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
@@ -1,4141 +1,4140 @@
 (* 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
   Linear_Algebra
   "HOL-Library.Set_Algebras"
 begin
 
 subsection \<open>Convexity\<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"
   apply (rule convexD)
   using assms
        apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
   done
 
 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)"
   unfolding convex_explicit
   apply safe
   subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto
   subgoal for t u
   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
     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"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
       by (auto simp: assms sum.If_cases if_distrib if_distrib_arg)
   qed
   done
 
 
 subsection \<open>Functions that are convex 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)"
 
 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 pos_is_convex: "convex {0 :: real <..}"
   unfolding convex_alt
 proof safe
   fix y x \<mu> :: real
   assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
   {
     assume "\<mu> = 0"
     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
       by simp
     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
       using * by simp
   }
   moreover
   {
     assume "\<mu> = 1"
     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
       using * by simp
   }
   moreover
   {
     assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
     then have "\<mu> > 0" "(1 - \<mu>) > 0"
       using * by auto
     then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
       using * by (auto simp: add_pos_pos)
   }
   ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
     by fastforce
 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"
   assumes "convex C"
   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[OF assms(1)]
   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"
     using convex_on_alt 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"
   shows "convex_on C f"
   using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function
   by fastforce
 
 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 pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
   show ?thesis
     by auto
 qed
 
 
 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)
 
 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_linear: "linear fst"
   unfolding linear_iff by (simp add: algebra_simps)
 
 lemma snd_linear: "linear snd"
   unfolding linear_iff by (simp add: algebra_simps)
 
 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_notmem:
   assumes "x \<notin> s"
   shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s"
     and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s"
   apply (rule_tac [!] sum.cong)
   using assms
   apply auto
   done
 
 lemma sum_delta'':
   fixes s::"'a::real_vector set"
   assumes "finite s"
   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 if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)"
   by (fact if_distrib)
 
 lemma dist_triangle_eq:
   fixes x y z :: "'a::real_inner"
   shows "dist x z = dist x y + dist y z \<longleftrightarrow>
     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>Affine set and affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> affine :: "'a::real_vector set \<Rightarrow> bool"
   where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
 
 lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
   unfolding affine_def by (metis eq_diff_eq')
 
 lemma affine_empty [iff]: "affine {}"
   unfolding affine_def by auto
 
 lemma affine_sing [iff]: "affine {x}"
   unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])
 
 lemma affine_UNIV [iff]: "affine UNIV"
   unfolding affine_def by auto
 
 lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)"
   unfolding affine_def by auto
 
 lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
   unfolding affine_def by auto
 
 lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)"
   apply (clarsimp simp add: affine_def)
   apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI)
   apply (auto simp: algebra_simps)
   done
 
 lemma affine_affine_hull [simp]: "affine(affine hull s)"
   unfolding hull_def
   using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto
 
 lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
   by (metis affine_affine_hull hull_same)
 
 lemma affine_hyperplane: "affine {x. a \<bullet> x = b}"
   by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some explicit formulations\<close>
 
 text "Formalized by Lars Schewe."
 
 lemma affine:
   fixes V::"'a::real_vector set"
   shows "affine V \<longleftrightarrow>
          (\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)"
 proof -
   have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)"
     and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v
   proof (cases "x = y")
     case True
     then show ?thesis
       using that by (metis scaleR_add_left scaleR_one)
   next
     case False
     then show ?thesis
       using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto
   qed
   moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
                 if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
                   and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u
   proof -
     define n where "n = card S"
     consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
     then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
     proof cases
       assume "card S = 1"
       then obtain a where "S={a}"
         by (auto simp: card_Suc_eq)
       then show ?thesis
         using that by simp
     next
       assume "card S = 2"
       then obtain a b where "S = {a, b}"
         by (metis Suc_1 card_1_singletonE card_Suc_eq)
       then show ?thesis
         using *[of a b] that
         by (auto simp: sum_clauses(2))
     next
       assume "card S > 2"
       then show ?thesis using that n_def
       proof (induct n arbitrary: u S)
         case 0
         then show ?case by auto
       next
         case (Suc n u S)
         have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)"
           using that unfolding card_eq_sum by auto
         with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force
         have c: "card (S - {x}) = card S - 1"
           by (simp add: Suc.prems(3) \<open>x \<in> S\<close>)
         have "sum u (S - {x}) = 1 - u x"
           by (simp add: Suc.prems sum_diff1 \<open>x \<in> S\<close>)
         with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
           by auto
         have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V"
         proof (cases "card (S - {x}) > 2")
           case True
           then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n"
             using Suc.prems c by force+
           show ?thesis
           proof (rule Suc.hyps)
             show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1"
               by (auto simp: eq1 sum_distrib_left[symmetric])
           qed (use S Suc.prems True in auto)
         next
           case False
           then have "card (S - {x}) = Suc (Suc 0)"
             using Suc.prems c by auto
           then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b"
             unfolding card_Suc_eq by auto
           then show ?thesis
             using eq1 \<open>S \<subseteq> V\<close>
             by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
         qed
         have "u x + (1 - u x) = 1 \<Longrightarrow>
           u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V"
           by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>)
         moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)"
           by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>)
         ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V"
           by (simp add: x)
       qed
     qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto)
   qed
   ultimately show ?thesis
     unfolding affine_def by meson
 qed
 
 
 lemma affine_hull_explicit:
   "affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
   (is "_ = ?rhs")
 proof (rule hull_unique)
   show "p \<subseteq> ?rhs"
   proof (intro subsetI CollectI exI conjI)
     show "\<And>x. sum (\<lambda>z. 1) {x} = 1"
       by auto
   qed auto
   show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T
     using that unfolding affine by blast
   show "affine ?rhs"
     unfolding affine_def
   proof clarify
     fix u v :: real and sx ux sy uy
     assume uv: "u + v = 1"
       and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)"
       and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)" 
     have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy"
       by auto
     show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and>
         sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
     proof (intro exI conjI)
       show "finite (sx \<union> sy)"
         using x y by auto
       show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1"
         using x y uv
         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
       have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i)
           = (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)"
         using x y
         unfolding scaleR_left_distrib scaleR_zero_left if_smult
         by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
       also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)"
         unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
       finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i) 
                   = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" .
     qed (use x y in auto)
   qed
 qed
 
 lemma affine_hull_finite:
   assumes "finite S"
   shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
 proof -
   have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x" 
     if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u
   proof -
     have "S \<inter> F = F"
       using that by auto
     show ?thesis
     proof (intro exI conjI)
       show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1"
         by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum)
       show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x"
         by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x)
     qed
   qed
   show ?thesis
     unfolding affine_hull_explicit using assms
     by (fastforce dest: *)
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems and hence small special cases\<close>
 
 lemma affine_hull_empty[simp]: "affine hull {} = {}"
   by simp
 
 lemma affine_hull_finite_step:
   fixes y :: "'a::real_vector"
   shows "finite S \<Longrightarrow>
       (\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow>
       (\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs")
 proof -
   assume fin: "finite S"
   show "?lhs = ?rhs"
   proof
     assume ?lhs
     then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y"
       by auto
     show ?rhs
     proof (cases "a \<in> S")
       case True
       then show ?thesis
         using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
     next
       case False
       show ?thesis
         by (rule exI [where x="u a"]) (use u fin False in auto)
     qed
   next
     assume ?rhs
     then obtain v u where vu: "sum u S = w - v"  "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a"
       by auto
     have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)"
       by auto
     show ?lhs
     proof (cases "a \<in> S")
       case True
       show ?thesis
         by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"])
            (simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
     next
       case False
       then show ?thesis
         apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI) 
         apply (simp add: vu sum_clauses(2)[OF fin] *)
         by (simp add: sum_delta_notmem(3) vu)
     qed
   qed
 qed
 
 lemma affine_hull_2:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}"
   (is "?lhs = ?rhs")
 proof -
   have *:
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
     using affine_hull_finite[of "{a,b}"] by auto
   also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
     by (simp add: affine_hull_finite_step[of "{b}" a])
   also have "\<dots> = ?rhs" unfolding * by auto
   finally show ?thesis by auto
 qed
 
 lemma affine_hull_3:
   fixes a b c :: "'a::real_vector"
   shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}"
 proof -
   have *:
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)"
     "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
   show ?thesis
     apply (simp add: affine_hull_finite affine_hull_finite_step)
     unfolding *
     apply safe
      apply (metis add.assoc)
     apply (rule_tac x=u in exI, force)
     done
 qed
 
 lemma mem_affine:
   assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
   using assms affine_def[of S] by auto
 
 lemma mem_affine_3:
   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1"
   shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S"
 proof -
   have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}"
     using affine_hull_3[of x y z] assms by auto
   moreover
   have "affine hull {x, y, z} \<subseteq> affine hull S"
     using hull_mono[of "{x, y, z}" "S"] assms by auto
   moreover
   have "affine hull S = S"
     using assms affine_hull_eq[of S] by auto
   ultimately show ?thesis by auto
 qed
 
 lemma mem_affine_3_minus:
   assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S"
   shows "x + v *\<^sub>R (y-z) \<in> S"
   using mem_affine_3[of S x y z 1 v "-v"] assms
   by (simp add: algebra_simps)
 
 corollary mem_affine_3_minus2:
     "\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S"
   by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some relations between affine hull and subspaces\<close>
 
 lemma affine_hull_insert_subset_span:
   "affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}"
 proof -
   have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)"
     if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x"
     for x F u
   proof -
     have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}"
       using that by auto
     show ?thesis
     proof (intro exI conjI)
       show "finite ((\<lambda>x. x - a) ` (F - {a}))"
         by (simp add: that(1))
       show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a"
         by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
             sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
     qed (use \<open>F \<subseteq> insert a S\<close> in auto)
   qed
   then show ?thesis
     unfolding affine_hull_explicit span_explicit by blast
 qed
 
 lemma affine_hull_insert_span:
   assumes "a \<notin> S"
   shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x.  x \<in> S}}"
 proof -
   have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
     if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v
   proof -
     from that
     obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y"
       unfolding span_explicit by auto
     define F where "F = (\<lambda>x. x + a) ` T"
     have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a"
       unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
     have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F"
       using F assms by auto
     show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y"
       apply (rule_tac x = "insert a F" in exI)
       apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI)
       using assms F
       apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
       done
   qed
   show ?thesis
     by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
 qed
 
 lemma affine_hull_span:
   assumes "a \<in> S"
   shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}"
   using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Parallel affine sets\<close>
 
 definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool"
   where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)"
 
 lemma affine_parallel_expl_aux:
   fixes S T :: "'a::real_vector set"
   assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T"
   shows "T = (\<lambda>x. a + x) ` S"
 proof -
   have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x
     using that
     by (simp add: image_iff) (metis add.commute diff_add_cancel assms)
   moreover have "T \<ge> (\<lambda>x. a + x) ` S"
     using assms by auto
   ultimately show ?thesis by auto
 qed
 
 lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)"
   by (auto simp add: affine_parallel_def)
     (use affine_parallel_expl_aux [of S _ T] in blast)
 
 lemma affine_parallel_reflex: "affine_parallel S S"
   unfolding affine_parallel_def
   using image_add_0 by blast
 
 lemma affine_parallel_commut:
   assumes "affine_parallel A B"
   shows "affine_parallel B A"
 proof -
   from assms obtain a where B: "B = (\<lambda>x. a + x) ` A"
     unfolding affine_parallel_def by auto
   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   from B show ?thesis
     using translation_galois [of B a A]
     unfolding affine_parallel_def by blast
 qed
 
 lemma affine_parallel_assoc:
   assumes "affine_parallel A B"
     and "affine_parallel B C"
   shows "affine_parallel A C"
 proof -
   from assms obtain ab where "B = (\<lambda>x. ab + x) ` A"
     unfolding affine_parallel_def by auto
   moreover
   from assms obtain bc where "C = (\<lambda>x. bc + x) ` B"
     unfolding affine_parallel_def by auto
   ultimately show ?thesis
     using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto
 qed
 
 lemma affine_translation_aux:
   fixes a :: "'a::real_vector"
   assumes "affine ((\<lambda>x. a + x) ` S)"
   shows "affine S"
 proof -
   {
     fix x y u v
     assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1"
     then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)"
       by auto
     then have h1: "u *\<^sub>R  (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S"
       using xy assms unfolding affine_def by auto
     have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)"
       by (simp add: algebra_simps)
     also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)"
       using \<open>u + v = 1\<close> by auto
     ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S"
       using h1 by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto
   }
   then show ?thesis unfolding affine_def by auto
 qed
 
 lemma affine_translation:
   "affine S \<longleftrightarrow> affine ((+) a ` S)" for a :: "'a::real_vector"
 proof
   show "affine ((+) a ` S)" if "affine S"
     using that translation_assoc [of "- a" a S]
     by (auto intro: affine_translation_aux [of "- a" "((+) a ` S)"])
   show "affine S" if "affine ((+) a ` S)"
     using that by (rule affine_translation_aux)
 qed
 
 lemma parallel_is_affine:
   fixes S T :: "'a::real_vector set"
   assumes "affine S" "affine_parallel S T"
   shows "affine T"
 proof -
   from assms obtain a where "T = (\<lambda>x. a + x) ` S"
     unfolding affine_parallel_def by auto
   then show ?thesis
     using affine_translation assms by auto
 qed
 
 lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s"
   unfolding subspace_def affine_def by auto
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Subspace parallel to an affine set\<close>
 
 lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S"
 proof -
   have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S"
     using subspace_imp_affine[of S] subspace_0 by auto
   {
     assume assm: "affine S \<and> 0 \<in> S"
     {
       fix c :: real
       fix x
       assume x: "x \<in> S"
       have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
       moreover
       have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S"
         using affine_alt[of S] assm x by auto
       ultimately have "c *\<^sub>R x \<in> S" by auto
     }
     then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto
 
     {
       fix x y
       assume xy: "x \<in> S" "y \<in> S"
       define u where "u = (1 :: real)/2"
       have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)"
         by auto
       moreover
       have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y"
         by (simp add: algebra_simps)
       moreover
       have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S"
         using affine_alt[of S] assm xy by auto
       ultimately
       have "(1/2) *\<^sub>R (x+y) \<in> S"
         using u_def by auto
       moreover
       have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))"
         by auto
       ultimately
       have "x + y \<in> S"
         using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
     }
     then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S"
       by auto
     then have "subspace S"
       using h1 assm unfolding subspace_def by auto
   }
   then show ?thesis using h0 by metis
 qed
 
 lemma affine_diffs_subspace:
   assumes "affine S" "a \<in> S"
   shows "subspace ((\<lambda>x. (-a)+x) ` S)"
 proof -
   have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff)
   have "affine ((\<lambda>x. (-a)+x) ` S)"
     using affine_translation assms by blast
   moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)"
     using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto
   ultimately show ?thesis using subspace_affine by auto
 qed
 
 lemma affine_diffs_subspace_subtract:
   "subspace ((\<lambda>x. x - a) ` S)" if "affine S" "a \<in> S"
   using that affine_diffs_subspace [of _ a] by simp
 
 lemma parallel_subspace_explicit:
   assumes "affine S"
     and "a \<in> S"
   assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}"
   shows "subspace L \<and> affine_parallel S L"
 proof -
   from assms have "L = plus (- a) ` S" by auto
   then have par: "affine_parallel S L"
     unfolding affine_parallel_def ..
   then have "affine L" using assms parallel_is_affine by auto
   moreover have "0 \<in> L"
     using assms by auto
   ultimately show ?thesis
     using subspace_affine par by auto
 qed
 
 lemma parallel_subspace_aux:
   assumes "subspace A"
     and "subspace B"
     and "affine_parallel A B"
   shows "A \<supseteq> B"
 proof -
   from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B"
     using affine_parallel_expl[of A B] by auto
   then have "-a \<in> A"
     using assms subspace_0[of B] by auto
   then have "a \<in> A"
     using assms subspace_neg[of A "-a"] by auto
   then show ?thesis
     using assms a unfolding subspace_def by auto
 qed
 
 lemma parallel_subspace:
   assumes "subspace A"
     and "subspace B"
     and "affine_parallel A B"
   shows "A = B"
 proof
   show "A \<supseteq> B"
     using assms parallel_subspace_aux by auto
   show "A \<subseteq> B"
     using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
 qed
 
 lemma affine_parallel_subspace:
   assumes "affine S" "S \<noteq> {}"
   shows "\<exists>!L. subspace L \<and> affine_parallel S L"
 proof -
   have ex: "\<exists>L. subspace L \<and> affine_parallel S L"
     using assms parallel_subspace_explicit by auto
   {
     fix L1 L2
     assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2"
     then have "affine_parallel L1 L2"
       using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
     then have "L1 = L2"
       using ass parallel_subspace by auto
   }
   then show ?thesis using ex by auto
 qed
 
 
 subsection \<open>Cones\<close>
 
 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"
   apply (rule hull_eq)
   using cone_Inter
   unfolding subset_eq
   apply auto
   done
 
 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"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain a where "a \<in> S" by auto
     then have "0 \<in> S"
       using assms mem_cone[of S a 0] by auto
   }
   then show ?thesis by auto
 qed
 
 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>cone S\<close> \<open>c > 0\<close>
           unfolding cone_def image_def \<open>c > 0\<close> by auto
       }
       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"
       apply auto
       apply (rule_tac x = 1 in exI, auto)
       done
     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
 
 
 subsection \<open>Affine dependence and consequential theorems\<close>
 
 text "Formalized by Lars Schewe."
 
 definition\<^marker>\<open>tag important\<close> affine_dependent :: "'a::real_vector set \<Rightarrow> bool"
   where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))"
 
 lemma affine_dependent_subset:
    "\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t"
 apply (simp add: affine_dependent_def Bex_def)
 apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]])
 done
 
 lemma affine_independent_subset:
   shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s"
 by (metis affine_dependent_subset)
 
 lemma affine_independent_Diff:
    "\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)"
 by (meson Diff_subset affine_dependent_subset)
 
 proposition affine_dependent_explicit:
   "affine_dependent p \<longleftrightarrow>
     (\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
 proof -
   have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0"
     if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u
   proof (intro exI conjI)
     have "x \<notin> S" 
       using that by auto
     then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0"
       using that by (simp add: sum_delta_notmem)
     show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0"
       using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong)
   qed (use that in auto)
   moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x"
     if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v
   proof (intro bexI exI conjI)
     have "S \<noteq> {v}"
       using that by auto
     then show "S - {v} \<noteq> {}"
       using that by auto
     show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1"
       unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that)
     show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v"
       unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
                 scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>] 
       using that by auto
     show "S - {v} \<subseteq> p - {v}"
       using that by auto
   qed (use that in auto)
   ultimately show ?thesis
     unfolding affine_dependent_def affine_hull_explicit by auto
 qed
 
 lemma affine_dependent_explicit_finite:
   fixes S :: "'a::real_vector set"
   assumes "finite S"
   shows "affine_dependent S \<longleftrightarrow>
     (\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)"
   (is "?lhs = ?rhs")
 proof
   have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)"
     by auto
   assume ?lhs
   then obtain t u v where
     "finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
     unfolding affine_dependent_explicit by auto
   then show ?rhs
     apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
     apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>])
     done
 next
   assume ?rhs
   then obtain u v where "sum u S = 0"  "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by auto
   then show ?lhs unfolding affine_dependent_explicit
     using assms by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close>
 
 lemma connectedD:
   "connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}"
   by (rule Topological_Spaces.topological_space_class.connectedD)
 
 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 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 add: atLeast_def[symmetric] convex_real_interval)
 
 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: convex_convex_hull 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
   by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel)
 
 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" "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)
         apply simp
         apply (rule sum_nonneg)
         using obt(1)
         apply auto
         done
     }
     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}"
         apply auto
         using f(1)[unfolded inj_on_def]
         by (metis One_nat_def atLeastAtMost_iff)
       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 (rule, case_tac[!] "a\<in>S")
   assume "a \<in> S"
   then have *: "insert a S = S" by auto
   assume ?lhs
   then show ?rhs
     unfolding *  by (rule_tac x=0 in exI, auto)
 next
   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
   assume "a \<notin> S"
   then show ?rhs
     apply (rule_tac x="u a" in exI)
     using u(1)[THEN bspec[where x=a]]
     apply simp
     apply (rule_tac x=u in exI)
     using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close>
     apply auto
     done
 next
   assume "a \<in> S"
   then have *: "insert a S = S" by auto
   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
   show ?lhs
     apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI)
     unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin]
     unfolding sum_clauses(2)[OF assms]
     using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close>
     apply auto
     done
 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 assume "a \<notin> S"
   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 \<open>a \<notin> S\<close>
     by (auto simp: intro!: sum.cong)
   ultimately show ?lhs
     by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms])
 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}"
 proof -
   have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b"
     by auto
   have **: "finite {b}" by auto
   show ?thesis
     apply (simp add: convex_hull_finite)
     unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
     apply auto
     apply (rule_tac x=v in exI)
     apply (rule_tac x="1 - v" in exI, simp)
     apply (rule_tac x=u in exI, simp)
     apply (rule_tac x="\<lambda>x. v" in exI, simp)
     done
 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)"
     unfolding *
     apply auto
     apply (rule_tac[!] x=u in exI)
     apply (auto simp: algebra_simps)
     done
 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 affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_affine subspace_span)
 
 lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)"
   by (metis hull_minimal span_superset subspace_imp_convex subspace_span)
 
 lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
   by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset)
 
 lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s"
   unfolding affine_dependent_def dependent_def
   using affine_hull_subset_span by auto
 
 lemma dependent_imp_affine_dependent:
   assumes "dependent {x - a| x . x \<in> s}"
     and "a \<notin> s"
   shows "affine_dependent (insert a s)"
 proof -
   from assms(1)[unfolded dependent_explicit] obtain S u v
     where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0"
     by auto
   define t where "t = (\<lambda>x. x + a) ` S"
 
   have inj: "inj_on (\<lambda>x. x + a) S"
     unfolding inj_on_def by auto
   have "0 \<notin> S"
     using obt(2) assms(2) unfolding subset_eq by auto
   have fin: "finite t" and "t \<subseteq> s"
     unfolding t_def using obt(1,2) by auto
   then have "finite (insert a t)" and "insert a t \<subseteq> insert a s"
     by auto
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
     apply (rule sum.cong)
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     apply auto
     done
   have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
     unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto
   moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
     using obt(3,4) \<open>0\<notin>S\<close>
     by (rule_tac x="v + a" in bexI) (auto simp: t_def)
   moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong)
   have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)"
     unfolding scaleR_left.sum
     unfolding t_def and sum.reindex[OF inj] and o_def
     using obt(5)
     by (auto simp: sum.distrib scaleR_right_distrib)
   then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
     unfolding sum_clauses(2)[OF fin]
     using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close>
     by (auto simp: *)
   ultimately show ?thesis
     unfolding affine_dependent_explicit
     apply (rule_tac x="insert a t" in exI, auto)
     done
 qed
 
 lemma convex_cone:
   "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x y
     assume "x\<in>s" "y\<in>s" and ?lhs
     then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s"
       unfolding cone_def by auto
     then have "x + y \<in> s"
       using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1]
       apply (erule_tac x="2*\<^sub>R x" in ballE)
       apply (erule_tac x="2*\<^sub>R y" in ballE)
       apply (erule_tac x="1/2" in allE, simp)
       apply (erule_tac x="1/2" in allE, auto)
       done
   }
   then show ?thesis
     unfolding convex_def cone_def by blast
 qed
 
 lemma affine_dependent_biggerset:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<ge> DIM('a) + 2"
   shows "affine_dependent s"
 proof -
   have "s \<noteq> {}" using assms by auto
   then obtain a where "a\<in>s" by auto
   have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})"
     by auto
   have "card {x - a |x. x \<in> s - {a}} = card (s - {a})"
     unfolding * by (simp add: card_image inj_on_def)
   also have "\<dots> > DIM('a)" using assms(2)
     unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto
   finally show ?thesis
     apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric])
     apply (rule dependent_imp_affine_dependent)
     apply (rule dependent_biggerset, auto)
     done
 qed
 
 lemma affine_dependent_biggerset_general:
   assumes "finite (S :: 'a::euclidean_space set)"
     and "card S \<ge> dim S + 2"
   shows "affine_dependent S"
 proof -
   from assms(2) have "S \<noteq> {}" by auto
   then obtain a where "a\<in>S" by auto
   have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})"
     by auto
   have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})"
     by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
   have "dim {x - a |x. x \<in> S - {a}} \<le> dim S"
     using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim)
   also have "\<dots> < dim S + 1" by auto
   also have "\<dots> \<le> card (S - {a})"
     using assms
     using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>]
     by auto
   finally show ?thesis
     apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric])
     apply (rule dependent_imp_affine_dependent)
     apply (rule dependent_biggerset_general)
     unfolding **
     apply auto
     done
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some Properties of Affine Dependent Sets\<close>
 
 lemma affine_independent_0 [simp]: "\<not> affine_dependent {}"
   by (simp add: affine_dependent_def)
 
 lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}"
   by (simp add: affine_dependent_def)
 
 lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}"
   by (simp add: affine_dependent_def insert_Diff_if hull_same)
 
 lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) `  S) = (\<lambda>x. a + x) ` (affine hull S)"
 proof -
   have "affine ((\<lambda>x. a + x) ` (affine hull S))"
     using affine_translation affine_affine_hull by blast
   moreover have "(\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
     using hull_subset[of S] by auto
   ultimately have h1: "affine hull ((\<lambda>x. a + x) `  S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)"
     by (metis hull_minimal)
   have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)))"
     using affine_translation affine_affine_hull by blast
   moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S))"
     using hull_subset[of "(\<lambda>x. a + x) `  S"] by auto
   moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) `  S"
     using translation_assoc[of "-a" a] by auto
   ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) `  S)) >= (affine hull S)"
     by (metis hull_minimal)
   then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)"
     by auto
   then show ?thesis using h1 by auto
 qed
 
 lemma affine_dependent_translation:
   assumes "affine_dependent S"
   shows "affine_dependent ((\<lambda>x. a + x) ` S)"
 proof -
   obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})"
     using assms affine_dependent_def by auto
   have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
     by auto
   then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})"
     using affine_hull_translation[of a "S - {x}"] x by auto
   moreover have "a + x \<in> (\<lambda>x. a + x) ` S"
     using x by auto
   ultimately show ?thesis
     unfolding affine_dependent_def by auto
 qed
 
 lemma affine_dependent_translation_eq:
   "affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)"
 proof -
   {
     assume "affine_dependent ((\<lambda>x. a + x) ` S)"
     then have "affine_dependent S"
       using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a]
       by auto
   }
   then show ?thesis
     using affine_dependent_translation by auto
 qed
 
 lemma affine_hull_0_dependent:
   assumes "0 \<in> affine hull S"
   shows "dependent S"
 proof -
   obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
     using assms affine_hull_explicit[of S] by auto
-  then have "\<exists>v\<in>s. u v \<noteq> 0"
-    using sum_not_0[of "u" "s"] by auto
+  then have "\<exists>v\<in>s. u v \<noteq> 0" by auto
   then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)"
     using s_u by auto
   then show ?thesis
     unfolding dependent_explicit[of S] by auto
 qed
 
 lemma affine_dependent_imp_dependent2:
   assumes "affine_dependent (insert 0 S)"
   shows "dependent S"
 proof -
   obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})"
     using affine_dependent_def[of "(insert 0 S)"] assms by blast
   then have "x \<in> span (insert 0 S - {x})"
     using affine_hull_subset_span by auto
   moreover have "span (insert 0 S - {x}) = span (S - {x})"
     using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
   ultimately have "x \<in> span (S - {x})" by auto
   then have "x \<noteq> 0 \<Longrightarrow> dependent S"
     using x dependent_def by auto
   moreover
   {
     assume "x = 0"
     then have "0 \<in> affine hull S"
       using x hull_mono[of "S - {0}" S] by auto
     then have "dependent S"
       using affine_hull_0_dependent by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma affine_dependent_iff_dependent:
   assumes "a \<notin> S"
   shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)"
 proof -
   have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto
   then show ?thesis
     using affine_dependent_translation_eq[of "(insert a S)" "-a"]
       affine_dependent_imp_dependent2 assms
       dependent_imp_affine_dependent[of a S]
     by (auto simp del: uminus_add_conv_diff)
 qed
 
 lemma affine_dependent_iff_dependent2:
   assumes "a \<in> S"
   shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))"
 proof -
   have "insert a (S - {a}) = S"
     using assms by auto
   then show ?thesis
     using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto
 qed
 
 lemma affine_hull_insert_span_gen:
   "affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)"
 proof -
   have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)"
     by auto
   {
     assume "a \<notin> s"
     then have ?thesis
       using affine_hull_insert_span[of a s] h1 by auto
   }
   moreover
   {
     assume a1: "a \<in> s"
     have "\<exists>x. x \<in> s \<and> -a+x=0"
       apply (rule exI[of _ a])
       using a1
       apply auto
       done
     then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s"
       by auto
     then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)"
       using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff)
     moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))"
       by auto
     moreover have "insert a (s - {a}) = insert a s"
       by auto
     ultimately have ?thesis
       using affine_hull_insert_span[of "a" "s-{a}"] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma affine_hull_span2:
   assumes "a \<in> s"
   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))"
   using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]]
   by auto
 
 lemma affine_hull_span_gen:
   assumes "a \<in> affine hull s"
   shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)"
 proof -
   have "affine hull (insert a s) = affine hull s"
     using hull_redundant[of a affine s] assms by auto
   then show ?thesis
     using affine_hull_insert_span_gen[of a "s"] by auto
 qed
 
 lemma affine_hull_span_0:
   assumes "0 \<in> affine hull S"
   shows "affine hull S = span S"
   using affine_hull_span_gen[of "0" S] assms by auto
 
 lemma extend_to_affine_basis_nonempty:
   fixes S V :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}"
   shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
 proof -
   obtain a where a: "a \<in> S"
     using assms by auto
   then have h0: "independent  ((\<lambda>x. -a + x) ` (S-{a}))"
     using affine_dependent_iff_dependent2 assms by auto
   obtain B where B:
     "(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B"
     using assms
     by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"])
   define T where "T = (\<lambda>x. a+x) ` insert 0 B"
   then have "T = insert a ((\<lambda>x. a+x) ` B)"
     by auto
   then have "affine hull T = (\<lambda>x. a+x) ` span B"
     using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B]
     by auto
   then have "V \<subseteq> affine hull T"
     using B assms translation_inverse_subset[of a V "span B"]
     by auto
   moreover have "T \<subseteq> V"
     using T_def B a assms by auto
   ultimately have "affine hull T = affine hull V"
     by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
   moreover have "S \<subseteq> T"
     using T_def B translation_inverse_subset[of a "S-{a}" B]
     by auto
   moreover have "\<not> affine_dependent T"
     using T_def affine_dependent_translation_eq[of "insert 0 B"]
       affine_dependent_imp_dependent2 B
     by auto
   ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto
 qed
 
 lemma affine_basis_exists:
   fixes V :: "'n::euclidean_space set"
   shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B"
 proof (cases "V = {}")
   case True
   then show ?thesis
     using affine_independent_0 by auto
 next
   case False
   then obtain x where "x \<in> V" by auto
   then show ?thesis
     using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V]
     by auto
 qed
 
 proposition extend_to_affine_basis:
   fixes S V :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent S" "S \<subseteq> V"
   obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V"
 proof (cases "S = {}")
   case True then show ?thesis
     using affine_basis_exists by (metis empty_subsetI that)
 next
   case False
   then show ?thesis by (metis assms extend_to_affine_basis_nonempty that)
 qed
 
 subsection \<open>Affine Dimension of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> aff_dim :: "('a::euclidean_space) set \<Rightarrow> int"
   where "aff_dim V =
   (SOME d :: int.
     \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)"
 
 lemma aff_dim_basis_exists:
   fixes V :: "('n::euclidean_space) set"
   shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
 proof -
   obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V"
     using affine_basis_exists[of V] by auto
   then show ?thesis
     unfolding aff_dim_def
       some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"]
     apply auto
     apply (rule exI[of _ "int (card B) - (1 :: int)"])
     apply (rule exI[of _ "B"], auto)
     done
 qed
 
 lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}"
 proof -
   have "S = {} \<Longrightarrow> affine hull S = {}"
     using affine_hull_empty by auto
   moreover have "affine hull S = {} \<Longrightarrow> S = {}"
     unfolding hull_def by auto
   ultimately show ?thesis by blast
 qed
 
 lemma aff_dim_parallel_subspace_aux:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B" "a \<in> B"
   shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))"
 proof -
   have "independent ((\<lambda>x. -a + x) ` (B-{a}))"
     using affine_dependent_iff_dependent2 assms by auto
   then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))"
     "finite ((\<lambda>x. -a + x) ` (B - {a}))"
     using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto
   show ?thesis
   proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}")
     case True
     have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))"
       using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
     then have "B = {a}" using True by auto
     then show ?thesis using assms fin by auto
   next
     case False
     then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0"
       using fin by auto
     moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})"
       by (rule card_image) (use translate_inj_on in blast)
     ultimately have "card (B-{a}) > 0" by auto
     then have *: "finite (B - {a})"
       using card_gt_0_iff[of "(B - {a})"] by auto
     then have "card (B - {a}) = card B - 1"
       using card_Diff_singleton assms by auto
     with * show ?thesis using fin h1 by auto
   qed
 qed
 
 lemma aff_dim_parallel_subspace:
   fixes V L :: "'n::euclidean_space set"
   assumes "V \<noteq> {}"
     and "subspace L"
     and "affine_parallel (affine hull V) L"
   shows "aff_dim V = int (dim L)"
 proof -
   obtain B where
     B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
     using aff_dim_basis_exists by auto
   then have "B \<noteq> {}"
     using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B]
     by auto
   then obtain a where a: "a \<in> B" by auto
   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
   moreover have "affine_parallel (affine hull B) Lb"
     using Lb_def B assms affine_hull_span2[of a B] a
       affine_parallel_commut[of "Lb" "(affine hull B)"]
     unfolding affine_parallel_def
     by auto
   moreover have "subspace Lb"
     using Lb_def subspace_span by auto
   moreover have "affine hull B \<noteq> {}"
     using assms B affine_hull_nonempty[of V] by auto
   ultimately have "L = Lb"
     using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
     by auto
   then have "dim L = dim Lb"
     by auto
   moreover have "card B - 1 = dim Lb" and "finite B"
     using Lb_def aff_dim_parallel_subspace_aux a B by auto
   ultimately show ?thesis
     using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
 qed
 
 lemma aff_independent_finite:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B"
   shows "finite B"
 proof -
   {
     assume "B \<noteq> {}"
     then obtain a where "a \<in> B" by auto
     then have ?thesis
       using aff_dim_parallel_subspace_aux assms by auto
   }
   then show ?thesis by auto
 qed
 
 lemmas independent_finite = independent_imp_finite
 
 lemma span_substd_basis:
   assumes d: "d \<subseteq> Basis"
   shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   (is "_ = ?B")
 proof -
   have "d \<subseteq> ?B"
     using d by (auto simp: inner_Basis)
   moreover have s: "subspace ?B"
     using subspace_substandard[of "\<lambda>i. i \<notin> d"] .
   ultimately have "span d \<subseteq> ?B"
     using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
   moreover have *: "card d \<le> dim (span d)"
     using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms]
       span_superset[of d]
     by auto
   moreover from * have "dim ?B \<le> dim (span d)"
     using dim_substandard[OF assms] by auto
   ultimately show ?thesis
     using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto
 qed
 
 lemma basis_to_substdbasis_subspace_isomorphism:
   fixes B :: "'a::euclidean_space set"
   assumes "independent B"
   shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and>
     f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
 proof -
   have B: "card B = dim B"
     using dim_unique[of B B "card B"] assms span_superset[of B] by auto
   have "dim B \<le> card (Basis :: 'a set)"
     using dim_subset_UNIV[of B] by simp
   from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B"
     by auto
   let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
   have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)"
   proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
     show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
       using d inner_not_same_Basis by blast
   qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
   with t \<open>card B = dim B\<close> d show ?thesis by auto
 qed
 
 lemma aff_dim_empty:
   fixes S :: "'n::euclidean_space set"
   shows "S = {} \<longleftrightarrow> aff_dim S = -1"
 proof -
   obtain B where *: "affine hull B = affine hull S"
     and "\<not> affine_dependent B"
     and "int (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by auto
   moreover
   from * have "S = {} \<longleftrightarrow> B = {}"
     using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
   ultimately show ?thesis
     using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
 qed
 
 lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
   by (simp add: aff_dim_empty [symmetric])
 
 lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
   unfolding aff_dim_def using hull_hull[of _ S] by auto
 
 lemma aff_dim_affine_hull2:
   assumes "affine hull S = affine hull T"
   shows "aff_dim S = aff_dim T"
   unfolding aff_dim_def using assms by auto
 
 lemma aff_dim_unique:
   fixes B V :: "'n::euclidean_space set"
   assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B"
   shows "of_nat (card B) = aff_dim V + 1"
 proof (cases "B = {}")
   case True
   then have "V = {}"
     using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms
     by auto
   then have "aff_dim V = (-1::int)"
     using aff_dim_empty by auto
   then show ?thesis
     using \<open>B = {}\<close> by auto
 next
   case False
   then obtain a where a: "a \<in> B" by auto
   define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))"
   have "affine_parallel (affine hull B) Lb"
     using Lb_def affine_hull_span2[of a B] a
       affine_parallel_commut[of "Lb" "(affine hull B)"]
     unfolding affine_parallel_def by auto
   moreover have "subspace Lb"
     using Lb_def subspace_span by auto
   ultimately have "aff_dim B = int(dim Lb)"
     using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto
   moreover have "(card B) - 1 = dim Lb" "finite B"
     using Lb_def aff_dim_parallel_subspace_aux a assms by auto
   ultimately have "of_nat (card B) = aff_dim B + 1"
     using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto
   then show ?thesis
     using aff_dim_affine_hull2 assms by auto
 qed
 
 lemma aff_dim_affine_independent:
   fixes B :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent B"
   shows "of_nat (card B) = aff_dim B + 1"
   using aff_dim_unique[of B B] assms by auto
 
 lemma affine_independent_iff_card:
     fixes s :: "'a::euclidean_space set"
     shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1"
   apply (rule iffI)
   apply (simp add: aff_dim_affine_independent aff_independent_finite)
   by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff)
 
 lemma aff_dim_sing [simp]:
   fixes a :: "'n::euclidean_space"
   shows "aff_dim {a} = 0"
   using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto
 
 lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)"
 proof (clarsimp)
   assume "a \<noteq> b"
   then have "aff_dim{a,b} = card{a,b} - 1"
     using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
   also have "\<dots> = 1"
     using \<open>a \<noteq> b\<close> by simp
   finally show "aff_dim {a, b} = 1" .
 qed
 
 lemma aff_dim_inner_basis_exists:
   fixes V :: "('n::euclidean_space) set"
   shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and>
     \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1"
 proof -
   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V"
     using affine_basis_exists[of V] by auto
   then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_le_card:
   fixes V :: "'n::euclidean_space set"
   assumes "finite V"
   shows "aff_dim V \<le> of_nat (card V) - 1"
 proof -
   obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1"
     using aff_dim_inner_basis_exists[of V] by auto
   then have "card B \<le> card V"
     using assms card_mono by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_parallel_eq:
   fixes S T :: "'n::euclidean_space set"
   assumes "affine_parallel (affine hull S) (affine hull T)"
   shows "aff_dim S = aff_dim T"
 proof -
   {
     assume "T \<noteq> {}" "S \<noteq> {}"
     then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L"
       using affine_parallel_subspace[of "affine hull T"]
         affine_affine_hull[of T] affine_hull_nonempty
       by auto
     then have "aff_dim T = int (dim L)"
       using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto
     moreover have *: "subspace L \<and> affine_parallel (affine hull S) L"
        using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
     moreover from * have "aff_dim S = int (dim L)"
       using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto
     ultimately have ?thesis by auto
   }
   moreover
   {
     assume "S = {}"
     then have "S = {}" and "T = {}"
       using assms affine_hull_nonempty
       unfolding affine_parallel_def
       by auto
     then have ?thesis using aff_dim_empty by auto
   }
   moreover
   {
     assume "T = {}"
     then have "S = {}" and "T = {}"
       using assms affine_hull_nonempty
       unfolding affine_parallel_def
       by auto
     then have ?thesis
       using aff_dim_empty by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma aff_dim_translation_eq:
   "aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space"
 proof -
   have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))"
     unfolding affine_parallel_def
     apply (rule exI[of _ "a"])
     using affine_hull_translation[of a S]
     apply auto
     done
   then show ?thesis
     using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto
 qed
 
 lemma aff_dim_translation_eq_subtract:
   "aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space"
   using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp)
 
 lemma aff_dim_affine:
   fixes S L :: "'n::euclidean_space set"
   assumes "S \<noteq> {}"
     and "affine S"
     and "subspace L"
     and "affine_parallel S L"
   shows "aff_dim S = int (dim L)"
 proof -
   have *: "affine hull S = S"
     using assms affine_hull_eq[of S] by auto
   then have "affine_parallel (affine hull S) L"
     using assms by (simp add: *)
   then show ?thesis
     using assms aff_dim_parallel_subspace[of S L] by blast
 qed
 
 lemma dim_affine_hull:
   fixes S :: "'n::euclidean_space set"
   shows "dim (affine hull S) = dim S"
 proof -
   have "dim (affine hull S) \<ge> dim S"
     using dim_subset by auto
   moreover have "dim (span S) \<ge> dim (affine hull S)"
     using dim_subset affine_hull_subset_span by blast
   moreover have "dim (span S) = dim S"
     using dim_span by auto
   ultimately show ?thesis by auto
 qed
 
 lemma aff_dim_subspace:
   fixes S :: "'n::euclidean_space set"
   assumes "subspace S"
   shows "aff_dim S = int (dim S)"
 proof (cases "S={}")
   case True with assms show ?thesis
     by (simp add: subspace_affine)
 next
   case False
   with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine
   show ?thesis by auto
 qed
 
 lemma aff_dim_zero:
   fixes S :: "'n::euclidean_space set"
   assumes "0 \<in> affine hull S"
   shows "aff_dim S = int (dim S)"
 proof -
   have "subspace (affine hull S)"
     using subspace_affine[of "affine hull S"] affine_affine_hull assms
     by auto
   then have "aff_dim (affine hull S) = int (dim (affine hull S))"
     using assms aff_dim_subspace[of "affine hull S"] by auto
   then show ?thesis
     using aff_dim_affine_hull[of S] dim_affine_hull[of S]
     by auto
 qed
 
 lemma aff_dim_eq_dim:
   "aff_dim S = int (dim ((+) (- a) ` S))" if "a \<in> affine hull S"
     for S :: "'n::euclidean_space set"
 proof -
   have "0 \<in> affine hull (+) (- a) ` S"
     unfolding affine_hull_translation
     using that by (simp add: ac_simps)
   with aff_dim_zero show ?thesis
     by (metis aff_dim_translation_eq)
 qed
 
 lemma aff_dim_eq_dim_subtract:
   "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))" if "a \<in> affine hull S"
     for S :: "'n::euclidean_space set"
   using aff_dim_eq_dim [of a] that by (simp cong: image_cong_simp)
 
 lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
   using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"]
     dim_UNIV[where 'a="'n::euclidean_space"]
   by auto
 
 lemma aff_dim_geq:
   fixes V :: "'n::euclidean_space set"
   shows "aff_dim V \<ge> -1"
 proof -
   obtain B where "affine hull B = affine hull V"
     and "\<not> affine_dependent B"
     and "int (card B) = aff_dim V + 1"
     using aff_dim_basis_exists by auto
   then show ?thesis by auto
 qed
 
 lemma aff_dim_negative_iff [simp]:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim S < 0 \<longleftrightarrow>S = {}"
 by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)
 
 lemma aff_lowdim_subset_hyperplane:
   fixes S :: "'a::euclidean_space set"
   assumes "aff_dim S < DIM('a)"
   obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}"
 proof (cases "S={}")
   case True
   moreover
   have "(SOME b. b \<in> Basis) \<noteq> 0"
     by (metis norm_some_Basis norm_zero zero_neq_one)
   ultimately show ?thesis
     using that by blast
 next
   case False
   then obtain c S' where "c \<notin> S'" "S = insert c S'"
     by (meson equals0I mk_disjoint_insert)
   have "dim ((+) (-c) ` S) < DIM('a)"
     by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
   then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}"
     using lowdim_subset_hyperplane by blast
   moreover
   have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w
   proof -
     have "w-c \<in> span ((+) (- c) ` S)"
       by (simp add: span_base \<open>w \<in> S\<close>)
     with that have "w-c \<in> {x. a \<bullet> x = 0}"
       by blast
     then show ?thesis
       by (auto simp: algebra_simps)
   qed
   ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}"
     by blast
   then show ?thesis
     by (rule that[OF \<open>a \<noteq> 0\<close>])
 qed
 
 lemma affine_independent_card_dim_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "\<not> affine_dependent S" "a \<in> S"
     shows "card S = dim {x - a|x. x \<in> S} + 1"
 proof -
   have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto
   have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x
   proof (cases "x = a")
     case True then show ?thesis by (simp add: span_clauses)
   next
     case False then show ?thesis
       using assms by (blast intro: span_base that)
   qed
   have "\<not> affine_dependent (insert a S)"
     by (simp add: assms insert_absorb)
   then have 3: "independent {b - a |b. b \<in> S - {a}}"
       using dependent_imp_affine_dependent by fastforce
   have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})"
     by blast
   then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))"
     by simp
   also have "\<dots> = card (S - {a})"
     by (metis (no_types, lifting) card_image diff_add_cancel inj_onI)
   also have "\<dots> = card S - 1"
     by (simp add: aff_independent_finite assms)
   finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" .
   have "finite S"
     by (meson assms aff_independent_finite)
   with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto
   moreover have "dim {x - a |x. x \<in> S} = card S - 1"
     using 2 by (blast intro: dim_unique [OF 1 _ 3 4])
   ultimately show ?thesis
     by auto
 qed
 
 lemma independent_card_le_aff_dim:
   fixes B :: "'n::euclidean_space set"
   assumes "B \<subseteq> V"
   assumes "\<not> affine_dependent B"
   shows "int (card B) \<le> aff_dim V + 1"
 proof -
   obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
     by (metis assms extend_to_affine_basis[of B V])
   then have "of_nat (card T) = aff_dim V + 1"
     using aff_dim_unique by auto
   then show ?thesis
     using T card_mono[of T B] aff_independent_finite[of T] by auto
 qed
 
 lemma aff_dim_subset:
   fixes S T :: "'n::euclidean_space set"
   assumes "S \<subseteq> T"
   shows "aff_dim S \<le> aff_dim T"
 proof -
   obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S"
     "of_nat (card B) = aff_dim S + 1"
     using aff_dim_inner_basis_exists[of S] by auto
   then have "int (card B) \<le> aff_dim T + 1"
     using assms independent_card_le_aff_dim[of B T] by auto
   with B show ?thesis by auto
 qed
 
 lemma aff_dim_le_DIM:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim S \<le> int (DIM('n))"
 proof -
   have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
     using aff_dim_UNIV by auto
   then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))"
     using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
 qed
 
 lemma affine_dim_equal:
   fixes S :: "'n::euclidean_space set"
   assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T"
   shows "S = T"
 proof -
   obtain a where "a \<in> S" using assms by auto
   then have "a \<in> T" using assms by auto
   define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}"
   then have ls: "subspace LS" "affine_parallel S LS"
     using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto
   then have h1: "int(dim LS) = aff_dim S"
     using assms aff_dim_affine[of S LS] by auto
   have "T \<noteq> {}" using assms by auto
   define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}"
   then have lt: "subspace LT \<and> affine_parallel T LT"
     using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto
   then have "int(dim LT) = aff_dim T"
     using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto
   then have "dim LS = dim LT"
     using h1 assms by auto
   moreover have "LS \<le> LT"
     using LS_def LT_def assms by auto
   ultimately have "LS = LT"
     using subspace_dim_equal[of LS LT] ls lt by auto
   moreover have "S = {x. \<exists>y \<in> LS. a+y=x}"
     using LS_def by auto
   moreover have "T = {x. \<exists>y \<in> LT. a+y=x}"
     using LT_def by auto
   ultimately show ?thesis by auto
 qed
 
 lemma aff_dim_eq_0:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by auto
 next
   case False
   then obtain a where "a \<in> S" by auto
   show ?thesis
   proof safe
     assume 0: "aff_dim S = 0"
     have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b
       by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
     then show "\<exists>a. S = {a}"
       using \<open>a \<in> S\<close> by blast
   qed auto
 qed
 
 lemma affine_hull_UNIV:
   fixes S :: "'n::euclidean_space set"
   assumes "aff_dim S = int(DIM('n))"
   shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
 proof -
   have "S \<noteq> {}"
     using assms aff_dim_empty[of S] by auto
   have h0: "S \<subseteq> affine hull S"
     using hull_subset[of S _] by auto
   have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S"
     using aff_dim_UNIV assms by auto
   then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)"
     using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto
   have h3: "aff_dim S \<le> aff_dim (affine hull S)"
     using h0 aff_dim_subset[of S "affine hull S"] assms by auto
   then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)"
     using h0 h1 h2 by auto
   then show ?thesis
     using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"]
       affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close>
     by auto
 qed
 
 lemma disjoint_affine_hull:
   fixes s :: "'n::euclidean_space set"
   assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}"
     shows "(affine hull t) \<inter> (affine hull u) = {}"
 proof -
   have "finite s" using assms by (simp add: aff_independent_finite)
   then have "finite t" "finite u" using assms finite_subset by blast+
   { fix y
     assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u"
     then obtain a b
            where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y"
              and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y"
       by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>)
     define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x
     have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto
     have "sum c s = 0"
       by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf)
     moreover have "\<not> (\<forall>v\<in>s. c v = 0)"
-      by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one)
+      by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum.neutral zero_neq_one)
     moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0"
       by (simp add: c_def if_smult sum_negf
              comm_monoid_add_class.sum.If_cases \<open>finite s\<close>)
     ultimately have False
       using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit)
   }
   then show ?thesis by blast
 qed
 
 lemma aff_dim_convex_hull:
   fixes S :: "'n::euclidean_space set"
   shows "aff_dim (convex hull S) = aff_dim S"
   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"
     apply (rule_tac ex_least_nat_le, auto)
     done
   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"
         apply (rule_tac sum_nonneg, auto)
         done
       then have "sum w s > 0"
         unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>]
         using as[THEN bspec[where x=v]]  \<open>v\<in>s\<close>  \<open>w v \<noteq> 0\<close> by auto
       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> unfolding t_def i_def
           apply (rule_tac Min_le)
           using obt(1) apply auto
           done
         then show ?thesis
           unfolding real_0_le_add_iff
           using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]]
           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
   show "?lhs \<subseteq> ?rhs"
     apply (subst convex_hull_caratheodory_aff_dim, clarify)
     apply (rule_tac x=s in exI)
     apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull])
     done
 next
   show "?rhs \<subseteq> ?lhs"
     using hull_mono by blast
 qed
 
 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"
     apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq)
     apply (erule ex_forward)+
     using aff_dim_le_DIM [of p]
     apply simp
     done
 next
   fix x
   assume "x \<in> ?rhs" then show "x \<in> ?lhs"
     by (auto simp: convex_hull_explicit)
 qed
 
 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"
     apply (rule_tac x=s in exI)
     using hull_subset[of s convex]
     using convex_convex_hull[simplified convex_explicit, of s,
       THEN spec[where x=s], THEN spec[where x=u]]
     apply auto
     done
 next
   fix x s
   assume  "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s"
   then show "x \<in> convex hull p"
     using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto
 qed
 
 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(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation)
 
 
 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 and sum.inter_restrict[OF assms(1), symmetric]
     apply (auto simp: Int_absorb1)
     done
 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"
         apply (rule_tac sum_mono, auto)
         done
       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
     apply (rule_tac conjI)
     apply (rule_tac sum_nonneg, auto)
     done
   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)
     apply (rule_tac[!] sum.mono_neutral_left, auto)
     done
   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"
     apply rule
     apply (rule mult_nonneg_nonneg)
     using *
     apply auto
     done
   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] and z_def
     apply (auto simp: sum_negf sum_distrib_left[symmetric])
     done
   moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x"
     apply rule
     apply (rule mult_nonneg_nonneg)
     using *
     apply auto
     done
   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] and z_def
     using *
     apply (auto simp: sum_negf sum_distrib_left[symmetric])
     done
   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
   then show ?thesis
     apply (rule_tac that[of p m])
     using s
     apply auto
     done
 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 mp(1)
         unfolding gh
         using inj_on_image_Int[OF True gh(3,4)]
         by auto
       have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h"
         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> {}"
   apply (rule Helly_induct)
   using assms
   apply auto
   done
 
 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
     apply (clarsimp simp: convex_def convex_on_def epigraph_def)
     apply (erule_tac x=x in allE)
     apply (erule_tac x="f x" in allE, safe)
     apply (erule_tac x=y in allE)
     apply (erule_tac x="f y" in allE)
     apply (auto simp: )
     done
 next
   assume "convex_on S f" "convex S"
   then show "convex (epigraph S f)"
     unfolding convex_def convex_on_def epigraph_def
     apply safe
      apply (rule_tac [2] y="u * f a + v * f aa" in order_trans)
       apply (auto intro!:mult_left_mono add_mono)
     done
 qed
 
 lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)"
   unfolding convex_epigraph by auto
 
 lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)"
   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})"
 
   unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
   unfolding fst_sum snd_sum fst_scaleR snd_scaleR
   apply safe
     apply (drule_tac x=k in spec)
     apply (drule_tac x=u in spec)
     apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
     apply simp
   using assms[unfolded convex] apply simp
   apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force)
    apply (rule sum_mono)
    apply (erule_tac x=i in allE)
   unfolding real_scaleR_def
    apply (rule mult_left_mono)
   using assms[unfolded convex] apply auto
   done
 
 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
     obt: "\<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] obt(2) mult_1
     apply (drule_tac meta_mp)
     apply (rule mult_left_mono)
     using assms(2) obt(1)
     apply auto
     done
   then show "f x \<le> b"
     using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
     unfolding obt(2-3)
     using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s]
     by auto
 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:
   shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))"
   apply (subst set_sum_alt [OF finite_Basis], safe)
   apply (fast intro: euclidean_representation [symmetric])
   apply (subst inner_sum_left)
 apply (rename_tac f)
   apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i")
   apply (drule (1) bspec)
   apply clarsimp
   apply (frule sum.remove [OF finite_Basis])
   apply (erule trans, simp)
   apply (rule sum.neutral, clarsimp)
   apply (frule_tac x=i in bspec, assumption)
   apply (drule_tac x=x in bspec, assumption, clarsimp)
   apply (cut_tac u=x and v=i in inner_Basis, assumption+)
   apply (rule ccontr, simp)
   done
 
 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
@@ -1,1975 +1,1972 @@
 (*  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 sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
-  by (rule ccontr) auto
-
 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, hide_lams) 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 \<open>Archimedean properties and useful consequences\<close>
 
 text\<open>Bernoulli's inequality\<close>
 proposition Bernoulli_inequality:
   fixes x :: real
   assumes "-1 \<le> x"
     shows "1 + n * x \<le> (1 + x) ^ n"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
     by (simp add: algebra_simps)
   also have "... = (1 + x) * (1 + n*x)"
     by (auto simp: power2_eq_square algebra_simps  of_nat_Suc)
   also have "... \<le> (1 + x) ^ Suc n"
     using Suc.hyps assms mult_left_mono by fastforce
   finally show ?case .
 qed
 
 corollary Bernoulli_inequality_even:
   fixes x :: real
   assumes "even n"
     shows "1 + n * x \<le> (1 + x) ^ n"
 proof (cases "-1 \<le> x \<or> n=0")
   case True
   then show ?thesis
     by (auto simp: Bernoulli_inequality)
 next
   case False
   then have "real n \<ge> 1"
     by simp
   with False have "n * x \<le> -1"
     by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
   then have "1 + n * x \<le> 0"
     by auto
   also have "... \<le> (1 + x) ^ n"
     using assms
     using zero_le_even_power by blast
   finally show ?thesis .
 qed
 
 corollary real_arch_pow:
   fixes x :: real
   assumes x: "1 < x"
   shows "\<exists>n. y < x^n"
 proof -
   from x have x0: "x - 1 > 0"
     by arith
   from reals_Archimedean3[OF x0, rule_format, of y]
   obtain n :: nat where n: "y < real n * (x - 1)" by metis
   from x0 have x00: "x- 1 \<ge> -1" by arith
   from Bernoulli_inequality[OF x00, of n] n
   have "y < x^n" by auto
   then show ?thesis by metis
 qed
 
 corollary real_arch_pow_inv:
   fixes x y :: real
   assumes y: "y > 0"
     and x1: "x < 1"
   shows "\<exists>n. x^n < y"
 proof (cases "x > 0")
   case True
   with x1 have ix: "1 < 1/x" by (simp add: field_simps)
   from real_arch_pow[OF ix, of "1/y"]
   obtain n where n: "1/y < (1/x)^n" by blast
   then show ?thesis using y \<open>x > 0\<close>
     by (auto simp add: field_simps)
 next
   case False
   with y x1 show ?thesis
     by (metis less_le_trans not_less power_one_right)
 qed
 
 lemma forall_pos_mono:
   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
     (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
   by (metis real_arch_inverse)
 
 lemma forall_pos_mono_1:
   "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
     (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
   apply (rule forall_pos_mono)
   apply auto
   apply (metis Suc_pred of_nat_Suc)
   done
 
 
 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)
 
 
 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
   fixes S :: "'a::euclidean_space set"
   assumes "independent S"
   shows independent_card_le:"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 add: dim_span)
   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
 
 
 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 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
 
 
 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 dim_span subspace_hyperplane
         subspace_span)
 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, Gram-Schmidt process, and related theorems\<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 sum.delta 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 simp: dim_span)
   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 simp: span_UNIV)
 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 dim_UNIV)
   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 add: dim_span)
   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 simp: subspace_span)
   also have "... = dim A + dim B"
     by (simp add: dim_span)
   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 simp: subspace_span)
   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 mult.commute)
       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