diff --git a/src/HOL/Analysis/Affine.thy b/src/HOL/Analysis/Affine.thy
--- a/src/HOL/Analysis/Affine.thy
+++ b/src/HOL/Analysis/Affine.thy
@@ -1,1657 +1,1638 @@
 section "Affine Sets"
 
 theory Affine
 imports Linear_Algebra
 begin
 
 lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
   by (fact if_distrib)
 
 lemma sum_delta_notmem:
   assumes "x \<notin> s"
   shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
   apply (rule_tac [!] sum.cong)
   using assms
   apply auto
   done
 
 lemmas independent_finite = independent_imp_finite
 
 lemma span_substd_basis:
   assumes d: "d \<subseteq> Basis"
   shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   (is "_ = ?B")
 proof -
   have "d \<subseteq> ?B"
     using d by (auto simp: inner_Basis)
   moreover have s: "subspace ?B"
     using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
   ultimately have "span d \<subseteq> ?B"
     using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
   moreover have *: "card d \<le> dim (span d)"
     using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
       span_superset[of d]
     by auto
   moreover from * have "dim ?B \<le> dim (span d)"
     using dim_substandard[OF assms] by auto
   ultimately show ?thesis
     using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
 qed
 
 lemma basis_to_substdbasis_subspace_isomorphism:
   fixes B :: "'a::euclidean_space set"
   assumes "independent B"
   shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
     f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
 proof -
   have B: "card B = dim B"
     using dim_unique[of B B "card B"] assms span_superset[of B] by auto
   have "dim B \<le> card (Basis :: 'a set)"
     using dim_subset_UNIV[of B] by simp
   from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
     by auto
   let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
   proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
     show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
       using d inner_not_same_Basis by blast
   qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
   with t \<open>card B = dim B\<close> d show ?thesis by auto
 qed
 
 subsection \<open>Affine set and affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> affine :: "'a::real_vector set \<Rightarrow> bool"
   where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
 
 lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
   unfolding affine_def by (metis eq_diff_eq')
 
 lemma affine_empty [iff]: "affine {}"
   unfolding affine_def by auto
 
 lemma affine_sing [iff]: "affine {x}"
   unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
 
 lemma affine_UNIV [iff]: "affine UNIV"
   unfolding affine_def by auto
 
 lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
   unfolding affine_def by auto
 
 lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   unfolding affine_def by auto
 
 lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
   apply (clarsimp simp add: affine_def)
   apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
   apply (auto simp: algebra_simps)
   done
 
 lemma affine_affine_hull [simp]: "affine(affine hull s)"
   unfolding hull_def
   using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
 
 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   by (metis affine_affine_hull hull_same)
 
 lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
   by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some explicit formulations\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma affine:
   fixes V::"'a::real_vector set"
   shows "affine V \<longleftrightarrow>
          (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
 proof -
   have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
     and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
   proof (cases "x = y")
     case True
     then show ?thesis
       using that by (metis scaleR_add_left scaleR_one)
   next
     case False
     then show ?thesis
       using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
   qed
   moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
                 if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
                   and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
   proof -
     define n where "n = card S"
     consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
     then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
     proof cases
       assume "card S = 1"
       then obtain a where "S={a}"
         by (auto simp: card_Suc_eq)
       then show ?thesis
         using that by simp
     next
       assume "card S = 2"
       then obtain a b where "S = {a, b}"
         by (metis Suc_1 card_1_singletonE card_Suc_eq)
       then show ?thesis
         using *[of a b] that
         by (auto simp: sum_clauses(2))
     next
       assume "card S > 2"
       then show ?thesis using that n_def
       proof (induct n arbitrary: u S)
         case 0
         then show ?case by auto
       next
         case (Suc n u S)
         have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
           using that unfolding card_eq_sum by auto
         with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
         have c: "card (S - {x}) = card S - 1"
           by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
         have "sum u (S - {x}) = 1 - u x"
           by (simp add: Suc.prems sum_diff1 \<open>x \<in> S\<close>)
         with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
           by auto
         have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
         proof (cases "card (S - {x}) > 2")
           case True
           then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
             using Suc.prems c by force+
           show ?thesis
           proof (rule Suc.hyps)
             show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
               by (auto simp: eq1 sum_distrib_left[symmetric])
           qed (use S Suc.prems True in auto)
         next
           case False
           then have "card (S - {x}) = Suc (Suc 0)"
             using Suc.prems c by auto
           then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
             unfolding card_Suc_eq by auto
           then show ?thesis
             using eq1 \<open>S \<subseteq> V\<close>
             by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
         qed
         have "u x + (1 - u x) = 1 \<Longrightarrow>
           u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
           by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
         moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
           by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
         ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
           by (simp add: x)
       qed
     qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
   qed
   ultimately show ?thesis
     unfolding affine_def by meson
 qed
 
 
 lemma affine_hull_explicit:
   "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?rhs")
 proof (rule hull_unique)
   show "p \<subseteq> ?rhs"
   proof (intro subsetI CollectI exI conjI)
     show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
       by auto
   qed auto
   show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
     using that unfolding affine by blast
   show "affine ?rhs"
     unfolding affine_def
   proof clarify
     fix u v :: real and sx ux sy uy
     assume uv: "u + v = 1"
       and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
       and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)" 
     have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
       by auto
     show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
         sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
     proof (intro exI conjI)
       show "finite (sx \<union> sy)"
         using x y by auto
       show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
         using x y uv
         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
       have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
           = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
         using x y
         unfolding scaleR_left_distrib scaleR_zero_left if_smult
         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
       also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
         unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
       finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i) 
                   = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
     qed (use x y in auto)
   qed
 qed
 
 lemma affine_hull_finite:
   assumes "finite S"
   shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
 proof -
   have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x" 
     if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
   proof -
     have "S \<inter> F = F"
       using that by auto
     show ?thesis
     proof (intro exI conjI)
       show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
         by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
       show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
         by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
     qed
   qed
   show ?thesis
     unfolding affine_hull_explicit using assms
     by (fastforce dest: *)
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems and hence small special cases\<close>
 
 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   by simp
 
 lemma affine_hull_finite_step:
   fixes y :: "'a::real_vector"
   shows "finite S \<Longrightarrow>
       (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
       (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
 proof -
   assume fin: "finite S"
   show "?lhs = ?rhs"
   proof
     assume ?lhs
     then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
       by auto
     show ?rhs
     proof (cases "a \<in> S")
       case True
       then show ?thesis
         using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
     next
       case False
       show ?thesis
         by (rule exI [where x="u a"]) (use u fin False in auto)
     qed
   next
     assume ?rhs
     then obtain v u where vu: "sum u S = w - v"  "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
       by auto
     have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
       by auto
     show ?lhs
     proof (cases "a \<in> S")
       case True
       show ?thesis
         by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
            (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
     next
       case False
       then show ?thesis
         apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI) 
         apply (simp add: vu sum_clauses(2)[OF fin] *)
         by (simp add: sum_delta_notmem(3) vu)
     qed
   qed
 qed
 
 lemma affine_hull_2:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
   (is "?lhs = ?rhs")
 proof -
   have *:
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
     using affine_hull_finite[of "{a,b}"] by auto
   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
     by (simp add: affine_hull_finite_step[of "{b}" a])
   also have "\<dots> = ?rhs" unfolding * by auto
   finally show ?thesis by auto
 qed
 
 lemma affine_hull_3:
   fixes a b c :: "'a::real_vector"
   shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
 proof -
   have *:
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   show ?thesis
     apply (simp add: affine_hull_finite affine_hull_finite_step)
     unfolding *
     apply safe
      apply (metis add.assoc)
     apply (rule_tac x=u in exI, force)
     done
 qed
 
 lemma mem_affine:
   assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
   using assms affine_def[of S] by auto
 
 lemma mem_affine_3:
   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
 proof -
   have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
     using affine_hull_3[of x y z] assms by auto
   moreover
   have "affine hull {x, y, z} \<subseteq> affine hull S"
     using hull_mono[of "{x, y, z}" "S"] assms by auto
   moreover
   have "affine hull S = S"
     using assms affine_hull_eq[of S] by auto
   ultimately show ?thesis by auto
 qed
 
 lemma mem_affine_3_minus:
   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
   shows "x + v *\<^sub>R (y-z) \<in> S"
   using mem_affine_3[of S x y z 1 v "-v"] assms
   by (simp add: algebra_simps)
 
 corollary%unimportant mem_affine_3_minus2:
     "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
   by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some relations between affine hull and subspaces\<close>
 
 lemma affine_hull_insert_subset_span:
   "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
 proof -
   have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
     if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
     for x F u
   proof -
     have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
       using that by auto
     show ?thesis
     proof (intro exI conjI)
       show "finite ((\<lambda>x. x - a) ` (F - {a}))"
         by (simp add: that(1))
       show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
         by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
             sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
     qed (use \<open>F \<subseteq> insert a S\<close> in auto)
   qed
   then show ?thesis
     unfolding affine_hull_explicit span_explicit by fast
 qed
 
 lemma affine_hull_insert_span:
   assumes "a \<notin> S"
   shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x.  x \<in> S}}"
 proof -
   have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
     if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
   proof -
     from that
     obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
       unfolding span_explicit by auto
     define F where "F = (\<lambda>x. x + a) ` T"
     have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
       unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
     have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
       using F assms by auto
     show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
       apply (rule_tac x = "insert a F" in exI)
       apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
       using assms F
       apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
       done
   qed
   show ?thesis
     by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
 qed
 
 lemma affine_hull_span:
   assumes "a \<in> S"
   shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
   using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Parallel affine sets\<close>
 
 definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
   where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
 
 lemma affine_parallel_expl_aux:
   fixes S T :: "'a::real_vector set"
   assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
   shows "T = (\<lambda>x. a + x) ` S"
 proof -
   have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
     using that
     by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
   moreover have "T \<ge> (\<lambda>x. a + x) ` S"
     using assms by auto
   ultimately show ?thesis by auto
 qed
 
 lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
   by (auto simp add: affine_parallel_def)
     (use affine_parallel_expl_aux [of S _ T] in blast)
 
 lemma affine_parallel_reflex: "affine_parallel S S"
   unfolding affine_parallel_def
   using image_add_0 by blast
 
 lemma affine_parallel_commut:
   assumes "affine_parallel A B"
   shows "affine_parallel B A"
 proof -
   from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
     unfolding affine_parallel_def by auto
   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   from B show ?thesis
     using translation_galois [of B a A]
     unfolding affine_parallel_def by blast
 qed
 
 lemma affine_parallel_assoc:
   assumes "affine_parallel A B"
     and "affine_parallel B C"
   shows "affine_parallel A C"
 proof -
   from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
     unfolding affine_parallel_def by auto
   moreover
   from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
     unfolding affine_parallel_def by auto
   ultimately show ?thesis
     using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
 qed
 
 lemma affine_translation_aux:
   fixes a :: "'a::real_vector"
   assumes "affine ((\<lambda>x. a + x) ` S)"
   shows "affine S"
 proof -
   {
     fix x y u v
     assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
     then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
       by auto
     then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
       using xy assms unfolding affine_def by auto
     have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
       by (simp add: algebra_simps)
     also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
       using \<open>u + v = 1\<close> by auto
     ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
       using h1 by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
   }
   then show ?thesis unfolding affine_def by auto
 qed
 
 lemma affine_translation:
   "affine S \<longleftrightarrow> affine ((+) a ` S)" for a :: "'a::real_vector"
 proof
   show "affine ((+) a ` S)" if "affine S"
     using that translation_assoc [of "- a" a S]
     by (auto intro: affine_translation_aux [of "- a" "((+) a ` S)"])
   show "affine S" if "affine ((+) a ` S)"
     using that by (rule affine_translation_aux)
 qed
 
 lemma parallel_is_affine:
   fixes S T :: "'a::real_vector set"
   assumes "affine S" "affine_parallel S T"
   shows "affine T"
 proof -
   from assms obtain a where "T = (\<lambda>x. a + x) ` S"
     unfolding affine_parallel_def by auto
   then show ?thesis
     using affine_translation assms by auto
 qed
 
 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
   unfolding subspace_def affine_def by auto
 
 lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Subspace parallel to an affine set\<close>
 
 lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
 proof -
   have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
     using subspace_imp_affine[of S] subspace_0 by auto
   {
     assume assm: "affine S \<and> 0 \<in> S"
     {
       fix c :: real
       fix x
       assume x: "x \<in> S"
       have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
       moreover
       have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
         using affine_alt[of S] assm x by auto
       ultimately have "c *\<^sub>R x \<in> S" by auto
     }
     then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
 
     {
       fix x y
       assume xy: "x \<in> S" "y \<in> S"
       define u where "u = (1 :: real)/2"
       have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
         by auto
       moreover
       have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
         by (simp add: algebra_simps)
       moreover
       have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
         using affine_alt[of S] assm xy by auto
       ultimately
       have "(1/2) *\<^sub>R (x+y) \<in> S"
         using u_def by auto
       moreover
       have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
         by auto
       ultimately
       have "x + y \<in> S"
         using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
     }
     then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
       by auto
     then have "subspace S"
       using h1 assm unfolding subspace_def by auto
   }
   then show ?thesis using h0 by metis
 qed
 
 lemma affine_diffs_subspace:
   assumes "affine S" "a \<in> S"
   shows "subspace ((\<lambda>x. (-a)+x) ` S)"
 proof -
   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   have "affine ((\<lambda>x. (-a)+x) ` S)"
     using affine_translation assms by blast
   moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
     using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
   ultimately show ?thesis using subspace_affine by auto
 qed
 
 lemma affine_diffs_subspace_subtract:
   "subspace ((\<lambda>x. x - a) ` S)" if "affine S" "a \<in> S"
   using that affine_diffs_subspace [of _ a] by simp
 
 lemma parallel_subspace_explicit:
   assumes "affine S"
     and "a \<in> S"
   assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
   shows "subspace L \<and> affine_parallel S L"
 proof -
   from assms have "L = plus (- a) ` S" by auto
   then have par: "affine_parallel S L"
     unfolding affine_parallel_def ..
   then have "affine L" using assms parallel_is_affine by auto
   moreover have "0 \<in> L"
     using assms by auto
   ultimately show ?thesis
     using subspace_affine par by auto
 qed
 
 lemma parallel_subspace_aux:
   assumes "subspace A"
     and "subspace B"
     and "affine_parallel A B"
   shows "A \<supseteq> B"
 proof -
   from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
     using affine_parallel_expl[of A B] by auto
   then have "-a \<in> A"
     using assms subspace_0[of B] by auto
   then have "a \<in> A"
     using assms subspace_neg[of A "-a"] by auto
   then show ?thesis
     using assms a unfolding subspace_def by auto
 qed
 
 lemma parallel_subspace:
   assumes "subspace A"
     and "subspace B"
     and "affine_parallel A B"
   shows "A = B"
 proof
   show "A \<supseteq> B"
     using assms parallel_subspace_aux by auto
   show "A \<subseteq> B"
     using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
 qed
 
 lemma affine_parallel_subspace:
   assumes "affine S" "S \<noteq> {}"
   shows "\<exists>!L. subspace L \<and> affine_parallel S L"
 proof -
   have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
     using assms parallel_subspace_explicit by auto
   {
     fix L1 L2
     assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
     then have "affine_parallel L1 L2"
       using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
     then have "L1 = L2"
       using ass parallel_subspace by auto
   }
   then show ?thesis using ex by auto
 qed
 
 
 subsection \<open>Affine Dependence\<close>
 
 text "Formalized by Lars Schewe."
 
 definition\<^marker>\<open>tag important\<close> affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
   where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
 
 lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
   unfolding affine_dependent_def dependent_def
   using affine_hull_subset_span by auto
 
 lemma affine_dependent_subset:
    "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
 apply (simp add: affine_dependent_def Bex_def)
 apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
 done
 
 lemma affine_independent_subset:
   shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
 by (metis affine_dependent_subset)
 
 lemma affine_independent_Diff:
    "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
 by (meson Diff_subset affine_dependent_subset)
 
 proposition affine_dependent_explicit:
   "affine_dependent p \<longleftrightarrow>
     (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
 proof -
   have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
     if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
   proof (intro exI conjI)
     have "x \<notin> S" 
       using that by auto
     then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
       using that by (simp add: sum_delta_notmem)
     show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
       using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
   qed (use that in auto)
   moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
     if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
   proof (intro bexI exI conjI)
     have "S \<noteq> {v}"
       using that by auto
     then show "S - {v} \<noteq> {}"
       using that by auto
     show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
       unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
     show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
       unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
                 scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>] 
       using that by auto
     show "S - {v} \<subseteq> p - {v}"
       using that by auto
   qed (use that in auto)
   ultimately show ?thesis
     unfolding affine_dependent_def affine_hull_explicit by auto
 qed
 
 lemma affine_dependent_explicit_finite:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows "affine_dependent S \<longleftrightarrow>
     (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
   (is "?lhs = ?rhs")
 proof
   have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
     by auto
   assume ?lhs
   then obtain t u v where
     "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
     unfolding affine_dependent_explicit by auto
   then show ?rhs
     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
     apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
     done
 next
   assume ?rhs
   then obtain u v where "sum u S = 0"  "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by auto
   then show ?lhs unfolding affine_dependent_explicit
     using assms by auto
 qed
 
 lemma dependent_imp_affine_dependent:
   assumes "dependent {x - a| x . x \<in> s}"
     and "a \<notin> s"
   shows "affine_dependent (insert a s)"
 proof -
   from assms(1)[unfolded dependent_explicit] obtain S u v
     where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by auto
   define t where "t = (\<lambda>x. x + a) ` S"
 
   have inj: "inj_on (\<lambda>x. x + a) S"
     unfolding inj_on_def by auto
   have "0 \<notin> S"
     using obt(2) assms(2) unfolding subset_eq by auto
   have fin: "finite t" and "t \<subseteq> s"
     unfolding t_def using obt(1,2) by auto
   then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
     by auto
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
     apply (rule sum.cong)
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     apply auto
     done
   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
     unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
   moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
     using obt(3,4) \<open>0\<notin>S\<close>
     by (rule_tac x="v + a" in bexI) (auto simp: t_def)
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
     unfolding scaleR_left.sum
     unfolding t_def and sum.reindex[OF inj] and o_def
     using obt(5)
     by (auto simp: sum.distrib scaleR_right_distrib)
   then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
     unfolding sum_clauses(2)[OF fin]
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     by (auto simp: *)
   ultimately show ?thesis
     unfolding affine_dependent_explicit
     apply (rule_tac x="insert a t" in exI, auto)
     done
 qed
 
 lemma affine_dependent_biggerset:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<ge> DIM('a) + 2"
   shows "affine_dependent s"
 proof -
   have "s \<noteq> {}" using assms by auto
   then obtain a where "a\<in>s" by auto
   have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
     by auto
   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
     unfolding * by (simp add: card_image inj_on_def)
   also have "\<dots> > DIM('a)" using assms(2)
     unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
   finally show ?thesis
     apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
     apply (rule dependent_imp_affine_dependent)
     apply (rule dependent_biggerset, auto)
     done
 qed
 
 lemma affine_dependent_biggerset_general:
   assumes "finite (S :: 'a::euclidean_space set)"
     and "card S \<ge> dim S + 2"
   shows "affine_dependent S"
 proof -
   from assms(2) have "S \<noteq> {}" by auto
   then obtain a where "a\<in>S" by auto
   have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
     by auto
   have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
     by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
   have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
     using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
   also have "\<dots> < dim S + 1" by auto
   also have "\<dots> \<le> card (S - {a})"
     using assms
     using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
     by auto
   finally show ?thesis
     apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
     apply (rule dependent_imp_affine_dependent)
     apply (rule dependent_biggerset_general)
     unfolding **
     apply auto
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some Properties of Affine Dependent Sets\<close>
 
 lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
   by (simp add: affine_dependent_def)
 
 lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
   by (simp add: affine_dependent_def)
 
 lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
   by (simp add: affine_dependent_def insert_Diff_if hull_same)
 
 lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
 proof -
   have "affine ((\<lambda>x. a + x) ` (affine hull S))"
     using affine_translation affine_affine_hull by blast
   moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
     using hull_subset[of S] by auto
   ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
     by (metis hull_minimal)
   have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
     using affine_translation affine_affine_hull by blast
   moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
     using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
   moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
     using translation_assoc[of "-a" a] by auto
   ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
     by (metis hull_minimal)
   then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
     by auto
   then show ?thesis using h1 by auto
 qed
 
 lemma affine_dependent_translation:
   assumes "affine_dependent S"
   shows "affine_dependent ((\<lambda>x. a + x) ` S)"
 proof -
   obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
     using assms affine_dependent_def by auto
   have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
     by auto
   then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
     using affine_hull_translation[of a "S - {x}"] x by auto
   moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
     using x by auto
   ultimately show ?thesis
     unfolding affine_dependent_def by auto
 qed
 
 lemma affine_dependent_translation_eq:
   "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
 proof -
   {
     assume "affine_dependent ((\<lambda>x. a + x) ` S)"
     then have "affine_dependent S"
       using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
       by auto
   }
   then show ?thesis
     using affine_dependent_translation by auto
 qed
 
 lemma affine_hull_0_dependent:
   assumes "0 \<in> affine hull S"
   shows "dependent S"
 proof -
   obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     using assms affine_hull_explicit[of S] by auto
   then have "\<exists>v\<in>s. u v \<noteq> 0" by auto
   then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
     using s_u by auto
   then show ?thesis
     unfolding dependent_explicit[of S] by auto
 qed
 
 lemma affine_dependent_imp_dependent2:
   assumes "affine_dependent (insert 0 S)"
   shows "dependent S"
 proof -
   obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
     using affine_dependent_def[of "(insert 0 S)"] assms by blast
   then have "x \<in> span (insert 0 S - {x})"
     using affine_hull_subset_span by auto
   moreover have "span (insert 0 S - {x}) = span (S - {x})"
     using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
   ultimately have "x \<in> span (S - {x})" by auto
   then have "x \<noteq> 0 \<Longrightarrow> dependent S"
     using x dependent_def by auto
   moreover
   {
     assume "x = 0"
     then have "0 \<in> affine hull S"
       using x hull_mono[of "S - {0}" S] by auto
     then have "dependent S"
       using affine_hull_0_dependent by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma affine_dependent_iff_dependent:
   assumes "a \<notin> S"
   shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
 proof -
   have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
   then show ?thesis
     using affine_dependent_translation_eq[of "(insert a S)" "-a"]
       affine_dependent_imp_dependent2 assms
       dependent_imp_affine_dependent[of a S]
     by (auto simp del: uminus_add_conv_diff)
 qed
 
 lemma affine_dependent_iff_dependent2:
   assumes "a \<in> S"
   shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
 proof -
   have "insert a (S - {a}) = S"
     using assms by auto
   then show ?thesis
     using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
 qed
 
 lemma affine_hull_insert_span_gen:
   "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
 proof -
   have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
     by auto
   {
     assume "a \<notin> s"
     then have ?thesis
       using affine_hull_insert_span[of a s] h1 by auto
   }
   moreover
   {
     assume a1: "a \<in> s"
     have "\<exists>x. x \<in> s \<and> -a+x=0"
       apply (rule exI[of _ a])
       using a1
       apply auto
       done
     then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
       by auto
     then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
       using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
     moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
       by auto
     moreover have "insert a (s - {a}) = insert a s"
       by auto
     ultimately have ?thesis
       using affine_hull_insert_span[of "a" "s-{a}"] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma affine_hull_span2:
   assumes "a \<in> s"
   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
   by auto
 
 lemma affine_hull_span_gen:
   assumes "a \<in> affine hull s"
   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
 proof -
   have "affine hull (insert a s) = affine hull s"
     using hull_redundant[of a affine s] assms by auto
   then show ?thesis
     using affine_hull_insert_span_gen[of a "s"] by auto
 qed
 
 lemma affine_hull_span_0:
   assumes "0 \<in> affine hull S"
   shows "affine hull S = span S"
   using affine_hull_span_gen[of "0" S] assms by auto
 
 lemma extend_to_affine_basis_nonempty:
   fixes S V :: "'n::real_vector set"
   assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
   shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
 proof -
   obtain a where a: "a \<in> S"
     using assms by auto
   then have h0: "independent  ((\<lambda>x. -a + x) ` (S-{a}))"
     using affine_dependent_iff_dependent2 assms by auto
   obtain B where B:
     "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
     using assms
     by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
   define T where "T = (\<lambda>x. a+x) ` insert 0 B"
   then have "T = insert a ((\<lambda>x. a+x) ` B)"
     by auto
   then have "affine hull T = (\<lambda>x. a+x) ` span B"
     using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
     by auto
   then have "V \<subseteq> affine hull T"
     using B assms translation_inverse_subset[of a V "span B"]
     by auto
   moreover have "T \<subseteq> V"
     using T_def B a assms by auto
   ultimately have "affine hull T = affine hull V"
     by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
   moreover have "S \<subseteq> T"
     using T_def B translation_inverse_subset[of a "S-{a}" B]
     by auto
   moreover have "\<not> affine_dependent T"
     using T_def affine_dependent_translation_eq[of "insert 0 B"]
       affine_dependent_imp_dependent2 B
     by auto
   ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
 qed
 
 lemma affine_basis_exists:
   fixes V :: "'n::real_vector set"
   shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
 proof (cases "V = {}")
   case True
   then show ?thesis
     using affine_independent_0 by auto
 next
   case False
   then obtain x where "x \<in> V" by auto
   then show ?thesis
     using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
     by auto
 qed
 
 proposition extend_to_affine_basis:
   fixes S V :: "'n::real_vector set"
   assumes "\<not> affine_dependent S" "S \<subseteq> V"
   obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
 proof (cases "S = {}")
   case True then show ?thesis
     using affine_basis_exists by (metis empty_subsetI that)
 next
   case False
   then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
 qed
 
 
 subsection \<open>Affine Dimension of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
   where "aff_dim V =
   (SOME d :: int.
     \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
 
 lemma aff_dim_basis_exists:
   fixes V :: "('n::euclidean_space) set"
   shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
 proof -
   obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
     using affine_basis_exists[of V] by auto
   then show ?thesis
     unfolding aff_dim_def
       some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
     apply auto
     apply (rule exI[of _ "int (card B) - (1 :: int)"])
     apply (rule exI[of _ "B"], auto)
     done
 qed
 
 lemma affine_hull_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
 by (metis affine_empty subset_empty subset_hull)
 
 lemma empty_eq_affine_hull[simp]: "{} = affine hull S \<longleftrightarrow> S = {}"
 by (metis affine_hull_eq_empty)
 
 lemma aff_dim_parallel_subspace_aux:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B" "a \<in> B"
   shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
 proof -
   have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
     using affine_dependent_iff_dependent2 assms by auto
   then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
     "finite ((\<lambda>x. -a + x) ` (B - {a}))"
     using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
   show ?thesis
   proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
     case True
     have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
       using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
     then have "B = {a}" using True by auto
     then show ?thesis using assms fin by auto
   next
     case False
     then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
       using fin by auto
     moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
       by (rule card_image) (use translate_inj_on in blast)
     ultimately have "card (B-{a}) > 0" by auto
     then have *: "finite (B - {a})"
       using card_gt_0_iff[of "(B - {a})"] by auto
     then have "card (B - {a}) = card B - 1"
       using card_Diff_singleton assms by auto
     with * show ?thesis using fin h1 by auto
   qed
 qed
 
 lemma aff_dim_parallel_subspace:
   fixes V L :: "'n::euclidean_space set"
   assumes "V \<noteq> {}"
     and "subspace L"
     and "affine_parallel (affine hull V) L"
   shows "aff_dim V = int (dim L)"
 proof -
   obtain B where
     B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
     using aff_dim_basis_exists by auto
   then have "B \<noteq> {}"
     using assms B
     by auto
   then obtain a where a: "a \<in> B" by auto
   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
   moreover have "affine_parallel (affine hull B) Lb"
     using Lb_def B assms affine_hull_span2[of a B] a
       affine_parallel_commut[of "Lb" "(affine hull B)"]
     unfolding affine_parallel_def
     by auto
   moreover have "subspace Lb"
     using Lb_def subspace_span by auto
   moreover have "affine hull B \<noteq> {}"
     using assms B by auto
   ultimately have "L = Lb"
     using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
     by auto
   then have "dim L = dim Lb"
     by auto
   moreover have "card B - 1 = dim Lb" and "finite B"
     using Lb_def aff_dim_parallel_subspace_aux a B by auto
   ultimately show ?thesis
     using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
 qed
 
 lemma aff_independent_finite:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B"
   shows "finite B"
 proof -
   {
     assume "B \<noteq> {}"
     then obtain a where "a \<in> B" by auto
     then have ?thesis
       using aff_dim_parallel_subspace_aux assms by auto
   }
   then show ?thesis by auto
 qed
 
 
 lemma aff_dim_empty:
   fixes S :: "'n::euclidean_space set"
   shows "S = {} \<longleftrightarrow> aff_dim S = -1"
 proof -
   obtain B where *: "affine hull B = affine hull S"
     and "\<not> affine_dependent B"
     and "int (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by auto
   moreover
   from * have "S = {} \<longleftrightarrow> B = {}"
     by auto
   ultimately show ?thesis
     using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
 qed
 
 lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
   by (simp add: aff_dim_empty [symmetric])
 
 lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
   unfolding aff_dim_def using hull_hull[of _ S] by auto
 
 lemma aff_dim_affine_hull2:
   assumes "affine hull S = affine hull T"
   shows "aff_dim S = aff_dim T"
   unfolding aff_dim_def using assms by auto
 
 lemma aff_dim_unique:
   fixes B V :: "'n::euclidean_space set"
   assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
   shows "of_nat (card B) = aff_dim V + 1"
 proof (cases "B = {}")
   case True
   then have "V = {}"
     using assms
     by auto
   then have "aff_dim V = (-1::int)"
     using aff_dim_empty by auto
   then show ?thesis
     using \<open>B = {}\<close> by auto
 next
   case False
   then obtain a where a: "a \<in> B" by auto
   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
   have "affine_parallel (affine hull B) Lb"
     using Lb_def affine_hull_span2[of a B] a
       affine_parallel_commut[of "Lb" "(affine hull B)"]
     unfolding affine_parallel_def by auto
   moreover have "subspace Lb"
     using Lb_def subspace_span by auto
   ultimately have "aff_dim B = int(dim Lb)"
     using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
   moreover have "(card B) - 1 = dim Lb" "finite B"
     using Lb_def aff_dim_parallel_subspace_aux a assms by auto
   ultimately have "of_nat (card B) = aff_dim B + 1"
     using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
   then show ?thesis
     using aff_dim_affine_hull2 assms by auto
 qed
 
 lemma aff_dim_affine_independent:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B"
   shows "of_nat (card B) = aff_dim B + 1"
   using aff_dim_unique[of B B] assms by auto
 
 lemma affine_independent_iff_card:
     fixes s :: "'a::euclidean_space set"
     shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
   apply (rule iffI)
   apply (simp add: aff_dim_affine_independent aff_independent_finite)
   by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
 
 lemma aff_dim_sing [simp]:
   fixes a :: "'n::euclidean_space"
   shows "aff_dim {a} = 0"
   using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
 
 lemma aff_dim_2 [simp]:
   fixes a :: "'n::euclidean_space"
   shows "aff_dim {a,b} = (if a = b then 0 else 1)"
 proof (clarsimp)
   assume "a \<noteq> b"
   then have "aff_dim{a,b} = card{a,b} - 1"
     using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
   also have "\<dots> = 1"
     using \<open>a \<noteq> b\<close> by simp
   finally show "aff_dim {a, b} = 1" .
 qed
 
 lemma aff_dim_inner_basis_exists:
   fixes V :: "('n::euclidean_space) set"
   shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
     \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
 proof -
   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
     using affine_basis_exists[of V] by auto
   then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_le_card:
   fixes V :: "'n::euclidean_space set"
   assumes "finite V"
   shows "aff_dim V \<le> of_nat (card V) - 1"
 proof -
   obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
     using aff_dim_inner_basis_exists[of V] by auto
   then have "card B \<le> card V"
     using assms card_mono by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_parallel_eq:
   fixes S T :: "'n::euclidean_space set"
   assumes "affine_parallel (affine hull S) (affine hull T)"
   shows "aff_dim S = aff_dim T"
 proof -
   {
     assume "T \<noteq> {}" "S \<noteq> {}"
     then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
       using affine_parallel_subspace[of "affine hull T"]
         affine_affine_hull[of T]
       by auto
     then have "aff_dim T = int (dim L)"
       using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
     moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
        using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
     moreover from * have "aff_dim S = int (dim L)"
       using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
     ultimately have ?thesis by auto
   }
   moreover
   {
     assume "S = {}"
     then have "S = {}" and "T = {}"
       using assms
       unfolding affine_parallel_def
       by auto
     then have ?thesis using aff_dim_empty by auto
   }
   moreover
   {
     assume "T = {}"
     then have "S = {}" and "T = {}"
       using assms
       unfolding affine_parallel_def
       by auto
     then have ?thesis
       using aff_dim_empty by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma aff_dim_translation_eq:
   "aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space"
 proof -
   have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
     unfolding affine_parallel_def
     apply (rule exI[of _ "a"])
     using affine_hull_translation[of a S]
     apply auto
     done
   then show ?thesis
     using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
 qed
 
 lemma aff_dim_translation_eq_subtract:
   "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space"
   using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp)
 
 lemma aff_dim_affine:
   fixes S L :: "'n::euclidean_space set"
   assumes "S \<noteq> {}"
     and "affine S"
     and "subspace L"
     and "affine_parallel S L"
   shows "aff_dim S = int (dim L)"
 proof -
   have *: "affine hull S = S"
     using assms affine_hull_eq[of S] by auto
   then have "affine_parallel (affine hull S) L"
     using assms by (simp add: *)
   then show ?thesis
     using assms aff_dim_parallel_subspace[of S L] by blast
 qed
 
 lemma dim_affine_hull:
   fixes S :: "'n::euclidean_space set"
   shows "dim (affine hull S) = dim S"
 proof -
   have "dim (affine hull S) \<ge> dim S"
     using dim_subset by auto
   moreover have "dim (span S) \<ge> dim (affine hull S)"
     using dim_subset affine_hull_subset_span by blast
   moreover have "dim (span S) = dim S"
     using dim_span by auto
   ultimately show ?thesis by auto
 qed
 
 lemma aff_dim_subspace:
   fixes S :: "'n::euclidean_space set"
   assumes "subspace S"
   shows "aff_dim S = int (dim S)"
 proof (cases "S={}")
   case True with assms show ?thesis
     by (simp add: subspace_affine)
 next
   case False
   with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
   show ?thesis by auto
 qed
 
 lemma aff_dim_zero:
   fixes S :: "'n::euclidean_space set"
   assumes "0 \<in> affine hull S"
   shows "aff_dim S = int (dim S)"
 proof -
   have "subspace (affine hull S)"
     using subspace_affine[of "affine hull S"] affine_affine_hull assms
     by auto
   then have "aff_dim (affine hull S) = int (dim (affine hull S))"
     using assms aff_dim_subspace[of "affine hull S"] by auto
   then show ?thesis
     using aff_dim_affine_hull[of S] dim_affine_hull[of S]
     by auto
 qed
 
 lemma aff_dim_eq_dim:
   "aff_dim S = int (dim ((+) (- a) ` S))" if "a \<in> affine hull S"
     for S :: "'n::euclidean_space set"
 proof -
   have "0 \<in> affine hull (+) (- a) ` S"
     unfolding affine_hull_translation
     using that by (simp add: ac_simps)
   with aff_dim_zero show ?thesis
     by (metis aff_dim_translation_eq)
 qed
 
 lemma aff_dim_eq_dim_subtract:
   "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))" if "a \<in> affine hull S"
     for S :: "'n::euclidean_space set"
   using aff_dim_eq_dim [of a] that by (simp cong: image_cong_simp)
 
 lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
   using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
     dim_UNIV[where 'a="'n::euclidean_space"]
   by auto
 
 lemma aff_dim_geq:
   fixes V :: "'n::euclidean_space set"
   shows "aff_dim V \<ge> -1"
 proof -
   obtain B where "affine hull B = affine hull V"
     and "\<not> affine_dependent B"
     and "int (card B) = aff_dim V + 1"
     using aff_dim_basis_exists by auto
   then show ?thesis by auto
 qed
 
 lemma aff_dim_negative_iff [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
 by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
 
 lemma aff_lowdim_subset_hyperplane:
   fixes S :: "'a::euclidean_space set"
   assumes "aff_dim S < DIM('a)"
   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
 proof (cases "S={}")
   case True
   moreover
   have "(SOME b. b \<in> Basis) \<noteq> 0"
     by (metis norm_some_Basis norm_zero zero_neq_one)
   ultimately show ?thesis
     using that by blast
 next
   case False
   then obtain c S' where "c \<notin> S'" "S = insert c S'"
     by (meson equals0I mk_disjoint_insert)
   have "dim ((+) (-c) ` S) < DIM('a)"
     by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
   then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
     using lowdim_subset_hyperplane by blast
   moreover
   have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
   proof -
     have "w-c \<in> span ((+) (- c) ` S)"
       by (simp add: span_base \<open>w \<in> S\<close>)
     with that have "w-c \<in> {x. a \<bullet> x = 0}"
       by blast
     then show ?thesis
       by (auto simp: algebra_simps)
   qed
   ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
     by blast
   then show ?thesis
     by (rule that[OF \<open>a \<noteq> 0\<close>])
 qed
 
 lemma affine_independent_card_dim_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "\<not> affine_dependent S" "a \<in> S"
-    shows "card S = dim {x - a|x. x \<in> S} + 1"
+    shows "card S = dim ((\<lambda>x. x - a) ` S) + 1"
 proof -
-  have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
-  have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
-  proof (cases "x = a")
-    case True then show ?thesis by (simp add: span_clauses)
-  next
-    case False then show ?thesis
-      using assms by (blast intro: span_base that)
-  qed
-  have "\<not> affine_dependent (insert a S)"
+  have non: "\<not> affine_dependent (insert a S)"
     by (simp add: assms insert_absorb)
-  then have 3: "independent {b - a |b. b \<in> S - {a}}"
-      using dependent_imp_affine_dependent by fastforce
-  have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
-    by blast
-  then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
-    by simp
-  also have "\<dots> = card (S - {a})"
-    by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
-  also have "\<dots> = card S - 1"
-    by (simp add: aff_independent_finite assms)
-  finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
   have "finite S"
     by (meson assms aff_independent_finite)
   with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
-  moreover have "dim {x - a |x. x \<in> S} = card S - 1"
-    using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
+  moreover have "dim ((\<lambda>x. x - a) ` S) = card S - 1"
+    using aff_dim_eq_dim_subtract aff_dim_unique \<open>a \<in> S\<close> hull_inc insert_absorb non by fastforce
   ultimately show ?thesis
     by auto
 qed
 
 lemma independent_card_le_aff_dim:
   fixes B :: "'n::euclidean_space set"
   assumes "B \<subseteq> V"
   assumes "\<not> affine_dependent B"
   shows "int (card B) \<le> aff_dim V + 1"
 proof -
   obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
     by (metis assms extend_to_affine_basis[of B V])
   then have "of_nat (card T) = aff_dim V + 1"
     using aff_dim_unique by auto
   then show ?thesis
     using T card_mono[of T B] aff_independent_finite[of T] by auto
 qed
 
 lemma aff_dim_subset:
   fixes S T :: "'n::euclidean_space set"
   assumes "S \<subseteq> T"
   shows "aff_dim S \<le> aff_dim T"
 proof -
   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
     "of_nat (card B) = aff_dim S + 1"
     using aff_dim_inner_basis_exists[of S] by auto
   then have "int (card B) \<le> aff_dim T + 1"
     using assms independent_card_le_aff_dim[of B T] by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_le_DIM:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim S \<le> int (DIM('n))"
 proof -
   have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
     using aff_dim_UNIV by auto
   then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
     using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
 qed
 
 lemma affine_dim_equal:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
   shows "S = T"
 proof -
   obtain a where "a \<in> S" using assms by auto
   then have "a \<in> T" using assms by auto
   define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
   then have ls: "subspace LS" "affine_parallel S LS"
     using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
   then have h1: "int(dim LS) = aff_dim S"
     using assms aff_dim_affine[of S LS] by auto
   have "T \<noteq> {}" using assms by auto
   define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
   then have lt: "subspace LT \<and> affine_parallel T LT"
     using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
   then have "int(dim LT) = aff_dim T"
     using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
   then have "dim LS = dim LT"
     using h1 assms by auto
   moreover have "LS \<le> LT"
     using LS_def LT_def assms by auto
   ultimately have "LS = LT"
     using subspace_dim_equal[of LS LT] ls lt by auto
   moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
     using LS_def by auto
   moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
     using LT_def by auto
   ultimately show ?thesis by auto
 qed
 
 lemma aff_dim_eq_0:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by auto
 next
   case False
   then obtain a where "a \<in> S" by auto
   show ?thesis
   proof safe
     assume 0: "aff_dim S = 0"
     have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
       by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
     then show "\<exists>a. S = {a}"
       using \<open>a \<in> S\<close> by blast
   qed auto
 qed
 
 lemma affine_hull_UNIV:
   fixes S :: "'n::euclidean_space set"
   assumes "aff_dim S = int(DIM('n))"
   shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
 proof -
   have "S \<noteq> {}"
     using assms aff_dim_empty[of S] by auto
   have h0: "S \<subseteq> affine hull S"
     using hull_subset[of S _] by auto
   have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
     using aff_dim_UNIV assms by auto
   then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
     using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
   have h3: "aff_dim S \<le> aff_dim (affine hull S)"
     using h0 aff_dim_subset[of S "affine hull S"] assms by auto
   then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
     using h0 h1 h2 by auto
   then show ?thesis
     using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
       affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
     by auto
 qed
 
 lemma disjoint_affine_hull:
   fixes s :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
     shows "(affine hull t) \<inter> (affine hull u) = {}"
 proof -
   have "finite s" using assms by (simp add: aff_independent_finite)
   then have "finite t" "finite u" using assms finite_subset by blast+
   { fix y
     assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
     then obtain a b
            where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
              and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
       by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
     define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
     have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
     have "sum c s = 0"
       by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
     moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
       by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum.neutral zero_neq_one)
     moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
       by (simp add: c_def if_smult sum_negf
              comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
     ultimately have False
       using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
   }
   then show ?thesis by blast
 qed
 
 end
\ No newline at end of file
diff --git a/src/HOL/Analysis/Starlike.thy b/src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy
+++ b/src/HOL/Analysis/Starlike.thy
@@ -1,6397 +1,6404 @@
 (* Title:      HOL/Analysis/Starlike.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 chapter \<open>Unsorted\<close>
 
 theory Starlike
   imports
     Convex_Euclidean_Space
     Line_Segment
 begin
 
 lemma affine_hull_closed_segment [simp]:
      "affine hull (closed_segment a b) = affine hull {a,b}"
   by (simp add: segment_convex_hull)
 
 lemma affine_hull_open_segment [simp]:
     fixes a :: "'a::euclidean_space"
     shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
 by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
 
 lemma rel_interior_closure_convex_segment:
   fixes S :: "_::euclidean_space set"
   assumes "convex S" "a \<in> rel_interior S" "b \<in> closure S"
     shows "open_segment a b \<subseteq> rel_interior S"
 proof
   fix x
   have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
     by (simp add: algebra_simps)
   assume "x \<in> open_segment a b"
   then show "x \<in> rel_interior S"
     unfolding closed_segment_def open_segment_def  using assms
     by (auto intro: rel_interior_closure_convex_shrink)
 qed
 
 lemma convex_hull_insert_segments:
    "convex hull (insert a S) =
     (if S = {} then {a} else  \<Union>x \<in> convex hull S. closed_segment a x)"
   by (force simp add: convex_hull_insert_alt in_segment)
 
 lemma Int_convex_hull_insert_rel_exterior:
   fixes z :: "'a::euclidean_space"
   assumes "convex C" "T \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
   shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
 proof
   have "T = {} \<Longrightarrow> z \<notin> S"
     using dis z by (auto simp add: disjnt_def)
   then show "?lhs \<subseteq> ?rhs"
   proof (clarsimp simp add: convex_hull_insert_segments)
     fix x y
     assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
     have "y \<in> closure C"
       by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
     moreover have "x \<notin> rel_interior C"
       by (meson \<open>x \<in> S\<close> dis disjnt_iff)
     moreover have "x \<in> open_segment z y \<union> {z, y}"
       using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
     ultimately show "x \<in> convex hull T"
       using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
       using y z by blast
   qed
   show "?rhs \<subseteq> ?lhs"
     by (meson hull_mono inf_mono subset_insertI subset_refl)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<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"
     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"
     using assms(2) unfolding mem_interior by auto
   show ?thesis
     unfolding mem_interior
   proof (intro exI subsetI conjI)
     fix y
     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 "c - ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = (1 / e) *\<^sub>R (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       using \<open>e > 0\<close>
       by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     then have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       by (simp add: dist_norm)
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
     finally have "(1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x \<in> S"
       using assms(3-5) d
       by (intro convexD_alt [OF \<open>convex S\<close>]) (auto intro: convexD_alt [OF \<open>convex S\<close>])
     with \<open>e > 0\<close> show "y \<in> S"
       by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
   qed (use \<open>e>0\<close> \<open>d>0\<close> in auto)
 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"
     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"
     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")
     case True
     then show ?thesis
       using \<open>e > 0\<close> \<open>d > 0\<close> by force
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
         using True \<open>0 < d\<close> by auto
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using islimpt_approachable x by blast
       then have "norm (y - x) * (1 - e) < e * d"
         by (metis "*" dist_norm mult_imp_div_pos_le not_less)
       then show ?thesis
         using \<open>y \<in> S\<close> by blast
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
-  have "z \<in> interior (ball c d)"
-    using y \<open>0 < e\<close> \<open>e \<le> 1\<close>
-    apply (simp add: interior_open[OF open_ball] z_def dist_norm)
-    by (simp add: field_simps norm_minus_commute)
+  have "(1 - e) * norm (x - y) / e < d"
+    using y \<open>0 < e\<close> by (simp add: field_simps norm_minus_commute)
+  then have "z \<in> interior (ball c d)"
+    using \<open>0 < e\<close> \<open>e \<le> 1\<close> by (simp add: interior_open[OF open_ball] z_def dist_norm)
   then have "z \<in> interior S"
     using d interiorI interior_ball by blast
   then show ?thesis
-    unfolding *
-    using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
+    unfolding * using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
 qed
 
 lemma in_interior_closure_convex_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
     shows "open_segment a b \<subseteq> interior S"
 proof (clarsimp simp: in_segment)
   fix u::real
   assume u: "0 < u" "u < 1"
   have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
     by (simp add: algebra_simps)
   also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
     by simp
   finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
 qed
 
 lemma convex_closure_interior:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and int: "interior S \<noteq> {}"
   shows "closure(interior S) = closure S"
 proof -
   obtain a where a: "a \<in> interior S"
     using int by auto
   have "closure S \<subseteq> closure(interior S)"
   proof
     fix x
     assume x: "x \<in> closure S"
     show "x \<in> closure (interior S)"
     proof (cases "x=a")
       case True
       then show ?thesis
         using \<open>a \<in> interior S\<close> closure_subset by blast
     next
       case False
       show ?thesis
       proof (clarsimp simp add: closure_def islimpt_approachable)
         fix e::real
         assume xnotS: "x \<notin> interior S" and "0 < e"
         show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
         proof (intro bexI conjI)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
             using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
           show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
             using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
             using \<open>0 < e\<close> False
             by (auto simp add: min_def a intro: mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
         qed
       qed
     qed
   qed
   then show ?thesis
     by (simp add: closure_mono interior_subset subset_antisym)
 qed
 
 lemma closure_convex_Int_superset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
   shows "closure(S \<inter> T) = closure S"
 proof -
   have "closure S \<subseteq> closure(interior S)"
     by (simp add: convex_closure_interior assms)
   also have "... \<subseteq> closure (S \<inter> T)"
     using interior_subset [of S] assms
     by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
   finally show ?thesis
     by (simp add: closure_mono dual_order.antisym)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<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}"
 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"
   shows "convex hull (insert 0 d) =
     {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
   (is "convex hull (insert 0 ?p) = ?s")
 proof -
   let ?D = d
   have "0 \<notin> ?p"
     using assms by (auto simp: image_def)
   from d have "finite d"
     by (blast intro: finite_subset finite_Basis)
   show ?thesis
     unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
   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 
     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
 
 lemma std_simplex:
   "convex hull (insert 0 Basis) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
   using substd_simplex[of Basis] by auto
 
 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}"
   unfolding set_eq_iff mem_interior std_simplex
 proof (intro allI iffI CollectI; clarify)
   fix x :: 'a
   fix e
   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 subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close> 
+      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>
+    have **: "dist x (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
       unfolding dist_norm
       by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
-    have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
+    have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
       x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
       by (auto simp: SOME_Basis inner_Basis inner_simps)
-    then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
+    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"
       by (auto simp: intro!: sum.cong)
-    have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
+    have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
       using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
     also have "\<dots> \<le> 1"
       using ** as by force
     finally show "sum ((\<bullet>) x) Basis < 1" by auto
   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. 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: "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"
       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"
       unfolding sum.distrib sum_constant
       by (auto simp add: Suc_le_eq)
     finally show "sum ((\<bullet>) y) Basis \<le> 1" .
     show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
         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)
     qed
   next
     have "Min (((\<bullet>) x) ` Basis) > 0"
       using as by simp
     moreover have "?d > 0"
       using as by (auto simp: Suc_le_eq)
     ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
       by linarith
   qed 
 qed
 
 lemma interior_std_simplex_nonempty:
   obtains a :: "'a::euclidean_space" where
     "a \<in> interior(convex hull (insert 0 Basis))"
 proof -
   let ?D = "Basis :: 'a set"
   let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
   {
     fix i :: 'a
     assume i: "i \<in> Basis"
     have "?a \<bullet> i = inverse (2 * real DIM('a))"
       by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
          (simp_all add: sum.If_cases i) }
   note ** = this
   show ?thesis
   proof
     show "?a \<in> interior(convex hull (insert 0 Basis))"
       unfolding interior_std_simplex mem_Collect_eq
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       show "0 < ?a \<bullet> i"
         unfolding **[OF i] by (auto simp add: Suc_le_eq)
     next
       have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
         by (auto intro: sum.cong)
       also have "\<dots> < 1"
         unfolding sum_constant divide_inverse[symmetric]
         by (auto simp add: field_simps)
       finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
     qed
   qed
 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)}"
-  (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
+         {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 "_ = ?s")
 proof -
   have "finite D"
     using D finite_Basis finite_subset by blast
   show ?thesis
   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)}"
+    have h0: "affine hull (convex hull (insert 0 D)) =
+              {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"
       by auto
     {
       fix x :: "'a::euclidean_space"
-      assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
+      assume x: "x \<in> rel_interior (convex hull (insert 0 D))"
       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 [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
+        "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 D)"
+        using mem_rel_interior_ball[of x "convex hull (insert 0 D)"] h0 by auto
+      then have as: "\<And>y. \<lbrakk>dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0)\<rbrakk> \<Longrightarrow>
+                            (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
+        using assms by (force simp: substd_simplex)
       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)"
       proof (intro conjI ballI)
         fix i :: 'a
         assume "i \<in> D"
-        then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
+        then have "\<forall>j\<in>D. 0 \<le> (x - (e/2) *\<^sub>R i) \<bullet> j"
           using D \<open>e > 0\<close> x0
-          by (force simp: dist_norm inner_simps inner_Basis intro!: as[THEN conjunct1])
+          by (intro as[THEN conjunct1]) (force simp: dist_norm inner_simps inner_Basis)
         then show "0 < x \<bullet> i"
           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
-        then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
-          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)"
+        then have **: "dist x (x + (e/2) *\<^sub>R a) < e"
+          using \<open>e > 0\<close> norm_Basis[of a] D by (auto simp: dist_norm)
+        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"
+        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)"
+        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"
+        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"]
+          using ** h1 as[rule_format, of "x + (e/2) *\<^sub>R a"]
           by auto
         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
     }
     moreover
     {
       fix x :: "'a::euclidean_space"
       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
       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"
         by metis
-      then have h2: "x \<in> convex hull (insert 0 ?p)"
-        using as assms
-        unfolding substd_simplex[OF assms] by fastforce
+      then have h2: "x \<in> convex hull (insert 0 D)"
+        using as assms by (force simp add: substd_simplex)
       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)
-      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 "\<exists>e>0. ball x e \<inter> {x. \<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0}
-          \<subseteq> convex hull insert 0 D"
+      define d where "d \<equiv> (1 - sum ((\<bullet>) x) D) / real (card D)"
+      have "\<exists>e>0. ball x e \<inter> {x. \<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0} \<subseteq> convex hull insert 0 D"
         unfolding substd_simplex[OF assms]
-        apply (rule_tac x="min (Min (((\<bullet>) x) ` D)) ?d" in exI)
-        apply (rule, rule h3)
-        apply safe
-        unfolding mem_ball
-      proof -
+      proof (intro exI; safe)
+        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)
+        moreover have "d > 0" 
+          using as \<open>0 < card D\<close> by (auto simp: d_def)
+        ultimately show "min (Min (((\<bullet>) x) ` D)) d > 0"
+          by auto
         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 \<in> ball x (min (Min ((\<bullet>) x ` D)) d)"
+        then have y: "dist x y < min (Min ((\<bullet>) x ` D)) d"
+          by auto
+        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"
             by auto
           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"
+          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
+          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 d_def using \<open>0 < card D\<close>
           by auto
         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")
           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>
-            by (simp add: card_gt_0_iff)
+            using y Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
+            by (simp add: dist_norm 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]
             by (auto simp: inner_simps)
         qed (use y2 in auto)
       qed
-      then have "x \<in> rel_interior (convex hull (insert 0 ?p))"
+      then have "x \<in> rel_interior (convex hull (insert 0 D))"
         using h0 h2 rel_interior_ball by force
     }
     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)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
 qed
 
 lemma rel_interior_substd_simplex_nonempty:
   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"
+  let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) D"
   have "finite D"
     using assms finite_Basis infinite_super by blast
   then have d1: "0 < real (card D)"
     using \<open>D \<noteq> {}\<close> by auto
   {
     fix i
     assume "i \<in> D"
-    have "?a \<bullet> i = sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"
+    have "?a \<bullet> i = sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) D"
       unfolding inner_sum_left
       using \<open>i \<in> D\<close> by (auto simp: inner_Basis subsetD[OF assms(2)] intro: sum.cong)
     also have "... = inverse (2 * real (card D))"
       using \<open>i \<in> D\<close> \<open>finite D\<close> by auto
     finally have "?a \<bullet> i = inverse (2 * real (card D))" .
   }
   note ** = this
   show ?thesis
   proof
     show "?a \<in> rel_interior (convex hull (insert 0 D))"
       unfolding rel_interior_substd_simplex[OF assms(2)] 
     proof safe
       fix i
       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>
         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]
         by (auto simp add: field_simps)
-      finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
+      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])
         {
           fix x :: "'a::euclidean_space"
           assume "x \<in> D"
           then have "x \<in> span D"
             using span_base[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"
           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
     qed
   qed
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex set\<close>
 
 lemma rel_interior_convex_nonempty_aux:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "0 \<in> S"
   shows "rel_interior S \<noteq> {}"
 proof (cases "S = {0}")
   case True
   then show ?thesis using rel_interior_sing by auto
 next
   case False
   obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
     using basis_exists[of S] by metis
   then have "B \<noteq> {}"
     using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
   have "insert 0 B \<le> span B"
     using subspace_span[of B] subspace_0[of "span B"]
       span_superset by auto
   then have "span (insert 0 B) \<le> span B"
     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
   then have "convex hull insert 0 B \<le> span B"
     using convex_hull_subset_span[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) \<le> span B"
     using span_span[of B]
       span_mono[of "convex hull insert 0 B" "span B"] by blast
   then have *: "span (convex hull insert 0 B) = span B"
     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) = span S"
     using B span_mono[of B S] span_mono[of S "span B"]
       span_span[of B] by auto
   moreover have "0 \<in> affine hull (convex hull insert 0 B)"
     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
       assms hull_subset[of S]
     by auto
   obtain d and f :: "'n \<Rightarrow> 'n" where
     fd: "card d = card B" "linear f" "f ` B = d"
       "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
     and d: "d \<subseteq> Basis"
     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
   then have "bounded_linear f"
     using linear_conv_bounded_linear by auto
   have "d \<noteq> {}"
     using fd B \<open>B \<noteq> {}\<close> by auto
   have "insert 0 d = f ` (insert 0 B)"
     using fd linear_0 by auto
   then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
     using convex_hull_linear_image[of f "(insert 0 d)"]
       convex_hull_linear_image[of f "(insert 0 B)"] \<open>linear f\<close>
     by auto
   moreover have "rel_interior (f ` (convex hull insert 0 B)) = f ` rel_interior (convex hull insert 0 B)"
   proof (rule rel_interior_injective_on_span_linear_image[OF \<open>bounded_linear f\<close>])
     show "inj_on f (span (convex hull insert 0 B))"
       using fd * by auto
   qed
   ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
     using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d] by fastforce
   moreover have "convex hull (insert 0 B) \<subseteq> S"
     using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
   ultimately show ?thesis
     using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 qed
 
 lemma rel_interior_eq_empty:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior S = {} \<longleftrightarrow> S = {}"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain a where "a \<in> S" by auto
     then have "0 \<in> (+) (-a) ` S"
       using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
     then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
       using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
         convex_translation[of S "-a"] assms
       by auto
     then have "rel_interior S \<noteq> {}"
       using rel_interior_translation [of "- a"] by simp
   }
   then show ?thesis by auto
 qed
 
 lemma interior_simplex_nonempty:
   fixes S :: "'N :: euclidean_space set"
   assumes "independent S" "finite S" "card S = DIM('N)"
   obtains a where "a \<in> interior (convex hull (insert 0 S))"
 proof -
   have "affine hull (insert 0 S) = UNIV"
     by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
          assms(1) assms(3) dim_eq_card_independent)
   moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
     using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
   ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
     by (simp add: rel_interior_interior)
   with that show ?thesis
     by auto
 qed
 
 lemma convex_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "convex (rel_interior S)"
 proof -
   {
     fix x y and u :: real
     assume assm: "x \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
     then have "x \<in> S"
       using rel_interior_subset by auto
     have "x - u *\<^sub>R (x-y) \<in> rel_interior S"
     proof (cases "0 = u")
       case False
       then have "0 < u" using assm by auto
       then show ?thesis
         using assm rel_interior_convex_shrink[of S y x u] assms \<open>x \<in> S\<close> by auto
     next
       case True
       then show ?thesis using assm by auto
     qed
     then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> rel_interior S"
       by (simp add: algebra_simps)
   }
   then show ?thesis
     unfolding convex_alt by auto
 qed
 
 lemma convex_closure_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "closure (rel_interior S) = closure S"
 proof -
   have h1: "closure (rel_interior S) \<le> closure S"
     using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
   show ?thesis
   proof (cases "S = {}")
     case False
     then obtain a where a: "a \<in> rel_interior S"
       using rel_interior_eq_empty assms by auto
     { fix x
       assume x: "x \<in> closure S"
       {
         assume "x = a"
         then have "x \<in> closure (rel_interior S)"
           using a unfolding closure_def by auto
       }
       moreover
       {
         assume "x \<noteq> a"
          {
            fix e :: real
            assume "e > 0"
            define e1 where "e1 = min 1 (e/norm (x - a))"
            then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
              using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
              by simp_all
            then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
              using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
              by auto
            have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
-              apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
-              using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
-              apply simp
-              done
+             using "*" \<open>x \<noteq> a\<close> e1 by force
         }
         then have "x islimpt rel_interior S"
           unfolding islimpt_approachable_le by auto
         then have "x \<in> closure(rel_interior S)"
           unfolding closure_def by auto
       }
       ultimately have "x \<in> closure(rel_interior S)" by auto
     }
     then show ?thesis using h1 by auto
-  next
-    case True
-    then have "rel_interior S = {}" by auto
-    then have "closure (rel_interior S) = {}"
-      using closure_empty by auto
-    with True show ?thesis by auto
-  qed
+  qed auto
 qed
 
 lemma rel_interior_same_affine_hull:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "affine hull (rel_interior S) = affine hull S"
   by (metis assms closure_same_affine_hull convex_closure_rel_interior)
 
 lemma rel_interior_aff_dim:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "aff_dim (rel_interior S) = aff_dim S"
   by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
 
 lemma rel_interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (rel_interior S) = rel_interior S"
 proof -
   have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
     using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
   then show ?thesis
     using rel_interior_def by auto
 qed
 
 lemma rel_interior_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_open (rel_interior S)"
   unfolding rel_open_def using rel_interior_rel_interior assms by auto
 
 lemma convex_rel_interior_closure_aux:
   fixes x y z :: "'n::euclidean_space"
   assumes "0 < a" "0 < b" "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
-  obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
+  obtains e where "0 < e" "e < 1" "z = y - e *\<^sub>R (y - x)"
 proof -
   define e where "e = a / (a + b)"
   have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
     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
   also have "\<dots> = y - e *\<^sub>R (y-x)"
     using e_def assms
-    apply (simp add: algebra_simps)
-    using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] add_divide_distrib[of a b "a+b"]
-    apply auto
-    done
+    by (simp add: divide_simps vector_fraction_eq_iff) (simp add: algebra_simps)
   finally have "z = y - e *\<^sub>R (y-x)"
     by auto
-  moreover have "e > 0" using e_def assms by auto
-  moreover have "e \<le> 1" using e_def assms by auto
+  moreover have "e > 0" "e < 1" using e_def assms by auto
   ultimately show ?thesis using that[of e] by auto
 qed
 
 lemma convex_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (closure S) = rel_interior S"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms rel_interior_eq_empty by auto
 next
   case False
   have "rel_interior (closure S) \<supseteq> rel_interior S"
     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
     by auto
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (closure S)"
     obtain x where x: "x \<in> rel_interior S"
       using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
     have "z \<in> rel_interior S"
     proof (cases "x = z")
       case True
       then show ?thesis using x by auto
     next
       case False
       obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S"
         using z rel_interior_cball[of "closure S"] by auto
       hence *: "0 < e/norm(z-x)" using e False by auto
       define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)"
       have yball: "y \<in> cball z e"
         using y_def dist_norm[of z y] e by auto
       have "x \<in> affine hull closure S"
         using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
       moreover have "z \<in> affine hull closure S"
         using z rel_interior_subset hull_subset[of "closure S"] by blast
       ultimately have "y \<in> affine hull closure S"
         using y_def affine_affine_hull[of "closure S"]
           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
       then have "y \<in> closure S" using e yball by auto
       have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
         using y_def by (simp add: algebra_simps)
-      then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>R (y - x)"
+      then obtain e1 where "0 < e1" "e1 < 1" "z = y - e1 *\<^sub>R (y - x)"
         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
         by (auto simp add: algebra_simps)
       then show ?thesis
         using rel_interior_closure_convex_shrink assms x \<open>y \<in> closure S\<close>
-        by auto
+        by fastforce
     qed
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "interior (closure S) = interior S"
   using closure_aff_dim[of S] interior_rel_interior_gen[of S]
     interior_rel_interior_gen[of "closure S"]
     convex_rel_interior_closure[of S] assms
   by auto
 
 lemma closure_eq_rel_interior_eq:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 = rel_interior S2"
   by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
 
 lemma closure_eq_between:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
   (is "?A \<longleftrightarrow> ?B")
 proof
   assume ?A
   then show ?B
     by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
 next
   assume ?B
   then have "closure S1 \<subseteq> closure S2"
     by (metis assms(1) convex_closure_rel_interior closure_mono)
   moreover from \<open>?B\<close> have "closure S1 \<supseteq> closure S2"
     by (metis closed_closure closure_minimal)
   ultimately show ?A ..
 qed
 
 lemma open_inter_closure_rel_interior:
   fixes S A :: "'n::euclidean_space set"
   assumes "convex S"
     and "open A"
   shows "A \<inter> closure S = {} \<longleftrightarrow> A \<inter> rel_interior S = {}"
   by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
 
 lemma rel_interior_open_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(open_segment a b) = open_segment a b"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then
   have "open_segment a b = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
     by (simp add: open_segment_as_ball)
   then show ?thesis
     unfolding rel_interior_eq openin_open
     by (metis Elementary_Metric_Spaces.open_ball False affine_hull_open_segment)
 qed
 
 lemma rel_interior_closed_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(closed_segment a b) =
          (if a = b then {a} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     by simp
        (metis closure_open_segment convex_open_segment convex_rel_interior_closure
               rel_interior_open_segment)
 qed
 
 lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
 
 subsection\<open>The relative frontier of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_frontier S = closure S - rel_interior S"
 
 lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_eq_empty:
     fixes S :: "'n::euclidean_space set"
     shows "rel_frontier S = {} \<longleftrightarrow> affine S"
   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"
     shows "rel_frontier {a} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_affine_hull:
   fixes S :: "'a::euclidean_space set"
   shows "rel_frontier S \<subseteq> affine hull S"
 using closure_affine_hull rel_frontier_def by fastforce
 
 lemma rel_frontier_cball [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
 proof (cases rule: linorder_cases [of r 0])
   case less then show ?thesis
     by (force simp: sphere_def)
 next
   case equal then show ?thesis by simp
 next
   case greater then show ?thesis
     by simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
 qed
 
 lemma rel_frontier_translation:
   fixes a :: "'a::euclidean_space"
   shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
   by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
 
 lemma rel_frontier_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"
   by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
 
 lemma rel_frontier_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull S = UNIV \<Longrightarrow> rel_frontier S = frontier S"
   by (simp add: frontier_def rel_frontier_def rel_interior_interior)
 
 lemma closest_point_in_rel_frontier:
    "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
    \<Longrightarrow> closest_point S x \<in> rel_frontier S"
   by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
 
 lemma closed_rel_frontier [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (rel_frontier S)"
 proof -
   have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
     by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
   show ?thesis
   proof (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
     show "closedin (top_of_set (affine hull S)) (rel_frontier S)"
       by (simp add: "*" rel_frontier_def)
   qed simp
 qed
 
 lemma closed_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "closed S \<Longrightarrow> closed(S - rel_interior S)"
   by (metis closed_rel_frontier closure_closed rel_frontier_def)
 
 lemma compact_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(S - rel_interior S)"
   by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
 
 lemma bounded_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> bounded(rel_frontier S)"
 by (simp add: bounded_closure bounded_diff rel_frontier_def)
 
 lemma compact_rel_frontier_bounded:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> compact(rel_frontier S)"
 using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
 
 lemma compact_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(rel_frontier S)"
 by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
 
 lemma convex_same_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> closure S = closure T"
 by (simp add: closure_eq_rel_interior_eq)
 
 lemma convex_same_rel_interior_closure_straddle:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
              rel_interior S \<subseteq> T \<and> T \<subseteq> closure S"
 by (simp add: closure_eq_between convex_same_rel_interior_closure)
 
 lemma convex_rel_frontier_aff_dim:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
     and "S2 \<noteq> {}"
     and "S1 \<le> rel_frontier S2"
   shows "aff_dim S1 < aff_dim S2"
 proof -
   have "S1 \<subseteq> closure S2"
     using assms unfolding rel_frontier_def by auto
   then have *: "affine hull S1 \<subseteq> affine hull S2"
     using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
   then have "aff_dim S1 \<le> aff_dim S2"
     using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
       aff_dim_subset[of "affine hull S1" "affine hull S2"]
     by auto
   moreover
   {
     assume eq: "aff_dim S1 = aff_dim S2"
     then have "S1 \<noteq> {}"
       using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
     have **: "affine hull S1 = affine hull S2"
       by (simp_all add: * eq \<open>S1 \<noteq> {}\<close> affine_dim_equal)
     obtain a where a: "a \<in> rel_interior S1"
       using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
     obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
        using mem_rel_interior[of a S1] a by auto
     then have "a \<in> T \<inter> closure S2"
       using a assms unfolding rel_frontier_def by auto
     then obtain b where b: "b \<in> T \<inter> rel_interior S2"
       using open_inter_closure_rel_interior[of S2 T] assms T by auto
     then have "b \<in> affine hull S1"
       using rel_interior_subset hull_subset[of S2] ** by auto
     then have "b \<in> S1"
       using T b by auto
     then have False
       using b assms unfolding rel_frontier_def by auto
   }
   ultimately show ?thesis
     using less_le by auto
 qed
 
 lemma convex_rel_interior_if:
   fixes S ::  "'n::euclidean_space set"
   assumes "convex S"
     and "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
 proof -
   obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
     using mem_rel_interior_cball[of z S] assms by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     {
       assume "x \<noteq> z"
       define m where "m = 1 + e1/norm(x-z)"
       hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         have "z \<in> affine hull S"
           using assms rel_interior_subset hull_subset[of S] by auto
         then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> affine hull S"
           using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
           by auto
         have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
           by (simp add: algebra_simps)
         also have "\<dots> = (e - 1) * norm (x-z)"
           using norm_scaleR e by auto
         also have "\<dots> \<le> (m - 1) * norm (x - z)"
           using e mult_right_mono[of _ _ "norm(x-z)"] by auto
         also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
           using m_def by auto
         also have "\<dots> = e1"
           using \<open>x \<noteq> z\<close> e1 by simp
         finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
           by auto
         have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
           using m_def **
           unfolding cball_def dist_norm
           by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
           using e * e1 by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
         using \<open>m> 1 \<close> by auto
     }
     moreover
     {
       assume "x = z"
       define m where "m = 1 + e1"
       then have "m > 1"
         using e1 by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
         using \<open>m > 1\<close> by auto
     }
     ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
       by blast
   }
   then show ?thesis by auto
 qed
 
 lemma convex_rel_interior_if2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   assumes "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
   using convex_rel_interior_if[of S z] assms by auto
 
 lemma convex_rel_interior_only_if:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
   shows "z \<in> rel_interior S"
 proof -
   obtain x where x: "x \<in> rel_interior S"
     using rel_interior_eq_empty assms by auto
   then have "x \<in> S"
     using rel_interior_subset by auto
   then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
     using assms by auto
   define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
   then have "y \<in> S" using e by auto
   define e1 where "e1 = 1/e"
   then have "0 < e1 \<and> e1 < 1" using e by auto
   then have "z  =y - (1 - e1) *\<^sub>R (y - x)"
     using e1_def y_def by (auto simp add: algebra_simps)
   then show ?thesis
     using rel_interior_convex_shrink[of S x y "1-e1"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
     by auto
 qed
 
 lemma convex_rel_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S "affine"]
     convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_rel_interior_iff2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   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"
     then have False
       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
       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"
         using r by auto
       define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
       then have x1: "x1 \<in> affine hull S"
         using e1 hull_subset[of S] by auto
       define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
       then have x2: "x2 \<in> affine hull S"
         using e2 hull_subset[of S] by auto
       have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
         using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
       then have "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
-        using x1_def x2_def
-        apply (auto simp add: algebra_simps)
-        using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
-        apply auto
-        done
+        by (simp add: x1_def x2_def algebra_simps) (simp add: "*" pth_8)
       then have z: "z \<in> affine hull S"
         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
           x1 x2 affine_affine_hull[of S] *
         by auto
       have "x1 - x2 = (e1 + e2) *\<^sub>R (x - z)"
         using x1_def x2_def by (auto simp add: algebra_simps)
       then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
         using e1 e2 by simp
       then have "x \<in> affine hull S"
         using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
           x1 x2 z affine_affine_hull[of S]
         by auto
     }
     then have "affine hull S = UNIV"
       by auto
     then have "aff_dim S = int DIM('n)"
       using aff_dim_affine_hull[of S] by (simp)
     then have False
       using False by auto
   }
   ultimately show ?thesis by auto
 next
   case True
   then have "S \<noteq> {}"
     using aff_dim_empty[of S] by auto
   have *: "affine hull S = UNIV"
     using True affine_hull_UNIV by auto
   {
     assume "z \<in> interior S"
     then have "z \<in> rel_interior S"
       using True interior_rel_interior_gen[of S] by auto
     then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
     fix x
     obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
       using **[rule_format, of "z-x"] by auto
     define e where [abs_def]: "e = e1 - 1"
     then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
       by (simp add: algebra_simps)
     then have "e > 0" "z + e *\<^sub>R x \<in> S"
       using e1 e_def by auto
     then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
       by auto
   }
   moreover
   {
     assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     {
       fix x
       obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
         using r[rule_format, of "z-x"] by auto
       define e where "e = e1 + 1"
       then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         by (simp add: algebra_simps)
       then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
         using e1 e_def by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
     }
     then have "z \<in> rel_interior S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
     then have "z \<in> interior S"
       using True interior_rel_interior_gen[of S] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior and closure under common operations\<close>
 
 lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
 proof -
   {
     fix y
     assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
       by auto
     {
       fix S
       assume "S \<in> I"
       then have "y \<in> S"
         using rel_interior_subset y by auto
     }
     then have "y \<in> \<Inter>I" by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
 proof -
   obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
     using assms by auto
   {
     fix y
     assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> closure S"
       by auto
     {
       assume "y = x"
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
     }
     moreover
     {
       assume "y \<noteq> x"
       { fix e :: real
         assume e: "e > 0"
         define e1 where "e1 = min 1 (e/norm (y - x))"
         then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
           using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
           by simp_all
         define z where "z = y - e1 *\<^sub>R (y - x)"
         {
           fix S
           assume "S \<in> I"
           then have "z \<in> rel_interior S"
             using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
             by auto
         }
         then have *: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
           by auto
         have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
           using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
           by (rule_tac x="z" in exI) auto
       }
       then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
         unfolding islimpt_approachable_le by blast
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         unfolding closure_def by auto
     }
     ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
       by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_inter_rel_interior_same_closure:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "convex (\<Inter>I)"
     using assms convex_Inter by auto
   moreover
   have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
     using assms convex_rel_interior by (force intro: convex_Inter)
   ultimately
   have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
     using convex_inter_rel_interior_same_closure assms
       closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by blast
   then show ?thesis
     using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
 qed
 
 lemma convex_rel_interior_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
     and "finite I"
   shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "\<Inter>I \<noteq> {}"
     using assms rel_interior_inter_aux[of I] by auto
   have "convex (\<Inter>I)"
     using convex_Inter assms by auto
   show ?thesis
   proof (cases "I = {}")
     case True
     then show ?thesis
       using Inter_empty rel_interior_UNIV by auto
   next
     case False
     {
       fix z
       assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
       {
         fix x
         assume x: "x \<in> \<Inter>I"
         {
           fix S
           assume S: "S \<in> I"
           then have "z \<in> rel_interior S" "x \<in> S"
             using z x by auto
           then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
             using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
         }
         then obtain mS where
           mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
         define e where "e = Min (mS ` I)"
         then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
         then have "e > 1" using mS by auto
         moreover have "\<forall>S\<in>I. e \<le> mS S"
           using e_def assms by auto
         ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
           using mS by auto
       }
       then have "z \<in> rel_interior (\<Inter>I)"
         using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
     }
     then show ?thesis
       using convex_rel_interior_inter[of I] assms by auto
   qed
 qed
 
 lemma convex_closure_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> closure T"
   using convex_closure_inter[of "{S,T}"] assms by auto
 
 lemma convex_rel_interior_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
   using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
 
 lemma convex_affine_closure_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma connected_component_1_gen:
   fixes S :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 unfolding connected_component_def
 by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
             ends_in_segment connected_convex_1_gen)
 
 lemma connected_component_1:
   fixes S :: "real set"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 by (simp add: connected_component_1_gen)
 
 lemma convex_affine_rel_interior_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma convex_affine_rel_frontier_Int:
    fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "interior S \<inter> T \<noteq> {}"
   shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
 using assms
-apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
-by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
+  unfolding rel_frontier_def  frontier_def
+  using convex_affine_closure_Int convex_affine_rel_interior_Int rel_interior_nonempty_interior by fastforce
 
 lemma rel_interior_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
     shows "rel_interior(S \<inter> T) = interior S \<inter> T"
 proof -
   obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
     using assms by force
   have "rel_interior S = interior S"
     by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
   then show ?thesis
     by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
 qed
 
 lemma closure_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
   shows "closure(S \<inter> T) = closure S \<inter> T"
 proof
   have "closure (S \<inter> T) \<subseteq> closure T"
     by (simp add: closure_mono)
   also have "... \<subseteq> T"
     by (simp add: affine_closed assms)
   finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
     by (simp add: closure_mono)
 next
   obtain a where "a \<in> rel_interior S" "a \<in> T"
     using assms by auto
   then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
     using affine_diffs_subspace rel_interior_subset assms by blast+
   show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
   proof
     fix x  assume "x \<in> closure S \<inter> T"
     show "x \<in> closure (S \<inter> T)"
     proof (cases "x = a")
       case True
       then show ?thesis
         using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
     next
       case False
       then have "x \<in> closure(open_segment a x)"
         by auto
       then show ?thesis
         using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
     qed
   qed
 qed
 
 lemma subset_rel_interior_convex:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<le> closure T"
     and "\<not> S \<subseteq> rel_frontier T"
   shows "rel_interior S \<subseteq> rel_interior T"
 proof -
   have *: "S \<inter> closure T = S"
     using assms by auto
   have "\<not> rel_interior S \<subseteq> rel_frontier T"
     using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
       closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
     by auto
   then have "rel_interior S \<inter> rel_interior (closure T) \<noteq> {}"
     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
     by auto
   then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> closure T)"
     using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
       convex_rel_interior_closure[of T]
     by auto
   also have "\<dots> = rel_interior S"
     using * by auto
   finally show ?thesis
     by auto
 qed
 
 lemma rel_interior_convex_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
   shows "f ` (rel_interior S) = rel_interior (f ` S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms by auto
 next
   case False
   interpret linear f by fact
   have *: "f ` (rel_interior S) \<subseteq> f ` S"
     unfolding image_mono using rel_interior_subset by auto
   have "f ` S \<subseteq> f ` (closure S)"
     unfolding image_mono using closure_subset by auto
   also have "\<dots> = f ` (closure (rel_interior S))"
     using convex_closure_rel_interior assms by auto
   also have "\<dots> \<subseteq> closure (f ` (rel_interior S))"
     using closure_linear_image_subset assms by auto
   finally have "closure (f ` S) = closure (f ` rel_interior S)"
     using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
       closure_mono[of "f ` rel_interior S" "f ` S"] *
     by auto
   then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
     using assms convex_rel_interior
       linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
       convex_linear_image[of _ "rel_interior S"]
       closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
     by auto
   then have "rel_interior (f ` S) \<subseteq> f ` rel_interior S"
     using rel_interior_subset by auto
   moreover
   {
     fix z
     assume "z \<in> f ` rel_interior S"
     then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
     {
       fix x
       assume "x \<in> f ` S"
       then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
         using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
       moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
       ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f ` S)"
       using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
         \<open>S \<noteq> {}\<close> convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "f -` (rel_interior S) \<noteq> {}"
   shows "rel_interior (f -` S) = f -` (rel_interior S)"
 proof -
   interpret linear f by fact
   have "S \<noteq> {}"
     using assms by auto
   have nonemp: "f -` S \<noteq> {}"
     by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
   then have "S \<inter> (range f) \<noteq> {}"
     by auto
   have conv: "convex (f -` S)"
     using convex_linear_vimage assms by auto
   then have "convex (S \<inter> range f)"
     by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
   {
     fix z
     assume "z \<in> f -` (rel_interior S)"
     then have z: "f z \<in> rel_interior S"
       by auto
     {
       fix x
       assume "x \<in> f -` S"
       then have "f x \<in> S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
         using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
       moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
         using \<open>linear f\<close> by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f -` S)"
       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
   }
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (f -` S)"
     {
       fix x
       assume "x \<in> S \<inter> range f"
       then obtain y where y: "f y = x" "y \<in> f -` S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
         using convex_rel_interior_iff[of "f -` S" z] z conv by auto
       moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
         using \<open>linear f\<close> y by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
         using e by auto
     }
     then have "f z \<in> rel_interior (S \<inter> range f)"
       using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
         convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
       by auto
     moreover have "affine (range f)"
       by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
     ultimately have "f z \<in> rel_interior S"
       using convex_affine_rel_interior_Int[of S "range f"] assms by auto
     then have "z \<in> f -` (rel_interior S)"
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
 proof (cases "S = {} \<or> T = {}")
   case True
   then show ?thesis 
     by auto
 next
   case False
   then have "S \<noteq> {}" "T \<noteq> {}"
     by auto
   then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
     using rel_interior_eq_empty assms by auto
   then have "fst -` rel_interior S \<noteq> {}"
     using fst_vimage_eq_Times[of "rel_interior S"] by auto
   then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
     using linear_fst \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
   then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
     by (simp add: fst_vimage_eq_Times)
   from ri have "snd -` rel_interior T \<noteq> {}"
     using snd_vimage_eq_Times[of "rel_interior T"] by auto
   then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
     using linear_snd \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
   then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
     by (simp add: snd_vimage_eq_Times)
   from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
       rel_interior S \<times> rel_interior T" by auto
   have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
     by auto
   then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
     by auto
   also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
     using * ri assms convex_Times
     by (subst convex_rel_interior_inter_two) auto
   finally show ?thesis using * by auto
 qed
 
 lemma rel_interior_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "c \<noteq> 0"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
     linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
   by auto
 
 lemma rel_interior_convex_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   by (metis assms linear_scaleR rel_interior_convex_linear_image)
 
 lemma convex_rel_open_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
   shows "convex (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
   by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 
 lemma convex_rel_open_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
     and "finite I"
   shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
 proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
   case True
   then have "\<Inter>I = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (\<Inter>I)"
     using assms unfolding rel_open_def
     using convex_rel_interior_finite_inter[of I]
     by auto
   then show ?thesis
     using convex_Inter assms by auto
 qed
 
 lemma convex_rel_open_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f ` S) \<and> rel_open (f ` S)"
   by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
 
 lemma convex_rel_open_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f -` S) \<and> rel_open (f -` S)"
 proof (cases "f -` (rel_interior S) = {}")
   case True
   then have "f -` S = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (f -` S)"
     using assms unfolding rel_open_def
     using rel_interior_convex_linear_preimage[of f S]
     by auto
   then show ?thesis
     using convex_linear_vimage assms
     by auto
 qed
 
 lemma rel_interior_projection:
   fixes S :: "('m::euclidean_space \<times> 'n::euclidean_space) set"
     and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
   assumes "convex S"
     and "f = (\<lambda>y. {z. (y, z) \<in> S})"
   shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
 proof -
   {
     fix y
     assume "y \<in> {y. f y \<noteq> {}}"
     then obtain z where "(y, z) \<in> S"
       using assms by auto
     then have "\<exists>x. x \<in> S \<and> y = fst x"
       by auto
     then obtain x where "x \<in> S" "y = fst x"
       by blast
     then have "y \<in> fst ` S"
       unfolding image_def by auto
   }
   then have "fst ` S = {y. f y \<noteq> {}}"
     unfolding fst_def using assms by auto
   then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
     using rel_interior_convex_linear_image[of fst S] assms linear_fst by auto
   {
     fix y
     assume "y \<in> rel_interior {y. f y \<noteq> {}}"
     then have "y \<in> fst ` rel_interior S"
       using h1 by auto
     then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
       by auto
     moreover have aff: "affine (fst -` {y})"
       unfolding affine_alt by (simp add: algebra_simps)
     ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
       using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
     have conv: "convex (S \<inter> fst -` {y})"
       using convex_Int assms aff affine_imp_convex by auto
     {
       fix x
       assume "x \<in> f y"
       then have "(y, x) \<in> S \<inter> (fst -` {y})"
         using assms by auto
       moreover have "x = snd (y, x)" by auto
       ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
         by blast
     }
     then have "snd ` (S \<inter> fst -` {y}) = f y"
       using assms by auto
     then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
       using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] linear_snd conv
       by auto
     {
       fix z
       assume "z \<in> rel_interior (f y)"
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         using *** by auto
       moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
         using * ** rel_interior_subset by auto
       ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         by force
       then have "(y,z) \<in> rel_interior S"
         using ** by auto
     }
     moreover
     {
       fix z
       assume "(y, z) \<in> rel_interior S"
       then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         using ** by auto
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         by (metis Range_iff snd_eq_Range)
       then have "z \<in> rel_interior (f y)"
         using *** by auto
     }
     ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
       by auto
   }
   then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
     (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
     by auto
   {
     fix y z
     assume asm: "(y, z) \<in> rel_interior S"
     then have "y \<in> fst ` rel_interior S"
       by (metis Domain_iff fst_eq_Domain)
     then have "y \<in> rel_interior {t. f t \<noteq> {}}"
       using h1 by auto
     then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> rel_interior (f y))"
       using h2 asm by auto
   }
   then show ?thesis using h2 by blast
 qed
 
 lemma rel_frontier_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_frontier S \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
     by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex cone\<close>
 
 lemma cone_rel_interior:
   fixes S :: "'m::euclidean_space set"
   assumes "cone S"
   shows "cone ({0} \<union> rel_interior S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_0)
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   then have *: "0 \<in> ({0} \<union> rel_interior S)"
     and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
     by (auto simp add: rel_interior_scaleR)
   then show ?thesis
     using cone_iff[of "{0} \<union> rel_interior S"] by auto
 qed
 
 lemma rel_interior_convex_cone_aux:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
     c > 0 \<and> x \<in> (((*\<^sub>R) c) ` (rel_interior S))"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_hull_empty)
 next
   case False
   then obtain s where "s \<in> S" by auto
   have conv: "convex ({(1 :: real)} \<times> S)"
     using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
     by auto
   define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
   then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
     (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
     using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
     by (subst rel_interior_projection) auto
   {
     fix y :: real
     assume "y \<ge> 0"
     then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
       using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
     then have "f y \<noteq> {}"
       using f_def by auto
   }
   then have "{y. f y \<noteq> {}} = {0..}"
     using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
   then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
     using rel_interior_real_semiline by auto
   {
     fix c :: real
     assume "c > 0"
     then have "f c = ((*\<^sub>R) c ` S)"
       using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
     then have "rel_interior (f c) = (*\<^sub>R) c ` rel_interior S"
       using rel_interior_convex_scaleR[of S c] assms by auto
   }
   then show ?thesis using * ** by auto
 qed
 
 lemma rel_interior_convex_cone:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (cone hull ({1 :: real} \<times> S)) =
     {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix z
     assume "z \<in> ?lhs"
     have *: "z = (fst z, snd z)"
       by auto
     then have "z \<in> ?rhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close> by fastforce
   }
   moreover
   {
     fix z
     assume "z \<in> ?rhs"
     then have "z \<in> ?lhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma convex_hull_finite_union:
   assumes "finite I"
   assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs \<supseteq> ?rhs"
   proof
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
       "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
     then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
       using hull_subset[of "\<Union>(S ` I)" convex] by auto
     then show "x \<in> ?lhs"
       unfolding *(1)[symmetric]
       using * assms convex_convex_hull
       by (subst convex_sum) auto
   qed
   {
     fix i
     assume "i \<in> I"
     with assms have "\<exists>p. p \<in> S i" by auto
   }
   then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
   {
     fix i
     assume "i \<in> I"
     {
       fix x
       assume "x \<in> S i"
       define c where "c j = (if j = i then 1::real else 0)" for j
       then have *: "sum c I = 1"
         using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
         by auto
       define s where "s j = (if j = i then x else p j)" for j
       then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
         using c_def by (auto simp add: algebra_simps)
       then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
         using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
         by auto
-      then have "x \<in> ?rhs"
-        apply auto
-        apply (rule_tac x = c in exI)
-        apply (rule_tac x = s in exI)
-        using * c_def s_def p \<open>x \<in> S i\<close>
-        apply auto
-        done
+      moreover have "(\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
+        using * c_def s_def p \<open>x \<in> S i\<close> by auto
+      ultimately have "x \<in> ?rhs"
+        by force
     }
     then have "?rhs \<supseteq> S i" by auto
   }
   then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
 
   {
     fix u v :: real
     assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
     fix x y
     assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
     from xy obtain c s where
       xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
       by auto
     from xy obtain d t where
       yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
       by auto
     define e where "e i = u * c i + v * d i" for i
     have ge0: "\<forall>i\<in>I. e i \<ge> 0"
       using e_def xc yc uv by simp
     have "sum (\<lambda>i. u * c i) I = u * sum c I"
       by (simp add: sum_distrib_left)
     moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
       by (simp add: sum_distrib_left)
     ultimately have sum1: "sum e I = 1"
       using e_def xc yc uv by (simp add: sum.distrib)
     define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
       for i
     {
       fix i
       assume i: "i \<in> I"
       have "q i \<in> S i"
       proof (cases "e i = 0")
         case True
         then show ?thesis using i p q_def by auto
       next
         case False
         then show ?thesis
           using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
             mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
             assms q_def e_def i False xc yc uv
           by (auto simp del: mult_nonneg_nonneg)
       qed
     }
     then have qs: "\<forall>i\<in>I. q i \<in> S i" by auto
     {
       fix i
       assume i: "i \<in> I"
       have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       proof (cases "e i = 0")
         case True
         have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
           using xc yc uv i by simp
         moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
           using True e_def i by simp
         ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
         with True show ?thesis by auto
       next
         case False
         then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
           using q_def by auto
         then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
                = (e i) *\<^sub>R (q i)" by auto
         with False show ?thesis by (simp add: algebra_simps)
       qed
     }
     then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       by auto
     have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
       using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
     also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
       using * by auto
     finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
       using ge0 sum1 qs by auto
   }
   then have "convex ?rhs" unfolding convex_def by auto
   then show ?thesis
     using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
     by blast
 qed
 
 lemma convex_hull_union_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
     and "convex T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) =
     {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T}"
   (is "?lhs = ?rhs")
 proof
   define I :: "nat set" where "I = {1, 2}"
   define s where "s i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(s ` I) = S \<union> T"
     using s_def I_def by auto
   then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(s ` I) =
     {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
       using assms s_def I_def
       by (subst convex_hull_finite_union) auto
   moreover have
     "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
     using s_def I_def by auto
   ultimately show "?lhs \<subseteq> ?rhs" by auto
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
       by auto
     then have "x \<in> convex hull {s, t}"
       using convex_hull_2[of s t] by auto
     then have "x \<in> convex hull (S \<union> T)"
       using * hull_mono[of "{s, t}" "S \<union> T"] by auto
   }
   then show "?lhs \<supseteq> ?rhs" by blast
 qed
 
 proposition ray_to_rel_frontier:
   fixes a :: "'a::real_inner"
   assumes "bounded S"
       and a: "a \<in> rel_interior S"
       and aff: "(a + l) \<in> affine hull S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
 proof -
   have aaff: "a \<in> affine hull S"
     by (meson a hull_subset rel_interior_subset rev_subsetD)
   let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
   obtain B where "B > 0" and B: "S \<subseteq> ball a B"
     using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
   have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
     by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
   with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
     using rel_interior_subset subsetCE by blast
   with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
     using divide_pos_pos zero_less_norm_iff by fastforce
   have bdd: "bdd_below ?D"
     by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
   have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
                     \<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
     using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
   define d where "d = Inf ?D"
   obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
   proof -
     obtain e where "e>0"
             and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex a by blast
     show thesis
     proof (rule_tac \<epsilon> = "e / norm l" in that)
       show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
     next
       show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
       proof (rule e)
         show "a + \<eta> *\<^sub>R l \<in> affine hull S"
           by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
         show "dist (a + \<eta> *\<^sub>R l) a < e"
           using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
       qed
     qed
   qed
   have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
     unfolding d_def using cInf_lower [OF _ bdd]
     by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
   have "\<epsilon> \<le> d"
     unfolding d_def
     using \<epsilon> dual_order.strict_implies_order le_less_linear 
     by (blast intro: cInf_greatest [OF nonMT])
   with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
   have "a + d *\<^sub>R l \<notin> rel_interior S"
   proof
     assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
     obtain e where "e > 0"
              and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex adl by blast
     have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
     proof (rule cInf_greatest [OF nonMT], clarsimp)
       fix x::real
       assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
       show "d + e / norm l \<le> x"
       proof (cases "x < d")
         case True with inint nonrel \<open>0 < x\<close>
           show ?thesis by auto
       next
         case False
           then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
             by (simp add: field_simps \<open>l \<noteq> 0\<close>)
           have ain: "a + x *\<^sub>R l \<in> affine hull S"
             by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
           show ?thesis
             using e [OF ain] nonrel dle by force
       qed
     qed
     then show False
       using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
   qed
   moreover have "a + d *\<^sub>R l \<in> closure S"
   proof (clarsimp simp: closure_approachable)
     fix \<eta>::real assume "0 < \<eta>"
     have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
-      apply (rule subsetD [OF rel_interior_subset inint])
-      using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
+    proof (rule subsetD [OF rel_interior_subset inint])
+      show "d - min d (\<eta> / 2 / norm l) < d"
+        using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
+    qed auto
     have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
       by (metis min_def mult_left_mono norm_ge_zero order_refl)
     also have "... < \<eta>"
       using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
     finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
     show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
-      apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
-      using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
-      done
+      using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> 
+      by (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI) (auto simp: algebra_simps)
   qed
   ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
     by (simp add: rel_frontier_def)
   show ?thesis
     by (rule that [OF \<open>0 < d\<close> infront inint])
 qed
 
 corollary ray_to_frontier:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
       and a: "a \<in> interior S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
 proof -
-  have "interior S = rel_interior S"
+  have \<section>: "interior S = rel_interior S"
     using a rel_interior_nonempty_interior by auto
   then have "a \<in> rel_interior S"
     using a by simp
-  then show thesis
-    apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
-     using a affine_hull_nonempty_interior apply blast
-    by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
+  moreover have "a + l \<in> affine hull S"
+    using a affine_hull_nonempty_interior by blast
+  ultimately show thesis
+    by (metis \<section> \<open>bounded S\<close> \<open>l \<noteq> 0\<close> frontier_def ray_to_rel_frontier rel_frontier_def that)
 qed
 
 
 lemma segment_to_rel_frontier_aux:
   fixes x :: "'a::euclidean_space"
   assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                    "open_segment x z \<subseteq> rel_interior S"
 proof -
   have "x + (y - x) \<in> affine hull S"
     using hull_inc [OF y] by auto
   then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
                   and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
     by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
   show ?thesis
   proof
     show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
       by (simp add: df)
   next
     have "open_segment x y \<subseteq> rel_interior S"
       using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
     moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
       using xy \<open>0 < d\<close> that by (force simp: in_segment algebra_simps)
     ultimately have "1 \<le> d"
       using df rel_frontier_def by fastforce
     moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
       by (metis \<open>0 < d\<close> add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
     ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
-      apply (simp add: in_segment)
-      apply (rule_tac x="1/d" in exI)
-      apply (auto simp: algebra_simps)
-      done
+      unfolding in_segment
+      by (rule_tac x="1/d" in exI) (auto simp: algebra_simps)
   next
     show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
     proof (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
       show "x + d *\<^sub>R (y - x) \<in> closure S"
         using df rel_frontier_def by auto
     qed
   qed
 qed
 
 lemma segment_to_rel_frontier:
   fixes x :: "'a::euclidean_space"
   assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
       and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                   "open_segment x z \<subseteq> rel_interior S"
 proof (cases "x=y")
   case True
   with xy have "S \<noteq> {x}"
     by blast
   with True show ?thesis
     by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
 next
   case False
   then show ?thesis
     using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
 proposition rel_frontier_not_sing:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
     shows "rel_frontier S \<noteq> {a}"
 proof (cases "S = {}")
   case True  then show ?thesis  by simp
 next
   case False
   then obtain z where "z \<in> S"
     by blast
   then show ?thesis
   proof (cases "S = {z}")
     case True then show ?thesis  by simp
   next
     case False
     then obtain w where "w \<in> S" "w \<noteq> z"
       using \<open>z \<in> S\<close> by blast
     show ?thesis
     proof
       assume "rel_frontier S = {a}"
       then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
         using \<open>w \<noteq> z\<close> by auto
       then show False
       proof cases
         case 1
         then have w: "w \<in> rel_interior S"
           using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "w + (w - z) \<in> affine hull S"
           by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
         moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       next
         case 2
         then have z: "z \<in> rel_interior S"
           using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "z + (z - w) \<in> affine hull S"
           by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
         moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       qed
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity on direct sums\<close>
 
 lemma closure_sum:
   fixes S T :: "'a::real_normed_vector set"
   shows "closure S + closure T \<subseteq> closure (S + T)"
   unfolding set_plus_image closure_Times [symmetric] split_def
   by (intro closure_bounded_linear_image_subset bounded_linear_add
     bounded_linear_fst bounded_linear_snd)
 
 lemma rel_interior_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S + T) = rel_interior S + rel_interior T"
 proof -
   have "rel_interior S + rel_interior T = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
     by (simp add: set_plus_image)
   also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
     using rel_interior_Times assms by auto
   also have "\<dots> = rel_interior (S + T)"
     using fst_snd_linear convex_Times assms
       rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp add: set_plus_image)
   finally show ?thesis ..
 qed
 
 lemma rel_interior_sum_gen:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i)"
   shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
   using rel_interior_sum rel_interior_sing[of "0"] assms
   by (subst sum_set_cond_linear[of convex], auto simp add: convex_set_plus)
 
 lemma convex_rel_open_direct_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S \<times> T) \<and> rel_open (S \<times> T)"
   by (metis assms convex_Times rel_interior_Times rel_open_def)
 
 lemma convex_rel_open_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S + T) \<and> rel_open (S + T)"
   by (metis assms convex_set_plus rel_interior_sum rel_open_def)
 
 lemma convex_hull_finite_union_cones:
   assumes "finite I"
     and "I \<noteq> {}"
   assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) = sum S I"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?lhs"
     then obtain c xs where
       x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
       using convex_hull_finite_union[of I S] assms by auto
     define s where "s i = c i *\<^sub>R xs i" for i
     have "\<forall>i\<in>I. s i \<in> S i"
         using s_def x assms by (simp add: mem_cone)
     moreover have "x = sum s I" using x s_def by auto
     ultimately have "x \<in> ?rhs"
       using set_sum_alt[of I S] assms by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
       using set_sum_alt[of I S] assms by auto
     define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
     then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
       using x assms by auto
     moreover have "\<forall>i\<in>I. xs i \<in> S i"
       using x xs_def assms by (simp add: cone_def)
     moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
       by auto
     moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
       using assms by auto
     ultimately have "x \<in> ?lhs"
       using assms
       apply (simp add: convex_hull_finite_union[of I S])
       by (rule_tac x = "(\<lambda>i. 1 / (card I))" in exI) auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_hull_union_cones_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "cone S"
     and "S \<noteq> {}"
   assumes "convex T"
     and "cone T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) = S + T"
 proof -
   define I :: "nat set" where "I = {1, 2}"
   define A where "A i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(A ` I) = S \<union> T"
     using A_def I_def by auto
   then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(A ` I) = sum A I"
     using A_def I_def
     by (metis assms convex_hull_finite_union_cones empty_iff finite.emptyI finite.insertI insertI1)
   moreover have "sum A I = S + T"
     using A_def I_def by (force simp add: set_plus_def)
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_hull_union:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "finite I"
     and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
   shows "rel_interior (convex hull (\<Union>(S ` I))) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
       (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
   (is "?lhs = ?rhs")
 proof (cases "I = {}")
   case True
   then show ?thesis
     using convex_hull_empty by auto
 next
   case False
   define C0 where "C0 = convex hull (\<Union>(S ` I))"
   have "\<forall>i\<in>I. C0 \<ge> S i"
     unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
   define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
   define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
   have "\<forall>i\<in>I. K i \<noteq> {}"
     unfolding K_def using assms
     by (simp add: cone_hull_empty_iff[symmetric])
   have convK: "\<forall>i\<in>I. convex (K i)"
     unfolding K_def
     by (simp add: assms(2) convex_Times convex_cone_hull)
   have "K0 \<supseteq> K i" if  "i \<in> I" for i
     unfolding K0_def K_def
     by (simp add: Sigma_mono \<open>\<forall>i\<in>I. S i \<subseteq> C0\<close> hull_mono that)
   then have "K0 \<supseteq> \<Union>(K ` I)" by auto
   moreover have "convex K0"
     unfolding K0_def by (simp add: C0_def convex_Times convex_cone_hull)
   ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
     using hull_minimal[of _ "K0" "convex"] by blast
   have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
     using K_def by (simp add: hull_subset)
   then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
     by auto
   then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
     by (simp add: hull_mono)
   then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
     unfolding C0_def
     using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
     by auto
   moreover have "cone (convex hull (\<Union>(K ` I)))"
     by (simp add: K_def cone_Union cone_cone_hull cone_convex_hull)
   ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
     unfolding K0_def
     using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
     by blast
   then have "K0 = convex hull (\<Union>(K ` I))"
     using geq by auto
   also have "\<dots> = sum K I"
     using assms False \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> cone_hull_eq convK 
     by (intro convex_hull_finite_union_cones; fastforce simp: K_def)
   finally have "K0 = sum K I" by auto
   then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
     using rel_interior_sum_gen[of I K] convK by auto
   {
     fix x
     assume "x \<in> ?lhs"
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
       by auto
     then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
       using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
     {
       fix i
       assume "i \<in> I"
       then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
         using k K_def assms by auto
       then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
         using rel_interior_convex_cone[of "S i"] by auto
     }
     then obtain c s where cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
       by metis
     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 cs by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
         (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
       by auto
     define k where "k i = (c i, c i *\<^sub>R s i)" for i
     {
       fix i assume "i \<in> I"
       then have "k i \<in> rel_interior (K i)"
         using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
         by auto
     }
     then have "(1, x) \<in> rel_interior K0"
-      using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
-      apply simp
-      apply (rule_tac x = k in exI)
-      apply (simp add: sum_prod)
-      done
+      using * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms cs
+      by (simp add: k_def) (metis (mono_tags, lifting) sum_prod)
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 
 lemma convex_le_Inf_differential:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex_on I f"
     and "x \<in> interior I"
     and "y \<in> I"
   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
   (is "_ \<ge> _ + Inf (?F x) * (y - x)")
 proof (cases rule: linorder_cases)
   assume "x < y"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
   ultimately have "x < t" "t < y" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps split: split_min)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   define K where "K = x - e / 2"
   with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
     by (auto simp: dist_real_def)
   then have "K \<in> I"
     using \<open>interior I \<subseteq> I\<close> e(2) by blast
 
   have "Inf (?F x) \<le> (f x - f y) / (x - y)"
   proof (intro bdd_belowI cInf_lower2)
     show "(f x - f t) / (x - t) \<in> ?F x"
       using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
     show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
       using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
       by (rule convex_on_diff)
   next
     fix y
     assume "y \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
     show "(f K - f x) / (K - x) \<le> y" by auto
   qed
   then show ?thesis
     using \<open>x < y\<close> by (simp add: field_simps)
 next
   assume "y < x"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = x + e / 2"
   ultimately have "x < t" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "(f x - f y) / (x - y) \<le> Inf (?F x)"
   proof (rule cInf_greatest)
     have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
       using \<open>y < x\<close> by (auto simp: field_simps)
     also
     fix z
     assume "z \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
     have "(f y - f x) / (y - x) \<le> z"
       by auto
     finally show "(f x - f y) / (x - y) \<le> z" .
   next
     have "x + e / 2 \<in> ball x e"
       using e by (auto simp: dist_real_def)
     with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
       by auto
     then show "?F x \<noteq> {}"
       by blast
   qed
   then show ?thesis
     using \<open>y < x\<close> by (simp add: field_simps)
 qed simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Explicit formulas for interior and relative interior of convex hull\<close>
 
 lemma at_within_cbox_finite:
   assumes "x \<in> box a b" "x \<notin> S" "finite S"
   shows "(at x within cbox a b - S) = at x"
 proof -
   have "interior (cbox a b - S) = box a b - S"
     using \<open>finite S\<close> by (simp add: interior_diff finite_imp_closed)
   then show ?thesis
     using at_within_interior assms by fastforce
 qed
 
 lemma affine_independent_convex_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S" "T \<subseteq> S"
     shows "convex hull T = affine hull T \<inter> convex hull S"
 proof -
   have fin: "finite S" "finite T" using assms aff_independent_finite finite_subset by auto
-    { fix u v x
-      assume uv: "sum u T = 1" "\<forall>x\<in>S. 0 \<le> v x" "sum v S = 1"
-                 "(\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>v\<in>T. u v *\<^sub>R v)" "x \<in> T"
-      then have S: "S = (S - T) \<union> T" \<comment> \<open>split into separate cases\<close>
-        using assms by auto
-      have [simp]: "(\<Sum>x\<in>T. v x *\<^sub>R x) + (\<Sum>x\<in>S - T. v x *\<^sub>R x) = (\<Sum>x\<in>T. u x *\<^sub>R x)"
-                   "sum v T + sum v (S - T) = 1"
-        using uv fin S
-        by (auto simp: sum.union_disjoint [symmetric] Un_commute)
-      have "(\<Sum>x\<in>S. if x \<in> T then v x - u x else v x) = 0"
-           "(\<Sum>x\<in>S. (if x \<in> T then v x - u x else v x) *\<^sub>R x) = 0"
-        using uv fin
-        by (subst S, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
-    } note [simp] = this
   have "convex hull T \<subseteq> affine hull T"
     using convex_hull_subset_affine_hull by blast
   moreover have "convex hull T \<subseteq> convex hull S"
     using assms hull_mono by blast
   moreover have "affine hull T \<inter> convex hull S \<subseteq> convex hull T"
   proof -
     have 0: "\<And>u. sum u S = 0 \<Longrightarrow> (\<forall>v\<in>S. u v = 0) \<or> (\<Sum>v\<in>S. u v *\<^sub>R v) \<noteq> 0"
       using affine_dependent_explicit_finite assms(1) fin(1) by auto
     show ?thesis
-      apply (clarsimp simp add: convex_hull_finite affine_hull_finite fin)
-      apply (rule_tac x=u in exI)
-      subgoal for u v
-        using 0 [of "\<lambda>x. if x \<in> T then v x - u x else v x"] \<open>T \<subseteq> S\<close> by force
-      done
+    proof (clarsimp simp add: affine_hull_finite fin)
+      fix u
+      assume S: "(\<Sum>v\<in>T. u v *\<^sub>R v) \<in> convex hull S"
+        and T1: "sum u T = 1"
+      then obtain v where v: "\<forall>x\<in>S. 0 \<le> v x" "sum v S = 1" "(\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>v\<in>T. u v *\<^sub>R v)"
+        by (auto simp add: convex_hull_finite fin)
+      { fix x
+        assume"x \<in> T"
+        then have S: "S = (S - T) \<union> T" \<comment> \<open>split into separate cases\<close>
+          using assms by auto
+        have [simp]: "(\<Sum>x\<in>T. v x *\<^sub>R x) + (\<Sum>x\<in>S - T. v x *\<^sub>R x) = (\<Sum>x\<in>T. u x *\<^sub>R x)"
+          "sum v T + sum v (S - T) = 1"
+          using v fin S
+          by (auto simp: sum.union_disjoint [symmetric] Un_commute)
+        have "(\<Sum>x\<in>S. if x \<in> T then v x - u x else v x) = 0"
+             "(\<Sum>x\<in>S. (if x \<in> T then v x - u x else v x) *\<^sub>R x) = 0"
+          using v fin T1
+          by (subst S, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
+      } note [simp] = this
+      have "(\<forall>x\<in>T. 0 \<le> u x)"
+        using 0 [of "\<lambda>x. if x \<in> T then v x - u x else v x"] \<open>T \<subseteq> S\<close> v(1) by fastforce
+      then show "(\<Sum>v\<in>T. u v *\<^sub>R v) \<in> convex hull T"
+        using 0 [of "\<lambda>x. if x \<in> T then v x - u x else v x"] \<open>T \<subseteq> S\<close> T1
+        by (fastforce simp add: convex_hull_finite fin)
+    qed
   qed
   ultimately show ?thesis
     by blast
 qed
 
 lemma affine_independent_span_eq:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S" "card S = Suc (DIM ('a))"
     shows "affine hull S = UNIV"
 proof (cases "S = {}")
   case True then show ?thesis
     using assms by simp
 next
   case False
     then obtain a T where T: "a \<notin> T" "S = insert a T"
       by blast
     then have fin: "finite T" using assms
       by (metis finite_insert aff_independent_finite)
-    then have "UNIV \<subseteq> (+) a ` span ((\<lambda>x. x - a) ` T)"
-      using assms T
-      apply (intro card_ge_dim_independent Fun.vimage_subsetD)
-        apply (auto simp: surj_def affine_dependent_iff_dependent card_image inj_on_def dim_subset_UNIV)
-      done
+    have "UNIV \<subseteq> (+) a ` span ((\<lambda>x. x - a) ` T)"
+    proof (intro card_ge_dim_independent Fun.vimage_subsetD)
+      show "independent ((\<lambda>x. x - a) ` T)"
+        using T affine_dependent_iff_dependent assms(1) by auto
+      show "dim ((+) a -` UNIV) \<le> card ((\<lambda>x. x - a) ` T)"
+        using assms T fin by (auto simp: card_image inj_on_def)
+    qed (use surj_plus in auto)
     then show ?thesis
       using T(2) affine_hull_insert_span_gen equalityI by fastforce
 qed
 
 lemma affine_independent_span_gt:
   fixes S :: "'a::euclidean_space set"
   assumes ind: "\<not> affine_dependent S" and dim: "DIM ('a) < card S"
     shows "affine hull S = UNIV"
 proof (intro affine_independent_span_eq [OF ind] antisym)
   show "card S \<le> Suc DIM('a)"
     using aff_independent_finite affine_dependent_biggerset ind by fastforce
   show "Suc DIM('a) \<le> card S"
     using Suc_leI dim by blast
 qed
 
 lemma empty_interior_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" and dim: "card S \<le> DIM ('a)"
     shows "interior(affine hull S) = {}"
   using assms
 proof (induct S rule: finite_induct)
   case (insert x S)
   then have "dim (span ((\<lambda>y. y - x) ` S)) < DIM('a)"
     by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
   then show ?case
     by (simp add: empty_interior_lowdim affine_hull_insert_span_gen interior_translation)
 qed auto
 
 lemma empty_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" and dim: "card S \<le> DIM ('a)"
     shows "interior(convex hull S) = {}"
   by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
             interior_mono empty_interior_affine_hull [OF assms])
 
 lemma explicit_subset_rel_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   shows "finite S
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> S. 0 < u x \<and> u x < 1) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}
              \<subseteq> rel_interior (convex hull S)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=S, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
   fixes S :: "'a::euclidean_space set"
   shows "finite S
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}
              \<subseteq> rel_interior (convex hull S)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=S, simplified])
 
 lemma rel_interior_convex_hull_explicit:
-  fixes s :: "'a::euclidean_space set"
-  assumes "\<not> affine_dependent s"
-  shows "rel_interior(convex hull s) =
-         {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent S"
+  shows "rel_interior(convex hull S) =
+         {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
          (is "?lhs = ?rhs")
 proof
   show "?rhs \<le> ?lhs"
     by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
 next
   show "?lhs \<le> ?rhs"
-  proof (cases "\<exists>a. s = {a}")
+  proof (cases "\<exists>a. S = {a}")
     case True then show "?lhs \<le> ?rhs"
       by force
   next
     case False
-    have fs: "finite s"
+    have fs: "finite S"
       using assms by (simp add: aff_independent_finite)
     { fix a b and d::real
-      assume ab: "a \<in> s" "b \<in> s" "a \<noteq> b"
-      then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
+      assume ab: "a \<in> S" "b \<in> S" "a \<noteq> b"
+      then have S: "S = (S - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
         by auto
-      have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
-           "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
+      have "(\<Sum>x\<in>S. if x = a then - d else if x = b then d else 0) = 0"
+           "(\<Sum>x\<in>S. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
         using ab fs
-        by (subst s, subst sum.union_disjoint, auto)+
+        by (subst S, subst sum.union_disjoint, auto)+
     } note [simp] = this
     { fix y
-      assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
-      { fix u T a
-        assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
-           and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
-           and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
+      assume y: "y \<in> convex hull S" "y \<notin> ?rhs"
+      have *: False if
+        ua: "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "\<not> 0 < u a" "a \<in> S"
+        and yT: "y = (\<Sum>x\<in>S. u x *\<^sub>R x)" "y \<in> T" "open T"
+        and sb: "T \<inter> affine hull S \<subseteq> {w. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = w}"
+      for u T a
+      proof -
         have ua0: "u a = 0"
           using ua by auto
-        obtain b where b: "b\<in>s" "a \<noteq> b"
+        obtain b where b: "b\<in>S" "a \<noteq> b"
           using ua False by auto
-        obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
+        obtain e where e: "0 < e" "ball (\<Sum>x\<in>S. u x *\<^sub>R x) e \<subseteq> T"
           using yT by (auto elim: openE)
         with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
           by (auto intro: that [of "e / 2 / norm(a-b)"])
-        have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
+        have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> affine hull S"
           using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
-        then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
+        then have "(\<Sum>x\<in>S. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull S"
           using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
-        then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
+        then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull S"
           using d e yT by auto
-        then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
-                            "sum v s = 1"
-                            "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
+        then obtain v where v: "\<forall>x\<in>S. 0 \<le> v x"
+          "sum v S = 1"
+          "(\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
           using subsetD [OF sb] yT
           by auto
-        then have False
-          using assms
-          apply (simp add: affine_dependent_explicit_finite fs)
-          apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
-          using ua b d
-          apply (auto simp: algebra_simps sum_subtractf sum.distrib)
-          done
-      } note * = this
-      have "y \<notin> rel_interior (convex hull s)"
+        have aff: "\<And>u. sum u S = 0 \<Longrightarrow> (\<forall>v\<in>S. u v = 0) \<or> (\<Sum>v\<in>S. u v *\<^sub>R v) \<noteq> 0"
+          using assms by (simp add: affine_dependent_explicit_finite fs)
+        show False
+          using ua b d v aff [of "\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)"]
+          by (auto simp: algebra_simps sum_subtractf sum.distrib)
+      qed
+      have "y \<notin> rel_interior (convex hull S)"
         using y
         apply (simp add: mem_rel_interior)
         apply (auto simp: convex_hull_finite [OF fs])
         apply (drule_tac x=u in spec)
         apply (auto intro: *)
         done
     } with rel_interior_subset show "?lhs \<le> ?rhs"
       by blast
   qed
 qed
 
 lemma interior_convex_hull_explicit_minimal:
-  fixes s :: "'a::euclidean_space set"
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent S"
   shows
-   "\<not> affine_dependent s
-        ==> interior(convex hull s) =
-             (if card(s) \<le> DIM('a) then {}
-              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
-  apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
-  apply (rule trans [of _ "rel_interior(convex hull s)"])
-  apply (simp add: affine_independent_span_gt rel_interior_interior)
-  by (simp add: rel_interior_convex_hull_explicit)
+   "interior(convex hull S) =
+             (if card(S) \<le> DIM('a) then {}
+              else {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"  
+   (is "_ = (if _ then _ else ?rhs)")
+proof (clarsimp simp: aff_independent_finite empty_interior_convex_hull assms)
+  assume S: "\<not> card S \<le> DIM('a)"
+  have "interior (convex hull S) = rel_interior(convex hull S)"
+    using assms S by (simp add: affine_independent_span_gt rel_interior_interior)
+  then show "interior(convex hull S) = ?rhs"
+    by (simp add: assms S rel_interior_convex_hull_explicit)
+qed
 
 lemma interior_convex_hull_explicit:
-  fixes s :: "'a::euclidean_space set"
-  assumes "\<not> affine_dependent s"
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent S"
   shows
-   "interior(convex hull s) =
-             (if card(s) \<le> DIM('a) then {}
-              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
+   "interior(convex hull S) =
+             (if card(S) \<le> DIM('a) then {}
+              else {y. \<exists>u. (\<forall>x \<in> S. 0 < u x \<and> u x < 1) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"
 proof -
   { fix u :: "'a \<Rightarrow> real" and a
-    assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
-    then have cs: "Suc 0 < card s"
+    assume "card Basis < card S" and u: "\<And>x. x\<in>S \<Longrightarrow> 0 < u x" "sum u S = 1" and a: "a \<in> S"
+    then have cs: "Suc 0 < card S"
       by (metis DIM_positive less_trans_Suc)
-    obtain b where b: "b \<in> s" "a \<noteq> b"
-    proof (cases "s \<le> {a}")
+    obtain b where b: "b \<in> S" "a \<noteq> b"
+    proof (cases "S \<le> {a}")
       case True
       then show thesis
         using cs subset_singletonD by fastforce
     qed blast
     have "u a + u b \<le> sum u {a,b}"
       using a b by simp
-    also have "... \<le> sum u s"
+    also have "... \<le> sum u S"
       using a b u
       by (intro Groups_Big.sum_mono2) (auto simp: less_imp_le aff_independent_finite assms)
     finally have "u a < 1"
-      using \<open>b \<in> s\<close> u by fastforce
+      using \<open>b \<in> S\<close> u by fastforce
   } note [simp] = this
   show ?thesis
     using assms by (force simp add: not_le interior_convex_hull_explicit_minimal)
 qed
 
 lemma interior_closed_segment_ge2:
   fixes a :: "'a::euclidean_space"
   assumes "2 \<le> DIM('a)"
     shows  "interior(closed_segment a b) = {}"
 using assms unfolding segment_convex_hull
 proof -
   have "card {a, b} \<le> DIM('a)"
     using assms
     by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
   then show "interior (convex hull {a, b}) = {}"
     by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
 qed
 
 lemma interior_open_segment:
   fixes a :: "'a::euclidean_space"
   shows  "interior(open_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (simp add: not_le, intro conjI impI)
   assume "2 \<le> DIM('a)"
   then show "interior (open_segment a b) = {}"
     using interior_closed_segment_ge2 interior_mono segment_open_subset_closed by blast
 next
   assume le2: "DIM('a) < 2"
   show "interior (open_segment a b) = open_segment a b"
   proof (cases "a = b")
     case True then show ?thesis by auto
   next
     case False
     with le2 have "affine hull (open_segment a b) = UNIV"
       by (simp add: False affine_independent_span_gt)
     then show "interior (open_segment a b) = open_segment a b"
       using rel_interior_interior rel_interior_open_segment by blast
   qed
 qed
 
 lemma interior_closed_segment:
   fixes a :: "'a::euclidean_space"
   shows "interior(closed_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by simp
 next
   case False
   then have "closure (open_segment a b) = closed_segment a b"
     by simp
   then show ?thesis
     by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
 qed
 
 lemmas interior_segment = interior_closed_segment interior_open_segment
 
 lemma closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b = closed_segment c d \<longleftrightarrow> {a,b} = {c,d}"
 proof
   assume abcd: "closed_segment a b = closed_segment c d"
   show "{a,b} = {c,d}"
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis by force
   next
     case False
     then have neq: "a \<noteq> b \<and> c \<noteq> d" by force
     have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
       using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
     have "b \<in> {c, d}"
     proof -
       have "insert b (closed_segment c d) = closed_segment c d"
         using abcd by blast
       then show ?thesis
         by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
     qed
     moreover have "a \<in> {c, d}"
       by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
     ultimately show "{a, b} = {c, d}"
       using neq by fastforce
   qed
 next
   assume "{a,b} = {c,d}"
   then show "closed_segment a b = closed_segment c d"
     by (simp add: segment_convex_hull)
 qed
 
 lemma closed_open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b \<noteq> open_segment c d"
 by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
 
 lemma open_closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<noteq> closed_segment c d"
 using closed_open_segment_eq by blast
 
 lemma open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b = open_segment c d \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
         (is "?lhs = ?rhs")
 proof
   assume abcd: ?lhs
   show ?rhs
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis
       using finite_open_segment by fastforce
   next
     case False
     then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
     with abcd show ?rhs
       unfolding open_segment_def
       by (metis (no_types) abcd closed_segment_eq closure_open_segment)
   qed
 next
   assume ?rhs
   then show ?lhs
     by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Similar results for closure and (relative or absolute) frontier\<close>
 
 lemma closure_convex_hull [simp]:
   fixes S :: "'a::euclidean_space set"
   shows "compact S ==> closure(convex hull S) = convex hull S"
   by (simp add: compact_imp_closed compact_convex_hull)
 
 lemma rel_frontier_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "rel_frontier(convex hull S) =
          {y. \<exists>u. (\<forall>x \<in> S. 0 \<le> u x) \<and> (\<exists>x \<in> S. u x = 0) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
-  have "\<And>u xa v.
-       \<lbrakk> xa \<in> S;
-        u xa = 0; sum u S = 1; \<forall>x\<in>S. 0 < v x;
-        sum v S = 1;
-        (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)\<rbrakk>
+  have "\<And>u y v.
+       \<lbrakk>y \<in> S; u y = 0; sum u S = 1; \<forall>x\<in>S. 0 < v x;
+        sum v S = 1; (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)\<rbrakk>
        \<Longrightarrow> \<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
     apply (force simp: sum_subtractf scaleR_diff_left)
     done
   then show ?thesis
     using fs assms
     apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit)
     apply (auto simp: convex_hull_finite)
-    apply (metis less_eq_real_def)
-    using affine_dependent_explicit_finite assms by blast
+    apply (metis less_eq_real_def) 
+    by (simp add: affine_dependent_explicit_finite)
 qed
 
 lemma frontier_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
          {y. \<exists>u. (\<forall>x \<in> S. 0 \<le> u x) \<and> (DIM ('a) < card S \<longrightarrow> (\<exists>x \<in> S. u x = 0)) \<and>
              sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
   proof (cases "DIM ('a) < card S")
     case True
     with assms fs show ?thesis
       by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
                     interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
   next
     case False
     then have "card S \<le> DIM ('a)"
       by linarith
     then show ?thesis
       using assms fs
       apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
       apply (simp add: convex_hull_finite)
       done
   qed
 qed
 
 lemma rel_frontier_convex_hull_cases:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "rel_frontier(convex hull S) = \<Union>{convex hull (S - {x}) |x. x \<in> S}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
   { fix u a
   have "\<forall>x\<in>S. 0 \<le> u x \<Longrightarrow> a \<in> S \<Longrightarrow> u a = 0 \<Longrightarrow> sum u S = 1 \<Longrightarrow>
             \<exists>x v. x \<in> S \<and>
                   (\<forall>x\<in>S - {x}. 0 \<le> v x) \<and>
                       sum v (S - {x}) = 1 \<and> (\<Sum>x\<in>S - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     apply (rule_tac x=a in exI)
     apply (rule_tac x=u in exI)
     apply (simp add: Groups_Big.sum_diff1 fs)
     done }
   moreover
   { fix a u
     have "a \<in> S \<Longrightarrow> \<forall>x\<in>S - {a}. 0 \<le> u x \<Longrightarrow> sum u (S - {a}) = 1 \<Longrightarrow>
             \<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and>
                  (\<exists>x\<in>S. v x = 0) \<and> sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S - {a}. u x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. if x = a then 0 else u x" in exI)
     apply (auto simp: sum.If_cases Diff_eq if_smult fs)
     done }
   ultimately show ?thesis
     using assms
     apply (simp add: rel_frontier_convex_hull_explicit)
-    apply (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
+    apply (auto simp add: convex_hull_finite fs Union_SetCompr_eq)
     done
 qed
 
 lemma frontier_convex_hull_eq_rel_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
            (if card S \<le> DIM ('a) then convex hull S else rel_frontier(convex hull S))"
   using assms
   unfolding rel_frontier_def frontier_def
   by (simp add: affine_independent_span_gt rel_interior_interior
                 finite_imp_compact empty_interior_convex_hull aff_independent_finite)
 
 lemma frontier_convex_hull_cases:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
            (if card S \<le> DIM ('a) then convex hull S else \<Union>{convex hull (S - {x}) |x. x \<in> S})"
 by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
 
 lemma in_frontier_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" "card S \<le> Suc (DIM ('a))" "x \<in> S"
   shows   "x \<in> frontier(convex hull S)"
 proof (cases "affine_dependent S")
   case True
-  with assms show ?thesis
-    apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
-    by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
+  with assms obtain y where "y \<in> S" and y: "y \<in> affine hull (S - {y})"
+    by (auto simp: affine_dependent_def)
+  moreover have "x \<in> closure (convex hull S)"
+    by (meson closure_subset hull_inc subset_eq \<open>x \<in> S\<close>)
+  moreover have "x \<notin> interior (convex hull S)"
+    using assms
+    by (metis Suc_mono affine_hull_convex_hull affine_hull_nonempty_interior \<open>y \<in> S\<close> y card.remove empty_iff empty_interior_affine_hull finite_Diff hull_redundant insert_Diff interior_UNIV not_less)
+  ultimately show ?thesis
+    unfolding frontier_def by blast
 next
   case False
   { assume "card S = Suc (card Basis)"
     then have cs: "Suc 0 < card S"
       by (simp)
     with subset_singletonD have "\<exists>y \<in> S. y \<noteq> x"
       by (cases "S \<le> {x}") fastforce+
   } note [dest!] = this
   show ?thesis using assms
     unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
     by (auto simp: le_Suc_eq hull_inc)
 qed
 
 lemma not_in_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" "card S \<le> Suc (DIM ('a))" "x \<in> S"
   shows   "x \<notin> interior(convex hull S)"
 using in_frontier_convex_hull [OF assms]
 by (metis Diff_iff frontier_def)
 
 lemma interior_convex_hull_eq_empty:
   fixes S :: "'a::euclidean_space set"
   assumes "card S = Suc (DIM ('a))"
   shows   "interior(convex hull S) = {} \<longleftrightarrow> affine_dependent S"
 proof 
   show "affine_dependent S \<Longrightarrow> interior (convex hull S) = {}"
   proof (clarsimp simp: affine_dependent_def)
     fix a b
     assume "b \<in> S" "b \<in> affine hull (S - {b})"
     then have "interior(affine hull S) = {}" using assms
       by (metis DIM_positive One_nat_def Suc_mono card.remove card.infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
     then show "interior (convex hull S) = {}" 
       using affine_hull_nonempty_interior by fastforce
   qed
 next
   show "interior (convex hull S) = {} \<Longrightarrow> affine_dependent S"
     by (metis affine_hull_convex_hull affine_hull_empty affine_independent_span_eq assms convex_convex_hull empty_not_UNIV rel_interior_eq_empty rel_interior_interior)
 qed
 
 
 subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> coplanar  where
    "coplanar S \<equiv> \<exists>u v w. S \<subseteq> affine hull {u,v,w}"
 
 lemma collinear_affine_hull:
   "collinear S \<longleftrightarrow> (\<exists>u v. S \<subseteq> affine hull {u,v})"
 proof (cases "S={}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x where x: "x \<in> S" by auto
   { fix u
     assume *: "\<And>x y. \<lbrakk>x\<in>S; y\<in>S\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
     have "\<And>y c. x - y = c *\<^sub>R u \<Longrightarrow> \<exists>a b. y = a *\<^sub>R x + b *\<^sub>R (x + u) \<and> a + b = 1"
       by (rule_tac x="1+c" in exI, rule_tac x="-c" in exI, simp add: algebra_simps)
     then have "\<exists>u v. S \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
       using * [OF x] by (rule_tac x=x in exI, rule_tac x="x+u" in exI, force)
   } moreover
   { fix u v x y
     assume *: "S \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
     have "\<exists>c. x - y = c *\<^sub>R (v-u)" if "x\<in>S" "y\<in>S"
     proof -
       obtain a r where "a + r = 1" "x = a *\<^sub>R u + r *\<^sub>R v"
         using "*" \<open>x \<in> S\<close> by blast
       moreover
       obtain b s where "b + s = 1" "y = b *\<^sub>R u + s *\<^sub>R v"
         using "*" \<open>y \<in> S\<close> by blast
       ultimately have "x - y = (r-s) *\<^sub>R (v-u)" 
         by (simp add: algebra_simps) (metis scaleR_left.add)
       then show ?thesis
         by blast
     qed
   } ultimately
   show ?thesis
   unfolding collinear_def affine_hull_2
     by blast
 qed
 
 lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
   by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
 
 lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
   unfolding open_segment_def
   by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
       convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
 
 lemma collinear_between_cases:
   fixes c :: "'a::euclidean_space"
   shows "collinear {a,b,c} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
     by (auto simp: collinear_alt)
   show ?rhs
     using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
 next
   assume ?rhs
   then show ?lhs
     unfolding between_mem_convex_hull
     by (metis (no_types, hide_lams) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
 qed
 
 
 lemma subset_continuous_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "continuous_on (closed_segment a b) f"
   shows "closed_segment (f a) (f b) \<subseteq> image f (closed_segment a b)"
 by (metis connected_segment convex_contains_segment ends_in_segment imageI
            is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
 
 lemma continuous_injective_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
   shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
 proof
   show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
     by (metis subset_continuous_image_segment_1 contf)
   show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
   proof (cases "a = b")
     case True
     then show ?thesis by auto
   next
     case False
     then have fnot: "f a \<noteq> f b"
       using inj_onD injf by fastforce
     moreover
     have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def)
       assume fa: "f a \<in> closed_segment (f c) (f b)"
       moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
         by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
       ultimately have "f a \<in> f ` closed_segment c b"
         by blast
       then have a: "a \<in> closed_segment c b"
         by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
       have cb: "closed_segment c b \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f a = f c"
       proof (rule between_antisym)
         show "between (f c, f b) (f a)"
           by (simp add: between_mem_segment fa)
         show "between (f a, f b) (f c)"
           by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
       qed
     qed
     moreover
     have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def fnot eq_commute)
       assume fb: "f b \<in> closed_segment (f a) (f c)"
       moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
         by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
       ultimately have "f b \<in> f ` closed_segment a c"
         by blast
       then have b: "b \<in> closed_segment a c"
         by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
       have ca: "closed_segment a c \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f b = f c"
       proof (rule between_antisym)
         show "between (f c, f a) (f b)"
           by (simp add: between_commute between_mem_segment fb)
         show "between (f b, f a) (f c)"
           by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
       qed
     qed
     ultimately show ?thesis
       by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
   qed
 qed
 
 lemma continuous_injective_image_open_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
     shows "f ` (open_segment a b) = open_segment (f a) (f b)"
 proof -
   have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
     by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
   also have "... = open_segment (f a) (f b)"
     using continuous_injective_image_segment_1 [OF assms]
     by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
   finally show ?thesis .
 qed
 
 lemma collinear_imp_coplanar:
   "collinear s ==> coplanar s"
 by (metis collinear_affine_hull coplanar_def insert_absorb2)
 
 lemma collinear_small:
   assumes "finite s" "card s \<le> 2"
     shows "collinear s"
 proof -
   have "card s = 0 \<or> card s = 1 \<or> card s = 2"
     using assms by linarith
   then show ?thesis using assms
     using card_eq_SucD numeral_2_eq_2 by (force simp: card_1_singleton_iff)
 qed
 
 lemma coplanar_small:
   assumes "finite s" "card s \<le> 3"
     shows "coplanar s"
 proof -
   consider "card s \<le> 2" | "card s = Suc (Suc (Suc 0))"
     using assms by linarith
   then show ?thesis
   proof cases
     case 1
     then show ?thesis
       by (simp add: \<open>finite s\<close> collinear_imp_coplanar collinear_small)
   next
     case 2
     then show ?thesis
       using hull_subset [of "{_,_,_}"]
       by (fastforce simp: coplanar_def dest!: card_eq_SucD)
   qed
 qed
 
 lemma coplanar_empty: "coplanar {}"
   by (simp add: coplanar_small)
 
 lemma coplanar_sing: "coplanar {a}"
   by (simp add: coplanar_small)
 
 lemma coplanar_2: "coplanar {a,b}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma coplanar_3: "coplanar {a,b,c}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma collinear_affine_hull_collinear: "collinear(affine hull s) \<longleftrightarrow> collinear s"
   unfolding collinear_affine_hull
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) \<longleftrightarrow> coplanar s"
   unfolding coplanar_def
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-  assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
+  assumes "coplanar S" "linear f" shows "coplanar(f ` S)"
 proof -
   { fix u v w
-    assume "s \<subseteq> affine hull {u, v, w}"
-    then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
+    assume "S \<subseteq> affine hull {u, v, w}"
+    then have "f ` S \<subseteq> f ` (affine hull {u, v, w})"
       by (simp add: image_mono)
-    then have "f ` s \<subseteq> affine hull (f ` {u, v, w})"
+    then have "f ` S \<subseteq> affine hull (f ` {u, v, w})"
       by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
   } then
   show ?thesis
     by auto (meson assms(1) coplanar_def)
 qed
 
-lemma coplanar_translation_imp: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
-  unfolding coplanar_def
-  apply clarify
-  apply (rule_tac x="u+a" in exI)
-  apply (rule_tac x="v+a" in exI)
-  apply (rule_tac x="w+a" in exI)
-  using affine_hull_translation [of a "{u,v,w}" for u v w]
-  apply (force simp: add.commute)
-  done
-
-lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
+lemma coplanar_translation_imp: 
+  assumes "coplanar S" shows "coplanar ((\<lambda>x. a + x) ` S)"
+proof -
+  obtain u v w where "S \<subseteq> affine hull {u,v,w}"
+    by (meson assms coplanar_def)
+  then have "(+) a ` S \<subseteq> affine hull {u + a, v + a, w + a}"
+    using affine_hull_translation [of a "{u,v,w}" for u v w]
+    by (force simp: add.commute)
+  then show ?thesis
+    unfolding coplanar_def by blast
+qed
+
+lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` S) \<longleftrightarrow> coplanar S"
     by (metis (no_types) coplanar_translation_imp translation_galois)
 
 lemma coplanar_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
+  assumes "linear f" "inj f" shows "coplanar(f ` S) = coplanar S"
 proof
-  assume "coplanar s"
-  then show "coplanar (f ` s)"
-    unfolding coplanar_def
-    using affine_hull_linear_image [of f "{u,v,w}" for u v w]  assms
-    by (meson coplanar_def coplanar_linear_image)
+  assume "coplanar S"
+  then show "coplanar (f ` S)"
+    using assms(1) coplanar_linear_image by blast
 next
   obtain g where g: "linear g" "g \<circ> f = id"
     using linear_injective_left_inverse [OF assms]
     by blast
-  assume "coplanar (f ` s)"
-  then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
-    by (auto simp: coplanar_def)
-  then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
-    by blast
-  then have "s \<subseteq> g ` (affine hull {u, v, w})"
-    using g by (simp add: Fun.image_comp)
-  then show "coplanar s"
-    unfolding coplanar_def
-    using affine_hull_linear_image [of g "{u,v,w}" for u v w]  \<open>linear g\<close> linear_conv_bounded_linear
-    by fastforce
-qed
-(*The HOL Light proof is simply
-    MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
-*)
-
-lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
+  assume "coplanar (f ` S)"
+  then show "coplanar S"
+    by (metis coplanar_linear_image g(1) g(2) id_apply image_comp image_id)
+qed
+
+lemma coplanar_subset: "\<lbrakk>coplanar t; S \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar S"
   by (meson coplanar_def order_trans)
 
 lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
   by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
 
 lemma collinear_3_imp_in_affine_hull:
   assumes "collinear {a,b,c}" "a \<noteq> b" shows "c \<in> affine hull {a,b}"
 proof -
   obtain u x y where "b - a = y *\<^sub>R u" "c - a = x *\<^sub>R u"
     using assms unfolding collinear_def by auto
-  with \<open>a \<noteq> b\<close> show ?thesis
-  apply (simp add: affine_hull_2)
-  apply (rule_tac x="1 - x/y" in exI)
-  apply (simp add: algebra_simps)
-  done
+  with \<open>a \<noteq> b\<close> have "\<exists>v. c = (1 - x / y) *\<^sub>R a + v *\<^sub>R b \<and> 1 - x / y + v = 1"
+    by (simp add: algebra_simps)
+  then show ?thesis
+    by (simp add: hull_inc mem_affine)
 qed
 
 lemma collinear_3_affine_hull:
   assumes "a \<noteq> b"
   shows "collinear {a,b,c} \<longleftrightarrow> c \<in> affine hull {a,b}"
   using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
 
 lemma collinear_3_eq_affine_dependent:
   "collinear{a,b,c} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
 proof (cases "a = b \<or> a = c \<or> b = c")
   case True
   then show ?thesis
     by (auto simp: insert_commute)
 next
   case False
-  then show ?thesis
-    apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
-     apply (metis collinear_3_affine_hull insert_commute)+
-    done 
+  then have "collinear{a,b,c}" if "affine_dependent {a,b,c}"
+    using that unfolding affine_dependent_def
+    by (auto simp: insert_Diff_if; metis affine_hull_3_imp_collinear insert_commute)
+  moreover
+  have "affine_dependent {a,b,c}" if "collinear{a,b,c}"
+    using False that by (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
+  ultimately
+  show ?thesis
+    using False by blast
 qed
 
 lemma affine_dependent_imp_collinear_3:
   "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
   by (simp add: collinear_3_eq_affine_dependent)
 
 lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
   by (auto simp add: collinear_def)
 
 lemma collinear_3_expand:
    "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
 proof -
   have "collinear{a,b,c} = collinear{a,c,b}"
     by (simp add: insert_commute)
   also have "... = collinear {0, a - c, b - c}"
     by (simp add: collinear_3)
   also have "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
     by (simp add: collinear_lemma)
   also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
     by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
   finally show ?thesis .
 qed
 
 lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
 proof
   assume "collinear S"
   then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
     by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
   then show "aff_dim S \<le> 1"
     using order_trans by fastforce
 next
   assume "aff_dim S \<le> 1"
   then have le1: "aff_dim (affine hull S) \<le> 1"
     by simp
   obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
     using affine_basis_exists [of S] by auto
   then have "finite B" "card B \<le> 2"
     using B le1 by (auto simp: affine_independent_iff_card)
   then have "collinear B"
     by (rule collinear_small)
   then show "collinear S"
     by (metis \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
 qed
 
-lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
-  apply (auto simp: collinear_3 collinear_lemma)
-  apply (drule_tac x="-1" in spec)
-  apply (simp add: algebra_simps)
-  done
+lemma collinear_midpoint: "collinear{a, midpoint a b, b}"
+proof -
+  have \<section>: "\<lbrakk>a \<noteq> midpoint a b; b - midpoint a b \<noteq> - 1 *\<^sub>R (a - midpoint a b)\<rbrakk> \<Longrightarrow> b = midpoint a b"
+    by (simp add: algebra_simps)
+  show ?thesis
+    by (auto simp: collinear_3 collinear_lemma intro: \<section>)
+qed
 
 lemma midpoint_collinear:
   fixes a b c :: "'a::real_normed_vector"
   assumes "a \<noteq> c"
     shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
 proof -
   have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
           "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
           "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
     by (auto simp: algebra_simps)
-  have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
+  have "b = midpoint a c \<Longrightarrow> collinear{a,b,c}"
     using collinear_midpoint by blast
-  moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
-    apply (auto simp: collinear_3_expand assms dist_midpoint)
-    apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
-    apply (simp add: algebra_simps)
-    done
+  moreover have "b = midpoint a c \<longleftrightarrow> dist a b = dist b c" if "collinear{a,b,c}"
+  proof -
+    consider "a = c" | u where "b = u *\<^sub>R a + (1 - u) *\<^sub>R c"
+      using \<open>collinear {a,b,c}\<close> unfolding collinear_3_expand by blast
+    then show ?thesis
+    proof cases
+      case 2
+      with assms have "dist a b = dist b c \<Longrightarrow> b = midpoint a c"
+        by (simp add: dist_norm * midpoint_def scaleR_add_right del: divide_const_simps)
+      then show ?thesis
+        by (auto simp: dist_midpoint)
+    qed (use assms in auto)
+  qed
   ultimately show ?thesis by blast
 qed
 
 lemma between_imp_collinear:
   fixes x :: "'a :: euclidean_space"
   assumes "between (a,b) x"
     shows "collinear {a,x,b}"
 proof (cases "x = a \<or> x = b \<or> a = b")
   case True with assms show ?thesis
     by (auto simp: dist_commute)
 next
-  case False with assms show ?thesis
-    apply (auto simp: collinear_3 collinear_lemma between_norm)
-    apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
-    apply (simp add: vector_add_divide_simps real_vector.scale_minus_right [symmetric])
-    done
+  case False 
+  then have False if "\<And>c. b - x \<noteq> c *\<^sub>R (a - x)"
+    using that [of "-(norm(b - x) / norm(x - a))"] assms
+    by (simp add: between_norm vector_add_divide_simps flip: real_vector.scale_minus_right)
+  then show ?thesis
+    by (auto simp: collinear_3 collinear_lemma)
 qed
 
 lemma midpoint_between:
   fixes a b :: "'a::euclidean_space"
   shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
 proof (cases "a = c")
-  case True then show ?thesis
-    by (auto simp: dist_commute)
-next
   case False
   show ?thesis
     using False between_imp_collinear between_midpoint(1) midpoint_collinear by blast
-qed
+qed (auto simp: dist_commute)
 
 lemma collinear_triples:
   assumes "a \<noteq> b"
     shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
           (is "?lhs = ?rhs")
 proof safe
   fix x
   assume ?lhs and "x \<in> S"
   then show "collinear {a, b, x}"
     using collinear_subset by force
 next
   assume ?rhs
   then have "\<forall>x \<in> S. collinear{a,x,b}"
     by (simp add: insert_commute)
-  then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
+  then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> insert a (insert b S)" for x
     using that assms collinear_3_expand by fastforce+
-  show ?lhs
-    unfolding collinear_def
-    apply (rule_tac x="b-a" in exI)
-    apply (clarify dest!: *)
-    by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
+  have "\<exists>c. x - y = c *\<^sub>R (b - a)" 
+    if x: "x \<in> insert a (insert b S)" and y: "y \<in> insert a (insert b S)" for x y
+  proof -
+    obtain u v where "x = u *\<^sub>R a + (1 - u) *\<^sub>R b" "y = v *\<^sub>R a + (1 - v) *\<^sub>R b"
+      using "*" x y by presburger
+    then have "x - y = (v - u) *\<^sub>R (b - a)"
+      by (simp add: scale_left_diff_distrib scale_right_diff_distrib)
+    then show ?thesis ..
+  qed
+  then show ?lhs
+    unfolding collinear_def by metis
 qed
 
 lemma collinear_4_3:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> collinear{a,b,d}"
   using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
 
 lemma collinear_3_trans:
   assumes "collinear{a,b,c}" "collinear{b,c,d}" "b \<noteq> c"
     shows "collinear{a,b,d}"
 proof -
   have "collinear{b,c,a,d}"
     by (metis (full_types) assms collinear_4_3 insert_commute)
   then show ?thesis
     by (simp add: collinear_subset)
 qed
 
 lemma affine_hull_2_alt:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
-apply (simp add: affine_hull_2, safe)
-apply (rule_tac x=v in image_eqI)
-apply (simp add: algebra_simps)
-apply (metis scaleR_add_left scaleR_one, simp)
-apply (rule_tac x="1-u" in exI)
-apply (simp add: algebra_simps)
-done
+proof -
+  have 1: "u *\<^sub>R a + v *\<^sub>R b = a + v *\<^sub>R (b - a)" if "u + v = 1" for u v
+    using that
+    by (simp add: algebra_simps flip: scaleR_add_left)
+  have 2: "a + u *\<^sub>R (b - a) = (1 - u) *\<^sub>R a + u *\<^sub>R b" for u
+    by (auto simp: algebra_simps)
+  show ?thesis
+    by (force simp add: affine_hull_2 dest: 1 intro!: 2)
+qed
 
 lemma interior_convex_hull_3_minimal:
   fixes a :: "'a::euclidean_space"
-  shows "\<lbrakk>\<not> collinear{a,b,c}; DIM('a) = 2\<rbrakk>
-         \<Longrightarrow> interior(convex hull {a,b,c}) =
-                {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
-                            x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
-apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
-apply (rule_tac x="u a" in exI, simp)
-apply (rule_tac x="u b" in exI, simp)
-apply (rule_tac x="u c" in exI, simp)
-apply (rename_tac uu x y z)
-apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
-apply simp
-done
+  assumes "\<not> collinear{a,b,c}" and 2: "DIM('a) = 2"
+  shows "interior(convex hull {a,b,c}) =
+         {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and> x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
+        (is "?lhs = ?rhs")
+proof
+  have abc: "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" "\<not> affine_dependent {a, b, c}"
+    using assms by (auto simp: collinear_3_eq_affine_dependent)
+  with 2 show "?lhs \<subseteq> ?rhs"
+    by (fastforce simp add: interior_convex_hull_explicit_minimal)
+  show "?rhs \<subseteq> ?lhs"
+    using abc 2
+    apply (clarsimp simp add: interior_convex_hull_explicit_minimal)
+    subgoal for x y z
+      by (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI) auto
+    done
+qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about hyperplanes and halfspaces\<close>
 
 lemma halfspace_Int_eq:
      "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
      "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
   by auto
 
 lemma hyperplane_eq_Ex:
   assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
   by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
 
 lemma hyperplane_eq_empty:
      "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
   using hyperplane_eq_Ex
   by (metis (mono_tags, lifting) empty_Collect_eq inner_zero_left)
 
 lemma hyperplane_eq_UNIV:
    "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
 proof -
   have "a = 0 \<and> b = 0" if "UNIV \<subseteq> {x. a \<bullet> x = b}"
     using subsetD [OF that, where c = "((b+1) / (a \<bullet> a)) *\<^sub>R a"]
     by (simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_lt:
    "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
 proof -
   have "a = 0 \<and> b \<le> 0" if "{x. a \<bullet> x < b} \<subseteq> {}"
     using subsetD [OF that, where c = "((b-1) / (a \<bullet> a)) *\<^sub>R a"]
     by (force simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_gt:
   "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
   using halfspace_eq_empty_lt [of "-a" "-b"]
   by simp
 
 lemma halfspace_eq_empty_le:
    "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
 proof -
   have "a = 0 \<and> b < 0" if "{x. a \<bullet> x \<le> b} \<subseteq> {}"
     using subsetD [OF that, where c = "((b-1) / (a \<bullet> a)) *\<^sub>R a"]
     by (force simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_ge:
   "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
   using halfspace_eq_empty_le [of "-a" "-b"] by simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Use set distance for an easy proof of separation properties\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separation_closures:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
   obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis by auto
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on UNIV f"
     unfolding f_def by (intro continuous_intros continuous_on_setdist)
   show ?thesis
   proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
     show "{x. 0 < f x} \<inter> {x. f x < 0} = {}"
       by auto
     show "open {x. 0 < f x}"
       by (simp add: open_Collect_less contf)
     show "open {x. f x < 0}"
       by (simp add: open_Collect_less contf)
     have "\<And>x. x \<in> S \<Longrightarrow> setdist {x} T \<noteq> 0" "\<And>x. x \<in> T \<Longrightarrow> setdist {x} S \<noteq> 0"
       by (meson False assms disjoint_iff setdist_eq_0_sing_1)+
     then show "S \<subseteq> {x. 0 < f x}" "T \<subseteq> {x. f x < 0}"
       using less_eq_real_def by (fastforce simp add: f_def setdist_sing_in_set)+
   qed
 qed
 
 lemma separation_normal:
   fixes S :: "'a::euclidean_space set"
   assumes "closed S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 using separation_closures [of S T]
 by (metis assms closure_closed disjnt_def inf_commute)
 
 lemma separation_normal_local:
   fixes S :: "'a::euclidean_space set"
   assumes US: "closedin (top_of_set U) S"
       and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
   obtains S' T' where "openin (top_of_set U) S'"
                       "openin (top_of_set U) T'"
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
     using UT US by (blast dest: closedin_subset)
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on U f"
     unfolding f_def by (intro continuous_intros)
   show ?thesis
   proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
     show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
       by auto
     show "openin (top_of_set U) (U \<inter> f -` {0<..})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     show "openin (top_of_set U) (U \<inter> f -` {..<0})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     have "S \<subseteq> U" "T \<subseteq> U"
       using closedin_imp_subset assms by blast+
     then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> f -` {..<0}"
       using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
   qed
 qed
 
 lemma separation_normal_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 proof -
   have "closed S" "bounded S"
     using assms by (auto simp: compact_eq_bounded_closed)
   then obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
     by (auto dest!: bounded_subset_ballD)
   have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
     using assms r by blast+
   then obtain U V where UV: "open U" "open V" "S \<subseteq> U" "T \<union> - ball 0 r \<subseteq> V" "U \<inter> V = {}"
     by (meson  \<open>closed S\<close> separation_normal)
   then have "compact(closure U)"
     by (meson bounded_ball bounded_subset compact_closure compl_le_swap2 disjoint_eq_subset_Compl le_sup_iff)
   with UV show thesis
     using that by auto
 qed
 
 subsection\<open>Connectedness of the intersection of a chain\<close>
 
 proposition connected_chain:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof (cases "\<F> = {}")
   case True then show ?thesis
     by auto
 next
   case False
   then have cf: "compact(\<Inter>\<F>)"
     by (simp add: cc compact_Inter)
   have False if AB: "closed A" "closed B" "A \<inter> B = {}"
                 and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
   proof -
     obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
       using separation_normal [OF AB] by metis
     obtain K where "K \<in> \<F>" "compact K"
       using cc False by blast
     then obtain N where "open N" and "K \<subseteq> N"
       by blast
     let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
     obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
     proof (rule compactE [OF \<open>compact K\<close>])
       show "K \<subseteq> \<Union>(insert (U \<union> V) ((-) N ` \<F>))"
         using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
       show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
         by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
     qed
     then have "finite(\<D> - {U \<union> V})"
       by blast
     moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
       using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
     ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
       using finite_subset_image by metis
     obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
     proof (cases "\<G> = {}")
       case True
       with \<open>\<F> \<noteq> {}\<close> that show ?thesis
         by auto
     next
       case False
       have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
         by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
       with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
       have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
       proof induction
         case (insert X \<H>)
         show ?case
         proof (cases "\<H> = {}")
           case True then show ?thesis by auto
         next
           case False
           then have "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
             by (simp add: insert.prems)
           with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
             by metis
           have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
             by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
           then show ?thesis
           proof
             assume "N - J \<subseteq> N - X" with J show ?thesis
               by auto
           next
             assume "N - X \<subseteq> N - J"
             with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
               by auto
             with \<open>J \<in> \<H>\<close> show ?thesis
               by blast
           qed
         qed
       qed simp
       with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
     qed
     have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
       using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
     also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
       by (metis (no_types, hide_lams) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
     finally have "J \<inter> K \<subseteq> U \<union> V"
       by blast
     moreover have "connected(J \<inter> K)"
       by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
     moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
     moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
     ultimately show False
         using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close>  by auto
   qed
   with cf show ?thesis
     by (auto simp: connected_closed_set compact_imp_closed)
 qed
 
 lemma connected_chain_gen:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes X: "X \<in> \<F>" "compact X"
       and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof -
   have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
     using X by blast
   moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
   proof (rule connected_chain)
     show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
       using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
     show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
       using local.linear by blast
   qed
   ultimately show ?thesis
     by metis
 qed
 
 lemma connected_nest:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
-  apply (rule connected_chain)
-  using S apply blast
+proof (rule connected_chain)
+  show "\<And>A T. A \<in> range S \<and> T \<in> range S \<Longrightarrow> A \<subseteq> T \<or> T \<subseteq> A"
   by (metis image_iff le_cases nest)
+qed (use S in blast)
 
 lemma connected_nest_gen:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
-  apply (rule connected_chain_gen [of "S k"])
-  using S apply auto
-  by (meson le_cases nest subsetCE)
+proof (rule connected_chain_gen [of "S k"])
+  show "\<And>A T. A \<in> range S \<and> T \<in> range S \<Longrightarrow> A \<subseteq> T \<or> T \<subseteq> A"
+    by (metis imageE le_cases nest)
+qed (use S in auto)
 
 subsection\<open>Proper maps, including projections out of compact sets\<close>
 
 lemma finite_indexed_bound:
   assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
     shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
 using A
 proof (induction A)
   case empty then show ?case by force
 next
   case (insert a A)
     then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
       by force
     then show ?case
       by (metis dual_order.trans insert_iff le_cases)
 qed
 
 proposition proper_map:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes "closedin (top_of_set S) K"
       and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
       and "f ` S \<subseteq> T"
     shows "closedin (top_of_set T) (f ` K)"
 proof -
   have "K \<subseteq> S"
     using assms closedin_imp_subset by metis
   obtain C where "closed C" and Keq: "K = S \<inter> C"
     using assms by (auto simp: closedin_closed)
   have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
   proof -
     obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
       using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
     then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
       by metis
     then have fX: "\<And>n. f (X n) = h n"
       by metis
+    define \<Psi> where "\<Psi> \<equiv> \<lambda>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))}"
     have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
     proof (intro closed_Int_compact [OF \<open>closed C\<close> com] compact_sequence_with_limit)
       show "insert y (range (\<lambda>i. f (X (n + i)))) \<subseteq> T"
         using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> by blast
       show "(\<lambda>i. f (X (n + i))) \<longlonglongrightarrow> y"
         by (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
     qed
-    then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
-      by (simp add: Keq Int_def conj_commute)
+    then have comf: "compact (\<Psi> n)" for n
+      by (simp add: Keq Int_def \<Psi>_def conj_commute)
     have ne: "\<Inter>\<F> \<noteq> {}"
              if "finite \<F>"
-                and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
-                           (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
+                and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow> (\<exists>n. t = \<Psi> n)"
              for \<F>
     proof -
-      obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
+      obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = \<Psi> k"
         by (rule exE [OF finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>]], force+)
       have "X m \<in> \<Inter>\<F>"
-        using X le_Suc_ex by (fastforce dest: m)
+        using X le_Suc_ex by (fastforce simp: \<Psi>_def dest: m)
       then show ?thesis by blast
     qed
-    have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
-               \<noteq> {}"
-      apply (rule compact_fip_Heine_Borel)
-       using comf apply force
-       using ne  apply (simp add: subset_iff del: insert_iff)
-      done
-    then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
-      by blast
+    have "(\<Inter>n. \<Psi> n) \<noteq> {}"
+    proof (rule compact_fip_Heine_Borel)
+      show "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> range \<Psi>\<rbrakk> \<Longrightarrow> \<Inter> \<F>' \<noteq> {}"
+        by (meson ne rangeE subset_eq)
+    qed (use comf in blast)
+    then obtain x where "x \<in> K" "\<And>n. (f x = y \<or> (\<exists>u. f x = h (n + u)))"
+      by (force simp add: \<Psi>_def fX)
     then show ?thesis
-      apply (simp add: image_iff fX)
-      by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
+      unfolding image_iff by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
   qed
   with assms closedin_subset show ?thesis
     by (force simp: closedin_limpt)
 qed
 
 
 lemma compact_continuous_image_eq:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes f: "inj_on f S"
   shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact(f ` T))"
            (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (metis continuous_on_subset compact_continuous_image)
 next
   assume RHS: ?rhs
   obtain g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     by (metis inv_into_f_f f)
   then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
     using that by fastforce
   have gfim: "g ` f ` S \<subseteq> S" using gf by auto
   have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
   proof -
     obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
       by (force simp: C RHS)
     moreover have "f ` C = (f ` S \<inter> g -` C)"
       using C gf by auto
     ultimately show ?thesis
       using C by auto
   qed
   show ?lhs
     using proper_map [OF _ _ gfim] **
     by (simp add: continuous_on_closed * closedin_imp_subset)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
 
 lemma convex_connected_collinear:
   fixes S :: "'a::euclidean_space set"
   assumes "collinear S"
     shows "convex S \<longleftrightarrow> connected S"
 proof
   assume "convex S"
   then show "connected S"
     using convex_connected by blast
 next
   assume S: "connected S"
   show "convex S"
   proof (cases "S = {}")
     case True
     then show ?thesis by simp
   next
     case False
     then obtain a where "a \<in> S" by auto
     have "collinear (affine hull S)"
       by (simp add: assms collinear_affine_hull_collinear)
     then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
       by (meson \<open>a \<in> S\<close> collinear hull_inc)
     then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>R z"
       by metis
     then have inj_f: "inj_on f (affine hull S)"
       by (metis diff_add_cancel inj_onI)
     have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
     proof -
       have "f x *\<^sub>R z = x - a"
         by (simp add: f hull_inc x)
       moreover have "f y *\<^sub>R z = y - a"
         by (simp add: f hull_inc y)
       ultimately show ?thesis
         by (simp add: scaleR_left.diff)
     qed
     have cont_f: "continuous_on (affine hull S) f"
-      apply (clarsimp simp: dist_norm continuous_on_iff diff)
-      by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
+    proof (clarsimp simp: dist_norm continuous_on_iff diff)
+      show "\<And>x e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>y \<in> affine hull S. \<bar>f y - f x\<bar> * norm z < d \<longrightarrow> \<bar>f y - f x\<bar> < e"
+        by (metis \<open>z \<noteq> 0\<close> mult_pos_pos real_mult_less_iff1 zero_less_norm_iff)
+    qed
     then have conn_fS: "connected (f ` S)"
       by (meson S connected_continuous_image continuous_on_subset hull_subset)
     show ?thesis
     proof (clarsimp simp: convex_contains_segment)
       fix x y z
       assume "x \<in> S" "y \<in> S" "z \<in> closed_segment x y"
       have False if "z \<notin> S"
       proof -
         have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
         proof (rule continuous_injective_image_segment_1)
           show "continuous_on (closed_segment x y) f"
             by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
           show "inj_on f (closed_segment x y)"
             by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
         qed
         then have fz: "f z \<in> closed_segment (f x) (f y)"
           using \<open>z \<in> closed_segment x y\<close> by blast
         have "z \<in> affine hull S"
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
         then have fz_notin: "f z \<notin> f ` S"
           using hull_subset inj_f inj_onD that by fastforce
         moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
         proof -
-          have "{..<f z} \<inter> f ` {x,y} \<noteq> {}"  "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
-            using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
-             apply (auto simp: closed_segment_eq_real_ivl less_eq_real_def split: if_split_asm)
-             apply (metis image_eqI)+
-            done
+          consider "f x \<le> f z \<and> f z \<le> f y" | "f y \<le> f z \<and> f z \<le> f x"
+            using fz
+            by (auto simp add: closed_segment_eq_real_ivl split: if_split_asm)
+          then have "{..<f z} \<inter> f ` {x,y} \<noteq> {} \<and> {f z<..} \<inter> f ` {x,y} \<noteq> {}"
+            by cases (use fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close> in \<open>auto simp: image_iff\<close>)
           then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
             using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
         qed
         ultimately show False
           using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
       qed
       then show "z \<in> S" by meson
     qed
   qed
 qed
 
 lemma compact_convex_collinear_segment_alt:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
   obtains a b where "S = closed_segment a b"
 proof -
   obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
   have "collinear (affine hull S)"
     by (simp add: assms collinear_affine_hull_collinear)
   then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
     by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
   then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
     by metis
   let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
   have gf: "?g (f x) = x" if "x \<in> affine hull S" for x
     by (metis diff_add_cancel f that)
   then have inj_f: "inj_on f (affine hull S)"
     by (metis inj_onI)
   have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
   proof -
     have "f x *\<^sub>R z = x - \<xi>"
       by (simp add: f hull_inc x)
     moreover have "f y *\<^sub>R z = y - \<xi>"
       by (simp add: f hull_inc y)
     ultimately show ?thesis
       by (simp add: scaleR_left.diff)
   qed
   have cont_f: "continuous_on (affine hull S) f"
-    apply (clarsimp simp: dist_norm continuous_on_iff diff)
-    by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
+  proof (clarsimp simp: dist_norm continuous_on_iff diff)
+    show "\<And>x e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>y \<in> affine hull S. \<bar>f y  - f x\<bar> * norm z < d \<longrightarrow> \<bar>f y  - f x\<bar> < e"
+      by (metis \<open>z \<noteq> 0\<close> mult_pos_pos real_mult_less_iff1 zero_less_norm_iff)
+  qed
   then have "connected (f ` S)"
     by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
   moreover have "compact (f ` S)"
     by (meson \<open>compact S\<close> compact_continuous_image_eq cont_f hull_subset inj_f)
   ultimately obtain x y where "f ` S = {x..y}"
     by (meson connected_compact_interval_1)
   then have fS_eq: "f ` S = closed_segment x y"
     using \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
   obtain a b where "a \<in> S" "f a = x" "b \<in> S" "f b = y"
     by (metis (full_types) ends_in_segment fS_eq imageE)
   have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
   proof (rule continuous_injective_image_segment_1)
     show "continuous_on (closed_segment a b) f"
       by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
     show "inj_on f (closed_segment a b)"
       by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
   qed
   then have "f ` (closed_segment a b) = f ` S"
     by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
   then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
     by simp
   moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
-    apply safe
-     apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
-    by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff subsetCE)
+    unfolding image_def using \<open>a \<in> S\<close> \<open>b \<in> S\<close>
+    by (safe; metis (mono_tags, lifting)  convex_affine_hull convex_contains_segment gf hull_subset subsetCE)
   ultimately have "closed_segment a b = S"
     using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
   then show ?thesis
     using that by blast
 qed
 
 lemma compact_convex_collinear_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
   obtains a b where "S = closed_segment a b"
   using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
 
 
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
           "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
 
 lemma proper_map_fst:
   assumes "compact T" "K \<subseteq> S" "compact K"
     shows "compact (S \<times> T \<inter> fst -` K)"
 proof -
   have "(S \<times> T \<inter> fst -` K) = K \<times> T"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_fst:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set S) (fst ` c)"
 proof -
   have *: "fst ` (S \<times> T) \<subseteq> S"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
 qed
 
 lemma proper_map_snd:
   assumes "compact S" "K \<subseteq> T" "compact K"
     shows "compact (S \<times> T \<inter> snd -` K)"
 proof -
   have "(S \<times> T \<inter> snd -` K) = S \<times> K"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_snd:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set T) (snd ` c)"
 proof -
   have *: "snd ` (S \<times> T) \<subseteq> T"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
 qed
 
 lemma closedin_compact_projection:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
     shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
 proof -
   have "U \<subseteq> S \<times> T"
     by (metis clo closedin_imp_subset)
   then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
     by force
   moreover have "closedin (top_of_set T) (snd ` U)"
     by (rule closed_map_snd [OF assms])
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma closed_compact_projection:
   fixes S :: "'a::euclidean_space set"
     and T :: "('a * 'b::euclidean_space) set"
   assumes "compact S" and clo: "closed T"
     shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
 proof -
   have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} = {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
     by auto
   show ?thesis
     unfolding *
     by (intro clo closedin_closed_Int closedin_closed_trans [OF _ closed_UNIV] closedin_compact_projection [OF \<open>compact S\<close>])
 qed
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_convex_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 proof
   show "affine hull (S \<inter> T) \<subseteq> affine hull S"
     by (simp add: hull_mono)
 next
   obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
     using assms interior_subset by blast
   then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
     using mem_interior_cball by blast
   have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
   proof (cases "x = a")
     case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
       by blast
   next
     case False
     define k where "k = min (1/2) (e / norm (x-a))"
     have k: "0 < k" "k < 1"
       using \<open>e > 0\<close> False by (auto simp: k_def)
     then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
       by simp
     have "e / norm (x - a) \<ge> k"
       using k_def by linarith
     then have "a + k *\<^sub>R (x - a) \<in> cball a e"
       using \<open>0 < k\<close> False
       by (simp add: dist_norm) (simp add: field_simps)
     then have T: "a + k *\<^sub>R (x - a) \<in> T"
       using e by blast
     have S: "a + k *\<^sub>R (x - a) \<in> S"
       using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
       by (simp add: algebra_simps)
     have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
       by (intro span_mul [OF span_base] image_eqI [where x = "a + k *\<^sub>R (x - a)"]) (auto simp: S T)
     with xa image_iff show ?thesis  by fastforce
   qed
   have "S \<subseteq> affine hull (S \<inter> T)"
     by (force simp: * \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
   then show "affine hull S \<subseteq> affine hull (S \<inter> T)"
     by (simp add: subset_hull)
 qed
 
 corollary affine_hull_convex_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
 
 corollary affine_hull_affine_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "S \<inter> interior T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
 
 corollary affine_hull_affine_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "open T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
 
 corollary affine_hull_convex_Int_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   using assms unfolding openin_open
   by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
   shows "affine hull S = affine hull T"
   using assms unfolding openin_open
   by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
 
 corollary affine_hull_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S" "S \<noteq> {}"
   shows "affine hull S = UNIV"
   by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
 
 lemma aff_dim_convex_Int_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> T) = aff_dim S"
   using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
 
 lemma aff_dim_convex_Int_open:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow>  aff_dim(S \<inter> T) = aff_dim S"
   using aff_dim_convex_Int_nonempty_interior interior_eq by blast
 
 lemma affine_hull_Diff:
   fixes S:: "'a::real_normed_vector set"
   assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
   shows "affine hull (S - F) = affine hull S"
 proof -
   have clo: "closedin (top_of_set S) F"
     using assms finite_imp_closedin by auto
   moreover have "S - F \<noteq> {}"
     using assms by auto
   ultimately show ?thesis
     by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
 qed
 
 lemma affine_hull_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x < r} = (if a = 0 \<and> r \<le> 0 then {} else UNIV)"
 using halfspace_eq_empty_lt [of a r]
 by (simp add: open_halfspace_lt affine_hull_open)
 
 lemma affine_hull_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<le> r} = (if a = 0 \<and> r < 0 then {} else UNIV)"
 proof (cases "a = 0")
   case True then show ?thesis by simp
 next
   case False
   then have "affine hull closure {x. a \<bullet> x < r} = UNIV"
     using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
   moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> r}"
     by (simp add: Collect_mono)
   ultimately show ?thesis using False antisym_conv hull_mono top_greatest
     by (metis affine_hull_halfspace_lt)
 qed
 
 lemma affine_hull_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x > r} = (if a = 0 \<and> r \<ge> 0 then {} else UNIV)"
 using halfspace_eq_empty_gt [of r a]
 by (simp add: open_halfspace_gt affine_hull_open)
 
 lemma affine_hull_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<ge> r} = (if a = 0 \<and> r > 0 then {} else UNIV)"
 using affine_hull_halfspace_le [of "-a" "-r"] by simp
 
 lemma aff_dim_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x < r} =
         (if a = 0 \<and> r \<le> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
 
 lemma aff_dim_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<le> r} =
         (if a = 0 \<and> r < 0 then -1 else DIM('a))"
 proof -
   have "int (DIM('a)) = aff_dim (UNIV::'a set)"
     by (simp)
   then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
     using that by (simp add: affine_hull_halfspace_le not_less)
   then show ?thesis
     by (force)
 qed
 
 lemma aff_dim_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x > r} =
         (if a = 0 \<and> r \<ge> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
 
 lemma aff_dim_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<ge> r} =
         (if a = 0 \<and> r > 0 then -1 else DIM('a))"
 using aff_dim_halfspace_le [of "-a" "-r"] by simp
 
 proposition aff_dim_eq_hyperplane:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
+  (is "?lhs = ?rhs")
 proof (cases "S = {}")
   case True then show ?thesis
     by (auto simp: dest: hyperplane_eq_Ex)
 next
   case False
   then obtain c where "c \<in> S" by blast
   show ?thesis
   proof (cases "c = 0")
-    case True show ?thesis
-      using span_zero [of S]
-        apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
-          del: One_nat_def)
-        apply (auto simp add: \<open>c = 0\<close>)
-        done
+    case True 
+    have "?lhs \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0})"
+      by (simp add: aff_dim_eq_dim [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane del: One_nat_def)
+    also have "... \<longleftrightarrow> ?rhs"
+      using span_zero [of S] True \<open>c \<in> S\<close> affine_hull_span_0 hull_inc  
+      by (fastforce simp add: affine_hull_span_gen [of c] \<open>c = 0\<close>)
+    finally show ?thesis .
   next
     case False
     have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
     proof -
       have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
         by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
       then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
         by blast
     qed
     have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
          if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
     proof -
       have "b = a \<bullet> c"
         using span_0 that by fastforce
       with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
         by simp
       then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
         by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
       also have "... = {x. a \<bullet> x = 0}"
         by (force simp: inner_distrib inner_diff_right
              intro: image_eqI [where x="x+c" for x])
       finally show ?thesis .
     qed
-    show ?thesis
-      apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
-                  del: One_nat_def cong: image_cong_simp, safe)
-      apply (fastforce simp add: inner_distrib intro: xc_im)
-      apply (force intro!: 2)
-      done
+    have "?lhs = (\<exists>a. a \<noteq> 0 \<and> span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0})"
+      by (simp add: aff_dim_eq_dim [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane del: One_nat_def)
+    also have "... = ?rhs"
+      by (fastforce simp add: affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc inner_distrib intro: xc_im intro!: 2)
+    finally show ?thesis .
   qed
 qed
 
 corollary aff_dim_hyperplane [simp]:
   fixes a :: "'a::euclidean_space"
   shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
 by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some stepping theorems\<close>
 
 lemma aff_dim_insert:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x s' where S: "S = insert x s'" "x \<notin> s'"
     by (meson Set.set_insert all_not_in_conv)
   show ?thesis using S
     by (force simp add: affine_hull_insert_span_gen span_zero insert_commute [of a] aff_dim_eq_dim [of x] dim_insert)
 qed
 
 lemma affine_dependent_choose:
   fixes a :: "'a :: euclidean_space"
   assumes "\<not>(affine_dependent S)"
   shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
         (is "?lhs = ?rhs")
 proof safe
   assume "affine_dependent (insert a S)" and "a \<in> S"
   then show "False"
     using \<open>a \<in> S\<close> assms insert_absorb by fastforce
 next
   assume lhs: "affine_dependent (insert a S)"
   then have "a \<notin> S"
     by (metis (no_types) assms insert_absorb)
   moreover have "finite S"
     using affine_independent_iff_card assms by blast
   moreover have "aff_dim (insert a S) \<noteq> int (card S)"
     using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
   ultimately show "a \<in> affine hull S"
     by (metis aff_dim_affine_independent aff_dim_insert assms)
 next
   assume "a \<notin> S" and "a \<in> affine hull S"
   show "affine_dependent (insert a S)"
     by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
 qed
 
 lemma affine_independent_insert:
   fixes a :: "'a :: euclidean_space"
   shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> affine_dependent(insert a S)"
   by (simp add: affine_dependent_choose)
 
 lemma subspace_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "subspace S"
     shows "bounded S \<longleftrightarrow> S = {0}"
 proof -
   have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
   proof -
     obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
       using \<open>bounded S\<close> by (force simp: bounded_pos_less)
     have "(B / norm x) *\<^sub>R x \<in> S"
       using assms subspace_mul \<open>x \<in> S\<close> by auto
     moreover have "norm ((B / norm x) *\<^sub>R x) = B"
       using that B by (simp add: algebra_simps)
     ultimately show False using B by force
   qed
   then have "bounded S \<Longrightarrow> S = {0}"
     using assms subspace_0 by fastforce
   then show ?thesis
     by blast
 qed
 
 lemma affine_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain b where "b \<in> S" by blast
   with False assms 
   have "bounded S \<Longrightarrow> S = {b}"
     using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
     by (metis (no_types, lifting) ab_group_add_class.ab_left_minus bounded_translation image_empty image_insert subspace_bounded_eq_trivial translation_invert)
   then show ?thesis by auto
 qed
 
 lemma affine_bounded_eq_lowdim:
   fixes S :: "'a::euclidean_space set"
   assumes "affine S"
   shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
 proof
   show "aff_dim S \<le> 0 \<Longrightarrow> bounded S"
   by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
 qed (use affine_bounded_eq_trivial assms in fastforce)
 
 
 lemma bounded_hyperplane_eq_trivial_0:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
   shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
 proof
   assume "bounded {x. a \<bullet> x = 0}"
   then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
   with assms show "DIM('a) = 1"
     by (simp add: le_Suc_eq)
 next
   assume "DIM('a) = 1"
   then show "bounded {x. a \<bullet> x = 0}"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
 qed
 
 lemma bounded_hyperplane_eq_trivial:
   fixes a :: "'a::euclidean_space"
   shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
 proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
   assume "r \<noteq> 0" "a \<noteq> 0"
   have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
     by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
   then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
     by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>General case without assuming closure and getting non-strict separation\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_closed_point_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
   obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
 proof -
   obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
     using distance_attains_inf [of S z] assms by auto
   then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
     using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
   show ?thesis
   proof (rule that [OF \<open>y \<in> S\<close> *])
     fix x
     assume "x \<in> S"
     have yz: "0 < (y - z) \<bullet> (y - z)"
       using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
     { assume 0: "0 < ((z - y) \<bullet> (x - y))"
       with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
       have False
         using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
     }
     then have "0 \<le> ((y - z) \<bullet> (x - y))"
       by (force simp: not_less inner_diff_left)
     with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
       by (simp add: algebra_simps)
     then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
   qed
 qed
 
 lemma separating_hyperplane_closed_0_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "0 \<notin> S"
   obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
   using separating_hyperplane_closed_point_inset [OF assms] by simp (metis \<open>0 \<notin> S\<close>)
 
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_set_0_inspan:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
   obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
 proof -
   define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
   have "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
           if f': "finite f'" "f' \<subseteq> k ` S" for f'
   proof -
     obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
       using finite_subset_image [OF f'] by blast
     obtain a where "a \<in> S" "a \<noteq> 0"
       using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
     then have "norm (a /\<^sub>R (norm a)) = 1"
       by simp
     moreover have "a /\<^sub>R (norm a) \<in> span S"
       by (simp add: \<open>a \<in> S\<close> span_scale span_base)
     ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
       by simp
     show ?thesis
     proof (cases "C = {}")
       case True with C ass show ?thesis
         by auto
     next
       case False
       have "closed (convex hull C)"
         using \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
       moreover have "convex hull C \<noteq> {}"
         by (simp add: False)
       moreover have "0 \<notin> convex hull C"
         by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
       ultimately obtain a b
             where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
                   and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
         using separating_hyperplane_closed_0_inset by blast
       then have "a \<in> S"
         by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
       moreover have "norm (a /\<^sub>R (norm a)) = 1"
         using \<open>a \<noteq> 0\<close> by simp
       moreover have "a /\<^sub>R (norm a) \<in> span S"
         by (simp add: \<open>a \<in> S\<close> span_scale span_base)
       ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
         by simp
       have "\<And>x. \<lbrakk>a \<noteq> 0; x \<in> C\<rbrakk> \<Longrightarrow> 0 \<le> x \<bullet> a"
         using ab \<open>0 < b\<close> by (metis hull_inc inner_commute less_eq_real_def less_trans)
       then have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
         by (auto simp add: field_split_simps)
       show ?thesis
         unfolding C k_def
         using ass aa Int_iff empty_iff by force
     qed
   qed
   moreover have "\<And>T. T \<in> k ` S \<Longrightarrow> closed T"
     using closed_halfspace_ge k_def by blast
   ultimately have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
     by (metis compact_imp_fip closed_Int_compact closed_span compact_cball compact_frontier)
   then show ?thesis
     unfolding set_eq_iff k_def
     by simp (metis inner_commute norm_eq_zero that zero_neq_one)
 qed
 
 
 lemma separating_hyperplane_set_point_inaff:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and zno: "z \<notin> S"
   obtains a b where "(z + a) \<in> affine hull (insert z S)"
                 and "a \<noteq> 0" and "a \<bullet> z \<le> b"
                 and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
 proof -
   from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
   have "convex ((+) (- z) ` S)"
     using \<open>convex S\<close> by simp
   moreover have "(+) (- z) ` S \<noteq> {}"
     by (simp add: \<open>S \<noteq> {}\<close>)
   moreover have "0 \<notin> (+) (- z) ` S"
     using zno by auto
   ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
                   and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
     using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
     by blast
   then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
     by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
   moreover
   have "z + a \<in> affine hull insert z S"
     using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen by blast
   ultimately show ?thesis
     using \<open>a \<noteq> 0\<close> szx that by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> supporting_hyperplane_rel_boundary:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 proof -
   obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
                   and "a \<noteq> 0" and "a \<bullet> x \<le> b"
                   and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> b"
     using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
     by (auto simp: rel_interior_eq_empty convex_rel_interior)
   have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
   proof -
     have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
       by (rule continuous_intros continuous_on_subset | blast)+
     have y: "y \<in> closure (rel_interior S)"
       using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
       by fastforce
     show ?thesis
       using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
       by fastforce
   qed
   have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
   proof -
     obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
       using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
     define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
     have "y' \<in> cball y e"
       unfolding y'_def using \<open>0 < e\<close> by force
     moreover have "y' \<in> affine hull S"
       unfolding y'_def
       by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
                 rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
     ultimately have "y' \<in> S"
       using e by auto
     have "a \<bullet> x \<le> a \<bullet> y"
       using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
     moreover have "a \<bullet> x \<noteq> a \<bullet> y"
       using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close> \<open>0 < e\<close> not_le
       by (fastforce simp add: y'_def inner_diff dot_square_norm power2_eq_square)
     ultimately show ?thesis by force
   qed
   show ?thesis
     by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
 qed
 
 lemma supporting_hyperplane_relative_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 using supporting_hyperplane_rel_boundary [of "closure S" x]
 by (metis assms convex_closure convex_rel_interior_closure)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
 
 lemma
-  fixes s :: "'a::euclidean_space set"
-  assumes "\<not> affine_dependent(s \<union> t)"
-    shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
-      and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent(S \<union> T)"
+    shows convex_hull_Int_subset: "convex hull S \<inter> convex hull T \<subseteq> convex hull (S \<inter> T)" (is ?C)
+      and affine_hull_Int_subset: "affine hull S \<inter> affine hull T \<subseteq> affine hull (S \<inter> T)" (is ?A)
 proof -
-  have [simp]: "finite s" "finite t"
+  have [simp]: "finite S" "finite T"
     using aff_independent_finite assms by blast+
-    have "sum u (s \<inter> t) = 1 \<and>
-          (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
-      if [simp]:  "sum u s = 1"
-                 "sum v t = 1"
-         and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
+    have "sum u (S \<inter> T) = 1 \<and>
+          (\<Sum>v\<in>S \<inter> T. u v *\<^sub>R v) = (\<Sum>v\<in>S. u v *\<^sub>R v)"
+      if [simp]:  "sum u S = 1"
+                 "sum v T = 1"
+         and eq: "(\<Sum>x\<in>T. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)" for u v
     proof -
-    define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
-    have "sum f (s \<union> t) = 0"
-      apply (simp add: f_def sum_Un sum_subtractf)
-      apply (simp add: Int_commute flip: sum.inter_restrict)
-      done
-    moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
-      apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
-      apply (simp add: if_smult Int_commute eq flip: sum.inter_restrict
-          cong del: if_weak_cong)
-      done
-    ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
-      using aff_independent_finite assms unfolding affine_dependent_explicit
-      by blast
-    then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
-      by (simp add: f_def) presburger
-    have "sum u (s \<inter> t) = sum u s"
-      by (simp add: sum.inter_restrict)
-    then have "sum u (s \<inter> t) = 1"
-      using that by linarith
-    moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
+      define f where "f x = (if x \<in> S then u x else 0) - (if x \<in> T then v x else 0)" for x
+      have "sum f (S \<union> T) = 0"
+        by (simp add: f_def sum_Un sum_subtractf flip: sum.inter_restrict)
+      moreover have "(\<Sum>x\<in>(S \<union> T). f x *\<^sub>R x) = 0"
+        by (simp add: eq f_def sum_Un scaleR_left_diff_distrib sum_subtractf if_smult flip: sum.inter_restrict cong: if_cong)
+      ultimately have "\<And>v. v \<in> S \<union> T \<Longrightarrow> f v = 0"
+        using aff_independent_finite assms unfolding affine_dependent_explicit
+        by blast
+      then have u [simp]: "\<And>x. x \<in> S \<Longrightarrow> u x = (if x \<in> T then v x else 0)"
+        by (simp add: f_def) presburger
+      have "sum u (S \<inter> T) = sum u S"
+        by (simp add: sum.inter_restrict)
+      then have "sum u (S \<inter> T) = 1"
+        using that by linarith
+      moreover have "(\<Sum>v\<in>S \<inter> T. u v *\<^sub>R v) = (\<Sum>v\<in>S. u v *\<^sub>R v)"
       by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
     ultimately show ?thesis
       by force
     qed
     then show ?A ?C
       by (auto simp: convex_hull_finite affine_hull_finite)
 qed
 
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_Int:
-  fixes s :: "'a::euclidean_space set"
-  assumes "\<not> affine_dependent(s \<union> t)"
-    shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent(S \<union> T)"
+    shows "affine hull (S \<inter> T) = affine hull S \<inter> affine hull T"
   by (simp add: affine_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition\<^marker>\<open>tag unimportant\<close> convex_hull_Int:
-  fixes s :: "'a::euclidean_space set"
-  assumes "\<not> affine_dependent(s \<union> t)"
-    shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
+  fixes S :: "'a::euclidean_space set"
+  assumes "\<not> affine_dependent(S \<union> T)"
+    shows "convex hull (S \<inter> T) = convex hull S \<inter> convex hull T"
   by (simp add: convex_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition\<^marker>\<open>tag unimportant\<close>
-  fixes s :: "'a::euclidean_space set set"
-  assumes "\<not> affine_dependent (\<Union>s)"
-    shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
-      and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
+  fixes S :: "'a::euclidean_space set set"
+  assumes "\<not> affine_dependent (\<Union>S)"
+    shows affine_hull_Inter: "affine hull (\<Inter>S) = (\<Inter>T\<in>S. affine hull T)" (is "?A")
+      and convex_hull_Inter: "convex hull (\<Inter>S) = (\<Inter>T\<in>S. convex hull T)" (is "?C")
 proof -
-  have "finite s"
+  have "finite S"
     using aff_independent_finite assms finite_UnionD by blast
   then have "?A \<and> ?C" using assms
-  proof (induction s rule: finite_induct)
+  proof (induction S rule: finite_induct)
     case empty then show ?case by auto
   next
-    case (insert t F)
+    case (insert T F)
     then show ?case
     proof (cases "F={}")
       case True then show ?thesis by simp
     next
       case False
-      with "insert.prems" have [simp]: "\<not> affine_dependent (t \<union> \<Inter>F)"
+      with "insert.prems" have [simp]: "\<not> affine_dependent (T \<union> \<Inter>F)"
         by (auto intro: affine_dependent_subset)
       have [simp]: "\<not> affine_dependent (\<Union>F)"
         using affine_independent_subset insert.prems by fastforce
       show ?thesis
         by (simp add: affine_hull_Int convex_hull_Int insert.IH)
     qed
   qed
   then show "?A" "?C"
     by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange_unique:
   fixes S :: "'a::euclidean_space set"
   assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
       and S: "T \<subseteq> S" "T' \<subseteq> S"
       and x:  "x \<in> convex hull (insert a T)"
       and x': "x \<in> convex hull (insert a T')"
     shows "x \<in> convex hull (insert a (T \<inter> T'))"
 proof (cases "a \<in> S")
   case True
   then have "\<not> affine_dependent (insert a T \<union> insert a T')"
     using affine_dependent_subset assms by auto
   then have "x \<in> convex hull (insert a T \<inter> insert a T')"
     by (metis IntI convex_hull_Int x x')
   then show ?thesis
     by simp
 next
   case False
   then have anot: "a \<notin> T" "a \<notin> T'"
     using assms by auto
   have [simp]: "finite S"
     by (simp add: aff_independent_finite assms)
   then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
                   and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
     using a by (auto simp: convex_hull_finite)
   have fin [simp]: "finite T" "finite T'"
     using assms infinite_super \<open>finite S\<close> by blast+
   then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
                      and c1: "sum c (insert a T) = 1"
                      and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
                      and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
                      and c'1: "sum c' (insert a T') = 1"
                      and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
     using x x' by (auto simp: convex_hull_finite)
   with fin anot
   have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
    and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
     by simp_all
   have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     using x'eq fin anot by simp
   define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
   define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
   define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
   have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
   have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
   have sum_dd0: "sum dd S = 0"
     unfolding dd_def  using S
     by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
                   algebra_simps sum_distrib_right [symmetric] b1)
   have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
     by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
   then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
     using aeq by blast
   have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
     unfolding dd_def using S
     by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
   then have dd0: "dd v = 0" if "v \<in> S" for v
     using naff [unfolded affine_dependent_explicit not_ex, rule_format, of S dd]
     using that sum_dd0 by force
   consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
   then show ?thesis
   proof cases
     case 1
     then have "sum cc S \<le> sum cc' S"
       by (simp add: sumSS')
     then have le: "cc x \<le> cc' x" if "x \<in> S" for x
       using dd0 [OF that] 1 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
         by (simp add: c0 cc_def)
       show "0 \<le> (cc(a := c a)) a"
         by (simp add: c0)
       have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c a + sum (cc(a := c a)) S"
         using \<open>T \<subseteq> S\<close> False cc0 cc_def \<open>a \<notin> S\<close> by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a + (1 - c a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS'(1))
       finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
         by simp
     qed
   next
     case 2
     then have "sum cc' S \<le> sum cc S"
       by (simp add: sumSS')
     then have le: "cc' x \<le> cc x" if "x \<in> S" for x
       using dd0 [OF that] 2 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc' x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c'0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
         by (simp add: c'0 cc'_def)
       show "0 \<le> (cc'(a := c' a)) a"
         by (simp add: c'0)
       have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c' a + sum (cc'(a := c' a)) S"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c' a + (1 - c' a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
         by simp
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> convex_hull_exchange_Int:
   fixes a  :: "'a::euclidean_space"
   assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
   shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
          convex hull (insert a (T \<inter> T'))" (is "?lhs = ?rhs")
 proof (rule subset_antisym)
   show "?lhs \<subseteq> ?rhs"
     using in_convex_hull_exchange_unique assms by blast
   show "?rhs \<subseteq> ?lhs"
     by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
 qed
 
 lemma Int_closed_segment:
   fixes b :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
     shows "closed_segment a b \<inter> closed_segment b c = {b}"
 proof (cases "c = a")
   case True
   then show ?thesis
     using assms collinear_3_eq_affine_dependent by fastforce
 next
   case False
   from assms show ?thesis
   proof
     assume "b \<in> closed_segment a c"
     moreover have "\<not> affine_dependent {a, c}"
       by (simp)
     ultimately show ?thesis
       using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
       by (simp add: segment_convex_hull insert_commute)
   next
     assume ncoll: "\<not> collinear {a, b, c}"
     have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
     proof -
       have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
         by auto
       with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> closed_segment b c"
         by force
       then have d: "collinear {a, d, b}"  "collinear {b, d, c}"
         by (auto simp:  between_mem_segment between_imp_collinear)
       have "collinear {a, b, c}"
         by (metis (full_types) \<open>b \<noteq> d\<close> collinear_3_trans d insert_commute)
       with ncoll show False ..
     qed
     then show ?thesis
       by blast
   qed
 qed
 
 lemma affine_hull_finite_intersection_hyperplanes:
   fixes S :: "'a::euclidean_space set"
   obtains \<F> where
      "finite \<F>"
      "of_nat (card \<F>) + aff_dim S = DIM('a)"
      "affine hull S = \<Inter>\<F>"
      "\<And>h. h \<in> \<F> \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
 proof -
   obtain b where "b \<subseteq> S"
              and indb: "\<not> affine_dependent b"
              and eq: "affine hull S = affine hull b"
     using affine_basis_exists by blast
   obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> c"
              and affc: "affine hull c = UNIV"
     by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
   then have "finite c"
     by (simp add: aff_independent_finite)
   then have fbc: "finite b" "card b \<le> card c"
     using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
   have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
     by blast
   have card_cb: "(card (c - b)) + aff_dim S = DIM('a)"
   proof -
     have aff: "aff_dim (UNIV::'a set) = aff_dim c"
       by (metis aff_dim_affine_hull affc)
     have "aff_dim b = aff_dim S"
       by (metis (no_types) aff_dim_affine_hull eq)
     then have "int (card b) = 1 + aff_dim S"
       by (simp add: aff_dim_affine_independent indb)
     then show ?thesis
       using fbc aff
       by (simp add: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent card_Diff_subset of_nat_diff)
   qed
   show ?thesis
   proof (cases "c = b")
     case True show ?thesis
     proof
       show "int (card {}) + aff_dim S = int DIM('a)"
         using True card_cb by auto
       show "affine hull S = \<Inter> {}"
         using True affc eq by blast
     qed auto
   next
     case False
     have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
       by (rule affine_independent_subset [OF indc]) auto
     have *: "1 + aff_dim (c - {t}) = int (DIM('a))" if t: "t \<in> c" for t
     proof -
       have "insert t c = c"
         using t by blast
       then show ?thesis
         by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
     qed
     let ?\<F> = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))"
     show ?thesis
     proof
       have "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
-        apply (rule card_image [OF inj_onI])
-        by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
+      proof (rule card_image)
+        show "inj_on (\<lambda>a. affine hull (c - {a})) (c - b)"
+          unfolding inj_on_def
+          by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
+      qed
       then show "int (card ?\<F>) + aff_dim S = int DIM('a)"
         by (simp add: imeq card_cb)
       show "affine hull S = \<Inter> ?\<F>"
-        using \<open>b \<subseteq> c\<close> False
-        apply (subst affine_hull_Inter [OF ind, symmetric])
-        apply (simp add: eq double_diff)
-        done
-      show "\<And>h. h \<in> ?\<F> \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
-        apply clarsimp
-        by (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
+        by (metis Diff_eq_empty_iff INT_simps(4) UN_singleton order_refl \<open>b \<subseteq> c\<close> False eq double_diff affine_hull_Inter [OF ind])
+      have "\<And>a. \<lbrakk>a \<in> c; a \<notin> b\<rbrakk> \<Longrightarrow> aff_dim (c - {a}) = int (DIM('a) - Suc 0)"
+        by (metis "*" DIM_ge_Suc0 One_nat_def add_diff_cancel_left' int_ops(2) of_nat_diff)
+      then show "\<And>h. h \<in> ?\<F> \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
+        by (auto simp only: One_nat_def aff_dim_eq_hyperplane [symmetric])
       qed (use \<open>finite c\<close> in auto)
   qed
 qed
 
 lemma affine_hyperplane_sums_eq_UNIV_0:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
      and "0 \<in> S" and "w \<in> S"
      and "a \<bullet> w \<noteq> 0"
    shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
 proof -
   have "subspace S"
     by (simp add: assms subspace_affine)
   have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>0 \<in> S\<close> add.left_neutral by (intro span_mono) force
   have "w \<notin> span {y. a \<bullet> y = 0}"
     using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
   moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>w \<in> S\<close>
     by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
   ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     by blast
   have "a \<noteq> 0" using assms inner_zero_left by blast
   then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
     by (simp add: dim_hyperplane)
   also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using span1 span2 by (blast intro: dim_psubset)
   finally have "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
   then have DD: "dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0} = DIM('a)"
     using antisym dim_subset_UNIV lowdim_subset_hyperplane not_le by fastforce
   have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
     using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
   moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
     using DD dim_eq_full by blast
   ultimately show ?thesis
     by (simp add: subs) (metis (lifting) span_eq_iff subs)
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hyperplane_sums_eq_UNIV:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
       and "S - {v. a \<bullet> v = b} \<noteq> {}"
     shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by (auto simp: if_splits)
 next
   case False
   obtain c where "c \<in> S" and c: "a \<bullet> c = b"
     using assms by force
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have "subspace ((+) (- c) ` S)" by blast
   then have aff: "affine ((+) (- c) ` S)"
     by (simp add: subspace_imp_affine)
   have 0: "0 \<in> (+) (- c) ` S"
     by (simp add: \<open>c \<in> S\<close>)
   obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
     using assms by auto
   then have adc: "a \<bullet> (d - c) \<noteq> 0"
     by (simp add: c inner_diff_right)
-  let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
-  have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
-              if "u \<in> S" "b = a \<bullet> v" for u v
-    apply (rule_tac x="u+v-c-c" in image_eqI)
-    apply (simp_all add: algebra_simps)
-    apply (rule_tac x="u-c" in exI)
-    apply (rule_tac x="v-c" in exI)
-    apply (simp add: algebra_simps that c)
-    done
+  define U where "U \<equiv> {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
+  have "u + v \<in> (+) (c+c) ` U"
+    if "u \<in> S" "b = a \<bullet> v" for u v
+  proof
+    show "u + v = c + c + (u + v - c - c)"
+      by (simp add: algebra_simps)
+    have "\<exists>x y. u + v - c - c = x + y \<and> (\<exists>xa\<in>S. x = xa - c) \<and> a \<bullet> y = 0"
+    proof (intro exI conjI)
+      show "u + v - c - c = (u-c) + (v-c)" "a \<bullet> (v - c) = 0"
+        by (simp_all add: algebra_simps c that)
+    qed (use that in auto)
+    then show "u + v - c - c \<in> U"
+      by (auto simp: U_def image_def)
+  qed
   moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
        \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
     by (metis add.left_commute c inner_right_distrib pth_d)
-  ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
-    by (fastforce simp: algebra_simps)
+  ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = (+) (c+c) ` U"
+    by (fastforce simp: algebra_simps U_def)
   also have "... = range ((+) (c + c))"
-    by (simp only: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
+    by (simp only: U_def affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
   also have "... = UNIV"
     by simp
   finally show ?thesis .
 qed
 
 lemma aff_dim_sums_Int_0:
   assumes "affine S"
       and "affine T"
       and "0 \<in> S" "0 \<in> T"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
     using assms by force
   then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
     by (metis (lifting) hull_inc)
   have sub: "subspace S"  "subspace T"
     using assms by (auto simp: subspace_affine)
   show ?thesis
     using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
 qed
 
 proposition aff_dim_sums_Int:
   assumes "affine S"
       and "affine T"
       and "S \<inter> T \<noteq> {}"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   obtain a where a: "a \<in> S" "a \<in> T" using assms by force
   have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
     using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
   have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
     using a assms by auto
   have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
       (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
     by (force simp: algebra_simps scaleR_2)
   moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
     by auto
   ultimately show ?thesis
     using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
     by (metis (lifting))
 qed
 
 lemma aff_dim_affine_Int_hyperplane:
   fixes a :: "'a::euclidean_space"
   assumes "affine S"
     shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
              (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
               else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S
               else aff_dim S - 1)"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by auto
 next
   case False
   then have "aff_dim (S \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
             if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
   proof -
     have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
       using affine_hyperplane_sums_eq_UNIV [OF assms non] that  by blast
     show ?thesis
       using aff_dim_sums_Int [OF assms affine_hyperplane non]
       by (simp add: of_nat_diff False)
   qed
   then show ?thesis
     by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
 qed
 
 lemma aff_dim_lt_full:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
 by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
 
 lemma aff_dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
   shows "aff_dim S = aff_dim T"
 proof -
   show ?thesis
   proof (rule order_antisym)
     show "aff_dim S \<le> aff_dim T"
       by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
   next
     obtain a where "a \<in> S"
       using \<open>S \<noteq> {}\<close> by blast
     have "S \<subseteq> T"
       using ope openin_imp_subset by auto
     then have "a \<in> T"
       using \<open>a \<in> S\<close> by auto
     then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
       using affine_diffs_subspace \<open>affine T\<close> by auto
     then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
                     and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
                     and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
                     and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
       by (rule orthonormal_basis_subspace) auto
     obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
       by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
     have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
       by (metis aff_dim_translation_eq)
     also have "... = dim ((\<lambda>x. - a + x) ` T)"
       using aff_dim_subspace subT' by blast
     also have "... = card B"
       by (simp add: cardB)
     also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
       using \<open>0 < e\<close>  by (force simp: inj_on_def card_image)
     also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
     proof (simp, rule independent_card_le_dim)
       have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
         using e by (auto simp: dist_norm norm_minus_commute subset_eq)
       have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
         using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
       then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
         using e' by blast
       have "inj_on ((*\<^sub>R) e) (span B)"
         using \<open>0 < e\<close> inj_on_def by fastforce
       then show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
         using linear_scale_self \<open>independent B\<close> linear_dependent_inj_imageD by blast
     qed
     also have "... = aff_dim S"
       using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
     finally show "aff_dim T \<le> aff_dim S" .
   qed
 qed
 
 lemma dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
   shows "dim S = dim T"
 proof (rule order_antisym)
   show "dim S \<le> dim T"
     by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
 next
   have "dim T = aff_dim S"
     using aff_dim_openin
     by (metis aff_dim_subspace \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
   also have "... \<le> dim S"
     by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
         subspace_span)
   finally show "dim T \<le> dim S" by simp
 qed
 
 subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
 
 proposition dense_complement_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
 proof -
   have "closure(S - U) = S" if "dim U < dim S" "U \<subseteq> S" for U
   proof -
     have "span U \<subset> span S"
       by (metis neq_iff psubsetI span_eq_dim span_mono that)
     then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
       using orthogonal_to_subspace_exists_gen by metis
     show ?thesis
     proof
       have "closed S"
         by (simp add: \<open>subspace S\<close> closed_subspace)
       then show "closure (S - U) \<subseteq> S"
         by (simp add: closure_minimal)
       show "S \<subseteq> closure (S - U)"
       proof (clarsimp simp: closure_approachable)
         fix x and e::real
         assume "x \<in> S" "0 < e"
         show "\<exists>y\<in>S - U. dist y x < e"
         proof (cases "x \<in> U")
           case True
           let ?y = "x + (e/2 / norm a) *\<^sub>R a"
           show ?thesis
           proof
             show "dist ?y x < e"
               using \<open>0 < e\<close> by (simp add: dist_norm)
           next
             have "?y \<in> S"
               by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
             moreover have "?y \<notin> U"
             proof -
               have "e/2 / norm a \<noteq> 0"
                 using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
               then show ?thesis
                 by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
             qed
             ultimately show "?y \<in> S - U" by blast
           qed
         next
           case False
           with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
         qed
       qed
     qed
   qed
   moreover have "S - S \<inter> T = S-T"
     by blast
   moreover have "dim (S \<inter> T) < dim S"
     by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
   ultimately show ?thesis
     by force
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_affine:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
 proof (cases "S \<inter> T = {}")
   case True
   then show ?thesis
     by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
 next
   case False
   then obtain z where z: "z \<in> S \<inter> T" by blast
   then have "subspace ((+) (- z) ` S)"
     by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
   moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
 thm aff_dim_eq_dim
     using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
   ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
     by (simp add: dense_complement_subspace)
   then show ?thesis
     by (metis closure_translation translation_diff translation_invert)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_openin_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S"
       and ope: "openin (top_of_set (affine hull S)) S"
     shows "closure(S - T) = closure S"
 proof -
   have "affine hull S - T \<subseteq> affine hull S"
     by blast
   then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (affine hull S - T))"
     by (rule closure_openin_Int_closure [OF ope])
   then show ?thesis
     by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S"
     shows "closure(S - T) = closure S"
 proof
   show "closure (S - T) \<subseteq> closure S"
     by (simp add: closure_mono)
   have "closure (rel_interior S - T) = closure (rel_interior S)"
     by (simp add: assms dense_complement_openin_affine_hull openin_rel_interior rel_interior_aff_dim rel_interior_same_affine_hull)
   then show "closure S \<subseteq> closure (S - T)"
     by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S" "closed S"
     shows "closure(S - T) = S"
   by (simp add: assms dense_complement_convex)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Parallel slices, etc\<close>
 
 text\<open> If we take a slice out of a set, we can do it perpendicularly,
   with the normal vector to the slice parallel to the affine hull.\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_parallel_slice:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
       and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
   obtains a' b' where "a' \<noteq> 0"
                    "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
                    "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
                    "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
 proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
   case True
   then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
     using assms by (auto simp: not_le)
   define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
   have "\<eta> \<in> S"
     by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
   moreover have "a \<bullet> \<eta> = b"
     using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
     by (simp add: \<eta>_def algebra_simps) (simp add: field_simps)
   ultimately have False
     using True by force
   then show ?thesis ..
 next
   case False
   then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
     using assms by auto
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have sub: "subspace ((+) (- z) ` S)" by blast
   then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
     by (auto simp: subspace_imp_affine)
   obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
                   and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
     using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
   then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
     by (simp add: span_base orthogonal_def)
   then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
     by (simp add: a inner_diff_right)
   then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
     by (simp add: inner_diff_left z)
-  have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
-    by (metis subspace_add a' span_eq_iff sub)
-  then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
-    by fastforce
   show ?thesis
   proof (cases "a' = 0")
     case True
     with a assms True a'' diff_zero less_irrefl show ?thesis
       by auto
   next
     case False
     show ?thesis
-      apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
-      apply (auto simp: a ba'' inner_left_distrib False Sclo)
-      done
+    proof
+      show "S \<inter> {x. a' \<bullet> x \<le> a' \<bullet> z} = S \<inter> {x. a \<bullet> x \<le> b}"
+        "S \<inter> {x. a' \<bullet> x = a' \<bullet> z} = S \<inter> {x. a \<bullet> x = b}"
+        by (auto simp: a ba'' inner_left_distrib)
+      have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
+        by (metis subspace_add a' span_eq_iff sub)
+      then show "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
+        by fastforce
+    qed (use False in auto)
   qed
 qed
 
 lemma diffs_affine_hull_span:
   assumes "a \<in> S"
-    shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
+    shows "(\<lambda>x. x - a) ` (affine hull S) = span ((\<lambda>x. x - a) ` S)"
 proof -
-  have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
+  have *: "((\<lambda>x. x - a) ` (S - {a})) = ((\<lambda>x. x - a) ` S) - {0}"
     by (auto simp: algebra_simps)
   show ?thesis
     by (auto simp add: algebra_simps affine_hull_span2 [OF assms] *)
 qed
 
 lemma aff_dim_dim_affine_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S" "a \<in> S"
-    shows "aff_dim S = dim {x - a |x. x \<in> S}"
+    shows "aff_dim S = dim ((\<lambda>x. x - a) ` S)"
 proof -
   obtain B where aff: "affine hull B = affine hull S"
              and ind: "\<not> affine_dependent B"
              and card: "of_nat (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by blast
   then have "B \<noteq> {}" using assms
     by (metis affine_hull_eq_empty ex_in_conv)
   then obtain c where "c \<in> B" by auto
   then have "c \<in> S"
     by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
   have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
     by (auto simp: algebra_simps)
-  have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
-    apply safe
-    apply (simp_all only: xy)
-    using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
-    done
+  have *: "(\<lambda>x. x - c) ` S = (\<lambda>x. x - a) ` S"
+    using assms \<open>c \<in> S\<close>
+    by (auto simp: image_iff xy; metis mem_affine_3_minus pth_1)
   have affS: "affine hull S = S"
     by (simp add: \<open>affine S\<close>)
   have "aff_dim S = of_nat (card B) - 1"
     using card by simp
-  also have "... = dim {x - c |x. x \<in> B}"
+  also have "... = dim ((\<lambda>x. x - c) ` B)"
+    using affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>]
     by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
-  also have "... = dim {x - c | x. x \<in> affine hull B}"
-     by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
-  also have "... = dim {x - a |x. x \<in> S}"
-     by (simp add: affS aff *)
-   finally show ?thesis .
+  also have "... = dim ((\<lambda>x. x - c) ` (affine hull B))"
+    by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
+  also have "... = dim ((\<lambda>x. x - a) ` S)"
+    by (simp add: affS aff *)
+  finally show ?thesis .
 qed
 
 lemma aff_dim_linear_image_le:
   assumes "linear f"
     shows "aff_dim(f ` S) \<le> aff_dim S"
 proof -
   have "aff_dim (f ` T) \<le> aff_dim T" if "affine T" for T
   proof (cases "T = {}")
     case True then show ?thesis by (simp add: aff_dim_geq)
   next
     case False
     then obtain a where "a \<in> T" by auto
     have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
       by auto
-    have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
+    have 2: "{x - f a| x. x \<in> f ` T} = f ` ((\<lambda>x. x - a) ` T)"
       by (force simp: linear_diff [OF assms])
     have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
       by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
-    also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
+    also have "... = int (dim (f ` ((\<lambda>x. x - a) ` T)))"
       by (force simp: linear_diff [OF assms] 2)
-    also have "... \<le> int (dim {x - a| x. x \<in> T})"
+    also have "... \<le> int (dim ((\<lambda>x. x - a) ` T))"
       by (simp add: dim_image_le [OF assms])
     also have "... \<le> aff_dim T"
       by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
     finally show ?thesis .
   qed
   then
   have "aff_dim (f ` (affine hull S)) \<le> aff_dim (affine hull S)"
     using affine_affine_hull [of S] by blast
   then show ?thesis
     using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
 qed
 
 lemma aff_dim_injective_linear_image [simp]:
   assumes "linear f" "inj f"
     shows "aff_dim (f ` S) = aff_dim S"
 proof (rule antisym)
   show "aff_dim (f ` S) \<le> aff_dim S"
     by (simp add: aff_dim_linear_image_le assms(1))
 next
   obtain g where "linear g" "g \<circ> f = id"
     using assms(1) assms(2) linear_injective_left_inverse by blast
   then have "aff_dim S \<le> aff_dim(g ` f ` S)"
     by (simp add: image_comp)
   also have "... \<le> aff_dim (f ` S)"
     by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
   finally show "aff_dim S \<le> aff_dim (f ` S)" .
 qed
 
 
 lemma choose_affine_subset:
   assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
   obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
 proof (cases "d = -1 \<or> S={}")
   case True with assms show ?thesis
     by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
 next
   case False
   with assms obtain a where "a \<in> S" "0 \<le> d" by auto
   with assms have ss: "subspace ((+) (- a) ` S)"
     by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
   have "nat d \<le> dim ((+) (- a) ` S)"
     by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
   then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
                   and Tdim: "dim T = nat d"
     using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
   then have "affine T"
     using subspace_affine by blast
   then have "affine ((+) a ` T)"
     by (metis affine_hull_eq affine_hull_translation)
   moreover have "(+) a ` T \<subseteq> S"
   proof -
     have "T \<subseteq> (+) (- a) ` S"
       by (metis (no_types) span_eq_iff Tsb ss)
     then show "(+) a ` T \<subseteq> S"
       using add_ac by auto
   qed
   moreover have "aff_dim ((+) a ` T) = d"
     by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
   ultimately show ?thesis
     by (rule that)
 qed
 
 subsection\<open>Paracompactness\<close>
 
 proposition paracompact:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. x \<in> S
                        \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
 proof (cases "S = {}")
   case True with that show ?thesis by blast
 next
   case False
   have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
   proof -
     obtain T where "x \<in> T" "T \<in> \<C>" "open T"
       using assms \<open>x \<in> S\<close> by blast
     then obtain e where "e > 0" "cball x e \<subseteq> T"
       by (force simp: open_contains_cball)
     then show ?thesis
       by (meson open_ball \<open>T \<in> \<C>\<close> ball_subset_cball centre_in_ball closed_cball closure_minimal dual_order.trans)
   qed
   then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
     and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
   if "x \<in> S" for x
     by metis
   then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
     using Lindelof [of "G ` S"] by (metis image_iff)
   then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
     by (metis countable_subset_image)
   with False Gin have "K \<noteq> {}" by force
   then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
     by (metis range_from_nat_into \<open>countable K\<close>)
   then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
     using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
   let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
   have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
   proof -
     have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
     then show ?thesis
       using clos K \<open>range a = K\<close> closure_subset by blast
   qed
   show ?thesis
   proof
     show "S \<subseteq> Union ?C"
     proof
       fix x assume "x \<in> S"
       define n where "n \<equiv> LEAST n. x \<in> F(a n)"
       have n: "x \<in> F(a n)"
         using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
       have notn: "x \<notin> F(a m)" if "m < n" for m
         using that not_less_Least by (force simp: n_def)
       then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
         using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
       with n show "x \<in> Union ?C"
         by blast
     qed
     show "\<And>U. U \<in> ?C \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
       using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close> by (auto simp: odif)
     show "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
     proof -
       obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
         using \<open>x \<in> S\<close> enum_S by auto
       have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
       proof clarsimp
         fix k  assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
         then have "k \<le> n"
           by auto (metis closure_subset not_le subsetCE)
         then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
                  \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
           by force
       qed
       moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
         by force
       ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
         using finite_subset by blast
       have "a n \<in> S"
         using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> by blast
       then show ?thesis
         by (blast intro: oG n *)
     qed
   qed
 qed
 
 corollary paracompact_closedin:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes cin: "closedin (top_of_set U) S"
       and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
                and "\<And>x. x \<in> U
                        \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
                                finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
 proof -
   have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
     using oin [OF that] by (auto simp: openin_open)
   then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
     by metis
   obtain K where K: "closed K" "U \<inter> K = S"
     using cin by (auto simp: closedin_closed)
   have 1: "U \<subseteq> \<Union>(insert (- K) (F ` \<C>))"
     by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
   have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
     using \<open>closed K\<close> by (auto simp: opF)
   obtain \<D> where "U \<subseteq> \<Union>\<D>"
              and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
              and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
     by (blast intro: paracompact [OF 1 2])
   let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
   show ?thesis
   proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
     show "S \<subseteq> \<Union>?C"
       using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
     show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
       using D1 intF by fastforce
     have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
              (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
       by blast
     show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
       if "x \<in> U" for x
     proof -
       from D2 [OF that] obtain V where "open V" "x \<in> V" "finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
         by auto
       with * show ?thesis
         by (rule_tac x="U \<inter> V" in exI) (auto intro: that finite_subset [OF *])
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> paracompact_closed:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "closed S"
       and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
                                finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
   by (rule paracompact_closedin [of UNIV S \<C>]) (auto simp: assms)
 
   
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Closed-graph characterization of continuity\<close>
 
 lemma continuous_closed_graph_gen:
   fixes T :: "'b::real_normed_vector set"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
     shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
 proof -
   have eq: "((\<lambda>x. Pair x (f x)) ` S) = (S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
     using fim by auto
   show ?thesis
-    apply (subst eq)
-    apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
-    by auto
+    unfolding eq
+    by (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf]) auto
 qed
 
 lemma continuous_closed_graph_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
          closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
 proof -
   have "?lhs" if ?rhs
   proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
     fix U
     assume U: "closedin (top_of_set T) U"
     have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
       by (force simp: image_iff)
     show "closedin (top_of_set S) (S \<inter> f -` U)"
       by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
   qed
   with continuous_closed_graph_gen assms show ?thesis by blast
 qed
 
 lemma continuous_closed_graph:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "closed S" and contf: "continuous_on S f"
   shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
 proof (rule closedin_closed_trans)
   show "closedin (top_of_set (S \<times> UNIV)) ((\<lambda>x. (x, f x)) ` S)"
     by (rule continuous_closed_graph_gen [OF contf subset_UNIV])
 qed (simp add: \<open>closed S\<close> closed_Times)
 
 lemma continuous_from_closed_graph:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
   shows "continuous_on S f"
     using fim clo
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
 
 lemma continuous_on_Un_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T f"
     shows "continuous_on (S \<union> T) f"
   using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
   by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
 
 lemma continuous_on_cases_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T g"
       and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
     shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
 proof -
   have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x"  "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
     by (simp_all add: fg)
   then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
     by (simp_all add: contf contg cong: continuous_on_cong)
   then show ?thesis
     by (rule continuous_on_Un_local_open [OF opS opT])
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The union of two collinear segments is another segment\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange:
   fixes a :: "'a::euclidean_space"
   assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
   obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
 proof (cases "a \<in> S")
   case True
   with xS insert_Diff that  show ?thesis by fastforce
 next
   case False
   show ?thesis
   proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
     case True
     then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
                     and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
         using a by (auto simp: convex_hull_finite)
     obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
                and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
       using True xS by (auto simp: convex_hull_finite)
     show ?thesis
     proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
       case True
       then obtain b where b: "b \<in> S" "v b = 0"
         by blast
       show ?thesis
       proof
         have fin: "finite (insert a (S - {b}))"
           using sum.infinite v1 by fastforce
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
                 (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
             using fin by (force intro: sum.mono_neutral_right)
           also have "... = (\<Sum>x \<in> S - {b}. v x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x)"
             by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
           finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
             by (simp add: v1)
           show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
             by (auto simp: v0)
           have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
                 (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
             using fin by (force intro: sum.mono_neutral_right)
           also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
           finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     next
       case False
       have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
         by (simp add: True that)
       have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
         using True v1 by (auto intro: Max_in)
       then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
         by blast
       then have "0 \<noteq> u b / v b"
         using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
         by (metis False eq_iff v0)
       then have  "0 < u b" "0 < v b"
         using False \<open>b \<in> S\<close> u0 v0 by force+
       have fin: "finite (insert a (S - {b}))"
         using sum.infinite v1 by fastforce
       show ?thesis
       proof
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
                 v b / u b + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  
             by (auto intro!: sum.cong split: if_split_asm)
           also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
             by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
           also have "... = (\<Sum>x\<in>S. v x)"
             using \<open>0 < u b\<close> True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
           finally show "sum (\<lambda>x. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
             by (simp add: v1)
           show "0 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
             if "i \<in> insert a (S - {b})" for i
             using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
             by (auto simp: field_simps split: if_split_asm)
           have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
                 (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
             by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             using \<open>0 < u b\<close> True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
           finally
           show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     qed
   next
     case False
     obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
       using xS by (auto simp: caratheodory [of S])
     with False obtain b where b: "b \<in> S" "b \<notin> T"
       by (metis antisym subsetI)
     show ?thesis
     proof
       show "x \<in> convex hull insert a (S - {b})"
         using  \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
     qed (rule \<open>b \<in> S\<close>)
   qed
 qed
 
 lemma convex_hull_exchange_Union:
   fixes a :: "'a::euclidean_space"
   assumes "a \<in> convex hull S"
   shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
     by (blast intro: in_convex_hull_exchange [OF assms])
   show "?rhs \<subseteq> ?lhs"
   proof clarify
     fix x b
     assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
     then show "x \<in> convex hull S" if "b \<in> S"
       by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
   qed
 qed
 
 lemma Un_closed_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c"
     shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
 proof (cases "c = a")
   case True
   with assms show ?thesis by simp
 next
   case False
   with assms have "convex hull {a, b} \<union> convex hull {b, c} = (\<Union>ba\<in>{a, c}. convex hull insert b ({a, c} - {ba}))"
     by (auto simp: insert_Diff_if insert_commute)
   then show ?thesis
     using convex_hull_exchange_Union
     by (metis assms segment_convex_hull)
 qed
 
 lemma Un_open_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> open_segment a c"
-  shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
+  shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c" (is "?lhs = ?rhs")
 proof -
   have b: "b \<in> closed_segment a c"
     by (simp add: assms open_closed_segment)
-  have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
-          if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
+  have *: "?rhs \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
+          if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> ?rhs"
   proof -
-    have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
+    have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (?rhs))"
       using that by (simp add: insert_commute)
     then show ?thesis
       by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
   qed
   show ?thesis
-    apply (rule antisym)
-    using Un_closed_segment [OF b] assms *
-    by (simp_all add: closed_segment_eq_open b subset_open_segment insert_commute)
+  proof
+    show "?lhs \<subseteq> ?rhs"
+      by (simp add: assms b subset_open_segment)
+    show "?rhs \<subseteq> ?lhs"
+      using Un_closed_segment [OF b] *
+      by (simp add: closed_segment_eq_open insert_commute)
+  qed
 qed
 
 subsection\<open>Covering an open set by a countable chain of compact sets\<close>
   
 proposition open_Union_compact_subsets:
   fixes S :: "'a::euclidean_space set"
   assumes "open S"
   obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
                   "\<And>n. C n \<subseteq> interior(C(Suc n))"
                   "\<Union>(range C) = S"
                   "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (rule_tac C = "\<lambda>n. {}" in that) auto
 next
   case False
   then obtain a where "a \<in> S"
     by auto
   let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
   have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
         if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
             and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
   proof -
     have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
       by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
     have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
       by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
     obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
     proof (rule compactE_image [OF \<open>compact K\<close>])
       show "K \<subseteq> (\<Union>n. interior (f n))"
         using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
     qed auto
     { fix n
       assume n: "Max I \<le> n"
       have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
         by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
       then have "K \<subseteq> f n"
         using I by auto
     }
     then show ?thesis
       by blast
   qed
   moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
                      ((\<Union>(range f) = S))"
   proof (intro exI conjI allI)
     show "\<And>n. compact (?C n)"
       by (auto simp: compact_diff open_sums)
     show "\<And>n. ?C n \<subseteq> S"
       by auto
     show "?C n \<subseteq> interior (?C (Suc n))" for n
     proof (simp add: interior_diff, rule Diff_mono)
       show "cball a (real n) \<subseteq> ball a (1 + real n)"
         by (simp add: cball_subset_ball_iff)
       have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
         using assms by (auto intro: closed_compact_sums)
       have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
             \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
         by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
       also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
-        apply (intro UN_mono order_refl)
-        apply (simp add: cball_subset_ball_iff field_split_simps)
-        done
+        by (simp add: cball_subset_ball_iff field_split_simps UN_mono)
       finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
                     \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
     qed
     have "S \<subseteq> \<Union> (range ?C)"
     proof
       fix x
       assume x: "x \<in> S"
       then obtain e where "e > 0" and e: "ball x e \<subseteq> S"
         using assms open_contains_ball by blast
       then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
         using reals_Archimedean2
         by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
       obtain N2 where N2: "norm(x - a) \<le> real N2"
         by (meson real_arch_simple)
       have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
         using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
       have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
       proof -
         have "e * real N1 < e * (1 + (real N1 + real N2))"
           by (simp add: \<open>0 < e\<close>)
         then have "1 / (1 + (real N1 + real N2)) < e"
           using N1 \<open>e > 0\<close>
           by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
         then have "x - z \<in> ball x e"
           using that by simp
         then have "x - z \<in> S"
           using e by blast
         with that show ?thesis
           by auto
       qed
       with N2 show "x \<in> \<Union> (range ?C)"
         by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
     qed
     then show "\<Union> (range ?C) = S" by auto
   qed
   ultimately show ?thesis
     using that by metis
 qed
 
 
 subsection\<open>Orthogonal complement\<close>
 
 definition\<^marker>\<open>tag important\<close> orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
   where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
 
 proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
   unfolding subspace_def orthogonal_comp_def orthogonal_def
   by (auto simp: inner_right_distrib)
 
 lemma orthogonal_comp_anti_mono:
   assumes "A \<subseteq> B"
   shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
 proof
   fix x assume x: "x \<in> B\<^sup>\<bottom>"
   show "x \<in> orthogonal_comp A" using x unfolding orthogonal_comp_def
     by (simp add: orthogonal_def, metis assms in_mono)
 qed
 
 lemma orthogonal_comp_null [simp]: "{0}\<^sup>\<bottom> = UNIV"
   by (auto simp: orthogonal_comp_def orthogonal_def)
 
 lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
   unfolding orthogonal_comp_def orthogonal_def
   by auto (use inner_eq_zero_iff in blast)
 
 lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
   by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
 
 lemma subspace_sum_minimal:
   assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
   shows "S + T \<subseteq> U"
 proof
   fix x
   assume "x \<in> S + T"
   then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
     by (meson set_plus_elim)
   then show "x \<in> U"
     by (meson assms subsetCE subspace_add)
 qed
 
 proposition subspace_sum_orthogonal_comp:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U + U\<^sup>\<bottom> = UNIV"
 proof -
   obtain B where "B \<subseteq> U"
     and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
     and "independent B" "card B = dim U" "span B = U"
     using orthonormal_basis_subspace [OF assms] by metis
   then have "finite B"
     by (simp add: indep_card_eq_dim_span)
   have *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
     using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
   { fix v
     let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
     have "v = ?u + (v - ?u)"
       by simp
     moreover have "?u \<in> U"
       by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
     moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
     proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
       fix y
       assume "y \<in> U"
       with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
       obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
         by auto
       have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
         using that \<open>finite B\<close>
         by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
       then show "y \<bullet> (v - ?u) = 0"
         by (simp add: u inner_sum_left)
     qed
     ultimately have "v \<in> U + U\<^sup>\<bottom>"
       using set_plus_intro by fastforce
   } then show ?thesis
     by auto
 qed
 
 lemma orthogonal_Int_0:
   assumes "subspace U"
   shows "U \<inter> U\<^sup>\<bottom> = {0}"
   using orthogonal_comp_def orthogonal_self
   by (force simp: assms subspace_0 subspace_orthogonal_comp)
 
 lemma orthogonal_comp_self:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
 proof
   have ssU': "subspace (U\<^sup>\<bottom>)"
     by (simp add: subspace_orthogonal_comp)
   have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
   proof -
     obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
       using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
     then have "u-v \<in> U\<^sup>\<bottom>"
       by simp
     moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
     then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       by (simp add: subspace_diff subspace_orthogonal_comp that)
     ultimately have "u-v = 0"
       using orthogonal_Int_0 ssU' by blast
     with \<open>v \<in> U\<close> show ?thesis
       by auto
   qed
   then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
     by auto
 qed (use orthogonal_comp_subset in auto)
 
 lemma ker_orthogonal_comp_adjoint:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows "f -` {0} = (range (adjoint f))\<^sup>\<bottom>"
-  apply (auto simp: orthogonal_comp_def orthogonal_def)
-  apply (simp add: adjoint_works assms(1) inner_commute)
-  by (metis adjoint_works all_zero_iff assms(1) inner_commute)
+proof -
+  have "\<And>x. \<lbrakk>\<forall>y. y \<bullet> f x = 0\<rbrakk> \<Longrightarrow> f x = 0"
+    using assms inner_commute all_zero_iff by metis
+  then show ?thesis
+    using assms 
+    by (auto simp: orthogonal_comp_def orthogonal_def adjoint_works inner_commute)
+qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A non-injective linear function maps into a hyperplane.\<close>
 
 lemma linear_surj_adj_imp_inj:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f" "surj (adjoint f)"
   shows "inj f"
 proof -
   have "\<exists>x. y = adjoint f x" for y
     using assms by (simp add: surjD)
   then show "inj f"
     using assms unfolding inj_on_def image_def
     by (metis (no_types) adjoint_works euclidean_eqI)
 qed
 
 \<comment> \<open>\<^url>\<open>https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\<close>\<close>
 lemma surj_adjoint_iff_inj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "surj (adjoint f) \<longleftrightarrow> inj f"
 proof
   assume "surj (adjoint f)"
   then show "inj f"
     by (simp add: assms linear_surj_adj_imp_inj)
 next
   assume "inj f"
   have "f -` {0} = {0}"
     using assms \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
   moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
     by (intro ker_orthogonal_comp_adjoint assms)
   ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
     by (metis orthogonal_comp_null)
   then show "surj (adjoint f)"
     using adjoint_linear \<open>linear f\<close>
     by (subst (asm) orthogonal_comp_self)
       (simp add: adjoint_linear linear_subspace_image)
 qed
 
 lemma inj_adjoint_iff_surj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "inj (adjoint f) \<longleftrightarrow> surj f"
 proof
   assume "inj (adjoint f)"
   have "(adjoint f) -` {0} = {0}"
     by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
   then have "(range(f))\<^sup>\<bottom> = {0}"
     by (metis (no_types, hide_lams) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
   then show "surj f"
     by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
 next
   assume "surj f"
   then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
     by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
   then have "{0} = adjoint f -` {0}"
     using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
   then show "inj (adjoint f)"
     by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
 qed
 
 lemma linear_singular_into_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f"
   shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
 proof
   assume "\<not>inj f"
   then show ?rhs
     using all_zero_iff
     by (metis (no_types, hide_lams) adjoint_clauses(2) adjoint_linear assms
         linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
 next
   assume ?rhs
   then show "\<not>inj f"
     by (metis assms linear_injective_isomorphism all_zero_iff)
 qed
 
 lemma linear_singular_image_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" "\<not>inj f"
   obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
   using assms by (fastforce simp add: linear_singular_into_hyperplane)
 
 end
diff --git a/src/HOL/Set.thy b/src/HOL/Set.thy
--- a/src/HOL/Set.thy
+++ b/src/HOL/Set.thy
@@ -1,2028 +1,2034 @@
 (*  Title:      HOL/Set.thy
     Author:     Tobias Nipkow
     Author:     Lawrence C Paulson
     Author:     Markus Wenzel
 *)
 
 section \<open>Set theory for higher-order logic\<close>
 
 theory Set
   imports Lattices
 begin
 
 subsection \<open>Sets as predicates\<close>
 
 typedecl 'a set
 
 axiomatization Collect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set" \<comment> \<open>comprehension\<close>
   and member :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>membership\<close>
   where mem_Collect_eq [iff, code_unfold]: "member a (Collect P) = P a"
     and Collect_mem_eq [simp]: "Collect (\<lambda>x. member x A) = A"
 
 notation
   member  ("'(\<in>')") and
   member  ("(_/ \<in> _)" [51, 51] 50)
 
 abbreviation not_member
   where "not_member x A \<equiv> \<not> (x \<in> A)" \<comment> \<open>non-membership\<close>
 notation
   not_member  ("'(\<notin>')") and
   not_member  ("(_/ \<notin> _)" [51, 51] 50)
 
 notation (ASCII)
   member  ("'(:')") and
   member  ("(_/ : _)" [51, 51] 50) and
   not_member  ("'(~:')") and
   not_member  ("(_/ ~: _)" [51, 51] 50)
 
 
 text \<open>Set comprehensions\<close>
 
 syntax
   "_Coll" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_./ _})")
 translations
   "{x. P}" \<rightleftharpoons> "CONST Collect (\<lambda>x. P)"
 
 syntax (ASCII)
   "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set"  ("(1{(_/: _)./ _})")
 syntax
   "_Collect" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a set"  ("(1{(_/ \<in> _)./ _})")
 translations
   "{p:A. P}" \<rightharpoonup> "CONST Collect (\<lambda>p. p \<in> A \<and> P)"
 
 lemma CollectI: "P a \<Longrightarrow> a \<in> {x. P x}"
   by simp
 
 lemma CollectD: "a \<in> {x. P x} \<Longrightarrow> P a"
   by simp
 
 lemma Collect_cong: "(\<And>x. P x = Q x) \<Longrightarrow> {x. P x} = {x. Q x}"
   by simp
 
 text \<open>
   Simproc for pulling \<open>x = t\<close> in \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>
   to the front (and similarly for \<open>t = x\<close>):
 \<close>
 
 simproc_setup defined_Collect ("{x. P x \<and> Q x}") = \<open>
   fn _ => Quantifier1.rearrange_Collect
     (fn ctxt =>
       resolve_tac ctxt @{thms Collect_cong} 1 THEN
       resolve_tac ctxt @{thms iffI} 1 THEN
       ALLGOALS
         (EVERY' [REPEAT_DETERM o eresolve_tac ctxt @{thms conjE},
           DEPTH_SOLVE_1 o (assume_tac ctxt ORELSE' resolve_tac ctxt @{thms conjI})]))
 \<close>
 
 lemmas CollectE = CollectD [elim_format]
 
 lemma set_eqI:
   assumes "\<And>x. x \<in> A \<longleftrightarrow> x \<in> B"
   shows "A = B"
 proof -
   from assms have "{x. x \<in> A} = {x. x \<in> B}"
     by simp
   then show ?thesis by simp
 qed
 
 lemma set_eq_iff: "A = B \<longleftrightarrow> (\<forall>x. x \<in> A \<longleftrightarrow> x \<in> B)"
   by (auto intro:set_eqI)
 
 lemma Collect_eqI:
   assumes "\<And>x. P x = Q x"
   shows "Collect P = Collect Q"
   using assms by (auto intro: set_eqI)
 
 text \<open>Lifting of predicate class instances\<close>
 
 instantiation set :: (type) boolean_algebra
 begin
 
 definition less_eq_set
   where "A \<le> B \<longleftrightarrow> (\<lambda>x. member x A) \<le> (\<lambda>x. member x B)"
 
 definition less_set
   where "A < B \<longleftrightarrow> (\<lambda>x. member x A) < (\<lambda>x. member x B)"
 
 definition inf_set
   where "A \<sqinter> B = Collect ((\<lambda>x. member x A) \<sqinter> (\<lambda>x. member x B))"
 
 definition sup_set
   where "A \<squnion> B = Collect ((\<lambda>x. member x A) \<squnion> (\<lambda>x. member x B))"
 
 definition bot_set
   where "\<bottom> = Collect \<bottom>"
 
 definition top_set
   where "\<top> = Collect \<top>"
 
 definition uminus_set
   where "- A = Collect (- (\<lambda>x. member x A))"
 
 definition minus_set
   where "A - B = Collect ((\<lambda>x. member x A) - (\<lambda>x. member x B))"
 
 instance
   by standard
     (simp_all add: less_eq_set_def less_set_def inf_set_def sup_set_def
       bot_set_def top_set_def uminus_set_def minus_set_def
       less_le_not_le sup_inf_distrib1 diff_eq set_eqI fun_eq_iff
       del: inf_apply sup_apply bot_apply top_apply minus_apply uminus_apply)
 
 end
 
 text \<open>Set enumerations\<close>
 
 abbreviation empty :: "'a set" ("{}")
   where "{} \<equiv> bot"
 
 definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
 
 syntax
   "_Finset" :: "args \<Rightarrow> 'a set"    ("{(_)}")
 translations
   "{x, xs}" \<rightleftharpoons> "CONST insert x {xs}"
   "{x}" \<rightleftharpoons> "CONST insert x {}"
 
 
 subsection \<open>Subsets and bounded quantifiers\<close>
 
 abbreviation subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "subset \<equiv> less"
 
 abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "subset_eq \<equiv> less_eq"
 
 notation
   subset  ("'(\<subset>')") and
   subset  ("(_/ \<subset> _)" [51, 51] 50) and
   subset_eq  ("'(\<subseteq>')") and
   subset_eq  ("(_/ \<subseteq> _)" [51, 51] 50)
 
 abbreviation (input)
   supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   "supset \<equiv> greater"
 
 abbreviation (input)
   supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where
   "supset_eq \<equiv> greater_eq"
 
 notation
   supset  ("'(\<supset>')") and
   supset  ("(_/ \<supset> _)" [51, 51] 50) and
   supset_eq  ("'(\<supseteq>')") and
   supset_eq  ("(_/ \<supseteq> _)" [51, 51] 50)
 
 notation (ASCII output)
   subset  ("'(<')") and
   subset  ("(_/ < _)" [51, 51] 50) and
   subset_eq  ("'(<=')") and
   subset_eq  ("(_/ <= _)" [51, 51] 50)
 
 definition Ball :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Ball A P \<longleftrightarrow> (\<forall>x. x \<in> A \<longrightarrow> P x)"   \<comment> \<open>bounded universal quantifiers\<close>
 
 definition Bex :: "'a set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Bex A P \<longleftrightarrow> (\<exists>x. x \<in> A \<and> P x)"   \<comment> \<open>bounded existential quantifiers\<close>
 
 syntax (ASCII)
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3ALL (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3EX (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3EX! (_/:_)./ _)" [0, 0, 10] 10)
   "_Bleast"     :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a"           ("(3LEAST (_/:_)./ _)" [0, 0, 10] 10)
 
 syntax (input)
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3! (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3? (_/:_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3?! (_/:_)./ _)" [0, 0, 10] 10)
 
 syntax
   "_Ball"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bex1"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>!(_/\<in>_)./ _)" [0, 0, 10] 10)
   "_Bleast"     :: "id \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> 'a"           ("(3LEAST(_/\<in>_)./ _)" [0, 0, 10] 10)
 
 translations
   "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball A (\<lambda>x. P)"
   "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex A (\<lambda>x. P)"
   "\<exists>!x\<in>A. P" \<rightharpoonup> "\<exists>!x. x \<in> A \<and> P"
   "LEAST x:A. P" \<rightharpoonup> "LEAST x. x \<in> A \<and> P"
 
 syntax (ASCII output)
   "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   "_setlessEx"  :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX _<_./ _)"  [0, 0, 10] 10)
   "_setleAll"   :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   "_setleEx"    :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX _<=_./ _)" [0, 0, 10] 10)
   "_setleEx1"   :: "[idt, 'a, bool] \<Rightarrow> bool"  ("(3EX! _<=_./ _)" [0, 0, 10] 10)
 
 syntax
   "_setlessAll" :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<forall>_\<subset>_./ _)"  [0, 0, 10] 10)
   "_setlessEx"  :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>_\<subset>_./ _)"  [0, 0, 10] 10)
   "_setleAll"   :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10)
   "_setleEx"    :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10)
   "_setleEx1"   :: "[idt, 'a, bool] \<Rightarrow> bool"   ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10)
 
 translations
  "\<forall>A\<subset>B. P" \<rightharpoonup> "\<forall>A. A \<subset> B \<longrightarrow> P"
  "\<exists>A\<subset>B. P" \<rightharpoonup> "\<exists>A. A \<subset> B \<and> P"
  "\<forall>A\<subseteq>B. P" \<rightharpoonup> "\<forall>A. A \<subseteq> B \<longrightarrow> P"
  "\<exists>A\<subseteq>B. P" \<rightharpoonup> "\<exists>A. A \<subseteq> B \<and> P"
  "\<exists>!A\<subseteq>B. P" \<rightharpoonup> "\<exists>!A. A \<subseteq> B \<and> P"
 
 print_translation \<open>
   let
     val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
     val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
     val impl = \<^const_syntax>\<open>HOL.implies\<close>;
     val conj = \<^const_syntax>\<open>HOL.conj\<close>;
     val sbset = \<^const_syntax>\<open>subset\<close>;
     val sbset_eq = \<^const_syntax>\<open>subset_eq\<close>;
 
     val trans =
      [((All_binder, impl, sbset), \<^syntax_const>\<open>_setlessAll\<close>),
       ((All_binder, impl, sbset_eq), \<^syntax_const>\<open>_setleAll\<close>),
       ((Ex_binder, conj, sbset), \<^syntax_const>\<open>_setlessEx\<close>),
       ((Ex_binder, conj, sbset_eq), \<^syntax_const>\<open>_setleEx\<close>)];
 
     fun mk v (v', T) c n P =
       if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n)
       then Syntax.const c $ Syntax_Trans.mark_bound_body (v', T) $ n $ P
       else raise Match;
 
     fun tr' q = (q, fn _ =>
       (fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, Type (\<^type_name>\<open>set\<close>, _)),
           Const (c, _) $
             (Const (d, _) $ (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', T)) $ n) $ P] =>
           (case AList.lookup (=) trans (q, c, d) of
             NONE => raise Match
           | SOME l => mk v (v', T) l n P)
         | _ => raise Match));
   in
     [tr' All_binder, tr' Ex_binder]
   end
 \<close>
 
 
 text \<open>
   \<^medskip>
   Translate between \<open>{e | x1\<dots>xn. P}\<close> and \<open>{u. \<exists>x1\<dots>xn. u = e \<and> P}\<close>;
   \<open>{y. \<exists>x1\<dots>xn. y = e \<and> P}\<close> is only translated if \<open>[0..n] \<subseteq> bvs e\<close>.
 \<close>
 
 syntax
   "_Setcompr" :: "'a \<Rightarrow> idts \<Rightarrow> bool \<Rightarrow> 'a set"    ("(1{_ |/_./ _})")
 
 parse_translation \<open>
   let
     val ex_tr = snd (Syntax_Trans.mk_binder_tr ("EX ", \<^const_syntax>\<open>Ex\<close>));
 
     fun nvars (Const (\<^syntax_const>\<open>_idts\<close>, _) $ _ $ idts) = nvars idts + 1
       | nvars _ = 1;
 
     fun setcompr_tr ctxt [e, idts, b] =
       let
         val eq = Syntax.const \<^const_syntax>\<open>HOL.eq\<close> $ Bound (nvars idts) $ e;
         val P = Syntax.const \<^const_syntax>\<open>HOL.conj\<close> $ eq $ b;
         val exP = ex_tr ctxt [idts, P];
       in Syntax.const \<^const_syntax>\<open>Collect\<close> $ absdummy dummyT exP end;
 
   in [(\<^syntax_const>\<open>_Setcompr\<close>, setcompr_tr)] end
 \<close>
 
 print_translation \<open>
  [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Ball\<close> \<^syntax_const>\<open>_Ball\<close>,
   Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Bex\<close> \<^syntax_const>\<open>_Bex\<close>]
 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
 
 print_translation \<open>
 let
   val ex_tr' = snd (Syntax_Trans.mk_binder_tr' (\<^const_syntax>\<open>Ex\<close>, "DUMMY"));
 
   fun setcompr_tr' ctxt [Abs (abs as (_, _, P))] =
     let
       fun check (Const (\<^const_syntax>\<open>Ex\<close>, _) $ Abs (_, _, P), n) = check (P, n + 1)
         | check (Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
               (Const (\<^const_syntax>\<open>HOL.eq\<close>, _) $ Bound m $ e) $ P, n) =
             n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
             subset (=) (0 upto (n - 1), add_loose_bnos (e, 0, []))
         | check _ = false;
 
         fun tr' (_ $ abs) =
           let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' ctxt [abs]
           in Syntax.const \<^syntax_const>\<open>_Setcompr\<close> $ e $ idts $ Q end;
     in
       if check (P, 0) then tr' P
       else
         let
           val (x as _ $ Free(xN, _), t) = Syntax_Trans.atomic_abs_tr' abs;
           val M = Syntax.const \<^syntax_const>\<open>_Coll\<close> $ x $ t;
         in
           case t of
             Const (\<^const_syntax>\<open>HOL.conj\<close>, _) $
               (Const (\<^const_syntax>\<open>Set.member\<close>, _) $
                 (Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (yN, _)) $ A) $ P =>
             if xN = yN then Syntax.const \<^syntax_const>\<open>_Collect\<close> $ x $ A $ P else M
           | _ => M
         end
     end;
   in [(\<^const_syntax>\<open>Collect\<close>, setcompr_tr')] end
 \<close>
 
 simproc_setup defined_Bex ("\<exists>x\<in>A. P x \<and> Q x") = \<open>
   fn _ => Quantifier1.rearrange_Bex
     (fn ctxt => unfold_tac ctxt @{thms Bex_def})
 \<close>
 
 simproc_setup defined_All ("\<forall>x\<in>A. P x \<longrightarrow> Q x") = \<open>
   fn _ => Quantifier1.rearrange_Ball
     (fn ctxt => unfold_tac ctxt @{thms Ball_def})
 \<close>
 
 lemma ballI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<Longrightarrow> \<forall>x\<in>A. P x"
   by (simp add: Ball_def)
 
 lemmas strip = impI allI ballI
 
 lemma bspec [dest?]: "\<forall>x\<in>A. P x \<Longrightarrow> x \<in> A \<Longrightarrow> P x"
   by (simp add: Ball_def)
 
 text \<open>Gives better instantiation for bound:\<close>
 setup \<open>
   map_theory_claset (fn ctxt =>
     ctxt addbefore ("bspec", fn ctxt' => dresolve_tac ctxt' @{thms bspec} THEN' assume_tac ctxt'))
 \<close>
 
 ML \<open>
 structure Simpdata =
 struct
   open Simpdata;
   val mksimps_pairs = [(\<^const_name>\<open>Ball\<close>, @{thms bspec})] @ mksimps_pairs;
 end;
 
 open Simpdata;
 \<close>
 
 declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))\<close>
 
 lemma ballE [elim]: "\<forall>x\<in>A. P x \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> Q"
   unfolding Ball_def by blast
 
 lemma bexI [intro]: "P x \<Longrightarrow> x \<in> A \<Longrightarrow> \<exists>x\<in>A. P x"
   \<comment> \<open>Normally the best argument order: \<open>P x\<close> constrains the choice of \<open>x \<in> A\<close>.\<close>
   unfolding Bex_def by blast
 
 lemma rev_bexI [intro?]: "x \<in> A \<Longrightarrow> P x \<Longrightarrow> \<exists>x\<in>A. P x"
   \<comment> \<open>The best argument order when there is only one \<open>x \<in> A\<close>.\<close>
   unfolding Bex_def by blast
 
 lemma bexCI: "(\<forall>x\<in>A. \<not> P x \<Longrightarrow> P a) \<Longrightarrow> a \<in> A \<Longrightarrow> \<exists>x\<in>A. P x"
   unfolding Bex_def by blast
 
 lemma bexE [elim!]: "\<exists>x\<in>A. P x \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q"
   unfolding Bex_def by blast
 
 lemma ball_triv [simp]: "(\<forall>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<longrightarrow> P)"
   \<comment> \<open>Trival rewrite rule.\<close>
   by (simp add: Ball_def)
 
 lemma bex_triv [simp]: "(\<exists>x\<in>A. P) \<longleftrightarrow> ((\<exists>x. x \<in> A) \<and> P)"
   \<comment> \<open>Dual form for existentials.\<close>
   by (simp add: Bex_def)
 
 lemma bex_triv_one_point1 [simp]: "(\<exists>x\<in>A. x = a) \<longleftrightarrow> a \<in> A"
   by blast
 
 lemma bex_triv_one_point2 [simp]: "(\<exists>x\<in>A. a = x) \<longleftrightarrow> a \<in> A"
   by blast
 
 lemma bex_one_point1 [simp]: "(\<exists>x\<in>A. x = a \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
   by blast
 
 lemma bex_one_point2 [simp]: "(\<exists>x\<in>A. a = x \<and> P x) \<longleftrightarrow> a \<in> A \<and> P a"
   by blast
 
 lemma ball_one_point1 [simp]: "(\<forall>x\<in>A. x = a \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)"
   by blast
 
 lemma ball_one_point2 [simp]: "(\<forall>x\<in>A. a = x \<longrightarrow> P x) \<longleftrightarrow> (a \<in> A \<longrightarrow> P a)"
   by blast
 
 lemma ball_conj_distrib: "(\<forall>x\<in>A. P x \<and> Q x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x)"
   by blast
 
 lemma bex_disj_distrib: "(\<exists>x\<in>A. P x \<or> Q x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x)"
   by blast
 
 
 text \<open>Congruence rules\<close>
 
 lemma ball_cong:
   "\<lbrakk> A = B;  \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
 by (simp add: Ball_def)
 
 lemma ball_cong_simp [cong]:
   "\<lbrakk> A = B;  \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<forall>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>B. Q x)"
 by (simp add: simp_implies_def Ball_def)
 
 lemma bex_cong:
   "\<lbrakk> A = B;  \<And>x. x \<in> B \<Longrightarrow> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
 by (simp add: Bex_def cong: conj_cong)
 
 lemma bex_cong_simp [cong]:
   "\<lbrakk> A = B;  \<And>x. x \<in> B =simp=> P x \<longleftrightarrow> Q x \<rbrakk> \<Longrightarrow>
     (\<exists>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>B. Q x)"
 by (simp add: simp_implies_def Bex_def cong: conj_cong)
 
 lemma bex1_def: "(\<exists>!x\<in>X. P x) \<longleftrightarrow> (\<exists>x\<in>X. P x) \<and> (\<forall>x\<in>X. \<forall>y\<in>X. P x \<longrightarrow> P y \<longrightarrow> x = y)"
   by auto
 
 
 subsection \<open>Basic operations\<close>
 
 subsubsection \<open>Subsets\<close>
 
 lemma subsetI [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B) \<Longrightarrow> A \<subseteq> B"
   by (simp add: less_eq_set_def le_fun_def)
 
 text \<open>
   \<^medskip>
   Map the type \<open>'a set \<Rightarrow> anything\<close> to just \<open>'a\<close>; for overloading constants
   whose first argument has type \<open>'a set\<close>.
 \<close>
 
 lemma subsetD [elim, intro?]: "A \<subseteq> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
   by (simp add: less_eq_set_def le_fun_def)
   \<comment> \<open>Rule in Modus Ponens style.\<close>
 
 lemma rev_subsetD [intro?,no_atp]: "c \<in> A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> c \<in> B"
   \<comment> \<open>The same, with reversed premises for use with @{method erule} -- cf. @{thm rev_mp}.\<close>
   by (rule subsetD)
 
 lemma subsetCE [elim,no_atp]: "A \<subseteq> B \<Longrightarrow> (c \<notin> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   \<comment> \<open>Classical elimination rule.\<close>
   by (auto simp add: less_eq_set_def le_fun_def)
 
 lemma subset_eq: "A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. x \<in> B)"
   by blast
 
 lemma contra_subsetD [no_atp]: "A \<subseteq> B \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<notin> A"
   by blast
 
 lemma subset_refl: "A \<subseteq> A"
   by (fact order_refl) (* already [iff] *)
 
 lemma subset_trans: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subseteq> C"
   by (fact order_trans)
 
 lemma subset_not_subset_eq [code]: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> \<not> B \<subseteq> A"
   by (fact less_le_not_le)
 
 lemma eq_mem_trans: "a = b \<Longrightarrow> b \<in> A \<Longrightarrow> a \<in> A"
   by simp
 
 lemmas basic_trans_rules [trans] =
   order_trans_rules rev_subsetD subsetD eq_mem_trans
 
 
 subsubsection \<open>Equality\<close>
 
 lemma subset_antisym [intro!]: "A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> A = B"
   \<comment> \<open>Anti-symmetry of the subset relation.\<close>
   by (iprover intro: set_eqI subsetD)
 
 text \<open>\<^medskip> Equality rules from ZF set theory -- are they appropriate here?\<close>
 
 lemma equalityD1: "A = B \<Longrightarrow> A \<subseteq> B"
   by simp
 
 lemma equalityD2: "A = B \<Longrightarrow> B \<subseteq> A"
   by simp
 
 text \<open>
   \<^medskip>
   Be careful when adding this to the claset as \<open>subset_empty\<close> is in the
   simpset: \<^prop>\<open>A = {}\<close> goes to \<^prop>\<open>{} \<subseteq> A\<close> and \<^prop>\<open>A \<subseteq> {}\<close>
   and then back to \<^prop>\<open>A = {}\<close>!
 \<close>
 
 lemma equalityE: "A = B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma equalityCE [elim]: "A = B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> (c \<notin> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
   by blast
 
 lemma eqset_imp_iff: "A = B \<Longrightarrow> x \<in> A \<longleftrightarrow> x \<in> B"
   by simp
 
 lemma eqelem_imp_iff: "x = y \<Longrightarrow> x \<in> A \<longleftrightarrow> y \<in> A"
   by simp
 
 
 subsubsection \<open>The empty set\<close>
 
 lemma empty_def: "{} = {x. False}"
   by (simp add: bot_set_def bot_fun_def)
 
 lemma empty_iff [simp]: "c \<in> {} \<longleftrightarrow> False"
   by (simp add: empty_def)
 
 lemma emptyE [elim!]: "a \<in> {} \<Longrightarrow> P"
   by simp
 
 lemma empty_subsetI [iff]: "{} \<subseteq> A"
   \<comment> \<open>One effect is to delete the ASSUMPTION \<^prop>\<open>{} \<subseteq> A\<close>\<close>
   by blast
 
 lemma equals0I: "(\<And>y. y \<in> A \<Longrightarrow> False) \<Longrightarrow> A = {}"
   by blast
 
 lemma equals0D: "A = {} \<Longrightarrow> a \<notin> A"
   \<comment> \<open>Use for reasoning about disjointness: \<open>A \<inter> B = {}\<close>\<close>
   by blast
 
 lemma ball_empty [simp]: "Ball {} P \<longleftrightarrow> True"
   by (simp add: Ball_def)
 
 lemma bex_empty [simp]: "Bex {} P \<longleftrightarrow> False"
   by (simp add: Bex_def)
 
 
 subsubsection \<open>The universal set -- UNIV\<close>
 
 abbreviation UNIV :: "'a set"
   where "UNIV \<equiv> top"
 
 lemma UNIV_def: "UNIV = {x. True}"
   by (simp add: top_set_def top_fun_def)
 
 lemma UNIV_I [simp]: "x \<in> UNIV"
   by (simp add: UNIV_def)
 
 declare UNIV_I [intro]  \<comment> \<open>unsafe makes it less likely to cause problems\<close>
 
 lemma UNIV_witness [intro?]: "\<exists>x. x \<in> UNIV"
   by simp
 
 lemma subset_UNIV: "A \<subseteq> UNIV"
   by (fact top_greatest) (* already simp *)
 
 text \<open>
   \<^medskip>
   Eta-contracting these two rules (to remove \<open>P\<close>) causes them
   to be ignored because of their interaction with congruence rules.
 \<close>
 
 lemma ball_UNIV [simp]: "Ball UNIV P \<longleftrightarrow> All P"
   by (simp add: Ball_def)
 
 lemma bex_UNIV [simp]: "Bex UNIV P \<longleftrightarrow> Ex P"
   by (simp add: Bex_def)
 
 lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A"
   by auto
 
 lemma UNIV_not_empty [iff]: "UNIV \<noteq> {}"
   by (blast elim: equalityE)
 
 lemma empty_not_UNIV[simp]: "{} \<noteq> UNIV"
   by blast
 
 
 subsubsection \<open>The Powerset operator -- Pow\<close>
 
 definition Pow :: "'a set \<Rightarrow> 'a set set"
   where Pow_def: "Pow A = {B. B \<subseteq> A}"
 
 lemma Pow_iff [iff]: "A \<in> Pow B \<longleftrightarrow> A \<subseteq> B"
   by (simp add: Pow_def)
 
 lemma PowI: "A \<subseteq> B \<Longrightarrow> A \<in> Pow B"
   by (simp add: Pow_def)
 
 lemma PowD: "A \<in> Pow B \<Longrightarrow> A \<subseteq> B"
   by (simp add: Pow_def)
 
 lemma Pow_bottom: "{} \<in> Pow B"
   by simp
 
 lemma Pow_top: "A \<in> Pow A"
   by simp
 
 lemma Pow_not_empty: "Pow A \<noteq> {}"
   using Pow_top by blast
 
 
 subsubsection \<open>Set complement\<close>
 
 lemma Compl_iff [simp]: "c \<in> - A \<longleftrightarrow> c \<notin> A"
   by (simp add: fun_Compl_def uminus_set_def)
 
 lemma ComplI [intro!]: "(c \<in> A \<Longrightarrow> False) \<Longrightarrow> c \<in> - A"
   by (simp add: fun_Compl_def uminus_set_def) blast
 
 text \<open>
   \<^medskip>
   This form, with negated conclusion, works well with the Classical prover.
   Negated assumptions behave like formulae on the right side of the
   notional turnstile \dots
 \<close>
 
 lemma ComplD [dest!]: "c \<in> - A \<Longrightarrow> c \<notin> A"
   by simp
 
 lemmas ComplE = ComplD [elim_format]
 
 lemma Compl_eq: "- A = {x. \<not> x \<in> A}"
   by blast
 
 
 subsubsection \<open>Binary intersection\<close>
 
 abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<inter>" 70)
   where "(\<inter>) \<equiv> inf"
 
 notation (ASCII)
   inter  (infixl "Int" 70)
 
 lemma Int_def: "A \<inter> B = {x. x \<in> A \<and> x \<in> B}"
   by (simp add: inf_set_def inf_fun_def)
 
 lemma Int_iff [simp]: "c \<in> A \<inter> B \<longleftrightarrow> c \<in> A \<and> c \<in> B"
   unfolding Int_def by blast
 
 lemma IntI [intro!]: "c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> c \<in> A \<inter> B"
   by simp
 
 lemma IntD1: "c \<in> A \<inter> B \<Longrightarrow> c \<in> A"
   by simp
 
 lemma IntD2: "c \<in> A \<inter> B \<Longrightarrow> c \<in> B"
   by simp
 
 lemma IntE [elim!]: "c \<in> A \<inter> B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
   by (fact mono_inf)
 
 
 subsubsection \<open>Binary union\<close>
 
 abbreviation union :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set"  (infixl "\<union>" 65)
   where "union \<equiv> sup"
 
 notation (ASCII)
   union  (infixl "Un" 65)
 
 lemma Un_def: "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
   by (simp add: sup_set_def sup_fun_def)
 
 lemma Un_iff [simp]: "c \<in> A \<union> B \<longleftrightarrow> c \<in> A \<or> c \<in> B"
   unfolding Un_def by blast
 
 lemma UnI1 [elim?]: "c \<in> A \<Longrightarrow> c \<in> A \<union> B"
   by simp
 
 lemma UnI2 [elim?]: "c \<in> B \<Longrightarrow> c \<in> A \<union> B"
   by simp
 
 text \<open>\<^medskip> Classical introduction rule: no commitment to \<open>A\<close> vs. \<open>B\<close>.\<close>
 lemma UnCI [intro!]: "(c \<notin> B \<Longrightarrow> c \<in> A) \<Longrightarrow> c \<in> A \<union> B"
   by auto
 
 lemma UnE [elim!]: "c \<in> A \<union> B \<Longrightarrow> (c \<in> A \<Longrightarrow> P) \<Longrightarrow> (c \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding Un_def by blast
 
 lemma insert_def: "insert a B = {x. x = a} \<union> B"
   by (simp add: insert_compr Un_def)
 
 lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
   by (fact mono_sup)
 
 
 subsubsection \<open>Set difference\<close>
 
 lemma Diff_iff [simp]: "c \<in> A - B \<longleftrightarrow> c \<in> A \<and> c \<notin> B"
   by (simp add: minus_set_def fun_diff_def)
 
 lemma DiffI [intro!]: "c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A - B"
   by simp
 
 lemma DiffD1: "c \<in> A - B \<Longrightarrow> c \<in> A"
   by simp
 
 lemma DiffD2: "c \<in> A - B \<Longrightarrow> c \<in> B \<Longrightarrow> P"
   by simp
 
 lemma DiffE [elim!]: "c \<in> A - B \<Longrightarrow> (c \<in> A \<Longrightarrow> c \<notin> B \<Longrightarrow> P) \<Longrightarrow> P"
   by simp
 
 lemma set_diff_eq: "A - B = {x. x \<in> A \<and> x \<notin> B}"
   by blast
 
 lemma Compl_eq_Diff_UNIV: "- A = (UNIV - A)"
   by blast
 
 
 subsubsection \<open>Augmenting a set -- \<^const>\<open>insert\<close>\<close>
 
 lemma insert_iff [simp]: "a \<in> insert b A \<longleftrightarrow> a = b \<or> a \<in> A"
   unfolding insert_def by blast
 
 lemma insertI1: "a \<in> insert a B"
   by simp
 
 lemma insertI2: "a \<in> B \<Longrightarrow> a \<in> insert b B"
   by simp
 
 lemma insertE [elim!]: "a \<in> insert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding insert_def by blast
 
 lemma insertCI [intro!]: "(a \<notin> B \<Longrightarrow> a = b) \<Longrightarrow> a \<in> insert b B"
   \<comment> \<open>Classical introduction rule.\<close>
   by auto
 
 lemma subset_insert_iff: "A \<subseteq> insert x B \<longleftrightarrow> (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
   by auto
 
 lemma set_insert:
   assumes "x \<in> A"
   obtains B where "A = insert x B" and "x \<notin> B"
 proof
   show "A = insert x (A - {x})" using assms by blast
   show "x \<notin> A - {x}" by blast
 qed
 
 lemma insert_ident: "x \<notin> A \<Longrightarrow> x \<notin> B \<Longrightarrow> insert x A = insert x B \<longleftrightarrow> A = B"
   by auto
 
 lemma insert_eq_iff:
   assumes "a \<notin> A" "b \<notin> B"
   shows "insert a A = insert b B \<longleftrightarrow>
     (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
     (is "?L \<longleftrightarrow> ?R")
 proof
   show ?R if ?L
   proof (cases "a = b")
     case True
     with assms \<open>?L\<close> show ?R
       by (simp add: insert_ident)
   next
     case False
     let ?C = "A - {b}"
     have "A = insert b ?C \<and> b \<notin> ?C \<and> B = insert a ?C \<and> a \<notin> ?C"
       using assms \<open>?L\<close> \<open>a \<noteq> b\<close> by auto
     then show ?R using \<open>a \<noteq> b\<close> by auto
   qed
   show ?L if ?R
     using that by (auto split: if_splits)
 qed
 
 lemma insert_UNIV: "insert x UNIV = UNIV"
   by auto
 
 
 subsubsection \<open>Singletons, using insert\<close>
 
 lemma singletonI [intro!]: "a \<in> {a}"
   \<comment> \<open>Redundant? But unlike \<open>insertCI\<close>, it proves the subgoal immediately!\<close>
   by (rule insertI1)
 
 lemma singletonD [dest!]: "b \<in> {a} \<Longrightarrow> b = a"
   by blast
 
 lemmas singletonE = singletonD [elim_format]
 
 lemma singleton_iff: "b \<in> {a} \<longleftrightarrow> b = a"
   by blast
 
 lemma singleton_inject [dest!]: "{a} = {b} \<Longrightarrow> a = b"
   by blast
 
 lemma singleton_insert_inj_eq [iff]: "{b} = insert a A \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
   by blast
 
 lemma singleton_insert_inj_eq' [iff]: "insert a A = {b} \<longleftrightarrow> a = b \<and> A \<subseteq> {b}"
   by blast
 
 lemma subset_singletonD: "A \<subseteq> {x} \<Longrightarrow> A = {} \<or> A = {x}"
   by fast
 
 lemma subset_singleton_iff: "X \<subseteq> {a} \<longleftrightarrow> X = {} \<or> X = {a}"
   by blast
 
 lemma subset_singleton_iff_Uniq: "(\<exists>a. A \<subseteq> {a}) \<longleftrightarrow> (\<exists>\<^sub>\<le>\<^sub>1x. x \<in> A)"
   unfolding Uniq_def by blast
 
 lemma singleton_conv [simp]: "{x. x = a} = {a}"
   by blast
 
 lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
   by blast
 
 lemma Diff_single_insert: "A - {x} \<subseteq> B \<Longrightarrow> A \<subseteq> insert x B"
   by blast
 
 lemma subset_Diff_insert: "A \<subseteq> B - insert x C \<longleftrightarrow> A \<subseteq> B - C \<and> x \<notin> A"
   by blast
 
 lemma doubleton_eq_iff: "{a, b} = {c, d} \<longleftrightarrow> a = c \<and> b = d \<or> a = d \<and> b = c"
   by (blast elim: equalityE)
 
 lemma Un_singleton_iff: "A \<union> B = {x} \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}"
   by auto
 
 lemma singleton_Un_iff: "{x} = A \<union> B \<longleftrightarrow> A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x}"
   by auto
 
 
 subsubsection \<open>Image of a set under a function\<close>
 
 text \<open>Frequently \<open>b\<close> does not have the syntactic form of \<open>f x\<close>.\<close>
 
 definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set"    (infixr "`" 90)
   where "f ` A = {y. \<exists>x\<in>A. y = f x}"
 
 lemma image_eqI [simp, intro]: "b = f x \<Longrightarrow> x \<in> A \<Longrightarrow> b \<in> f ` A"
   unfolding image_def by blast
 
 lemma imageI: "x \<in> A \<Longrightarrow> f x \<in> f ` A"
   by (rule image_eqI) (rule refl)
 
 lemma rev_image_eqI: "x \<in> A \<Longrightarrow> b = f x \<Longrightarrow> b \<in> f ` A"
   \<comment> \<open>This version's more effective when we already have the required \<open>x\<close>.\<close>
   by (rule image_eqI)
 
 lemma imageE [elim!]:
   assumes "b \<in> (\<lambda>x. f x) ` A"  \<comment> \<open>The eta-expansion gives variable-name preservation.\<close>
   obtains x where "b = f x" and "x \<in> A"
   using assms unfolding image_def by blast
 
 lemma Compr_image_eq: "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
   by auto
 
 lemma image_Un: "f ` (A \<union> B) = f ` A \<union> f ` B"
   by blast
 
 lemma image_iff: "z \<in> f ` A \<longleftrightarrow> (\<exists>x\<in>A. z = f x)"
   by blast
 
 lemma image_subsetI: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` A \<subseteq> B"
   \<comment> \<open>Replaces the three steps \<open>subsetI\<close>, \<open>imageE\<close>,
     \<open>hypsubst\<close>, but breaks too many existing proofs.\<close>
   by blast
 
 lemma image_subset_iff: "f ` A \<subseteq> B \<longleftrightarrow> (\<forall>x\<in>A. f x \<in> B)"
   \<comment> \<open>This rewrite rule would confuse users if made default.\<close>
   by blast
 
 lemma subset_imageE:
   assumes "B \<subseteq> f ` A"
   obtains C where "C \<subseteq> A" and "B = f ` C"
 proof -
   from assms have "B = f ` {a \<in> A. f a \<in> B}" by fast
   moreover have "{a \<in> A. f a \<in> B} \<subseteq> A" by blast
   ultimately show thesis by (blast intro: that)
 qed
 
 lemma subset_image_iff: "B \<subseteq> f ` A \<longleftrightarrow> (\<exists>AA\<subseteq>A. B = f ` AA)"
   by (blast elim: subset_imageE)
 
 lemma image_ident [simp]: "(\<lambda>x. x) ` Y = Y"
   by blast
 
 lemma image_empty [simp]: "f ` {} = {}"
   by blast
 
 lemma image_insert [simp]: "f ` insert a B = insert (f a) (f ` B)"
   by blast
 
 lemma image_constant: "x \<in> A \<Longrightarrow> (\<lambda>x. c) ` A = {c}"
   by auto
 
 lemma image_constant_conv: "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
   by auto
 
 lemma image_image: "f ` (g ` A) = (\<lambda>x. f (g x)) ` A"
   by blast
 
 lemma insert_image [simp]: "x \<in> A \<Longrightarrow> insert (f x) (f ` A) = f ` A"
   by blast
 
 lemma image_is_empty [iff]: "f ` A = {} \<longleftrightarrow> A = {}"
   by blast
 
 lemma empty_is_image [iff]: "{} = f ` A \<longleftrightarrow> A = {}"
   by blast
 
 lemma image_Collect: "f ` {x. P x} = {f x | x. P x}"
   \<comment> \<open>NOT suitable as a default simp rule: the RHS isn't simpler than the LHS,
       with its implicit quantifier and conjunction.  Also image enjoys better
       equational properties than does the RHS.\<close>
   by blast
 
 lemma if_image_distrib [simp]:
   "(\<lambda>x. if P x then f x else g x) ` S = f ` (S \<inter> {x. P x}) \<union> g ` (S \<inter> {x. \<not> P x})"
   by auto
 
 lemma image_cong:
   "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N \<Longrightarrow> f x = g x"
   using that by (simp add: image_def)
 
 lemma image_cong_simp [cong]:
   "f ` M = g ` N" if "M = N" "\<And>x. x \<in> N =simp=> f x = g x"
   using that image_cong [of M N f g] by (simp add: simp_implies_def)
 
 lemma image_Int_subset: "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
   by blast
 
 lemma image_diff_subset: "f ` A - f ` B \<subseteq> f ` (A - B)"
   by blast
 
 lemma Setcompr_eq_image: "{f x |x. x \<in> A} = f ` A"
   by blast
 
 lemma setcompr_eq_image: "{f x |x. P x} = f ` {x. P x}"
   by auto
 
 lemma ball_imageD: "\<forall>x\<in>f ` A. P x \<Longrightarrow> \<forall>x\<in>A. P (f x)"
   by simp
 
 lemma bex_imageD: "\<exists>x\<in>f ` A. P x \<Longrightarrow> \<exists>x\<in>A. P (f x)"
   by auto
 
 lemma image_add_0 [simp]: "(+) (0::'a::comm_monoid_add) ` S = S"
   by auto
 
 
 text \<open>\<^medskip> Range of a function -- just an abbreviation for image!\<close>
 
 abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set"  \<comment> \<open>of function\<close>
   where "range f \<equiv> f ` UNIV"
 
 lemma range_eqI: "b = f x \<Longrightarrow> b \<in> range f"
   by simp
 
 lemma rangeI: "f x \<in> range f"
   by simp
 
 lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> P) \<Longrightarrow> P"
   by (rule imageE)
 
 lemma full_SetCompr_eq: "{u. \<exists>x. u = f x} = range f"
   by auto
 
 lemma range_composition: "range (\<lambda>x. f (g x)) = f ` range g"
   by auto
 
 lemma range_constant [simp]: "range (\<lambda>_. x) = {x}"
   by (simp add: image_constant)
 
 lemma range_eq_singletonD: "range f = {a} \<Longrightarrow> f x = a"
   by auto
 
 
 subsubsection \<open>Some rules with \<open>if\<close>\<close>
 
 text \<open>Elimination of \<open>{x. \<dots> \<and> x = t \<and> \<dots>}\<close>.\<close>
 
 lemma Collect_conv_if: "{x. x = a \<and> P x} = (if P a then {a} else {})"
   by auto
 
 lemma Collect_conv_if2: "{x. a = x \<and> P x} = (if P a then {a} else {})"
   by auto
 
 text \<open>
   Rewrite rules for boolean case-splitting: faster than \<open>if_split [split]\<close>.
 \<close>
 
 lemma if_split_eq1: "(if Q then x else y) = b \<longleftrightarrow> (Q \<longrightarrow> x = b) \<and> (\<not> Q \<longrightarrow> y = b)"
   by (rule if_split)
 
 lemma if_split_eq2: "a = (if Q then x else y) \<longleftrightarrow> (Q \<longrightarrow> a = x) \<and> (\<not> Q \<longrightarrow> a = y)"
   by (rule if_split)
 
 text \<open>
   Split ifs on either side of the membership relation.
   Not for \<open>[simp]\<close> -- can cause goals to blow up!
 \<close>
 
 lemma if_split_mem1: "(if Q then x else y) \<in> b \<longleftrightarrow> (Q \<longrightarrow> x \<in> b) \<and> (\<not> Q \<longrightarrow> y \<in> b)"
   by (rule if_split)
 
 lemma if_split_mem2: "(a \<in> (if Q then x else y)) \<longleftrightarrow> (Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y)"
   by (rule if_split [where P = "\<lambda>S. a \<in> S"])
 
 lemmas split_ifs = if_bool_eq_conj if_split_eq1 if_split_eq2 if_split_mem1 if_split_mem2
 
 (*Would like to add these, but the existing code only searches for the
   outer-level constant, which in this case is just Set.member; we instead need
   to use term-nets to associate patterns with rules.  Also, if a rule fails to
   apply, then the formula should be kept.
   [("uminus", Compl_iff RS iffD1), ("minus", [Diff_iff RS iffD1]),
    ("Int", [IntD1,IntD2]),
    ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
  *)
 
 
 subsection \<open>Further operations and lemmas\<close>
 
 subsubsection \<open>The ``proper subset'' relation\<close>
 
 lemma psubsetI [intro!]: "A \<subseteq> B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<subset> B"
   unfolding less_le by blast
 
 lemma psubsetE [elim!]: "A \<subset> B \<Longrightarrow> (A \<subseteq> B \<Longrightarrow> \<not> B \<subseteq> A \<Longrightarrow> R) \<Longrightarrow> R"
   unfolding less_le by blast
 
 lemma psubset_insert_iff:
   "A \<subset> insert x B \<longleftrightarrow> (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)"
   by (auto simp add: less_le subset_insert_iff)
 
 lemma psubset_eq: "A \<subset> B \<longleftrightarrow> A \<subseteq> B \<and> A \<noteq> B"
   by (simp only: less_le)
 
 lemma psubset_imp_subset: "A \<subset> B \<Longrightarrow> A \<subseteq> B"
   by (simp add: psubset_eq)
 
 lemma psubset_trans: "A \<subset> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
   unfolding less_le by (auto dest: subset_antisym)
 
 lemma psubsetD: "A \<subset> B \<Longrightarrow> c \<in> A \<Longrightarrow> c \<in> B"
   unfolding less_le by (auto dest: subsetD)
 
 lemma psubset_subset_trans: "A \<subset> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<subset> C"
   by (auto simp add: psubset_eq)
 
 lemma subset_psubset_trans: "A \<subseteq> B \<Longrightarrow> B \<subset> C \<Longrightarrow> A \<subset> C"
   by (auto simp add: psubset_eq)
 
 lemma psubset_imp_ex_mem: "A \<subset> B \<Longrightarrow> \<exists>b. b \<in> B - A"
   unfolding less_le by blast
 
 lemma atomize_ball: "(\<And>x. x \<in> A \<Longrightarrow> P x) \<equiv> Trueprop (\<forall>x\<in>A. P x)"
   by (simp only: Ball_def atomize_all atomize_imp)
 
 lemmas [symmetric, rulify] = atomize_ball
   and [symmetric, defn] = atomize_ball
 
 lemma image_Pow_mono: "f ` A \<subseteq> B \<Longrightarrow> image f ` Pow A \<subseteq> Pow B"
   by blast
 
 lemma image_Pow_surj: "f ` A = B \<Longrightarrow> image f ` Pow A = Pow B"
   by (blast elim: subset_imageE)
 
 
 subsubsection \<open>Derived rules involving subsets.\<close>
 
 text \<open>\<open>insert\<close>.\<close>
 
 lemma subset_insertI: "B \<subseteq> insert a B"
   by (rule subsetI) (erule insertI2)
 
 lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B"
   by blast
 
 lemma subset_insert: "x \<notin> A \<Longrightarrow> A \<subseteq> insert x B \<longleftrightarrow> A \<subseteq> B"
   by blast
 
 
 text \<open>\<^medskip> Finite Union -- the least upper bound of two sets.\<close>
 
 lemma Un_upper1: "A \<subseteq> A \<union> B"
   by (fact sup_ge1)
 
 lemma Un_upper2: "B \<subseteq> A \<union> B"
   by (fact sup_ge2)
 
 lemma Un_least: "A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> A \<union> B \<subseteq> C"
   by (fact sup_least)
 
 
 text \<open>\<^medskip> Finite Intersection -- the greatest lower bound of two sets.\<close>
 
 lemma Int_lower1: "A \<inter> B \<subseteq> A"
   by (fact inf_le1)
 
 lemma Int_lower2: "A \<inter> B \<subseteq> B"
   by (fact inf_le2)
 
 lemma Int_greatest: "C \<subseteq> A \<Longrightarrow> C \<subseteq> B \<Longrightarrow> C \<subseteq> A \<inter> B"
   by (fact inf_greatest)
 
 
 text \<open>\<^medskip> Set difference.\<close>
 
 lemma Diff_subset[simp]: "A - B \<subseteq> A"
   by blast
 
 lemma Diff_subset_conv: "A - B \<subseteq> C \<longleftrightarrow> A \<subseteq> B \<union> C"
   by blast
 
 
 subsubsection \<open>Equalities involving union, intersection, inclusion, etc.\<close>
 
 text \<open>\<open>{}\<close>.\<close>
 
 lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})"
   \<comment> \<open>supersedes \<open>Collect_False_empty\<close>\<close>
   by auto
 
 lemma subset_empty [simp]: "A \<subseteq> {} \<longleftrightarrow> A = {}"
   by (fact bot_unique)
 
 lemma not_psubset_empty [iff]: "\<not> (A < {})"
   by (fact not_less_bot) (* FIXME: already simp *)
 
 lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
 
 lemma Collect_empty_eq [simp]: "Collect P = {} \<longleftrightarrow> (\<forall>x. \<not> P x)"
   by blast
 
 lemma empty_Collect_eq [simp]: "{} = Collect P \<longleftrightarrow> (\<forall>x. \<not> P x)"
   by blast
 
 lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}"
   by blast
 
 lemma Collect_disj_eq: "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
   by blast
 
 lemma Collect_imp_eq: "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
   by blast
 
 lemma Collect_conj_eq: "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
   by blast
 
 lemma Collect_mono_iff: "Collect P \<subseteq> Collect Q \<longleftrightarrow> (\<forall>x. P x \<longrightarrow> Q x)"
   by blast
 
 
 text \<open>\<^medskip> \<open>insert\<close>.\<close>
 
 lemma insert_is_Un: "insert a A = {a} \<union> A"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a {}\<close>\<close>
   by blast
 
 lemma insert_not_empty [simp]: "insert a A \<noteq> {}"
   and empty_not_insert [simp]: "{} \<noteq> insert a A"
   by blast+
 
 lemma insert_absorb: "a \<in> A \<Longrightarrow> insert a A = A"
   \<comment> \<open>\<open>[simp]\<close> causes recursive calls when there are nested inserts\<close>
   \<comment> \<open>with \<^emph>\<open>quadratic\<close> running time\<close>
   by blast
 
 lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A"
   by blast
 
 lemma insert_commute: "insert x (insert y A) = insert y (insert x A)"
   by blast
 
 lemma insert_subset [simp]: "insert x A \<subseteq> B \<longleftrightarrow> x \<in> B \<and> A \<subseteq> B"
   by blast
 
 lemma mk_disjoint_insert: "a \<in> A \<Longrightarrow> \<exists>B. A = insert a B \<and> a \<notin> B"
   \<comment> \<open>use new \<open>B\<close> rather than \<open>A - {a}\<close> to avoid infinite unfolding\<close>
   by (rule exI [where x = "A - {a}"]) blast
 
 lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a \<longrightarrow> P u}"
   by auto
 
 lemma insert_inter_insert [simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)"
   by blast
 
 lemma insert_disjoint [simp]:
   "insert a A \<inter> B = {} \<longleftrightarrow> a \<notin> B \<and> A \<inter> B = {}"
   "{} = insert a A \<inter> B \<longleftrightarrow> a \<notin> B \<and> {} = A \<inter> B"
   by auto
 
 lemma disjoint_insert [simp]:
   "B \<inter> insert a A = {} \<longleftrightarrow> a \<notin> B \<and> B \<inter> A = {}"
   "{} = A \<inter> insert b B \<longleftrightarrow> b \<notin> A \<and> {} = A \<inter> B"
   by auto
 
 
 text \<open>\<^medskip> \<open>Int\<close>\<close>
 
 lemma Int_absorb: "A \<inter> A = A"
   by (fact inf_idem) (* already simp *)
 
 lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
   by (fact inf_left_idem)
 
 lemma Int_commute: "A \<inter> B = B \<inter> A"
   by (fact inf_commute)
 
 lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
   by (fact inf_left_commute)
 
 lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
   by (fact inf_assoc)
 
 lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
   \<comment> \<open>Intersection is an AC-operator\<close>
 
 lemma Int_absorb1: "B \<subseteq> A \<Longrightarrow> A \<inter> B = B"
   by (fact inf_absorb2)
 
 lemma Int_absorb2: "A \<subseteq> B \<Longrightarrow> A \<inter> B = A"
   by (fact inf_absorb1)
 
 lemma Int_empty_left: "{} \<inter> B = {}"
   by (fact inf_bot_left) (* already simp *)
 
 lemma Int_empty_right: "A \<inter> {} = {}"
   by (fact inf_bot_right) (* already simp *)
 
 lemma disjoint_eq_subset_Compl: "A \<inter> B = {} \<longleftrightarrow> A \<subseteq> - B"
   by blast
 
 lemma disjoint_iff: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x. x\<in>A \<longrightarrow> x \<notin> B)"
   by blast
 
 lemma disjoint_iff_not_equal: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
   by blast
 
 lemma Int_UNIV_left: "UNIV \<inter> B = B"
   by (fact inf_top_left) (* already simp *)
 
 lemma Int_UNIV_right: "A \<inter> UNIV = A"
   by (fact inf_top_right) (* already simp *)
 
 lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
   by (fact inf_sup_distrib1)
 
 lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
   by (fact inf_sup_distrib2)
 
 lemma Int_UNIV [simp]: "A \<inter> B = UNIV \<longleftrightarrow> A = UNIV \<and> B = UNIV"
   by (fact inf_eq_top_iff) (* already simp *)
 
 lemma Int_subset_iff [simp]: "C \<subseteq> A \<inter> B \<longleftrightarrow> C \<subseteq> A \<and> C \<subseteq> B"
   by (fact le_inf_iff)
 
 lemma Int_Collect: "x \<in> A \<inter> {x. P x} \<longleftrightarrow> x \<in> A \<and> P x"
   by blast
 
 
 text \<open>\<^medskip> \<open>Un\<close>.\<close>
 
 lemma Un_absorb: "A \<union> A = A"
   by (fact sup_idem) (* already simp *)
 
 lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
   by (fact sup_left_idem)
 
 lemma Un_commute: "A \<union> B = B \<union> A"
   by (fact sup_commute)
 
 lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
   by (fact sup_left_commute)
 
 lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
   by (fact sup_assoc)
 
 lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
   \<comment> \<open>Union is an AC-operator\<close>
 
 lemma Un_absorb1: "A \<subseteq> B \<Longrightarrow> A \<union> B = B"
   by (fact sup_absorb2)
 
 lemma Un_absorb2: "B \<subseteq> A \<Longrightarrow> A \<union> B = A"
   by (fact sup_absorb1)
 
 lemma Un_empty_left: "{} \<union> B = B"
   by (fact sup_bot_left) (* already simp *)
 
 lemma Un_empty_right: "A \<union> {} = A"
   by (fact sup_bot_right) (* already simp *)
 
 lemma Un_UNIV_left: "UNIV \<union> B = UNIV"
   by (fact sup_top_left) (* already simp *)
 
 lemma Un_UNIV_right: "A \<union> UNIV = UNIV"
   by (fact sup_top_right) (* already simp *)
 
 lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
   by blast
 
 lemma Un_insert_right [simp]: "A \<union> (insert a B) = insert a (A \<union> B)"
   by blast
 
 lemma Int_insert_left: "(insert a B) \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
   by auto
 
 lemma Int_insert_left_if0 [simp]: "a \<notin> C \<Longrightarrow> (insert a B) \<inter> C = B \<inter> C"
   by auto
 
 lemma Int_insert_left_if1 [simp]: "a \<in> C \<Longrightarrow> (insert a B) \<inter> C = insert a (B \<inter> C)"
   by auto
 
 lemma Int_insert_right: "A \<inter> (insert a B) = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
   by auto
 
 lemma Int_insert_right_if0 [simp]: "a \<notin> A \<Longrightarrow> A \<inter> (insert a B) = A \<inter> B"
   by auto
 
 lemma Int_insert_right_if1 [simp]: "a \<in> A \<Longrightarrow> A \<inter> (insert a B) = insert a (A \<inter> B)"
   by auto
 
 lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
   by (fact sup_inf_distrib1)
 
 lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
   by (fact sup_inf_distrib2)
 
 lemma Un_Int_crazy: "(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
   by blast
 
 lemma subset_Un_eq: "A \<subseteq> B \<longleftrightarrow> A \<union> B = B"
   by (fact le_iff_sup)
 
 lemma Un_empty [iff]: "A \<union> B = {} \<longleftrightarrow> A = {} \<and> B = {}"
   by (fact sup_eq_bot_iff) (* FIXME: already simp *)
 
 lemma Un_subset_iff [simp]: "A \<union> B \<subseteq> C \<longleftrightarrow> A \<subseteq> C \<and> B \<subseteq> C"
   by (fact le_sup_iff)
 
 lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
   by blast
 
 lemma Diff_Int2: "A \<inter> C - B \<inter> C = A \<inter> C - B"
   by blast
 
 lemma subset_UnE: 
   assumes "C \<subseteq> A \<union> B"
   obtains A' B' where "A' \<subseteq> A" "B' \<subseteq> B" "C = A' \<union> B'"
 proof
   show "C \<inter> A \<subseteq> A" "C \<inter> B \<subseteq> B" "C = (C \<inter> A) \<union> (C \<inter> B)"
     using assms by blast+
 qed
 
+lemma Un_Int_eq [simp]: "(S \<union> T) \<inter> S = S" "(S \<union> T) \<inter> T = T" "S \<inter> (S \<union> T) = S" "T \<inter> (S \<union> T) = T"
+  by auto
+
+lemma Int_Un_eq [simp]: "(S \<inter> T) \<union> S = S" "(S \<inter> T) \<union> T = T" "S \<union> (S \<inter> T) = S" "T \<union> (S \<inter> T) = T"
+  by auto
+
 text \<open>\<^medskip> Set complement\<close>
 
 lemma Compl_disjoint [simp]: "A \<inter> - A = {}"
   by (fact inf_compl_bot)
 
 lemma Compl_disjoint2 [simp]: "- A \<inter> A = {}"
   by (fact compl_inf_bot)
 
 lemma Compl_partition: "A \<union> - A = UNIV"
   by (fact sup_compl_top)
 
 lemma Compl_partition2: "- A \<union> A = UNIV"
   by (fact compl_sup_top)
 
 lemma double_complement: "- (-A) = A" for A :: "'a set"
   by (fact double_compl) (* already simp *)
 
 lemma Compl_Un: "- (A \<union> B) = (- A) \<inter> (- B)"
   by (fact compl_sup) (* already simp *)
 
 lemma Compl_Int: "- (A \<inter> B) = (- A) \<union> (- B)"
   by (fact compl_inf) (* already simp *)
 
 lemma subset_Compl_self_eq: "A \<subseteq> - A \<longleftrightarrow> A = {}"
   by blast
 
 lemma Un_Int_assoc_eq: "(A \<inter> B) \<union> C = A \<inter> (B \<union> C) \<longleftrightarrow> C \<subseteq> A"
   \<comment> \<open>Halmos, Naive Set Theory, page 16.\<close>
   by blast
 
 lemma Compl_UNIV_eq: "- UNIV = {}"
   by (fact compl_top_eq) (* already simp *)
 
 lemma Compl_empty_eq: "- {} = UNIV"
   by (fact compl_bot_eq) (* already simp *)
 
 lemma Compl_subset_Compl_iff [iff]: "- A \<subseteq> - B \<longleftrightarrow> B \<subseteq> A"
   by (fact compl_le_compl_iff) (* FIXME: already simp *)
 
 lemma Compl_eq_Compl_iff [iff]: "- A = - B \<longleftrightarrow> A = B"
   for A B :: "'a set"
   by (fact compl_eq_compl_iff) (* FIXME: already simp *)
 
 lemma Compl_insert: "- insert x A = (- A) - {x}"
   by blast
 
 text \<open>\<^medskip> Bounded quantifiers.
 
   The following are not added to the default simpset because
   (a) they duplicate the body and (b) there are no similar rules for \<open>Int\<close>.
 \<close>
 
 lemma ball_Un: "(\<forall>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x)"
   by blast
 
 lemma bex_Un: "(\<exists>x \<in> A \<union> B. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x)"
   by blast
 
 
 text \<open>\<^medskip> Set difference.\<close>
 
 lemma Diff_eq: "A - B = A \<inter> (- B)"
   by blast
 
 lemma Diff_eq_empty_iff [simp]: "A - B = {} \<longleftrightarrow> A \<subseteq> B"
   by blast
 
 lemma Diff_cancel [simp]: "A - A = {}"
   by blast
 
 lemma Diff_idemp [simp]: "(A - B) - B = A - B"
   for A B :: "'a set"
   by blast
 
 lemma Diff_triv: "A \<inter> B = {} \<Longrightarrow> A - B = A"
   by (blast elim: equalityE)
 
 lemma empty_Diff [simp]: "{} - A = {}"
   by blast
 
 lemma Diff_empty [simp]: "A - {} = A"
   by blast
 
 lemma Diff_UNIV [simp]: "A - UNIV = {}"
   by blast
 
 lemma Diff_insert0 [simp]: "x \<notin> A \<Longrightarrow> A - insert x B = A - B"
   by blast
 
 lemma Diff_insert: "A - insert a B = A - B - {a}"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
   by blast
 
 lemma Diff_insert2: "A - insert a B = A - {a} - B"
   \<comment> \<open>NOT SUITABLE FOR REWRITING since \<open>{a} \<equiv> insert a 0\<close>\<close>
   by blast
 
 lemma insert_Diff_if: "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
   by auto
 
 lemma insert_Diff1 [simp]: "x \<in> B \<Longrightarrow> insert x A - B = A - B"
   by blast
 
 lemma insert_Diff_single[simp]: "insert a (A - {a}) = insert a A"
   by blast
 
 lemma insert_Diff: "a \<in> A \<Longrightarrow> insert a (A - {a}) = A"
   by blast
 
 lemma Diff_insert_absorb: "x \<notin> A \<Longrightarrow> (insert x A) - {x} = A"
   by auto
 
 lemma Diff_disjoint [simp]: "A \<inter> (B - A) = {}"
   by blast
 
 lemma Diff_partition: "A \<subseteq> B \<Longrightarrow> A \<union> (B - A) = B"
   by blast
 
 lemma double_diff: "A \<subseteq> B \<Longrightarrow> B \<subseteq> C \<Longrightarrow> B - (C - A) = A"
   by blast
 
 lemma Un_Diff_cancel [simp]: "A \<union> (B - A) = A \<union> B"
   by blast
 
 lemma Un_Diff_cancel2 [simp]: "(B - A) \<union> A = B \<union> A"
   by blast
 
 lemma Diff_Un: "A - (B \<union> C) = (A - B) \<inter> (A - C)"
   by blast
 
 lemma Diff_Int: "A - (B \<inter> C) = (A - B) \<union> (A - C)"
   by blast
 
 lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"
   by blast
 
 lemma Un_Diff: "(A \<union> B) - C = (A - C) \<union> (B - C)"
   by blast
 
 lemma Int_Diff: "(A \<inter> B) - C = A \<inter> (B - C)"
   by blast
 
 lemma Diff_Int_distrib: "C \<inter> (A - B) = (C \<inter> A) - (C \<inter> B)"
   by blast
 
 lemma Diff_Int_distrib2: "(A - B) \<inter> C = (A \<inter> C) - (B \<inter> C)"
   by blast
 
 lemma Diff_Compl [simp]: "A - (- B) = A \<inter> B"
   by auto
 
 lemma Compl_Diff_eq [simp]: "- (A - B) = - A \<union> B"
   by blast
 
 lemma subset_Compl_singleton [simp]: "A \<subseteq> - {b} \<longleftrightarrow> b \<notin> A"
   by blast
 
 text \<open>\<^medskip> Quantification over type \<^typ>\<open>bool\<close>.\<close>
 
 lemma bool_induct: "P True \<Longrightarrow> P False \<Longrightarrow> P x"
   by (cases x) auto
 
 lemma all_bool_eq: "(\<forall>b. P b) \<longleftrightarrow> P True \<and> P False"
   by (auto intro: bool_induct)
 
 lemma bool_contrapos: "P x \<Longrightarrow> \<not> P False \<Longrightarrow> P True"
   by (cases x) auto
 
 lemma ex_bool_eq: "(\<exists>b. P b) \<longleftrightarrow> P True \<or> P False"
   by (auto intro: bool_contrapos)
 
 lemma UNIV_bool: "UNIV = {False, True}"
   by (auto intro: bool_induct)
 
 text \<open>\<^medskip> \<open>Pow\<close>\<close>
 
 lemma Pow_empty [simp]: "Pow {} = {{}}"
   by (auto simp add: Pow_def)
 
 lemma Pow_singleton_iff [simp]: "Pow X = {Y} \<longleftrightarrow> X = {} \<and> Y = {}"
   by blast  (* somewhat slow *)
 
 lemma Pow_insert: "Pow (insert a A) = Pow A \<union> (insert a ` Pow A)"
   by (blast intro: image_eqI [where ?x = "u - {a}" for u])
 
 lemma Pow_Compl: "Pow (- A) = {- B | B. A \<in> Pow B}"
   by (blast intro: exI [where ?x = "- u" for u])
 
 lemma Pow_UNIV [simp]: "Pow UNIV = UNIV"
   by blast
 
 lemma Un_Pow_subset: "Pow A \<union> Pow B \<subseteq> Pow (A \<union> B)"
   by blast
 
 lemma Pow_Int_eq [simp]: "Pow (A \<inter> B) = Pow A \<inter> Pow B"
   by blast
 
 
 text \<open>\<^medskip> Miscellany.\<close>
 
 lemma set_eq_subset: "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
   by blast
 
 lemma subset_iff: "A \<subseteq> B \<longleftrightarrow> (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
   by blast
 
 lemma subset_iff_psubset_eq: "A \<subseteq> B \<longleftrightarrow> A \<subset> B \<or> A = B"
   unfolding less_le by blast
 
 lemma all_not_in_conv [simp]: "(\<forall>x. x \<notin> A) \<longleftrightarrow> A = {}"
   by blast
 
 lemma ex_in_conv: "(\<exists>x. x \<in> A) \<longleftrightarrow> A \<noteq> {}"
   by blast
 
 lemma ball_simps [simp, no_atp]:
   "\<And>A P Q. (\<forall>x\<in>A. P x \<or> Q) \<longleftrightarrow> ((\<forall>x\<in>A. P x) \<or> Q)"
   "\<And>A P Q. (\<forall>x\<in>A. P \<or> Q x) \<longleftrightarrow> (P \<or> (\<forall>x\<in>A. Q x))"
   "\<And>A P Q. (\<forall>x\<in>A. P \<longrightarrow> Q x) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x\<in>A. Q x))"
   "\<And>A P Q. (\<forall>x\<in>A. P x \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<longrightarrow> Q)"
   "\<And>P. (\<forall>x\<in>{}. P x) \<longleftrightarrow> True"
   "\<And>P. (\<forall>x\<in>UNIV. P x) \<longleftrightarrow> (\<forall>x. P x)"
   "\<And>a B P. (\<forall>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<and> (\<forall>x\<in>B. P x))"
   "\<And>P Q. (\<forall>x\<in>Collect Q. P x) \<longleftrightarrow> (\<forall>x. Q x \<longrightarrow> P x)"
   "\<And>A P f. (\<forall>x\<in>f`A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P (f x))"
   "\<And>A P. (\<not> (\<forall>x\<in>A. P x)) \<longleftrightarrow> (\<exists>x\<in>A. \<not> P x)"
   by auto
 
 lemma bex_simps [simp, no_atp]:
   "\<And>A P Q. (\<exists>x\<in>A. P x \<and> Q) \<longleftrightarrow> ((\<exists>x\<in>A. P x) \<and> Q)"
   "\<And>A P Q. (\<exists>x\<in>A. P \<and> Q x) \<longleftrightarrow> (P \<and> (\<exists>x\<in>A. Q x))"
   "\<And>P. (\<exists>x\<in>{}. P x) \<longleftrightarrow> False"
   "\<And>P. (\<exists>x\<in>UNIV. P x) \<longleftrightarrow> (\<exists>x. P x)"
   "\<And>a B P. (\<exists>x\<in>insert a B. P x) \<longleftrightarrow> (P a \<or> (\<exists>x\<in>B. P x))"
   "\<And>P Q. (\<exists>x\<in>Collect Q. P x) \<longleftrightarrow> (\<exists>x. Q x \<and> P x)"
   "\<And>A P f. (\<exists>x\<in>f`A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
   "\<And>A P. (\<not>(\<exists>x\<in>A. P x)) \<longleftrightarrow> (\<forall>x\<in>A. \<not> P x)"
   by auto
 
 lemma ex_image_cong_iff [simp, no_atp]:
   "(\<exists>x. x\<in>f`A) \<longleftrightarrow> A \<noteq> {}" "(\<exists>x. x\<in>f`A \<and> P x) \<longleftrightarrow> (\<exists>x\<in>A. P (f x))"
   by auto
 
 subsubsection \<open>Monotonicity of various operations\<close>
 
 lemma image_mono: "A \<subseteq> B \<Longrightarrow> f ` A \<subseteq> f ` B"
   by blast
 
 lemma Pow_mono: "A \<subseteq> B \<Longrightarrow> Pow A \<subseteq> Pow B"
   by blast
 
 lemma insert_mono: "C \<subseteq> D \<Longrightarrow> insert a C \<subseteq> insert a D"
   by blast
 
 lemma Un_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<union> B \<subseteq> C \<union> D"
   by (fact sup_mono)
 
 lemma Int_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A \<inter> B \<subseteq> C \<inter> D"
   by (fact inf_mono)
 
 lemma Diff_mono: "A \<subseteq> C \<Longrightarrow> D \<subseteq> B \<Longrightarrow> A - B \<subseteq> C - D"
   by blast
 
 lemma Compl_anti_mono: "A \<subseteq> B \<Longrightarrow> - B \<subseteq> - A"
   by (fact compl_mono)
 
 text \<open>\<^medskip> Monotonicity of implications.\<close>
 
 lemma in_mono: "A \<subseteq> B \<Longrightarrow> x \<in> A \<longrightarrow> x \<in> B"
   by (rule impI) (erule subsetD)
 
 lemma conj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<and> P2) \<longrightarrow> (Q1 \<and> Q2)"
   by iprover
 
 lemma disj_mono: "P1 \<longrightarrow> Q1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<or> P2) \<longrightarrow> (Q1 \<or> Q2)"
   by iprover
 
 lemma imp_mono: "Q1 \<longrightarrow> P1 \<Longrightarrow> P2 \<longrightarrow> Q2 \<Longrightarrow> (P1 \<longrightarrow> P2) \<longrightarrow> (Q1 \<longrightarrow> Q2)"
   by iprover
 
 lemma imp_refl: "P \<longrightarrow> P" ..
 
 lemma not_mono: "Q \<longrightarrow> P \<Longrightarrow> \<not> P \<longrightarrow> \<not> Q"
   by iprover
 
 lemma ex_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)"
   by iprover
 
 lemma all_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"
   by iprover
 
 lemma Collect_mono: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> Collect P \<subseteq> Collect Q"
   by blast
 
 lemma Int_Collect_mono: "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x) \<Longrightarrow> A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
   by blast
 
 lemmas basic_monos =
   subset_refl imp_refl disj_mono conj_mono ex_mono Collect_mono in_mono
 
 lemma eq_to_mono: "a = b \<Longrightarrow> c = d \<Longrightarrow> b \<longrightarrow> d \<Longrightarrow> a \<longrightarrow> c"
   by iprover
 
 
 subsubsection \<open>Inverse image of a function\<close>
 
 definition vimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a set"  (infixr "-`" 90)
   where "f -` B \<equiv> {x. f x \<in> B}"
 
 lemma vimage_eq [simp]: "a \<in> f -` B \<longleftrightarrow> f a \<in> B"
   unfolding vimage_def by blast
 
 lemma vimage_singleton_eq: "a \<in> f -` {b} \<longleftrightarrow> f a = b"
   by simp
 
 lemma vimageI [intro]: "f a = b \<Longrightarrow> b \<in> B \<Longrightarrow> a \<in> f -` B"
   unfolding vimage_def by blast
 
 lemma vimageI2: "f a \<in> A \<Longrightarrow> a \<in> f -` A"
   unfolding vimage_def by fast
 
 lemma vimageE [elim!]: "a \<in> f -` B \<Longrightarrow> (\<And>x. f a = x \<Longrightarrow> x \<in> B \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding vimage_def by blast
 
 lemma vimageD: "a \<in> f -` A \<Longrightarrow> f a \<in> A"
   unfolding vimage_def by fast
 
 lemma vimage_empty [simp]: "f -` {} = {}"
   by blast
 
 lemma vimage_Compl: "f -` (- A) = - (f -` A)"
   by blast
 
 lemma vimage_Un [simp]: "f -` (A \<union> B) = (f -` A) \<union> (f -` B)"
   by blast
 
 lemma vimage_Int [simp]: "f -` (A \<inter> B) = (f -` A) \<inter> (f -` B)"
   by fast
 
 lemma vimage_Collect_eq [simp]: "f -` Collect P = {y. P (f y)}"
   by blast
 
 lemma vimage_Collect: "(\<And>x. P (f x) = Q x) \<Longrightarrow> f -` (Collect P) = Collect Q"
   by blast
 
 lemma vimage_insert: "f -` (insert a B) = (f -` {a}) \<union> (f -` B)"
   \<comment> \<open>NOT suitable for rewriting because of the recurrence of \<open>{a}\<close>.\<close>
   by blast
 
 lemma vimage_Diff: "f -` (A - B) = (f -` A) - (f -` B)"
   by blast
 
 lemma vimage_UNIV [simp]: "f -` UNIV = UNIV"
   by blast
 
 lemma vimage_mono: "A \<subseteq> B \<Longrightarrow> f -` A \<subseteq> f -` B"
   \<comment> \<open>monotonicity\<close>
   by blast
 
 lemma vimage_image_eq: "f -` (f ` A) = {y. \<exists>x\<in>A. f x = f y}"
   by (blast intro: sym)
 
 lemma image_vimage_subset: "f ` (f -` A) \<subseteq> A"
   by blast
 
 lemma image_vimage_eq [simp]: "f ` (f -` A) = A \<inter> range f"
   by blast
 
 lemma image_subset_iff_subset_vimage: "f ` A \<subseteq> B \<longleftrightarrow> A \<subseteq> f -` B"
   by blast
 
 lemma vimage_const [simp]: "((\<lambda>x. c) -` A) = (if c \<in> A then UNIV else {})"
   by auto
 
 lemma vimage_if [simp]: "((\<lambda>x. if x \<in> B then c else d) -` A) =
    (if c \<in> A then (if d \<in> A then UNIV else B)
     else if d \<in> A then - B else {})"
   by (auto simp add: vimage_def)
 
 lemma vimage_inter_cong: "(\<And> w. w \<in> S \<Longrightarrow> f w = g w) \<Longrightarrow> f -` y \<inter> S = g -` y \<inter> S"
   by auto
 
 lemma vimage_ident [simp]: "(\<lambda>x. x) -` Y = Y"
   by blast
 
 
 subsubsection \<open>Singleton sets\<close>
 
 definition is_singleton :: "'a set \<Rightarrow> bool"
   where "is_singleton A \<longleftrightarrow> (\<exists>x. A = {x})"
 
 lemma is_singletonI [simp, intro!]: "is_singleton {x}"
   unfolding is_singleton_def by simp
 
 lemma is_singletonI': "A \<noteq> {} \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y) \<Longrightarrow> is_singleton A"
   unfolding is_singleton_def by blast
 
 lemma is_singletonE: "is_singleton A \<Longrightarrow> (\<And>x. A = {x} \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding is_singleton_def by blast
 
 
 subsubsection \<open>Getting the contents of a singleton set\<close>
 
 definition the_elem :: "'a set \<Rightarrow> 'a"
   where "the_elem X = (THE x. X = {x})"
 
 lemma the_elem_eq [simp]: "the_elem {x} = x"
   by (simp add: the_elem_def)
 
 lemma is_singleton_the_elem: "is_singleton A \<longleftrightarrow> A = {the_elem A}"
   by (auto simp: is_singleton_def)
 
 lemma the_elem_image_unique:
   assumes "A \<noteq> {}"
     and *: "\<And>y. y \<in> A \<Longrightarrow> f y = f x"
   shows "the_elem (f ` A) = f x"
   unfolding the_elem_def
 proof (rule the1_equality)
   from \<open>A \<noteq> {}\<close> obtain y where "y \<in> A" by auto
   with * have "f x = f y" by simp
   with \<open>y \<in> A\<close> have "f x \<in> f ` A" by blast
   with * show "f ` A = {f x}" by auto
   then show "\<exists>!x. f ` A = {x}" by auto
 qed
 
 
 subsubsection \<open>Least value operator\<close>
 
 lemma Least_mono: "mono f \<Longrightarrow> \<exists>x\<in>S. \<forall>y\<in>S. x \<le> y \<Longrightarrow> (LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
   for f :: "'a::order \<Rightarrow> 'b::order"
   \<comment> \<open>Courtesy of Stephan Merz\<close>
   apply clarify
   apply (erule_tac P = "\<lambda>x. x \<in> S" in LeastI2_order)
    apply fast
   apply (rule LeastI2_order)
     apply (auto elim: monoD intro!: order_antisym)
   done
 
 
 subsubsection \<open>Monad operation\<close>
 
 definition bind :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"
   where "bind A f = {x. \<exists>B \<in> f`A. x \<in> B}"
 
 hide_const (open) bind
 
 lemma bind_bind: "Set.bind (Set.bind A B) C = Set.bind A (\<lambda>x. Set.bind (B x) C)"
   for A :: "'a set"
   by (auto simp: bind_def)
 
 lemma empty_bind [simp]: "Set.bind {} f = {}"
   by (simp add: bind_def)
 
 lemma nonempty_bind_const: "A \<noteq> {} \<Longrightarrow> Set.bind A (\<lambda>_. B) = B"
   by (auto simp: bind_def)
 
 lemma bind_const: "Set.bind A (\<lambda>_. B) = (if A = {} then {} else B)"
   by (auto simp: bind_def)
 
 lemma bind_singleton_conv_image: "Set.bind A (\<lambda>x. {f x}) = f ` A"
   by (auto simp: bind_def)
 
 
 subsubsection \<open>Operations for execution\<close>
 
 definition is_empty :: "'a set \<Rightarrow> bool"
   where [code_abbrev]: "is_empty A \<longleftrightarrow> A = {}"
 
 hide_const (open) is_empty
 
 definition remove :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where [code_abbrev]: "remove x A = A - {x}"
 
 hide_const (open) remove
 
 lemma member_remove [simp]: "x \<in> Set.remove y A \<longleftrightarrow> x \<in> A \<and> x \<noteq> y"
   by (simp add: remove_def)
 
 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set"
   where [code_abbrev]: "filter P A = {a \<in> A. P a}"
 
 hide_const (open) filter
 
 lemma member_filter [simp]: "x \<in> Set.filter P A \<longleftrightarrow> x \<in> A \<and> P x"
   by (simp add: filter_def)
 
 instantiation set :: (equal) equal
 begin
 
 definition "HOL.equal A B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A"
 
 instance by standard (auto simp add: equal_set_def)
 
 end
 
 
 text \<open>Misc\<close>
 
 definition pairwise :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
   where "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y \<in> S. x \<noteq> y \<longrightarrow> R x y)"
 
 lemma pairwise_alt: "pairwise R S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S-{x}. R x y)"
 by (auto simp add: pairwise_def)
 
 lemma pairwise_trivial [simp]: "pairwise (\<lambda>i j. j \<noteq> i) I"
   by (auto simp: pairwise_def)
 
 lemma pairwiseI [intro?]:
   "pairwise R S" if "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> R x y"
   using that by (simp add: pairwise_def)
 
 lemma pairwiseD:
   "R x y" and "R y x"
   if "pairwise R S" "x \<in> S" and "y \<in> S" and "x \<noteq> y"
   using that by (simp_all add: pairwise_def)
 
 lemma pairwise_empty [simp]: "pairwise P {}"
   by (simp add: pairwise_def)
 
 lemma pairwise_singleton [simp]: "pairwise P {A}"
   by (simp add: pairwise_def)
 
 lemma pairwise_insert:
   "pairwise r (insert x s) \<longleftrightarrow> (\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s"
   by (force simp: pairwise_def)
 
 lemma pairwise_subset: "pairwise P S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> pairwise P T"
   by (force simp: pairwise_def)
 
 lemma pairwise_mono: "\<lbrakk>pairwise P A; \<And>x y. P x y \<Longrightarrow> Q x y; B \<subseteq> A\<rbrakk> \<Longrightarrow> pairwise Q B"
   by (fastforce simp: pairwise_def)
 
 lemma pairwise_imageI:
   "pairwise P (f ` A)"
   if "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x \<noteq> f y \<Longrightarrow> P (f x) (f y)"
   using that by (auto intro: pairwiseI)
 
 lemma pairwise_image: "pairwise r (f ` s) \<longleftrightarrow> pairwise (\<lambda>x y. (f x \<noteq> f y) \<longrightarrow> r (f x) (f y)) s"
   by (force simp: pairwise_def)
 
 definition disjnt :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
   where "disjnt A B \<longleftrightarrow> A \<inter> B = {}"
 
 lemma disjnt_self_iff_empty [simp]: "disjnt S S \<longleftrightarrow> S = {}"
   by (auto simp: disjnt_def)
 
 lemma disjnt_iff: "disjnt A B \<longleftrightarrow> (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
   by (force simp: disjnt_def)
 
 lemma disjnt_sym: "disjnt A B \<Longrightarrow> disjnt B A"
   using disjnt_iff by blast
 
 lemma disjnt_empty1 [simp]: "disjnt {} A" and disjnt_empty2 [simp]: "disjnt A {}"
   by (auto simp: disjnt_def)
 
 lemma disjnt_insert1 [simp]: "disjnt (insert a X) Y \<longleftrightarrow> a \<notin> Y \<and> disjnt X Y"
   by (simp add: disjnt_def)
 
 lemma disjnt_insert2 [simp]: "disjnt Y (insert a X) \<longleftrightarrow> a \<notin> Y \<and> disjnt Y X"
   by (simp add: disjnt_def)
 
 lemma disjnt_subset1 : "\<lbrakk>disjnt X Y; Z \<subseteq> X\<rbrakk> \<Longrightarrow> disjnt Z Y"
   by (auto simp: disjnt_def)
 
 lemma disjnt_subset2 : "\<lbrakk>disjnt X Y; Z \<subseteq> Y\<rbrakk> \<Longrightarrow> disjnt X Z"
   by (auto simp: disjnt_def)
 
 lemma disjnt_Un1 [simp]: "disjnt (A \<union> B) C \<longleftrightarrow> disjnt A C \<and> disjnt B C"
   by (auto simp: disjnt_def)
 
 lemma disjnt_Un2 [simp]: "disjnt C (A \<union> B) \<longleftrightarrow> disjnt C A \<and> disjnt C B"
   by (auto simp: disjnt_def)
 
 lemma disjoint_image_subset: "\<lbrakk>pairwise disjnt \<A>; \<And>X. X \<in> \<A> \<Longrightarrow> f X \<subseteq> X\<rbrakk> \<Longrightarrow> pairwise disjnt (f `\<A>)"
   unfolding disjnt_def pairwise_def by fast
 
 lemma pairwise_disjnt_iff: "pairwise disjnt \<A> \<longleftrightarrow> (\<forall>x. \<exists>\<^sub>\<le>\<^sub>1 X. X \<in> \<A> \<and> x \<in> X)"
   by (auto simp: Uniq_def disjnt_iff pairwise_def)
 
 lemma Int_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> False) \<Longrightarrow> A \<inter> B = {}"
   by blast
 
 lemma in_image_insert_iff:
   assumes "\<And>C. C \<in> B \<Longrightarrow> x \<notin> C"
   shows "A \<in> insert x ` B \<longleftrightarrow> x \<in> A \<and> A - {x} \<in> B" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P then show ?Q
     using assms by auto
 next
   assume ?Q
   then have "x \<in> A" and "A - {x} \<in> B"
     by simp_all
   from \<open>A - {x} \<in> B\<close> have "insert x (A - {x}) \<in> insert x ` B"
     by (rule imageI)
   also from \<open>x \<in> A\<close>
   have "insert x (A - {x}) = A"
     by auto
   finally show ?P .
 qed
 
 hide_const (open) member not_member
 
 lemmas equalityI = subset_antisym
 lemmas set_mp = subsetD
 lemmas set_rev_mp = rev_subsetD
 
 ML \<open>
 val Ball_def = @{thm Ball_def}
 val Bex_def = @{thm Bex_def}
 val CollectD = @{thm CollectD}
 val CollectE = @{thm CollectE}
 val CollectI = @{thm CollectI}
 val Collect_conj_eq = @{thm Collect_conj_eq}
 val Collect_mem_eq = @{thm Collect_mem_eq}
 val IntD1 = @{thm IntD1}
 val IntD2 = @{thm IntD2}
 val IntE = @{thm IntE}
 val IntI = @{thm IntI}
 val Int_Collect = @{thm Int_Collect}
 val UNIV_I = @{thm UNIV_I}
 val UNIV_witness = @{thm UNIV_witness}
 val UnE = @{thm UnE}
 val UnI1 = @{thm UnI1}
 val UnI2 = @{thm UnI2}
 val ballE = @{thm ballE}
 val ballI = @{thm ballI}
 val bexCI = @{thm bexCI}
 val bexE = @{thm bexE}
 val bexI = @{thm bexI}
 val bex_triv = @{thm bex_triv}
 val bspec = @{thm bspec}
 val contra_subsetD = @{thm contra_subsetD}
 val equalityCE = @{thm equalityCE}
 val equalityD1 = @{thm equalityD1}
 val equalityD2 = @{thm equalityD2}
 val equalityE = @{thm equalityE}
 val equalityI = @{thm equalityI}
 val imageE = @{thm imageE}
 val imageI = @{thm imageI}
 val image_Un = @{thm image_Un}
 val image_insert = @{thm image_insert}
 val insert_commute = @{thm insert_commute}
 val insert_iff = @{thm insert_iff}
 val mem_Collect_eq = @{thm mem_Collect_eq}
 val rangeE = @{thm rangeE}
 val rangeI = @{thm rangeI}
 val range_eqI = @{thm range_eqI}
 val subsetCE = @{thm subsetCE}
 val subsetD = @{thm subsetD}
 val subsetI = @{thm subsetI}
 val subset_refl = @{thm subset_refl}
 val subset_trans = @{thm subset_trans}
 val vimageD = @{thm vimageD}
 val vimageE = @{thm vimageE}
 val vimageI = @{thm vimageI}
 val vimageI2 = @{thm vimageI2}
 val vimage_Collect = @{thm vimage_Collect}
 val vimage_Int = @{thm vimage_Int}
 val vimage_Un = @{thm vimage_Un}
 \<close>
 
 end