diff --git a/src/HOL/Analysis/Convex.thy b/src/HOL/Analysis/Convex.thy
--- a/src/HOL/Analysis/Convex.thy
+++ b/src/HOL/Analysis/Convex.thy
@@ -1,2371 +1,2430 @@
 (* Title:      HOL/Analysis/Convex.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 
 section \<open>Convex Sets and Functions\<close>
 
 theory Convex
 imports
   Affine
   "HOL-Library.Set_Algebras"
 begin
 
 subsection \<open>Convex Sets\<close>
 
 definition\<^marker>\<open>tag important\<close> convex :: "'a::real_vector set \<Rightarrow> bool"
   where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
 
 lemma convexI:
   assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
   shows "convex s"
   using assms unfolding convex_def by fast
 
 lemma convexD:
   assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
   using assms unfolding convex_def by fast
 
 lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
   (is "_ \<longleftrightarrow> ?alt")
 proof
   show "convex s" if alt: ?alt
   proof -
     {
       fix x y and u v :: real
       assume mem: "x \<in> s" "y \<in> s"
       assume "0 \<le> u" "0 \<le> v"
       moreover
       assume "u + v = 1"
       then have "u = 1 - v" by auto
       ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
         using alt [rule_format, OF mem] by auto
     }
     then show ?thesis
       unfolding convex_def by auto
   qed
   show ?alt if "convex s"
     using that by (auto simp: convex_def)
 qed
 
 lemma convexD_alt:
   assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
   shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s"
   using assms unfolding convex_alt by auto
 
 lemma mem_convex_alt:
   assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0"
   shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
   using assms
   by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric])
 
 lemma convex_empty[intro,simp]: "convex {}"
   unfolding convex_def by simp
 
 lemma convex_singleton[intro,simp]: "convex {a}"
   unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric])
 
 lemma convex_UNIV[intro,simp]: "convex UNIV"
   unfolding convex_def by auto
 
 lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)"
   unfolding convex_def by auto
 
 lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)"
   unfolding convex_def by auto
 
 lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)"
   unfolding convex_def by auto
 
 lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)"
   unfolding convex_def by auto
 
 lemma convex_halfspace_le: "convex {x. inner a x \<le> b}"
   unfolding convex_def
   by (auto simp: inner_add intro!: convex_bound_le)
 
 lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}"
 proof -
   have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}"
     by auto
   show ?thesis
     unfolding * using convex_halfspace_le[of "-a" "-b"] by auto
 qed
 
 lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}"
 proof -
   have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}"
     by auto
   show ?thesis
     unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le)
 qed
 
 lemma convex_hyperplane: "convex {x. inner a x = b}"
 proof -
   have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}"
     by auto
   show ?thesis using convex_halfspace_le convex_halfspace_ge
     by (auto intro!: convex_Int simp: *)
 qed
 
 lemma convex_halfspace_lt: "convex {x. inner a x < b}"
   unfolding convex_def
   by (auto simp: convex_bound_lt inner_add)
 
 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
   using convex_halfspace_lt[of "-a" "-b"] by auto
 
 lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
   using convex_halfspace_ge[of b "1::complex"] by simp
 
 lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
   using convex_halfspace_le[of "1::complex" b] by simp
 
 lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
   using convex_halfspace_ge[of b \<i>] by simp
 
 lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
   using convex_halfspace_le[of \<i> b] by simp
 
 lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
   using convex_halfspace_gt[of b "1::complex"] by simp
 
 lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
   using convex_halfspace_lt[of "1::complex" b] by simp
 
 lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
   using convex_halfspace_gt[of b \<i>] by simp
 
 lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
   using convex_halfspace_lt[of \<i> b] by simp
 
 lemma convex_real_interval [iff]:
   fixes a b :: "real"
   shows "convex {a..}" and "convex {..b}"
     and "convex {a<..}" and "convex {..<b}"
     and "convex {a..b}" and "convex {a<..b}"
     and "convex {a..<b}" and "convex {a<..<b}"
 proof -
   have "{a..} = {x. a \<le> inner 1 x}"
     by auto
   then show 1: "convex {a..}"
     by (simp only: convex_halfspace_ge)
   have "{..b} = {x. inner 1 x \<le> b}"
     by auto
   then show 2: "convex {..b}"
     by (simp only: convex_halfspace_le)
   have "{a<..} = {x. a < inner 1 x}"
     by auto
   then show 3: "convex {a<..}"
     by (simp only: convex_halfspace_gt)
   have "{..<b} = {x. inner 1 x < b}"
     by auto
   then show 4: "convex {..<b}"
     by (simp only: convex_halfspace_lt)
   have "{a..b} = {a..} \<inter> {..b}"
     by auto
   then show "convex {a..b}"
     by (simp only: convex_Int 1 2)
   have "{a<..b} = {a<..} \<inter> {..b}"
     by auto
   then show "convex {a<..b}"
     by (simp only: convex_Int 3 2)
   have "{a..<b} = {a..} \<inter> {..<b}"
     by auto
   then show "convex {a..<b}"
     by (simp only: convex_Int 1 4)
   have "{a<..<b} = {a<..} \<inter> {..<b}"
     by auto
   then show "convex {a<..<b}"
     by (simp only: convex_Int 3 4)
 qed
 
 lemma convex_Reals: "convex \<real>"
   by (simp add: convex_def scaleR_conv_of_real)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expressions for convexity in terms of arbitrary sums\<close>
 
 lemma convex_sum:
   fixes C :: "'a::real_vector set"
   assumes "finite S"
     and "convex C"
     and "(\<Sum> i \<in> S. a i) = 1"
   assumes "\<And>i. i \<in> S \<Longrightarrow> a i \<ge> 0"
     and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C"
   shows "(\<Sum> j \<in> S. a j *\<^sub>R y j) \<in> C"
   using assms(1,3,4,5)
 proof (induct arbitrary: a set: finite)
   case empty
   then show ?case by simp
 next
   case (insert i S) note IH = this(3)
   have "a i + sum a S = 1"
     and "0 \<le> a i"
     and "\<forall>j\<in>S. 0 \<le> a j"
     and "y i \<in> C"
     and "\<forall>j\<in>S. y j \<in> C"
     using insert.hyps(1,2) insert.prems by simp_all
   then have "0 \<le> sum a S"
     by (simp add: sum_nonneg)
   have "a i *\<^sub>R y i + (\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
   proof (cases "sum a S = 0")
     case True
     with \<open>a i + sum a S = 1\<close> have "a i = 1"
       by simp
     from sum_nonneg_0 [OF \<open>finite S\<close> _ True] \<open>\<forall>j\<in>S. 0 \<le> a j\<close> have "\<forall>j\<in>S. a j = 0"
       by simp
     show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>S. a j = 0\<close> and \<open>y i \<in> C\<close>
       by simp
   next
     case False
     with \<open>0 \<le> sum a S\<close> have "0 < sum a S"
       by simp
     then have "(\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C"
       using \<open>\<forall>j\<in>S. 0 \<le> a j\<close> and \<open>\<forall>j\<in>S. y j \<in> C\<close>
       by (simp add: IH sum_divide_distrib [symmetric])
     from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close>
       and \<open>0 \<le> sum a S\<close> and \<open>a i + sum a S = 1\<close>
     have "a i *\<^sub>R y i + sum a S *\<^sub>R (\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C"
       by (rule convexD)
     then show ?thesis
       by (simp add: scaleR_sum_right False)
   qed
   then show ?case using \<open>finite S\<close> and \<open>i \<notin> S\<close>
     by simp
 qed
 
 lemma convex:
   "convex S \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>S) \<and> (sum u {1..k} = 1)
       \<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> S)"
 proof safe
   fix k :: nat
   fix u :: "nat \<Rightarrow> real"
   fix x
   assume "convex S"
     "\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S"
     "sum u {1..k} = 1"
   with convex_sum[of "{1 .. k}" S] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> S"
     by auto
 next
   assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1
     \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> S"
   {
     fix \<mu> :: real
     fix x y :: 'a
     assume xy: "x \<in> S" "y \<in> S"
     assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1"
     let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>"
     let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y"
     have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}"
       by auto
     then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1"
       by simp
     then have "sum ?u {1 .. 2} = 1"
       using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"]
       by auto
     with *[rule_format, of "2" ?u ?x] have S: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> S"
       using mu xy by auto
     have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y"
       using sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto
     from sum.atLeast_Suc_atMost[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this]
     have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
       by auto
     then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> S"
       using S by (auto simp: add.commute)
   }
   then show "convex S"
     unfolding convex_alt by auto
 qed
 
 
 lemma convex_explicit:
   fixes S :: "'a::real_vector set"
   shows "convex S \<longleftrightarrow>
     (\<forall>t u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> S)"
 proof safe
   fix t
   fix u :: "'a \<Rightarrow> real"
   assume "convex S"
     and "finite t"
     and "t \<subseteq> S" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1"
   then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S"
     using convex_sum[of t S u "\<lambda> x. x"] by auto
 next
   assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and>
     sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S"
   show "convex S"
     unfolding convex_alt
   proof safe
     fix x y
     fix \<mu> :: real
     assume **: "x \<in> S" "y \<in> S" "0 \<le> \<mu>" "\<mu> \<le> 1"
     show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> S"
     proof (cases "x = y")
       case False
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
         by auto
     next
       case True
       then show ?thesis
         using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
         by (auto simp: field_simps real_vector.scale_left_diff_distrib)
     qed
   qed
 qed
 
 lemma convex_finite:
   assumes "finite S"
   shows "convex S \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) S \<in> S)"
        (is "?lhs = ?rhs")
 proof 
   { have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
       by simp
     fix T :: "'a set" and u :: "'a \<Rightarrow> real"
     assume sum: "\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> S"
     assume *: "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1"
     assume "T \<subseteq> S"
     then have "S \<inter> T = T" by auto
     with sum[THEN spec[where x="\<lambda>x. if x\<in>T then u x else 0"]] * have "(\<Sum>x\<in>T. u x *\<^sub>R x) \<in> S"
       by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) }
   moreover assume ?rhs
   ultimately show ?lhs
     unfolding convex_explicit by auto
 qed (auto simp: convex_explicit assms)
 
 
 subsection \<open>Convex Functions on a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
   where "convex_on S f \<longleftrightarrow>
     (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
 
+definition\<^marker>\<open>tag important\<close> concave_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
+  where "concave_on S f \<equiv> convex_on S (\<lambda>x. - f x)"
+
+lemma concave_on_iff:
+  "concave_on S f \<longleftrightarrow>
+    (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<ge> u * f x + v * f y)"
+  by (auto simp: concave_on_def convex_on_def algebra_simps)
+
 lemma convex_onI [intro?]:
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   shows "convex_on A f"
   unfolding convex_on_def
 proof clarify
   fix x y
   fix u v :: real
   assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
   from A(5) have [simp]: "v = 1 - u"
     by (simp add: algebra_simps)
   from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
     using assms[of u y x]
     by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
 qed
 
 lemma convex_on_linorderI [intro?]:
   fixes A :: "('a::{linorder,real_vector}) set"
   assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   shows "convex_on A f"
 proof
   fix x y
   fix t :: real
   assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
   with assms [of t x y] assms [of "1 - t" y x]
   show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
     by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
 qed
 
 lemma convex_onD:
   assumes "convex_on A f"
   shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   using assms by (auto simp: convex_on_def)
 
 lemma convex_onD_Icc:
   assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
   shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
     f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
   using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
 
 lemma convex_on_subset: "convex_on t f \<Longrightarrow> S \<subseteq> t \<Longrightarrow> convex_on S f"
   unfolding convex_on_def by auto
 
 lemma convex_on_add [intro]:
   assumes "convex_on S f"
     and "convex_on S g"
   shows "convex_on S (\<lambda>x. f x + g x)"
 proof -
   {
     fix x y
     assume "x \<in> S" "y \<in> S"
     moreover
     fix u v :: real
     assume "0 \<le> u" "0 \<le> v" "u + v = 1"
     ultimately
     have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)"
       using assms unfolding convex_on_def by (auto simp: add_mono)
     then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)"
       by (simp add: field_simps)
   }
   then show ?thesis
     unfolding convex_on_def by auto
 qed
 
 lemma convex_on_cmul [intro]:
   fixes c :: real
   assumes "0 \<le> c"
     and "convex_on S f"
   shows "convex_on S (\<lambda>x. c * f x)"
 proof -
   have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
     for u c fx v fy :: real
     by (simp add: field_simps)
   show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
     unfolding convex_on_def and * by auto
 qed
 
 lemma convex_lower:
   assumes "convex_on S f"
     and "x \<in> S"
     and "y \<in> S"
     and "0 \<le> u"
     and "0 \<le> v"
     and "u + v = 1"
   shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)"
 proof -
   let ?m = "max (f x) (f y)"
   have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)"
     using assms(4,5) by (auto simp: mult_left_mono add_mono)
   also have "\<dots> = max (f x) (f y)"
     using assms(6) by (simp add: distrib_right [symmetric])
   finally show ?thesis
     using assms unfolding convex_on_def by fastforce
 qed
 
 lemma convex_on_dist [intro]:
   fixes S :: "'a::real_normed_vector set"
   shows "convex_on S (\<lambda>x. dist a x)"
 proof (auto simp: convex_on_def dist_norm)
   fix x y
   assume "x \<in> S" "y \<in> S"
   fix u v :: real
   assume "0 \<le> u"
   assume "0 \<le> v"
   assume "u + v = 1"
   have "a = u *\<^sub>R a + v *\<^sub>R a"
     unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp
   then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))"
     by (auto simp: algebra_simps)
   show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)"
     unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"]
     using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic operations on sets preserve convexity\<close>
 
 lemma convex_linear_image:
   assumes "linear f"
     and "convex S"
   shows "convex (f ` S)"
 proof -
   interpret f: linear f by fact
   from \<open>convex S\<close> show "convex (f ` S)"
     by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric])
 qed
 
 lemma convex_linear_vimage:
   assumes "linear f"
     and "convex S"
   shows "convex (f -` S)"
 proof -
   interpret f: linear f by fact
   from \<open>convex S\<close> show "convex (f -` S)"
     by (simp add: convex_def f.add f.scaleR)
 qed
 
 lemma convex_scaling:
   assumes "convex S"
   shows "convex ((\<lambda>x. c *\<^sub>R x) ` S)"
 proof -
   have "linear (\<lambda>x. c *\<^sub>R x)"
     by (simp add: linearI scaleR_add_right)
   then show ?thesis
     using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_scaled:
   assumes "convex S"
   shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
 proof -
   have "linear (\<lambda>x. x *\<^sub>R c)"
     by (simp add: linearI scaleR_add_left)
   then show ?thesis
     using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_negations:
   assumes "convex S"
   shows "convex ((\<lambda>x. - x) ` S)"
 proof -
   have "linear (\<lambda>x. - x)"
     by (simp add: linearI)
   then show ?thesis
     using \<open>convex S\<close> by (rule convex_linear_image)
 qed
 
 lemma convex_sums:
   assumes "convex S"
     and "convex T"
   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   have "linear (\<lambda>(x, y). x + y)"
     by (auto intro: linearI simp: scaleR_add_right)
   with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))"
     by (intro convex_linear_image convex_Times)
   also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by auto
   finally show ?thesis .
 qed
 
 lemma convex_differences:
   assumes "convex S" "convex T"
   shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}"
     by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff)
   then show ?thesis
     using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto
 qed
 
 lemma convex_translation:
   "convex ((+) a ` S)" if "convex S"
 proof -
   have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (+) a ` S"
     by auto
   then show ?thesis
     using convex_sums [OF convex_singleton [of a] that] by auto
 qed
 
 lemma convex_translation_subtract:
   "convex ((\<lambda>b. b - a) ` S)" if "convex S"
   using convex_translation [of S "- a"] that by (simp cong: image_cong_simp)
 
 lemma convex_affinity:
   assumes "convex S"
   shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
   have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S"
     by auto
   then show ?thesis
     using convex_translation[OF convex_scaling[OF assms], of a c] by auto
 qed
 
 lemma convex_on_sum:
   fixes a :: "'a \<Rightarrow> real"
     and y :: "'a \<Rightarrow> 'b::real_vector"
     and f :: "'b \<Rightarrow> real"
   assumes "finite s" "s \<noteq> {}"
     and "convex_on C f"
     and "convex C"
     and "(\<Sum> i \<in> s. a i) = 1"
     and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0"
     and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C"
   shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))"
   using assms
 proof (induct s arbitrary: a rule: finite_ne_induct)
   case (singleton i)
   then have ai: "a i = 1"
     by auto
   then show ?case
     by auto
 next
   case (insert i s)
   then have "convex_on C f"
     by simp
   from this[unfolded convex_on_def, rule_format]
   have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by simp
   show ?case
   proof (cases "a i = 1")
     case True
     then have "(\<Sum> j \<in> s. a j) = 0"
       using insert by auto
     then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0"
       using insert by (fastforce simp: sum_nonneg_eq_0_iff)
     then show ?thesis
       using insert by auto
   next
     case False
     from insert have yai: "y i \<in> C" "a i \<ge> 0"
       by auto
     have fis: "finite (insert i s)"
       using insert by auto
     then have ai1: "a i \<le> 1"
       using sum_nonneg_leq_bound[of "insert i s" a] insert by simp
     then have "a i < 1"
       using False by auto
     then have i0: "1 - a i > 0"
       by auto
     let ?a = "\<lambda>j. a j / (1 - a i)"
     have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j
       using i0 insert that by fastforce
     have "(\<Sum> j \<in> insert i s. a j) = 1"
       using insert by auto
     then have "(\<Sum> j \<in> s. a j) = 1 - a i"
       using sum.insert insert by fastforce
     then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1"
       using i0 by auto
     then have a1: "(\<Sum> j \<in> s. ?a j) = 1"
       unfolding sum_divide_distrib by simp
     have "convex C" using insert by auto
     then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
       using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
     have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
       using a_nonneg a1 insert by blast
     have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert
       by (auto simp only: add.commute)
     also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       using i0 by auto
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)"
       using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric]
       by (auto simp: algebra_simps)
     also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)"
       by (auto simp: divide_inverse)
     also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)"
       using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
       by (auto simp: add.commute)
     also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
       using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
             OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
       by simp
     also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
       unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
       using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
       using i0 by auto
     also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
       using insert by auto
     finally show ?thesis
       by simp
   qed
 qed
 
 lemma convex_on_alt:
   fixes C :: "'a::real_vector set"
   shows "convex_on C f \<longleftrightarrow>
     (\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow>
       f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)"
 proof safe
   fix x y
   fix \<mu> :: real
   assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
   from this[unfolded convex_on_def, rule_format]
   have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
     by auto
   from this [of "\<mu>" "1 - \<mu>", simplified] *
   show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by auto
 next
   assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow>
     f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
   {
     fix x y
     fix u v :: real
     assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
     then have[simp]: "1 - u = v" by auto
     from *[rule_format, of x y u]
     have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
       using ** by auto
   }
   then show "convex_on C f"
     unfolding convex_on_def by auto
 qed
 
 lemma convex_on_diff:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "convex_on I f"
     and I: "x \<in> I" "y \<in> I"
     and t: "x < t" "t < y"
   shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
     and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
 proof -
   define a where "a \<equiv> (t - y) / (x - y)"
   with t have "0 \<le> a" "0 \<le> 1 - a"
     by (auto simp: field_simps)
   with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y"
     by (auto simp: convex_on_def)
   have "a * x + (1 - a) * y = a * (x - y) + y"
     by (simp add: field_simps)
   also have "\<dots> = t"
     unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp
   finally have "f t \<le> a * f x + (1 - a) * f y"
     using cvx by simp
   also have "\<dots> = a * (f x - f y) + f y"
     by (simp add: field_simps)
   finally have "f t - f y \<le> a * (f x - f y)"
     by simp
   with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
     by (simp add: le_divide_eq divide_le_eq field_simps a_def)
   with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
     by (simp add: le_divide_eq divide_le_eq field_simps)
 qed
 
 lemma pos_convex_function:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
     and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x"
   shows "convex_on C f"
   unfolding convex_on_alt
   using assms
 proof safe
   fix x y \<mu> :: real
   let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y"
   assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   then have "1 - \<mu> \<ge> 0" by auto
   then have xpos: "?x \<in> C"
     using * unfolding convex_alt by fastforce
   have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
       \<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
     using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
         mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
     by auto
   then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
     by (auto simp: field_simps)
   then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
     by auto
 qed
 
 lemma atMostAtLeast_subset_convex:
   fixes C :: "real set"
   assumes "convex C"
     and "x \<in> C" "y \<in> C" "x < y"
   shows "{x .. y} \<subseteq> C"
 proof safe
   fix z assume z: "z \<in> {x .. y}"
   have less: "z \<in> C" if *: "x < z" "z < y"
   proof -
     let ?\<mu> = "(y - z) / (y - x)"
     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
       using assms * by (auto simp: field_simps)
     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
       by (simp add: algebra_simps)
     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
       by (auto simp: field_simps)
     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
       using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
     also have "\<dots> = z"
       using assms by (auto simp: field_simps)
     finally show ?thesis
       using comb by auto
   qed
   show "z \<in> C"
     using z less assms by (auto simp: le_less)
 qed
 
 lemma f''_imp_f':
   fixes f :: "real \<Rightarrow> real"
   assumes "convex C"
     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
     and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
     and x: "x \<in> C"
     and y: "y \<in> C"
   shows "f' x * (y - x) \<le> f y - f x"
   using assms
 proof -
   have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
     if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
   proof -
     from * have ge: "y - x > 0" "y - x \<ge> 0"
       by auto
     from * have le: "x - y < 0" "x - y \<le> 0"
       by auto
     then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
           THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
       by auto
     then have "z1 \<in> C"
       using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
       by fastforce
     from z1 have z1': "f x - f y = (x - y) * f' z1"
       by (simp add: field_simps)
     obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
           THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
       using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
           THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
       by auto
     have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
       using * z1' by auto
     also have "\<dots> = (y - z1) * f'' z3"
       using z3 by auto
     finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3"
       by simp
     have A': "y - z1 \<ge> 0"
       using z1 by auto
     have "z3 \<in> C"
       using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>
       by fastforce
     then have B': "f'' z3 \<ge> 0"
       using assms by auto
     from A' B' have "(y - z1) * f'' z3 \<ge> 0"
       by auto
     from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0"
       by auto
     from mult_right_mono_neg[OF this le(2)]
     have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)"
       by (simp add: algebra_simps)
     then have "f' y * (x - y) - (f x - f y) \<le> 0"
       using le by auto
     then have res: "f' y * (x - y) \<le> f x - f y"
       by auto
     have "(f y - f x) / (y - x) - f' x = f' z1 - f' x"
       using * z1 by auto
     also have "\<dots> = (z1 - x) * f'' z2"
       using z2 by auto
     finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2"
       by simp
     have A: "z1 - x \<ge> 0"
       using z1 by auto
     have "z2 \<in> C"
       using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>
       by fastforce
     then have B: "f'' z2 \<ge> 0"
       using assms by auto
     from A B have "(z1 - x) * f'' z2 \<ge> 0"
       by auto
     with cool have "(f y - f x) / (y - x) - f' x \<ge> 0"
       by auto
     from mult_right_mono[OF this ge(2)]
     have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)"
       by (simp add: algebra_simps)
     then have "f y - f x - f' x * (y - x) \<ge> 0"
       using ge by auto
     then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
       using res by auto
   qed
   show ?thesis
   proof (cases "x = y")
     case True
     with x y show ?thesis by auto
   next
     case False
     with less_imp x y show ?thesis
       by (auto simp: neq_iff)
   qed
 qed
 
 lemma f''_ge0_imp_convex:
   fixes f :: "real \<Rightarrow> real"
   assumes conv: "convex C"
     and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
     and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
-    and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
+    and 0: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
   shows "convex_on C f"
-  using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
+  using f''_imp_f'[OF conv f' f'' 0] assms pos_convex_function
   by fastforce
 
+lemma f''_le0_imp_concave:
+  fixes f :: "real \<Rightarrow> real"
+  assumes "convex C"
+    and "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
+    and "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
+    and "\<And>x. x \<in> C \<Longrightarrow> f'' x \<le> 0"
+  shows "concave_on C f"
+  unfolding concave_on_def
+  by (rule assms f''_ge0_imp_convex derivative_eq_intros | simp)+
+
+lemma log_concave:
+  fixes b :: real
+  assumes "b > 1"
+  shows "concave_on {0<..} (\<lambda> x. log b x)"
+  using assms
+  by (intro f''_le0_imp_concave derivative_eq_intros | simp)+
+
+lemma ln_concave: "concave_on {0<..} ln"
+  unfolding log_ln by (simp add: log_concave)
+
 lemma minus_log_convex:
   fixes b :: real
   assumes "b > 1"
   shows "convex_on {0 <..} (\<lambda> x. - log b x)"
-proof -
-  have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)"
-    using DERIV_log by auto
-  then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
-    by (auto simp: DERIV_minus)
-  have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
-    using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
-  from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
-  have "\<And>z::real. z > 0 \<Longrightarrow>
-    DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
-    by auto
-  then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
-    DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)"
-    unfolding inverse_eq_divide by (auto simp: mult.assoc)
-  have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
-    using \<open>b > 1\<close> by (auto intro!: less_imp_le)
-  from f''_ge0_imp_convex[OF convex_real_interval(3), unfolded greaterThan_iff, OF f' f''0 f''_ge0]
-  show ?thesis
-    by auto
-qed
+  using assms concave_on_def log_concave by blast
+
+lemma powr_convex: 
+  assumes "p \<ge> 1" shows "convex_on {0<..} (\<lambda>x. x powr p)"
+  using assms
+  by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+
+
+lemma exp_convex: "convex_on UNIV exp"
+  by (intro f''_ge0_imp_convex derivative_eq_intros | simp)+
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of real functions\<close>
 
 lemma convex_on_realI:
   assumes "connected A"
     and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
     and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
   shows "convex_on A f"
 proof (rule convex_on_linorderI)
   fix t x y :: real
   assume t: "t > 0" "t < 1"
   assume xy: "x \<in> A" "y \<in> A" "x < y"
   define z where "z = (1 - t) * x + t * y"
   with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
     using connected_contains_Icc by blast
 
   from xy t have xz: "z > x"
     by (simp add: z_def algebra_simps)
   have "y - z = (1 - t) * (y - x)"
     by (simp add: z_def algebra_simps)
   also from xy t have "\<dots> > 0"
     by (intro mult_pos_pos) simp_all
   finally have yz: "z < y"
     by simp
 
   from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
     by auto
   from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
     by auto
 
   from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
   also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
     by auto
   with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
     by (intro assms(3)) auto
   also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
   finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
     using xz yz by (simp add: field_simps)
   also have "z - x = t * (y - x)"
     by (simp add: z_def algebra_simps)
   also have "y - z = (1 - t) * (y - x)"
     by (simp add: z_def algebra_simps)
   finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
     using xy by simp
   then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
     by (simp add: z_def algebra_simps)
 qed
 
 lemma convex_on_inverse:
   assumes "A \<subseteq> {0<..}"
   shows "convex_on A (inverse :: real \<Rightarrow> real)"
 proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
   fix u v :: real
   assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
   with assms show "-inverse (u^2) \<le> -inverse (v^2)"
     by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
 qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square)
 
 lemma convex_onD_Icc':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
   shows "f c \<le> (f y - f x) / d * (c - x) + f x"
 proof (cases x y rule: linorder_cases)
   case less
   then have d: "d > 0"
     by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
     by (simp_all add: d_def field_split_simps)
   have "f c = f (x + (c - x) * 1)"
     by simp
   also from less have "1 = ((y - x) / d)"
     by (simp add: d_def)
   also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
     using assms less by (intro convex_onD_Icc) simp_all
   also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
     by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
 lemma convex_onD_Icc'':
   assumes "convex_on {x..y} f" "c \<in> {x..y}"
   defines "d \<equiv> y - x"
   shows "f c \<le> (f x - f y) / d * (y - c) + f y"
 proof (cases x y rule: linorder_cases)
   case less
   then have d: "d > 0"
     by (simp add: d_def)
   from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
     by (simp_all add: d_def field_split_simps)
   have "f c = f (y - (y - c) * 1)"
     by simp
   also from less have "1 = ((y - x) / d)"
     by (simp add: d_def)
   also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
     using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
   also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
     by (simp add: field_simps)
   finally show ?thesis .
 qed (insert assms(2), simp_all)
 
+subsection \<open>Some inequalities\<close>
+
+lemma Youngs_inequality_0:
+  fixes a::real
+  assumes "0 \<le> \<alpha>" "0 \<le> \<beta>" "\<alpha>+\<beta> = 1" "a>0" "b>0"
+  shows "a powr \<alpha> * b powr \<beta> \<le> \<alpha>*a + \<beta>*b"
+proof -
+  have "\<alpha> * ln a + \<beta> * ln b \<le> ln (\<alpha> * a + \<beta> * b)"
+    using assms ln_concave by (simp add: concave_on_iff)
+  moreover have "0 < \<alpha> * a + \<beta> * b"
+    using assms by (smt (verit) mult_pos_pos split_mult_pos_le)
+  ultimately show ?thesis
+    using assms by (simp add: powr_def mult_exp_exp flip: ln_ge_iff)
+qed
+
+lemma Youngs_inequality:
+  fixes p::real
+  assumes "p>1" "q>1" "1/p + 1/q = 1" "a\<ge>0" "b\<ge>0"
+  shows "a * b \<le> a powr p / p + b powr q / q"
+proof (cases "a=0 \<or> b=0")
+  case False
+  then show ?thesis 
+  using Youngs_inequality_0 [of "1/p" "1/q" "a powr p" "b powr q"] assms
+  by (simp add: powr_powr)
+qed (use assms in auto)
+
+lemma Cauchy_Schwarz_ineq_sum:
+  fixes a :: "'a \<Rightarrow> 'b::linordered_field"
+  shows "(\<Sum>i\<in>I. a i * b i)\<^sup>2 \<le> (\<Sum>i\<in>I. (a i)\<^sup>2) * (\<Sum>i\<in>I. (b i)\<^sup>2)"
+proof (cases "(\<Sum>i\<in>I. (b i)\<^sup>2) > 0")
+  case False
+  then consider "\<And>i. i\<in>I \<Longrightarrow> b i = 0" | "infinite I"
+    by (metis (mono_tags, lifting) sum_pos2 zero_le_power2 zero_less_power2)
+  thus ?thesis
+    by fastforce
+next
+  case True
+  define r where "r \<equiv> (\<Sum>i\<in>I. a i * b i) / (\<Sum>i\<in>I. (b i)\<^sup>2)"
+  with True have *: "(\<Sum>i\<in>I. a i * b i) = r * (\<Sum>i\<in>I. (b i)\<^sup>2)"
+    by simp
+  have "0 \<le> (\<Sum>i\<in>I. (a i - r * b i)\<^sup>2)"
+    by (meson sum_nonneg zero_le_power2)
+  also have "... = (\<Sum>i\<in>I. (a i)\<^sup>2) - 2 * r * (\<Sum>i\<in>I. a i * b i) + r\<^sup>2 * (\<Sum>i\<in>I. (b i)\<^sup>2)"
+    by (simp add: algebra_simps power2_eq_square sum_distrib_left flip: sum.distrib)
+  also have "\<dots> = (\<Sum>i\<in>I. (a i)\<^sup>2) - (\<Sum>i\<in>I. a i * b i) * r"
+    by (simp add: * power2_eq_square)
+  also have "\<dots> = (\<Sum>i\<in>I. (a i)\<^sup>2) - ((\<Sum>i\<in>I. a i * b i))\<^sup>2 / (\<Sum>i\<in>I. (b i)\<^sup>2)"
+    by (simp add: r_def power2_eq_square)
+  finally have "0 \<le> (\<Sum>i\<in>I. (a i)\<^sup>2) - ((\<Sum>i\<in>I. a i * b i))\<^sup>2 / (\<Sum>i\<in>I. (b i)\<^sup>2)" .
+  hence "((\<Sum>i\<in>I. a i * b i))\<^sup>2 / (\<Sum>i\<in>I. (b i)\<^sup>2) \<le> (\<Sum>i\<in>I. (a i)\<^sup>2)"
+    by (simp add: le_diff_eq)
+  thus "((\<Sum>i\<in>I. a i * b i))\<^sup>2 \<le> (\<Sum>i\<in>I. (a i)\<^sup>2) * (\<Sum>i\<in>I. (b i)\<^sup>2)"
+    by (simp add: pos_divide_le_eq True)
+qed
+
+subsection \<open>Misc related lemmas\<close>
+
 lemma convex_translation_eq [simp]:
   "convex ((+) a ` s) \<longleftrightarrow> convex s"
   by (metis convex_translation translation_galois)
 
 lemma convex_translation_subtract_eq [simp]:
   "convex ((\<lambda>b. b - a) ` s) \<longleftrightarrow> convex s"
   using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp)
 
 lemma convex_linear_image_eq [simp]:
     fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
     shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
     by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
 
-lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
-  unfolding linear_iff by (simp add: algebra_simps)
-
 lemma vector_choose_size:
   assumes "0 \<le> c"
   obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c"
 proof -
   obtain a::'a where "a \<noteq> 0"
     using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce
   then show ?thesis
     by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms)
 qed
 
 lemma vector_choose_dist:
   assumes "0 \<le> c"
   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'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
   shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
 proof -
   have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)"
     by auto
   show ?thesis
     unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
 qed
 
-lemma dist_triangle_eq:
-  fixes x y z :: "'a::real_inner"
-  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
-    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
-proof -
-  have *: "x - y + (y - z) = x - z" by auto
-  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
-    by (auto simp:norm_minus_commute)
-qed
-
-
-
 
 subsection \<open>Cones\<close>
 
 definition\<^marker>\<open>tag important\<close> cone :: "'a::real_vector set \<Rightarrow> bool"
   where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
 
 lemma cone_empty[intro, simp]: "cone {}"
   unfolding cone_def by auto
 
 lemma cone_univ[intro, simp]: "cone UNIV"
   unfolding cone_def by auto
 
 lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)"
   unfolding cone_def by auto
 
 lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S"
   by (simp add: cone_def subspace_scale)
 
 
 subsubsection \<open>Conic hull\<close>
 
 lemma cone_cone_hull: "cone (cone hull S)"
   unfolding hull_def by auto
 
 lemma cone_hull_eq: "cone hull S = S \<longleftrightarrow> cone S"
   by (metis cone_cone_hull hull_same)
 
 lemma mem_cone:
   assumes "cone S" "x \<in> S" "c \<ge> 0"
   shows "c *\<^sub>R x \<in> S"
   using assms cone_def[of S] by auto
 
 lemma cone_contains_0:
   assumes "cone S"
   shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S"
   using assms mem_cone by fastforce
 
 lemma cone_0: "cone {0}"
   unfolding cone_def by auto
 
 lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)"
   unfolding cone_def by blast
 
 lemma cone_iff:
   assumes "S \<noteq> {}"
   shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
 proof -
   {
     assume "cone S"
     {
       fix c :: real
       assume "c > 0"
       {
         fix x
         assume "x \<in> S"
         then have "x \<in> ((*\<^sub>R) c) ` S"
           unfolding image_def
           using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"]
             exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"]
           by auto
       }
       moreover
       {
         fix x
         assume "x \<in> ((*\<^sub>R) c) ` S"
         then have "x \<in> S"
           using \<open>0 < c\<close> \<open>cone S\<close> mem_cone by fastforce
       }
       ultimately have "((*\<^sub>R) c) ` S = S" by blast
     }
     then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
       using \<open>cone S\<close> cone_contains_0[of S] assms by auto
   }
   moreover
   {
     assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)"
     {
       fix x
       assume "x \<in> S"
       fix c1 :: real
       assume "c1 \<ge> 0"
       then have "c1 = 0 \<or> c1 > 0" by auto
       then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto
     }
     then have "cone S" unfolding cone_def by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma cone_hull_empty: "cone hull {} = {}"
   by (metis cone_empty cone_hull_eq)
 
 lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}"
   by (metis bot_least cone_hull_empty hull_subset xtrans(5))
 
 lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S"
   using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S]
   by auto
 
 lemma mem_cone_hull:
   assumes "x \<in> S" "c \<ge> 0"
   shows "c *\<^sub>R x \<in> cone hull S"
   by (metis assms cone_cone_hull hull_inc mem_cone)
 
 proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
       by auto
     fix c :: real
     assume c: "c \<ge> 0"
     then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx"
       using x by (simp add: algebra_simps)
     moreover
     have "c * cx \<ge> 0" using c x by auto
     ultimately
     have "c *\<^sub>R x \<in> ?rhs" using x by auto
   }
   then have "cone ?rhs"
     unfolding cone_def by auto
   then have "?rhs \<in> Collect cone"
     unfolding mem_Collect_eq by auto
   {
     fix x
     assume "x \<in> S"
     then have "1 *\<^sub>R x \<in> ?rhs"
       using zero_le_one by blast
     then have "x \<in> ?rhs" by auto
   }
   then have "S \<subseteq> ?rhs" by auto
   then have "?lhs \<subseteq> ?rhs"
     using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
       by auto
     then have "xx \<in> cone hull S"
       using hull_subset[of S] by auto
     then have "x \<in> ?lhs"
       using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
   }
   ultimately show ?thesis by auto
 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
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close>
 
 lemma convex_connected:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S"
   shows "connected S"
 proof (rule connectedI)
   fix A B
   assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
   moreover
   assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}"
   then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto
   define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u
   then have "continuous_on {0 .. 1} f"
     by (auto intro!: continuous_intros)
   then have "connected (f ` {0 .. 1})"
     by (auto intro!: connected_continuous_image)
   note connectedD[OF this, of A B]
   moreover have "a \<in> A \<inter> f ` {0 .. 1}"
     using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def)
   moreover have "b \<in> B \<inter> f ` {0 .. 1}"
     using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def)
   moreover have "f ` {0 .. 1} \<subseteq> S"
     using \<open>convex S\<close> a b unfolding convex_def f_def by auto
   ultimately show False by auto
 qed
 
 corollary%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
 by (simp add: convex_connected)
 
 lemma convex_prod:
   assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}"
   shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}"
   using assms unfolding convex_def
   by (auto simp: inner_add_left)
 
 lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}"
 by (rule convex_prod) (simp flip: atLeast_def)
 
 subsection \<open>Convex hull\<close>
 
 lemma convex_convex_hull [iff]: "convex (convex hull s)"
   unfolding hull_def
   using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"]
   by auto
 
 lemma convex_hull_subset:
     "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
   by (simp add: subset_hull)
 
 lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
   by (metis convex_convex_hull hull_same)
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Convex hull is "preserved" by a linear function\<close>
 
 lemma convex_hull_linear_image:
   assumes f: "linear f"
   shows "f ` (convex hull s) = convex hull (f ` s)"
 proof
   show "convex hull (f ` s) \<subseteq> f ` (convex hull s)"
     by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull)
   show "f ` (convex hull s) \<subseteq> convex hull (f ` s)"
   proof (unfold image_subset_iff_subset_vimage, rule hull_minimal)
     show "s \<subseteq> f -` (convex hull (f ` s))"
       by (fast intro: hull_inc)
     show "convex (f -` (convex hull (f ` s)))"
       by (intro convex_linear_vimage [OF f] convex_convex_hull)
   qed
 qed
 
 lemma in_convex_hull_linear_image:
   assumes "linear f"
     and "x \<in> convex hull s"
   shows "f x \<in> convex hull (f ` s)"
   using convex_hull_linear_image[OF assms(1)] assms(2) by auto
 
 lemma convex_hull_Times:
   "convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)"
 proof
   show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)"
     by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull)
   have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y
   proof (rule hull_induct [OF x], rule hull_induct [OF y])
     fix x y assume "x \<in> s" and "y \<in> t"
     then show "(x, y) \<in> convex hull (s \<times> t)"
       by (simp add: hull_inc)
   next
     fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))"
     have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
     also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}"
       by (auto simp: image_def Bex_def)
     finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" .
   next
     show "convex {x. (x, y) \<in> convex hull s \<times> t}"
     proof -
       fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))"
       have "convex ?S"
       by (intro convex_linear_vimage convex_translation convex_convex_hull,
         simp add: linear_iff)
       also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}"
         by (auto simp: image_def Bex_def)
       finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" .
     qed
   qed
   then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)"
     unfolding subset_eq split_paired_Ball_Sigma by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems for convex hulls of finite sets\<close>
 
 lemma convex_hull_empty[simp]: "convex hull {} = {}"
   by (rule hull_unique) auto
 
 lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
   by (rule hull_unique) auto
 
 lemma convex_hull_insert:
   fixes S :: "'a::real_vector set"
   assumes "S \<noteq> {}"
   shows "convex hull (insert a S) =
          {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}"
   (is "_ = ?hull")
 proof (intro equalityI hull_minimal subsetI)
   fix x
   assume "x \<in> insert a S"
   then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)"
   unfolding insert_iff
   proof
     assume "x = a"
     then show ?thesis
       by (rule_tac x=1 in exI) (use assms hull_subset in fastforce)
   next
     assume "x \<in> S"
     with hull_subset[of S convex] show ?thesis
       by force
   qed
   then show "x \<in> ?hull"
     by simp
 next
   fix x
   assume "x \<in> ?hull"
   then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b"
     by auto
   have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S"
     using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4)
     by auto
   then show "x \<in> convex hull insert a S"
     unfolding obt(5) using obt(1-3)
     by (rule convexD [OF convex_convex_hull])
 next
   show "convex ?hull"
   proof (rule convexI)
     fix x y u v
     assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull"
     from x obtain u1 v1 b1 where
       obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1"
       by auto
     from y obtain u2 v2 b2 where
       obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2"
       by auto
     have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
       by (auto simp: algebra_simps)
     have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y =
       (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
     proof (cases "u * v1 + v * v2 = 0")
       case True
       have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x"
         by (auto simp: algebra_simps)
       have eq0: "u * v1 = 0" "v * v2 = 0"
         using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>]
         by arith+
       then have "u * u1 + v * u2 = 1"
         using as(3) obt1(3) obt2(3) by auto
       then show ?thesis
         using "*" eq0 as obt1(4) xeq yeq by auto
     next
       case False
       have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)"
         using as(3) obt1(3) obt2(3) by (auto simp: field_simps)
       also have "\<dots> = u * v1 + v * v2"
         by simp
       finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
       let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2"
       have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2"
         using as(1,2) obt1(1,2) obt2(1,2) by auto
       show ?thesis
       proof
         show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)"
           unfolding xeq yeq * **
           using False by (auto simp: scaleR_left_distrib scaleR_right_distrib)
         show "?b \<in> convex hull S"
           using False zeroes obt1(4) obt2(4)
           by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib  add_divide_distrib[symmetric]  zero_le_divide_iff)
       qed
     qed
     then obtain b where b: "b \<in> convex hull S" 
        "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" ..
 
     have u1: "u1 \<le> 1"
       unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto
     have u2: "u2 \<le> 1"
       unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto
     have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v"
     proof (rule add_mono)
       show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v"
         by (simp_all add: as mult_right_mono)
     qed
     also have "\<dots> \<le> 1"
       unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto
     finally have le1: "u1 * u + u2 * v \<le> 1" .    
     show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
     proof (intro CollectI exI conjI)
       show "0 \<le> u * u1 + v * u2"
         by (simp add: as(1) as(2) obt1(1) obt2(1))
       show "0 \<le> 1 - u * u1 - v * u2"
         by (simp add: le1 diff_diff_add mult.commute)
     qed (use b in \<open>auto simp: algebra_simps\<close>)
   qed
 qed
 
 lemma convex_hull_insert_alt:
    "convex hull (insert a S) =
      (if S = {} then {a}
       else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
   apply (auto simp: convex_hull_insert)
   using diff_eq_eq apply fastforce
   using diff_add_cancel diff_ge_0_iff_ge by blast
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expression for convex hull\<close>
 
 proposition convex_hull_indexed:
   fixes S :: "'a::real_vector set"
   shows "convex hull S =
     {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and>
                 (sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
     (is "?xyz = ?hull")
 proof (rule hull_unique [OF _ convexI])
   show "S \<subseteq> ?hull" 
     by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto)
 next
   fix T
   assume "S \<subseteq> T" "convex T"
   then show "?hull \<subseteq> T"
     by (blast intro: convex_sum)
 next
   fix x y u v
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
   assume xy: "x \<in> ?hull" "y \<in> ?hull"
   from xy obtain k1 u1 x1 where
     x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S" 
                       "sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x"
     by auto
   from xy obtain k2 u2 x2 where
     y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S" 
                      "sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y"
     by auto
   have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)"
           "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
     by auto
   have inj: "inj_on (\<lambda>i. i + k1) {1..k2}"
     unfolding inj_on_def by auto
   let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)"
   let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)"
   show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull"
   proof (intro CollectI exI conjI ballI)
     show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i
       using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1))
     show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1"  "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y"
       unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]]
         sum.reindex[OF inj] Collect_mem_eq o_def
       unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric]
       by (simp_all add: sum_distrib_left[symmetric]  x(2,3) y(2,3) uv(3))
   qed 
 qed
 
 lemma convex_hull_finite:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
   (is "?HULL = _")
 proof (rule hull_unique [OF _ convexI]; clarify)
   fix x
   assume "x \<in> S"
   then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x"
     by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms])
 next
   fix u v :: real
   assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1"
   fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)"
   fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)"
   have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x
     by (simp add: that uv ux(1) uy(1))
   moreover
   have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1"
     unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2)
     using uv(3) by auto
   moreover
   have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
     unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
     by auto
   ultimately
   show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and>
              (\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)"
     by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto)
 qed (use assms in \<open>auto simp: convex_explicit\<close>)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Another formulation\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma convex_hull_explicit:
   fixes p :: "'a::real_vector set"
   shows "convex hull p =
     {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x\<in>?lhs"
     then obtain k u y where
         obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
       unfolding convex_hull_indexed by auto
 
     have fin: "finite {1..k}" by auto
     have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
     {
       fix j
       assume "j\<in>{1..k}"
       then have "y j \<in> p \<and> 0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
         using obt(1)[THEN bspec[where x=j]] and obt(2)
         by (metis (no_types, lifting) One_nat_def atLeastAtMost_iff mem_Collect_eq obt(1) sum_nonneg)
     }
     moreover
     have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1"
       unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto
     moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
       using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric]
       unfolding scaleR_left.sum using obt(3) by auto
     ultimately
     have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
       apply (rule_tac x="y ` {1..k}" in exI)
       apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto)
       done
     then have "x\<in>?rhs" by auto
   }
   moreover
   {
     fix y
     assume "y\<in>?rhs"
     then obtain S u where
       obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
       by auto
 
     obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S"
       using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
     {
       fix i :: nat
       assume "i\<in>{1..card S}"
       then have "f i \<in> S"
         using f(2) by blast
       then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto
     }
     moreover have *: "finite {1..card S}" by auto
     {
       fix y
       assume "y\<in>S"
       then obtain i where "i\<in>{1..card S}" "f i = y"
         using f using image_iff[of y f "{1..card S}"]
         by auto
       then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}"
         using f(1) inj_onD by fastforce
       then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto
       then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y"
           "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
         by (auto simp: sum_constant_scaleR)
     }
     then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y"
       unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f]
         and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f]
       unfolding f
       using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
       using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u]
       unfolding obt(4,5)
       by auto
     ultimately
     have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and>
         (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
       apply (rule_tac x="card S" in exI)
       apply (rule_tac x="u \<circ> f" in exI)
       apply (rule_tac x=f in exI, fastforce)
       done
     then have "y \<in> ?lhs"
       unfolding convex_hull_indexed by auto
   }
   ultimately show ?thesis
     unfolding set_eq_iff by blast
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>A stepping theorem for that expansion\<close>
 
 lemma convex_hull_finite_step:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows
     "(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y)
       \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)"
   (is "?lhs = ?rhs")
 proof (cases "a \<in> S")
   case True
   then have *: "insert a S = S" by auto
   show ?thesis
   proof
     assume ?lhs
     then show ?rhs
       unfolding * by force
   next
     have fin: "finite (insert a S)" using assms by auto
     assume ?rhs
     then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
       by auto
     then show ?lhs
       using uv True assms
       apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
       apply (auto simp: sum_clauses scaleR_left_distrib sum.distrib sum_delta''[OF fin])
       done
   qed
 next
   case False
   show ?thesis
   proof
     assume ?lhs
     then obtain u where u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
       by auto
     then show ?rhs
       using u \<open>a\<notin>S\<close> by (rule_tac x="u a" in exI) (auto simp: sum_clauses assms)
   next
     assume ?rhs
     then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
       by auto
     moreover
     have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S"  "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
       using False by (auto intro!: sum.cong)
     ultimately show ?lhs
       using False by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
   qed
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence some special cases\<close>
 
 lemma convex_hull_2: "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
        (is "?lhs = ?rhs")
 proof -
   have **: "finite {b}" by auto
   have "\<And>x v u. \<lbrakk>0 \<le> v; v \<le> 1; (1 - v) *\<^sub>R b = x - v *\<^sub>R a\<rbrakk>
                 \<Longrightarrow> \<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1"
     by (metis add.commute diff_add_cancel diff_ge_0_iff_ge)
   moreover
   have "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk>
                \<Longrightarrow> \<exists>p\<ge>0. \<exists>q. 0 \<le> q b \<and> q b = 1 - p \<and> q b *\<^sub>R b = u *\<^sub>R a + v *\<^sub>R b - p *\<^sub>R a"
     apply (rule_tac x=u in exI, simp)
     apply (rule_tac x="\<lambda>x. v" in exI, simp)
     done
   ultimately show ?thesis
     using convex_hull_finite_step[OF **, of a 1]
     by (auto simp add: convex_hull_finite)
 qed
 
 lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
   unfolding convex_hull_2
 proof (rule Collect_cong)
   have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y"
     by auto
   fix x
   show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow>
     (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
     apply (simp add: *)
     by (rule ex_cong1) (auto simp: algebra_simps)
 qed
 
 lemma convex_hull_3:
   "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
 proof -
   have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}"
     by auto
   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
     by (auto simp: field_simps)
   show ?thesis
     unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and *
     unfolding convex_hull_finite_step[OF fin(3)]
     apply (rule Collect_cong, simp)
     apply auto
     apply (rule_tac x=va in exI)
     apply (rule_tac x="u c" in exI, simp)
     apply (rule_tac x="1 - v - w" in exI, simp)
     apply (rule_tac x=v in exI, simp)
     apply (rule_tac x="\<lambda>x. w" in exI, simp)
     done
 qed
 
 lemma convex_hull_3_alt:
   "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
 proof -
   have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
     by auto
   show ?thesis
     unfolding convex_hull_3
     apply (auto simp: *)
     apply (rule_tac x=v in exI)
     apply (rule_tac x=w in exI)
     apply (simp add: algebra_simps)
     apply (rule_tac x=u in exI)
     apply (rule_tac x=v in exI)
     apply (simp add: algebra_simps)
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relations among closure notions and corresponding hulls\<close>
 
 lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
   unfolding affine_def convex_def by auto
 
 lemma convex_affine_hull [simp]: "convex (affine hull S)"
   by (simp add: affine_imp_convex)
 
 lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s"
   using subspace_imp_affine affine_imp_convex by auto
 
 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 aff_dim_convex_hull:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (convex hull S) = aff_dim S"
   using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S]
     hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"]
     aff_dim_subset[of "convex hull S" "affine hull S"]
   by auto
 
 
 subsection \<open>Caratheodory's theorem\<close>
 
 lemma convex_hull_caratheodory_aff_dim:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
     {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
         (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
 proof (intro allI iffI)
   fix y
   let ?P = "\<lambda>n. \<exists>S u. finite S \<and> card S = n \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and>
     sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
   assume "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
   then obtain N where "?P N" by auto
   then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n"
     by (rule_tac ex_least_nat_le, auto)
   then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k"
     by blast
   then obtain S u where obt: "finite S" "card S = n" "S\<subseteq>p" "\<forall>x\<in>S. 0 \<le> u x"
     "sum u S = 1"  "(\<Sum>v\<in>S. u v *\<^sub>R v) = y" by auto
 
   have "card S \<le> aff_dim p + 1"
   proof (rule ccontr, simp only: not_le)
     assume "aff_dim p + 1 < card S"
     then have "affine_dependent S"
       using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3)
       by blast
     then obtain w v where wv: "sum w S = 0" "v\<in>S" "w v \<noteq> 0" "(\<Sum>v\<in>S. w v *\<^sub>R v) = 0"
       using affine_dependent_explicit_finite[OF obt(1)] by auto
     define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>S. w v < 0}"
     define t where "t = Min i"
     have "\<exists>x\<in>S. w x < 0"
     proof (rule ccontr, simp add: not_less)
       assume as:"\<forall>x\<in>S. 0 \<le> w x"
       then have "sum w (S - {v}) \<ge> 0"
         by (meson Diff_iff sum_nonneg)
       then have "sum w S > 0"
         using as obt(1) sum_nonneg_eq_0_iff wv by blast
       then show False using wv(1) by auto
     qed
     then have "i \<noteq> {}" unfolding i_def by auto
     then have "t \<ge> 0"
       using Min_ge_iff[of i 0] and obt(1)
       unfolding t_def i_def
       using obt(4)[unfolded le_less]
       by (auto simp: divide_le_0_iff)
     have t: "\<forall>v\<in>S. u v + t * w v \<ge> 0"
     proof
       fix v
       assume "v \<in> S"
       then have v: "0 \<le> u v"
         using obt(4)[THEN bspec[where x=v]] by auto
       show "0 \<le> u v + t * w v"
       proof (cases "w v < 0")
         case False
         thus ?thesis using v \<open>t\<ge>0\<close> by auto
       next
         case True
         then have "t \<le> u v / (- w v)"
           using \<open>v\<in>S\<close> obt unfolding t_def i_def by (auto intro: Min_le)
         then show ?thesis
           unfolding real_0_le_add_iff
           using True neg_le_minus_divide_eq by auto
       qed
     qed
     obtain a where "a \<in> S" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
       using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto
     then have a: "a \<in> S" "u a + t * w a = 0" by auto
     have *: "\<And>f. sum f (S - {a}) = sum f S - ((f a)::'b::ab_group_add)"
       unfolding sum.remove[OF obt(1) \<open>a\<in>S\<close>] by auto
     have "(\<Sum>v\<in>S. u v + t * w v) = 1"
       unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto
     moreover have "(\<Sum>v\<in>S. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
       unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4)
       using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
     ultimately have "?P (n - 1)"
       apply (rule_tac x="(S - {a})" in exI)
       apply (rule_tac x="\<lambda>v. u v + t * w v" in exI)
       using obt(1-3) and t and a
       apply (auto simp: * scaleR_left_distrib)
       done
     then show False
       using smallest[THEN spec[where x="n - 1"]] by auto
   qed
   then show "\<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and>
       (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y"
     using obt by auto
 qed auto
 
 lemma caratheodory_aff_dim:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and> x \<in> convex hull S}"
         (is "?lhs = ?rhs")
 proof
   have "\<And>x S u. \<lbrakk>finite S; S \<subseteq> p; int (card S) \<le> aff_dim p + 1; \<forall>x\<in>S. 0 \<le> u x; sum u S = 1\<rbrakk>
                 \<Longrightarrow> (\<Sum>v\<in>S. u v *\<^sub>R v) \<in> convex hull S"
     by (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
   then show "?lhs \<subseteq> ?rhs"
     by (subst convex_hull_caratheodory_aff_dim, auto)
 qed (use hull_mono in auto)
 
 lemma convex_hull_caratheodory:
   fixes p :: "('a::euclidean_space) set"
   shows "convex hull p =
             {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and>
               (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
         (is "?lhs = ?rhs")
 proof (intro set_eqI iffI)
   fix x
   assume "x \<in> ?lhs" then show "x \<in> ?rhs"
     unfolding convex_hull_caratheodory_aff_dim 
     using aff_dim_le_DIM [of p] by fastforce
 qed (auto simp: convex_hull_explicit)
 
 theorem caratheodory:
   "convex hull p =
     {x::'a::euclidean_space. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}"
 proof safe
   fix x
   assume "x \<in> convex hull p"
   then obtain S u where "finite S" "S \<subseteq> p" "card S \<le> DIM('a) + 1"
     "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = x"
     unfolding convex_hull_caratheodory by auto
   then show "\<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S"
     using convex_hull_finite by fastforce
 qed (use hull_mono in force)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some Properties of subset of standard basis\<close>
 
 lemma affine_hull_substd_basis:
   assumes "d \<subseteq> Basis"
   shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   (is "affine hull (insert 0 ?A) = ?B")
 proof -
   have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A"
     by auto
   show ?thesis
     unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * ..
 qed
 
 lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S"
   by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Moving and scaling convex hulls\<close>
 
 lemma convex_hull_set_plus:
   "convex hull (S + T) = convex hull S + convex hull T"
   unfolding set_plus_image 
   apply (subst convex_hull_linear_image [symmetric])
   apply (simp add: linear_iff scaleR_right_distrib)
   apply (simp add: convex_hull_Times)
   done
 
 lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T"
   unfolding set_plus_def by auto
 
 lemma convex_hull_translation:
   "convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)"
   unfolding translation_eq_singleton_plus
   by (simp only: convex_hull_set_plus convex_hull_singleton)
 
 lemma convex_hull_scaling:
   "convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)"
   using linear_scaleR by (rule convex_hull_linear_image [symmetric])
 
 lemma convex_hull_affinity:
   "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)"
   by (metis convex_hull_scaling convex_hull_translation image_image)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of cone hulls\<close>
 
 lemma convex_cone_hull:
   assumes "convex S"
   shows "convex (cone hull S)"
 proof (rule convexI)
   fix x y
   assume xy: "x \<in> cone hull S" "y \<in> cone hull S"
   then have "S \<noteq> {}"
     using cone_hull_empty_iff[of S] by auto
   fix u v :: real
   assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1"
   then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S"
     using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto
   from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S"
     using cone_hull_expl[of S] by auto
   from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S"
     using cone_hull_expl[of S] by auto
   {
     assume "cx + cy \<le> 0"
     then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0"
       using x y by auto
     then have "u *\<^sub>R x + v *\<^sub>R y = 0"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
       using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto
   }
   moreover
   {
     assume "cx + cy > 0"
     then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S"
       using assms mem_convex_alt[of S xx yy cx cy] x y by auto
     then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S"
       using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close>
       by (auto simp: scaleR_right_distrib)
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S"
       using x y by auto
   }
   moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto
   ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast
 qed
 
 lemma cone_convex_hull:
   assumes "cone S"
   shows "cone (convex hull S)"
 proof (cases "S = {}")
   case True
   then show ?thesis by auto
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   {
     fix c :: real
     assume "c > 0"
     then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)"
       using convex_hull_scaling[of _ S] by auto
     also have "\<dots> = convex hull S"
       using * \<open>c > 0\<close> by auto
     finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S"
       by auto
   }
   then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)"
     using * hull_subset[of S convex] by auto
   then show ?thesis
     using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto
 qed
 
 subsection \<open>Radon's theorem\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma Radon_ex_lemma:
   assumes "finite c" "affine_dependent c"
   shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0"
 proof -
   from assms(2)[unfolded affine_dependent_explicit]
   obtain S u where
       "finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by blast
   then show ?thesis
     apply (rule_tac x="\<lambda>v. if v\<in>S then u v else 0" in exI)
     unfolding if_smult scaleR_zero_left 
     by (auto simp: Int_absorb1 sum.inter_restrict[OF \<open>finite c\<close>, symmetric])
 qed
 
 lemma Radon_s_lemma:
   assumes "finite S"
     and "sum f S = (0::real)"
   shows "sum f {x\<in>S. 0 < f x} = - sum f {x\<in>S. f x < 0}"
 proof -
   have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x"
     by auto
   show ?thesis
     unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)]
       and sum.distrib[symmetric] and *
     using assms(2)
     by assumption
 qed
 
 lemma Radon_v_lemma:
   assumes "finite S"
     and "sum f S = 0"
     and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
   shows "(sum f {x\<in>S. 0 < g x}) = - sum f {x\<in>S. g x < 0}"
 proof -
   have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x"
     using assms(3) by auto
   show ?thesis
     unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)]
       and sum.distrib[symmetric] and *
     using assms(2)
     apply assumption
     done
 qed
 
 lemma Radon_partition:
   assumes "finite C" "affine_dependent C"
   shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = C \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}"
 proof -
   obtain u v where uv: "sum u C = 0" "v\<in>C" "u v \<noteq> 0"  "(\<Sum>v\<in>C. u v *\<^sub>R v) = 0"
     using Radon_ex_lemma[OF assms] by auto
   have fin: "finite {x \<in> C. 0 < u x}" "finite {x \<in> C. 0 > u x}"
     using assms(1) by auto
   define z  where "z = inverse (sum u {x\<in>C. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>C. u x > 0}"
   have "sum u {x \<in> C. 0 < u x} \<noteq> 0"
   proof (cases "u v \<ge> 0")
     case False
     then have "u v < 0" by auto
     then show ?thesis
     proof (cases "\<exists>w\<in>{x \<in> C. 0 < u x}. u w > 0")
       case True
       then show ?thesis
         using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
     next
       case False
       then have "sum u C \<le> sum (\<lambda>x. if x=v then u v else 0) C"
         by (rule_tac sum_mono, auto)
       then show ?thesis
         unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto
     qed
   qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
 
   then have *: "sum u {x\<in>C. u x > 0} > 0"
     unfolding less_le by (metis (no_types, lifting) mem_Collect_eq sum_nonneg)
   moreover have "sum u ({x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}) = sum u C"
     "(\<Sum>x\<in>{x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>C. u x *\<^sub>R x)"
     using assms(1)
     by (rule_tac[!] sum.mono_neutral_left, auto)
   then have "sum u {x \<in> C. 0 < u x} = - sum u {x \<in> C. 0 > u x}"
     "(\<Sum>x\<in>{x \<in> C. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> C. 0 > u x}. u x *\<^sub>R x)"
     unfolding eq_neg_iff_add_eq_0
     using uv(1,4)
     by (auto simp: sum.union_inter_neutral[OF fin, symmetric])
   moreover have "\<forall>x\<in>{v \<in> C. u v < 0}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * - u x"
     using * by (fastforce intro: mult_nonneg_nonneg)
   ultimately have "z \<in> convex hull {v \<in> C. u v \<le> 0}"
     unfolding convex_hull_explicit mem_Collect_eq
     apply (rule_tac x="{v \<in> C. u v < 0}" in exI)
     apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * - u y" in exI)
     using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] 
     by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
   moreover have "\<forall>x\<in>{v \<in> C. 0 < u v}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * u x"
     using * by (fastforce intro: mult_nonneg_nonneg)
   then have "z \<in> convex hull {v \<in> C. u v > 0}"
     unfolding convex_hull_explicit mem_Collect_eq
     apply (rule_tac x="{v \<in> C. 0 < u v}" in exI)
     apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * u y" in exI)
     using assms(1)
     unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric]
     using * by (auto simp: z_def sum_negf sum_distrib_left[symmetric])
   ultimately show ?thesis
     apply (rule_tac x="{v\<in>C. u v \<le> 0}" in exI)
     apply (rule_tac x="{v\<in>C. u v > 0}" in exI, auto)
     done
 qed
 
 theorem Radon:
   assumes "affine_dependent c"
   obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
 proof -
   from assms[unfolded affine_dependent_explicit]
   obtain S u where
       "finite S" "S \<subseteq> c" "sum u S = 0" "\<exists>v\<in>S. u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by blast
   then have *: "finite S" "affine_dependent S" and S: "S \<subseteq> c"
     unfolding affine_dependent_explicit by auto
   from Radon_partition[OF *]
   obtain m p where "m \<inter> p = {}" "m \<union> p = S" "convex hull m \<inter> convex hull p \<noteq> {}"
     by blast
   with S show ?thesis
     by (force intro: that[of p m])
 qed
 
 
 subsection \<open>Helly's theorem\<close>
 
 lemma Helly_induct:
   fixes f :: "'a::euclidean_space set set"
   assumes "card f = n"
     and "n \<ge> DIM('a) + 1"
     and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
   using assms
 proof (induction n arbitrary: f)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   have "finite f"
     using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite)
   show "\<Inter>f \<noteq> {}"
   proof (cases "n = DIM('a)")
     case True
     then show ?thesis
       by (simp add: Suc.prems(1) Suc.prems(4))
   next
     case False
     have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s
     proof (rule Suc.IH[rule_format])
       show "card (f - {s}) = n"
         by (simp add: Suc.prems(1) \<open>finite f\<close> that)
       show "DIM('a) + 1 \<le> n"
         using False Suc.prems(2) by linarith
       show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
         by (simp add: Suc.prems(4) subset_Diff_insert)
     qed (use Suc in auto)
     then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})"
       by blast
     then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})"
       by metis
     show ?thesis
     proof (cases "inj_on X f")
       case False
       then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t"
         unfolding inj_on_def by auto
       then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto
       show ?thesis
         by (metis "*" X disjoint_iff_not_equal st)
     next
       case True
       then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
         using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
         unfolding card_image[OF True] and \<open>card f = Suc n\<close>
         using Suc(3) \<open>finite f\<close> and False
         by auto
       have "m \<subseteq> X ` f" "p \<subseteq> X ` f"
         using mp(2) by auto
       then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f"
         unfolding subset_image_iff by auto
       then have "f \<union> (g \<union> h) = f" by auto
       then have f: "f = g \<union> h"
         using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
         unfolding mp(2)[unfolded image_Un[symmetric] gh]
         by auto
       have *: "g \<inter> h = {}"
         using gh(1) gh(2) local.mp(1) by blast
       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
         by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+
       then show ?thesis
         unfolding f using mp(3)[unfolded gh] by blast
     qed
   qed 
 qed
 
 theorem Helly:
   fixes f :: "'a::euclidean_space set set"
   assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
     and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}"
   shows "\<Inter>f \<noteq> {}"
   using Helly_induct assms by blast
 
 subsection \<open>Epigraphs of convex functions\<close>
 
 definition\<^marker>\<open>tag important\<close> "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
 
 lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y"
   unfolding epigraph_def by auto
 
 lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S"
 proof safe
   assume L: "convex (epigraph S f)"
   then show "convex_on S f"
     by (auto simp: convex_def convex_on_def epigraph_def)
   show "convex S"
     using L by (fastforce simp: convex_def convex_on_def epigraph_def)
 next
   assume "convex_on S f" "convex S"
   then show "convex (epigraph S f)"
     unfolding convex_def convex_on_def epigraph_def
     apply safe
      apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
       apply (auto intro!:mult_left_mono add_mono)
     done
 qed
 
 lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
   unfolding convex_epigraph by auto
 
 lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
   by (simp add: convex_epigraph)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Use this to derive general bound property of convex function\<close>
 
 
 lemma convex_on:
   assumes "convex S"
   shows "convex_on S f \<longleftrightarrow>
     (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow>
       f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})"
   (is "?lhs = (\<forall>k u x. ?rhs k u x)")
 proof
   assume ?lhs 
   then have \<section>: "convex {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
     by (metis assms convex_epigraph epigraph_def)
   show "\<forall>k u x. ?rhs k u x"
   proof (intro allI)
     fix k u x
     show "?rhs k u x"
       using \<section>
       unfolding  convex mem_Collect_eq fst_sum snd_sum 
       apply safe
       apply (drule_tac x=k in spec)
       apply (drule_tac x=u in spec)
       apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
       apply simp
       done
   qed
 next
   assume "\<forall>k u x. ?rhs k u x"
   then show ?lhs
   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq fst_sum snd_sum
   using assms[unfolded convex] apply clarsimp
   apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
   by (auto simp add: mult_left_mono intro: sum_mono)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within a convex hull\<close>
 
 lemma convex_on_convex_hull_bound:
   assumes "convex_on (convex hull S) f"
     and "\<forall>x\<in>S. f x \<le> b"
   shows "\<forall>x\<in> convex hull S. f x \<le> b"
 proof
   fix x
   assume "x \<in> convex hull S"
   then obtain k u v where
     u: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> S" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
     unfolding convex_hull_indexed mem_Collect_eq by auto
   have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
     using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
     unfolding sum_distrib_right[symmetric] u(2) mult_1
     using assms(2) mult_left_mono u(1) by blast
   then show "f x \<le> b"
     using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
     using hull_inc u by fastforce
 qed
 
 lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1"
   by (simp add: inner_sum_left sum.If_cases inner_Basis)
 
 lemma convex_set_plus:
   assumes "convex S" and "convex T" shows "convex (S + T)"
 proof -
   have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     using assms by (rule convex_sums)
   moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T"
     unfolding set_plus_def by auto
   finally show "convex (S + T)" .
 qed
 
 lemma convex_set_sum:
   assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)"
   shows "convex (\<Sum>i\<in>A. B i)"
 proof (cases "finite A")
   case True then show ?thesis using assms
     by induct (auto simp: convex_set_plus)
 qed auto
 
 lemma finite_set_sum:
   assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)"
   using assms by (induct set: finite, simp, simp add: finite_set_plus)
 
 lemma box_eq_set_sum_Basis:
   "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (B i))" (is "?lhs = ?rhs")
 proof -
   have "\<And>x. \<forall>i\<in>Basis. x \<bullet> i \<in> B i \<Longrightarrow>
          \<exists>s. x = sum s Basis \<and> (\<forall>i\<in>Basis. s i \<in> (\<lambda>x. x *\<^sub>R i) ` B i)"
     by (metis (mono_tags, lifting) euclidean_representation image_iff)
   moreover
   have "sum f Basis \<bullet> i \<in> B i" if "i \<in> Basis" and f: "\<forall>i\<in>Basis. f i \<in> (\<lambda>x. x *\<^sub>R i) ` B i" for i f
   proof -
     have "(\<Sum>x\<in>Basis - {i}. f x \<bullet> i) = 0"
     proof (rule sum.neutral, intro strip)
       show "f x \<bullet> i = 0" if "x \<in> Basis - {i}" for x
         using that f \<open>i \<in> Basis\<close> inner_Basis that by fastforce
     qed
     then have "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i"
       by (metis (no_types) \<open>i \<in> Basis\<close> add.right_neutral sum.remove [OF finite_Basis])
     then have "(\<Sum>x\<in>Basis. f x \<bullet> i) \<in> B i"
       using f that(1) by auto
     then show ?thesis
       by (simp add: inner_sum_left)
   qed
   ultimately show ?thesis
     by (subst set_sum_alt [OF finite_Basis]) auto
 qed
 
 lemma convex_hull_set_sum:
   "convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))"
 proof (cases "finite A")
   assume "finite A" then show ?thesis
     by (induct set: finite, simp, simp add: convex_hull_set_plus)
 qed simp
 
 
 end
\ No newline at end of file
diff --git a/src/HOL/Analysis/Linear_Algebra.thy b/src/HOL/Analysis/Linear_Algebra.thy
--- a/src/HOL/Analysis/Linear_Algebra.thy
+++ b/src/HOL/Analysis/Linear_Algebra.thy
@@ -1,1907 +1,1916 @@
 (*  Title:      HOL/Analysis/Linear_Algebra.thy
     Author:     Amine Chaieb, University of Cambridge
 *)
 
 section \<open>Elementary Linear Algebra on Euclidean Spaces\<close>
 
 theory Linear_Algebra
 imports
   Euclidean_Space
   "HOL-Library.Infinite_Set"
 begin
 
 lemma linear_simps:
   assumes "bounded_linear f"
   shows
     "f (a + b) = f a + f b"
     "f (a - b) = f a - f b"
     "f 0 = 0"
     "f (- a) = - f a"
     "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
 proof -
   interpret f: bounded_linear f by fact
   show "f (a + b) = f a + f b" by (rule f.add)
   show "f (a - b) = f a - f b" by (rule f.diff)
   show "f 0 = 0" by (rule f.zero)
   show "f (- a) = - f a" by (rule f.neg)
   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale)
 qed
 
 lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
   using finite finite_image_set by blast
 
 lemma substdbasis_expansion_unique:
   includes inner_syntax
   assumes d: "d \<subseteq> Basis"
   shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
     (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
 proof -
   have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
     by auto
   have **: "finite d"
     by (auto intro: finite_subset[OF assms])
   have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
     using d
     by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
   show ?thesis
     unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
 qed
 
 lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
   by (rule independent_mono[OF independent_Basis])
 
 lemma subset_translation_eq [simp]:
     fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
   by auto
 
 lemma translate_inj_on:
   fixes A :: "'a::ab_group_add set"
   shows "inj_on (\<lambda>x. a + x) A"
   unfolding inj_on_def by auto
 
 lemma translation_assoc:
   fixes a b :: "'a::ab_group_add"
   shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
   by auto
 
 lemma translation_invert:
   fixes a :: "'a::ab_group_add"
   assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
   shows "A = B"
 proof -
   have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
     using assms by auto
   then show ?thesis
     using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
 qed
 
 lemma translation_galois:
   fixes a :: "'a::ab_group_add"
   shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
   using translation_assoc[of "-a" a S]
   apply auto
   using translation_assoc[of a "-a" T]
   apply auto
   done
 
 lemma translation_inverse_subset:
   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
   shows "V \<le> ((\<lambda>x. a + x) ` S)"
 proof -
   {
     fix x
     assume "x \<in> V"
     then have "x-a \<in> S" using assms by auto
     then have "x \<in> {a + v |v. v \<in> S}"
       apply auto
       apply (rule exI[of _ "x-a"], simp)
       done
     then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
   }
   then show ?thesis by auto
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>More interesting properties of the norm\<close>
 
 unbundle inner_syntax
 
 text\<open>Equality of vectors in terms of \<^term>\<open>(\<bullet>)\<close> products.\<close>
 
 lemma linear_componentwise:
   fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
   assumes lf: "linear f"
   shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
 proof -
   interpret linear f by fact
   have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
     by (simp add: inner_sum_left)
   then show ?thesis
     by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
 qed
 
 lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then show ?rhs by simp
 next
   assume ?rhs
   then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
     by simp
   then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
     by (simp add: inner_diff inner_commute)
   then have "(x - y) \<bullet> (x - y) = 0"
     by (simp add: field_simps inner_diff inner_commute)
   then show "x = y" by simp
 qed
 
 lemma norm_triangle_half_r:
   "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
   using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
 
 lemma norm_triangle_half_l:
   assumes "norm (x - y) < e / 2"
     and "norm (x' - y) < e / 2"
   shows "norm (x - x') < e"
   using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
   unfolding dist_norm[symmetric] .
 
 lemma abs_triangle_half_r:
   fixes y :: "'a::linordered_field"
   shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e"
   by linarith
 
 lemma abs_triangle_half_l:
   fixes y :: "'a::linordered_field"
   assumes "abs (x - y) < e / 2"
     and "abs (x' - y) < e / 2"
   shows "abs (x - x') < e"
   using assms by linarith
 
 lemma sum_clauses:
   shows "sum f {} = 0"
     and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)"
   by (auto simp add: insert_absorb)
 
 lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
 proof
   assume "\<forall>x. x \<bullet> y = x \<bullet> z"
   then have "\<forall>x. x \<bullet> (y - z) = 0"
     by (simp add: inner_diff)
   then have "(y - z) \<bullet> (y - z) = 0" ..
   then show "y = z" by simp
 qed simp
 
 lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
 proof
   assume "\<forall>z. x \<bullet> z = y \<bullet> z"
   then have "\<forall>z. (x - y) \<bullet> z = 0"
     by (simp add: inner_diff)
   then have "(x - y) \<bullet> (x - y) = 0" ..
   then show "x = y" by simp
 qed simp
 
 subsection \<open>Substandard Basis\<close>
 
 lemma ex_card:
   assumes "n \<le> card A"
   shows "\<exists>S\<subseteq>A. card S = n"
 proof (cases "finite A")
   case True
   from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
   moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
     by (auto simp: bij_betw_def intro: subset_inj_on)
   ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
     by (auto simp: bij_betw_def card_image)
   then show ?thesis by blast
 next
   case False
   with \<open>n \<le> card A\<close> show ?thesis by force
 qed
 
 lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
   by (auto simp: subspace_def inner_add_left)
 
 lemma dim_substandard:
   assumes d: "d \<subseteq> Basis"
   shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
 proof (rule dim_unique)
   from d show "d \<subseteq> ?A"
     by (auto simp: inner_Basis)
   from d show "independent d"
     by (rule independent_mono [OF independent_Basis])
   have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
   proof -
     have "finite d"
       by (rule finite_subset [OF d finite_Basis])
     then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
       by (simp add: span_sum span_clauses)
     also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
       by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
     finally show "x \<in> span d"
       by (simp only: euclidean_representation)
   qed
   then show "?A \<subseteq> span d" by auto
 qed simp
 
 
 subsection \<open>Orthogonality\<close>
 
 definition\<^marker>\<open>tag important\<close> (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
 
 context real_inner
 begin
 
 lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0"
   by (simp add: orthogonal_def)
 
 lemma orthogonal_clauses:
   "orthogonal a 0"
   "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
   "orthogonal a x \<Longrightarrow> orthogonal a (- x)"
   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
   "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
   "orthogonal 0 a"
   "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
   "orthogonal x a \<Longrightarrow> orthogonal (- x) a"
   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
   "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
   unfolding orthogonal_def inner_add inner_diff by auto
 
 end
 
 lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
   by (simp add: orthogonal_def inner_commute)
 
 lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x"
   by (rule ext) (simp add: orthogonal_def)
 
 lemma pairwise_ortho_scaleR:
     "pairwise (\<lambda>i j. orthogonal (f i) (g j)) B
     \<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B"
   by (auto simp: pairwise_def orthogonal_clauses)
 
 lemma orthogonal_rvsum:
     "\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)"
   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
 
 lemma orthogonal_lvsum:
     "\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y"
   by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)
 
 lemma norm_add_Pythagorean:
   assumes "orthogonal a b"
     shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2"
 proof -
   from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)"
     by (simp add: algebra_simps orthogonal_def inner_commute)
   then show ?thesis
     by (simp add: power2_norm_eq_inner)
 qed
 
 lemma norm_sum_Pythagorean:
   assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I"
     shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)"
 using assms
 proof (induction I rule: finite_induct)
   case empty then show ?case by simp
 next
   case (insert x I)
   then have "orthogonal (f x) (sum f I)"
     by (metis pairwise_insert orthogonal_rvsum)
   with insert show ?case
     by (simp add: pairwise_insert norm_add_Pythagorean)
 qed
 
 
 subsection  \<open>Orthogonality of a transformation\<close>
 
 definition\<^marker>\<open>tag important\<close>  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation:
   "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
   unfolding orthogonal_transformation_def
   apply auto
   apply (erule_tac x=v in allE)+
   apply (simp add: norm_eq_sqrt_inner)
   apply (simp add: dot_norm linear_add[symmetric])
   done
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
   by (simp add: linear_iff orthogonal_transformation_def)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_orthogonal_transformation:
     "orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
   by (simp add: orthogonal_def orthogonal_transformation_def)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_compose:
    "\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
   by (auto simp: orthogonal_transformation_def linear_compose)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_neg:
   "orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
   by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
   by (simp add: linear_iff orthogonal_transformation_def)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_linear:
    "orthogonal_transformation f \<Longrightarrow> linear f"
   by (simp add: orthogonal_transformation_def)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_inj:
   "orthogonal_transformation f \<Longrightarrow> inj f"
   unfolding orthogonal_transformation_def inj_on_def
   by (metis vector_eq)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_surj:
   "orthogonal_transformation f \<Longrightarrow> surj f"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_bij:
   "orthogonal_transformation f \<Longrightarrow> bij f"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_inv:
   "orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
   for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
   by (metis (no_types, opaque_lifting) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
 
 lemma\<^marker>\<open>tag unimportant\<close>  orthogonal_transformation_norm:
   "orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
   by (metis orthogonal_transformation)
 
 
 subsection \<open>Bilinear functions\<close>
 
 definition\<^marker>\<open>tag important\<close>
 bilinear :: "('a::real_vector \<Rightarrow> 'b::real_vector \<Rightarrow> 'c::real_vector) \<Rightarrow> bool" where
 "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
 
 lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
   by (simp add: bilinear_def linear_iff)
 
 lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
   by (simp add: bilinear_def linear_iff)
 
 lemma bilinear_times:
   fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
   by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
 
 lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_iff)
 
 lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
   by (simp add: bilinear_def linear_iff)
 
 lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
   by (drule bilinear_lmul [of _ "- 1"]) simp
 
 lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
   by (drule bilinear_rmul [of _ _ "- 1"]) simp
 
 lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
   using add_left_imp_eq[of x y 0] by auto
 
 lemma bilinear_lzero:
   assumes "bilinear h"
   shows "h 0 x = 0"
   using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
 
 lemma bilinear_rzero:
   assumes "bilinear h"
   shows "h x 0 = 0"
   using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
 
 lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
   using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)
 
 lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
   using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
 
 lemma bilinear_sum:
   assumes "bilinear h"
   shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
 proof -
   interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
   interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
   have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
     by (simp add: l.sum)
   also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
     by (rule sum.cong) (simp_all add: r.sum)
   finally show ?thesis
     unfolding sum.cartesian_product .
 qed
 
 
 subsection \<open>Adjoints\<close>
 
 definition\<^marker>\<open>tag important\<close> adjoint :: "(('a::real_inner) \<Rightarrow> ('b::real_inner)) \<Rightarrow> 'b \<Rightarrow> 'a" where
 "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
 
 lemma adjoint_unique:
   assumes "\<forall>x y. inner (f x) y = inner x (g y)"
   shows "adjoint f = g"
   unfolding adjoint_def
 proof (rule some_equality)
   show "\<forall>x y. inner (f x) y = inner x (g y)"
     by (rule assms)
 next
   fix h
   assume "\<forall>x y. inner (f x) y = inner x (h y)"
   then have "\<forall>x y. inner x (g y) = inner x (h y)"
     using assms by simp
   then have "\<forall>x y. inner x (g y - h y) = 0"
     by (simp add: inner_diff_right)
   then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
     by simp
   then have "\<forall>y. h y = g y"
     by simp
   then show "h = g" by (simp add: ext)
 qed
 
 text \<open>TODO: The following lemmas about adjoints should hold for any
   Hilbert space (i.e. complete inner product space).
   (see \<^url>\<open>https://en.wikipedia.org/wiki/Hermitian_adjoint\<close>)
 \<close>
 
 lemma adjoint_works:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
   shows "x \<bullet> adjoint f y = f x \<bullet> y"
 proof -
   interpret linear f by fact
   have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
   proof (intro allI exI)
     fix y :: "'m" and x
     let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
     have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
       by (simp add: euclidean_representation)
     also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
       by (simp add: sum scale)
     finally show "f x \<bullet> y = x \<bullet> ?w"
       by (simp add: inner_sum_left inner_sum_right mult.commute)
   qed
   then show ?thesis
     unfolding adjoint_def choice_iff
     by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
 qed
 
 lemma adjoint_clauses:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
   shows "x \<bullet> adjoint f y = f x \<bullet> y"
     and "adjoint f y \<bullet> x = y \<bullet> f x"
   by (simp_all add: adjoint_works[OF lf] inner_commute)
 
 lemma adjoint_linear:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
   shows "linear (adjoint f)"
   by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]
     adjoint_clauses[OF lf] inner_distrib)
 
 lemma adjoint_adjoint:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes lf: "linear f"
   shows "adjoint (adjoint f) = f"
   by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Euclidean Spaces as Typeclass\<close>
 
 lemma independent_Basis: "independent Basis"
   by (rule independent_Basis)
 
 lemma span_Basis [simp]: "span Basis = UNIV"
   by (rule span_Basis)
 
 lemma in_span_Basis: "x \<in> span Basis"
   unfolding span_Basis ..
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Linearity and Bilinearity continued\<close>
 
 lemma linear_bounded:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes lf: "linear f"
   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
 proof
   interpret linear f by fact
   let ?B = "\<Sum>b\<in>Basis. norm (f b)"
   show "\<forall>x. norm (f x) \<le> ?B * norm x"
   proof
     fix x :: 'a
     let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
     have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
       unfolding euclidean_representation ..
     also have "\<dots> = norm (sum ?g Basis)"
       by (simp add: sum scale)
     finally have th0: "norm (f x) = norm (sum ?g Basis)" .
     have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
     proof -
       from Basis_le_norm[OF that, of x]
       show "norm (?g i) \<le> norm (f i) * norm x"
         unfolding norm_scaleR  by (metis mult.commute mult_left_mono norm_ge_zero)
     qed
     from sum_norm_le[of _ ?g, OF th]
     show "norm (f x) \<le> ?B * norm x"
       unfolding th0 sum_distrib_right by metis
   qed
 qed
 
 lemma linear_conv_bounded_linear:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   shows "linear f \<longleftrightarrow> bounded_linear f"
 proof
   assume "linear f"
   then interpret f: linear f .
   show "bounded_linear f"
   proof
     have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
       using \<open>linear f\<close> by (rule linear_bounded)
     then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
       by (simp add: mult.commute)
   qed
 next
   assume "bounded_linear f"
   then interpret f: bounded_linear f .
   show "linear f" ..
 qed
 
 lemmas linear_linear = linear_conv_bounded_linear[symmetric]
 
 lemma inj_linear_imp_inv_bounded_linear:
   fixes f::"'a::euclidean_space \<Rightarrow> 'a"
   shows "\<lbrakk>bounded_linear f; inj f\<rbrakk> \<Longrightarrow> bounded_linear (inv f)"
   by (simp add: inj_linear_imp_inv_linear linear_linear)
 
 lemma linear_bounded_pos:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes lf: "linear f"
  obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x"
 proof -
   have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B"
     using lf unfolding linear_conv_bounded_linear
     by (rule bounded_linear.pos_bounded)
   with that show ?thesis
     by (auto simp: mult.commute)
 qed
 
 lemma linear_invertible_bounded_below_pos:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "linear g" "g \<circ> f = id"
   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
 proof -
   obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x"
     using linear_bounded_pos [OF \<open>linear g\<close>] by blast
   show thesis
   proof
     show "0 < 1/B"
       by (simp add: \<open>B > 0\<close>)
     show "1/B * norm x \<le> norm (f x)" for x
     proof -
       have "1/B * norm x = 1/B * norm (g (f x))"
         using assms by (simp add: pointfree_idE)
       also have "\<dots> \<le> norm (f x)"
         using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq)
       finally show ?thesis .
     qed
   qed
 qed
 
 lemma linear_inj_bounded_below_pos:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
   obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
   using linear_injective_left_inverse [OF assms]
     linear_invertible_bounded_below_pos assms by blast
 
 lemma bounded_linearI':
   fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "\<And>x y. f (x + y) = f x + f y"
     and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
   shows "bounded_linear f"
   using assms linearI linear_conv_bounded_linear by blast
 
 lemma bilinear_bounded:
   fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
   assumes bh: "bilinear h"
   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
 proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
   fix x :: 'm
   fix y :: 'n
   have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
     by (simp add: euclidean_representation)
   also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
     unfolding bilinear_sum[OF bh] ..
   finally have th: "norm (h x y) = \<dots>" .
   have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
            \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
     by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
   then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
     unfolding sum_distrib_right th sum.cartesian_product
     by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
       field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
 qed
 
 lemma bilinear_conv_bounded_bilinear:
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
   shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
 proof
   assume "bilinear h"
   show "bounded_bilinear h"
   proof
     fix x y z
     show "h (x + y) z = h x z + h y z"
       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
   next
     fix x y z
     show "h x (y + z) = h x y + h x z"
       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
   next
     show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
       using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
       by simp_all
   next
     have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
       using \<open>bilinear h\<close> by (rule bilinear_bounded)
     then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
       by (simp add: ac_simps)
   qed
 next
   assume "bounded_bilinear h"
   then interpret h: bounded_bilinear h .
   show "bilinear h"
     unfolding bilinear_def linear_conv_bounded_linear
     using h.bounded_linear_left h.bounded_linear_right by simp
 qed
 
 lemma bilinear_bounded_pos:
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
   assumes bh: "bilinear h"
   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
 proof -
   have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B"
     using bh [unfolded bilinear_conv_bounded_bilinear]
     by (rule bounded_bilinear.pos_bounded)
   then show ?thesis
     by (simp only: ac_simps)
 qed
 
 lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
   by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
       dest: bounded_linear.linear)
 
 lemma linear_imp_has_derivative:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   shows "linear f \<Longrightarrow> (f has_derivative f) net"
   by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
 
 lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
   using bounded_linear_imp_has_derivative differentiable_def by blast
 
 lemma linear_imp_differentiable:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   shows "linear f \<Longrightarrow> f differentiable net"
   by (metis linear_imp_has_derivative differentiable_def)
 
 lemma of_real_differentiable [simp,derivative_intros]: "of_real differentiable F"
   by (simp add: bounded_linear_imp_differentiable bounded_linear_of_real)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>We continue\<close>
 
 lemma independent_bound:
   fixes S :: "'a::euclidean_space set"
   shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
   by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
 
 lemmas independent_imp_finite = finiteI_independent
 
 corollary\<^marker>\<open>tag unimportant\<close> independent_card_le:
   fixes S :: "'a::euclidean_space set"
   assumes "independent S"
   shows "card S \<le> DIM('a)"
   using assms independent_bound by auto
 
 lemma dependent_biggerset:
   fixes S :: "'a::euclidean_space set"
   shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
   by (metis independent_bound not_less)
 
 text \<open>Picking an orthogonal replacement for a spanning set.\<close>
 
 lemma vector_sub_project_orthogonal:
   fixes b x :: "'a::euclidean_space"
   shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
   unfolding inner_simps by auto
 
 lemma pairwise_orthogonal_insert:
   assumes "pairwise orthogonal S"
     and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
   shows "pairwise orthogonal (insert x S)"
   using assms unfolding pairwise_def
   by (auto simp add: orthogonal_commute)
 
 lemma basis_orthogonal:
   fixes B :: "'a::real_inner set"
   assumes fB: "finite B"
   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
   (is " \<exists>C. ?P B C")
   using fB
 proof (induct rule: finite_induct)
   case empty
   then show ?case
     apply (rule exI[where x="{}"])
     apply (auto simp add: pairwise_def)
     done
 next
   case (insert a B)
   note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
   from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
   obtain C where C: "finite C" "card C \<le> card B"
     "span C = span B" "pairwise orthogonal C" by blast
   let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
   let ?C = "insert ?a C"
   from C(1) have fC: "finite ?C"
     by simp
   from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
     by (simp add: card_insert_if)
   {
     fix x k
     have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
       by (simp add: field_simps)
     have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
       apply (simp only: scaleR_right_diff_distrib th0)
       apply (rule span_add_eq)
       apply (rule span_scale)
       apply (rule span_sum)
       apply (rule span_scale)
       apply (rule span_base)
       apply assumption
       done
   }
   then have SC: "span ?C = span (insert a B)"
     unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
   {
     fix y
     assume yC: "y \<in> C"
     then have Cy: "C = insert y (C - {y})"
       by blast
     have fth: "finite (C - {y})"
       using C by simp
     have "orthogonal ?a y"
       unfolding orthogonal_def
       unfolding inner_diff inner_sum_left right_minus_eq
       unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>]
       apply (clarsimp simp add: inner_commute[of y a])
       apply (rule sum.neutral)
       apply clarsimp
       apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
       using \<open>y \<in> C\<close> by auto
   }
   with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C"
     by (rule pairwise_orthogonal_insert)
   from fC cC SC CPO have "?P (insert a B) ?C"
     by blast
   then show ?case by blast
 qed
 
 lemma orthogonal_basis_exists:
   fixes V :: "('a::euclidean_space) set"
   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
   (card B = dim V) \<and> pairwise orthogonal B"
 proof -
   from basis_exists[of V] obtain B where
     B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
     by force
   from B have fB: "finite B" "card B = dim V"
     using independent_bound by auto
   from basis_orthogonal[OF fB(1)] obtain C where
     C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
     by blast
   from C B have CSV: "C \<subseteq> span V"
     by (metis span_superset span_mono subset_trans)
   from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
     by (simp add: span_span)
   from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
   have iC: "independent C"
     by (simp)
   from C fB have "card C \<le> dim V"
     by simp
   moreover have "dim V \<le> card C"
     using span_card_ge_dim[OF CSV SVC C(1)]
     by simp
   ultimately have CdV: "card C = dim V"
     using C(1) by simp
   from C B CSV CdV iC show ?thesis
     by auto
 qed
 
 text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
 
 lemma span_not_univ_orthogonal:
   fixes S :: "'a::euclidean_space set"
   assumes sU: "span S \<noteq> UNIV"
   shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
 proof -
   from sU obtain a where a: "a \<notin> span S"
     by blast
   from orthogonal_basis_exists obtain B where
     B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
     "card B = dim S" "pairwise orthogonal B"
     by blast
   from B have fB: "finite B" "card B = dim S"
     using independent_bound by auto
   from span_mono[OF B(2)] span_mono[OF B(3)]
   have sSB: "span S = span B"
     by (simp add: span_span)
   let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
   have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
     unfolding sSB
     apply (rule span_sum)
     apply (rule span_scale)
     apply (rule span_base)
     apply assumption
     done
   with a have a0:"?a  \<noteq> 0"
     by auto
   have "?a \<bullet> x = 0" if "x\<in>span B" for x
   proof (rule span_induct [OF that])
     show "subspace {x. ?a \<bullet> x = 0}"
       by (auto simp add: subspace_def inner_add)
   next
     {
       fix x
       assume x: "x \<in> B"
       from x have B': "B = insert x (B - {x})"
         by blast
       have fth: "finite (B - {x})"
         using fB by simp
       have "?a \<bullet> x = 0"
         apply (subst B')
         using fB fth
         unfolding sum_clauses(2)[OF fth]
         apply simp unfolding inner_simps
         apply (clarsimp simp add: inner_add inner_sum_left)
         apply (rule sum.neutral, rule ballI)
         apply (simp only: inner_commute)
         apply (auto simp add: x field_simps
           intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
         done
     }
     then show "?a \<bullet> x = 0" if "x \<in> B" for x
       using that by blast
     qed
   with a0 show ?thesis
     unfolding sSB by (auto intro: exI[where x="?a"])
 qed
 
 lemma span_not_univ_subset_hyperplane:
   fixes S :: "'a::euclidean_space set"
   assumes SU: "span S \<noteq> UNIV"
   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
   using span_not_univ_orthogonal[OF SU] by auto
 
 lemma lowdim_subset_hyperplane:
   fixes S :: "'a::euclidean_space set"
   assumes d: "dim S < DIM('a)"
   shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
 proof -
   {
     assume "span S = UNIV"
     then have "dim (span S) = dim (UNIV :: ('a) set)"
       by simp
     then have "dim S = DIM('a)"
       by (metis Euclidean_Space.dim_UNIV dim_span)
     with d have False by arith
   }
   then have th: "span S \<noteq> UNIV"
     by blast
   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
 qed
 
 lemma linear_eq_stdbasis:
   fixes f :: "'a::euclidean_space \<Rightarrow> _"
   assumes lf: "linear f"
     and lg: "linear g"
     and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
   shows "f = g"
   using linear_eq_on_span[OF lf lg, of Basis] fg
   by auto
 
 
 text \<open>Similar results for bilinear functions.\<close>
 
 lemma bilinear_eq:
   assumes bf: "bilinear f"
     and bg: "bilinear g"
     and SB: "S \<subseteq> span B"
     and TC: "T \<subseteq> span C"
     and "x\<in>S" "y\<in>T"
     and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y"
   shows "f x y = g x y"
 proof -
   let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
   from bf bg have sp: "subspace ?P"
     unfolding bilinear_def linear_iff subspace_def bf bg
     by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
         span_add Ball_def
       intro: bilinear_ladd[OF bf])
   have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}"
     apply (auto simp add: subspace_def)
     using bf bg unfolding bilinear_def linear_iff
       apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
         span_add Ball_def
       intro: bilinear_ladd[OF bf])
     done
   have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x
     apply (rule span_induct [OF that sp])
     using fg sfg span_induct by blast
   then show ?thesis
     using SB TC assms by auto
 qed
 
 lemma bilinear_eq_stdbasis:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
   assumes bf: "bilinear f"
     and bg: "bilinear g"
     and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j"
   shows "f = g"
   using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast
 
 
 subsection \<open>Infinity norm\<close>
 
 definition\<^marker>\<open>tag important\<close> "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
 
 lemma infnorm_set_image:
   fixes x :: "'a::euclidean_space"
   shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
   by blast
 
 lemma infnorm_Max:
   fixes x :: "'a::euclidean_space"
   shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
   by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
 
 lemma infnorm_set_lemma:
   fixes x :: "'a::euclidean_space"
   shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
     and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
   unfolding infnorm_set_image
   by auto
 
 lemma infnorm_pos_le:
   fixes x :: "'a::euclidean_space"
   shows "0 \<le> infnorm x"
   by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
 
 lemma infnorm_triangle:
   fixes x :: "'a::euclidean_space"
   shows "infnorm (x + y) \<le> infnorm x + infnorm y"
 proof -
   have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
     by simp
   show ?thesis
     by (auto simp: infnorm_Max inner_add_left intro!: *)
 qed
 
 lemma infnorm_eq_0:
   fixes x :: "'a::euclidean_space"
   shows "infnorm x = 0 \<longleftrightarrow> x = 0"
 proof -
   have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
     unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
   then show ?thesis
     using infnorm_pos_le[of x] by simp
 qed
 
 lemma infnorm_0: "infnorm 0 = 0"
   by (simp add: infnorm_eq_0)
 
 lemma infnorm_neg: "infnorm (- x) = infnorm x"
   unfolding infnorm_def by simp
 
 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
   by (metis infnorm_neg minus_diff_eq)
 
 lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
 proof -
   have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
     by arith
   show ?thesis
   proof (rule *)
     from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
     show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x"
       by (simp_all add: field_simps infnorm_neg)
   qed
 qed
 
 lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
   using infnorm_pos_le[of x] by arith
 
 lemma Basis_le_infnorm:
   fixes x :: "'a::euclidean_space"
   shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
   by (simp add: infnorm_Max)
 
 lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
   unfolding infnorm_Max
 proof (safe intro!: Max_eqI)
   let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
   { fix b :: 'a
     assume "b \<in> Basis"
     then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
       by (simp add: abs_mult mult_left_mono)
   next
     from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
       by (auto simp del: Max_in)
     then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
       by (intro image_eqI[where x=b]) (auto simp: abs_mult)
   }
 qed simp
 
 lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
   unfolding infnorm_mul ..
 
 lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
   using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
 
 text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
 
 lemma infnorm_le_norm: "infnorm x \<le> norm x"
   by (simp add: Basis_le_norm infnorm_Max)
 
 lemma norm_le_infnorm:
   fixes x :: "'a::euclidean_space"
   shows "norm x \<le> sqrt DIM('a) * infnorm x"
   unfolding norm_eq_sqrt_inner id_def
 proof (rule real_le_lsqrt[OF inner_ge_zero])
   show "sqrt DIM('a) * infnorm x \<ge> 0"
     by (simp add: zero_le_mult_iff infnorm_pos_le)
   have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))"
     by (metis euclidean_inner order_refl)
   also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2"
     by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
   also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2"
     by (simp add: power_mult_distrib)
   finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" .
 qed
 
 lemma tendsto_infnorm [tendsto_intros]:
   assumes "(f \<longlongrightarrow> a) F"
   shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
 proof (rule tendsto_compose [OF LIM_I assms])
   fix r :: real
   assume "r > 0"
   then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
     by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
 qed
 
 text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
 
 lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof (cases "x=0")
   case True
   then show ?thesis
     by auto
 next
   case False
   from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
   have "?rhs \<longleftrightarrow>
       (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
         norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
     using False unfolding inner_simps
     by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
   also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)"
     using False  by (simp add: field_simps inner_commute)
   also have "\<dots> \<longleftrightarrow> ?lhs"
     using False by auto
   finally show ?thesis by metis
 qed
 
 lemma norm_cauchy_schwarz_abs_eq:
   "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
     norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof -
   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
     by arith
   have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
     by simp
   also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
     unfolding norm_cauchy_schwarz_eq[symmetric]
     unfolding norm_minus_cancel norm_scaleR ..
   also have "\<dots> \<longleftrightarrow> ?lhs"
     unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
     by auto
   finally show ?thesis ..
 qed
 
 lemma norm_triangle_eq:
   fixes x y :: "'a::real_inner"
   shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
 proof (cases "x = 0 \<or> y = 0")
   case True
   then show ?thesis
     by force
 next
   case False
   then have n: "norm x > 0" "norm y > 0"
     by auto
   have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
     by simp
   also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
     unfolding norm_cauchy_schwarz_eq[symmetric]
     unfolding power2_norm_eq_inner inner_simps
     by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
   finally show ?thesis .
 qed
 
+lemma dist_triangle_eq:
+  fixes x y z :: "'a::real_inner"
+  shows "dist x z = dist x y + dist y z \<longleftrightarrow>
+    norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
+proof -
+  have *: "x - y + (y - z) = x - z" by auto
+  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
+    by (auto simp:norm_minus_commute)
+qed
 
 subsection \<open>Collinearity\<close>
 
 definition\<^marker>\<open>tag important\<close> collinear :: "'a::real_vector set \<Rightarrow> bool"
   where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
 
 lemma collinear_alt:
      "collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
 next
   assume ?rhs
   then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
     by (auto simp: )
   have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y
         by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left)
   then show ?lhs
     using collinear_def by blast
 qed
 
 lemma collinear:
   fixes S :: "'a::{perfect_space,real_vector} set"
   shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))"
 proof -
   have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)"
     if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u
   proof -
     have "\<forall>x\<in>S. \<forall>y\<in>S. x = y"
       using that by auto
     moreover
     obtain v::'a where "v \<noteq> 0"
       using UNIV_not_singleton [of 0] by auto
     ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v"
       by auto
     then show ?thesis
       using \<open>v \<noteq> 0\<close> by blast
   qed
   then show ?thesis
     apply (clarsimp simp: collinear_def)
     by (metis scaleR_zero_right vector_fraction_eq_iff)
 qed
 
 lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"
   by (meson collinear_def subsetCE)
 
 lemma collinear_empty [iff]: "collinear {}"
   by (simp add: collinear_def)
 
 lemma collinear_sing [iff]: "collinear {x}"
   by (simp add: collinear_def)
 
 lemma collinear_2 [iff]: "collinear {x, y}"
   apply (simp add: collinear_def)
   apply (rule exI[where x="x - y"])
   by (metis minus_diff_eq scaleR_left.minus scaleR_one)
 
 lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof (cases "x = 0 \<or> y = 0")
   case True
   then show ?thesis
     by (auto simp: insert_commute)
 next
   case False
   show ?thesis
   proof
     assume h: "?lhs"
     then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
       unfolding collinear_def by blast
     from u[rule_format, of x 0] u[rule_format, of y 0]
     obtain cx and cy where
       cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
       by auto
     from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto
     let ?d = "cy / cx"
     from cx cy cx0 have "y = ?d *\<^sub>R x"
       by simp
     then show ?rhs using False by blast
   next
     assume h: "?rhs"
     then obtain c where c: "y = c *\<^sub>R x"
       using False by blast
     show ?lhs
       unfolding collinear_def c
       apply (rule exI[where x=x])
       apply auto
           apply (rule exI[where x="- 1"], simp)
          apply (rule exI[where x= "-c"], simp)
         apply (rule exI[where x=1], simp)
        apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
       apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
       done
   qed
 qed
 
 lemma collinear_iff_Reals: "collinear {0::complex,w,z} \<longleftrightarrow> z/w \<in> \<real>"
 proof
   show "z/w \<in> \<real> \<Longrightarrow> collinear {0,w,z}"
     by (metis Reals_cases collinear_lemma nonzero_divide_eq_eq scaleR_conv_of_real)
 qed (auto simp: collinear_lemma scaleR_conv_of_real)
 
 lemma collinear_scaleR_iff: "collinear {0, \<alpha> *\<^sub>R w, \<beta> *\<^sub>R z} \<longleftrightarrow> collinear {0,w,z} \<or> \<alpha>=0 \<or> \<beta>=0"
   (is "?lhs = ?rhs")
 proof (cases "\<alpha>=0 \<or> \<beta>=0")
   case False
   then have "(\<exists>c. \<beta> *\<^sub>R z = (c * \<alpha>) *\<^sub>R w) = (\<exists>c. z = c *\<^sub>R w)"
     by (metis mult.commute scaleR_scaleR vector_fraction_eq_iff)
   then show ?thesis
     by (auto simp add: collinear_lemma)
 qed (auto simp: collinear_lemma)
 
 lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
 proof (cases "x=0")
   case True
   then show ?thesis
     by (auto simp: insert_commute)
 next
   case False
   then have nnz: "norm x \<noteq> 0"
     by auto
   show ?thesis
   proof
     assume "\<bar>x \<bullet> y\<bar> = norm x * norm y"
     then show "collinear {0, x, y}"
       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
       by (meson eq_vector_fraction_iff nnz)
   next
     assume "collinear {0, x, y}"
     with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y"
       unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
   qed
 qed
 
 lemma norm_triangle_eq_imp_collinear:
   fixes x y :: "'a::real_inner"
   assumes "norm (x + y) = norm x + norm y"
   shows "collinear{0,x,y}"
 proof (cases "x = 0 \<or> y = 0")
   case False
   with assms show ?thesis
     by (meson norm_cauchy_schwarz_abs_eq norm_cauchy_schwarz_equal norm_triangle_eq)
 qed (use collinear_lemma in blast)
 
 
 subsection\<open>Properties of special hyperplanes\<close>
 
 lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
   by (simp add: subspace_def inner_right_distrib)
 
 lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
   by (simp add: inner_commute inner_right_distrib subspace_def)
 
 lemma special_hyperplane_span:
   fixes S :: "'n::euclidean_space set"
   assumes "k \<in> Basis"
   shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
 proof -
   have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
   proof -
     have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
       by (simp add: euclidean_representation)
     also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
       by (auto simp: sum.remove [of _ k] inner_commute assms that)
     finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
     then show ?thesis
       by (simp add: span_finite)
   qed
   show ?thesis
     apply (rule span_subspace [symmetric])
     using assms
     apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
     done
 qed
 
 lemma dim_special_hyperplane:
   fixes k :: "'n::euclidean_space"
   shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
 apply (simp add: special_hyperplane_span)
 apply (rule dim_unique [OF subset_refl])
 apply (auto simp: independent_substdbasis)
 apply (metis member_remove remove_def span_base)
 done
 
 proposition dim_hyperplane:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
     shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
 proof -
   have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
     by (rule span_unique) (auto simp: subspace_hyperplane)
   then obtain B where "independent B"
               and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
               and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
               and card0: "(card B = dim {x. a \<bullet> x = 0})"
               and ortho: "pairwise orthogonal B"
     using orthogonal_basis_exists by metis
   with assms have "a \<notin> span B"
     by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
   then have ind: "independent (insert a B)"
     by (simp add: \<open>independent B\<close> independent_insert)
   have "finite B"
     using \<open>independent B\<close> independent_bound by blast
   have "UNIV \<subseteq> span (insert a B)"
   proof fix y::'a
     obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
       apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
       using assms
       by (auto simp: algebra_simps)
     show "y \<in> span (insert a B)"
       by (metis (mono_tags, lifting) z Bsub span_eq_iff
          add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
   qed
   then have dima: "DIM('a) = dim(insert a B)"
     by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
   then show ?thesis
     by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0
         card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
         subspB)
 qed
 
 lemma lowdim_eq_hyperplane:
   fixes S :: "'a::euclidean_space set"
   assumes "dim S = DIM('a) - 1"
   obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
 proof -
   have dimS: "dim S < DIM('a)"
     by (simp add: assms)
   then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
     using lowdim_subset_hyperplane [of S] by fastforce
   show ?thesis
     apply (rule that[OF b(1)])
     apply (rule subspace_dim_equal)
     by (auto simp: assms b dim_hyperplane subspace_hyperplane)
 qed
 
 lemma dim_eq_hyperplane:
   fixes S :: "'n::euclidean_space set"
   shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
 by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
 
 
 subsection\<open> Orthogonal bases and Gram-Schmidt process\<close>
 
 lemma pairwise_orthogonal_independent:
   assumes "pairwise orthogonal S" and "0 \<notin> S"
     shows "independent S"
 proof -
   have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     using assms by (simp add: pairwise_def orthogonal_def)
   have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
   proof -
     obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
       using a by (force simp: span_explicit)
     then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
       by simp
     also have "... = 0"
       apply (simp add: inner_sum_right)
       apply (rule comm_monoid_add_class.sum.neutral)
       by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
     finally show ?thesis
       using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
   qed
   then show ?thesis
     by (force simp: dependent_def)
 qed
 
 lemma pairwise_orthogonal_imp_finite:
   fixes S :: "'a::euclidean_space set"
   assumes "pairwise orthogonal S"
     shows "finite S"
 proof -
   have "independent (S - {0})"
     apply (rule pairwise_orthogonal_independent)
      apply (metis Diff_iff assms pairwise_def)
     by blast
   then show ?thesis
     by (meson independent_imp_finite infinite_remove)
 qed
 
 lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
   by (simp add: subspace_def orthogonal_clauses)
 
 lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
   by (simp add: subspace_def orthogonal_clauses)
 
 lemma orthogonal_to_span:
   assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
     shows "orthogonal x a"
   by (metis a orthogonal_clauses(1,2,4)
       span_induct_alt x)
 
 proposition Gram_Schmidt_step:
   fixes S :: "'a::euclidean_space set"
   assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
     shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
 proof -
   have "finite S"
     by (simp add: S pairwise_orthogonal_imp_finite)
   have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
        if "x \<in> S" for x
   proof -
     have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
       by (simp add: \<open>finite S\<close> inner_commute that)
     also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
       apply (rule sum.cong [OF refl], simp)
       by (meson S orthogonal_def pairwise_def that)
    finally show ?thesis
      by (simp add: orthogonal_def algebra_simps inner_sum_left)
   qed
   then show ?thesis
     using orthogonal_to_span orthogonal_commute x by blast
 qed
 
 
 lemma orthogonal_extension_aux:
   fixes S :: "'a::euclidean_space set"
   assumes "finite T" "finite S" "pairwise orthogonal S"
     shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
 using assms
 proof (induction arbitrary: S)
   case empty then show ?case
     by simp (metis sup_bot_right)
 next
   case (insert a T)
   have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     using insert by (simp add: pairwise_def orthogonal_def)
   define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
   obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
              and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
     by (rule exE [OF insert.IH [of "insert a' S"]])
       (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
         pairwise_orthogonal_insert span_clauses)
   have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
     apply (simp add: a'_def)
     using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
     apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
     done
   have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
     using spanU by simp
   also have "... = span (insert a (S \<union> T))"
     apply (rule eq_span_insert_eq)
     apply (simp add: a'_def span_neg span_sum span_base span_mul)
     done
   also have "... = span (S \<union> insert a T)"
     by simp
   finally show ?case
     by (rule_tac x="insert a' U" in exI) (use orthU in auto)
 qed
 
 
 proposition orthogonal_extension:
   fixes S :: "'a::euclidean_space set"
   assumes S: "pairwise orthogonal S"
   obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
 proof -
   obtain B where "finite B" "span B = span T"
     using basis_subspace_exists [of "span T"] subspace_span by metis
   with orthogonal_extension_aux [of B S]
   obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
     using assms pairwise_orthogonal_imp_finite by auto
   with \<open>span B = span T\<close> show ?thesis
     by (rule_tac U=U in that) (auto simp: span_Un)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> orthogonal_extension_strong:
   fixes S :: "'a::euclidean_space set"
   assumes S: "pairwise orthogonal S"
   obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
                   "span (S \<union> U) = span (S \<union> T)"
 proof -
   obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
     using orthogonal_extension assms by blast
   then show ?thesis
     apply (rule_tac U = "U - (insert 0 S)" in that)
       apply blast
      apply (force simp: pairwise_def)
     apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
     done
 qed
 
 subsection\<open>Decomposing a vector into parts in orthogonal subspaces\<close>
 
 text\<open>existence of orthonormal basis for a subspace.\<close>
 
 lemma orthogonal_spanningset_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes "subspace S"
   obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
 proof -
   obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
     using basis_exists by blast
   with orthogonal_extension [of "{}" B]
   show ?thesis
     by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
 qed
 
 lemma orthogonal_basis_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes "subspace S"
   obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
                   "card B = dim S" "span B = S"
 proof -
   obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
     using assms orthogonal_spanningset_subspace by blast
   then show ?thesis
     apply (rule_tac B = "B - {0}" in that)
     apply (auto simp: indep_card_eq_dim_span pairwise_subset pairwise_orthogonal_independent elim: pairwise_subset)
     done
 qed
 
 proposition orthonormal_basis_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes "subspace S"
   obtains B where "B \<subseteq> S" "pairwise orthogonal B"
               and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
               and "independent B" "card B = dim S" "span B = S"
 proof -
   obtain B where "0 \<notin> B" "B \<subseteq> S"
              and orth: "pairwise orthogonal B"
              and "independent B" "card B = dim S" "span B = S"
     by (blast intro: orthogonal_basis_subspace [OF assms])
   have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
     using \<open>span B = S\<close> span_superset span_mul by fastforce
   have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
     using orth by (force simp: pairwise_def orthogonal_clauses)
   have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
     by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
   have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
     by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
   have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
   proof
     fix x y
     assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
     moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
       using 3 by blast
     ultimately show "x = y"
       by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
   qed
   then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
     by (metis \<open>card B = dim S\<close> card_image)
   have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
     by (metis "1" "4" "5" assms card_eq_dim independent_imp_finite span_subspace)
   show ?thesis
     by (rule that [OF 1 2 3 4 5 6])
 qed
 
 
 proposition\<^marker>\<open>tag unimportant\<close> orthogonal_to_subspace_exists_gen:
   fixes S :: "'a :: euclidean_space set"
   assumes "span S \<subset> span T"
   obtains x where "x \<noteq> 0" "x \<in> span T" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
 proof -
   obtain B where "B \<subseteq> span S" and orthB: "pairwise orthogonal B"
              and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
              and "independent B" "card B = dim S" "span B = span S"
     by (rule orthonormal_basis_subspace [of "span S", OF subspace_span]) (auto)
   with assms obtain u where spanBT: "span B \<subseteq> span T" and "u \<notin> span B" "u \<in> span T"
     by auto
   obtain C where orthBC: "pairwise orthogonal (B \<union> C)" and spanBC: "span (B \<union> C) = span (B \<union> {u})"
     by (blast intro: orthogonal_extension [OF orthB])
   show thesis
   proof (cases "C \<subseteq> insert 0 B")
     case True
     then have "C \<subseteq> span B"
       using span_eq
       by (metis span_insert_0 subset_trans)
     moreover have "u \<in> span (B \<union> C)"
       using \<open>span (B \<union> C) = span (B \<union> {u})\<close> span_superset by force
     ultimately show ?thesis
       using True \<open>u \<notin> span B\<close>
       by (metis Un_insert_left span_insert_0 sup.orderE)
   next
     case False
     then obtain x where "x \<in> C" "x \<noteq> 0" "x \<notin> B"
       by blast
     then have "x \<in> span T"
       by (metis (no_types, lifting) Un_insert_right Un_upper2 \<open>u \<in> span T\<close> spanBT spanBC
           \<open>u \<in> span T\<close> insert_subset span_superset span_mono
           span_span subsetCE subset_trans sup_bot.comm_neutral)
     moreover have "orthogonal x y" if "y \<in> span B" for y
       using that
     proof (rule span_induct)
       show "subspace {a. orthogonal x a}"
         by (simp add: subspace_orthogonal_to_vector)
       show "\<And>b. b \<in> B \<Longrightarrow> orthogonal x b"
         by (metis Un_iff \<open>x \<in> C\<close> \<open>x \<notin> B\<close> orthBC pairwise_def)
     qed
     ultimately show ?thesis
       using \<open>x \<noteq> 0\<close> that \<open>span B = span S\<close> by auto
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> orthogonal_to_subspace_exists:
   fixes S :: "'a :: euclidean_space set"
   assumes "dim S < DIM('a)"
   obtains x where "x \<noteq> 0" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
 proof -
   have "span S \<subset> UNIV"
   by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
       mem_Collect_eq top.extremum_strict top.not_eq_extremum)
   with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
     by (auto)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> orthogonal_to_vector_exists:
   fixes x :: "'a :: euclidean_space"
   assumes "2 \<le> DIM('a)"
   obtains y where "y \<noteq> 0" "orthogonal x y"
 proof -
   have "dim {x} < DIM('a)"
     using assms by auto
   then show thesis
     by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> orthogonal_subspace_decomp_exists:
   fixes S :: "'a :: euclidean_space set"
   obtains y z
   where "y \<in> span S"
     and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w"
     and "x = y + z"
 proof -
   obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T"
     "card T = dim (span S)" "span T = span S"
     using orthogonal_basis_subspace subspace_span by blast
   let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
   have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
     by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close>
         orthogonal_commute that)
   show ?thesis
     apply (rule_tac y = "?a" and z = "x - ?a" in that)
       apply (meson \<open>T \<subseteq> span S\<close> span_scale span_sum subsetCE)
      apply (fact orth, simp)
     done
 qed
 
 lemma orthogonal_subspace_decomp_unique:
   fixes S :: "'a :: euclidean_space set"
   assumes "x + y = x' + y'"
       and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
       and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
   shows "x = x' \<and> y = y'"
 proof -
   have "x + y - y' = x'"
     by (simp add: assms)
   moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
     by (meson orth orthogonal_commute orthogonal_to_span)
   ultimately have "0 = x' - x"
     by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
   with assms show ?thesis by auto
 qed
 
 lemma vector_in_orthogonal_spanningset:
   fixes a :: "'a::euclidean_space"
   obtains S where "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
   by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
       pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
 
 lemma vector_in_orthogonal_basis:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
   obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S"
                   "span S = UNIV" "card S = DIM('a)"
 proof -
   obtain S where S: "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
     using vector_in_orthogonal_spanningset .
   show thesis
   proof
     show "pairwise orthogonal (S - {0})"
       using pairwise_mono S(2) by blast
     show "independent (S - {0})"
       by (simp add: \<open>pairwise orthogonal (S - {0})\<close> pairwise_orthogonal_independent)
     show "finite (S - {0})"
       using \<open>independent (S - {0})\<close> independent_imp_finite by blast
     show "card (S - {0}) = DIM('a)"
       using span_delete_0 [of S] S
       by (simp add: \<open>independent (S - {0})\<close> indep_card_eq_dim_span)
   qed (use S \<open>a \<noteq> 0\<close> in auto)
 qed
 
 lemma vector_in_orthonormal_basis:
   fixes a :: "'a::euclidean_space"
   assumes "norm a = 1"
   obtains S where "a \<in> S" "pairwise orthogonal S" "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
     "independent S" "card S = DIM('a)" "span S = UNIV"
 proof -
   have "a \<noteq> 0"
     using assms by auto
   then obtain S where "a \<in> S" "0 \<notin> S" "finite S"
           and S: "pairwise orthogonal S" "independent S" "span S = UNIV" "card S = DIM('a)"
     by (metis vector_in_orthogonal_basis)
   let ?S = "(\<lambda>x. x /\<^sub>R norm x) ` S"
   show thesis
   proof
     show "a \<in> ?S"
       using \<open>a \<in> S\<close> assms image_iff by fastforce
   next
     show "pairwise orthogonal ?S"
       using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
     show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
       using \<open>0 \<notin> S\<close> by (auto simp: field_split_simps)
     then show "independent ?S"
       by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
     have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
       unfolding inj_on_def
       by (metis (full_types) S(1) \<open>0 \<notin> S\<close> inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
     then show "card ?S = DIM('a)"
       by (simp add: card_image S)
     show "span ?S = UNIV"
       by (metis (no_types) \<open>0 \<notin> S\<close> \<open>finite S\<close> \<open>span S = UNIV\<close>
           field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
           zero_less_norm_iff)
   qed
 qed
 
 proposition dim_orthogonal_sum:
   fixes A :: "'a::euclidean_space set"
   assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     shows "dim(A \<union> B) = dim A + dim B"
 proof -
   have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
   have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     using 1 by (simp add: span_induct [OF _ subspace_hyperplane])
   then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
     by simp
   have "dim(A \<union> B) = dim (span (A \<union> B))"
     by (simp)
   also have "span (A \<union> B) = ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
     by (auto simp add: span_Un image_def)
   also have "dim \<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
     by (auto intro!: arg_cong [where f=dim])
   also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
     by (auto simp: dest: 0)
   also have "... = dim (span A) + dim (span B)"
     by (rule dim_sums_Int) (auto)
   also have "... = dim A + dim B"
     by (simp)
   finally show ?thesis .
 qed
 
 lemma dim_subspace_orthogonal_to_vectors:
   fixes A :: "'a::euclidean_space set"
   assumes "subspace A" "subspace B" "A \<subseteq> B"
     shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
 proof -
   have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
   proof (rule arg_cong [where f=dim, OF subset_antisym])
     show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
       by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
   next
     have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
          if "x \<in> B" for x
     proof -
       obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
         using orthogonal_subspace_decomp_exists [of A x] that by auto
       have "y \<in> span B"
         using \<open>y \<in> span A\<close> assms(3) span_mono by blast
       then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
         apply simp
         using \<open>x = y + z\<close> assms(1) assms(2) orth orthogonal_commute span_add_eq
           span_eq_iff that by blast
       then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
         by (meson span_superset subset_iff)
       then show ?thesis
         apply (auto simp: span_Un image_def  \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
         using \<open>y \<in> span A\<close> add.commute by blast
     qed
     show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
       by (rule span_minimal) (auto intro: * span_minimal)
   qed
   then show ?thesis
     by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq
         orthogonal_commute orthogonal_def)
 qed
 
 subsection\<open>Linear functions are (uniformly) continuous on any set\<close>
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Topological properties of linear functions\<close>
 
 lemma linear_lim_0:
   assumes "bounded_linear f"
   shows "(f \<longlongrightarrow> 0) (at (0))"
 proof -
   interpret f: bounded_linear f by fact
   have "(f \<longlongrightarrow> f 0) (at 0)"
     using tendsto_ident_at by (rule f.tendsto)
   then show ?thesis unfolding f.zero .
 qed
 
 lemma linear_continuous_at:
   assumes "bounded_linear f"
   shows "continuous (at a) f"
   unfolding continuous_at using assms
   apply (rule bounded_linear.tendsto)
   apply (rule tendsto_ident_at)
   done
 
 lemma linear_continuous_within:
   "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
   using continuous_at_imp_continuous_at_within linear_continuous_at by blast
 
 lemma linear_continuous_on:
   "bounded_linear f \<Longrightarrow> continuous_on s f"
   using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
 
 lemma Lim_linear:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and h :: "'b \<Rightarrow> 'c::real_normed_vector"
   assumes "(f \<longlongrightarrow> l) F" "linear h"
   shows "((\<lambda>x. h(f x)) \<longlongrightarrow> h l) F"
 proof -
   obtain B where B: "B > 0" "\<And>x. norm (h x) \<le> B * norm x"
     using linear_bounded_pos [OF \<open>linear h\<close>] by blast
   show ?thesis
     unfolding tendsto_iff
   proof (intro allI impI)
     show "\<forall>\<^sub>F x in F. dist (h (f x)) (h l) < e" if "e > 0" for e
     proof -
       have "\<forall>\<^sub>F x in F. dist (f x) l < e/B"
         by (simp add: \<open>0 < B\<close> assms(1) tendstoD that)
       then show ?thesis
         unfolding dist_norm
       proof (rule eventually_mono)
         show "norm (h (f x) - h l) < e" if "norm (f x - l) < e / B" for x
           using that B
           apply (simp add: field_split_simps)
           by (metis \<open>linear h\<close> le_less_trans linear_diff)
       qed
     qed
   qed
 qed
 
 lemma linear_continuous_compose:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and g :: "'b \<Rightarrow> 'c::real_normed_vector"
   assumes "continuous F f" "linear g"
   shows "continuous F (\<lambda>x. g(f x))"
   using assms unfolding continuous_def by (rule Lim_linear)
 
 lemma linear_continuous_on_compose:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" and g :: "'b \<Rightarrow> 'c::real_normed_vector"
   assumes "continuous_on S f" "linear g"
   shows "continuous_on S (\<lambda>x. g(f x))"
   using assms by (simp add: continuous_on_eq_continuous_within linear_continuous_compose)
 
 text\<open>Also bilinear functions, in composition form\<close>
 
 lemma bilinear_continuous_compose:
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
   assumes "continuous F f" "continuous F g" "bilinear h"
   shows "continuous F (\<lambda>x. h (f x) (g x))"
   using assms bilinear_conv_bounded_bilinear bounded_bilinear.continuous by blast
 
 lemma bilinear_continuous_on_compose:
   fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
     and f :: "'d::t2_space \<Rightarrow> 'a"
   assumes "continuous_on S f" "continuous_on S g" "bilinear h"
   shows "continuous_on S (\<lambda>x. h (f x) (g x))"
   using assms by (simp add: continuous_on_eq_continuous_within bilinear_continuous_compose)
 
 end
diff --git a/src/HOL/Analysis/Starlike.thy b/src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy
+++ b/src/HOL/Analysis/Starlike.thy
@@ -1,6345 +1,6348 @@
 (* Title:      HOL/Analysis/Starlike.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 chapter \<open>Unsorted\<close>
 
 theory Starlike
   imports
     Convex_Euclidean_Space
     Line_Segment
 begin
 
 lemma affine_hull_closed_segment [simp]:
      "affine hull (closed_segment a b) = affine hull {a,b}"
   by (simp add: segment_convex_hull)
 
 lemma affine_hull_open_segment [simp]:
     fixes a :: "'a::euclidean_space"
     shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
 by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
 
 lemma rel_interior_closure_convex_segment:
   fixes S :: "_::euclidean_space set"
   assumes "convex S" "a \<in> rel_interior S" "b \<in> closure S"
     shows "open_segment a b \<subseteq> rel_interior S"
 proof
   fix x
   have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
     by (simp add: algebra_simps)
   assume "x \<in> open_segment a b"
   then show "x \<in> rel_interior S"
     unfolding closed_segment_def open_segment_def  using assms
     by (auto intro: rel_interior_closure_convex_shrink)
 qed
 
 lemma convex_hull_insert_segments:
    "convex hull (insert a S) =
     (if S = {} then {a} else  \<Union>x \<in> convex hull S. closed_segment a x)"
   by (force simp add: convex_hull_insert_alt in_segment)
 
 lemma Int_convex_hull_insert_rel_exterior:
   fixes z :: "'a::euclidean_space"
   assumes "convex C" "T \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
   shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
 proof
   have "T = {} \<Longrightarrow> z \<notin> S"
     using dis z by (auto simp add: disjnt_def)
   then show "?lhs \<subseteq> ?rhs"
   proof (clarsimp simp add: convex_hull_insert_segments)
     fix x y
     assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
     have "y \<in> closure C"
       by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
     moreover have "x \<notin> rel_interior C"
       by (meson \<open>x \<in> S\<close> dis disjnt_iff)
     moreover have "x \<in> open_segment z y \<union> {z, y}"
       using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
     ultimately show "x \<in> convex hull T"
       using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
       using y z by blast
   qed
   show "?rhs \<subseteq> ?lhs"
     by (meson hull_mono inf_mono subset_insertI subset_refl)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Shrinking towards the interior of a convex set\<close>
 
 lemma mem_interior_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   show ?thesis
     unfolding mem_interior
   proof (intro exI subsetI conjI)
     fix y
     assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
     then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
       by simp
     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
       using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "c - ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = (1 / e) *\<^sub>R (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       using \<open>e > 0\<close>
       by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     then have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       by (simp add: dist_norm)
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
     finally have "(1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x \<in> S"
       using assms(3-5) d
       by (intro convexD_alt [OF \<open>convex S\<close>]) (auto intro: convexD_alt [OF \<open>convex S\<close>])
     with \<open>e > 0\<close> show "y \<in> S"
       by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
   qed (use \<open>e>0\<close> \<open>d>0\<close> in auto)
 qed
 
 lemma mem_interior_closure_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> closure S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   have "\<exists>y\<in>S. norm (y - x) * (1 - e) < e * d"
   proof (cases "x \<in> S")
     case True
     then show ?thesis
       using \<open>e > 0\<close> \<open>d > 0\<close> by force
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
         using True \<open>0 < d\<close> by auto
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using islimpt_approachable x by blast
       then have "norm (y - x) * (1 - e) < e * d"
         by (metis "*" dist_norm mult_imp_div_pos_le not_less)
       then show ?thesis
         using \<open>y \<in> S\<close> by blast
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
   have "(1 - e) * norm (x - y) / e < d"
     using y \<open>0 < e\<close> by (simp add: field_simps norm_minus_commute)
   then have "z \<in> interior (ball c d)"
     using \<open>0 < e\<close> \<open>e \<le> 1\<close> by (simp add: interior_open[OF open_ball] z_def dist_norm)
   then have "z \<in> interior S"
     using d interiorI interior_ball by blast
   then show ?thesis
     unfolding * using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
 qed
 
 lemma in_interior_closure_convex_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
     shows "open_segment a b \<subseteq> interior S"
 proof (clarsimp simp: in_segment)
   fix u::real
   assume u: "0 < u" "u < 1"
   have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
     by (simp add: algebra_simps)
   also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
     by simp
   finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
 qed
 
 lemma convex_closure_interior:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and int: "interior S \<noteq> {}"
   shows "closure(interior S) = closure S"
 proof -
   obtain a where a: "a \<in> interior S"
     using int by auto
   have "closure S \<subseteq> closure(interior S)"
   proof
     fix x
     assume x: "x \<in> closure S"
     show "x \<in> closure (interior S)"
     proof (cases "x=a")
       case True
       then show ?thesis
         using \<open>a \<in> interior S\<close> closure_subset by blast
     next
       case False
       show ?thesis
       proof (clarsimp simp add: closure_def islimpt_approachable)
         fix e::real
         assume xnotS: "x \<notin> interior S" and "0 < e"
         show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
         proof (intro bexI conjI)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
             using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
           show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
             using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
             using \<open>0 < e\<close> False
             by (auto simp add: min_def a intro: mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
         qed
       qed
     qed
   qed
   then show ?thesis
     by (simp add: closure_mono interior_subset subset_antisym)
 qed
 
 lemma closure_convex_Int_superset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
   shows "closure(S \<inter> T) = closure S"
 proof -
   have "closure S \<subseteq> closure(interior S)"
     by (simp add: convex_closure_interior assms)
   also have "... \<subseteq> closure (S \<inter> T)"
     using interior_subset [of S] assms
     by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
   finally show ?thesis
     by (simp add: closure_mono dual_order.antisym)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some obvious but surprisingly hard simplex lemmas\<close>
 
 lemma simplex:
   assumes "finite S"
     and "0 \<notin> S"
   shows "convex hull (insert 0 S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof (simp add: convex_hull_finite set_eq_iff assms, safe)
   fix x and u :: "'a \<Rightarrow> real"
   assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
   then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by force
 next
   fix x and u :: "'a \<Rightarrow> real"
   assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
   then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
 qed
 
 lemma substd_simplex:
   assumes d: "d \<subseteq> Basis"
   shows "convex hull (insert 0 d) =
     {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
   (is "convex hull (insert 0 ?p) = ?s")
 proof -
   let ?D = d
   have "0 \<notin> ?p"
     using assms by (auto simp: image_def)
   from d have "finite d"
     by (blast intro: finite_subset finite_Basis)
   show ?thesis
     unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
   proof (intro set_eqI; safe)
     fix u :: "'a \<Rightarrow> real"
     assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" 
     let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
     have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
       and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
       using substdbasis_expansion_unique[OF assms] by blast+
     then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
       using assms by (auto intro!: sum.cong)
     show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
       using as(1) ind notind that by fastforce
     show "sum ((\<bullet>) ?x) ?D \<le> 1"
       using "**" as(2) by linarith
     show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
       using notind that by blast
   next
     fix x 
     assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
     with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
       unfolding substdbasis_expansion_unique[OF assms] 
       by (rule_tac x="inner x" in exI) auto
   qed
 qed
 
 lemma std_simplex:
   "convex hull (insert 0 Basis) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
   using substd_simplex[of Basis] by auto
 
 lemma interior_std_simplex:
   "interior (convex hull (insert 0 Basis)) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
   unfolding set_eq_iff mem_interior std_simplex
 proof (intro allI iffI CollectI; clarify)
   fix x :: 'a
   fix e
   assume "e > 0" and as: "ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
   proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     then show "0 < x \<bullet> i"
       using as[THEN subsetD[where c="x - (e/2) *\<^sub>R i"]] and \<open>e > 0\<close> 
       by (force simp add: inner_simps)
   next
     have **: "dist x (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
       unfolding dist_norm
       by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
     have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
       x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
       by (auto simp: SOME_Basis inner_Basis inner_simps)
     then have *: "sum ((\<bullet>) (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
       sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
       by (auto simp: intro!: sum.cong)
     have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e/2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
       using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
     also have "\<dots> \<le> 1"
       using ** as by force
     finally show "sum ((\<bullet>) x) Basis < 1" by auto
   qed 
 next
   fix x :: 'a
   assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
   obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
   let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
   show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
     fix y
     assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
     have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
     proof (rule sum_mono)
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
         by (metis Basis_le_norm i inner_commute inner_diff_right)
       also have "... < ?d"
         using y by (simp add: dist_norm norm_minus_commute)
       finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
       unfolding sum.distrib sum_constant
       by (auto simp add: Suc_le_eq)
     finally show "sum ((\<bullet>) y) Basis \<le> 1" .
     show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
         using y by (auto simp: dist_norm less_eq_real_def)
       also have "... \<le> x\<bullet>i"
         using i by auto
       finally have "norm (x - y) < x\<bullet>i" .
       then show "0 \<le> y\<bullet>i"
         using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
         by (auto simp: inner_simps)
     qed
   next
     have "Min (((\<bullet>) x) ` Basis) > 0"
       using as by simp
     moreover have "?d > 0"
       using as by (auto simp: Suc_le_eq)
     ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
       by linarith
   qed 
 qed
 
 lemma interior_std_simplex_nonempty:
   obtains a :: "'a::euclidean_space" where
     "a \<in> interior(convex hull (insert 0 Basis))"
 proof -
   let ?D = "Basis :: 'a set"
   let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
   {
     fix i :: 'a
     assume i: "i \<in> Basis"
     have "?a \<bullet> i = inverse (2 * real DIM('a))"
       by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
          (simp_all add: sum.If_cases i) }
   note ** = this
   show ?thesis
   proof
     show "?a \<in> interior(convex hull (insert 0 Basis))"
       unfolding interior_std_simplex mem_Collect_eq
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       show "0 < ?a \<bullet> i"
         unfolding **[OF i] by (auto simp add: Suc_le_eq)
     next
       have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
         by (auto intro: sum.cong)
       also have "\<dots> < 1"
         unfolding sum_constant divide_inverse[symmetric]
         by (auto simp add: field_simps)
       finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
     qed
   qed
 qed
 
 lemma rel_interior_substd_simplex:
   assumes D: "D \<subseteq> Basis"
   shows "rel_interior (convex hull (insert 0 D)) =
          {x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
      (is "_ = ?s")
 proof -
   have "finite D"
     using D finite_Basis finite_subset by blast
   show ?thesis
   proof (cases "D = {}")
     case True
     then show ?thesis
       using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
   next
     case False
     have h0: "affine hull (convex hull (insert 0 D)) =
               {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
       using affine_hull_convex_hull affine_hull_substd_basis assms by auto
     have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
       by auto
     {
       fix x :: "'a::euclidean_space"
       assume x: "x \<in> rel_interior (convex hull (insert 0 D))"
       then obtain e where "e > 0" and
         "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 D)"
         using mem_rel_interior_ball[of x "convex hull (insert 0 D)"] h0 by auto
       then have as: "\<And>y. \<lbrakk>dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0)\<rbrakk> \<Longrightarrow>
                             (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
         using assms by (force simp: substd_simplex)
       have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
         using x rel_interior_subset  substd_simplex[OF assms] by auto
       have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
       proof (intro conjI ballI)
         fix i :: 'a
         assume "i \<in> D"
         then have "\<forall>j\<in>D. 0 \<le> (x - (e/2) *\<^sub>R i) \<bullet> j"
           using D \<open>e > 0\<close> x0
           by (intro as[THEN conjunct1]) (force simp: dist_norm inner_simps inner_Basis)
         then show "0 < x \<bullet> i"
           using \<open>e > 0\<close> \<open>i \<in> D\<close> D  by (force simp: inner_simps inner_Basis)
       next
         obtain a where a: "a \<in> D"
           using \<open>D \<noteq> {}\<close> by auto
         then have **: "dist x (x + (e/2) *\<^sub>R a) < e"
           using \<open>e > 0\<close> norm_Basis[of a] D by (auto simp: dist_norm)
         have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e/2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
           using a D by (auto simp: inner_simps inner_Basis)
         then have *: "sum ((\<bullet>) (x + (e/2) *\<^sub>R a)) D = sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
           using D by (intro sum.cong) auto
         have "a \<in> Basis"
           using \<open>a \<in> D\<close> D by auto
         then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e/2) *\<^sub>R a) \<bullet> i = 0)"
           using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
         have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e/2) *\<^sub>R a)) D"
           using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
         also have "\<dots> \<le> 1"
           using ** h1 as[rule_format, of "x + (e/2) *\<^sub>R a"]
           by auto
         finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
           using x0 by auto
       qed
     }
     moreover
     {
       fix x :: "'a::euclidean_space"
       assume as: "x \<in> ?s"
       have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
         by auto
       moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
       ultimately
       have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
         by metis
       then have h2: "x \<in> convex hull (insert 0 D)"
         using as assms by (force simp add: substd_simplex)
       obtain a where a: "a \<in> D"
         using \<open>D \<noteq> {}\<close> by auto
       define d where "d \<equiv> (1 - sum ((\<bullet>) x) D) / real (card D)"
       have "\<exists>e>0. ball x e \<inter> {x. \<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0} \<subseteq> convex hull insert 0 D"
         unfolding substd_simplex[OF assms]
       proof (intro exI; safe)
         have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
           by (simp add: card_gt_0_iff)
         have "Min (((\<bullet>) x) ` D) > 0"
           using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp)
         moreover have "d > 0" 
           using as \<open>0 < card D\<close> by (auto simp: d_def)
         ultimately show "min (Min (((\<bullet>) x) ` D)) d > 0"
           by auto
         fix y :: 'a
         assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
         assume "y \<in> ball x (min (Min ((\<bullet>) x ` D)) d)"
         then have y: "dist x y < min (Min ((\<bullet>) x ` D)) d"
           by auto
         have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + d) D"
         proof (rule sum_mono)
           fix i
           assume "i \<in> D"
           with D have i: "i \<in> Basis"
             by auto
           have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
             by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
           also have "... < d"
             by (metis dist_norm min_less_iff_conj norm_minus_commute y)
           finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < d" .
           then show "y \<bullet> i \<le> x \<bullet> i + d" by auto
         qed
         also have "\<dots> \<le> 1"
           unfolding sum.distrib sum_constant d_def using \<open>0 < card D\<close>
           by auto
         finally show "sum ((\<bullet>) y) D \<le> 1" .
 
         fix i :: 'a
         assume i: "i \<in> Basis"
         then show "0 \<le> y\<bullet>i"
         proof (cases "i\<in>D")
           case True
           have "norm (x - y) < x\<bullet>i"
             using y Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
             by (simp add: dist_norm card_gt_0_iff)
           then show "0 \<le> y\<bullet>i"
             using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
             by (auto simp: inner_simps)
         qed (use y2 in auto)
       qed
       then have "x \<in> rel_interior (convex hull (insert 0 D))"
         using h0 h2 rel_interior_ball by force
     }
     ultimately have
       "\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
         x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
 qed
 
 lemma rel_interior_substd_simplex_nonempty:
   assumes "D \<noteq> {}"
     and "D \<subseteq> Basis"
   obtains a :: "'a::euclidean_space"
     where "a \<in> rel_interior (convex hull (insert 0 D))"
 proof -
   let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) D"
   have "finite D"
     using assms finite_Basis infinite_super by blast
   then have d1: "0 < real (card D)"
     using \<open>D \<noteq> {}\<close> by auto
   {
     fix i
     assume "i \<in> D"
     have "?a \<bullet> i = sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) D"
       unfolding inner_sum_left
       using \<open>i \<in> D\<close> by (auto simp: inner_Basis subsetD[OF assms(2)] intro: sum.cong)
     also have "... = inverse (2 * real (card D))"
       using \<open>i \<in> D\<close> \<open>finite D\<close> by auto
     finally have "?a \<bullet> i = inverse (2 * real (card D))" .
   }
   note ** = this
   show ?thesis
   proof
     show "?a \<in> rel_interior (convex hull (insert 0 D))"
       unfolding rel_interior_substd_simplex[OF assms(2)] 
     proof safe
       fix i
       assume "i \<in> D"
       have "0 < inverse (2 * real (card D))"
         using d1 by auto
       also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
         by auto
       finally show "0 < ?a \<bullet> i" by auto
     next
       have "sum ((\<bullet>) ?a) D = sum (\<lambda>i. inverse (2 * real (card D))) D"
         by (rule sum.cong) (rule refl, rule **)
       also have "\<dots> < 1"
         unfolding sum_constant divide_real_def[symmetric]
         by (auto simp add: field_simps)
       finally show "sum ((\<bullet>) ?a) D < 1" by auto
     next
       fix i
       assume "i \<in> Basis" and "i \<notin> D"
       have "?a \<in> span D"
       proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
         {
           fix x :: "'a::euclidean_space"
           assume "x \<in> D"
           then have "x \<in> span D"
             using span_base[of _ "D"] by auto
           then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
             using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
         }
         then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
           by auto
       qed
       then show "?a \<bullet> i = 0 "
         using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
     qed
   qed
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex set\<close>
 
 lemma rel_interior_convex_nonempty_aux:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "0 \<in> S"
   shows "rel_interior S \<noteq> {}"
 proof (cases "S = {0}")
   case True
   then show ?thesis using rel_interior_sing by auto
 next
   case False
   obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
     using basis_exists[of S] by metis
   then have "B \<noteq> {}"
     using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
   have "insert 0 B \<le> span B"
     using subspace_span[of B] subspace_0[of "span B"]
       span_superset by auto
   then have "span (insert 0 B) \<le> span B"
     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
   then have "convex hull insert 0 B \<le> span B"
     using convex_hull_subset_span[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) \<le> span B"
     using span_span[of B]
       span_mono[of "convex hull insert 0 B" "span B"] by blast
   then have *: "span (convex hull insert 0 B) = span B"
     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) = span S"
     using B span_mono[of B S] span_mono[of S "span B"]
       span_span[of B] by auto
   moreover have "0 \<in> affine hull (convex hull insert 0 B)"
     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
       assms hull_subset[of S]
     by auto
   obtain d and f :: "'n \<Rightarrow> 'n" where
     fd: "card d = card B" "linear f" "f ` B = d"
       "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
     and d: "d \<subseteq> Basis"
     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
   then have "bounded_linear f"
     using linear_conv_bounded_linear by auto
   have "d \<noteq> {}"
     using fd B \<open>B \<noteq> {}\<close> by auto
   have "insert 0 d = f ` (insert 0 B)"
     using fd linear_0 by auto
   then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
     using convex_hull_linear_image[of f "(insert 0 d)"]
       convex_hull_linear_image[of f "(insert 0 B)"] \<open>linear f\<close>
     by auto
   moreover have "rel_interior (f ` (convex hull insert 0 B)) = f ` rel_interior (convex hull insert 0 B)"
   proof (rule rel_interior_injective_on_span_linear_image[OF \<open>bounded_linear f\<close>])
     show "inj_on f (span (convex hull insert 0 B))"
       using fd * by auto
   qed
   ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
     using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d] by fastforce
   moreover have "convex hull (insert 0 B) \<subseteq> S"
     using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
   ultimately show ?thesis
     using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 qed
 
 lemma rel_interior_eq_empty:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior S = {} \<longleftrightarrow> S = {}"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain a where "a \<in> S" by auto
     then have "0 \<in> (+) (-a) ` S"
       using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
     then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
       using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
         convex_translation[of S "-a"] assms
       by auto
     then have "rel_interior S \<noteq> {}"
       using rel_interior_translation [of "- a"] by simp
   }
   then show ?thesis by auto
 qed
 
 lemma interior_simplex_nonempty:
   fixes S :: "'N :: euclidean_space set"
   assumes "independent S" "finite S" "card S = DIM('N)"
   obtains a where "a \<in> interior (convex hull (insert 0 S))"
 proof -
   have "affine hull (insert 0 S) = UNIV"
     by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
          assms(1) assms(3) dim_eq_card_independent)
   moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
     using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
   ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
     by (simp add: rel_interior_interior)
   with that show ?thesis
     by auto
 qed
 
 lemma convex_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "convex (rel_interior S)"
 proof -
   {
     fix x y and u :: real
     assume assm: "x \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
     then have "x \<in> S"
       using rel_interior_subset by auto
     have "x - u *\<^sub>R (x-y) \<in> rel_interior S"
     proof (cases "0 = u")
       case False
       then have "0 < u" using assm by auto
       then show ?thesis
         using assm rel_interior_convex_shrink[of S y x u] assms \<open>x \<in> S\<close> by auto
     next
       case True
       then show ?thesis using assm by auto
     qed
     then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> rel_interior S"
       by (simp add: algebra_simps)
   }
   then show ?thesis
     unfolding convex_alt by auto
 qed
 
 lemma convex_closure_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "closure (rel_interior S) = closure S"
 proof -
   have h1: "closure (rel_interior S) \<le> closure S"
     using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
   show ?thesis
   proof (cases "S = {}")
     case False
     then obtain a where a: "a \<in> rel_interior S"
       using rel_interior_eq_empty assms by auto
     { fix x
       assume x: "x \<in> closure S"
       {
         assume "x = a"
         then have "x \<in> closure (rel_interior S)"
           using a unfolding closure_def by auto
       }
       moreover
       {
         assume "x \<noteq> a"
          {
            fix e :: real
            assume "e > 0"
            define e1 where "e1 = min 1 (e/norm (x - a))"
            then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
              using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
              by simp_all
            then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
              using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
              by auto
            have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
              using "*" \<open>x \<noteq> a\<close> e1 by force
         }
         then have "x islimpt rel_interior S"
           unfolding islimpt_approachable_le by auto
         then have "x \<in> closure(rel_interior S)"
           unfolding closure_def by auto
       }
       ultimately have "x \<in> closure(rel_interior S)" by auto
     }
     then show ?thesis using h1 by auto
   qed auto
 qed
 
 lemma rel_interior_same_affine_hull:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "affine hull (rel_interior S) = affine hull S"
   by (metis assms closure_same_affine_hull convex_closure_rel_interior)
 
 lemma rel_interior_aff_dim:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "aff_dim (rel_interior S) = aff_dim S"
   by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
 
 lemma rel_interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (rel_interior S) = rel_interior S"
 proof -
   have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
     using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
   then show ?thesis
     using rel_interior_def by auto
 qed
 
 lemma rel_interior_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_open (rel_interior S)"
   unfolding rel_open_def using rel_interior_rel_interior assms by auto
 
 lemma convex_rel_interior_closure_aux:
   fixes x y z :: "'n::euclidean_space"
   assumes "0 < a" "0 < b" "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
   obtains e where "0 < e" "e < 1" "z = y - e *\<^sub>R (y - x)"
 proof -
   define e where "e = a / (a + b)"
   have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
     using assms  by (simp add: eq_vector_fraction_iff)
   also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
     using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
     by auto
   also have "\<dots> = y - e *\<^sub>R (y-x)"
     using e_def assms
     by (simp add: divide_simps vector_fraction_eq_iff) (simp add: algebra_simps)
   finally have "z = y - e *\<^sub>R (y-x)"
     by auto
   moreover have "e > 0" "e < 1" using e_def assms by auto
   ultimately show ?thesis using that[of e] by auto
 qed
 
 lemma convex_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (closure S) = rel_interior S"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms rel_interior_eq_empty by auto
 next
   case False
   have "rel_interior (closure S) \<supseteq> rel_interior S"
     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
     by auto
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (closure S)"
     obtain x where x: "x \<in> rel_interior S"
       using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
     have "z \<in> rel_interior S"
     proof (cases "x = z")
       case True
       then show ?thesis using x by auto
     next
       case False
       obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S"
         using z rel_interior_cball[of "closure S"] by auto
       hence *: "0 < e/norm(z-x)" using e False by auto
       define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)"
       have yball: "y \<in> cball z e"
         using y_def dist_norm[of z y] e by auto
       have "x \<in> affine hull closure S"
         using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
       moreover have "z \<in> affine hull closure S"
         using z rel_interior_subset hull_subset[of "closure S"] by blast
       ultimately have "y \<in> affine hull closure S"
         using y_def affine_affine_hull[of "closure S"]
           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
       then have "y \<in> closure S" using e yball by auto
       have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
         using y_def by (simp add: algebra_simps)
       then obtain e1 where "0 < e1" "e1 < 1" "z = y - e1 *\<^sub>R (y - x)"
         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
         by (auto simp add: algebra_simps)
       then show ?thesis
         using rel_interior_closure_convex_shrink assms x \<open>y \<in> closure S\<close>
         by fastforce
     qed
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "interior (closure S) = interior S"
   using closure_aff_dim[of S] interior_rel_interior_gen[of S]
     interior_rel_interior_gen[of "closure S"]
     convex_rel_interior_closure[of S] assms
   by auto
 
 lemma closure_eq_rel_interior_eq:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 = rel_interior S2"
   by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
 
 lemma closure_eq_between:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
   (is "?A \<longleftrightarrow> ?B")
 proof
   assume ?A
   then show ?B
     by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
 next
   assume ?B
   then have "closure S1 \<subseteq> closure S2"
     by (metis assms(1) convex_closure_rel_interior closure_mono)
   moreover from \<open>?B\<close> have "closure S1 \<supseteq> closure S2"
     by (metis closed_closure closure_minimal)
   ultimately show ?A ..
 qed
 
 lemma open_inter_closure_rel_interior:
   fixes S A :: "'n::euclidean_space set"
   assumes "convex S"
     and "open A"
   shows "A \<inter> closure S = {} \<longleftrightarrow> A \<inter> rel_interior S = {}"
   by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
 
 lemma rel_interior_open_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(open_segment a b) = open_segment a b"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then
   have "open_segment a b = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
     by (simp add: open_segment_as_ball)
   then show ?thesis
     unfolding rel_interior_eq openin_open
     by (metis Elementary_Metric_Spaces.open_ball False affine_hull_open_segment)
 qed
 
 lemma rel_interior_closed_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(closed_segment a b) =
          (if a = b then {a} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     by simp
        (metis closure_open_segment convex_open_segment convex_rel_interior_closure
               rel_interior_open_segment)
 qed
 
 lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
 
 subsection\<open>The relative frontier of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_frontier S = closure S - rel_interior S"
 
 lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_eq_empty:
     fixes S :: "'n::euclidean_space set"
     shows "rel_frontier S = {} \<longleftrightarrow> affine S"
   unfolding rel_frontier_def
   using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])
 
 lemma rel_frontier_sing [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier {a} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_affine_hull:
   fixes S :: "'a::euclidean_space set"
   shows "rel_frontier S \<subseteq> affine hull S"
 using closure_affine_hull rel_frontier_def by fastforce
 
 lemma rel_frontier_cball [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
 proof (cases rule: linorder_cases [of r 0])
   case less then show ?thesis
     by (force simp: sphere_def)
 next
   case equal then show ?thesis by simp
 next
   case greater then show ?thesis
     by simp (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
 qed
 
 lemma rel_frontier_translation:
   fixes a :: "'a::euclidean_space"
   shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
   by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
 
 lemma rel_frontier_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"
   by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
 
 lemma rel_frontier_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull S = UNIV \<Longrightarrow> rel_frontier S = frontier S"
   by (simp add: frontier_def rel_frontier_def rel_interior_interior)
 
 lemma closest_point_in_rel_frontier:
    "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
    \<Longrightarrow> closest_point S x \<in> rel_frontier S"
   by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
 
 lemma closed_rel_frontier [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (rel_frontier S)"
 proof -
   have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
     by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
   show ?thesis
   proof (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
     show "closedin (top_of_set (affine hull S)) (rel_frontier S)"
       by (simp add: "*" rel_frontier_def)
   qed simp
 qed
 
 lemma closed_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "closed S \<Longrightarrow> closed(S - rel_interior S)"
   by (metis closed_rel_frontier closure_closed rel_frontier_def)
 
 lemma compact_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(S - rel_interior S)"
   by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
 
 lemma bounded_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> bounded(rel_frontier S)"
 by (simp add: bounded_closure bounded_diff rel_frontier_def)
 
 lemma compact_rel_frontier_bounded:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> compact(rel_frontier S)"
 using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
 
 lemma compact_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(rel_frontier S)"
 by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
 
 lemma convex_same_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> closure S = closure T"
 by (simp add: closure_eq_rel_interior_eq)
 
 lemma convex_same_rel_interior_closure_straddle:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
              rel_interior S \<subseteq> T \<and> T \<subseteq> closure S"
 by (simp add: closure_eq_between convex_same_rel_interior_closure)
 
 lemma convex_rel_frontier_aff_dim:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
     and "S2 \<noteq> {}"
     and "S1 \<le> rel_frontier S2"
   shows "aff_dim S1 < aff_dim S2"
 proof -
   have "S1 \<subseteq> closure S2"
     using assms unfolding rel_frontier_def by auto
   then have *: "affine hull S1 \<subseteq> affine hull S2"
     using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
   then have "aff_dim S1 \<le> aff_dim S2"
     using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
       aff_dim_subset[of "affine hull S1" "affine hull S2"]
     by auto
   moreover
   {
     assume eq: "aff_dim S1 = aff_dim S2"
     then have "S1 \<noteq> {}"
       using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
     have **: "affine hull S1 = affine hull S2"
       by (simp_all add: * eq \<open>S1 \<noteq> {}\<close> affine_dim_equal)
     obtain a where a: "a \<in> rel_interior S1"
       using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
     obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
        using mem_rel_interior[of a S1] a by auto
     then have "a \<in> T \<inter> closure S2"
       using a assms unfolding rel_frontier_def by auto
     then obtain b where b: "b \<in> T \<inter> rel_interior S2"
       using open_inter_closure_rel_interior[of S2 T] assms T by auto
     then have "b \<in> affine hull S1"
       using rel_interior_subset hull_subset[of S2] ** by auto
     then have "b \<in> S1"
       using T b by auto
     then have False
       using b assms unfolding rel_frontier_def by auto
   }
   ultimately show ?thesis
     using less_le by auto
 qed
 
 lemma convex_rel_interior_if:
   fixes S ::  "'n::euclidean_space set"
   assumes "convex S"
     and "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
 proof -
   obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
     using mem_rel_interior_cball[of z S] assms by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     {
       assume "x \<noteq> z"
       define m where "m = 1 + e1/norm(x-z)"
       hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         have "z \<in> affine hull S"
           using assms rel_interior_subset hull_subset[of S] by auto
         then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> affine hull S"
           using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
           by auto
         have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
           by (simp add: algebra_simps)
         also have "\<dots> = (e - 1) * norm (x-z)"
           using norm_scaleR e by auto
         also have "\<dots> \<le> (m - 1) * norm (x - z)"
           using e mult_right_mono[of _ _ "norm(x-z)"] by auto
         also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
           using m_def by auto
         also have "\<dots> = e1"
           using \<open>x \<noteq> z\<close> e1 by simp
         finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
           by auto
         have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
           using m_def **
           unfolding cball_def dist_norm
           by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
           using e * e1 by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
         using \<open>m> 1 \<close> by auto
     }
     moreover
     {
       assume "x = z"
       define m where "m = 1 + e1"
       then have "m > 1"
         using e1 by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
         using \<open>m > 1\<close> by auto
     }
     ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
       by blast
   }
   then show ?thesis by auto
 qed
 
 lemma convex_rel_interior_if2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   assumes "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
   using convex_rel_interior_if[of S z] assms by auto
 
 lemma convex_rel_interior_only_if:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
   shows "z \<in> rel_interior S"
 proof -
   obtain x where x: "x \<in> rel_interior S"
     using rel_interior_eq_empty assms by auto
   then have "x \<in> S"
     using rel_interior_subset by auto
   then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
     using assms by auto
   define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
   then have "y \<in> S" using e by auto
   define e1 where "e1 = 1/e"
   then have "0 < e1 \<and> e1 < 1" using e by auto
   then have "z  =y - (1 - e1) *\<^sub>R (y - x)"
     using e1_def y_def by (auto simp add: algebra_simps)
   then show ?thesis
     using rel_interior_convex_shrink[of S x y "1-e1"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
     by auto
 qed
 
 lemma convex_rel_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S "affine"]
     convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_rel_interior_iff2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "z \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
 proof (cases "aff_dim S = int DIM('n)")
   case False
   { assume "z \<in> interior S"
     then have False
       using False interior_rel_interior_gen[of S] by auto }
   moreover
   { assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     { fix x
       obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
         using r by auto
       obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
         using r by auto
       define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
       then have x1: "x1 \<in> affine hull S"
         using e1 hull_subset[of S] by auto
       define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
       then have x2: "x2 \<in> affine hull S"
         using e2 hull_subset[of S] by auto
       have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
         using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
       then have "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
         by (simp add: x1_def x2_def algebra_simps) (simp add: "*" pth_8)
       then have z: "z \<in> affine hull S"
         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
           x1 x2 affine_affine_hull[of S] *
         by auto
       have "x1 - x2 = (e1 + e2) *\<^sub>R (x - z)"
         using x1_def x2_def by (auto simp add: algebra_simps)
       then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
         using e1 e2 by simp
       then have "x \<in> affine hull S"
         using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
           x1 x2 z affine_affine_hull[of S]
         by auto
     }
     then have "affine hull S = UNIV"
       by auto
     then have "aff_dim S = int DIM('n)"
       using aff_dim_affine_hull[of S] by (simp)
     then have False
       using False by auto
   }
   ultimately show ?thesis by auto
 next
   case True
   then have "S \<noteq> {}"
     using aff_dim_empty[of S] by auto
   have *: "affine hull S = UNIV"
     using True affine_hull_UNIV by auto
   {
     assume "z \<in> interior S"
     then have "z \<in> rel_interior S"
       using True interior_rel_interior_gen[of S] by auto
     then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
     fix x
     obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
       using **[rule_format, of "z-x"] by auto
     define e where [abs_def]: "e = e1 - 1"
     then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
       by (simp add: algebra_simps)
     then have "e > 0" "z + e *\<^sub>R x \<in> S"
       using e1 e_def by auto
     then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
       by auto
   }
   moreover
   {
     assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     {
       fix x
       obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
         using r[rule_format, of "z-x"] by auto
       define e where "e = e1 + 1"
       then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         by (simp add: algebra_simps)
       then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
         using e1 e_def by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
     }
     then have "z \<in> rel_interior S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
     then have "z \<in> interior S"
       using True interior_rel_interior_gen[of S] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior and closure under common operations\<close>
 
 lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
 proof -
   {
     fix y
     assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
       by auto
     {
       fix S
       assume "S \<in> I"
       then have "y \<in> S"
         using rel_interior_subset y by auto
     }
     then have "y \<in> \<Inter>I" by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
 proof -
   obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
     using assms by auto
   {
     fix y
     assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> closure S"
       by auto
     {
       assume "y = x"
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
     }
     moreover
     {
       assume "y \<noteq> x"
       { fix e :: real
         assume e: "e > 0"
         define e1 where "e1 = min 1 (e/norm (y - x))"
         then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
           using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
           by simp_all
         define z where "z = y - e1 *\<^sub>R (y - x)"
         {
           fix S
           assume "S \<in> I"
           then have "z \<in> rel_interior S"
             using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
             by auto
         }
         then have *: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
           by auto
         have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
           using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
           by (rule_tac x="z" in exI) auto
       }
       then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
         unfolding islimpt_approachable_le by blast
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         unfolding closure_def by auto
     }
     ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
       by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_inter_rel_interior_same_closure:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "convex (\<Inter>I)"
     using assms convex_Inter by auto
   moreover
   have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
     using assms convex_rel_interior by (force intro: convex_Inter)
   ultimately
   have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
     using convex_inter_rel_interior_same_closure assms
       closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by blast
   then show ?thesis
     using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
 qed
 
 lemma convex_rel_interior_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
     and "finite I"
   shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "\<Inter>I \<noteq> {}"
     using assms rel_interior_inter_aux[of I] by auto
   have "convex (\<Inter>I)"
     using convex_Inter assms by auto
   show ?thesis
   proof (cases "I = {}")
     case True
     then show ?thesis
       using Inter_empty rel_interior_UNIV by auto
   next
     case False
     {
       fix z
       assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
       {
         fix x
         assume x: "x \<in> \<Inter>I"
         {
           fix S
           assume S: "S \<in> I"
           then have "z \<in> rel_interior S" "x \<in> S"
             using z x by auto
           then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
             using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
         }
         then obtain mS where
           mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
         define e where "e = Min (mS ` I)"
         then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
         then have "e > 1" using mS by auto
         moreover have "\<forall>S\<in>I. e \<le> mS S"
           using e_def assms by auto
         ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
           using mS by auto
       }
       then have "z \<in> rel_interior (\<Inter>I)"
         using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
     }
     then show ?thesis
       using convex_rel_interior_inter[of I] assms by auto
   qed
 qed
 
 lemma convex_closure_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> closure T"
   using convex_closure_inter[of "{S,T}"] assms by auto
 
 lemma convex_rel_interior_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
   using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
 
 lemma convex_affine_closure_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> T"
   by (metis affine_imp_convex assms convex_closure_inter_two rel_interior_affine rel_interior_eq_closure)
 
 lemma connected_component_1_gen:
   fixes S :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 unfolding connected_component_def
 by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
             ends_in_segment connected_convex_1_gen)
 
 lemma connected_component_1:
   fixes S :: "real set"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 by (simp add: connected_component_1_gen)
 
 lemma convex_affine_rel_interior_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
   by (simp add: affine_imp_convex assms convex_rel_interior_inter_two rel_interior_affine)
 
 lemma convex_affine_rel_frontier_Int:
    fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "interior S \<inter> T \<noteq> {}"
   shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
 using assms
   unfolding rel_frontier_def  frontier_def
   using convex_affine_closure_Int convex_affine_rel_interior_Int rel_interior_nonempty_interior by fastforce
 
 lemma rel_interior_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
   shows "rel_interior(S \<inter> T) = interior S \<inter> T"
   by (metis Int_emptyI assms convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
 
 lemma subset_rel_interior_convex:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<le> closure T"
     and "\<not> S \<subseteq> rel_frontier T"
   shows "rel_interior S \<subseteq> rel_interior T"
 proof -
   have *: "S \<inter> closure T = S"
     using assms by auto
   have "\<not> rel_interior S \<subseteq> rel_frontier T"
     using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
       closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
     by auto
   then have "rel_interior S \<inter> rel_interior (closure T) \<noteq> {}"
     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
     by auto
   then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> closure T)"
     using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
       convex_rel_interior_closure[of T]
     by auto
   also have "\<dots> = rel_interior S"
     using * by auto
   finally show ?thesis
     by auto
 qed
 
 lemma rel_interior_convex_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
   shows "f ` (rel_interior S) = rel_interior (f ` S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms by auto
 next
   case False
   interpret linear f by fact
   have *: "f ` (rel_interior S) \<subseteq> f ` S"
     unfolding image_mono using rel_interior_subset by auto
   have "f ` S \<subseteq> f ` (closure S)"
     unfolding image_mono using closure_subset by auto
   also have "\<dots> = f ` (closure (rel_interior S))"
     using convex_closure_rel_interior assms by auto
   also have "\<dots> \<subseteq> closure (f ` (rel_interior S))"
     using closure_linear_image_subset assms by auto
   finally have "closure (f ` S) = closure (f ` rel_interior S)"
     using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
       closure_mono[of "f ` rel_interior S" "f ` S"] *
     by auto
   then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
     using assms convex_rel_interior
       linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
       convex_linear_image[of _ "rel_interior S"]
       closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
     by auto
   then have "rel_interior (f ` S) \<subseteq> f ` rel_interior S"
     using rel_interior_subset by auto
   moreover
   {
     fix z
     assume "z \<in> f ` rel_interior S"
     then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
     {
       fix x
       assume "x \<in> f ` S"
       then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
         using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
       moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
       ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f ` S)"
       using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
         \<open>S \<noteq> {}\<close> convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "f -` (rel_interior S) \<noteq> {}"
   shows "rel_interior (f -` S) = f -` (rel_interior S)"
 proof -
   interpret linear f by fact
   have "S \<noteq> {}"
     using assms by auto
   have nonemp: "f -` S \<noteq> {}"
     by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
   then have "S \<inter> (range f) \<noteq> {}"
     by auto
   have conv: "convex (f -` S)"
     using convex_linear_vimage assms by auto
   then have "convex (S \<inter> range f)"
     by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
   {
     fix z
     assume "z \<in> f -` (rel_interior S)"
     then have z: "f z \<in> rel_interior S"
       by auto
     {
       fix x
       assume "x \<in> f -` S"
       then have "f x \<in> S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
         using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
       moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
         using \<open>linear f\<close> by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f -` S)"
       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
   }
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (f -` S)"
     {
       fix x
       assume "x \<in> S \<inter> range f"
       then obtain y where y: "f y = x" "y \<in> f -` S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
         using convex_rel_interior_iff[of "f -` S" z] z conv by auto
       moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
         using \<open>linear f\<close> y by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
         using e by auto
     }
     then have "f z \<in> rel_interior (S \<inter> range f)"
       using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
         convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
       by auto
     moreover have "affine (range f)"
       by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
     ultimately have "f z \<in> rel_interior S"
       using convex_affine_rel_interior_Int[of S "range f"] assms by auto
     then have "z \<in> f -` (rel_interior S)"
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
 proof (cases "S = {} \<or> T = {}")
   case True
   then show ?thesis 
     by auto
 next
   case False
   then have "S \<noteq> {}" "T \<noteq> {}"
     by auto
   then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
     using rel_interior_eq_empty assms by auto
   then have "fst -` rel_interior S \<noteq> {}"
     using fst_vimage_eq_Times[of "rel_interior S"] by auto
   then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
     using linear_fst \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
   then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
     by (simp add: fst_vimage_eq_Times)
   from ri have "snd -` rel_interior T \<noteq> {}"
     using snd_vimage_eq_Times[of "rel_interior T"] by auto
   then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
     using linear_snd \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
   then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
     by (simp add: snd_vimage_eq_Times)
   from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
       rel_interior S \<times> rel_interior T" by auto
   have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
     by auto
   then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
     by auto
   also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
     using * ri assms convex_Times
     by (subst convex_rel_interior_inter_two) auto
   finally show ?thesis using * by auto
 qed
 
 lemma rel_interior_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "c \<noteq> 0"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
     linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
   by auto
 
 lemma rel_interior_convex_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   by (metis assms linear_scaleR rel_interior_convex_linear_image)
 
 lemma convex_rel_open_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
   shows "convex (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
   by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 
 lemma convex_rel_open_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
     and "finite I"
   shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
 proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
   case True
   then have "\<Inter>I = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (\<Inter>I)"
     using assms unfolding rel_open_def
     using convex_rel_interior_finite_inter[of I]
     by auto
   then show ?thesis
     using convex_Inter assms by auto
 qed
 
 lemma convex_rel_open_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f ` S) \<and> rel_open (f ` S)"
   by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
 
 lemma convex_rel_open_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f -` S) \<and> rel_open (f -` S)"
 proof (cases "f -` (rel_interior S) = {}")
   case True
   then have "f -` S = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (f -` S)"
     using assms unfolding rel_open_def
     using rel_interior_convex_linear_preimage[of f S]
     by auto
   then show ?thesis
     using convex_linear_vimage assms
     by auto
 qed
 
 lemma rel_interior_projection:
   fixes S :: "('m::euclidean_space \<times> 'n::euclidean_space) set"
     and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
   assumes "convex S"
     and "f = (\<lambda>y. {z. (y, z) \<in> S})"
   shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
 proof -
   {
     fix y
     assume "y \<in> {y. f y \<noteq> {}}"
     then obtain z where "(y, z) \<in> S"
       using assms by auto
     then have "\<exists>x. x \<in> S \<and> y = fst x"
       by auto
     then obtain x where "x \<in> S" "y = fst x"
       by blast
     then have "y \<in> fst ` S"
       unfolding image_def by auto
   }
   then have "fst ` S = {y. f y \<noteq> {}}"
     unfolding fst_def using assms by auto
   then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
     using rel_interior_convex_linear_image[of fst S] assms linear_fst by auto
   {
     fix y
     assume "y \<in> rel_interior {y. f y \<noteq> {}}"
     then have "y \<in> fst ` rel_interior S"
       using h1 by auto
     then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
       by auto
     moreover have aff: "affine (fst -` {y})"
       unfolding affine_alt by (simp add: algebra_simps)
     ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
       using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
     have conv: "convex (S \<inter> fst -` {y})"
       using convex_Int assms aff affine_imp_convex by auto
     {
       fix x
       assume "x \<in> f y"
       then have "(y, x) \<in> S \<inter> (fst -` {y})"
         using assms by auto
       moreover have "x = snd (y, x)" by auto
       ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
         by blast
     }
     then have "snd ` (S \<inter> fst -` {y}) = f y"
       using assms by auto
     then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
       using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] linear_snd conv
       by auto
     {
       fix z
       assume "z \<in> rel_interior (f y)"
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         using *** by auto
       moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
         using * ** rel_interior_subset by auto
       ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         by force
       then have "(y,z) \<in> rel_interior S"
         using ** by auto
     }
     moreover
     {
       fix z
       assume "(y, z) \<in> rel_interior S"
       then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         using ** by auto
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         by (metis Range_iff snd_eq_Range)
       then have "z \<in> rel_interior (f y)"
         using *** by auto
     }
     ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
       by auto
   }
   then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
     (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
     by auto
   {
     fix y z
     assume asm: "(y, z) \<in> rel_interior S"
     then have "y \<in> fst ` rel_interior S"
       by (metis Domain_iff fst_eq_Domain)
     then have "y \<in> rel_interior {t. f t \<noteq> {}}"
       using h1 by auto
     then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> rel_interior (f y))"
       using h2 asm by auto
   }
   then show ?thesis using h2 by blast
 qed
 
 lemma rel_frontier_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_frontier S \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
     by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex cone\<close>
 
 lemma cone_rel_interior:
   fixes S :: "'m::euclidean_space set"
   assumes "cone S"
   shows "cone ({0} \<union> rel_interior S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_0)
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   then have *: "0 \<in> ({0} \<union> rel_interior S)"
     and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
     by (auto simp add: rel_interior_scaleR)
   then show ?thesis
     using cone_iff[of "{0} \<union> rel_interior S"] by auto
 qed
 
 lemma rel_interior_convex_cone_aux:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
     c > 0 \<and> x \<in> (((*\<^sub>R) c) ` (rel_interior S))"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_hull_empty)
 next
   case False
   then obtain s where "s \<in> S" by auto
   have conv: "convex ({(1 :: real)} \<times> S)"
     using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
     by auto
   define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
   then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
     (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
     using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
     by (subst rel_interior_projection) auto
   {
     fix y :: real
     assume "y \<ge> 0"
     then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
       using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
     then have "f y \<noteq> {}"
       using f_def by auto
   }
   then have "{y. f y \<noteq> {}} = {0..}"
     using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
   then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
     using rel_interior_real_semiline by auto
   {
     fix c :: real
     assume "c > 0"
     then have "f c = ((*\<^sub>R) c ` S)"
       using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
     then have "rel_interior (f c) = (*\<^sub>R) c ` rel_interior S"
       using rel_interior_convex_scaleR[of S c] assms by auto
   }
   then show ?thesis using * ** by auto
 qed
 
 lemma rel_interior_convex_cone:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (cone hull ({1 :: real} \<times> S)) =
     {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix z
     assume "z \<in> ?lhs"
     have *: "z = (fst z, snd z)"
       by auto
     then have "z \<in> ?rhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close> by fastforce
   }
   moreover
   {
     fix z
     assume "z \<in> ?rhs"
     then have "z \<in> ?lhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma convex_hull_finite_union:
   assumes "finite I"
   assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs \<supseteq> ?rhs"
   proof
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
       "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
     then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
       using hull_subset[of "\<Union>(S ` I)" convex] by auto
     then show "x \<in> ?lhs"
       unfolding *(1)[symmetric]
       using * assms convex_convex_hull
       by (subst convex_sum) auto
   qed
   {
     fix i
     assume "i \<in> I"
     with assms have "\<exists>p. p \<in> S i" by auto
   }
   then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
   {
     fix i
     assume "i \<in> I"
     {
       fix x
       assume "x \<in> S i"
       define c where "c j = (if j = i then 1::real else 0)" for j
       then have *: "sum c I = 1"
         using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
         by auto
       define s where "s j = (if j = i then x else p j)" for j
       then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
         using c_def by (auto simp add: algebra_simps)
       then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
         using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
         by auto
       moreover have "(\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
         using * c_def s_def p \<open>x \<in> S i\<close> by auto
       ultimately have "x \<in> ?rhs"
         by force
     }
     then have "?rhs \<supseteq> S i" by auto
   }
   then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
 
   {
     fix u v :: real
     assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
     fix x y
     assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
     from xy obtain c s where
       xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
       by auto
     from xy obtain d t where
       yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
       by auto
     define e where "e i = u * c i + v * d i" for i
     have ge0: "\<forall>i\<in>I. e i \<ge> 0"
       using e_def xc yc uv by simp
     have "sum (\<lambda>i. u * c i) I = u * sum c I"
       by (simp add: sum_distrib_left)
     moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
       by (simp add: sum_distrib_left)
     ultimately have sum1: "sum e I = 1"
       using e_def xc yc uv by (simp add: sum.distrib)
     define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
       for i
     {
       fix i
       assume i: "i \<in> I"
       have "q i \<in> S i"
       proof (cases "e i = 0")
         case True
         then show ?thesis using i p q_def by auto
       next
         case False
         then show ?thesis
           using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
             mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
             assms q_def e_def i False xc yc uv
           by (auto simp del: mult_nonneg_nonneg)
       qed
     }
     then have qs: "\<forall>i\<in>I. q i \<in> S i" by auto
     {
       fix i
       assume i: "i \<in> I"
       have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       proof (cases "e i = 0")
         case True
         have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
           using xc yc uv i by simp
         moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
           using True e_def i by simp
         ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
         with True show ?thesis by auto
       next
         case False
         then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
           using q_def by auto
         then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
                = (e i) *\<^sub>R (q i)" by auto
         with False show ?thesis by (simp add: algebra_simps)
       qed
     }
     then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       by auto
     have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
       using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
     also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
       using * by auto
     finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
       using ge0 sum1 qs by auto
   }
   then have "convex ?rhs" unfolding convex_def by auto
   then show ?thesis
     using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
     by blast
 qed
 
 lemma convex_hull_union_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
     and "convex T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) =
     {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T}"
   (is "?lhs = ?rhs")
 proof
   define I :: "nat set" where "I = {1, 2}"
   define s where "s i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(s ` I) = S \<union> T"
     using s_def I_def by auto
   then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(s ` I) =
     {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
       using assms s_def I_def
       by (subst convex_hull_finite_union) auto
   moreover have
     "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
     using s_def I_def by auto
   ultimately show "?lhs \<subseteq> ?rhs" by auto
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
       by auto
     then have "x \<in> convex hull {s, t}"
       using convex_hull_2[of s t] by auto
     then have "x \<in> convex hull (S \<union> T)"
       using * hull_mono[of "{s, t}" "S \<union> T"] by auto
   }
   then show "?lhs \<supseteq> ?rhs" by blast
 qed
 
 proposition ray_to_rel_frontier:
   fixes a :: "'a::real_inner"
   assumes "bounded S"
       and a: "a \<in> rel_interior S"
       and aff: "(a + l) \<in> affine hull S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
 proof -
   have aaff: "a \<in> affine hull S"
     by (meson a hull_subset rel_interior_subset rev_subsetD)
   let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
   obtain B where "B > 0" and B: "S \<subseteq> ball a B"
     using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
   have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
     by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
   with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
     using rel_interior_subset subsetCE by blast
   with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
     using divide_pos_pos zero_less_norm_iff by fastforce
   have bdd: "bdd_below ?D"
     by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
   have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
                     \<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
     using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
   define d where "d = Inf ?D"
   obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
   proof -
     obtain e where "e>0"
             and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex a by blast
     show thesis
     proof (rule_tac \<epsilon> = "e / norm l" in that)
       show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
     next
       show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
       proof (rule e)
         show "a + \<eta> *\<^sub>R l \<in> affine hull S"
           by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
         show "dist (a + \<eta> *\<^sub>R l) a < e"
           using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
       qed
     qed
   qed
   have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
     unfolding d_def using cInf_lower [OF _ bdd]
     by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
   have "\<epsilon> \<le> d"
     unfolding d_def
     using \<epsilon> dual_order.strict_implies_order le_less_linear 
     by (blast intro: cInf_greatest [OF nonMT])
   with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
   have "a + d *\<^sub>R l \<notin> rel_interior S"
   proof
     assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
     obtain e where "e > 0"
              and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex adl by blast
     have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
     proof (rule cInf_greatest [OF nonMT], clarsimp)
       fix x::real
       assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
       show "d + e / norm l \<le> x"
       proof (cases "x < d")
         case True with inint nonrel \<open>0 < x\<close>
           show ?thesis by auto
       next
         case False
           then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
             by (simp add: field_simps \<open>l \<noteq> 0\<close>)
           have ain: "a + x *\<^sub>R l \<in> affine hull S"
             by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
           show ?thesis
             using e [OF ain] nonrel dle by force
       qed
     qed
     then show False
       using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
   qed
   moreover have "a + d *\<^sub>R l \<in> closure S"
   proof (clarsimp simp: closure_approachable)
     fix \<eta>::real assume "0 < \<eta>"
     have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
     proof (rule subsetD [OF rel_interior_subset inint])
       show "d - min d (\<eta> / 2 / norm l) < d"
         using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
     qed auto
     have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
       by (metis min_def mult_left_mono norm_ge_zero order_refl)
     also have "... < \<eta>"
       using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
     finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
     show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
       using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> 
       by (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI) (auto simp: algebra_simps)
   qed
   ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
     by (simp add: rel_frontier_def)
   show ?thesis
     by (rule that [OF \<open>0 < d\<close> infront inint])
 qed
 
 corollary ray_to_frontier:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
       and a: "a \<in> interior S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
 proof -
   have \<section>: "interior S = rel_interior S"
     using a rel_interior_nonempty_interior by auto
   then have "a \<in> rel_interior S"
     using a by simp
   moreover have "a + l \<in> affine hull S"
     using a affine_hull_nonempty_interior by blast
   ultimately show thesis
     by (metis \<section> \<open>bounded S\<close> \<open>l \<noteq> 0\<close> frontier_def ray_to_rel_frontier rel_frontier_def that)
 qed
 
 
 lemma segment_to_rel_frontier_aux:
   fixes x :: "'a::euclidean_space"
   assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                    "open_segment x z \<subseteq> rel_interior S"
 proof -
   have "x + (y - x) \<in> affine hull S"
     using hull_inc [OF y] by auto
   then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
                   and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
     by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
   show ?thesis
   proof
     show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
       by (simp add: df)
   next
     have "open_segment x y \<subseteq> rel_interior S"
       using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
     moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
       using xy \<open>0 < d\<close> that by (force simp: in_segment algebra_simps)
     ultimately have "1 \<le> d"
       using df rel_frontier_def by fastforce
     moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
       by (metis \<open>0 < d\<close> add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
     ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
       unfolding in_segment
       by (rule_tac x="1/d" in exI) (auto simp: algebra_simps)
   next
     show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
     proof (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
       show "x + d *\<^sub>R (y - x) \<in> closure S"
         using df rel_frontier_def by auto
     qed
   qed
 qed
 
 lemma segment_to_rel_frontier:
   fixes x :: "'a::euclidean_space"
   assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
       and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                   "open_segment x z \<subseteq> rel_interior S"
 proof (cases "x=y")
   case True
   with xy have "S \<noteq> {x}"
     by blast
   with True show ?thesis
     by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
 next
   case False
   then show ?thesis
     using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
 proposition rel_frontier_not_sing:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
     shows "rel_frontier S \<noteq> {a}"
 proof (cases "S = {}")
   case True  then show ?thesis  by simp
 next
   case False
   then obtain z where "z \<in> S"
     by blast
   then show ?thesis
   proof (cases "S = {z}")
     case True then show ?thesis  by simp
   next
     case False
     then obtain w where "w \<in> S" "w \<noteq> z"
       using \<open>z \<in> S\<close> by blast
     show ?thesis
     proof
       assume "rel_frontier S = {a}"
       then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
         using \<open>w \<noteq> z\<close> by auto
       then show False
       proof cases
         case 1
         then have w: "w \<in> rel_interior S"
           using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "w + (w - z) \<in> affine hull S"
           by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
         moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       next
         case 2
         then have z: "z \<in> rel_interior S"
           using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "z + (z - w) \<in> affine hull S"
           by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
         moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       qed
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity on direct sums\<close>
 
 lemma closure_sum:
   fixes S T :: "'a::real_normed_vector set"
   shows "closure S + closure T \<subseteq> closure (S + T)"
   unfolding set_plus_image closure_Times [symmetric] split_def
   by (intro closure_bounded_linear_image_subset bounded_linear_add
     bounded_linear_fst bounded_linear_snd)
 
+lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)"
+  unfolding linear_iff by (simp add: algebra_simps)
+
 lemma rel_interior_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S + T) = rel_interior S + rel_interior T"
 proof -
   have "rel_interior S + rel_interior T = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
     by (simp add: set_plus_image)
   also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
     using rel_interior_Times assms by auto
   also have "\<dots> = rel_interior (S + T)"
     using fst_snd_linear convex_Times assms
       rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp add: set_plus_image)
   finally show ?thesis ..
 qed
 
 lemma rel_interior_sum_gen:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i)"
   shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
   using rel_interior_sum rel_interior_sing[of "0"] assms
   by (subst sum_set_cond_linear[of convex], auto simp add: convex_set_plus)
 
 lemma convex_rel_open_direct_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S \<times> T) \<and> rel_open (S \<times> T)"
   by (metis assms convex_Times rel_interior_Times rel_open_def)
 
 lemma convex_rel_open_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S + T) \<and> rel_open (S + T)"
   by (metis assms convex_set_plus rel_interior_sum rel_open_def)
 
 lemma convex_hull_finite_union_cones:
   assumes "finite I"
     and "I \<noteq> {}"
   assumes "\<And>i. i\<in>I \<Longrightarrow> convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) = sum S I"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?lhs"
     then obtain c xs where
       x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
       using convex_hull_finite_union[of I S] assms by auto
     define s where "s i = c i *\<^sub>R xs i" for i
     have "\<forall>i\<in>I. s i \<in> S i"
         using s_def x assms by (simp add: mem_cone)
     moreover have "x = sum s I" using x s_def by auto
     ultimately have "x \<in> ?rhs"
       using set_sum_alt[of I S] assms by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
       using set_sum_alt[of I S] assms by auto
     define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
     then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
       using x assms by auto
     moreover have "\<forall>i\<in>I. xs i \<in> S i"
       using x xs_def assms by (simp add: cone_def)
     moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
       by auto
     moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
       using assms by auto
     ultimately have "x \<in> ?lhs"
       using assms
       apply (simp add: convex_hull_finite_union[of I S])
       by (rule_tac x = "(\<lambda>i. 1 / (card I))" in exI) auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_hull_union_cones_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "cone S"
     and "S \<noteq> {}"
   assumes "convex T"
     and "cone T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) = S + T"
 proof -
   define I :: "nat set" where "I = {1, 2}"
   define A where "A i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(A ` I) = S \<union> T"
     using A_def I_def by auto
   then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(A ` I) = sum A I"
     using A_def I_def
     by (metis assms convex_hull_finite_union_cones empty_iff finite.emptyI finite.insertI insertI1)
   moreover have "sum A I = S + T"
     using A_def I_def by (force simp add: set_plus_def)
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_hull_union:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "finite I"
     and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
   shows "rel_interior (convex hull (\<Union>(S ` I))) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
       (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
   (is "?lhs = ?rhs")
 proof (cases "I = {}")
   case True
   then show ?thesis
     using convex_hull_empty by auto
 next
   case False
   define C0 where "C0 = convex hull (\<Union>(S ` I))"
   have "\<forall>i\<in>I. C0 \<ge> S i"
     unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
   define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
   define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
   have "\<forall>i\<in>I. K i \<noteq> {}"
     unfolding K_def using assms
     by (simp add: cone_hull_empty_iff[symmetric])
   have convK: "\<forall>i\<in>I. convex (K i)"
     unfolding K_def
     by (simp add: assms(2) convex_Times convex_cone_hull)
   have "K0 \<supseteq> K i" if  "i \<in> I" for i
     unfolding K0_def K_def
     by (simp add: Sigma_mono \<open>\<forall>i\<in>I. S i \<subseteq> C0\<close> hull_mono that)
   then have "K0 \<supseteq> \<Union>(K ` I)" by auto
   moreover have "convex K0"
     unfolding K0_def by (simp add: C0_def convex_Times convex_cone_hull)
   ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
     using hull_minimal[of _ "K0" "convex"] by blast
   have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
     using K_def by (simp add: hull_subset)
   then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
     by auto
   then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
     by (simp add: hull_mono)
   then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
     unfolding C0_def
     using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
     by auto
   moreover have "cone (convex hull (\<Union>(K ` I)))"
     by (simp add: K_def cone_Union cone_cone_hull cone_convex_hull)
   ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
     unfolding K0_def
     using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
     by blast
   then have "K0 = convex hull (\<Union>(K ` I))"
     using geq by auto
   also have "\<dots> = sum K I"
     using assms False \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> cone_hull_eq convK 
     by (intro convex_hull_finite_union_cones; fastforce simp: K_def)
   finally have "K0 = sum K I" by auto
   then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
     using rel_interior_sum_gen[of I K] convK by auto
   {
     fix x
     assume "x \<in> ?lhs"
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
       by auto
     then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
       using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
     {
       fix i
       assume "i \<in> I"
       then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
         using k K_def assms by auto
       then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
         using rel_interior_convex_cone[of "S i"] by auto
     }
     then obtain c s where cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
       by metis
     then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
       using k by (simp add: sum_prod)
     then have "x \<in> ?rhs"
       using k cs by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
         (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
       by auto
     define k where "k i = (c i, c i *\<^sub>R s i)" for i
     {
       fix i assume "i \<in> I"
       then have "k i \<in> rel_interior (K i)"
         using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
         by auto
     }
     then have "(1, x) \<in> rel_interior K0"
       using * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms cs
       by (simp add: k_def) (metis (mono_tags, lifting) sum_prod)
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 
 lemma convex_le_Inf_differential:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex_on I f"
     and "x \<in> interior I"
     and "y \<in> I"
   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
   (is "_ \<ge> _ + Inf (?F x) * (y - x)")
 proof (cases rule: linorder_cases)
   assume "x < y"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
   ultimately have "x < t" "t < y" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps split: split_min)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   define K where "K = x - e / 2"
   with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
     by (auto simp: dist_real_def)
   then have "K \<in> I"
     using \<open>interior I \<subseteq> I\<close> e(2) by blast
 
   have "Inf (?F x) \<le> (f x - f y) / (x - y)"
   proof (intro bdd_belowI cInf_lower2)
     show "(f x - f t) / (x - t) \<in> ?F x"
       using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
     show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
       using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
       by (rule convex_on_diff)
   next
     fix y
     assume "y \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
     show "(f K - f x) / (K - x) \<le> y" by auto
   qed
   then show ?thesis
     using \<open>x < y\<close> by (simp add: field_simps)
 next
   assume "y < x"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = x + e / 2"
   ultimately have "x < t" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "(f x - f y) / (x - y) \<le> Inf (?F x)"
   proof (rule cInf_greatest)
     have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
       using \<open>y < x\<close> by (auto simp: field_simps)
     also
     fix z
     assume "z \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
     have "(f y - f x) / (y - x) \<le> z"
       by auto
     finally show "(f x - f y) / (x - y) \<le> z" .
   next
     have "x + e / 2 \<in> ball x e"
       using e by (auto simp: dist_real_def)
     with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
       by auto
     then show "?F x \<noteq> {}"
       by blast
   qed
   then show ?thesis
     using \<open>y < x\<close> by (simp add: field_simps)
 qed simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Explicit formulas for interior and relative interior of convex hull\<close>
 
 lemma at_within_cbox_finite:
   assumes "x \<in> box a b" "x \<notin> S" "finite S"
   shows "(at x within cbox a b - S) = at x"
 proof -
   have "interior (cbox a b - S) = box a b - S"
     using \<open>finite S\<close> by (simp add: interior_diff finite_imp_closed)
   then show ?thesis
     using at_within_interior assms by fastforce
 qed
 
 lemma affine_independent_convex_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S" "T \<subseteq> S"
     shows "convex hull T = affine hull T \<inter> convex hull S"
 proof -
   have fin: "finite S" "finite T" using assms aff_independent_finite finite_subset by auto
   have "convex hull T \<subseteq> affine hull T"
     using convex_hull_subset_affine_hull by blast
   moreover have "convex hull T \<subseteq> convex hull S"
     using assms hull_mono by blast
   moreover have "affine hull T \<inter> convex hull S \<subseteq> convex hull T"
   proof -
     have 0: "\<And>u. sum u S = 0 \<Longrightarrow> (\<forall>v\<in>S. u v = 0) \<or> (\<Sum>v\<in>S. u v *\<^sub>R v) \<noteq> 0"
       using affine_dependent_explicit_finite assms(1) fin(1) by auto
     show ?thesis
     proof (clarsimp simp add: affine_hull_finite fin)
       fix u
       assume S: "(\<Sum>v\<in>T. u v *\<^sub>R v) \<in> convex hull S"
         and T1: "sum u T = 1"
       then obtain v where v: "\<forall>x\<in>S. 0 \<le> v x" "sum v S = 1" "(\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>v\<in>T. u v *\<^sub>R v)"
         by (auto simp add: convex_hull_finite fin)
       { fix x
         assume"x \<in> T"
         then have S: "S = (S - T) \<union> T" \<comment> \<open>split into separate cases\<close>
           using assms by auto
         have [simp]: "(\<Sum>x\<in>T. v x *\<^sub>R x) + (\<Sum>x\<in>S - T. v x *\<^sub>R x) = (\<Sum>x\<in>T. u x *\<^sub>R x)"
           "sum v T + sum v (S - T) = 1"
           using v fin S
           by (auto simp: sum.union_disjoint [symmetric] Un_commute)
         have "(\<Sum>x\<in>S. if x \<in> T then v x - u x else v x) = 0"
              "(\<Sum>x\<in>S. (if x \<in> T then v x - u x else v x) *\<^sub>R x) = 0"
           using v fin T1
           by (subst S, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
       } note [simp] = this
       have "(\<forall>x\<in>T. 0 \<le> u x)"
         using 0 [of "\<lambda>x. if x \<in> T then v x - u x else v x"] \<open>T \<subseteq> S\<close> v(1) by fastforce
       then show "(\<Sum>v\<in>T. u v *\<^sub>R v) \<in> convex hull T"
         using 0 [of "\<lambda>x. if x \<in> T then v x - u x else v x"] \<open>T \<subseteq> S\<close> T1
         by (fastforce simp add: convex_hull_finite fin)
     qed
   qed
   ultimately show ?thesis
     by blast
 qed
 
 lemma affine_independent_span_eq:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S" "card S = Suc (DIM ('a))"
     shows "affine hull S = UNIV"
 proof (cases "S = {}")
   case True then show ?thesis
     using assms by simp
 next
   case False
     then obtain a T where T: "a \<notin> T" "S = insert a T"
       by blast
     then have fin: "finite T" using assms
       by (metis finite_insert aff_independent_finite)
     have "UNIV \<subseteq> (+) a ` span ((\<lambda>x. x - a) ` T)"
     proof (intro card_ge_dim_independent Fun.vimage_subsetD)
       show "independent ((\<lambda>x. x - a) ` T)"
         using T affine_dependent_iff_dependent assms(1) by auto
       show "dim ((+) a -` UNIV) \<le> card ((\<lambda>x. x - a) ` T)"
         using assms T fin by (auto simp: card_image inj_on_def)
     qed (use surj_plus in auto)
     then show ?thesis
       using T(2) affine_hull_insert_span_gen equalityI by fastforce
 qed
 
 lemma affine_independent_span_gt:
   fixes S :: "'a::euclidean_space set"
   assumes ind: "\<not> affine_dependent S" and dim: "DIM ('a) < card S"
     shows "affine hull S = UNIV"
 proof (intro affine_independent_span_eq [OF ind] antisym)
   show "card S \<le> Suc DIM('a)"
     using aff_independent_finite affine_dependent_biggerset ind by fastforce
   show "Suc DIM('a) \<le> card S"
     using Suc_leI dim by blast
 qed
 
 lemma empty_interior_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" and dim: "card S \<le> DIM ('a)"
     shows "interior(affine hull S) = {}"
   using assms
 proof (induct S rule: finite_induct)
   case (insert x S)
   then have "dim (span ((\<lambda>y. y - x) ` S)) < DIM('a)"
     by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
   then show ?case
     by (simp add: empty_interior_lowdim affine_hull_insert_span_gen interior_translation)
 qed auto
 
 lemma empty_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" and dim: "card S \<le> DIM ('a)"
     shows "interior(convex hull S) = {}"
   by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
             interior_mono empty_interior_affine_hull [OF assms])
 
 lemma explicit_subset_rel_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   shows "finite S
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> S. 0 < u x \<and> u x < 1) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}
              \<subseteq> rel_interior (convex hull S)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=S, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
   fixes S :: "'a::euclidean_space set"
   shows "finite S
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}
              \<subseteq> rel_interior (convex hull S)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=S, simplified])
 
 lemma rel_interior_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "rel_interior(convex hull S) =
          {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
          (is "?lhs = ?rhs")
 proof
   show "?rhs \<le> ?lhs"
     by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
 next
   show "?lhs \<le> ?rhs"
   proof (cases "\<exists>a. S = {a}")
     case True then show "?lhs \<le> ?rhs"
       by force
   next
     case False
     have fs: "finite S"
       using assms by (simp add: aff_independent_finite)
     { fix a b and d::real
       assume ab: "a \<in> S" "b \<in> S" "a \<noteq> b"
       then have S: "S = (S - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
         by auto
       have "(\<Sum>x\<in>S. if x = a then - d else if x = b then d else 0) = 0"
            "(\<Sum>x\<in>S. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
         using ab fs
         by (subst S, subst sum.union_disjoint, auto)+
     } note [simp] = this
     { fix y
       assume y: "y \<in> convex hull S" "y \<notin> ?rhs"
       have *: False if
         ua: "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "\<not> 0 < u a" "a \<in> S"
         and yT: "y = (\<Sum>x\<in>S. u x *\<^sub>R x)" "y \<in> T" "open T"
         and sb: "T \<inter> affine hull S \<subseteq> {w. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = w}"
       for u T a
       proof -
         have ua0: "u a = 0"
           using ua by auto
         obtain b where b: "b\<in>S" "a \<noteq> b"
           using ua False by auto
         obtain e where e: "0 < e" "ball (\<Sum>x\<in>S. u x *\<^sub>R x) e \<subseteq> T"
           using yT by (auto elim: openE)
         with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
           by (auto intro: that [of "e / 2 / norm(a-b)"])
         have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> affine hull S"
           using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
         then have "(\<Sum>x\<in>S. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull S"
           using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
         then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull S"
           using d e yT by auto
         then obtain v where v: "\<forall>x\<in>S. 0 \<le> v x"
           "sum v S = 1"
           "(\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
           using subsetD [OF sb] yT
           by auto
         have aff: "\<And>u. sum u S = 0 \<Longrightarrow> (\<forall>v\<in>S. u v = 0) \<or> (\<Sum>v\<in>S. u v *\<^sub>R v) \<noteq> 0"
           using assms by (simp add: affine_dependent_explicit_finite fs)
         show False
           using ua b d v aff [of "\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)"]
           by (auto simp: algebra_simps sum_subtractf sum.distrib)
       qed
       have "y \<notin> rel_interior (convex hull S)"
         using y
         apply (simp add: mem_rel_interior)
         apply (auto simp: convex_hull_finite [OF fs])
         apply (drule_tac x=u in spec)
         apply (auto intro: *)
         done
     } with rel_interior_subset show "?lhs \<le> ?rhs"
       by blast
   qed
 qed
 
 lemma interior_convex_hull_explicit_minimal:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows
    "interior(convex hull S) =
              (if card(S) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> S. 0 < u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"  
    (is "_ = (if _ then _ else ?rhs)")
 proof (clarsimp simp: aff_independent_finite empty_interior_convex_hull assms)
   assume S: "\<not> card S \<le> DIM('a)"
   have "interior (convex hull S) = rel_interior(convex hull S)"
     using assms S by (simp add: affine_independent_span_gt rel_interior_interior)
   then show "interior(convex hull S) = ?rhs"
     by (simp add: assms S rel_interior_convex_hull_explicit)
 qed
 
 lemma interior_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows
    "interior(convex hull S) =
              (if card(S) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> S. 0 < u x \<and> u x < 1) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y})"
 proof -
   { fix u :: "'a \<Rightarrow> real" and a
     assume "card Basis < card S" and u: "\<And>x. x\<in>S \<Longrightarrow> 0 < u x" "sum u S = 1" and a: "a \<in> S"
     then have cs: "Suc 0 < card S"
       by (metis DIM_positive less_trans_Suc)
     obtain b where b: "b \<in> S" "a \<noteq> b"
     proof (cases "S \<le> {a}")
       case True
       then show thesis
         using cs subset_singletonD by fastforce
     qed blast
     have "u a + u b \<le> sum u {a,b}"
       using a b by simp
     also have "... \<le> sum u S"
       using a b u
       by (intro Groups_Big.sum_mono2) (auto simp: less_imp_le aff_independent_finite assms)
     finally have "u a < 1"
       using \<open>b \<in> S\<close> u by fastforce
   } note [simp] = this
   show ?thesis
     using assms by (force simp add: not_le interior_convex_hull_explicit_minimal)
 qed
 
 lemma interior_closed_segment_ge2:
   fixes a :: "'a::euclidean_space"
   assumes "2 \<le> DIM('a)"
     shows  "interior(closed_segment a b) = {}"
 using assms unfolding segment_convex_hull
 proof -
   have "card {a, b} \<le> DIM('a)"
     using assms
     by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
   then show "interior (convex hull {a, b}) = {}"
     by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
 qed
 
 lemma interior_open_segment:
   fixes a :: "'a::euclidean_space"
   shows  "interior(open_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (simp add: not_le, intro conjI impI)
   assume "2 \<le> DIM('a)"
   then show "interior (open_segment a b) = {}"
     using interior_closed_segment_ge2 interior_mono segment_open_subset_closed by blast
 next
   assume le2: "DIM('a) < 2"
   show "interior (open_segment a b) = open_segment a b"
   proof (cases "a = b")
     case True then show ?thesis by auto
   next
     case False
     with le2 have "affine hull (open_segment a b) = UNIV"
       by (simp add: False affine_independent_span_gt)
     then show "interior (open_segment a b) = open_segment a b"
       using rel_interior_interior rel_interior_open_segment by blast
   qed
 qed
 
 lemma interior_closed_segment:
   fixes a :: "'a::euclidean_space"
   shows "interior(closed_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by simp
 next
   case False
   then have "closure (open_segment a b) = closed_segment a b"
     by simp
   then show ?thesis
     by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
 qed
 
 lemmas interior_segment = interior_closed_segment interior_open_segment
 
 lemma closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b = closed_segment c d \<longleftrightarrow> {a,b} = {c,d}"
 proof
   assume abcd: "closed_segment a b = closed_segment c d"
   show "{a,b} = {c,d}"
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis by force
   next
     case False
     then have neq: "a \<noteq> b \<and> c \<noteq> d" by force
     have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
       using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
     have "b \<in> {c, d}"
     proof -
       have "insert b (closed_segment c d) = closed_segment c d"
         using abcd by blast
       then show ?thesis
         by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
     qed
     moreover have "a \<in> {c, d}"
       by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
     ultimately show "{a, b} = {c, d}"
       using neq by fastforce
   qed
 next
   assume "{a,b} = {c,d}"
   then show "closed_segment a b = closed_segment c d"
     by (simp add: segment_convex_hull)
 qed
 
 lemma closed_open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b \<noteq> open_segment c d"
 by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
 
 lemma open_closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<noteq> closed_segment c d"
 using closed_open_segment_eq by blast
 
 lemma open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b = open_segment c d \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
         (is "?lhs = ?rhs")
 proof
   assume abcd: ?lhs
   show ?rhs
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis
       using finite_open_segment by fastforce
   next
     case False
     then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
     with abcd show ?rhs
       unfolding open_segment_def
       by (metis (no_types) abcd closed_segment_eq closure_open_segment)
   qed
 next
   assume ?rhs
   then show ?lhs
     by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Similar results for closure and (relative or absolute) frontier\<close>
 
 lemma closure_convex_hull [simp]:
   fixes S :: "'a::euclidean_space set"
   shows "compact S ==> closure(convex hull S) = convex hull S"
   by (simp add: compact_imp_closed compact_convex_hull)
 
 lemma rel_frontier_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "rel_frontier(convex hull S) =
          {y. \<exists>u. (\<forall>x \<in> S. 0 \<le> u x) \<and> (\<exists>x \<in> S. u x = 0) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
   have "\<And>u y v.
        \<lbrakk>y \<in> S; u y = 0; sum u S = 1; \<forall>x\<in>S. 0 < v x;
         sum v S = 1; (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)\<rbrakk>
        \<Longrightarrow> \<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
     apply (force simp: sum_subtractf scaleR_diff_left)
     done
   then show ?thesis
     using fs assms
     apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit)
     apply (auto simp: convex_hull_finite)
     apply (metis less_eq_real_def) 
     by (simp add: affine_dependent_explicit_finite)
 qed
 
 lemma frontier_convex_hull_explicit:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
          {y. \<exists>u. (\<forall>x \<in> S. 0 \<le> u x) \<and> (DIM ('a) < card S \<longrightarrow> (\<exists>x \<in> S. u x = 0)) \<and>
              sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
   proof (cases "DIM ('a) < card S")
     case True
     with assms fs show ?thesis
       by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
                     interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
   next
     case False
     then have "card S \<le> DIM ('a)"
       by linarith
     then show ?thesis
       using assms fs
       apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
       apply (simp add: convex_hull_finite)
       done
   qed
 qed
 
 lemma rel_frontier_convex_hull_cases:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "rel_frontier(convex hull S) = \<Union>{convex hull (S - {x}) |x. x \<in> S}"
 proof -
   have fs: "finite S"
     using assms by (simp add: aff_independent_finite)
   { fix u a
   have "\<forall>x\<in>S. 0 \<le> u x \<Longrightarrow> a \<in> S \<Longrightarrow> u a = 0 \<Longrightarrow> sum u S = 1 \<Longrightarrow>
             \<exists>x v. x \<in> S \<and>
                   (\<forall>x\<in>S - {x}. 0 \<le> v x) \<and>
                       sum v (S - {x}) = 1 \<and> (\<Sum>x\<in>S - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     apply (rule_tac x=a in exI)
     apply (rule_tac x=u in exI)
     apply (simp add: Groups_Big.sum_diff1 fs)
     done }
   moreover
   { fix a u
     have "a \<in> S \<Longrightarrow> \<forall>x\<in>S - {a}. 0 \<le> u x \<Longrightarrow> sum u (S - {a}) = 1 \<Longrightarrow>
             \<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and>
                  (\<exists>x\<in>S. v x = 0) \<and> sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S - {a}. u x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. if x = a then 0 else u x" in exI)
     apply (auto simp: sum.If_cases Diff_eq if_smult fs)
     done }
   ultimately show ?thesis
     using assms
     apply (simp add: rel_frontier_convex_hull_explicit)
     apply (auto simp add: convex_hull_finite fs Union_SetCompr_eq)
     done
 qed
 
 lemma frontier_convex_hull_eq_rel_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
            (if card S \<le> DIM ('a) then convex hull S else rel_frontier(convex hull S))"
   using assms
   unfolding rel_frontier_def frontier_def
   by (simp add: affine_independent_span_gt rel_interior_interior
                 finite_imp_compact empty_interior_convex_hull aff_independent_finite)
 
 lemma frontier_convex_hull_cases:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent S"
   shows "frontier(convex hull S) =
            (if card S \<le> DIM ('a) then convex hull S else \<Union>{convex hull (S - {x}) |x. x \<in> S})"
 by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
 
 lemma in_frontier_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" "card S \<le> Suc (DIM ('a))" "x \<in> S"
   shows   "x \<in> frontier(convex hull S)"
 proof (cases "affine_dependent S")
   case True
   with assms obtain y where "y \<in> S" and y: "y \<in> affine hull (S - {y})"
     by (auto simp: affine_dependent_def)
   moreover have "x \<in> closure (convex hull S)"
     by (meson closure_subset hull_inc subset_eq \<open>x \<in> S\<close>)
   moreover have "x \<notin> interior (convex hull S)"
     using assms
     by (metis Suc_mono affine_hull_convex_hull affine_hull_nonempty_interior \<open>y \<in> S\<close> y card.remove empty_iff empty_interior_affine_hull finite_Diff hull_redundant insert_Diff interior_UNIV not_less)
   ultimately show ?thesis
     unfolding frontier_def by blast
 next
   case False
   { assume "card S = Suc (card Basis)"
     then have cs: "Suc 0 < card S"
       by (simp)
     with subset_singletonD have "\<exists>y \<in> S. y \<noteq> x"
       by (cases "S \<le> {x}") fastforce+
   } note [dest!] = this
   show ?thesis using assms
     unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
     by (auto simp: le_Suc_eq hull_inc)
 qed
 
 lemma not_in_interior_convex_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "finite S" "card S \<le> Suc (DIM ('a))" "x \<in> S"
   shows   "x \<notin> interior(convex hull S)"
 using in_frontier_convex_hull [OF assms]
 by (metis Diff_iff frontier_def)
 
 lemma interior_convex_hull_eq_empty:
   fixes S :: "'a::euclidean_space set"
   assumes "card S = Suc (DIM ('a))"
   shows   "interior(convex hull S) = {} \<longleftrightarrow> affine_dependent S"
 proof 
   show "affine_dependent S \<Longrightarrow> interior (convex hull S) = {}"
   proof (clarsimp simp: affine_dependent_def)
     fix a b
     assume "b \<in> S" "b \<in> affine hull (S - {b})"
     then have "interior(affine hull S) = {}" using assms
       by (metis DIM_positive One_nat_def Suc_mono card.remove card.infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
     then show "interior (convex hull S) = {}" 
       using affine_hull_nonempty_interior by fastforce
   qed
 next
   show "interior (convex hull S) = {} \<Longrightarrow> affine_dependent S"
     by (metis affine_hull_convex_hull affine_hull_empty affine_independent_span_eq assms convex_convex_hull empty_not_UNIV rel_interior_eq_empty rel_interior_interior)
 qed
 
 
 subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> coplanar  where
    "coplanar S \<equiv> \<exists>u v w. S \<subseteq> affine hull {u,v,w}"
 
 lemma collinear_affine_hull:
   "collinear S \<longleftrightarrow> (\<exists>u v. S \<subseteq> affine hull {u,v})"
 proof (cases "S={}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x where x: "x \<in> S" by auto
   { fix u
     assume *: "\<And>x y. \<lbrakk>x\<in>S; y\<in>S\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
     have "\<And>y c. x - y = c *\<^sub>R u \<Longrightarrow> \<exists>a b. y = a *\<^sub>R x + b *\<^sub>R (x + u) \<and> a + b = 1"
       by (rule_tac x="1+c" in exI, rule_tac x="-c" in exI, simp add: algebra_simps)
     then have "\<exists>u v. S \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
       using * [OF x] by (rule_tac x=x in exI, rule_tac x="x+u" in exI, force)
   } moreover
   { fix u v x y
     assume *: "S \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
     have "\<exists>c. x - y = c *\<^sub>R (v-u)" if "x\<in>S" "y\<in>S"
     proof -
       obtain a r where "a + r = 1" "x = a *\<^sub>R u + r *\<^sub>R v"
         using "*" \<open>x \<in> S\<close> by blast
       moreover
       obtain b s where "b + s = 1" "y = b *\<^sub>R u + s *\<^sub>R v"
         using "*" \<open>y \<in> S\<close> by blast
       ultimately have "x - y = (r-s) *\<^sub>R (v-u)" 
         by (simp add: algebra_simps) (metis scaleR_left.add)
       then show ?thesis
         by blast
     qed
   } ultimately
   show ?thesis
   unfolding collinear_def affine_hull_2
     by blast
 qed
 
 lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
   by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
 
 lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
   unfolding open_segment_def
   by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
       convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
 
 lemma collinear_between_cases:
   fixes c :: "'a::euclidean_space"
   shows "collinear {a,b,c} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
     by (auto simp: collinear_alt)
   show ?rhs
     using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
 next
   assume ?rhs
   then show ?lhs
     unfolding between_mem_convex_hull
     by (metis (no_types, opaque_lifting) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
 qed
 
 
 lemma subset_continuous_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "continuous_on (closed_segment a b) f"
   shows "closed_segment (f a) (f b) \<subseteq> image f (closed_segment a b)"
 by (metis connected_segment convex_contains_segment ends_in_segment imageI
            is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
 
 lemma continuous_injective_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
   shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
 proof
   show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
     by (metis subset_continuous_image_segment_1 contf)
   show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
   proof (cases "a = b")
     case True
     then show ?thesis by auto
   next
     case False
     then have fnot: "f a \<noteq> f b"
       using inj_onD injf by fastforce
     moreover
     have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def)
       assume fa: "f a \<in> closed_segment (f c) (f b)"
       moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
         by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
       ultimately have "f a \<in> f ` closed_segment c b"
         by blast
       then have a: "a \<in> closed_segment c b"
         by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
       have cb: "closed_segment c b \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f a = f c"
       proof (rule between_antisym)
         show "between (f c, f b) (f a)"
           by (simp add: between_mem_segment fa)
         show "between (f a, f b) (f c)"
           by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
       qed
     qed
     moreover
     have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def fnot eq_commute)
       assume fb: "f b \<in> closed_segment (f a) (f c)"
       moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
         by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
       ultimately have "f b \<in> f ` closed_segment a c"
         by blast
       then have b: "b \<in> closed_segment a c"
         by (meson ends_in_segment inj_on_image_mem_iff injf subset_closed_segment that)
       have ca: "closed_segment a c \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f b = f c"
       proof (rule between_antisym)
         show "between (f c, f a) (f b)"
           by (simp add: between_commute between_mem_segment fb)
         show "between (f b, f a) (f c)"
           by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
       qed
     qed
     ultimately show ?thesis
       by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
   qed
 qed
 
 lemma continuous_injective_image_open_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
     shows "f ` (open_segment a b) = open_segment (f a) (f b)"
 proof -
   have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
     by (metis (no_types, opaque_lifting) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
   also have "... = open_segment (f a) (f b)"
     using continuous_injective_image_segment_1 [OF assms]
     by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
   finally show ?thesis .
 qed
 
 lemma collinear_imp_coplanar:
   "collinear s ==> coplanar s"
 by (metis collinear_affine_hull coplanar_def insert_absorb2)
 
 lemma collinear_small:
   assumes "finite s" "card s \<le> 2"
     shows "collinear s"
 proof -
   have "card s = 0 \<or> card s = 1 \<or> card s = 2"
     using assms by linarith
   then show ?thesis using assms
     using card_eq_SucD numeral_2_eq_2 by (force simp: card_1_singleton_iff)
 qed
 
 lemma coplanar_small:
   assumes "finite s" "card s \<le> 3"
     shows "coplanar s"
 proof -
   consider "card s \<le> 2" | "card s = Suc (Suc (Suc 0))"
     using assms by linarith
   then show ?thesis
   proof cases
     case 1
     then show ?thesis
       by (simp add: \<open>finite s\<close> collinear_imp_coplanar collinear_small)
   next
     case 2
     then show ?thesis
       using hull_subset [of "{_,_,_}"]
       by (fastforce simp: coplanar_def dest!: card_eq_SucD)
   qed
 qed
 
 lemma coplanar_empty: "coplanar {}"
   by (simp add: coplanar_small)
 
 lemma coplanar_sing: "coplanar {a}"
   by (simp add: coplanar_small)
 
 lemma coplanar_2: "coplanar {a,b}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma coplanar_3: "coplanar {a,b,c}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma collinear_affine_hull_collinear: "collinear(affine hull s) \<longleftrightarrow> collinear s"
   unfolding collinear_affine_hull
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) \<longleftrightarrow> coplanar s"
   unfolding coplanar_def
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "coplanar S" "linear f" shows "coplanar(f ` S)"
 proof -
   { fix u v w
     assume "S \<subseteq> affine hull {u, v, w}"
     then have "f ` S \<subseteq> f ` (affine hull {u, v, w})"
       by (simp add: image_mono)
     then have "f ` S \<subseteq> affine hull (f ` {u, v, w})"
       by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
   } then
   show ?thesis
     by auto (meson assms(1) coplanar_def)
 qed
 
 lemma coplanar_translation_imp: 
   assumes "coplanar S" shows "coplanar ((\<lambda>x. a + x) ` S)"
 proof -
   obtain u v w where "S \<subseteq> affine hull {u,v,w}"
     by (meson assms coplanar_def)
   then have "(+) a ` S \<subseteq> affine hull {u + a, v + a, w + a}"
     using affine_hull_translation [of a "{u,v,w}" for u v w]
     by (force simp: add.commute)
   then show ?thesis
     unfolding coplanar_def by blast
 qed
 
 lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` S) \<longleftrightarrow> coplanar S"
     by (metis (no_types) coplanar_translation_imp translation_galois)
 
 lemma coplanar_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f" shows "coplanar(f ` S) = coplanar S"
 proof
   assume "coplanar S"
   then show "coplanar (f ` S)"
     using assms(1) coplanar_linear_image by blast
 next
   obtain g where g: "linear g" "g \<circ> f = id"
     using linear_injective_left_inverse [OF assms]
     by blast
   assume "coplanar (f ` S)"
   then show "coplanar S"
     by (metis coplanar_linear_image g(1) g(2) id_apply image_comp image_id)
 qed
 
 lemma coplanar_subset: "\<lbrakk>coplanar t; S \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar S"
   by (meson coplanar_def order_trans)
 
 lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
   by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
 
 lemma collinear_3_imp_in_affine_hull:
   assumes "collinear {a,b,c}" "a \<noteq> b" shows "c \<in> affine hull {a,b}"
 proof -
   obtain u x y where "b - a = y *\<^sub>R u" "c - a = x *\<^sub>R u"
     using assms unfolding collinear_def by auto
   with \<open>a \<noteq> b\<close> have "\<exists>v. c = (1 - x / y) *\<^sub>R a + v *\<^sub>R b \<and> 1 - x / y + v = 1"
     by (simp add: algebra_simps)
   then show ?thesis
     by (simp add: hull_inc mem_affine)
 qed
 
 lemma collinear_3_affine_hull:
   assumes "a \<noteq> b"
   shows "collinear {a,b,c} \<longleftrightarrow> c \<in> affine hull {a,b}"
   using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
 
 lemma collinear_3_eq_affine_dependent:
   "collinear{a,b,c} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
 proof (cases "a = b \<or> a = c \<or> b = c")
   case True
   then show ?thesis
     by (auto simp: insert_commute)
 next
   case False
   then have "collinear{a,b,c}" if "affine_dependent {a,b,c}"
     using that unfolding affine_dependent_def
     by (auto simp: insert_Diff_if; metis affine_hull_3_imp_collinear insert_commute)
   moreover
   have "affine_dependent {a,b,c}" if "collinear{a,b,c}"
     using False that by (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
   ultimately
   show ?thesis
     using False by blast
 qed
 
 lemma affine_dependent_imp_collinear_3:
   "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
   by (simp add: collinear_3_eq_affine_dependent)
 
 lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
   by (auto simp add: collinear_def)
 
 lemma collinear_3_expand:
    "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
 proof -
   have "collinear{a,b,c} = collinear{a,c,b}"
     by (simp add: insert_commute)
   also have "... = collinear {0, a - c, b - c}"
     by (simp add: collinear_3)
   also have "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
     by (simp add: collinear_lemma)
   also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
     by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
   finally show ?thesis .
 qed
 
 lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
 proof
   assume "collinear S"
   then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
     by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
   then show "aff_dim S \<le> 1"
     using order_trans by fastforce
 next
   assume "aff_dim S \<le> 1"
   then have le1: "aff_dim (affine hull S) \<le> 1"
     by simp
   obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
     using affine_basis_exists [of S] by auto
   then have "finite B" "card B \<le> 2"
     using B le1 by (auto simp: affine_independent_iff_card)
   then have "collinear B"
     by (rule collinear_small)
   then show "collinear S"
     by (metis \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
 qed
 
 lemma collinear_midpoint: "collinear{a, midpoint a b, b}"
 proof -
   have \<section>: "\<lbrakk>a \<noteq> midpoint a b; b - midpoint a b \<noteq> - 1 *\<^sub>R (a - midpoint a b)\<rbrakk> \<Longrightarrow> b = midpoint a b"
     by (simp add: algebra_simps)
   show ?thesis
     by (auto simp: collinear_3 collinear_lemma intro: \<section>)
 qed
 
 lemma midpoint_collinear:
   fixes a b c :: "'a::real_normed_vector"
   assumes "a \<noteq> c"
     shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
 proof -
   have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
           "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
           "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
     by (auto simp: algebra_simps)
   have "b = midpoint a c \<Longrightarrow> collinear{a,b,c}"
     using collinear_midpoint by blast
   moreover have "b = midpoint a c \<longleftrightarrow> dist a b = dist b c" if "collinear{a,b,c}"
   proof -
     consider "a = c" | u where "b = u *\<^sub>R a + (1 - u) *\<^sub>R c"
       using \<open>collinear {a,b,c}\<close> unfolding collinear_3_expand by blast
     then show ?thesis
     proof cases
       case 2
       with assms have "dist a b = dist b c \<Longrightarrow> b = midpoint a c"
         by (simp add: dist_norm * midpoint_def scaleR_add_right del: divide_const_simps)
       then show ?thesis
         by (auto simp: dist_midpoint)
     qed (use assms in auto)
   qed
   ultimately show ?thesis by blast
 qed
 
 lemma between_imp_collinear:
   fixes x :: "'a :: euclidean_space"
   assumes "between (a,b) x"
     shows "collinear {a,x,b}"
 proof (cases "x = a \<or> x = b \<or> a = b")
   case True with assms show ?thesis
     by (auto simp: dist_commute)
 next
   case False 
   then have False if "\<And>c. b - x \<noteq> c *\<^sub>R (a - x)"
     using that [of "-(norm(b - x) / norm(x - a))"] assms
     by (simp add: between_norm vector_add_divide_simps flip: real_vector.scale_minus_right)
   then show ?thesis
     by (auto simp: collinear_3 collinear_lemma)
 qed
 
 lemma midpoint_between:
   fixes a b :: "'a::euclidean_space"
   shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
 proof (cases "a = c")
   case False
   show ?thesis
     using False between_imp_collinear between_midpoint(1) midpoint_collinear by blast
 qed (auto simp: dist_commute)
 
 lemma collinear_triples:
   assumes "a \<noteq> b"
     shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
           (is "?lhs = ?rhs")
 proof safe
   fix x
   assume ?lhs and "x \<in> S"
   then show "collinear {a, b, x}"
     using collinear_subset by force
 next
   assume ?rhs
   then have "\<forall>x \<in> S. collinear{a,x,b}"
     by (simp add: insert_commute)
   then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> insert a (insert b S)" for x
     using that assms collinear_3_expand by fastforce+
   have "\<exists>c. x - y = c *\<^sub>R (b - a)" 
     if x: "x \<in> insert a (insert b S)" and y: "y \<in> insert a (insert b S)" for x y
   proof -
     obtain u v where "x = u *\<^sub>R a + (1 - u) *\<^sub>R b" "y = v *\<^sub>R a + (1 - v) *\<^sub>R b"
       using "*" x y by presburger
     then have "x - y = (v - u) *\<^sub>R (b - a)"
       by (simp add: scale_left_diff_distrib scale_right_diff_distrib)
     then show ?thesis ..
   qed
   then show ?lhs
     unfolding collinear_def by metis
 qed
 
 lemma collinear_4_3:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> collinear{a,b,d}"
   using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
 
 lemma collinear_3_trans:
   assumes "collinear{a,b,c}" "collinear{b,c,d}" "b \<noteq> c"
     shows "collinear{a,b,d}"
 proof -
   have "collinear{b,c,a,d}"
     by (metis (full_types) assms collinear_4_3 insert_commute)
   then show ?thesis
     by (simp add: collinear_subset)
 qed
 
 lemma affine_hull_2_alt:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
 proof -
   have 1: "u *\<^sub>R a + v *\<^sub>R b = a + v *\<^sub>R (b - a)" if "u + v = 1" for u v
     using that
     by (simp add: algebra_simps flip: scaleR_add_left)
   have 2: "a + u *\<^sub>R (b - a) = (1 - u) *\<^sub>R a + u *\<^sub>R b" for u
     by (auto simp: algebra_simps)
   show ?thesis
     by (force simp add: affine_hull_2 dest: 1 intro!: 2)
 qed
 
 lemma interior_convex_hull_3_minimal:
   fixes a :: "'a::euclidean_space"
   assumes "\<not> collinear{a,b,c}" and 2: "DIM('a) = 2"
   shows "interior(convex hull {a,b,c}) =
          {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and> x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
         (is "?lhs = ?rhs")
 proof
   have abc: "a \<noteq> b" "a \<noteq> c" "b \<noteq> c" "\<not> affine_dependent {a, b, c}"
     using assms by (auto simp: collinear_3_eq_affine_dependent)
   with 2 show "?lhs \<subseteq> ?rhs"
     by (fastforce simp add: interior_convex_hull_explicit_minimal)
   show "?rhs \<subseteq> ?lhs"
     using abc 2
     apply (clarsimp simp add: interior_convex_hull_explicit_minimal)
     subgoal for x y z
       by (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI) auto
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about hyperplanes and halfspaces\<close>
 
 lemma halfspace_Int_eq:
      "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
      "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
   by auto
 
 lemma hyperplane_eq_Ex:
   assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
   by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
 
 lemma hyperplane_eq_empty:
      "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
   using hyperplane_eq_Ex
   by (metis (mono_tags, lifting) empty_Collect_eq inner_zero_left)
 
 lemma hyperplane_eq_UNIV:
    "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
 proof -
   have "a = 0 \<and> b = 0" if "UNIV \<subseteq> {x. a \<bullet> x = b}"
     using subsetD [OF that, where c = "((b+1) / (a \<bullet> a)) *\<^sub>R a"]
     by (simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_lt:
    "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
 proof -
   have "a = 0 \<and> b \<le> 0" if "{x. a \<bullet> x < b} \<subseteq> {}"
     using subsetD [OF that, where c = "((b-1) / (a \<bullet> a)) *\<^sub>R a"]
     by (force simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_gt:
   "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
   using halfspace_eq_empty_lt [of "-a" "-b"]
   by simp
 
 lemma halfspace_eq_empty_le:
    "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
 proof -
   have "a = 0 \<and> b < 0" if "{x. a \<bullet> x \<le> b} \<subseteq> {}"
     using subsetD [OF that, where c = "((b-1) / (a \<bullet> a)) *\<^sub>R a"]
     by (force simp add: field_split_simps split: if_split_asm)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_ge:
   "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
   using halfspace_eq_empty_le [of "-a" "-b"] by simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Use set distance for an easy proof of separation properties\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separation_closures:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
   obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis by auto
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on UNIV f"
     unfolding f_def by (intro continuous_intros continuous_on_setdist)
   show ?thesis
   proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
     show "{x. 0 < f x} \<inter> {x. f x < 0} = {}"
       by auto
     show "open {x. 0 < f x}"
       by (simp add: open_Collect_less contf)
     show "open {x. f x < 0}"
       by (simp add: open_Collect_less contf)
     have "\<And>x. x \<in> S \<Longrightarrow> setdist {x} T \<noteq> 0" "\<And>x. x \<in> T \<Longrightarrow> setdist {x} S \<noteq> 0"
       by (meson False assms disjoint_iff setdist_eq_0_sing_1)+
     then show "S \<subseteq> {x. 0 < f x}" "T \<subseteq> {x. f x < 0}"
       using less_eq_real_def by (fastforce simp add: f_def setdist_sing_in_set)+
   qed
 qed
 
 lemma separation_normal:
   fixes S :: "'a::euclidean_space set"
   assumes "closed S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 using separation_closures [of S T]
 by (metis assms closure_closed disjnt_def inf_commute)
 
 lemma separation_normal_local:
   fixes S :: "'a::euclidean_space set"
   assumes US: "closedin (top_of_set U) S"
       and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
   obtains S' T' where "openin (top_of_set U) S'"
                       "openin (top_of_set U) T'"
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
     using UT US by (blast dest: closedin_subset)
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on U f"
     unfolding f_def by (intro continuous_intros)
   show ?thesis
   proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
     show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
       by auto
     show "openin (top_of_set U) (U \<inter> f -` {0<..})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     show "openin (top_of_set U) (U \<inter> f -` {..<0})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     have "S \<subseteq> U" "T \<subseteq> U"
       using closedin_imp_subset assms by blast+
     then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> f -` {..<0}"
       using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
   qed
 qed
 
 lemma separation_normal_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 proof -
   have "closed S" "bounded S"
     using assms by (auto simp: compact_eq_bounded_closed)
   then obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
     by (auto dest!: bounded_subset_ballD)
   have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
     using assms r by blast+
   then obtain U V where UV: "open U" "open V" "S \<subseteq> U" "T \<union> - ball 0 r \<subseteq> V" "U \<inter> V = {}"
     by (meson  \<open>closed S\<close> separation_normal)
   then have "compact(closure U)"
     by (meson bounded_ball bounded_subset compact_closure compl_le_swap2 disjoint_eq_subset_Compl le_sup_iff)
   with UV show thesis
     using that by auto
 qed
 
 subsection\<open>Connectedness of the intersection of a chain\<close>
 
 proposition connected_chain:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof (cases "\<F> = {}")
   case True then show ?thesis
     by auto
 next
   case False
   then have cf: "compact(\<Inter>\<F>)"
     by (simp add: cc compact_Inter)
   have False if AB: "closed A" "closed B" "A \<inter> B = {}"
                 and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
   proof -
     obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
       using separation_normal [OF AB] by metis
     obtain K where "K \<in> \<F>" "compact K"
       using cc False by blast
     then obtain N where "open N" and "K \<subseteq> N"
       by blast
     let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
     obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
     proof (rule compactE [OF \<open>compact K\<close>])
       show "K \<subseteq> \<Union>(insert (U \<union> V) ((-) N ` \<F>))"
         using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
       show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
         by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
     qed
     then have "finite(\<D> - {U \<union> V})"
       by blast
     moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
       using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
     ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
       using finite_subset_image by metis
     obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
     proof (cases "\<G> = {}")
       case True
       with \<open>\<F> \<noteq> {}\<close> that show ?thesis
         by auto
     next
       case False
       have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
         by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
       with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
       have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
       proof induction
         case (insert X \<H>)
         show ?case
         proof (cases "\<H> = {}")
           case True then show ?thesis by auto
         next
           case False
           then have "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
             by (simp add: insert.prems)
           with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
             by metis
           have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
             by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
           then show ?thesis
           proof
             assume "N - J \<subseteq> N - X" with J show ?thesis
               by auto
           next
             assume "N - X \<subseteq> N - J"
             with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
               by auto
             with \<open>J \<in> \<H>\<close> show ?thesis
               by blast
           qed
         qed
       qed simp
       with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
     qed
     have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
       using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
     also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
       by (metis (no_types, opaque_lifting) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
     finally have "J \<inter> K \<subseteq> U \<union> V"
       by blast
     moreover have "connected(J \<inter> K)"
       by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
     moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
     moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
     ultimately show False
         using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close>  by auto
   qed
   with cf show ?thesis
     by (auto simp: connected_closed_set compact_imp_closed)
 qed
 
 lemma connected_chain_gen:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes X: "X \<in> \<F>" "compact X"
       and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof -
   have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
     using X by blast
   moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
   proof (rule connected_chain)
     show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
       using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
     show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
       using local.linear by blast
   qed
   ultimately show ?thesis
     by metis
 qed
 
 lemma connected_nest:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
 proof (rule connected_chain)
   show "\<And>A T. A \<in> range S \<and> T \<in> range S \<Longrightarrow> A \<subseteq> T \<or> T \<subseteq> A"
   by (metis image_iff le_cases nest)
 qed (use S in blast)
 
 lemma connected_nest_gen:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
 proof (rule connected_chain_gen [of "S k"])
   show "\<And>A T. A \<in> range S \<and> T \<in> range S \<Longrightarrow> A \<subseteq> T \<or> T \<subseteq> A"
     by (metis imageE le_cases nest)
 qed (use S in auto)
 
 subsection\<open>Proper maps, including projections out of compact sets\<close>
 
 lemma finite_indexed_bound:
   assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
     shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
 using A
 proof (induction A)
   case empty then show ?case by force
 next
   case (insert a A)
     then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
       by force
     then show ?case
       by (metis dual_order.trans insert_iff le_cases)
 qed
 
 proposition proper_map:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes "closedin (top_of_set S) K"
       and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
       and "f ` S \<subseteq> T"
     shows "closedin (top_of_set T) (f ` K)"
 proof -
   have "K \<subseteq> S"
     using assms closedin_imp_subset by metis
   obtain C where "closed C" and Keq: "K = S \<inter> C"
     using assms by (auto simp: closedin_closed)
   have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
   proof -
     obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
       using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
     then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
       by metis
     then have fX: "\<And>n. f (X n) = h n"
       by metis
     define \<Psi> where "\<Psi> \<equiv> \<lambda>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))}"
     have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
     proof (intro closed_Int_compact [OF \<open>closed C\<close> com] compact_sequence_with_limit)
       show "insert y (range (\<lambda>i. f (X (n + i)))) \<subseteq> T"
         using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> by blast
       show "(\<lambda>i. f (X (n + i))) \<longlonglongrightarrow> y"
         by (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
     qed
     then have comf: "compact (\<Psi> n)" for n
       by (simp add: Keq Int_def \<Psi>_def conj_commute)
     have ne: "\<Inter>\<F> \<noteq> {}"
              if "finite \<F>"
                 and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow> (\<exists>n. t = \<Psi> n)"
              for \<F>
     proof -
       obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = \<Psi> k"
         by (rule exE [OF finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>]], force+)
       have "X m \<in> \<Inter>\<F>"
         using X le_Suc_ex by (fastforce simp: \<Psi>_def dest: m)
       then show ?thesis by blast
     qed
     have "(\<Inter>n. \<Psi> n) \<noteq> {}"
     proof (rule compact_fip_Heine_Borel)
       show "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> range \<Psi>\<rbrakk> \<Longrightarrow> \<Inter> \<F>' \<noteq> {}"
         by (meson ne rangeE subset_eq)
     qed (use comf in blast)
     then obtain x where "x \<in> K" "\<And>n. (f x = y \<or> (\<exists>u. f x = h (n + u)))"
       by (force simp add: \<Psi>_def fX)
     then show ?thesis
       unfolding image_iff by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
   qed
   with assms closedin_subset show ?thesis
     by (force simp: closedin_limpt)
 qed
 
 
 lemma compact_continuous_image_eq:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes f: "inj_on f S"
   shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact(f ` T))"
            (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (metis continuous_on_subset compact_continuous_image)
 next
   assume RHS: ?rhs
   obtain g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     by (metis inv_into_f_f f)
   then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
     using that by fastforce
   have gfim: "g ` f ` S \<subseteq> S" using gf by auto
   have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
   proof -
     obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
       by (force simp: C RHS)
     moreover have "f ` C = (f ` S \<inter> g -` C)"
       using C gf by auto
     ultimately show ?thesis
       using C by auto
   qed
   show ?lhs
     using proper_map [OF _ _ gfim] **
     by (simp add: continuous_on_closed * closedin_imp_subset)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
 
 lemma convex_connected_collinear:
   fixes S :: "'a::euclidean_space set"
   assumes "collinear S"
     shows "convex S \<longleftrightarrow> connected S"
 proof
   assume "convex S"
   then show "connected S"
     using convex_connected by blast
 next
   assume S: "connected S"
   show "convex S"
   proof (cases "S = {}")
     case True
     then show ?thesis by simp
   next
     case False
     then obtain a where "a \<in> S" by auto
     have "collinear (affine hull S)"
       by (simp add: assms collinear_affine_hull_collinear)
     then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
       by (meson \<open>a \<in> S\<close> collinear hull_inc)
     then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>R z"
       by metis
     then have inj_f: "inj_on f (affine hull S)"
       by (metis diff_add_cancel inj_onI)
     have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
     proof -
       have "f x *\<^sub>R z = x - a"
         by (simp add: f hull_inc x)
       moreover have "f y *\<^sub>R z = y - a"
         by (simp add: f hull_inc y)
       ultimately show ?thesis
         by (simp add: scaleR_left.diff)
     qed
     have cont_f: "continuous_on (affine hull S) f"
     proof (clarsimp simp: dist_norm continuous_on_iff diff)
       show "\<And>x e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>y \<in> affine hull S. \<bar>f y - f x\<bar> * norm z < d \<longrightarrow> \<bar>f y - f x\<bar> < e"
         by (metis \<open>z \<noteq> 0\<close> mult_pos_pos mult_less_iff1 zero_less_norm_iff)
     qed
     then have conn_fS: "connected (f ` S)"
       by (meson S connected_continuous_image continuous_on_subset hull_subset)
     show ?thesis
     proof (clarsimp simp: convex_contains_segment)
       fix x y z
       assume "x \<in> S" "y \<in> S" "z \<in> closed_segment x y"
       have False if "z \<notin> S"
       proof -
         have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
         proof (rule continuous_injective_image_segment_1)
           show "continuous_on (closed_segment x y) f"
             by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
           show "inj_on f (closed_segment x y)"
             by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
         qed
         then have fz: "f z \<in> closed_segment (f x) (f y)"
           using \<open>z \<in> closed_segment x y\<close> by blast
         have "z \<in> affine hull S"
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
         then have fz_notin: "f z \<notin> f ` S"
           using hull_subset inj_f inj_onD that by fastforce
         moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
         proof -
           consider "f x \<le> f z \<and> f z \<le> f y" | "f y \<le> f z \<and> f z \<le> f x"
             using fz
             by (auto simp add: closed_segment_eq_real_ivl split: if_split_asm)
           then have "{..<f z} \<inter> f ` {x,y} \<noteq> {} \<and> {f z<..} \<inter> f ` {x,y} \<noteq> {}"
             by cases (use fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close> in \<open>auto simp: image_iff\<close>)
           then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
             using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
         qed
         ultimately show False
           using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
       qed
       then show "z \<in> S" by meson
     qed
   qed
 qed
 
 lemma compact_convex_collinear_segment_alt:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
   obtains a b where "S = closed_segment a b"
 proof -
   obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
   have "collinear (affine hull S)"
     by (simp add: assms collinear_affine_hull_collinear)
   then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
     by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
   then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
     by metis
   let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
   have gf: "?g (f x) = x" if "x \<in> affine hull S" for x
     by (metis diff_add_cancel f that)
   then have inj_f: "inj_on f (affine hull S)"
     by (metis inj_onI)
   have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
   proof -
     have "f x *\<^sub>R z = x - \<xi>"
       by (simp add: f hull_inc x)
     moreover have "f y *\<^sub>R z = y - \<xi>"
       by (simp add: f hull_inc y)
     ultimately show ?thesis
       by (simp add: scaleR_left.diff)
   qed
   have cont_f: "continuous_on (affine hull S) f"
   proof (clarsimp simp: dist_norm continuous_on_iff diff)
     show "\<And>x e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>y \<in> affine hull S. \<bar>f y  - f x\<bar> * norm z < d \<longrightarrow> \<bar>f y  - f x\<bar> < e"
       by (metis \<open>z \<noteq> 0\<close> mult_pos_pos mult_less_iff1 zero_less_norm_iff)
   qed
   then have "connected (f ` S)"
     by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
   moreover have "compact (f ` S)"
     by (meson \<open>compact S\<close> compact_continuous_image_eq cont_f hull_subset inj_f)
   ultimately obtain x y where "f ` S = {x..y}"
     by (meson connected_compact_interval_1)
   then have fS_eq: "f ` S = closed_segment x y"
     using \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
   obtain a b where "a \<in> S" "f a = x" "b \<in> S" "f b = y"
     by (metis (full_types) ends_in_segment fS_eq imageE)
   have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
   proof (rule continuous_injective_image_segment_1)
     show "continuous_on (closed_segment a b) f"
       by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
     show "inj_on f (closed_segment a b)"
       by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
   qed
   then have "f ` (closed_segment a b) = f ` S"
     by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
   then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
     by simp
   moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
     unfolding image_def using \<open>a \<in> S\<close> \<open>b \<in> S\<close>
     by (safe; metis (mono_tags, lifting)  convex_affine_hull convex_contains_segment gf hull_subset subsetCE)
   ultimately have "closed_segment a b = S"
     using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
   then show ?thesis
     using that by blast
 qed
 
 lemma compact_convex_collinear_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
   obtains a b where "S = closed_segment a b"
   using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
 
 
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
           "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
 
 lemma proper_map_fst:
   assumes "compact T" "K \<subseteq> S" "compact K"
     shows "compact (S \<times> T \<inter> fst -` K)"
 proof -
   have "(S \<times> T \<inter> fst -` K) = K \<times> T"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_fst:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set S) (fst ` c)"
 proof -
   have *: "fst ` (S \<times> T) \<subseteq> S"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
 qed
 
 lemma proper_map_snd:
   assumes "compact S" "K \<subseteq> T" "compact K"
     shows "compact (S \<times> T \<inter> snd -` K)"
 proof -
   have "(S \<times> T \<inter> snd -` K) = S \<times> K"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_snd:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set T) (snd ` c)"
 proof -
   have *: "snd ` (S \<times> T) \<subseteq> T"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
 qed
 
 lemma closedin_compact_projection:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
     shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
 proof -
   have "U \<subseteq> S \<times> T"
     by (metis clo closedin_imp_subset)
   then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
     by force
   moreover have "closedin (top_of_set T) (snd ` U)"
     by (rule closed_map_snd [OF assms])
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma closed_compact_projection:
   fixes S :: "'a::euclidean_space set"
     and T :: "('a * 'b::euclidean_space) set"
   assumes "compact S" and clo: "closed T"
     shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
 proof -
   have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} = {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
     by auto
   show ?thesis
     unfolding *
     by (intro clo closedin_closed_Int closedin_closed_trans [OF _ closed_UNIV] closedin_compact_projection [OF \<open>compact S\<close>])
 qed
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_convex_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 proof
   show "affine hull (S \<inter> T) \<subseteq> affine hull S"
     by (simp add: hull_mono)
 next
   obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
     using assms interior_subset by blast
   then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
     using mem_interior_cball by blast
   have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
   proof (cases "x = a")
     case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
       by blast
   next
     case False
     define k where "k = min (1/2) (e / norm (x-a))"
     have k: "0 < k" "k < 1"
       using \<open>e > 0\<close> False by (auto simp: k_def)
     then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
       by simp
     have "e / norm (x - a) \<ge> k"
       using k_def by linarith
     then have "a + k *\<^sub>R (x - a) \<in> cball a e"
       using \<open>0 < k\<close> False
       by (simp add: dist_norm) (simp add: field_simps)
     then have T: "a + k *\<^sub>R (x - a) \<in> T"
       using e by blast
     have S: "a + k *\<^sub>R (x - a) \<in> S"
       using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
       by (simp add: algebra_simps)
     have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
       by (intro span_mul [OF span_base] image_eqI [where x = "a + k *\<^sub>R (x - a)"]) (auto simp: S T)
     with xa image_iff show ?thesis  by fastforce
   qed
   have "S \<subseteq> affine hull (S \<inter> T)"
     by (force simp: * \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
   then show "affine hull S \<subseteq> affine hull (S \<inter> T)"
     by (simp add: subset_hull)
 qed
 
 corollary affine_hull_convex_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
 
 corollary affine_hull_affine_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "S \<inter> interior T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
 
 corollary affine_hull_affine_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "open T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
 
 corollary affine_hull_convex_Int_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
   shows "affine hull (S \<inter> T) = affine hull S"
   using assms unfolding openin_open
   by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
   shows "affine hull S = affine hull T"
   using assms unfolding openin_open
   by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
 
 corollary affine_hull_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S" "S \<noteq> {}"
   shows "affine hull S = UNIV"
   by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
 
 lemma aff_dim_convex_Int_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> T) = aff_dim S"
   using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
 
 lemma aff_dim_convex_Int_open:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow>  aff_dim(S \<inter> T) = aff_dim S"
   using aff_dim_convex_Int_nonempty_interior interior_eq by blast
 
 lemma affine_hull_Diff:
   fixes S:: "'a::real_normed_vector set"
   assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
   shows "affine hull (S - F) = affine hull S"
 proof -
   have clo: "closedin (top_of_set S) F"
     using assms finite_imp_closedin by auto
   moreover have "S - F \<noteq> {}"
     using assms by auto
   ultimately show ?thesis
     by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
 qed
 
 lemma affine_hull_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x < r} = (if a = 0 \<and> r \<le> 0 then {} else UNIV)"
 using halfspace_eq_empty_lt [of a r]
 by (simp add: open_halfspace_lt affine_hull_open)
 
 lemma affine_hull_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<le> r} = (if a = 0 \<and> r < 0 then {} else UNIV)"
 proof (cases "a = 0")
   case True then show ?thesis by simp
 next
   case False
   then have "affine hull closure {x. a \<bullet> x < r} = UNIV"
     using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
   moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> r}"
     by (simp add: Collect_mono)
   ultimately show ?thesis using False antisym_conv hull_mono top_greatest
     by (metis affine_hull_halfspace_lt)
 qed
 
 lemma affine_hull_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x > r} = (if a = 0 \<and> r \<ge> 0 then {} else UNIV)"
 using halfspace_eq_empty_gt [of r a]
 by (simp add: open_halfspace_gt affine_hull_open)
 
 lemma affine_hull_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<ge> r} = (if a = 0 \<and> r > 0 then {} else UNIV)"
 using affine_hull_halfspace_le [of "-a" "-r"] by simp
 
 lemma aff_dim_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x < r} =
         (if a = 0 \<and> r \<le> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
 
 lemma aff_dim_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<le> r} =
         (if a = 0 \<and> r < 0 then -1 else DIM('a))"
 proof -
   have "int (DIM('a)) = aff_dim (UNIV::'a set)"
     by (simp)
   then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
     using that by (simp add: affine_hull_halfspace_le not_less)
   then show ?thesis
     by (force)
 qed
 
 lemma aff_dim_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x > r} =
         (if a = 0 \<and> r \<ge> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
 
 lemma aff_dim_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<ge> r} =
         (if a = 0 \<and> r > 0 then -1 else DIM('a))"
 using aff_dim_halfspace_le [of "-a" "-r"] by simp
 
 proposition aff_dim_eq_hyperplane:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
   (is "?lhs = ?rhs")
 proof (cases "S = {}")
   case True then show ?thesis
     by (auto simp: dest: hyperplane_eq_Ex)
 next
   case False
   then obtain c where "c \<in> S" by blast
   show ?thesis
   proof (cases "c = 0")
     case True 
     have "?lhs \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0})"
       by (simp add: aff_dim_eq_dim [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane del: One_nat_def)
     also have "... \<longleftrightarrow> ?rhs"
       using span_zero [of S] True \<open>c \<in> S\<close> affine_hull_span_0 hull_inc  
       by (fastforce simp add: affine_hull_span_gen [of c] \<open>c = 0\<close>)
     finally show ?thesis .
   next
     case False
     have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
     proof -
       have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
         by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
       then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
         by blast
     qed
     have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
          if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
     proof -
       have "b = a \<bullet> c"
         using span_0 that by fastforce
       with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
         by simp
       then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
         by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
       also have "... = {x. a \<bullet> x = 0}"
         by (force simp: inner_distrib inner_diff_right
              intro: image_eqI [where x="x+c" for x])
       finally show ?thesis .
     qed
     have "?lhs = (\<exists>a. a \<noteq> 0 \<and> span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0})"
       by (simp add: aff_dim_eq_dim [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane del: One_nat_def)
     also have "... = ?rhs"
       by (fastforce simp add: affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc inner_distrib intro: xc_im intro!: 2)
     finally show ?thesis .
   qed
 qed
 
 corollary aff_dim_hyperplane [simp]:
   fixes a :: "'a::euclidean_space"
   shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
 by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some stepping theorems\<close>
 
 lemma aff_dim_insert:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x s' where S: "S = insert x s'" "x \<notin> s'"
     by (meson Set.set_insert all_not_in_conv)
   show ?thesis using S
     by (force simp add: affine_hull_insert_span_gen span_zero insert_commute [of a] aff_dim_eq_dim [of x] dim_insert)
 qed
 
 lemma affine_dependent_choose:
   fixes a :: "'a :: euclidean_space"
   assumes "\<not>(affine_dependent S)"
   shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
         (is "?lhs = ?rhs")
 proof safe
   assume "affine_dependent (insert a S)" and "a \<in> S"
   then show "False"
     using \<open>a \<in> S\<close> assms insert_absorb by fastforce
 next
   assume lhs: "affine_dependent (insert a S)"
   then have "a \<notin> S"
     by (metis (no_types) assms insert_absorb)
   moreover have "finite S"
     using affine_independent_iff_card assms by blast
   moreover have "aff_dim (insert a S) \<noteq> int (card S)"
     using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
   ultimately show "a \<in> affine hull S"
     by (metis aff_dim_affine_independent aff_dim_insert assms)
 next
   assume "a \<notin> S" and "a \<in> affine hull S"
   show "affine_dependent (insert a S)"
     by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
 qed
 
 lemma affine_independent_insert:
   fixes a :: "'a :: euclidean_space"
   shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> affine_dependent(insert a S)"
   by (simp add: affine_dependent_choose)
 
 lemma subspace_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "subspace S"
     shows "bounded S \<longleftrightarrow> S = {0}"
 proof -
   have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
   proof -
     obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
       using \<open>bounded S\<close> by (force simp: bounded_pos_less)
     have "(B / norm x) *\<^sub>R x \<in> S"
       using assms subspace_mul \<open>x \<in> S\<close> by auto
     moreover have "norm ((B / norm x) *\<^sub>R x) = B"
       using that B by (simp add: algebra_simps)
     ultimately show False using B by force
   qed
   then have "bounded S \<Longrightarrow> S = {0}"
     using assms subspace_0 by fastforce
   then show ?thesis
     by blast
 qed
 
 lemma affine_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain b where "b \<in> S" by blast
   with False assms 
   have "bounded S \<Longrightarrow> S = {b}"
     using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
     by (metis (no_types, lifting) ab_group_add_class.ab_left_minus bounded_translation image_empty image_insert subspace_bounded_eq_trivial translation_invert)
   then show ?thesis by auto
 qed
 
 lemma affine_bounded_eq_lowdim:
   fixes S :: "'a::euclidean_space set"
   assumes "affine S"
   shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
 proof
   show "aff_dim S \<le> 0 \<Longrightarrow> bounded S"
   by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
 qed (use affine_bounded_eq_trivial assms in fastforce)
 
 
 lemma bounded_hyperplane_eq_trivial_0:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
   shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
 proof
   assume "bounded {x. a \<bullet> x = 0}"
   then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
   with assms show "DIM('a) = 1"
     by (simp add: le_Suc_eq)
 next
   assume "DIM('a) = 1"
   then show "bounded {x. a \<bullet> x = 0}"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
 qed
 
 lemma bounded_hyperplane_eq_trivial:
   fixes a :: "'a::euclidean_space"
   shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
 proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
   assume "r \<noteq> 0" "a \<noteq> 0"
   have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
     by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
   then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
     by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>General case without assuming closure and getting non-strict separation\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_closed_point_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
   obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
 proof -
   obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
     using distance_attains_inf [of S z] assms by auto
   then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
     using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
   show ?thesis
   proof (rule that [OF \<open>y \<in> S\<close> *])
     fix x
     assume "x \<in> S"
     have yz: "0 < (y - z) \<bullet> (y - z)"
       using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
     { assume 0: "0 < ((z - y) \<bullet> (x - y))"
       with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
       have False
         using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
     }
     then have "0 \<le> ((y - z) \<bullet> (x - y))"
       by (force simp: not_less inner_diff_left)
     with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
       by (simp add: algebra_simps)
     then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
   qed
 qed
 
 lemma separating_hyperplane_closed_0_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "0 \<notin> S"
   obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
   using separating_hyperplane_closed_point_inset [OF assms] by simp (metis \<open>0 \<notin> S\<close>)
 
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_set_0_inspan:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
   obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
 proof -
   define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
   have "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
           if f': "finite f'" "f' \<subseteq> k ` S" for f'
   proof -
     obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
       using finite_subset_image [OF f'] by blast
     obtain a where "a \<in> S" "a \<noteq> 0"
       using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
     then have "norm (a /\<^sub>R (norm a)) = 1"
       by simp
     moreover have "a /\<^sub>R (norm a) \<in> span S"
       by (simp add: \<open>a \<in> S\<close> span_scale span_base)
     ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
       by simp
     show ?thesis
     proof (cases "C = {}")
       case True with C ass show ?thesis
         by auto
     next
       case False
       have "closed (convex hull C)"
         using \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
       moreover have "convex hull C \<noteq> {}"
         by (simp add: False)
       moreover have "0 \<notin> convex hull C"
         by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
       ultimately obtain a b
             where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
                   and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
         using separating_hyperplane_closed_0_inset by blast
       then have "a \<in> S"
         by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
       moreover have "norm (a /\<^sub>R (norm a)) = 1"
         using \<open>a \<noteq> 0\<close> by simp
       moreover have "a /\<^sub>R (norm a) \<in> span S"
         by (simp add: \<open>a \<in> S\<close> span_scale span_base)
       ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
         by simp
       have "\<And>x. \<lbrakk>a \<noteq> 0; x \<in> C\<rbrakk> \<Longrightarrow> 0 \<le> x \<bullet> a"
         using ab \<open>0 < b\<close> by (metis hull_inc inner_commute less_eq_real_def less_trans)
       then have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
         by (auto simp add: field_split_simps)
       show ?thesis
         unfolding C k_def
         using ass aa Int_iff empty_iff by force
     qed
   qed
   moreover have "\<And>T. T \<in> k ` S \<Longrightarrow> closed T"
     using closed_halfspace_ge k_def by blast
   ultimately have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
     by (metis compact_imp_fip closed_Int_compact closed_span compact_cball compact_frontier)
   then show ?thesis
     unfolding set_eq_iff k_def
     by simp (metis inner_commute norm_eq_zero that zero_neq_one)
 qed
 
 
 lemma separating_hyperplane_set_point_inaff:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and zno: "z \<notin> S"
   obtains a b where "(z + a) \<in> affine hull (insert z S)"
                 and "a \<noteq> 0" and "a \<bullet> z \<le> b"
                 and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
 proof -
   from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
   have "convex ((+) (- z) ` S)"
     using \<open>convex S\<close> by simp
   moreover have "(+) (- z) ` S \<noteq> {}"
     by (simp add: \<open>S \<noteq> {}\<close>)
   moreover have "0 \<notin> (+) (- z) ` S"
     using zno by auto
   ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
                   and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
     using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
     by blast
   then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
     by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
   moreover
   have "z + a \<in> affine hull insert z S"
     using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen by blast
   ultimately show ?thesis
     using \<open>a \<noteq> 0\<close> szx that by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> supporting_hyperplane_rel_boundary:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 proof -
   obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
                   and "a \<noteq> 0" and "a \<bullet> x \<le> b"
                   and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> b"
     using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
     by (auto simp: rel_interior_eq_empty convex_rel_interior)
   have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
   proof -
     have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
       by (rule continuous_intros continuous_on_subset | blast)+
     have y: "y \<in> closure (rel_interior S)"
       using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
       by fastforce
     show ?thesis
       using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
       by fastforce
   qed
   have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
   proof -
     obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
       using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
     define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
     have "y' \<in> cball y e"
       unfolding y'_def using \<open>0 < e\<close> by force
     moreover have "y' \<in> affine hull S"
       unfolding y'_def
       by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
                 rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
     ultimately have "y' \<in> S"
       using e by auto
     have "a \<bullet> x \<le> a \<bullet> y"
       using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
     moreover have "a \<bullet> x \<noteq> a \<bullet> y"
       using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close> \<open>0 < e\<close> not_le
       by (fastforce simp add: y'_def inner_diff dot_square_norm power2_eq_square)
     ultimately show ?thesis by force
   qed
   show ?thesis
     by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
 qed
 
 lemma supporting_hyperplane_relative_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 using supporting_hyperplane_rel_boundary [of "closure S" x]
 by (metis assms convex_closure convex_rel_interior_closure)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
 
 lemma
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(S \<union> T)"
     shows convex_hull_Int_subset: "convex hull S \<inter> convex hull T \<subseteq> convex hull (S \<inter> T)" (is ?C)
       and affine_hull_Int_subset: "affine hull S \<inter> affine hull T \<subseteq> affine hull (S \<inter> T)" (is ?A)
 proof -
   have [simp]: "finite S" "finite T"
     using aff_independent_finite assms by blast+
     have "sum u (S \<inter> T) = 1 \<and>
           (\<Sum>v\<in>S \<inter> T. u v *\<^sub>R v) = (\<Sum>v\<in>S. u v *\<^sub>R v)"
       if [simp]:  "sum u S = 1"
                  "sum v T = 1"
          and eq: "(\<Sum>x\<in>T. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)" for u v
     proof -
       define f where "f x = (if x \<in> S then u x else 0) - (if x \<in> T then v x else 0)" for x
       have "sum f (S \<union> T) = 0"
         by (simp add: f_def sum_Un sum_subtractf flip: sum.inter_restrict)
       moreover have "(\<Sum>x\<in>(S \<union> T). f x *\<^sub>R x) = 0"
         by (simp add: eq f_def sum_Un scaleR_left_diff_distrib sum_subtractf if_smult flip: sum.inter_restrict cong: if_cong)
       ultimately have "\<And>v. v \<in> S \<union> T \<Longrightarrow> f v = 0"
         using aff_independent_finite assms unfolding affine_dependent_explicit
         by blast
       then have u [simp]: "\<And>x. x \<in> S \<Longrightarrow> u x = (if x \<in> T then v x else 0)"
         by (simp add: f_def) presburger
       have "sum u (S \<inter> T) = sum u S"
         by (simp add: sum.inter_restrict)
       then have "sum u (S \<inter> T) = 1"
         using that by linarith
       moreover have "(\<Sum>v\<in>S \<inter> T. u v *\<^sub>R v) = (\<Sum>v\<in>S. u v *\<^sub>R v)"
       by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
     ultimately show ?thesis
       by force
     qed
     then show ?A ?C
       by (auto simp: convex_hull_finite affine_hull_finite)
 qed
 
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_Int:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(S \<union> T)"
     shows "affine hull (S \<inter> T) = affine hull S \<inter> affine hull T"
   by (simp add: affine_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition\<^marker>\<open>tag unimportant\<close> convex_hull_Int:
   fixes S :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(S \<union> T)"
     shows "convex hull (S \<inter> T) = convex hull S \<inter> convex hull T"
   by (simp add: convex_hull_Int_subset assms hull_mono subset_antisym)
 
 proposition\<^marker>\<open>tag unimportant\<close>
   fixes S :: "'a::euclidean_space set set"
   assumes "\<not> affine_dependent (\<Union>S)"
     shows affine_hull_Inter: "affine hull (\<Inter>S) = (\<Inter>T\<in>S. affine hull T)" (is "?A")
       and convex_hull_Inter: "convex hull (\<Inter>S) = (\<Inter>T\<in>S. convex hull T)" (is "?C")
 proof -
   have "finite S"
     using aff_independent_finite assms finite_UnionD by blast
   then have "?A \<and> ?C" using assms
   proof (induction S rule: finite_induct)
     case empty then show ?case by auto
   next
     case (insert T F)
     then show ?case
     proof (cases "F={}")
       case True then show ?thesis by simp
     next
       case False
       with "insert.prems" have [simp]: "\<not> affine_dependent (T \<union> \<Inter>F)"
         by (auto intro: affine_dependent_subset)
       have [simp]: "\<not> affine_dependent (\<Union>F)"
         using affine_independent_subset insert.prems by fastforce
       show ?thesis
         by (simp add: affine_hull_Int convex_hull_Int insert.IH)
     qed
   qed
   then show "?A" "?C"
     by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange_unique:
   fixes S :: "'a::euclidean_space set"
   assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
       and S: "T \<subseteq> S" "T' \<subseteq> S"
       and x:  "x \<in> convex hull (insert a T)"
       and x': "x \<in> convex hull (insert a T')"
     shows "x \<in> convex hull (insert a (T \<inter> T'))"
 proof (cases "a \<in> S")
   case True
   then have "\<not> affine_dependent (insert a T \<union> insert a T')"
     using affine_dependent_subset assms by auto
   then have "x \<in> convex hull (insert a T \<inter> insert a T')"
     by (metis IntI convex_hull_Int x x')
   then show ?thesis
     by simp
 next
   case False
   then have anot: "a \<notin> T" "a \<notin> T'"
     using assms by auto
   have [simp]: "finite S"
     by (simp add: aff_independent_finite assms)
   then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
                   and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
     using a by (auto simp: convex_hull_finite)
   have fin [simp]: "finite T" "finite T'"
     using assms infinite_super \<open>finite S\<close> by blast+
   then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
                      and c1: "sum c (insert a T) = 1"
                      and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
                      and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
                      and c'1: "sum c' (insert a T') = 1"
                      and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
     using x x' by (auto simp: convex_hull_finite)
   with fin anot
   have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
    and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
     by simp_all
   have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     using x'eq fin anot by simp
   define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
   define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
   define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
   have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
   have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
   have sum_dd0: "sum dd S = 0"
     unfolding dd_def  using S
     by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
                   algebra_simps sum_distrib_right [symmetric] b1)
   have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
     by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
   then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
     using aeq by blast
   have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
     unfolding dd_def using S
     by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
   then have dd0: "dd v = 0" if "v \<in> S" for v
     using naff [unfolded affine_dependent_explicit not_ex, rule_format, of S dd]
     using that sum_dd0 by force
   consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
   then show ?thesis
   proof cases
     case 1
     then have "sum cc S \<le> sum cc' S"
       by (simp add: sumSS')
     then have le: "cc x \<le> cc' x" if "x \<in> S" for x
       using dd0 [OF that] 1 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
         by (simp add: c0 cc_def)
       show "0 \<le> (cc(a := c a)) a"
         by (simp add: c0)
       have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c a + sum (cc(a := c a)) S"
         using \<open>T \<subseteq> S\<close> False cc0 cc_def \<open>a \<notin> S\<close> by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a + (1 - c a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS'(1))
       finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
         by simp
     qed
   next
     case 2
     then have "sum cc' S \<le> sum cc S"
       by (simp add: sumSS')
     then have le: "cc' x \<le> cc x" if "x \<in> S" for x
       using dd0 [OF that] 2 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc' x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c'0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
         by (simp add: c'0 cc'_def)
       show "0 \<le> (cc'(a := c' a)) a"
         by (simp add: c'0)
       have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c' a + sum (cc'(a := c' a)) S"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c' a + (1 - c' a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
         using \<open>T \<subseteq> S\<close> False cc0 by (fastforce intro!: sum.mono_neutral_left split: if_split_asm)
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
         by simp
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> convex_hull_exchange_Int:
   fixes a  :: "'a::euclidean_space"
   assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
   shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
          convex hull (insert a (T \<inter> T'))" (is "?lhs = ?rhs")
 proof (rule subset_antisym)
   show "?lhs \<subseteq> ?rhs"
     using in_convex_hull_exchange_unique assms by blast
   show "?rhs \<subseteq> ?lhs"
     by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
 qed
 
 lemma Int_closed_segment:
   fixes b :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
     shows "closed_segment a b \<inter> closed_segment b c = {b}"
 proof (cases "c = a")
   case True
   then show ?thesis
     using assms collinear_3_eq_affine_dependent by fastforce
 next
   case False
   from assms show ?thesis
   proof
     assume "b \<in> closed_segment a c"
     moreover have "\<not> affine_dependent {a, c}"
       by (simp)
     ultimately show ?thesis
       using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
       by (simp add: segment_convex_hull insert_commute)
   next
     assume ncoll: "\<not> collinear {a, b, c}"
     have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
     proof -
       have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
         by auto
       with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> closed_segment b c"
         by force
       then have d: "collinear {a, d, b}"  "collinear {b, d, c}"
         by (auto simp:  between_mem_segment between_imp_collinear)
       have "collinear {a, b, c}"
         by (metis (full_types) \<open>b \<noteq> d\<close> collinear_3_trans d insert_commute)
       with ncoll show False ..
     qed
     then show ?thesis
       by blast
   qed
 qed
 
 lemma affine_hull_finite_intersection_hyperplanes:
   fixes S :: "'a::euclidean_space set"
   obtains \<F> where
      "finite \<F>"
      "of_nat (card \<F>) + aff_dim S = DIM('a)"
      "affine hull S = \<Inter>\<F>"
      "\<And>h. h \<in> \<F> \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
 proof -
   obtain b where "b \<subseteq> S"
              and indb: "\<not> affine_dependent b"
              and eq: "affine hull S = affine hull b"
     using affine_basis_exists by blast
   obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> c"
              and affc: "affine hull c = UNIV"
     by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
   then have "finite c"
     by (simp add: aff_independent_finite)
   then have fbc: "finite b" "card b \<le> card c"
     using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
   have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
     by blast
   have card_cb: "(card (c - b)) + aff_dim S = DIM('a)"
   proof -
     have aff: "aff_dim (UNIV::'a set) = aff_dim c"
       by (metis aff_dim_affine_hull affc)
     have "aff_dim b = aff_dim S"
       by (metis (no_types) aff_dim_affine_hull eq)
     then have "int (card b) = 1 + aff_dim S"
       by (simp add: aff_dim_affine_independent indb)
     then show ?thesis
       using fbc aff
       by (simp add: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent card_Diff_subset of_nat_diff)
   qed
   show ?thesis
   proof (cases "c = b")
     case True show ?thesis
     proof
       show "int (card {}) + aff_dim S = int DIM('a)"
         using True card_cb by auto
       show "affine hull S = \<Inter> {}"
         using True affc eq by blast
     qed auto
   next
     case False
     have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
       by (rule affine_independent_subset [OF indc]) auto
     have *: "1 + aff_dim (c - {t}) = int (DIM('a))" if t: "t \<in> c" for t
     proof -
       have "insert t c = c"
         using t by blast
       then show ?thesis
         by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
     qed
     let ?\<F> = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))"
     show ?thesis
     proof
       have "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
       proof (rule card_image)
         show "inj_on (\<lambda>a. affine hull (c - {a})) (c - b)"
           unfolding inj_on_def
           by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
       qed
       then show "int (card ?\<F>) + aff_dim S = int DIM('a)"
         by (simp add: imeq card_cb)
       show "affine hull S = \<Inter> ?\<F>"
         by (metis Diff_eq_empty_iff INT_simps(4) UN_singleton order_refl \<open>b \<subseteq> c\<close> False eq double_diff affine_hull_Inter [OF ind])
       have "\<And>a. \<lbrakk>a \<in> c; a \<notin> b\<rbrakk> \<Longrightarrow> aff_dim (c - {a}) = int (DIM('a) - Suc 0)"
         by (metis "*" DIM_ge_Suc0 One_nat_def add_diff_cancel_left' int_ops(2) of_nat_diff)
       then show "\<And>h. h \<in> ?\<F> \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
         by (auto simp only: One_nat_def aff_dim_eq_hyperplane [symmetric])
       qed (use \<open>finite c\<close> in auto)
   qed
 qed
 
 lemma affine_hyperplane_sums_eq_UNIV_0:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
      and "0 \<in> S" and "w \<in> S"
      and "a \<bullet> w \<noteq> 0"
    shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
 proof -
   have "subspace S"
     by (simp add: assms subspace_affine)
   have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>0 \<in> S\<close> add.left_neutral by (intro span_mono) force
   have "w \<notin> span {y. a \<bullet> y = 0}"
     using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
   moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>w \<in> S\<close>
     by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
   ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     by blast
   have "a \<noteq> 0" using assms inner_zero_left by blast
   then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
     by (simp add: dim_hyperplane)
   also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using span1 span2 by (blast intro: dim_psubset)
   finally have "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
   then have DD: "dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0} = DIM('a)"
     using antisym dim_subset_UNIV lowdim_subset_hyperplane not_le by fastforce
   have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
     using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
   moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
     using DD dim_eq_full by blast
   ultimately show ?thesis
     by (simp add: subs) (metis (lifting) span_eq_iff subs)
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hyperplane_sums_eq_UNIV:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
       and "S - {v. a \<bullet> v = b} \<noteq> {}"
     shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by (auto simp: if_splits)
 next
   case False
   obtain c where "c \<in> S" and c: "a \<bullet> c = b"
     using assms by force
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have "subspace ((+) (- c) ` S)" by blast
   then have aff: "affine ((+) (- c) ` S)"
     by (simp add: subspace_imp_affine)
   have 0: "0 \<in> (+) (- c) ` S"
     by (simp add: \<open>c \<in> S\<close>)
   obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
     using assms by auto
   then have adc: "a \<bullet> (d - c) \<noteq> 0"
     by (simp add: c inner_diff_right)
   define U where "U \<equiv> {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
   have "u + v \<in> (+) (c+c) ` U"
     if "u \<in> S" "b = a \<bullet> v" for u v
   proof
     show "u + v = c + c + (u + v - c - c)"
       by (simp add: algebra_simps)
     have "\<exists>x y. u + v - c - c = x + y \<and> (\<exists>xa\<in>S. x = xa - c) \<and> a \<bullet> y = 0"
     proof (intro exI conjI)
       show "u + v - c - c = (u-c) + (v-c)" "a \<bullet> (v - c) = 0"
         by (simp_all add: algebra_simps c that)
     qed (use that in auto)
     then show "u + v - c - c \<in> U"
       by (auto simp: U_def image_def)
   qed
   moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
        \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
     by (metis add.left_commute c inner_right_distrib pth_d)
   ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = (+) (c+c) ` U"
     by (fastforce simp: algebra_simps U_def)
   also have "... = range ((+) (c + c))"
     by (simp only: U_def affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
   also have "... = UNIV"
     by simp
   finally show ?thesis .
 qed
 
 lemma aff_dim_sums_Int_0:
   assumes "affine S"
       and "affine T"
       and "0 \<in> S" "0 \<in> T"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
     using assms by force
   then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
     by (metis (lifting) hull_inc)
   have sub: "subspace S"  "subspace T"
     using assms by (auto simp: subspace_affine)
   show ?thesis
     using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
 qed
 
 proposition aff_dim_sums_Int:
   assumes "affine S"
       and "affine T"
       and "S \<inter> T \<noteq> {}"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   obtain a where a: "a \<in> S" "a \<in> T" using assms by force
   have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
     using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
   have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
     using a assms by auto
   have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
       (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
     by (force simp: algebra_simps scaleR_2)
   moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
     by auto
   ultimately show ?thesis
     using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
     by (metis (lifting))
 qed
 
 lemma aff_dim_affine_Int_hyperplane:
   fixes a :: "'a::euclidean_space"
   assumes "affine S"
     shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
              (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
               else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S
               else aff_dim S - 1)"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by auto
 next
   case False
   then have "aff_dim (S \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
             if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
   proof -
     have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
       using affine_hyperplane_sums_eq_UNIV [OF assms non] that  by blast
     show ?thesis
       using aff_dim_sums_Int [OF assms affine_hyperplane non]
       by (simp add: of_nat_diff False)
   qed
   then show ?thesis
     by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
 qed
 
 lemma aff_dim_lt_full:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
 by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
 
 lemma aff_dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
   shows "aff_dim S = aff_dim T"
 proof -
   show ?thesis
   proof (rule order_antisym)
     show "aff_dim S \<le> aff_dim T"
       by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
   next
     obtain a where "a \<in> S"
       using \<open>S \<noteq> {}\<close> by blast
     have "S \<subseteq> T"
       using ope openin_imp_subset by auto
     then have "a \<in> T"
       using \<open>a \<in> S\<close> by auto
     then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
       using affine_diffs_subspace \<open>affine T\<close> by auto
     then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
                     and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
                     and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
                     and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
       by (rule orthonormal_basis_subspace) auto
     obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
       by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
     have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
       by (metis aff_dim_translation_eq)
     also have "... = dim ((\<lambda>x. - a + x) ` T)"
       using aff_dim_subspace subT' by blast
     also have "... = card B"
       by (simp add: cardB)
     also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
       using \<open>0 < e\<close>  by (force simp: inj_on_def card_image)
     also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
     proof (simp, rule independent_card_le_dim)
       have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
         using e by (auto simp: dist_norm norm_minus_commute subset_eq)
       have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
         using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
       then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
         using e' by blast
       have "inj_on ((*\<^sub>R) e) (span B)"
         using \<open>0 < e\<close> inj_on_def by fastforce
       then show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
         using linear_scale_self \<open>independent B\<close> linear_dependent_inj_imageD by blast
     qed
     also have "... = aff_dim S"
       using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
     finally show "aff_dim T \<le> aff_dim S" .
   qed
 qed
 
 lemma dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
   shows "dim S = dim T"
 proof (rule order_antisym)
   show "dim S \<le> dim T"
     by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
 next
   have "dim T = aff_dim S"
     using aff_dim_openin
     by (metis aff_dim_subspace \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
   also have "... \<le> dim S"
     by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
         subspace_span)
   finally show "dim T \<le> dim S" by simp
 qed
 
 subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
 
 proposition dense_complement_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
 proof -
   have "closure(S - U) = S" if "dim U < dim S" "U \<subseteq> S" for U
   proof -
     have "span U \<subset> span S"
       by (metis neq_iff psubsetI span_eq_dim span_mono that)
     then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
       using orthogonal_to_subspace_exists_gen by metis
     show ?thesis
     proof
       have "closed S"
         by (simp add: \<open>subspace S\<close> closed_subspace)
       then show "closure (S - U) \<subseteq> S"
         by (simp add: closure_minimal)
       show "S \<subseteq> closure (S - U)"
       proof (clarsimp simp: closure_approachable)
         fix x and e::real
         assume "x \<in> S" "0 < e"
         show "\<exists>y\<in>S - U. dist y x < e"
         proof (cases "x \<in> U")
           case True
           let ?y = "x + (e/2 / norm a) *\<^sub>R a"
           show ?thesis
           proof
             show "dist ?y x < e"
               using \<open>0 < e\<close> by (simp add: dist_norm)
           next
             have "?y \<in> S"
               by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
             moreover have "?y \<notin> U"
             proof -
               have "e/2 / norm a \<noteq> 0"
                 using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
               then show ?thesis
                 by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
             qed
             ultimately show "?y \<in> S - U" by blast
           qed
         next
           case False
           with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
         qed
       qed
     qed
   qed
   moreover have "S - S \<inter> T = S-T"
     by blast
   moreover have "dim (S \<inter> T) < dim S"
     by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
   ultimately show ?thesis
     by force
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_affine:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
 proof (cases "S \<inter> T = {}")
   case True
   then show ?thesis
     by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
 next
   case False
   then obtain z where z: "z \<in> S \<inter> T" by blast
   then have "subspace ((+) (- z) ` S)"
     by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
   moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
 thm aff_dim_eq_dim
     using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
   ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
     by (simp add: dense_complement_subspace)
   then show ?thesis
     by (metis closure_translation translation_diff translation_invert)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_openin_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S"
       and ope: "openin (top_of_set (affine hull S)) S"
     shows "closure(S - T) = closure S"
 proof -
   have "affine hull S - T \<subseteq> affine hull S"
     by blast
   then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (affine hull S - T))"
     by (rule closure_openin_Int_closure [OF ope])
   then show ?thesis
     by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S"
     shows "closure(S - T) = closure S"
 proof
   show "closure (S - T) \<subseteq> closure S"
     by (simp add: closure_mono)
   have "closure (rel_interior S - T) = closure (rel_interior S)"
     by (simp add: assms dense_complement_openin_affine_hull openin_rel_interior rel_interior_aff_dim rel_interior_same_affine_hull)
   then show "closure S \<subseteq> closure (S - T)"
     by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S" "closed S"
     shows "closure(S - T) = S"
   by (simp add: assms dense_complement_convex)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Parallel slices, etc\<close>
 
 text\<open> If we take a slice out of a set, we can do it perpendicularly,
   with the normal vector to the slice parallel to the affine hull.\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_parallel_slice:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
       and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
   obtains a' b' where "a' \<noteq> 0"
                    "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
                    "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
                    "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
 proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
   case True
   then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
     using assms by (auto simp: not_le)
   define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
   have "\<eta> \<in> S"
     by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
   moreover have "a \<bullet> \<eta> = b"
     using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
     by (simp add: \<eta>_def algebra_simps) (simp add: field_simps)
   ultimately have False
     using True by force
   then show ?thesis ..
 next
   case False
   then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
     using assms by auto
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have sub: "subspace ((+) (- z) ` S)" by blast
   then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
     by (auto simp: subspace_imp_affine)
   obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
                   and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
     using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
   then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
     by (simp add: span_base orthogonal_def)
   then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
     by (simp add: a inner_diff_right)
   then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
     by (simp add: inner_diff_left z)
   show ?thesis
   proof (cases "a' = 0")
     case True
     with a assms True a'' diff_zero less_irrefl show ?thesis
       by auto
   next
     case False
     show ?thesis
     proof
       show "S \<inter> {x. a' \<bullet> x \<le> a' \<bullet> z} = S \<inter> {x. a \<bullet> x \<le> b}"
         "S \<inter> {x. a' \<bullet> x = a' \<bullet> z} = S \<inter> {x. a \<bullet> x = b}"
         by (auto simp: a ba'' inner_left_distrib)
       have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
         by (metis subspace_add a' span_eq_iff sub)
       then show "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
         by fastforce
     qed (use False in auto)
   qed
 qed
 
 lemma diffs_affine_hull_span:
   assumes "a \<in> S"
     shows "(\<lambda>x. x - a) ` (affine hull S) = span ((\<lambda>x. x - a) ` S)"
 proof -
   have *: "((\<lambda>x. x - a) ` (S - {a})) = ((\<lambda>x. x - a) ` S) - {0}"
     by (auto simp: algebra_simps)
   show ?thesis
     by (auto simp add: algebra_simps affine_hull_span2 [OF assms] *)
 qed
 
 lemma aff_dim_dim_affine_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S" "a \<in> S"
     shows "aff_dim S = dim ((\<lambda>x. x - a) ` S)"
 proof -
   obtain B where aff: "affine hull B = affine hull S"
              and ind: "\<not> affine_dependent B"
              and card: "of_nat (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by blast
   then have "B \<noteq> {}" using assms
     by (metis affine_hull_eq_empty ex_in_conv)
   then obtain c where "c \<in> B" by auto
   then have "c \<in> S"
     by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
   have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
     by (auto simp: algebra_simps)
   have *: "(\<lambda>x. x - c) ` S = (\<lambda>x. x - a) ` S"
     using assms \<open>c \<in> S\<close>
     by (auto simp: image_iff xy; metis mem_affine_3_minus pth_1)
   have affS: "affine hull S = S"
     by (simp add: \<open>affine S\<close>)
   have "aff_dim S = of_nat (card B) - 1"
     using card by simp
   also have "... = dim ((\<lambda>x. x - c) ` B)"
     using affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>]
     by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
   also have "... = dim ((\<lambda>x. x - c) ` (affine hull B))"
     by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
   also have "... = dim ((\<lambda>x. x - a) ` S)"
     by (simp add: affS aff *)
   finally show ?thesis .
 qed
 
 lemma aff_dim_linear_image_le:
   assumes "linear f"
     shows "aff_dim(f ` S) \<le> aff_dim S"
 proof -
   have "aff_dim (f ` T) \<le> aff_dim T" if "affine T" for T
   proof (cases "T = {}")
     case True then show ?thesis by (simp add: aff_dim_geq)
   next
     case False
     then obtain a where "a \<in> T" by auto
     have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
       by auto
     have 2: "{x - f a| x. x \<in> f ` T} = f ` ((\<lambda>x. x - a) ` T)"
       by (force simp: linear_diff [OF assms])
     have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
       by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
     also have "... = int (dim (f ` ((\<lambda>x. x - a) ` T)))"
       by (force simp: linear_diff [OF assms] 2)
     also have "... \<le> int (dim ((\<lambda>x. x - a) ` T))"
       by (simp add: dim_image_le [OF assms])
     also have "... \<le> aff_dim T"
       by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
     finally show ?thesis .
   qed
   then
   have "aff_dim (f ` (affine hull S)) \<le> aff_dim (affine hull S)"
     using affine_affine_hull [of S] by blast
   then show ?thesis
     using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
 qed
 
 lemma aff_dim_injective_linear_image [simp]:
   assumes "linear f" "inj f"
     shows "aff_dim (f ` S) = aff_dim S"
 proof (rule antisym)
   show "aff_dim (f ` S) \<le> aff_dim S"
     by (simp add: aff_dim_linear_image_le assms(1))
 next
   obtain g where "linear g" "g \<circ> f = id"
     using assms(1) assms(2) linear_injective_left_inverse by blast
   then have "aff_dim S \<le> aff_dim(g ` f ` S)"
     by (simp add: image_comp)
   also have "... \<le> aff_dim (f ` S)"
     by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
   finally show "aff_dim S \<le> aff_dim (f ` S)" .
 qed
 
 
 lemma choose_affine_subset:
   assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
   obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
 proof (cases "d = -1 \<or> S={}")
   case True with assms show ?thesis
     by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
 next
   case False
   with assms obtain a where "a \<in> S" "0 \<le> d" by auto
   with assms have ss: "subspace ((+) (- a) ` S)"
     by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
   have "nat d \<le> dim ((+) (- a) ` S)"
     by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
   then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
                   and Tdim: "dim T = nat d"
     using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
   then have "affine T"
     using subspace_affine by blast
   then have "affine ((+) a ` T)"
     by (metis affine_hull_eq affine_hull_translation)
   moreover have "(+) a ` T \<subseteq> S"
   proof -
     have "T \<subseteq> (+) (- a) ` S"
       by (metis (no_types) span_eq_iff Tsb ss)
     then show "(+) a ` T \<subseteq> S"
       using add_ac by auto
   qed
   moreover have "aff_dim ((+) a ` T) = d"
     by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
   ultimately show ?thesis
     by (rule that)
 qed
 
 subsection\<open>Paracompactness\<close>
 
 proposition paracompact:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. x \<in> S
                        \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
 proof (cases "S = {}")
   case True with that show ?thesis by blast
 next
   case False
   have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
   proof -
     obtain T where "x \<in> T" "T \<in> \<C>" "open T"
       using assms \<open>x \<in> S\<close> by blast
     then obtain e where "e > 0" "cball x e \<subseteq> T"
       by (force simp: open_contains_cball)
     then show ?thesis
       by (meson open_ball \<open>T \<in> \<C>\<close> ball_subset_cball centre_in_ball closed_cball closure_minimal dual_order.trans)
   qed
   then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
     and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
   if "x \<in> S" for x
     by metis
   then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
     using Lindelof [of "G ` S"] by (metis image_iff)
   then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
     by (metis countable_subset_image)
   with False Gin have "K \<noteq> {}" by force
   then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
     by (metis range_from_nat_into \<open>countable K\<close>)
   then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
     using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
   let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
   have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
   proof -
     have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
     then show ?thesis
       using clos K \<open>range a = K\<close> closure_subset by blast
   qed
   show ?thesis
   proof
     show "S \<subseteq> Union ?C"
     proof
       fix x assume "x \<in> S"
       define n where "n \<equiv> LEAST n. x \<in> F(a n)"
       have n: "x \<in> F(a n)"
         using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
       have notn: "x \<notin> F(a m)" if "m < n" for m
         using that not_less_Least by (force simp: n_def)
       then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
         using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
       with n show "x \<in> Union ?C"
         by blast
     qed
     show "\<And>U. U \<in> ?C \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
       using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close> by (auto simp: odif)
     show "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
     proof -
       obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
         using \<open>x \<in> S\<close> enum_S by auto
       have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
       proof clarsimp
         fix k  assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
         then have "k \<le> n"
           by auto (metis closure_subset not_le subsetCE)
         then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
                  \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
           by force
       qed
       moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
         by force
       ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
         using finite_subset by blast
       have "a n \<in> S"
         using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> by blast
       then show ?thesis
         by (blast intro: oG n *)
     qed
   qed
 qed
 
 corollary paracompact_closedin:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes cin: "closedin (top_of_set U) S"
       and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
                and "\<And>x. x \<in> U
                        \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
                                finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
 proof -
   have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
     using oin [OF that] by (auto simp: openin_open)
   then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
     by metis
   obtain K where K: "closed K" "U \<inter> K = S"
     using cin by (auto simp: closedin_closed)
   have 1: "U \<subseteq> \<Union>(insert (- K) (F ` \<C>))"
     by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
   have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
     using \<open>closed K\<close> by (auto simp: opF)
   obtain \<D> where "U \<subseteq> \<Union>\<D>"
              and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
              and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
     by (blast intro: paracompact [OF 1 2])
   let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
   show ?thesis
   proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
     show "S \<subseteq> \<Union>?C"
       using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
     show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
       using D1 intF by fastforce
     have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
              (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
       by blast
     show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
       if "x \<in> U" for x
     proof -
       from D2 [OF that] obtain V where "open V" "x \<in> V" "finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
         by auto
       with * show ?thesis
         by (rule_tac x="U \<inter> V" in exI) (auto intro: that finite_subset [OF *])
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> paracompact_closed:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "closed S"
       and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
                                finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
   by (rule paracompact_closedin [of UNIV S \<C>]) (auto simp: assms)
 
   
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Closed-graph characterization of continuity\<close>
 
 lemma continuous_closed_graph_gen:
   fixes T :: "'b::real_normed_vector set"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
     shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
 proof -
   have eq: "((\<lambda>x. Pair x (f x)) ` S) = (S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
     using fim by auto
   show ?thesis
     unfolding eq
     by (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf]) auto
 qed
 
 lemma continuous_closed_graph_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
          closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
 proof -
   have "?lhs" if ?rhs
   proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
     fix U
     assume U: "closedin (top_of_set T) U"
     have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
       by (force simp: image_iff)
     show "closedin (top_of_set S) (S \<inter> f -` U)"
       by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
   qed
   with continuous_closed_graph_gen assms show ?thesis by blast
 qed
 
 lemma continuous_closed_graph:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "closed S" and contf: "continuous_on S f"
   shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
 proof (rule closedin_closed_trans)
   show "closedin (top_of_set (S \<times> UNIV)) ((\<lambda>x. (x, f x)) ` S)"
     by (rule continuous_closed_graph_gen [OF contf subset_UNIV])
 qed (simp add: \<open>closed S\<close> closed_Times)
 
 lemma continuous_from_closed_graph:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
   shows "continuous_on S f"
     using fim clo
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
 
 lemma continuous_on_Un_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T f"
     shows "continuous_on (S \<union> T) f"
   using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
   by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
 
 lemma continuous_on_cases_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T g"
       and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
     shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
 proof -
   have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x"  "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
     by (simp_all add: fg)
   then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
     by (simp_all add: contf contg cong: continuous_on_cong)
   then show ?thesis
     by (rule continuous_on_Un_local_open [OF opS opT])
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The union of two collinear segments is another segment\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange:
   fixes a :: "'a::euclidean_space"
   assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
   obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
 proof (cases "a \<in> S")
   case True
   with xS insert_Diff that  show ?thesis by fastforce
 next
   case False
   show ?thesis
   proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
     case True
     then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
                     and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
         using a by (auto simp: convex_hull_finite)
     obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
                and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
       using True xS by (auto simp: convex_hull_finite)
     show ?thesis
     proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
       case True
       then obtain b where b: "b \<in> S" "v b = 0"
         by blast
       show ?thesis
       proof
         have fin: "finite (insert a (S - {b}))"
           using sum.infinite v1 by fastforce
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
                 (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
             using fin by (force intro: sum.mono_neutral_right)
           also have "... = (\<Sum>x \<in> S - {b}. v x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x)"
             by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
           finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
             by (simp add: v1)
           show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
             by (auto simp: v0)
           have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
                 (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
             using fin by (force intro: sum.mono_neutral_right)
           also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
           finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     next
       case False
       have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
         by (simp add: True that)
       have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
         using True v1 by (auto intro: Max_in)
       then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
         by blast
       then have "0 \<noteq> u b / v b"
         using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
         by (metis False eq_iff v0)
       then have  "0 < u b" "0 < v b"
         using False \<open>b \<in> S\<close> u0 v0 by force+
       have fin: "finite (insert a (S - {b}))"
         using sum.infinite v1 by fastforce
       show ?thesis
       proof
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
                 v b / u b + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  
             by (auto intro!: sum.cong split: if_split_asm)
           also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
             by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
           also have "... = (\<Sum>x\<in>S. v x)"
             using \<open>0 < u b\<close> True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
           finally show "sum (\<lambda>x. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
             by (simp add: v1)
           show "0 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
             if "i \<in> insert a (S - {b})" for i
             using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
             by (auto simp: field_simps split: if_split_asm)
           have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
                 (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
             by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             using \<open>0 < u b\<close> True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
           finally
           show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     qed
   next
     case False
     obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
       using xS by (auto simp: caratheodory [of S])
     with False obtain b where b: "b \<in> S" "b \<notin> T"
       by (metis antisym subsetI)
     show ?thesis
     proof
       show "x \<in> convex hull insert a (S - {b})"
         using  \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
     qed (rule \<open>b \<in> S\<close>)
   qed
 qed
 
 lemma convex_hull_exchange_Union:
   fixes a :: "'a::euclidean_space"
   assumes "a \<in> convex hull S"
   shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
     by (blast intro: in_convex_hull_exchange [OF assms])
   show "?rhs \<subseteq> ?lhs"
   proof clarify
     fix x b
     assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
     then show "x \<in> convex hull S" if "b \<in> S"
       by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
   qed
 qed
 
 lemma Un_closed_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c"
     shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
 proof (cases "c = a")
   case True
   with assms show ?thesis by simp
 next
   case False
   with assms have "convex hull {a, b} \<union> convex hull {b, c} = (\<Union>ba\<in>{a, c}. convex hull insert b ({a, c} - {ba}))"
     by (auto simp: insert_Diff_if insert_commute)
   then show ?thesis
     using convex_hull_exchange_Union
     by (metis assms segment_convex_hull)
 qed
 
 lemma Un_open_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> open_segment a c"
   shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c" (is "?lhs = ?rhs")
 proof -
   have b: "b \<in> closed_segment a c"
     by (simp add: assms open_closed_segment)
   have *: "?rhs \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
           if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> ?rhs"
   proof -
     have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (?rhs))"
       using that by (simp add: insert_commute)
     then show ?thesis
       by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
   qed
   show ?thesis
   proof
     show "?lhs \<subseteq> ?rhs"
       by (simp add: assms b subset_open_segment)
     show "?rhs \<subseteq> ?lhs"
       using Un_closed_segment [OF b] *
       by (simp add: closed_segment_eq_open insert_commute)
   qed
 qed
 
 subsection\<open>Covering an open set by a countable chain of compact sets\<close>
   
 proposition open_Union_compact_subsets:
   fixes S :: "'a::euclidean_space set"
   assumes "open S"
   obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
                   "\<And>n. C n \<subseteq> interior(C(Suc n))"
                   "\<Union>(range C) = S"
                   "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (rule_tac C = "\<lambda>n. {}" in that) auto
 next
   case False
   then obtain a where "a \<in> S"
     by auto
   let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
   have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
         if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
             and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
   proof -
     have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
       by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
     have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
       by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
     obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
     proof (rule compactE_image [OF \<open>compact K\<close>])
       show "K \<subseteq> (\<Union>n. interior (f n))"
         using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
     qed auto
     { fix n
       assume n: "Max I \<le> n"
       have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
         by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
       then have "K \<subseteq> f n"
         using I by auto
     }
     then show ?thesis
       by blast
   qed
   moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
                      ((\<Union>(range f) = S))"
   proof (intro exI conjI allI)
     show "\<And>n. compact (?C n)"
       by (auto simp: compact_diff open_sums)
     show "\<And>n. ?C n \<subseteq> S"
       by auto
     show "?C n \<subseteq> interior (?C (Suc n))" for n
     proof (simp add: interior_diff, rule Diff_mono)
       show "cball a (real n) \<subseteq> ball a (1 + real n)"
         by (simp add: cball_subset_ball_iff)
       have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
         using assms by (auto intro: closed_compact_sums)
       have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
             \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
         by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
       also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
         by (simp add: cball_subset_ball_iff field_split_simps UN_mono)
       finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
                     \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
     qed
     have "S \<subseteq> \<Union> (range ?C)"
     proof
       fix x
       assume x: "x \<in> S"
       then obtain e where "e > 0" and e: "ball x e \<subseteq> S"
         using assms open_contains_ball by blast
       then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
         using reals_Archimedean2
         by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
       obtain N2 where N2: "norm(x - a) \<le> real N2"
         by (meson real_arch_simple)
       have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
         using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
       have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
       proof -
         have "e * real N1 < e * (1 + (real N1 + real N2))"
           by (simp add: \<open>0 < e\<close>)
         then have "1 / (1 + (real N1 + real N2)) < e"
           using N1 \<open>e > 0\<close>
           by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
         then have "x - z \<in> ball x e"
           using that by simp
         then have "x - z \<in> S"
           using e by blast
         with that show ?thesis
           by auto
       qed
       with N2 show "x \<in> \<Union> (range ?C)"
         by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
     qed
     then show "\<Union> (range ?C) = S" by auto
   qed
   ultimately show ?thesis
     using that by metis
 qed
 
 
 subsection\<open>Orthogonal complement\<close>
 
 definition\<^marker>\<open>tag important\<close> orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
   where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
 
 proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
   unfolding subspace_def orthogonal_comp_def orthogonal_def
   by (auto simp: inner_right_distrib)
 
 lemma orthogonal_comp_anti_mono:
   assumes "A \<subseteq> B"
   shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
 proof
   fix x assume x: "x \<in> B\<^sup>\<bottom>"
   show "x \<in> orthogonal_comp A" using x unfolding orthogonal_comp_def
     by (simp add: orthogonal_def, metis assms in_mono)
 qed
 
 lemma orthogonal_comp_null [simp]: "{0}\<^sup>\<bottom> = UNIV"
   by (auto simp: orthogonal_comp_def orthogonal_def)
 
 lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
   unfolding orthogonal_comp_def orthogonal_def
   by auto (use inner_eq_zero_iff in blast)
 
 lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
   by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
 
 lemma subspace_sum_minimal:
   assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
   shows "S + T \<subseteq> U"
 proof
   fix x
   assume "x \<in> S + T"
   then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
     by (meson set_plus_elim)
   then show "x \<in> U"
     by (meson assms subsetCE subspace_add)
 qed
 
 proposition subspace_sum_orthogonal_comp:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U + U\<^sup>\<bottom> = UNIV"
 proof -
   obtain B where "B \<subseteq> U"
     and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
     and "independent B" "card B = dim U" "span B = U"
     using orthonormal_basis_subspace [OF assms] by metis
   then have "finite B"
     by (simp add: indep_card_eq_dim_span)
   have *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
     using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
   { fix v
     let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
     have "v = ?u + (v - ?u)"
       by simp
     moreover have "?u \<in> U"
       by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
     moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
     proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
       fix y
       assume "y \<in> U"
       with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
       obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
         by auto
       have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
         using that \<open>finite B\<close>
         by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
       then show "y \<bullet> (v - ?u) = 0"
         by (simp add: u inner_sum_left)
     qed
     ultimately have "v \<in> U + U\<^sup>\<bottom>"
       using set_plus_intro by fastforce
   } then show ?thesis
     by auto
 qed
 
 lemma orthogonal_Int_0:
   assumes "subspace U"
   shows "U \<inter> U\<^sup>\<bottom> = {0}"
   using orthogonal_comp_def orthogonal_self
   by (force simp: assms subspace_0 subspace_orthogonal_comp)
 
 lemma orthogonal_comp_self:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
 proof
   have ssU': "subspace (U\<^sup>\<bottom>)"
     by (simp add: subspace_orthogonal_comp)
   have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
   proof -
     obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
       using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
     then have "u-v \<in> U\<^sup>\<bottom>"
       by simp
     moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
     then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       by (simp add: subspace_diff subspace_orthogonal_comp that)
     ultimately have "u-v = 0"
       using orthogonal_Int_0 ssU' by blast
     with \<open>v \<in> U\<close> show ?thesis
       by auto
   qed
   then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
     by auto
 qed (use orthogonal_comp_subset in auto)
 
 lemma ker_orthogonal_comp_adjoint:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows "f -` {0} = (range (adjoint f))\<^sup>\<bottom>"
 proof -
   have "\<And>x. \<lbrakk>\<forall>y. y \<bullet> f x = 0\<rbrakk> \<Longrightarrow> f x = 0"
     using assms inner_commute all_zero_iff by metis
   then show ?thesis
     using assms 
     by (auto simp: orthogonal_comp_def orthogonal_def adjoint_works inner_commute)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A non-injective linear function maps into a hyperplane.\<close>
 
 lemma linear_surj_adj_imp_inj:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f" "surj (adjoint f)"
   shows "inj f"
 proof -
   have "\<exists>x. y = adjoint f x" for y
     using assms by (simp add: surjD)
   then show "inj f"
     using assms unfolding inj_on_def image_def
     by (metis (no_types) adjoint_works euclidean_eqI)
 qed
 
 \<comment> \<open>\<^url>\<open>https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\<close>\<close>
 lemma surj_adjoint_iff_inj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "surj (adjoint f) \<longleftrightarrow> inj f"
 proof
   assume "surj (adjoint f)"
   then show "inj f"
     by (simp add: assms linear_surj_adj_imp_inj)
 next
   assume "inj f"
   have "f -` {0} = {0}"
     using assms \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
   moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
     by (intro ker_orthogonal_comp_adjoint assms)
   ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
     by (metis orthogonal_comp_null)
   then show "surj (adjoint f)"
     using adjoint_linear \<open>linear f\<close>
     by (subst (asm) orthogonal_comp_self)
       (simp add: adjoint_linear linear_subspace_image)
 qed
 
 lemma inj_adjoint_iff_surj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "inj (adjoint f) \<longleftrightarrow> surj f"
 proof
   assume "inj (adjoint f)"
   have "(adjoint f) -` {0} = {0}"
     by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
   then have "(range(f))\<^sup>\<bottom> = {0}"
     by (metis (no_types, opaque_lifting) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
   then show "surj f"
     by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
 next
   assume "surj f"
   then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
     by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
   then have "{0} = adjoint f -` {0}"
     using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
   then show "inj (adjoint f)"
     by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
 qed
 
 lemma linear_singular_into_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f"
   shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
 proof
   assume "\<not>inj f"
   then show ?rhs
     using all_zero_iff
     by (metis (no_types, opaque_lifting) adjoint_clauses(2) adjoint_linear assms
         linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
 next
   assume ?rhs
   then show "\<not>inj f"
     by (metis assms linear_injective_isomorphism all_zero_iff)
 qed
 
 lemma linear_singular_image_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" "\<not>inj f"
   obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
   using assms by (fastforce simp add: linear_singular_into_hyperplane)
 
 end