] vector_component_simps * mem_box_cart
-      using not_le by auto
-  next
-    assume as: "x$1 = -1"
+  proof cases
+    case 1
     then have *: "f (x $ 1) $ 1 = - 1"
       using assms(5) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
       using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
-      unfolding as negatex_def vector_2
-      by auto
+      by (auto simp: negatex_def 1)
     moreover
     from x1 have "g (x $ 2) \<in> cbox (-1) 1"
       using assms(2) by blast
     ultimately show False
-      unfolding lem3[OF nz] vector_component_simps * mem_box_cart
+      unfolding iff[OF nz] vector_component_simps * mem_box_cart
       using not_le by auto
   next
-    assume as: "x$2 = 1"
+    case 2
+    then have *: "f (x $ 1) $ 1 = 1"
+      using assms(6) by auto
+    have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]] 2
+      by (auto simp: negatex_def)
+    moreover have "g (x $ 2) \<in> cbox (-1) 1"
+      using assms(2) x1 by blast
+    ultimately show False
+      unfolding iff[OF nz] vector_component_simps * mem_box_cart
+      using not_le by auto
+  next
+    case 3
+    then have *: "g (x $ 2) $ 2 = - 1"
+      using assms(7) by auto
+    have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] 3 by (auto simp: negatex_def)
+    moreover
+    from x1 have "f (x $ 1) \<in> cbox (-1) 1"
+      using assms(1) by blast
+    ultimately show False
+      unfolding iff[OF nz] vector_component_simps * mem_box_cart
+      by (erule_tac x=2 in allE) auto
+  next
+    case 4
     then have *: "g (x $ 2) $ 2 = 1"
       using assms(8) by auto
     have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
-      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
-      unfolding as negatex_def vector_2
-      by auto
+      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] 4 by (auto simp: negatex_def)
     moreover
     from x1 have "f (x $ 1) \<in> cbox (-1) 1"
-      apply -
-      apply (rule assms(1)[unfolded subset_eq,rule_format])
-      apply auto
-      done
+      using assms(1) by blast
     ultimately show False
-      unfolding lem3[OF nz] vector_component_simps * mem_box_cart
-      apply (erule_tac x=2 in allE)
-      apply auto
-      done
-  next
-    assume as: "x$2 = -1"
-    then have *: "g (x $ 2) $ 2 = - 1"
-      using assms(7) by auto
-    have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
-      using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
-      unfolding as negatex_def vector_2
-      by auto
-    moreover
-    from x1 have "f (x $ 1) \<in> cbox (-1) 1"
-      apply -
-      apply (rule assms(1)[unfolded subset_eq,rule_format])
-      apply auto
-      done
-    ultimately show False
-      unfolding lem3[OF nz] vector_component_simps * mem_box_cart
-      apply (erule_tac x=2 in allE)
-      apply auto
-      done
-  qed auto
+      unfolding iff[OF nz] vector_component_simps * mem_box_cart
+      by (erule_tac x=2 in allE) auto
+  qed 
 qed
 
 lemma fashoda_unit_path:
@@ -533,14 +461,12 @@
     using as
     by auto
   show thesis
-    apply (rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
-    apply (subst zf)
-    defer
-    apply (subst zg)
-    unfolding o_def interval_bij_bij_cart[OF *] path_image_def
-    using zf(1) zg(1)
-    apply auto
-    done
+  proof (rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
+    show "interval_bij (- 1, 1) (a, b) z \<in> path_image f"
+      using zf by (simp add: interval_bij_bij_cart[OF *] path_image_def)
+    show "interval_bij (- 1, 1) (a, b) z \<in> path_image g"
+      using zg by (simp add: interval_bij_bij_cart[OF *] path_image_def)
+  qed
 qed
 
 
@@ -714,8 +640,8 @@
   fixes a :: "real^2"
   assumes "path f"
     and "path g"
-    and "path_image f \<subseteq> cbox a b"
-    and "path_image g \<subseteq> cbox a b"
+    and paf: "path_image f \<subseteq> cbox a b"
+    and pag: "path_image g \<subseteq> cbox a b"
     and "(pathstart f)$2 = a$2"
     and "(pathfinish f)$2 = a$2"
     and "(pathstart g)$2 = a$2"
@@ -776,30 +702,9 @@
   proof -
     show "path ?P1" and "path ?P2"
       using assms by auto
-    have "path_image ?P1 \<subseteq> cbox ?a ?b"
-      unfolding P1P2 path_image_linepath
-      apply (rule Un_least)+
-      defer 3
-      apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format])
-      unfolding mem_box_cart forall_2 vector_2
-      using ab startfin abab assms(3)
-      using assms(9-)
-      unfolding assms
-      apply (auto simp add: field_simps box_def)
-      done
-    then show "path_image ?P1 \<subseteq> cbox ?a ?b" .
-    have "path_image ?P2 \<subseteq> cbox ?a ?b"
-      unfolding P1P2 path_image_linepath
-      apply (rule Un_least)+
-      defer 2
-      apply (rule_tac[1-4] convex_box(1)[unfolded convex_contains_segment,rule_format])
-      unfolding mem_box_cart forall_2 vector_2
-      using ab startfin abab assms(4)
-      using assms(9-)
-      unfolding assms
-      apply (auto simp add: field_simps box_def)
-      done
-    then show "path_image ?P2 \<subseteq> cbox ?a ?b" .
+    show "path_image ?P1 \<subseteq> cbox ?a ?b" "path_image ?P2 \<subseteq> cbox ?a ?b"
+      unfolding P1P2 path_image_linepath using startfin paf pag
+      by (auto simp: mem_box_cart segment_horizontal segment_vertical forall_2)
     show "a $ 1 - 2 = a $ 1 - 2"
       and "b $ 1 + 2 = b $ 1 + 2"
       and "pathstart g $ 2 - 3 = a $ 2 - 3"
@@ -808,8 +713,7 @@
   qed
   note z=this[unfolded P1P2 path_image_linepath]
   show thesis
-    apply (rule that[of z])
-  proof -
+  proof (rule that[of z])
     have "(z \<in> closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) \<or>
       z \<in> closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) \<or>
       z \<in> closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) \<or>
@@ -838,32 +742,26 @@
         using prems(2) using assms ab
         by (auto simp add: field_simps)
       ultimately have *: "z$2 = a$2 - 2"
-        using prems(1)
-        by auto
+        using prems(1) by auto
       have "z$1 \<noteq> pathfinish g$1"
-        using prems(2)
-        using assms ab
+        using prems(2) assms ab
         by (auto simp add: field_simps *)
       moreover have "pathstart g \<in> cbox a b"
         using assms(4) pathstart_in_path_image[of g]
         by auto
       note this[unfolded mem_box_cart forall_2]
       then have "z$1 \<noteq> pathstart g$1"
-        using prems(1)
-        using assms ab
+        using prems(1) assms ab
         by (auto simp add: field_simps *)
       ultimately have "a $ 2 - 1 \<le> z $ 2 \<and> z $ 2 \<le> b $ 2 + 3 \<or> b $ 2 + 3 \<le> z $ 2 \<and> z $ 2 \<le> a $ 2 - 1"
-        using prems(2)
-        unfolding * assms
-        by (auto simp add: field_simps)
+        using prems(2)  unfolding * assms by (auto simp add: field_simps)
       then show False
         unfolding * using ab by auto
     qed
     then have "z \<in> path_image f \<or> z \<in> path_image g"
       using z unfolding Un_iff by blast
     then have z': "z \<in> cbox a b"
-      using assms(3-4)
-      by auto
+      using assms(3-4) by auto
     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow>
       z = pathstart f \<or> z = pathfinish f"
       unfolding vec_eq_iff forall_2 assms
@@ -871,11 +769,7 @@
     with z' show "z \<in> path_image f"
       using z(1)
       unfolding Un_iff mem_box_cart forall_2
-      apply -
-      apply (simp only: segment_vertical segment_horizontal vector_2)
-      unfolding assms
-      apply auto
-      done
+      by (simp only: segment_vertical segment_horizontal vector_2) (auto simp: assms)
     have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow>
       z = pathstart g \<or> z = pathfinish g"
       unfolding vec_eq_iff forall_2 assms
@@ -883,10 +777,7 @@
     with z' show "z \<in> path_image g"
       using z(2)
       unfolding Un_iff mem_box_cart forall_2
-      apply (simp only: segment_vertical segment_horizontal vector_2)
-      unfolding assms
-      apply auto
-      done
+      by (simp only: segment_vertical segment_horizontal vector_2) (auto simp: assms)
   qed
 qed
 
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed May 02 12:48:08 2018 +0100
@@ -128,20 +128,13 @@
   assumes "k \<in> Basis"
   shows "content (cbox a b) = content(cbox a b \<inter> {x. x\<bullet>k \<le> c}) + content(cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
   \<comment> \<open>Prove using measure theory\<close>
-proof cases
+proof (cases "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i")
+  case True
+  have 1: "\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
+    by (simp add: if_distrib prod.delta_remove assms)
   note simps = interval_split[OF assms] content_cbox_cases
-  have *: "Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
-    using assms by auto
-  have *: "\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
-    "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
-    apply (subst *(1))
-    defer
-    apply (subst *(1))
-    unfolding prod.insert[OF *(2-)]
-    apply auto
-    done
-  assume as: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
-  moreover
+  have 2: "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
+    by (metis (no_types, lifting) assms finite_Basis mult.commute prod.remove)
   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
     x * (b\<bullet>k - a\<bullet>k) = x * (max (a \<bullet> k) c - a \<bullet> k) + x * (b \<bullet> k - max (a \<bullet> k) c)"
     by  (auto simp add: field_simps)
@@ -152,17 +145,12 @@
       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
     by (auto intro!: prod.cong)
   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
-    unfolding not_le
-    using as[unfolded ,rule_format,of k] assms
-    by auto
+    unfolding not_le using True assms by auto
   ultimately show ?thesis
-    using assms
-    unfolding simps **
-    unfolding *(1)[of "\<lambda>i x. b\<bullet>i - x"] *(1)[of "\<lambda>i x. x - a\<bullet>i"]
-    unfolding *(2)
+    using assms unfolding simps ** 1[of "\<lambda>i x. b\<bullet>i - x"] 1[of "\<lambda>i x. x - a\<bullet>i"] 2
     by auto
 next
-  assume "\<not> (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
+  case False
   then have "cbox a b = {}"
     unfolding box_eq_empty by (auto simp: not_le)
   then show ?thesis
@@ -2585,24 +2573,22 @@
       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
 proof (cases "content (cbox a b) = 0")
   case True
-  show ?thesis
+  have "\<And>e p. p tagged_division_of cbox a b \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) \<le> e * content (cbox a b)"
+    unfolding sum_content_null[OF True] True by force
+  moreover have "i = 0" 
+    if "\<And>e. e > 0 \<Longrightarrow> \<exists>d. gauge d \<and>
+              (\<forall>p. p tagged_division_of cbox a b \<and>
+                   d fine p \<longrightarrow>
+                   norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content (cbox a b))"
+    using that [of 1]
+    by (force simp add: True sum_content_null[OF True] intro: fine_division_exists[of _ a b])
+  ultimately show ?thesis
     unfolding has_integral_null_eq[OF True]
-    apply safe
-    apply (rule, rule, rule gauge_trivial, safe)
-    unfolding sum_content_null[OF True] True
-    defer
-    apply (erule_tac x=1 in allE)
-    apply safe
-    defer
-    apply (rule fine_division_exists[of _ a b])
-    apply assumption
-    apply (erule_tac x=p in allE)
-    unfolding sum_content_null[OF True]
-    apply auto
-    done
+    by (force simp add: )
 next
   case False
-  note F = this[unfolded content_lt_nz[symmetric]]
+  then have F: "0 < content (cbox a b)"
+    using zero_less_measure_iff by blast
   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
     (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
   show ?thesis
@@ -2610,28 +2596,18 @@
   proof safe
     fix e :: real
     assume e: "e > 0"
-    {
-      assume "\<forall>e>0. ?P e (<)"
+    { assume "\<forall>e>0. ?P e (<)"
       then show "?P (e * content (cbox a b)) (\<le>)"
-        apply (erule_tac x="e * content (cbox a b)" in allE)
-        apply (erule impE)
-        defer
-        apply (erule exE,rule_tac x=d in exI)
-        using F e
-        apply (auto simp add:field_simps)
-        done
-    }
-    {
-      assume "\<forall>e>0. ?P (e * content (cbox a b)) (\<le>)"
+        apply (rule allE [where x="e * content (cbox a b)"])
+        apply (elim impE ex_forward conj_forward)
+        using F e apply (auto simp add: algebra_simps)
+        done }
+    { assume "\<forall>e>0. ?P (e * content (cbox a b)) (\<le>)"
       then show "?P e (<)"
-        apply (erule_tac x="e/2 / content (cbox a b)" in allE)
-        apply (erule impE)
-        defer
-        apply (erule exE,rule_tac x=d in exI)
-        using F e
-        apply (auto simp add: field_simps)
-        done
-    }
+        apply (rule allE [where x="e/2 / content (cbox a b)"])
+        apply (elim impE ex_forward conj_forward)
+        using F e apply (auto simp add: algebra_simps)
+        done }
   qed
 qed
 
@@ -2995,19 +2971,22 @@
 
 lemma integrable_subinterval:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
-  assumes "f integrable_on cbox a b"
-    and "cbox c d \<subseteq> cbox a b"
+  assumes f: "f integrable_on cbox a b"
+    and cd: "cbox c d \<subseteq> cbox a b"
   shows "f integrable_on cbox c d"
 proof -
   interpret operative conj True "\<lambda>i. f integrable_on i"
     using order_refl by (rule operative_integrableI)
   show ?thesis
-    apply (cases "cbox c d = {}")
-     defer
-     apply (rule partial_division_extend_1[OF assms(2)],assumption)
-    using division [symmetric] assms(1)
-     apply (auto simp: comm_monoid_set_F_and)
-    done
+  proof (cases "cbox c d = {}")
+    case True
+    then show ?thesis
+      using division [symmetric] f by (auto simp: comm_monoid_set_F_and)
+  next
+    case False
+    then show ?thesis
+      by (metis cd comm_monoid_set_F_and division division_of_finite f partial_division_extend_1)
+  qed
 qed
 
 lemma integrable_subinterval_real:
@@ -4261,31 +4240,29 @@
 lemma has_derivative_zero_unique_strong_interval:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "finite k"
-    and "continuous_on {a..b} f"
+    and contf: "continuous_on {a..b} f"
     and "f a = y"
-    and "\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
+    and fder: "\<And>x. x \<in> {a..b} - k \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within {a..b})"
+    and x: "x \<in> {a..b}"
   shows "f x = y"
 proof -
-  have ab: "a \<le> b"
+  have "a \<le> b" "a \<le> x"
     using assms by auto
-  have *: "a \<le> x"
-    using assms(5) by auto
   have "((\<lambda>x. 0::'a) has_integral f x - f a) {a..x}"
-    apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
-    apply (rule continuous_on_subset[OF assms(2)])
-    defer
-    apply safe
-    unfolding has_vector_derivative_def
-    apply (subst has_derivative_within_open[symmetric])
-    apply assumption
-    apply (rule open_greaterThanLessThan)
-    apply (rule has_derivative_within_subset[where s="{a..b}"])
-    using assms(4) assms(5)
-    apply (auto simp: mem_box)
-    done
-  note this[unfolded *]
-  note has_integral_unique[OF has_integral_0 this]
-  then show ?thesis
+  proof (rule fundamental_theorem_of_calculus_interior_strong[OF \<open>finite k\<close> \<open>a \<le> x\<close>]; clarify?)
+    have "{a..x} \<subseteq> {a..b}"
+      using x by auto
+    then show "continuous_on {a..x} f"
+      by (rule continuous_on_subset[OF contf])
+    show "(f has_vector_derivative 0) (at z)" if z: "z \<in> {a<..<x}" and notin: "z \<notin> k" for z
+      unfolding has_vector_derivative_def
+    proof (simp add: has_derivative_within_open[OF z, symmetric])
+      show "(f has_derivative (\<lambda>x. 0)) (at z within {a<..<x})"
+        by (rule has_derivative_within_subset [OF fder]) (use x z notin in auto)
+    qed
+  qed
+  from has_integral_unique[OF has_integral_0 this]
+  show ?thesis
     unfolding assms by auto
 qed
 
@@ -4297,7 +4274,7 @@
   assumes "convex S" "finite K"
     and contf: "continuous_on S f"
     and "c \<in> S" "f c = y"
-    and derf: "\<And>x. x \<in> (S - K) \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
+    and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof (cases "x = c")
@@ -4307,8 +4284,10 @@
   case False
   let ?\<phi> = "\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x"
   have contf': "continuous_on {0 ..1} (f \<circ> ?\<phi>)"
-    apply (rule continuous_intros continuous_on_subset[OF contf])+
-    using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> by (auto simp add: convex_alt algebra_simps)
+  proof (rule continuous_intros continuous_on_subset[OF contf])+
+    show "(\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x) ` {0..1} \<subseteq> S"
+      using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> by (auto simp add: convex_alt algebra_simps)
+  qed
   have "t = u" if "?\<phi> t = ?\<phi> u" for t u
   proof -
     from that have "(t - u) *\<^sub>R x = (t - u) *\<^sub>R c"
@@ -4353,7 +4332,7 @@
     and contf: "continuous_on S f"
     and "c \<in> S"
     and "f c = y"
-    and derf: "\<And>x. x \<in> (S - K) \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
+    and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof -
@@ -4486,10 +4465,7 @@
   note has_integral_restrict_open_subinterval[OF assms]
   note * = has_integral_spike[OF negligible_frontier_interval _ this]
   show ?thesis
-    apply (rule *[of c d])
-    using box_subset_cbox[of c d]
-    apply auto
-    done
+    by (rule *[of c d]) (use box_subset_cbox[of c d] in auto)
 qed
 
 lemma has_integral_restrict_closed_subintervals_eq:
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Linear_Algebra.thy
--- a/src/HOL/Analysis/Linear_Algebra.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Linear_Algebra.thy	Wed May 02 12:48:08 2018 +0100
@@ -27,13 +27,6 @@
   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
 qed
 
-lemma bounded_linearI:
-  assumes "\<And>x y. f (x + y) = f x + f y"
-    and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
-    and "\<And>x. norm (f x) \<le> norm x * K"
-  shows "bounded_linear f"
-  using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
-
 subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
 
 definition%important hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "hull" 75)
@@ -72,7 +65,7 @@
 lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
   unfolding hull_def by auto
 
-lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
+lemma hull_induct: "\<lbrakk>a \<in> Q hull S; \<And>x. x\<in> S \<Longrightarrow> P x; Q {x. P x}\<rbrakk> \<Longrightarrow> P a"
   using hull_minimal[of S "{x. P x}" Q]
   by (auto simp add: subset_eq)
 
@@ -339,20 +332,12 @@
   unfolding dependent_def by auto
 
 lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
-  apply (clarsimp simp add: dependent_def span_mono)
-  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
-  apply force
-  apply (rule span_mono)
-  apply auto
-  done
+  unfolding dependent_def span_mono
+  by (metis insert_Diff local.span_mono subsetCE subset_insert_iff) 
 
 lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
   by (metis order_antisym span_def hull_minimal)
 
-lemma (in real_vector) span_induct':
-  "\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x"
-  unfolding span_def by (rule hull_induct) auto
-
 inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
 where
   span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
@@ -1063,19 +1048,19 @@
   done
 
 lemma linear_eq_0_span:
-  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
-  shows "\<forall>x \<in> span B. f x = 0"
-  using f0 subspace_kernel[OF lf]
-  by (rule span_induct')
-
-lemma linear_eq_0: "linear f \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = 0 \<Longrightarrow> \<forall>x\<in>S. f x = 0"
-  using linear_eq_0_span[of f B] by auto
-
-lemma linear_eq_span:  "linear f \<Longrightarrow> linear g \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x \<in> span B. f x = g x"
-  using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub)
-
-lemma linear_eq: "linear f \<Longrightarrow> linear g \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x\<in>S. f x = g x"
-  using linear_eq_span[of f g B] by auto
+  assumes x: "x \<in> span B" and lf: "linear f" and f0: "\<And>x. x\<in>B \<Longrightarrow> f x = 0"
+  shows "f x = 0"
+  using x f0 subspace_kernel[OF lf] span_induct
+  by blast
+
+lemma linear_eq_0: "\<lbrakk>x \<in> S; linear f; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = 0\<rbrakk> \<Longrightarrow> f x = 0"
+  using linear_eq_0_span[of x B f] by auto
+
+lemma linear_eq_span: "\<lbrakk>x \<in> span B; linear f; linear g; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
+  using linear_eq_0_span[of x B "\<lambda>x. f x - g x"]  by (auto simp: linear_compose_sub)
+
+lemma linear_eq: "\<lbrakk>x \<in> S; linear f; linear g; S \<subseteq> span B; \<And>x. x\<in>B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> f x = g x"
+  using linear_eq_span[of _ B f g] by auto
 
 text \<open>The degenerate case of the Exchange Lemma.\<close>
 
@@ -1174,13 +1159,13 @@
   have fB: "inj_on f B"
     using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
 
-  have "\<forall>x\<in>span B. g (f x) = x"
-  proof (intro linear_eq_span)
+  have "g (f x) = x" if "x \<in> span B" for x
+  proof (rule linear_eq_span)
     show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
       using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
-    show "\<forall>x \<in> B. g (f x) = x"
-      using g fi \<open>span S = span B\<close> by (auto simp: fB)
-  qed
+    show "g (f x) = x" if "x \<in> B" for x
+      using g fi \<open>span S = span B\<close>   by (simp add: fB that)
+  qed (rule that)
   moreover
   have "inv_into B f ` f ` B \<subseteq> B"
     by (auto simp: fB)
@@ -1210,8 +1195,7 @@
   ultimately have "\<forall>x\<in>B. f (g x) = x"
     by auto
   then have "\<forall>x\<in>span B. f (g x) = x"
-    using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>]
-    by (intro linear_eq_span) (auto simp: id_def comp_def)
+    using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>] linear_eq_span by fastforce
   moreover have "inv_into (span S) f ` B \<subseteq> span S"
     using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
   then have "range g \<subseteq> span S"
@@ -2457,8 +2441,8 @@
     done
   with a have a0:"?a  \<noteq> 0"
     by auto
-  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
-  proof (rule span_induct')
+  have "?a \<bullet> x = 0" if "x\<in>span B" for x
+  proof (rule span_induct [OF that])
     show "subspace {x. ?a \<bullet> x = 0}"
       by (auto simp add: subspace_def inner_add)
   next
@@ -2481,9 +2465,9 @@
           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
         done
     }
-    then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
-      by blast
-  qed
+    then show "?a \<bullet> x = 0" if "x \<in> B" for x
+      using that by blast
+    qed
   with a0 show ?thesis
     unfolding sSB by (auto intro: exI[where x="?a"])
 qed
@@ -2523,35 +2507,27 @@
     and fx: "f x = 0"
   shows "x = 0"
   using fB ifB fi xsB fx
-proof (induct arbitrary: x rule: finite_induct[OF fB])
-  case 1
+proof (induction B arbitrary: x rule: finite_induct)
+  case empty
   then show ?case by auto
 next
-  case (2 a b x)
-  have fb: "finite b" using "2.prems" by simp
+  case (insert a b x)
   have th0: "f ` b \<subseteq> f ` (insert a b)"
-    apply (rule image_mono)
-    apply blast
-    done
-  from independent_mono[ OF "2.prems"(2) th0]
-  have ifb: "independent (f ` b)"  .
+    by (simp add: subset_insertI)
+  have ifb: "independent (f ` b)"
+    using independent_mono insert.prems(1) th0 by blast  
   have fib: "inj_on f b"
-    apply (rule subset_inj_on [OF "2.prems"(3)])
-    apply blast
-    done
-  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
+    using insert.prems(2) by blast
+  from span_breakdown[of a "insert a b", simplified, OF insert.prems(3)]
   obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
     by blast
   have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
     unfolding span_linear_image[OF lf]
-    apply (rule imageI)
-    using k span_mono[of "b - {a}" b]
-    apply blast
-    done
+    using "insert.hyps"(2) k by auto
   then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
     by (simp add: linear_diff[OF lf] linear_cmul[OF lf])
   then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
-    using "2.prems"(5) by simp
+    using insert.prems(4) by simp
   have xsb: "x \<in> span b"
   proof (cases "k = 0")
     case True
@@ -2560,19 +2536,18 @@
       by blast
   next
     case False
-    with span_mul[OF th, of "- 1/ k"]
-    have th1: "f a \<in> span (f ` b)"
-      by auto
-    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
-    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
-    from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
-    have "f a \<notin> span (f ` b)" using tha
-      using "2.hyps"(2)
-      "2.prems"(3) by auto
-    with th1 have False by blast
+    from inj_on_image_set_diff[OF insert.prems(2), of "insert a b " "{a}", symmetric]
+    have "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
+    then have "f a \<notin> span (f ` b)" 
+      using dependent_def insert.hyps(2) insert.prems(1) by fastforce
+    moreover have "f a \<in> span (f ` b)"
+      using False span_mul[OF th, of "- 1/ k"] by auto
+    ultimately have False
+      by blast
     then show ?thesis by blast
   qed
-  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
+  show "x = 0" 
+    using ifb fib xsb insert.IH insert.prems(4) by blast
 qed
 
 text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
@@ -2635,9 +2610,9 @@
   fixes f :: "'a::euclidean_space \<Rightarrow> _"
   assumes lf: "linear f"
     and lg: "linear g"
-    and fg: "\<forall>b\<in>Basis. f b = g b"
+    and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
   shows "f = g"
-  using linear_eq[OF lf lg, of _ Basis] fg by auto
+  using linear_eq[OF _ lf lg, of _ _ Basis] fg by auto
 
 text \<open>Similar results for bilinear functions.\<close>
 
@@ -2646,36 +2621,36 @@
     and bg: "bilinear g"
     and SB: "S \<subseteq> span B"
     and TC: "T \<subseteq> span C"
-    and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
-  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
+    and "x\<in>S" "y\<in>T"
+    and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
+  shows "f x y = g x y"
 proof -
   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
   from bf bg have sp: "subspace ?P"
     unfolding bilinear_def linear_iff subspace_def bf bg
-    by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
+    by (auto simp add: bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add 
       intro: bilinear_ladd[OF bf])
-
-  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
-    apply (rule span_induct' [OF _ sp])
-    apply (rule ballI)
-    apply (rule span_induct')
-    apply (simp add: fg)
+  have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
     apply (auto simp add: subspace_def)
     using bf bg unfolding bilinear_def linear_iff
-    apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
+    apply (auto simp add: bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add 
       intro: bilinear_ladd[OF bf])
     done
+  have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
+    apply (rule span_induct [OF that sp])
+    using fg sfg span_induct by blast
   then show ?thesis
-    using SB TC by auto
+    using SB TC assms by auto
 qed
 
 lemma bilinear_eq_stdbasis:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
   assumes bf: "bilinear f"
     and bg: "bilinear g"
-    and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
+    and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
   shows "f = g"
-  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
+  using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg
+  by blast
 
 text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close>
 
@@ -2695,13 +2670,7 @@
     apply (rule card_ge_dim_independent)
     apply blast
     apply (rule independent_injective_image[OF B(2) lf fi])
-    apply (rule order_eq_refl)
-    apply (rule sym)
-    unfolding d
-    apply (rule card_image)
-    apply (rule subset_inj_on[OF fi])
-    apply blast
-    done
+    by (metis card_image d fi inj_on_subset order_refl top_greatest)
   from th show ?thesis
     unfolding span_linear_image[OF lf] surj_def
     using B(3) by blast
@@ -2718,11 +2687,8 @@
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume h: "?lhs"
-  {
-    fix x y
-    assume x: "x \<in> S"
-    assume y: "y \<in> S"
-    assume f: "f x = f y"
+  { fix x y
+    assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
     from x fS have S0: "card S \<noteq> 0"
       by auto
     have "x = y"
@@ -2730,15 +2696,13 @@
       assume xy: "\<not> ?thesis"
       have th: "card S \<le> card (f ` (S - {y}))"
         unfolding c
-        apply (rule card_mono)
-        apply (rule finite_imageI)
-        using fS apply simp
-        using h xy x y f unfolding subset_eq image_iff
-        apply auto
-        apply (case_tac "xa = f x")
-        apply (rule bexI[where x=x])
-        apply auto
-        done
+      proof (rule card_mono)
+        show "finite (f ` (S - {y}))"
+          by (simp add: fS)
+        show "T \<subseteq> f ` (S - {y})"
+          using h xy x y f unfolding subset_eq image_iff
+          by (metis member_remove remove_def)
+      qed
       also have " \<dots> \<le> card (S - {y})"
         apply (rule card_image_le)
         using fS by simp
@@ -2751,17 +2715,13 @@
 next
   assume h: ?rhs
   have "f ` S = T"
-    apply (rule card_subset_eq[OF fT ST])
-    unfolding card_image[OF h]
-    apply (rule c)
-    done
+    by (simp add: ST c card_image card_subset_eq fT h)
   then show ?lhs by blast
 qed
 
 lemma linear_surjective_imp_injective:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
-  assumes lf: "linear f"
-    and sf: "surj f"
+  assumes lf: "linear f" and sf: "surj f"
   shows "inj f"
 proof -
   let ?U = "UNIV :: 'a set"
@@ -2770,46 +2730,29 @@
     by blast
   {
     fix x
-    assume x: "x \<in> span B"
-    assume fx: "f x = 0"
+    assume x: "x \<in> span B" and fx: "f x = 0"
     from B(2) have fB: "finite B"
       using independent_bound by auto
+    have Uspan: "UNIV \<subseteq> span (f ` B)"
+      by (simp add: B(3) lf sf spanning_surjective_image)
     have fBi: "independent (f ` B)"
-      apply (rule card_le_dim_spanning[of "f ` B" ?U])
-      apply blast
-      using sf B(3)
-      unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
-      apply blast
-      using fB apply blast
-      unfolding d[symmetric]
-      apply (rule card_image_le)
-      apply (rule fB)
-      done
+    proof (rule card_le_dim_spanning)
+      show "card (f ` B) \<le> dim ?U"
+        using card_image_le d fB by fastforce
+    qed (use fB Uspan in auto)
     have th0: "dim ?U \<le> card (f ` B)"
-      apply (rule span_card_ge_dim)
-      apply blast
-      unfolding span_linear_image[OF lf]
-      apply (rule subset_trans[where B = "f ` UNIV"])
-      using sf unfolding surj_def
-      apply blast
-      apply (rule image_mono)
-      apply (rule B(3))
-      apply (metis finite_imageI fB)
-      done
+      by (rule span_card_ge_dim) (use Uspan fB in auto)
     moreover have "card (f ` B) \<le> card B"
       by (rule card_image_le, rule fB)
     ultimately have th1: "card B = card (f ` B)"
       unfolding d by arith
     have fiB: "inj_on f B"
-      unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
-      by blast
+      by (simp add: eq_card_imp_inj_on fB th1)
     from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
     have "x = 0" by blast
   }
   then show ?thesis
-    unfolding linear_injective_0[OF lf]
-    using B(3)
-    by blast
+    unfolding linear_injective_0[OF lf] using B(3) by blast
 qed
 
 text \<open>Hence either is enough for isomorphism.\<close>
@@ -2865,9 +2808,7 @@
     assume lf: "linear f" "linear f'"
     assume f: "f \<circ> f' = id"
     from f have sf: "surj f"
-      apply (auto simp add: o_def id_def surj_def)
-      apply metis
-      done
+      by (auto simp add: o_def id_def surj_def) metis
     from linear_surjective_isomorphism[OF lf(1) sf] lf f
     have "f' \<circ> f = id"
       unfolding fun_eq_iff o_def id_def by metis
@@ -2885,18 +2826,13 @@
   shows "linear g"
 proof -
   from gf have fi: "inj f"
-    apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
-    apply metis
-    done
+    by (auto simp add: inj_on_def o_def id_def fun_eq_iff) metis
   from linear_injective_isomorphism[OF lf fi]
-  obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
+  obtain h :: "'a \<Rightarrow> 'a" where "linear h" and h: "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
     by blast
   have "h = g"
-    apply (rule ext) using gf h(2,3)
-    apply (simp add: o_def id_def fun_eq_iff)
-    apply metis
-    done
-  with h(1) show ?thesis by blast
+    by (metis gf h isomorphism_expand left_right_inverse_eq)
+  with \<open>linear h\<close> show ?thesis by blast
 qed
 
 lemma inj_linear_imp_inv_linear:
@@ -2955,28 +2891,21 @@
   by (simp add: infnorm_eq_0)
 
 lemma infnorm_neg: "infnorm (- x) = infnorm x"
-  unfolding infnorm_def
-  apply (rule cong[of "Sup" "Sup"])
-  apply blast
-  apply auto
-  done
+  unfolding infnorm_def by simp
 
 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
-proof -
-  have "y - x = - (x - y)" by simp
-  then show ?thesis
-    by (metis infnorm_neg)
-qed
-
-lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
+  by (metis infnorm_neg minus_diff_eq)
+
+lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
 proof -
-  have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
+  have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
     by arith
-  from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
-  have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
-    "infnorm y \<le> infnorm (x - y) + infnorm x"
-    by (simp_all add: field_simps infnorm_neg)
-  from th[OF ths] show ?thesis .
+  show ?thesis
+  proof (rule *)
+    from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
+    show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
+      by (simp_all add: field_simps infnorm_neg)
+  qed
 qed
 
 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
@@ -2991,8 +2920,7 @@
   unfolding infnorm_Max
 proof (safe intro!: Max_eqI)
   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
-  {
-    fix b :: 'a
+  { fix b :: 'a
     assume "b \<in> Basis"
     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
       by (simp add: abs_mult mult_left_mono)
@@ -3018,27 +2946,17 @@
 lemma norm_le_infnorm:
   fixes x :: "'a::euclidean_space"
   shows "norm x \<le> sqrt DIM('a) * infnorm x"
-proof -
-  let ?d = "DIM('a)"
-  have "real ?d \<ge> 0"
-    by simp
-  then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
-    by (auto intro: real_sqrt_pow2)
-  have th: "sqrt (real ?d) * infnorm x \<ge> 0"
+  unfolding norm_eq_sqrt_inner id_def 
+proof (rule real_le_lsqrt[OF inner_ge_zero])
+  show "sqrt DIM('a) * infnorm x \<ge> 0"
     by (simp add: zero_le_mult_iff infnorm_pos_le)
-  have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
-    unfolding power_mult_distrib d2
-    apply (subst euclidean_inner)
-    apply (subst power2_abs[symmetric])
-    apply (rule order_trans[OF sum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]])
-    apply (auto simp add: power2_eq_square[symmetric])
-    apply (subst power2_abs[symmetric])
-    apply (rule power_mono)
-    apply (auto simp: infnorm_Max)
-    done
-  from real_le_lsqrt[OF inner_ge_zero th th1]
-  show ?thesis
-    unfolding norm_eq_sqrt_inner id_def .
+  have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
+    by (metis euclidean_inner order_refl)
+  also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
+    by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
+  also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
+    by (simp add: power_mult_distrib)
+  finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
 qed
 
 lemma tendsto_infnorm [tendsto_intros]:
@@ -3048,46 +2966,30 @@
   fix r :: real
   assume "r > 0"
   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
-    by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
+    by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
 qed
 
 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
 
 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
   (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  {
-    assume h: "x = 0"
-    then have ?thesis by simp
-  }
-  moreover
-  {
-    assume h: "y = 0"
-    then have ?thesis by simp
-  }
-  moreover
-  {
-    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
-    have "?rhs \<longleftrightarrow>
+proof (cases "x=0")
+  case True
+  then show ?thesis 
+    by auto
+next
+  case False
+  from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
+  have "?rhs \<longleftrightarrow>
       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
-      using x y
-      unfolding inner_simps
-      unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq
-      apply (simp add: inner_commute)
-      apply (simp add: field_simps)
-      apply metis
-      done
-    also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
-      by (simp add: field_simps inner_commute)
-    also have "\<dots> \<longleftrightarrow> ?lhs" using x y
-      apply simp
-      apply metis
-      done
-    finally have ?thesis by blast
-  }
-  ultimately show ?thesis by blast
+    using False unfolding inner_simps
+    by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
+  also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" 
+    using False  by (simp add: field_simps inner_commute)
+  also have "\<dots> \<longleftrightarrow> ?lhs" 
+    using False by auto
+  finally show ?thesis by metis
 qed
 
 lemma norm_cauchy_schwarz_abs_eq:
@@ -3099,7 +3001,7 @@
     by arith
   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
     by simp
-  also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
+  also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
     unfolding norm_cauchy_schwarz_eq[symmetric]
     unfolding norm_minus_cancel norm_scaleR ..
   also have "\<dots> \<longleftrightarrow> ?lhs"
@@ -3111,33 +3013,21 @@
 lemma norm_triangle_eq:
   fixes x y :: "'a::real_inner"
   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-proof -
-  {
-    assume x: "x = 0 \<or> y = 0"
-    then have ?thesis
-      by (cases "x = 0") simp_all
-  }
-  moreover
-  {
-    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    then have "norm x \<noteq> 0" "norm y \<noteq> 0"
-      by simp_all
-    then have n: "norm x > 0" "norm y > 0"
-      using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
-    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
-      by algebra
-    have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
-      apply (rule th)
-      using n norm_ge_zero[of "x + y"]
-      apply arith
-      done
-    also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-      unfolding norm_cauchy_schwarz_eq[symmetric]
-      unfolding power2_norm_eq_inner inner_simps
-      by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
-    finally have ?thesis .
-  }
-  ultimately show ?thesis by blast
+proof (cases "x = 0 \<or> y = 0")
+  case True
+  then show ?thesis 
+    by force
+next
+  case False
+  then have n: "norm x > 0" "norm y > 0"
+    by auto
+  have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
+    by simp
+  also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+    unfolding norm_cauchy_schwarz_eq[symmetric]
+    unfolding power2_norm_eq_inner inner_simps
+    by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
+  finally show ?thesis .
 qed
 
 
@@ -3196,81 +3086,67 @@
 lemma collinear_2 [iff]: "collinear {x, y}"
   apply (simp add: collinear_def)
   apply (rule exI[where x="x - y"])
-  apply auto
-  apply (rule exI[where x=1], simp)
-  apply (rule exI[where x="- 1"], simp)
-  done
+  by (metis minus_diff_eq scaleR_left.minus scaleR_one)
 
 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
   (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
-  {
-    assume "x = 0 \<or> y = 0"
-    then have ?thesis
-      by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
-  }
-  moreover
-  {
-    assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
-    have ?thesis
-    proof
-      assume h: "?lhs"
-      then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
-        unfolding collinear_def by blast
-      from u[rule_format, of x 0] u[rule_format, of y 0]
-      obtain cx and cy where
-        cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
-        by auto
-      from cx x have cx0: "cx \<noteq> 0" by auto
-      from cy y have cy0: "cy \<noteq> 0" by auto
-      let ?d = "cy / cx"
-      from cx cy cx0 have "y = ?d *\<^sub>R x"
-        by simp
-      then show ?rhs using x y by blast
-    next
-      assume h: "?rhs"
-      then obtain c where c: "y = c *\<^sub>R x"
-        using x y by blast
-      show ?lhs
-        unfolding collinear_def c
-        apply (rule exI[where x=x])
-        apply auto
-        apply (rule exI[where x="- 1"], simp)
-        apply (rule exI[where x= "-c"], simp)
+proof (cases "x = 0 \<or> y = 0")
+  case True
+  then show ?thesis
+    by (auto simp: insert_commute)
+next
+  case False
+  show ?thesis 
+  proof
+    assume h: "?lhs"
+    then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
+      unfolding collinear_def by blast
+    from u[rule_format, of x 0] u[rule_format, of y 0]
+    obtain cx and cy where
+      cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
+      by auto
+    from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
+    let ?d = "cy / cx"
+    from cx cy cx0 have "y = ?d *\<^sub>R x"
+      by simp
+    then show ?rhs using False by blast
+  next
+    assume h: "?rhs"
+    then obtain c where c: "y = c *\<^sub>R x"
+      using False by blast
+    show ?lhs
+      unfolding collinear_def c
+      apply (rule exI[where x=x])
+      apply auto
+          apply (rule exI[where x="- 1"], simp)
+         apply (rule exI[where x= "-c"], simp)
         apply (rule exI[where x=1], simp)
-        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
-        apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
-        done
-    qed
-  }
-  ultimately show ?thesis by blast
+       apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
+      apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
+      done
+  qed
 qed
 
 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
-  unfolding norm_cauchy_schwarz_abs_eq
-  apply (cases "x=0", simp_all)
-  apply (cases "y=0", simp_all add: insert_commute)
-  unfolding collinear_lemma
-  apply simp
-  apply (subgoal_tac "norm x \<noteq> 0")
-  apply (subgoal_tac "norm y \<noteq> 0")
-  apply (rule iffI)
-  apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
-  apply (rule exI[where x="(1/norm x) * norm y"])
-  apply (drule sym)
-  unfolding scaleR_scaleR[symmetric]
-  apply (simp add: field_simps)
-  apply (rule exI[where x="(1/norm x) * - norm y"])
-  apply clarify
-  apply (drule sym)
-  unfolding scaleR_scaleR[symmetric]
-  apply (simp add: field_simps)
-  apply (erule exE)
-  apply (erule ssubst)
-  unfolding scaleR_scaleR
-  unfolding norm_scaleR
-  apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
-  apply (auto simp add: field_simps)
-  done
+proof (cases "x=0")
+  case True
+  then show ?thesis
+    by (auto simp: insert_commute)
+next
+  case False
+  then have nnz: "norm x \<noteq> 0"
+    by auto
+  show ?thesis
+  proof
+    assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
+    then show "collinear {0, x, y}"
+      unfolding norm_cauchy_schwarz_abs_eq collinear_lemma 
+      by (meson eq_vector_fraction_iff nnz)
+  next
+    assume "collinear {0, x, y}"
+    with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
+      unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
+  qed
+qed
 
 end
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Analysis/Starlike.thy	Wed May 02 12:48:08 2018 +0100
@@ -19,10 +19,7 @@
   where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
 
 lemma midpoint_idem [simp]: "midpoint x x = x"
-  unfolding midpoint_def
-  unfolding scaleR_right_distrib
-  unfolding scaleR_left_distrib[symmetric]
-  by auto
+  unfolding midpoint_def  by simp
 
 lemma midpoint_sym: "midpoint a b = midpoint b a"
   unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
@@ -924,61 +921,49 @@
 proof (cases "a = b")
   case True
   then show ?thesis
-    unfolding between_def split_conv
-    by (auto simp add: dist_commute)
+    by (auto simp add: between_def dist_commute)
 next
   case False
   then have Fal: "norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0"
     by auto
   have *: "\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)"
     by (auto simp add: algebra_simps)
-  show ?thesis
-    unfolding between_def split_conv closed_segment_def mem_Collect_eq
-    apply rule
-    apply (elim exE conjE)
-    apply (subst dist_triangle_eq)
+  have "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)" if "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" for u
   proof -
-    fix u
-    assume as: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
-    then have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
-      unfolding as(1) by (auto simp add:algebra_simps)
+    have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
+      unfolding that(1) by (auto simp add:algebra_simps)
     show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
-      unfolding norm_minus_commute[of x a] * using as(2,3)
+      unfolding norm_minus_commute[of x a] * using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>
       by (auto simp add: field_simps)
-  next
-    assume as: "dist a b = dist a x + dist x b"
-    have "norm (a - x) / norm (a - b) \<le> 1"
-      using Fal2 unfolding as[unfolded dist_norm] norm_ge_zero by auto
-    then show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
-      apply (rule_tac x="dist a x / dist a b" in exI)
-      unfolding dist_norm
-      apply (subst euclidean_eq_iff)
-      apply rule
-      defer
-      apply rule
-      prefer 3
-      apply rule
-    proof -
-      fix i :: 'a
-      assume i: "i \<in> Basis"
-      have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i =
-        ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
-        using Fal by (auto simp add: field_simps inner_simps)
-      also have "\<dots> = x\<bullet>i"
-        apply (rule divide_eq_imp[OF Fal])
-        unfolding as[unfolded dist_norm]
-        using as[unfolded dist_triangle_eq]
-        apply -
-        apply (subst (asm) euclidean_eq_iff)
-        using i
-        apply (erule_tac x=i in ballE)
-        apply (auto simp add: field_simps inner_simps)
-        done
-      finally show "x \<bullet> i =
-        ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
-        by auto
-    qed (insert Fal2, auto)
   qed
+  moreover have "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" if "dist a b = dist a x + dist x b" 
+  proof -
+    let ?\<beta> = "norm (a - x) / norm (a - b)"
+    show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
+    proof (intro exI conjI)
+      show "?\<beta> \<le> 1"
+        using Fal2 unfolding that[unfolded dist_norm] norm_ge_zero by auto
+      show "x = (1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b"
+      proof (subst euclidean_eq_iff; intro ballI)
+        fix i :: 'a
+        assume i: "i \<in> Basis"
+        have "((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> i 
+              = ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
+          using Fal by (auto simp add: field_simps inner_simps)
+        also have "\<dots> = x\<bullet>i"
+          apply (rule divide_eq_imp[OF Fal])
+          unfolding that[unfolded dist_norm]
+          using that[unfolded dist_triangle_eq] i
+          apply (subst (asm) euclidean_eq_iff)
+           apply (auto simp add: field_simps inner_simps)
+          done
+        finally show "x \<bullet> i = ((1 - ?\<beta>) *\<^sub>R a + (?\<beta>) *\<^sub>R b) \<bullet> i"
+          by auto
+      qed
+    qed (use Fal2 in auto)
+  qed
+  ultimately show ?thesis
+    by (force simp add: between_def closed_segment_def dist_triangle_eq)
 qed
 
 lemma between_midpoint:
@@ -990,10 +975,7 @@
     by auto
   show ?t1 ?t2
     unfolding between midpoint_def dist_norm
-    apply(rule_tac[!] *)
-    unfolding euclidean_eq_iff[where 'a='a]
-    apply (auto simp add: field_simps inner_simps)
-    done
+    by (auto simp add: field_simps inner_simps euclidean_eq_iff[where 'a='a] intro!: *)
 qed
 
 lemma between_mem_convex_hull:
@@ -1058,25 +1040,23 @@
 subsection%unimportant \<open>Shrinking towards the interior of a convex set\<close>
 
 lemma mem_interior_convex_shrink:
-  fixes s :: "'a::euclidean_space set"
-  assumes "convex s"
-    and "c \<in> interior s"
-    and "x \<in> s"
+  fixes S :: "'a::euclidean_space set"
+  assumes "convex S"
+    and "c \<in> interior S"
+    and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
-  shows "x - e *\<^sub>R (x - c) \<in> interior s"
-proof -
-  obtain d where "d > 0" and d: "ball c d \<subseteq> s"
+  shows "x - e *\<^sub>R (x - c) \<in> interior S"
+proof -
+  obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   show ?thesis
     unfolding mem_interior
-    apply (rule_tac x="e*d" in exI)
-    apply rule
-    defer
-    unfolding subset_eq Ball_def mem_ball
-  proof (rule, rule)
+  proof (intro exI subsetI conjI)
     fix y
-    assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
+    assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
+    then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
+      by simp
     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 add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     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)"
@@ -1090,7 +1070,7 @@
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
-    finally show "y \<in> s"
+    finally show "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
       apply (rule d[unfolded subset_eq,rule_format])
@@ -1102,18 +1082,18 @@
 qed
 
 lemma mem_interior_closure_convex_shrink:
-  fixes s :: "'a::euclidean_space set"
-  assumes "convex s"
-    and "c \<in> interior s"
-    and "x \<in> closure s"
+  fixes S :: "'a::euclidean_space set"
+  assumes "convex S"
+    and "c \<in> interior S"
+    and "x \<in> closure S"
     and "0 < e"
     and "e \<le> 1"
-  shows "x - e *\<^sub>R (x - c) \<in> interior s"
-proof -
-  obtain d where "d > 0" and d: "ball c d \<subseteq> s"
+  shows "x - e *\<^sub>R (x - c) \<in> interior S"
+proof -
+  obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
-  have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d"
-  proof (cases "x \<in> s")
+  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>
@@ -1122,12 +1102,12 @@
       done
   next
     case False
-    then have x: "x islimpt s"
+    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"
+      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)
@@ -1139,7 +1119,7 @@
       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)"
+      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)
@@ -1149,24 +1129,20 @@
         done
     qed
   qed
-  then obtain y where "y \<in> s" and y: "norm (y - x) * (1 - e) < e * d"
+  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 add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
-  have "z \<in> interior s"
+  have "z \<in> interior S"
     apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
     apply (auto simp add:field_simps norm_minus_commute)
     done
   then show ?thesis
     unfolding *
-    apply -
-    apply (rule mem_interior_convex_shrink)
-    using assms(1,4-5) \<open>y\<in>s\<close>
-    apply auto
-    done
+    using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
 qed
 
 lemma in_interior_closure_convex_segment:
@@ -1252,23 +1228,20 @@
 subsection%unimportant \<open>Some obvious but surprisingly hard simplex lemmas\<close>
 
 lemma simplex:
-  assumes "finite s"
-    and "0 \<notin> s"
-  shows "convex hull (insert 0 s) =
-    {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y)}"
-  unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
-  apply (rule set_eqI, rule)
-  unfolding mem_Collect_eq
-  apply (erule_tac[!] exE)
-  apply (erule_tac[!] conjE)+
-  unfolding sum_clauses(2)[OF \<open>finite s\<close>]
-  apply (rule_tac x=u in exI)
-  defer
-  apply (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u s else u x" in exI)
-  using assms(2)
-  unfolding if_smult and sum_delta_notmem[OF assms(2)]
-  apply auto
-  done
+  assumes "finite S"
+    and "0 \<notin> S"
+  shows "convex hull (insert 0 S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
+proof (simp add: convex_hull_finite set_eq_iff assms, safe)
+  fix x and u :: "'a \<Rightarrow> real"
+  assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
+  then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
+    by force
+next
+  fix x and u :: "'a \<Rightarrow> real"
+  assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
+  then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
+    by (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
+qed
 
 lemma substd_simplex:
   assumes d: "d \<subseteq> Basis"
@@ -1283,50 +1256,27 @@
     by (blast intro: finite_subset finite_Basis)
   show ?thesis
     unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
-    apply (rule set_eqI)
-    unfolding mem_Collect_eq
-    apply rule
-    apply (elim exE conjE)
-    apply (erule_tac[2] conjE)+
-  proof -
-    fix x :: "'a::euclidean_space"
-    fix u
-    assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" "(\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
-    have *: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = x\<bullet>i"
-      and "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
-      using as(3)
-      unfolding substdbasis_expansion_unique[OF assms]
-      by auto
-    then have **: "sum u ?D = sum ((\<bullet>) x) ?D"
-      apply -
-      apply (rule sum.cong)
-      using assms
-      apply auto
-      done
-    have "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum ((\<bullet>) x) ?D \<le> 1"
-    proof (rule,rule)
-      fix i :: 'a
-      assume i: "i \<in> Basis"
-      have "i \<in> d \<Longrightarrow> 0 \<le> x\<bullet>i"
-        unfolding *[rule_format,OF i,symmetric]
-         apply (rule_tac as(1)[rule_format])
-         apply auto
-         done
-      moreover have "i \<notin> d \<Longrightarrow> 0 \<le> x\<bullet>i"
-        using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close>[rule_format, OF i] by auto
-      ultimately show "0 \<le> x\<bullet>i" by auto
-    qed (insert as(2)[unfolded **], auto)
-    then show "(\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum ((\<bullet>) x) ?D \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
-      using \<open>(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)\<close> by auto
+  proof (intro set_eqI; safe)
+    fix u :: "'a \<Rightarrow> real"
+    assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" 
+    let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
+    have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
+      and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
+      using substdbasis_expansion_unique[OF assms] by blast+
+    then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
+      using assms by (auto intro!: sum.cong)
+    show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
+      using as(1) ind notind that by fastforce
+    show "sum ((\<bullet>) ?x) ?D \<le> 1"
+      using "**" as(2) by linarith
+    show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
+      using notind that by blast
   next
-    fix x :: "'a::euclidean_space"
-    assume as: "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
-    show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
-      using as d
-      unfolding substdbasis_expansion_unique[OF assms]
-      apply (rule_tac x="inner x" in exI)
-      apply auto
-      done
+    fix x 
+    assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
+    with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
+      unfolding substdbasis_expansion_unique[OF assms] 
+      by (rule_tac x="inner x" in exI) auto
   qed
 qed
 
@@ -1338,27 +1288,18 @@
 lemma interior_std_simplex:
   "interior (convex hull (insert 0 Basis)) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
-  apply (rule set_eqI)
-  unfolding mem_interior std_simplex
-  unfolding subset_eq mem_Collect_eq Ball_def mem_ball
-  unfolding Ball_def[symmetric]
-  apply rule
-  apply (elim exE conjE)
-  defer
-  apply (erule conjE)
-proof -
+  unfolding set_eq_iff mem_interior std_simplex
+proof (intro allI iffI CollectI; clarify)
   fix x :: 'a
   fix e
-  assume "e > 0" and as: "\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x\<in>Basis. 0 \<le> xa \<bullet> x) \<and> sum ((\<bullet>) xa) Basis \<le> 1"
-  show "(\<forall>xa\<in>Basis. 0 < x \<bullet> xa) \<and> sum ((\<bullet>) x) Basis < 1"
-    apply safe
-  proof -
+  assume "e > 0" and as: "ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
+  show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
+  proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     then show "0 < x \<bullet> i"
-      using as[THEN spec[where x="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close>
-      unfolding dist_norm
-      by (auto elim!: ballE[where x=i] simp: inner_simps)
+      using as[THEN subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close> 
+      by (force simp add: inner_simps)
   next
     have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
       unfolding dist_norm
@@ -1368,42 +1309,31 @@
       by (auto simp: SOME_Basis inner_Basis inner_simps)
     then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
       sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
-      apply (rule_tac sum.cong)
-      apply auto
-      done
+      by (auto simp: intro!: sum.cong)
     have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
-      unfolding * sum.distrib
-      using \<open>e > 0\<close> DIM_positive[where 'a='a]
-      apply (subst sum.delta')
-      apply (auto simp: SOME_Basis)
-      done
+      using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
     also have "\<dots> \<le> 1"
-      using **
-      apply (drule_tac as[rule_format])
-      apply auto
-      done
+      using ** as by force
     finally show "sum ((\<bullet>) x) Basis < 1" by auto
-  qed
+  qed 
 next
   fix x :: 'a
   assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
   obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
   let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
-  show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) Basis \<le> 1"
-  proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI impI allI)
+  show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
+  proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
     fix y
-    assume y: "dist x y < min (Min ((\<bullet>) x ` Basis)) ?d"
+    assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
     have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
     proof (rule sum_mono)
       fix i :: 'a
       assume i: "i \<in> Basis"
-      then have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
-        apply -
-        apply (rule le_less_trans)
-        using Basis_le_norm[OF i, of "y - x"]
-        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
-        apply (auto simp add: norm_minus_commute inner_diff_left)
-        done
+      have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
+        by (metis Basis_le_norm i inner_commute inner_diff_right)
+      also have "... < ?d"
+        using y by (simp add: dist_norm norm_minus_commute)
+      finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
@@ -1414,12 +1344,11 @@
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
-      have "norm (x - y) < x\<bullet>i"
-        apply (rule less_le_trans)
-        apply (rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1])
-        using i
-        apply auto
-        done
+      have "norm (x - y) < MINIMUM Basis ((\<bullet>) x)"
+        using y by (auto simp: dist_norm less_eq_real_def)
+      also have "... \<le> x\<bullet>i"
+        using i by auto
+      finally have "norm (x - y) < x\<bullet>i" .
       then show "0 \<le> y\<bullet>i"
         using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
         by (auto simp: inner_simps)
@@ -1469,82 +1398,70 @@
 qed
 
 lemma rel_interior_substd_simplex:
-  assumes d: "d \<subseteq> Basis"
-  shows "rel_interior (convex hull (insert 0 d)) =
-    {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
+  assumes D: "D \<subseteq> Basis"
+  shows "rel_interior (convex hull (insert 0 D)) =
+    {x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
 proof -
-  have "finite d"
-    apply (rule finite_subset)
-    using assms
-    apply auto
-    done
+  have "finite D"
+    using D finite_Basis finite_subset by blast
   show ?thesis
-  proof (cases "d = {}")
+  proof (cases "D = {}")
     case True
     then show ?thesis
       using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
   next
     case False
     have h0: "affine hull (convex hull (insert 0 ?p)) =
-      {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
+      {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
       using affine_hull_convex_hull affine_hull_substd_basis assms by auto
-    have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>d. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
+    have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
       by auto
     {
       fix x :: "'a::euclidean_space"
       assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
-      then obtain e where e0: "e > 0" and
-        "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
+      then obtain e where "e > 0" and
+        "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
         using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
-      then have as: "\<forall>xa. dist x xa < e \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> xa\<bullet>i = 0) \<longrightarrow>
-        (\<forall>i\<in>d. 0 \<le> xa \<bullet> i) \<and> sum ((\<bullet>) xa) d \<le> 1"
+      then have as [rule_format]: "\<And>y. dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0) \<longrightarrow>
+        (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
         unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
-      have x0: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+      have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
         using x rel_interior_subset  substd_simplex[OF assms] by auto
-      have "(\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
-        apply rule
-        apply rule
-      proof -
+      have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
+      proof (intro conjI ballI)
         fix i :: 'a
-        assume "i \<in> d"
-        then have "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> ia"
+        assume "i \<in> D"
+        then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
           apply -
-          apply (rule as[rule_format,THEN conjunct1])
-          unfolding dist_norm
-          using d \<open>e > 0\<close> x0
-          apply (auto simp: inner_simps inner_Basis)
+          apply (rule as[THEN conjunct1])
+          using D \<open>e > 0\<close> x0
+          apply (auto simp: dist_norm inner_simps inner_Basis)
           done
         then show "0 < x \<bullet> i"
-          apply (erule_tac x=i in ballE)
-          using \<open>e > 0\<close> \<open>i \<in> d\<close> d
-          apply (auto simp: inner_simps inner_Basis)
-          done
+          using \<open>e > 0\<close> \<open>i \<in> D\<close> D  by (force simp: inner_simps inner_Basis)
       next
-        obtain a where a: "a \<in> d"
-          using \<open>d \<noteq> {}\<close> by auto
+        obtain a where a: "a \<in> D"
+          using \<open>D \<noteq> {}\<close> by auto
         then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
-          using \<open>e > 0\<close> norm_Basis[of a] d
+          using \<open>e > 0\<close> norm_Basis[of a] D
           unfolding dist_norm
           by auto
         have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
-          using a d by (auto simp: inner_simps inner_Basis)
-        then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) d =
-          sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) d"
-          using d by (intro sum.cong) auto
+          using a D by (auto simp: inner_simps inner_Basis)
+        then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D =
+          sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
+          using D by (intro sum.cong) auto
         have "a \<in> Basis"
-          using \<open>a \<in> d\<close> d by auto
-        then have h1: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
-          using x0 d \<open>a\<in>d\<close> by (auto simp add: inner_add_left inner_Basis)
-        have "sum ((\<bullet>) x) d < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) d"
-          unfolding * sum.distrib
-          using \<open>e > 0\<close> \<open>a \<in> d\<close>
-          using \<open>finite d\<close>
-          by (auto simp add: sum.delta')
+          using \<open>a \<in> D\<close> D by auto
+        then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
+          using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
+        have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D"
+          using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
         also have "\<dots> \<le> 1"
           using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
           by auto
-        finally show "sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)"
+        finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
           using x0 by auto
       qed
     }
@@ -1554,63 +1471,62 @@
       assume as: "x \<in> ?s"
       have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
         by auto
-      moreover have "\<forall>i. i \<in> d \<or> i \<notin> d" by auto
+      moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
       ultimately
-      have "\<forall>i. (\<forall>i\<in>d. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> d \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
+      have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
         by metis
       then have h2: "x \<in> convex hull (insert 0 ?p)"
         using as assms
         unfolding substd_simplex[OF assms] by fastforce
-      obtain a where a: "a \<in> d"
-        using \<open>d \<noteq> {}\<close> by auto
-      let ?d = "(1 - sum ((\<bullet>) x) d) / real (card d)"
-      have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
+      obtain a where a: "a \<in> D"
+        using \<open>D \<noteq> {}\<close> by auto
+      let ?d = "(1 - sum ((\<bullet>) x) D) / real (card D)"
+      have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
         by (simp add: card_gt_0_iff)
-      have "Min (((\<bullet>) x) ` d) > 0"
-        using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_gr_iff)
-      moreover have "?d > 0" using as using \<open>0 < card d\<close> by auto
-      ultimately have h3: "min (Min (((\<bullet>) x) ` d)) ?d > 0"
+      have "Min (((\<bullet>) x) ` D) > 0"
+        using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp add: Min_gr_iff)
+      moreover have "?d > 0" using as using \<open>0 < card D\<close> by auto
+      ultimately have h3: "min (Min (((\<bullet>) x) ` D)) ?d > 0"
         by auto
 
       have "x \<in> rel_interior (convex hull (insert 0 ?p))"
         unfolding rel_interior_ball mem_Collect_eq h0
         apply (rule,rule h2)
         unfolding substd_simplex[OF assms]
-        apply (rule_tac x="min (Min (((\<bullet>) x) ` d)) ?d" in exI)
+        apply (rule_tac x="min (Min (((\<bullet>) x) ` D)) ?d" in exI)
         apply (rule, rule h3)
         apply safe
         unfolding mem_ball
       proof -
         fix y :: 'a
-        assume y: "dist x y < min (Min ((\<bullet>) x ` d)) ?d"
-        assume y2: "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> y\<bullet>i = 0"
-        have "sum ((\<bullet>) y) d \<le> sum (\<lambda>i. x\<bullet>i + ?d) d"
+        assume y: "dist x y < min (Min ((\<bullet>) x ` D)) ?d"
+        assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
+        have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + ?d) D"
         proof (rule sum_mono)
           fix i
-          assume "i \<in> d"
-          with d have i: "i \<in> Basis"
+          assume "i \<in> D"
+          with D have i: "i \<in> Basis"
             by auto
-          have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d"
-            apply (rule le_less_trans)
-            using Basis_le_norm[OF i, of "y - x"]
-            using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
-            apply (auto simp add: norm_minus_commute inner_simps)
-            done
+          have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
+            by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
+          also have "... < ?d"
+            by (metis dist_norm min_less_iff_conj norm_minus_commute y)
+          finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
           then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
         qed
         also have "\<dots> \<le> 1"
-          unfolding sum.distrib sum_constant  using \<open>0 < card d\<close>
+          unfolding sum.distrib sum_constant  using \<open>0 < card D\<close>
           by auto
-        finally show "sum ((\<bullet>) y) d \<le> 1" .
+        finally show "sum ((\<bullet>) y) D \<le> 1" .
 
         fix i :: 'a
         assume i: "i \<in> Basis"
         then show "0 \<le> y\<bullet>i"
-        proof (cases "i\<in>d")
+        proof (cases "i\<in>D")
           case True
           have "norm (x - y) < x\<bullet>i"
             using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
-            using Min_gr_iff[of "(\<bullet>) x ` d" "norm (x - y)"] \<open>0 < card d\<close> \<open>i \<in> d\<close>
+            using Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
             by (simp add: card_gt_0_iff)
           then show "0 \<le> y\<bullet>i"
             using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
@@ -1619,36 +1535,36 @@
       qed
     }
     ultimately have
-      "\<And>x. x \<in> rel_interior (convex hull insert 0 d) \<longleftrightarrow>
-        x \<in> {x. (\<forall>i\<in>d. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) d < 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)}"
+      "\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
+        x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
 qed
 
 lemma rel_interior_substd_simplex_nonempty:
-  assumes "d \<noteq> {}"
-    and "d \<subseteq> Basis"
+  assumes "D \<noteq> {}"
+    and "D \<subseteq> Basis"
   obtains a :: "'a::euclidean_space"
-    where "a \<in> rel_interior (convex hull (insert 0 d))"
-proof -
-  let ?D = d
-  let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) ?D"
-  have "finite d"
+    where "a \<in> rel_interior (convex hull (insert 0 D))"
+proof -
+  let ?D = D
+  let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) ?D"
+  have "finite D"
     apply (rule finite_subset)
     using assms(2)
     apply auto
     done
-  then have d1: "0 < real (card d)"
-    using \<open>d \<noteq> {}\<close> by auto
+  then have d1: "0 < real (card D)"
+    using \<open>D \<noteq> {}\<close> by auto
   {
     fix i
-    assume "i \<in> d"
-    have "?a \<bullet> i = inverse (2 * real (card d))"
-      apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"])
+    assume "i \<in> D"
+    have "?a \<bullet> i = inverse (2 * real (card D))"
+      apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"])
       unfolding inner_sum_left
       apply (rule sum.cong)
-      using \<open>i \<in> d\<close> \<open>finite d\<close> sum.delta'[of d i "(\<lambda>k. inverse (2 * real (card d)))"]
+      using \<open>i \<in> D\<close> \<open>finite D\<close> sum.delta'[of D i "(\<lambda>k. inverse (2 * real (card D)))"]
         d1 assms(2)
       by (auto simp: inner_Basis set_rev_mp[OF _ assms(2)])
   }
@@ -1658,14 +1574,14 @@
     unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
   proof safe
     fix i
-    assume "i \<in> d"
-    have "0 < inverse (2 * real (card d))"
+    assume "i \<in> D"
+    have "0 < inverse (2 * real (card D))"
       using d1 by auto
-    also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> d\<close>
+    also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
       by auto
     finally show "0 < ?a \<bullet> i" by auto
   next
-    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card d))) ?D"
+    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card D))) ?D"
       by (rule sum.cong) (rule refl, rule **)
     also have "\<dots> < 1"
       unfolding sum_constant divide_real_def[symmetric]
@@ -1673,22 +1589,22 @@
     finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
   next
     fix i
-    assume "i \<in> Basis" and "i \<notin> d"
-    have "?a \<in> span d"
-    proof (rule span_sum[of d "(\<lambda>b. b /\<^sub>R (2 * real (card d)))" d])
+    assume "i \<in> Basis" and "i \<notin> D"
+    have "?a \<in> span D"
+    proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
       {
         fix x :: "'a::euclidean_space"
-        assume "x \<in> d"
-        then have "x \<in> span d"
-          using span_superset[of _ "d"] by auto
-        then have "x /\<^sub>R (2 * real (card d)) \<in> span d"
-          using span_mul[of x "d" "(inverse (real (card d)) / 2)"] by auto
+        assume "x \<in> D"
+        then have "x \<in> span D"
+          using span_superset[of _ "D"] by auto
+        then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
+          using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
       }
-      then show "\<And>x. x\<in>d \<Longrightarrow> x /\<^sub>R (2 * real (card d)) \<in> span d"
+      then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
         by auto
     qed
     then show "?a \<bullet> i = 0 "
-      using \<open>i \<notin> d\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
+      using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
   qed
 qed
 
@@ -1916,10 +1832,7 @@
 proof -
   define e where "e = a / (a + b)"
   have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
-    apply auto
-    using assms
-    apply simp
-    done
+    using assms  by (simp add: eq_vector_fraction_iff)
   also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
     using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
     by auto
@@ -2089,9 +2002,8 @@
 lemma rel_frontier_eq_empty:
     fixes S :: "'n::euclidean_space set"
     shows "rel_frontier S = {} \<longleftrightarrow> affine S"
-  apply (simp add: rel_frontier_def)
-  apply (simp add: rel_interior_eq_closure [symmetric])
-  using rel_interior_subset_closure by blast
+  unfolding rel_frontier_def
+  using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])
 
 lemma rel_frontier_sing [simp]:
     fixes a :: "'n::euclidean_space"
@@ -2365,16 +2277,12 @@
   shows "z \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
 proof (cases "aff_dim S = int DIM('n)")
   case False
-  {
-    assume "z \<in> interior S"
+  { assume "z \<in> interior S"
     then have False
-      using False interior_rel_interior_gen[of S] by auto
-  }
+      using False interior_rel_interior_gen[of S] by auto }
   moreover
-  {
-    assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
-    {
-      fix x
+  { assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
+    { fix x
       obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
         using r by auto
       obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
@@ -3679,13 +3587,7 @@
     then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
       using k by (simp add: sum_prod)
     then have "x \<in> ?rhs"
-      using k
-      apply auto
-      apply (rule_tac x = c in exI)
-      apply (rule_tac x = s in exI)
-      using cs
-      apply auto
-      done
+      using k cs by auto
   }
   moreover
   {
@@ -3709,7 +3611,7 @@
       done
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
-      by (auto simp add: convex_convex_hull)
+      by auto
   }
   ultimately show ?thesis by blast
 qed
@@ -5105,10 +5007,7 @@
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
-    apply safe
-    using UT closedin_subset apply blast
-    using US closedin_subset apply blast
-    done
+    using UT US by (blast dest: closedin_subset)
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Library/Adhoc_Overloading.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Adhoc_Overloading.thy	Wed May 02 12:48:08 2018 +0100
@@ -0,0 +1,15 @@
+(*  Title:      HOL/Library/Adhoc_Overloading.thy
+    Author:     Alexander Krauss, TU Muenchen
+    Author:     Christian Sternagel, University of Innsbruck
+*)
+
+section \<open>Adhoc overloading of constants based on their types\<close>
+
+theory Adhoc_Overloading
+  imports Main
+  keywords "adhoc_overloading" "no_adhoc_overloading" :: thy_decl
+begin
+
+ML_file "adhoc_overloading.ML"
+
+end
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Library/Library.thy	Wed May 02 12:48:08 2018 +0100
@@ -2,6 +2,7 @@
 theory Library
 imports
   AList
+  Adhoc_Overloading
   BigO
   Bit
   BNF_Axiomatization
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Library/Monad_Syntax.thy
--- a/src/HOL/Library/Monad_Syntax.thy	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Library/Monad_Syntax.thy	Wed May 02 12:48:08 2018 +0100
@@ -1,11 +1,12 @@
-(* Author: Alexander Krauss, TU Muenchen
-   Author: Christian Sternagel, University of Innsbruck
+(*  Title:      HOL/Library/Monad_Syntax.thy
+    Author:     Alexander Krauss, TU Muenchen
+    Author:     Christian Sternagel, University of Innsbruck
 *)
 
 section \<open>Monad notation for arbitrary types\<close>
 
 theory Monad_Syntax
-imports Main "~~/src/Tools/Adhoc_Overloading"
+  imports Adhoc_Overloading
 begin
 
 text \<open>
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Library/adhoc_overloading.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/adhoc_overloading.ML	Wed May 02 12:48:08 2018 +0100
@@ -0,0 +1,246 @@
+(*  Author:     Alexander Krauss, TU Muenchen
+    Author:     Christian Sternagel, University of Innsbruck
+
+Adhoc overloading of constants based on their types.
+*)
+
+signature ADHOC_OVERLOADING =
+sig
+  val is_overloaded: Proof.context -> string -> bool
+  val generic_add_overloaded: string -> Context.generic -> Context.generic
+  val generic_remove_overloaded: string -> Context.generic -> Context.generic
+  val generic_add_variant: string -> term -> Context.generic -> Context.generic
+  (*If the list of variants is empty at the end of "generic_remove_variant", then
+  "generic_remove_overloaded" is called implicitly.*)
+  val generic_remove_variant: string -> term -> Context.generic -> Context.generic
+  val show_variants: bool Config.T
+end
+
+structure Adhoc_Overloading: ADHOC_OVERLOADING =
+struct
+
+val show_variants = Attrib.setup_config_bool @{binding show_variants} (K false);
+
+
+(* errors *)
+
+fun err_duplicate_variant oconst =
+  error ("Duplicate variant of " ^ quote oconst);
+
+fun err_not_a_variant oconst =
+  error ("Not a variant of " ^  quote oconst);
+
+fun err_not_overloaded oconst =
+  error ("Constant " ^ quote oconst ^ " is not declared as overloaded");
+
+fun err_unresolved_overloading ctxt0 (c, T) t instances =
+  let
+    val ctxt = Config.put show_variants true ctxt0
+    val const_space = Proof_Context.const_space ctxt
+    val prt_const =
+      Pretty.block [Name_Space.pretty ctxt const_space c, Pretty.str " ::", Pretty.brk 1,
+        Pretty.quote (Syntax.pretty_typ ctxt T)]
+  in
+    error (Pretty.string_of (Pretty.chunks [
+      Pretty.block [
+        Pretty.str "Unresolved adhoc overloading of constant", Pretty.brk 1,
+        prt_const, Pretty.brk 1, Pretty.str "in term", Pretty.brk 1,
+        Pretty.block [Pretty.quote (Syntax.pretty_term ctxt t)]],
+      Pretty.block (
+        (if null instances then [Pretty.str "no instances"]
+        else Pretty.fbreaks (
+          Pretty.str "multiple instances:" ::
+          map (Pretty.mark Markup.item o Syntax.pretty_term ctxt) instances)))]))
+  end;
+
+
+(* generic data *)
+
+fun variants_eq ((v1, T1), (v2, T2)) =
+  Term.aconv_untyped (v1, v2) andalso T1 = T2;
+
+structure Overload_Data = Generic_Data
+(
+  type T =
+    {variants : (term * typ) list Symtab.table,
+     oconsts : string Termtab.table};
+  val empty = {variants = Symtab.empty, oconsts = Termtab.empty};
+  val extend = I;
+
+  fun merge
+    ({variants = vtab1, oconsts = otab1},
+     {variants = vtab2, oconsts = otab2}) : T =
+    let
+      fun merge_oconsts _ (oconst1, oconst2) =
+        if oconst1 = oconst2 then oconst1
+        else err_duplicate_variant oconst1;
+    in
+      {variants = Symtab.merge_list variants_eq (vtab1, vtab2),
+       oconsts = Termtab.join merge_oconsts (otab1, otab2)}
+    end;
+);
+
+fun map_tables f g =
+  Overload_Data.map (fn {variants = vtab, oconsts = otab} =>
+    {variants = f vtab, oconsts = g otab});
+
+val is_overloaded = Symtab.defined o #variants o Overload_Data.get o Context.Proof;
+val get_variants = Symtab.lookup o #variants o Overload_Data.get o Context.Proof;
+val get_overloaded = Termtab.lookup o #oconsts o Overload_Data.get o Context.Proof;
+
+fun generic_add_overloaded oconst context =
+  if is_overloaded (Context.proof_of context) oconst then context
+  else map_tables (Symtab.update (oconst, [])) I context;
+
+fun generic_remove_overloaded oconst context =
+  let
+    fun remove_oconst_and_variants context oconst =
+      let
+        val remove_variants =
+          (case get_variants (Context.proof_of context) oconst of
+            NONE => I
+          | SOME vs => fold (Termtab.remove (op =) o rpair oconst o fst) vs);
+      in map_tables (Symtab.delete_safe oconst) remove_variants context end;
+  in
+    if is_overloaded (Context.proof_of context) oconst then remove_oconst_and_variants context oconst
+    else err_not_overloaded oconst
+  end;
+
+local
+  fun generic_variant add oconst t context =
+    let
+      val ctxt = Context.proof_of context;
+      val _ = if is_overloaded ctxt oconst then () else err_not_overloaded oconst;
+      val T = t |> fastype_of;
+      val t' = Term.map_types (K dummyT) t;
+    in
+      if add then
+        let
+          val _ =
+            (case get_overloaded ctxt t' of
+              NONE => ()
+            | SOME oconst' => err_duplicate_variant oconst');
+        in
+          map_tables (Symtab.cons_list (oconst, (t', T))) (Termtab.update (t', oconst)) context
+        end
+      else
+        let
+          val _ =
+            if member variants_eq (the (get_variants ctxt oconst)) (t', T) then ()
+            else err_not_a_variant oconst;
+        in
+          map_tables (Symtab.map_entry oconst (remove1 variants_eq (t', T)))
+            (Termtab.delete_safe t') context
+          |> (fn context =>
+            (case get_variants (Context.proof_of context) oconst of
+              SOME [] => generic_remove_overloaded oconst context
+            | _ => context))
+        end
+    end;
+in
+  val generic_add_variant = generic_variant true;
+  val generic_remove_variant = generic_variant false;
+end;
+
+
+(* check / uncheck *)
+
+fun unifiable_with thy T1 T2 =
+  let
+    val maxidx1 = Term.maxidx_of_typ T1;
+    val T2' = Logic.incr_tvar (maxidx1 + 1) T2;
+    val maxidx2 = Term.maxidx_typ T2' maxidx1;
+  in can (Sign.typ_unify thy (T1, T2')) (Vartab.empty, maxidx2) end;
+
+fun get_candidates ctxt (c, T) =
+  get_variants ctxt c
+  |> Option.map (map_filter (fn (t, T') =>
+    if unifiable_with (Proof_Context.theory_of ctxt) T T' then SOME t
+    else NONE));
+
+fun insert_variants ctxt t (oconst as Const (c, T)) =
+      (case get_candidates ctxt (c, T) of
+        SOME [] => err_unresolved_overloading ctxt (c, T) t []
+      | SOME [variant] => variant
+      | _ => oconst)
+  | insert_variants _ _ oconst = oconst;
+
+fun insert_overloaded ctxt =
+  let
+    fun proc t =
+      Term.map_types (K dummyT) t
+      |> get_overloaded ctxt
+      |> Option.map (Const o rpair (Term.type_of t));
+  in
+    Pattern.rewrite_term_top (Proof_Context.theory_of ctxt) [] [proc]
+  end;
+
+fun check ctxt =
+  map (fn t => Term.map_aterms (insert_variants ctxt t) t);
+
+fun uncheck ctxt ts =
+  if Config.get ctxt show_variants orelse exists (is_none o try Term.type_of) ts then ts
+  else map (insert_overloaded ctxt) ts;
+
+fun reject_unresolved ctxt =
+  let
+    val the_candidates = the o get_candidates ctxt;
+    fun check_unresolved t =
+      (case filter (is_overloaded ctxt o fst) (Term.add_consts t []) of
+        [] => t
+      | (cT :: _) => err_unresolved_overloading ctxt cT t (the_candidates cT));
+  in map check_unresolved end;
+
+
+(* setup *)
+
+val _ = Context.>>
+  (Syntax_Phases.term_check 0 "adhoc_overloading" check
+   #> Syntax_Phases.term_check 1 "adhoc_overloading_unresolved_check" reject_unresolved
+   #> Syntax_Phases.term_uncheck 0 "adhoc_overloading" uncheck);
+
+
+(* commands *)
+
+fun generic_adhoc_overloading_cmd add =
+  if add then
+    fold (fn (oconst, ts) =>
+      generic_add_overloaded oconst
+      #> fold (generic_add_variant oconst) ts)
+  else
+    fold (fn (oconst, ts) =>
+      fold (generic_remove_variant oconst) ts);
+
+fun adhoc_overloading_cmd' add args phi =
+  let val args' = args
+    |> map (apsnd (map_filter (fn t =>
+         let val t' = Morphism.term phi t;
+         in if Term.aconv_untyped (t, t') then SOME t' else NONE end)));
+  in generic_adhoc_overloading_cmd add args' end;
+
+fun adhoc_overloading_cmd add raw_args lthy =
+  let
+    fun const_name ctxt =
+      fst o dest_Const o Proof_Context.read_const {proper = false, strict = false} ctxt;  (* FIXME {proper = true, strict = true} (!?) *)
+    fun read_term ctxt = singleton (Variable.polymorphic ctxt) o Syntax.read_term ctxt;
+    val args =
+      raw_args
+      |> map (apfst (const_name lthy))
+      |> map (apsnd (map (read_term lthy)));
+  in
+    Local_Theory.declaration {syntax = true, pervasive = false}
+      (adhoc_overloading_cmd' add args) lthy
+  end;
+
+val _ =
+  Outer_Syntax.local_theory @{command_keyword adhoc_overloading}
+    "add adhoc overloading for constants / fixed variables"
+    (Parse.and_list1 (Parse.const -- Scan.repeat Parse.term) >> adhoc_overloading_cmd true);
+
+val _ =
+  Outer_Syntax.local_theory @{command_keyword no_adhoc_overloading}
+    "delete adhoc overloading for constants / fixed variables"
+    (Parse.and_list1 (Parse.const -- Scan.repeat Parse.term) >> adhoc_overloading_cmd false);
+
+end;
+
diff -r b249fab48c76 -r d2daeef3ce47 src/HOL/Tools/SMT/z3_proof.ML
--- a/src/HOL/Tools/SMT/z3_proof.ML	Wed May 02 12:47:56 2018 +0100
+++ b/src/HOL/Tools/SMT/z3_proof.ML	Wed May 02 12:48:08 2018 +0100
@@ -216,9 +216,15 @@
   let
     val match = Sign.typ_match (Proof_Context.theory_of ctxt)
 
+    fun objT_of bound =
+      (case Symtab.lookup env bound of
+        SOME objT => objT
+      | NONE => raise Fail ("Replaying the proof trace produced by Z3 failed: " ^
+          "the bound " ^ quote bound ^ " is undeclared; this indicates a bug in Z3"))
+
     val t' = singleton (Variable.polymorphic ctxt) t
     val patTs = map snd (Term.strip_qnt_vars @{const_name Pure.all} t')
-    val objTs = map (the o Symtab.lookup env) bounds
+    val objTs = map objT_of bounds
     val subst = subst_of (fold match (patTs ~~ objTs) Vartab.empty)
   in Same.commit (Term_Subst.map_types_same (substTs_same subst)) t' end
 
diff -r b249fab48c76 -r d2daeef3ce47 src/Tools/Adhoc_Overloading.thy
--- a/src/Tools/Adhoc_Overloading.thy	Wed May 02 12:47:56 2018 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,16 +0,0 @@
-(* Author: Alexander Krauss, TU Muenchen
-   Author: Christian Sternagel, University of Innsbruck
-*)
-
-section \<open>Adhoc overloading of constants based on their types\<close>
-
-theory Adhoc_Overloading
-imports Pure
-keywords
-  "adhoc_overloading" "no_adhoc_overloading" :: thy_decl
-begin
-
-ML_file "adhoc_overloading.ML"
-
-end
-
diff -r b249fab48c76 -r d2daeef3ce47 src/Tools/adhoc_overloading.ML
--- a/src/Tools/adhoc_overloading.ML	Wed May 02 12:47:56 2018 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,246 +0,0 @@
-(*  Author:     Alexander Krauss, TU Muenchen
-    Author:     Christian Sternagel, University of Innsbruck
-
-Adhoc overloading of constants based on their types.
-*)
-
-signature ADHOC_OVERLOADING =
-sig
-  val is_overloaded: Proof.context -> string -> bool
-  val generic_add_overloaded: string -> Context.generic -> Context.generic
-  val generic_remove_overloaded: string -> Context.generic -> Context.generic
-  val generic_add_variant: string -> term -> Context.generic -> Context.generic
-  (*If the list of variants is empty at the end of "generic_remove_variant", then
-  "generic_remove_overloaded" is called implicitly.*)
-  val generic_remove_variant: string -> term -> Context.generic -> Context.generic
-  val show_variants: bool Config.T
-end
-
-structure Adhoc_Overloading: ADHOC_OVERLOADING =
-struct
-
-val show_variants = Attrib.setup_config_bool @{binding show_variants} (K false);
-
-
-(* errors *)
-
-fun err_duplicate_variant oconst =
-  error ("Duplicate variant of " ^ quote oconst);
-
-fun err_not_a_variant oconst =
-  error ("Not a variant of " ^  quote oconst);
-
-fun err_not_overloaded oconst =
-  error ("Constant " ^ quote oconst ^ " is not declared as overloaded");
-
-fun err_unresolved_overloading ctxt0 (c, T) t instances =
-  let
-    val ctxt = Config.put show_variants true ctxt0
-    val const_space = Proof_Context.const_space ctxt
-    val prt_const =
-      Pretty.block [Name_Space.pretty ctxt const_space c, Pretty.str " ::", Pretty.brk 1,
-        Pretty.quote (Syntax.pretty_typ ctxt T)]
-  in
-    error (Pretty.string_of (Pretty.chunks [
-      Pretty.block [
-        Pretty.str "Unresolved adhoc overloading of constant", Pretty.brk 1,
-        prt_const, Pretty.brk 1, Pretty.str "in term", Pretty.brk 1,
-        Pretty.block [Pretty.quote (Syntax.pretty_term ctxt t)]],
-      Pretty.block (
-        (if null instances then [Pretty.str "no instances"]
-        else Pretty.fbreaks (
-          Pretty.str "multiple instances:" ::
-          map (Pretty.mark Markup.item o Syntax.pretty_term ctxt) instances)))]))
-  end;
-
-
-(* generic data *)
-
-fun variants_eq ((v1, T1), (v2, T2)) =
-  Term.aconv_untyped (v1, v2) andalso T1 = T2;
-
-structure Overload_Data = Generic_Data
-(
-  type T =
-    {variants : (term * typ) list Symtab.table,
-     oconsts : string Termtab.table};
-  val empty = {variants = Symtab.empty, oconsts = Termtab.empty};
-  val extend = I;
-
-  fun merge
-    ({variants = vtab1, oconsts = otab1},
-     {variants = vtab2, oconsts = otab2}) : T =
-    let
-      fun merge_oconsts _ (oconst1, oconst2) =
-        if oconst1 = oconst2 then oconst1
-        else err_duplicate_variant oconst1;
-    in
-      {variants = Symtab.merge_list variants_eq (vtab1, vtab2),
-       oconsts = Termtab.join merge_oconsts (otab1, otab2)}
-    end;
-);
-
-fun map_tables f g =
-  Overload_Data.map (fn {variants = vtab, oconsts = otab} =>
-    {variants = f vtab, oconsts = g otab});
-
-val is_overloaded = Symtab.defined o #variants o Overload_Data.get o Context.Proof;
-val get_variants = Symtab.lookup o #variants o Overload_Data.get o Context.Proof;
-val get_overloaded = Termtab.lookup o #oconsts o Overload_Data.get o Context.Proof;
-
-fun generic_add_overloaded oconst context =
-  if is_overloaded (Context.proof_of context) oconst then context
-  else map_tables (Symtab.update (oconst, [])) I context;
-
-fun generic_remove_overloaded oconst context =
-  let
-    fun remove_oconst_and_variants context oconst =
-      let
-        val remove_variants =
-          (case get_variants (Context.proof_of context) oconst of
-            NONE => I
-          | SOME vs => fold (Termtab.remove (op =) o rpair oconst o fst) vs);
-      in map_tables (Symtab.delete_safe oconst) remove_variants context end;
-  in
-    if is_overloaded (Context.proof_of context) oconst then remove_oconst_and_variants context oconst
-    else err_not_overloaded oconst
-  end;
-
-local
-  fun generic_variant add oconst t context =
-    let
-      val ctxt = Context.proof_of context;
-      val _ = if is_overloaded ctxt oconst then () else err_not_overloaded oconst;
-      val T = t |> fastype_of;
-      val t' = Term.map_types (K dummyT) t;
-    in
-      if add then
-        let
-          val _ =
-            (case get_overloaded ctxt t' of
-              NONE => ()
-            | SOME oconst' => err_duplicate_variant oconst');
-        in
-          map_tables (Symtab.cons_list (oconst, (t', T))) (Termtab.update (t', oconst)) context
-        end
-      else
-        let
-          val _ =
-            if member variants_eq (the (get_variants ctxt oconst)) (t', T) then ()
-            else err_not_a_variant oconst;
-        in
-          map_tables (Symtab.map_entry oconst (remove1 variants_eq (t', T)))
-            (Termtab.delete_safe t') context
-          |> (fn context =>
-            (case get_variants (Context.proof_of context) oconst of
-              SOME [] => generic_remove_overloaded oconst context
-            | _ => context))
-        end
-    end;
-in
-  val generic_add_variant = generic_variant true;
-  val generic_remove_variant = generic_variant false;
-end;
-
-
-(* check / uncheck *)
-
-fun unifiable_with thy T1 T2 =
-  let
-    val maxidx1 = Term.maxidx_of_typ T1;
-    val T2' = Logic.incr_tvar (maxidx1 + 1) T2;
-    val maxidx2 = Term.maxidx_typ T2' maxidx1;
-  in can (Sign.typ_unify thy (T1, T2')) (Vartab.empty, maxidx2) end;
-
-fun get_candidates ctxt (c, T) =
-  get_variants ctxt c
-  |> Option.map (map_filter (fn (t, T') =>
-    if unifiable_with (Proof_Context.theory_of ctxt) T T' then SOME t
-    else NONE));
-
-fun insert_variants ctxt t (oconst as Const (c, T)) =
-      (case get_candidates ctxt (c, T) of
-        SOME [] => err_unresolved_overloading ctxt (c, T) t []
-      | SOME [variant] => variant
-      | _ => oconst)
-  | insert_variants _ _ oconst = oconst;
-
-fun insert_overloaded ctxt =
-  let
-    fun proc t =
-      Term.map_types (K dummyT) t
-      |> get_overloaded ctxt
-      |> Option.map (Const o rpair (Term.type_of t));
-  in
-    Pattern.rewrite_term_top (Proof_Context.theory_of ctxt) [] [proc]
-  end;
-
-fun check ctxt =
-  map (fn t => Term.map_aterms (insert_variants ctxt t) t);
-
-fun uncheck ctxt ts =
-  if Config.get ctxt show_variants orelse exists (is_none o try Term.type_of) ts then ts
-  else map (insert_overloaded ctxt) ts;
-
-fun reject_unresolved ctxt =
-  let
-    val the_candidates = the o get_candidates ctxt;
-    fun check_unresolved t =
-      (case filter (is_overloaded ctxt o fst) (Term.add_consts t []) of
-        [] => t
-      | (cT :: _) => err_unresolved_overloading ctxt cT t (the_candidates cT));
-  in map check_unresolved end;
-
-
-(* setup *)
-
-val _ = Context.>>
-  (Syntax_Phases.term_check 0 "adhoc_overloading" check
-   #> Syntax_Phases.term_check 1 "adhoc_overloading_unresolved_check" reject_unresolved
-   #> Syntax_Phases.term_uncheck 0 "adhoc_overloading" uncheck);
-
-
-(* commands *)
-
-fun generic_adhoc_overloading_cmd add =
-  if add then
-    fold (fn (oconst, ts) =>
-      generic_add_overloaded oconst
-      #> fold (generic_add_variant oconst) ts)
-  else
-    fold (fn (oconst, ts) =>
-      fold (generic_remove_variant oconst) ts);
-
-fun adhoc_overloading_cmd' add args phi =
-  let val args' = args
-    |> map (apsnd (map_filter (fn t =>
-         let val t' = Morphism.term phi t;
-         in if Term.aconv_untyped (t, t') then SOME t' else NONE end)));
-  in generic_adhoc_overloading_cmd add args' end;
-
-fun adhoc_overloading_cmd add raw_args lthy =
-  let
-    fun const_name ctxt =
-      fst o dest_Const o Proof_Context.read_const {proper = false, strict = false} ctxt;  (* FIXME {proper = true, strict = true} (!?) *)
-    fun read_term ctxt = singleton (Variable.polymorphic ctxt) o Syntax.read_term ctxt;
-    val args =
-      raw_args
-      |> map (apfst (const_name lthy))
-      |> map (apsnd (map (read_term lthy)));
-  in
-    Local_Theory.declaration {syntax = true, pervasive = false}
-      (adhoc_overloading_cmd' add args) lthy
-  end;
-
-val _ =
-  Outer_Syntax.local_theory @{command_keyword adhoc_overloading}
-    "add adhoc overloading for constants / fixed variables"
-    (Parse.and_list1 (Parse.const -- Scan.repeat Parse.term) >> adhoc_overloading_cmd true);
-
-val _ =
-  Outer_Syntax.local_theory @{command_keyword no_adhoc_overloading}
-    "delete adhoc overloading for constants / fixed variables"
-    (Parse.and_list1 (Parse.const -- Scan.repeat Parse.term) >> adhoc_overloading_cmd false);
-
-end;
-
diff -r b249fab48c76 -r d2daeef3ce47 src/Tools/jEdit/src/jedit_lib.scala
--- a/src/Tools/jEdit/src/jedit_lib.scala	Wed May 02 12:47:56 2018 +0100
+++ b/src/Tools/jEdit/src/jedit_lib.scala	Wed May 02 12:48:08 2018 +0100
@@ -129,6 +129,13 @@
 
   def buffer_file(buffer: Buffer): Option[JFile] = check_file(buffer_name(buffer))
 
+  def buffer_undo_in_progress[A](buffer: JEditBuffer, body: => A): A =
+  {
+    val undo_in_progress = buffer.isUndoInProgress
+    def set(b: Boolean) { Untyped.set[Boolean](buffer, "undoInProgress", b) }
+    try { set(true); body } finally { set(undo_in_progress) }
+  }
+
 
   /* main jEdit components */
 
diff -r b249fab48c76 -r d2daeef3ce47 src/Tools/jEdit/src/pretty_text_area.scala
--- a/src/Tools/jEdit/src/pretty_text_area.scala	Wed May 02 12:47:56 2018 +0100
+++ b/src/Tools/jEdit/src/pretty_text_area.scala	Wed May 02 12:48:08 2018 +0100
@@ -144,7 +144,7 @@
             JEdit_Lib.buffer_edit(getBuffer) {
               rich_text_area.active_reset()
               getBuffer.setFoldHandler(new Fold_Handling.Document_Fold_Handler(rendering))
-              setText(text)
+              JEdit_Lib.buffer_undo_in_progress(getBuffer, setText(text))
               setCaretPosition(0)
             }
           }