separated Affine theory from Convex

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Affine.thy	Thu Dec 05 21:03:06 2019 +0100
@@ -0,0 +1,1653 @@
+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 blast
+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::euclidean_space 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::euclidean_space 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::euclidean_space 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]: "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"
+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)"
+    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])
+  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

--- a/src/HOL/Analysis/Convex.thy	Thu Dec 05 19:53:41 2019 +0100
+++ b/src/HOL/Analysis/Convex.thy	Thu Dec 05 21:03:06 2019 +0100
@@ -10,7 +10,7 @@
 
 theory Convex
 imports
-  Linear_Algebra
+  Affine
   "HOL-Library.Set_Algebras"
 begin
 
@@ -1073,17 +1073,6 @@
   obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c"
 by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size)
 
-lemma sum_delta_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
-
 lemma sum_delta'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
@@ -1095,9 +1084,6 @@
     unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
 qed
 
-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 dist_triangle_eq:
   fixes x y z :: "'a::real_inner"
   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
@@ -1109,596 +1095,6 @@
 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 blast
-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
-
-
-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>Cones\<close>
@@ -1864,85 +1260,25 @@
   ultimately show ?thesis 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_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)"
+lemma convex_cone:
+  "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
+  (is "?lhs = ?rhs")
 proof -
-  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
+  {
+    fix x y
+    assume "x\<in>s" "y\<in>s" and ?lhs
+    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
+      unfolding cone_def by auto
+    then have "x + y \<in> s"
+      using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
+      apply (erule_tac x="2*\<^sub>R x" in ballE)
+      apply (erule_tac x="2*\<^sub>R y" in ballE)
+      apply (erule_tac x="1/2" in allE, simp)
+      apply (erule_tac x="1/2" in allE, auto)
+      done
+  }
+  then show ?thesis
+    unfolding convex_def cone_def by blast
 qed
 
 
@@ -2502,993 +1838,12 @@
 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
   using subspace_imp_affine affine_imp_convex by auto
 
-lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
-  by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
-
 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
 
 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
 
-lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
-  unfolding affine_dependent_def dependent_def
-  using affine_hull_subset_span by auto
-
-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 convex_cone:
-  "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
-  (is "?lhs = ?rhs")
-proof -
-  {
-    fix x y
-    assume "x\<in>s" "y\<in>s" and ?lhs
-    then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
-      unfolding cone_def by auto
-    then have "x + y \<in> s"
-      using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
-      apply (erule_tac x="2*\<^sub>R x" in ballE)
-      apply (erule_tac x="2*\<^sub>R y" in ballE)
-      apply (erule_tac x="1/2" in allE, simp)
-      apply (erule_tac x="1/2" in allE, auto)
-      done
-  }
-  then show ?thesis
-    unfolding convex_def cone_def by blast
-qed
-
-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::euclidean_space 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::euclidean_space 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::euclidean_space 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
-
-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
-
-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]: "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"
-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)"
-    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])
-  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
-
 lemma aff_dim_convex_hull:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (convex hull S) = aff_dim S"
@@ -3497,6 +1852,7 @@
     aff_dim_subset[of "convex hull S" "affine hull S"]
   by auto
 
+
 subsection \<open>Caratheodory's theorem\<close>
 
 lemma convex_hull_caratheodory_aff_dim: